Science, Faculty of
Mathematics, Department of
DSpace
UBCV
Pelz, Merlin
Doctor of Philosophy - PhD
University of British Columbia
Originating from the pioneering study of Alan Turing, the bifurcation analysis predicting spatial pattern formation from a spatially uniform state for diffusing morphogens or chemical species that interact through nonlinear reactions is a central problem in many chemical and biological systems. From a mathematical viewpoint, one key challenge with this theory for two component systems is that stable spatial patterns typically occur from spatially uniform states when a slowly diffusing "activator" species reacts with a much faster diffusing "inhibitor" species. Thus, one key long-standing question is how to robustly obtain pattern formation in the biologically often realistic case where the time scales for diffusion of the interacting species are comparable. For introduced finite and infinite 1-D and 2-D coupled PDE-ODE bulk-cell models, we investigate symmetry-breaking bifurcations that can emerge when spatially segregated dynamically active compartments, the "cells", are coupled by a PDE bulk diffusion field with comparable diffusivities and degradation rates that is both produced and absorbed by the entire cell population. In the singular limit of a small common cell radius, we construct steady-state solutions and corresponding eigenperturbations and formulate a nonlinear matrix eigenvalue problem that determines the linear stability properties of the steady-states. As regulated by the ratio of the membrane reaction rates on the cell boundaries, we show for various specific prototypical intracellular reactions that our 1-D and 2-D coupled PDE-ODE models admit symmetry-breaking bifurcations from symmetric steady-states defined by a common intracellular steady-state for all cells, leading to linearly stable asymmetric patterns (cell specialization), even when the bulk diffusing species have comparable or possibly equal diffusivities. For a version of our proposed model in ℝ² and ℝ³ with one diffusing species, a hybrid asymptotic-numerical theory is developed by deriving a nonlocal reduced integro-ODE system for all coupled intracellular concentrations along with a numerical marching scheme. This provides a new theoretical and computationally efficient approach for studying how oscillatory dynamics are regulated by the diffusive coupling, leading to phase synchronization, mixed-mode oscillations, and illustrations of quorum-sensing. The study of oscillator synchronization in a PDE diffusion field was one of the initial aims of Yoshiki Kuramoto's foundational work.
Symmetry-Breaking Bifurcations inCompartmental-Reaction Diffusion Systems withComparable DiffusivitiesbyMerlin PelzB.Sc. in Technomathematics, University of Bayreuth, 2016M.Sc. in Mathematics, Technical University of Munich, 2019A THESIS SUBMITTED IN PARTIAL FULFILLMENTOF THE REQUIREMENTS FOR THE DEGREE OFDoctor of PhilosophyinTHE FACULTY OF GRADUATE AND POSTDOCTORALSTUDIES(Mathematics)The University of British Columbia(Vancouver)July 2024© Merlin Pelz, 2024The following individuals certify that they have read, and recommend to theFaculty of Graduate and Postdoctoral Studies for acceptance, the thesis entitled:Symmetry-Breaking Bifurcations in Compartmental-ReactionDiffusion Systems with Comparable Diffusivitiessubmitted by Merlin Pelz in partial fulfillment of the requirements for the degreeof Doctor of Philosophy in Mathematics.Examining Committee:Michael Jeffrey Ward, Professor, Mathematics, UBCSupervisorMiranda Holmes-Cerfon, Associate Professor, Mathematics, UBCSupervisory Committee MemberEric Cytrynbaum, Associate Professor, Mathematics, UBCUniversity ExaminerCarl Ollivier-Gooch, Professor, Mechanical Engineering, UBCUniversity ExaminerAdditional Supervisory Committee Members:Daniel Coombs, Professor, Mathematics, UBCCo-SupervisorAnotida Madzvamuse, Professor, Mathematics, UBCSupervisory Committee MemberBrian Wetton, Professor, Mathematics, UBCSupervisory Committee MemberiiAbstractOriginating from the pioneering study of Alan Turing, the bifurcation analysispredicting spatial pattern formation from a spatially uniform state for diffusingmorphogens or chemical species that interact through nonlinear reactions is acentral problem in many chemical and biological systems. From a mathematicalviewpoint, one key challenge with this theory for two component systems isthat stable spatial patterns typically occur from spatially uniform states when aslowly diffusing “activator” species reacts with a much faster diffusing “inhibitor”species. Thus, one key long-standing question is how to robustly obtain patternformation in the biologically often realistic case where the time scales for diffusionof the interacting species are comparable. For introduced finite and infinite1-D and 2-D coupled PDE-ODE bulk-cell models, we investigate symmetry-breaking bifurcations that can emerge when spatially segregated dynamicallyactive compartments, the “cells“, are coupled by a PDE bulk diffusion field withcomparable diffusivities and degradation rates that is both produced and absorbedby the entire cell population. In the singular limit of a small common cell radius,we construct steady-state solutions and corresponding eigenperturbations andformulate a nonlinear matrix eigenvalue problem that determines the linear stabilityproperties of the steady-states. As regulated by the ratio of the membrane reactionrates on the cell boundaries, we show for various specific prototypical intracellularreactions that our 1-D and 2-D coupled PDE-ODE models admit symmetry-breakingbifurcations from symmetric steady-states defined by a common intracellularsteady-state for all cells, leading to linearly stable asymmetric patterns (cellspecialization), even when the bulk diffusing species have comparable or possiblyequal diffusivities. For a version of our proposed model in R2 and R3 with oneiiidiffusing species, a hybrid asymptotic-numerical theory is developed by derivinga nonlocal reduced integro-ODE system for all coupled intracellular concentrationsalong with a numerical marching scheme. This provides a new theoretical andcomputationally efficient approach for studying how oscillatory dynamics areregulated by the diffusive coupling, leading to phase synchronization, mixed-mode oscillations, and illustrations of quorum-sensing. The study of oscillatorsynchronization in a PDE diffusion field was one of the initial aims of YoshikiKuramoto’s foundational work.ivLay SummaryThis dissertation contributes to the fundamental understanding of how spatialpatterns are forming in biological, physical, and chemical cell systems. Suchpatterns are caused by the specialization of cells in their cell colony in theenvironment they all share. By means of constructing approximate solutions whoseprecision can be rigorously increased that solve an introduced model of such cellsystems, precise conditions are derived that robustly lead either to individual cellsspecializing to different time-independent cell-states or to (a)synchronization ofeach cell’s intracellular chemical concentration oscillations with the other cells’oscillations. Such synchronization behavior can also be viewed as specializationto time-dependent, dynamic cell-states. All mathematical theory is checked withnumerical results obtained from either a commercial solver or self-developednumerical computer simulations. The theoretical framework and results hereincan potentially be used by theoretical natural scientists to engineer cell systems inwhich cells are specializing in certain predefined ways.vPrefaceThis dissertation is an original intellectual piece of writing of the author, MerlinPelz, under the supervision of Prof. Michael J. Ward. It is comprised of twopublished articles and three work-in-progress and thus not yet published articles,all written jointly with my supervisor Michael J. Ward.Chapter 2 is a version of our published article The emergence of spatialpatterns for compartmental reaction kinetics coupled by two bulk diffusing specieswith comparable diffusivities1. Michael J. Ward and I contributed equally to thisarticle. The numerical code was written by me, and, with it, all figures werecreated. The theoretical extension §2.4 was developed and written by me. Adiscussion with Prof. Brian Wetton helped me with implementing the IMEXmethod on a centered grid of Appendix A.2. The sections were drafted by both meand Michael J. Ward, after which we merged both of our versions.Chapter 3 is to a major part a version of our published article Symmetry-breaking bifurcations for compartmental reaction kinetics coupled by two bulkdiffusing species with comparable diffusivities in 2-D2. Michael J. Ward and Icontributed equally to this article. The numerical code, besides the one for the1M. Pelz, M. J. Ward. The emergence of spatial patterns for compartmental reaction kineticscoupled by two bulk diffusing species with comparable diffusivities. Philosophical Transactions ofthe Royal Society A. 2023 Apr 17;381(2245):20220089.2M. Pelz, M. J. Ward. Symmetry-breaking bifurcations for compartmental reaction kineticscoupled by two bulk diffusing species with comparable diffusivities in 2-D. Frontiers in AppliedMathematics and Statistics. 2023 Feb 9;9:1110497.viopen-source software MATCONT3 and commercial software FlexPDE4 we used,was written by me, and all figures were created by me. The sections were draftedby both Michael J. Ward and me, after which we merged both of our versions.Section 3.5 on compartments centered on a Bravais lattice will be extendedand published soon. This section, including its theory, was developed and writtenby me in consultation with Michael J. Ward.Chapter 4 will be only slightly modified and then sent to a journal forpublication using the same title. Michael J. Ward and I contributed equallyto this article. The numerical code was written by both Michael J. Ward andme in respectively two different coding languages, Fortran5 and Julia6, andwe successfully checked that both of us obtained the same results. Both ofus developed and checked the theory together, and Michael J. Ward wrote themajor part of the sections and created most of the figures. I have independentlydeveloped and written §4.7.Chapter 5 will be published using the same title. Michael J. Ward developedthe theory and wrote the sections, and I checked and re-derived all theory and wrotethe introduction and discussion.3A. Dhooge, W. Govaerts, Y. A. Kuznetsov. MATCONT: a MATLAB package for numericalbifurcation analysis of ODEs. ACM Transactions on Mathematical Software (TOMS). 2003 Jun1;29(2):141-64.4FlexPDE 6.50/L64 3D Professional. PDE Solutions Inc. http://www.pdesolutions.com. 2015.Website accessed on 2024 May 2.5Fortran. https://fortran-lang.org. Website accessed on 2024 May 2.6Julia. https://julialang.org. Website accessed on 2024 May 2.viiTable of ContentsAbstract . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . iiiLay Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . vPreface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . viTable of Contents . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . viiiList of Tables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xiiList of Figures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xvList of Abbreviations . . . . . . . . . . . . . . . . . . . . . . . . . . . . xxiiiAcknowledgments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xxivDedication . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xxv1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12 Compartmental Reaction Kinetics Coupled by Two Bulk DiffusingSpecies with Comparable Diffusivities in 1-D . . . . . . . . . . . . . 102.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102.2 Two Diffusion-Coupled Reaction Compartments . . . . . . . . . 122.2.1 Symmetry-breaking . . . . . . . . . . . . . . . . . . . . . 142.2.2 Compartments with FitzHugh-Nagumo kinetics . . . . . . 182.2.3 Compartments with Gierer-Meinhardt kinetics . . . . . . 28viii2.2.4 Compartments with Rauch-Millonas kinetics . . . . . . . 312.3 A Periodic Compartmental-Reaction Diffusion Model . . . . . . . 352.3.1 Compartments with general reaction kinetics . . . . . . . 382.3.2 Two compartments with g(µ,η) = g1(µ)−g2η . . . . . . 442.3.3 FitzHugh-Nagumo kinetics . . . . . . . . . . . . . . . . . 462.3.4 Gierer-Meinhardt kinetics . . . . . . . . . . . . . . . . . 462.3.5 Rauch-Millonas kinetics . . . . . . . . . . . . . . . . . . 492.4 Extension: Introduction of Randomness into the Kinetics . . . . . 522.5 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 613 Compartmental Reaction Kinetics Coupled by Two Bulk DiffusingSpecies with Comparable Diffusivities in 2-D . . . . . . . . . . . . . 643.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 643.2 Compartmental-Reaction Diffusion System in 2-D . . . . . . . . . 663.2.1 Model formulation . . . . . . . . . . . . . . . . . . . . . 663.2.2 Asymptotic construction of the steady-states . . . . . . . 683.2.3 Symmetry-breaking bifurcations . . . . . . . . . . . . . . 743.2.4 A symmetric cell arrangement with two cells . . . . . . . 763.3 The Linear Stability Analysis . . . . . . . . . . . . . . . . . . . . 773.3.1 Re-formulation of the linear stability problem . . . . . . . 813.4 Illustrations of the Theory: A Ring Pattern of Cells . . . . . . . . 843.4.1 Gierer-Meinhardt reaction kinetics . . . . . . . . . . . . . 853.4.2 Rauch-Millonas reaction kinetics . . . . . . . . . . . . . 963.4.3 FitzHugh-Nagumo reaction kinetics . . . . . . . . . . . . 973.4.4 Numerical experiments with closely-spaced cells: GMkinetics . . . . . . . . . . . . . . . . . . . . . . . . . . . 1013.5 Compartments Centered on a Bravais Lattice . . . . . . . . . . . 1073.5.1 Construction of the theory . . . . . . . . . . . . . . . . . 1083.5.2 Illustrations of the theory . . . . . . . . . . . . . . . . . . 1173.5.3 The finite lattice system . . . . . . . . . . . . . . . . . . 1243.6 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 129ix4 Synchronized Memory-Dependent Intracellular Oscillations for aBulk-Cell ODE-PDE Model in R2 . . . . . . . . . . . . . . . . . . . 1344.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1344.2 Transient Analysis Near the Cells . . . . . . . . . . . . . . . . . 1404.3 Derivation of ODE System for Intracellular Kinetics . . . . . . . . 1444.3.1 Matching to the transient solution . . . . . . . . . . . . . 1484.4 Steady-State and Linear Stability Analysis . . . . . . . . . . . . . 1504.4.1 Linear stability analysis . . . . . . . . . . . . . . . . . . 1524.5 Time-Marching Scheme for the Integro-Differential System . . . . 1554.5.1 Sum-of-exponentials approximation for D j(t) and C jk(t) . 1574.5.2 Time-marching scheme for D j(t) . . . . . . . . . . . . . 1614.5.3 Time-marching scheme for C jk(t) . . . . . . . . . . . . . 1654.5.4 Marching scheme for integro-differential system . . . . . 1674.6 Numerical Results: Sel’kov Intracellular Dynamics . . . . . . . . 1704.6.1 An isolated cell . . . . . . . . . . . . . . . . . . . . . . . 1714.6.2 Steady-states and their stability with bulk-cell coupling . . 1724.6.3 Two-cell configurations . . . . . . . . . . . . . . . . . . . 1764.6.4 A ring configuration of cells . . . . . . . . . . . . . . . . 1904.6.5 A ring configuration with a center cell . . . . . . . . . . . 1984.6.6 Measuring phase synchronization . . . . . . . . . . . . . 2034.7 Two Diffusing Species . . . . . . . . . . . . . . . . . . . . . . . 2084.8 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2155 Extensions: Theory for R3 . . . . . . . . . . . . . . . . . . . . . . . 2215.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2215.2 Transient Analysis Near the Cells . . . . . . . . . . . . . . . . . 2255.3 Derivation of ODE System for Intracellular Kinetics . . . . . . . . 2265.3.1 Matching to the transient solution . . . . . . . . . . . . . 2305.4 Steady-State and Linear Stability Analysis . . . . . . . . . . . . . 2315.4.1 Linear stability analysis . . . . . . . . . . . . . . . . . . 2345.4.2 Perturbation theory to the linear stability analysis . . . . . 2385.5 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 240x6 Discussion & Prospects . . . . . . . . . . . . . . . . . . . . . . . . . 242Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 250A Supporting Materials . . . . . . . . . . . . . . . . . . . . . . . . . . 264A.1 CN-RK4 IMEX method on a non-centered grid . . . . . . . . . . 264A.2 BE-RK4 IMEX method on a centered grid . . . . . . . . . . . . . 268A.3 Non-dimensionalization of the 2-D bulk-cell model . . . . . . . . 273A.4 Reduced-wave Green’s function for the unit disk . . . . . . . . . 275A.5 Non-dimensionalization of the 3-D bulk-cell model . . . . . . . . 277A.6 Long-time dynamics of the transient solution . . . . . . . . . . . 278A.7 Improved approximation for B j(∆t) . . . . . . . . . . . . . . . . 279A.8 An artificial boundary condition for full PDE simulations . . . . . 280xiList of TablesTable 2.1 Numerical values (rounded to 5th decimal place) of thesubcritical pitchfork bifurcation point ρp for the symmetricbranch and the two fold bifurcation points ρ f along theasymmetric branches. . . . . . . . . . . . . . . . . . . . . . . 31Table 2.2 Numerical values (rounded to 5th decimal place) of thesubcritical (or supercritical) pitchfork bifurcation point ρp, thetwo fold bifurcation points ρ f , and the corresponding values forthe symmetric (µe) and one of the asymmetric (µe1 , µe2) solutionbranches. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33Table 2.3 Numerical values (rounded to 5th decimal place) of thesubcritical pitchfork bifurcation point ρp for the symmetricbranch and the fold points ρ f along the asymmetric branches. . 49Table 2.4 Similar caption as in Table 2.3 except that the bifurcationquantities for ρ = βv/βu are tabulated for various values of βuwhen Dv = Du = 1. . . . . . . . . . . . . . . . . . . . . . . . 52Table 3.1 Numerical values (rounded to 5th decimal place) of thesubcritical (or supercritical) pitchfork bifurcation point ρp, thefold bifurcation point ρs, and the associated values for thesymmetric µe and one of the asymmetric (µe1, µe2) solutionbranches. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 90xiiTable 3.2 Decreasing the ratio Dv/Du does not trigger hysteresis whendu = 0.09, but rather there is a minimum threshold ofthe diffusivity ratio where the symmetry-breaking pitchforkbifurcation point exists. . . . . . . . . . . . . . . . . . . . . . 94Table 3.3 Increasing the diffusivity ratio Dv/Du when du = 0.08 doesnot eliminate hysteresis, as the symmetry-breaking bifurcationpoint remains subcritical. . . . . . . . . . . . . . . . . . . . . 95Table 4.1 The number 2n+1 of terms needed to approximate the 1-D heatkernel G1D(x, t) := (4πt)−1/2e−x2/(4t) with Laplace transformL [G1(x, t)] = 12√s e−√s|x| to a precision ε f corresponding to(4.82) for the three time intervals I1 = [10−3,1], I2 = [10−3,103]and I3 = [10−5,104]. This table essentially reproduces Table 1in [66] with a few additional explicit error values. . . . . . . . 160Table 4.2 Number 2n+ 1 of terms needed to approximate the 2-D heatkernel with degradation G2D(x, t) = (4πt)−1e−x2/(4t)−σt withLaplace transform L [G2D(x, t)] = (2π)−1K0(x√s+σ) to aprecision ε f for σ = 1 corresponding to (4.82) for the three timeintervals I1 = [10−3,1], I2 = [10−3,103] and I3 = [10−5,104]. . 161Table 4.3 Number 2n+1 of terms needed to approximate the exponentialintegral E1(σt) for σ = 1 with Laplace transformL [E1(σt)] =log(1+ s/σ)/s to a precision ε f corresponding to (4.82) forthe three time intervals I1 = [10−3,1], I2 = [10−3,103] and I3 =[10−5,104]. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 161Table 4.4 Moduli of the components c j of the eigenvector c = (c1,c2),normalized by cHc = 1, and of the normalized components ofKc at some specific points along the horizontal slice through thephase diagram in Fig. 4.15a for the two roots of detM (λ ) = 0with the largest real part. . . . . . . . . . . . . . . . . . . . . 187Table 4.5 Averaged Kuramoto order parameter Qave from (4.124) forthe two-ring configuration of Fig. 4.30a for various bulkdiffusivities D when either σ = 1 or σ = 0.5. . . . . . . . . . . 205xiiiTable 4.6 Averaged Kuramoto order parameter Qave from (4.124) for thefinite hexagonal lattice configuration of Fig. 4.30b for variousbulk diffusivities D. . . . . . . . . . . . . . . . . . . . . . . . 207xivList of FiguresFigure 2.1 The domain [0,L] with two diffusion-coupled compartments(cells) and intracellular reaction kinetics on its boundary. . . . 13Figure 2.2 The instability boundary tr(DF(µe,ηe)) = 0 (magenta curve)in the (q,z)-plane for a unique steady-state of (2.17) with ε =0.7, and a generic curve (blue curve) where the steady-state islinearly stable. . . . . . . . . . . . . . . . . . . . . . . . . . 19Figure 2.3 Bifurcation diagram for the two-cell system with FN kinetics . 21Figure 2.4 Zero-eigenvalue crossing of the eigenvalue λ of the linearizedsystem (2.23) to the asymmetric eigenvector q2 = (1,−1)T . . 22Figure 2.5 Symmetry-breaking pitchfork bifurcation thresholds zP,1 andzP,2 versus ρ corresponding to zero-eigenvalue crossingsfor the anti-phase mode where the asymmetric steady-statebifurcates from the symmetric branch. . . . . . . . . . . . . . 23Figure 2.6 For an initial condition near the unstable symmetric branch,and for z = 4.52211 and ρ = 80, we predict that theasymmetric solution branch is linearly stable since zP,1 < z <zH,1 (see Fig. 2.5). . . . . . . . . . . . . . . . . . . . . . . . 24Figure 2.7 The complex eigenvalues to the symmetric eigenperturbationof the linearized system (2.23) have a positive real part insidethe Hopf bubble depicted in Figure 2.5. . . . . . . . . . . . . 27xvFigure 2.8 3-D bifurcation diagram for µe1 and µe2 with bifurcationparameter ρ = βv/βu for steady-states of the GMcompartmental-reaction diffusion system as computedfrom (2.34) using MatCont [28]. . . . . . . . . . . . . . . . . 30Figure 2.9 Full numerical computations of (2.1) by the CN-RK4 IMEXmethod of Appendix A.1 for GM kinetics. . . . . . . . . . . . 32Figure 2.10 Full numerical computations of (2.1) by the CN-RK4 IMEXmethod for GM kinetics. . . . . . . . . . . . . . . . . . . . . 34Figure 2.11 3-D bifurcation diagrams using MatCont [28] for the two-cell compartmental-reaction diffusion system with Rauch-Millonas (RM) kinetics (2.36). . . . . . . . . . . . . . . . . . 35Figure 2.12 Zero-eigenvalue crossing for the growth rate λ of theasymmetric eigenperturbation in direction q2 = (1,−1)Tbetween the two supercritical pitchfork points wP,1v and wP,2v ,as obtained as roots ofF− of (2.30). . . . . . . . . . . . . . . 36Figure 2.13 Full numerical computations of (2.1) by the CN-RK4 IMEXmethod of Appendix A.1 for Rauch-Millonas kinetics (2.36). . 37Figure 2.14 The periodic compartmental-reaction diffusion model on[0,nL] with compartments centered at the blue dots. Two bulkdiffusing species provide the inter-compartment coupling. . . 37Figure 2.15 FN kinetics with n = 2: For an initial condition near theunstable symmetric branch, and for z = zP,2 ≈ 6.63675 andρ = βv/βu = 60, the full time-dependent solution computedusing the BE-RK4 IMEX scheme of Appendix A.2 convergesto a stable asymmetric steady-state. . . . . . . . . . . . . . . 47Figure 2.16 FN kinetics with n= 3: Full-numerical time-dependent resultsfrom the BE-RK4 IMEX scheme of Appendix A.2. . . . . . . 48Figure 2.17 3-D Bifurcation diagram, exhibiting hysteresis, withbifurcation parameter ρ = βv/βu for steady-states withGM kinetics and n = 2 computed from (2.53) using MatCont[28]. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50Figure 2.18 GM kinetics with n= 2. Full numerical time-dependent resultsfrom the BE-RK4 IMEX scheme. . . . . . . . . . . . . . . . 51xviFigure 2.19 GM kinetics with n= 2. Full numerical time-dependent resultsfrom the BE-RK4 IMEX scheme. . . . . . . . . . . . . . . . 53Figure 2.20 3-D Bifurcation diagrams for Rauch-Millonas kinetics (2.36)with n = 2 computed from (2.53) using MatCont [28]. . . . . 54Figure 2.21 Rauch-Millonas kinetics with n = 2. For an initial conditionnear the unstable symmetric branch, and for ρ = 15 and wv =wP,2v , the full time-dependent solution computed using the BE-RK4-IMEX scheme of Appendix A.2 converges to a stableasymmetric steady-state. . . . . . . . . . . . . . . . . . . . . 55Figure 2.22 Full stochastic numerical simulation results of (2.37) in Julia[15]. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59Figure 3.1 A 2-D bounded domain with four diffusion-coupled circularcells of a common radius. In the jth cell, activator-inhibitorreaction kinetics occur for the activator µ j and inhibitor η j. . . 69Figure 3.2 A schematic plot of a ring pattern in the unit disk with two cells. 85Figure 3.3 Bifurcation diagram and r-dependence of bifurcation point forGM kinetics . . . . . . . . . . . . . . . . . . . . . . . . . . . 87Figure 3.4 Full numerical PDE simulation results of (3.3) with FlexPDE[39] for GM kinetics (3.55). . . . . . . . . . . . . . . . . . . 88Figure 3.5 3-D Bifurcation diagram, computed from (3.56), forsymmetric and asymmetric steady-states of a two-cell ringpattern with ring radius r = 0.5 and two different values ofdu with GM kinetics (3.55). . . . . . . . . . . . . . . . . . . 89Figure 3.6 Plots of the numerically computed largest roots of σ1(λ ) = 0and σ2(λ ) = 0 versus ρ , as defined in (3.51), that determinethe linear stability properties to either in-phase e = (1,1)T oranti-phase q2 = (1,−1)T eigenperturbations of the symmetricsteady-state solution, respectively. . . . . . . . . . . . . . . . 91Figure 3.7 Zero-crossings of det(N (λ )) = 0, as defined in (3.49),determine the linear stability properties of an asymmetricsteady-state solution with two cells. . . . . . . . . . . . . . . 91xviiFigure 3.8 Full numerical simulation results of (3.3) with FlexPDE [39]for GM kinetics (3.55). . . . . . . . . . . . . . . . . . . . . . 92Figure 3.9 Full numerical simulation results of (3.3) with FlexPDE [39]for GM kinetics (3.55). . . . . . . . . . . . . . . . . . . . . . 93Figure 3.10 Effect of the diffusivity ratio Dv/Du of the two bulk species onthe extent of the hysteresis when du = 0.08, as measured by thedistance between the fold bifurcation points and the subcriticalpitchfork bifurcation point (left) and by the distance of the twoasymmetric equilibria µe1 and µe2 from each other (right). . . 95Figure 3.11 Effect of the diffusivity ratio Dv/Du on the existence of thepitchfork point when du = 0.08 (left) and du = 0.09 (right). . . 96Figure 3.12 3-D Bifurcation diagram, computed from (3.60) usingMatCont [28], for symmetric and asymmetric steady-states ofa two-cell ring pattern with ring radius r = 0.5 and with RMkinetics (3.59). . . . . . . . . . . . . . . . . . . . . . . . . . 98Figure 3.13 For the two-cell system with RM kinetics (3.59) andparameters as in the caption of Figure 3.12 with wv = wP,2v ,we plot the eigenvalue λ , satisfying σ2(λ ) = 0 in (3.51),versus ρ that determines the linear stability of the symmetricsteady-state solution branch to eigenperturbations of the formq2 = (1,−1)T . . . . . . . . . . . . . . . . . . . . . . . . . . 99Figure 3.14 3-D Bifurcation diagram, computed from (3.64) usingMatCont [28], for symmetric and asymmetric steady-states ofa two-cell ring pattern with ring radius r = 0.5 and with FNkinetics (3.63). . . . . . . . . . . . . . . . . . . . . . . . . . 101Figure 3.15 For the two-cell system with FN kinetics (3.63) and parametersas in the caption of Figure 3.14 with z = zP,2, we plot thenumerically computed eigenvalue λ , satisfying σ2(λ ) = 0 in(3.51), versus ρ . . . . . . . . . . . . . . . . . . . . . . . . . 102Figure 3.16 Bifurcation diagram to FN kinetics and zero-crossing ofeigenvalue λ . . . . . . . . . . . . . . . . . . . . . . . . . . 103xviiiFigure 3.17 Full numerical simulation results of (3.3) with FlexPDE [39]for GM kinetics (3.55) with two closely spaced cells centeredon a ring of radius r= 0.031 and with minimum cell separationof 0.002. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 104Figure 3.18 Full numerical simulation results of (3.3) with FlexPDE [39]for GM kinetics (3.55) with three closely spaced cells withcell centers located on the vertices of an equilateral trianglecentered at the origin. . . . . . . . . . . . . . . . . . . . . . . 105Figure 3.19 Full numerical simulation results of (3.3) with FlexPDE [39]for GM kinetics (3.55) in four close cells. . . . . . . . . . . . 106Figure 3.20 A disk-shaped excerpt of the considered lattice cell system inwhich cells of small radius ε are centered in a well-separatedmanner on a Bravais lattice with generators l1 and l2. . . . . . 109Figure 3.21 Growth rate distribution in Brillouin zone for GM kinetics . . 120Figure 3.22 Growth rate discrepancy from origin to vertices and walkingalong the boundary . . . . . . . . . . . . . . . . . . . . . . . 121Figure 3.23 Growth rate distribution in Brillouin zone for FN kinetics . . . 123Figure 3.24 Growth rate distribution in Brillouin zone for RM kinetics . . 125Figure 3.25 A finite version of the Bravais lattice compartment system. . . 126Figure 4.1 Compartmental-reaction diffusion system in R2 . . . . . . . . 137Figure 4.2 Plot of the hyperbolic-shaped contour Γ in the Laplacetransform plane defined by (4.74) . . . . . . . . . . . . . . . 162Figure 4.3 Hopf bubble for isolated cell with Sel’kov kinetics. . . . . . . 171Figure 4.4 Hopf bubble for single cell in R2 without compartment influx. 172Figure 4.5 Two-cell, four-cell and ring-center-cell arrangements. . . . . . 175Figure 4.6 Hopf bubbles for two diffusion-coupled cells with sameproperties. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 176Figure 4.7 Hopf bubbles for two diffusion-coupled cells with oneactivated cell. . . . . . . . . . . . . . . . . . . . . . . . . . . 177Figure 4.8 Dominant eigenvalue zero-transition for system with twodiffusion-coupled cells. . . . . . . . . . . . . . . . . . . . . . 178xixFigure 4.9 Comparison of the numerical solutions computed from theintegro-differential system (4.32) using the algorithm in §4.5.4with that computed from the bulk-cell model (4.5) usingFlexPDE [39] with the artificial boundary condition given in(A.15) of Appendix A.8. . . . . . . . . . . . . . . . . . . . . 180Figure 4.10 Comparison of the amplitudes and periods of the numericalsolutions computed from the integro-differential system (4.32)using the algorithm in §4.5.4 with that computed from thebulk-cell model (4.5) using FlexPDE [39] with the artificialboundary condition given in (A.15) of Appendix A.8. . . . . . 181Figure 4.11 Comparison of the numerical solutions computed from theintegro-differential system (4.32) using the algorithm in §4.5.4with that computed from the bulk-cell model (4.5) usingFlexPDE [39] with the artificial boundary condition given in(A.15) of Appendix A.8. . . . . . . . . . . . . . . . . . . . . 182Figure 4.12 Comparison of the amplitudes and periods of the numericalsolutions computed from the integro-differential system (4.32)using the algorithm in §4.5.4 with that computed from thebulk-cell model (4.5) using FlexPDE [39] with the artificialboundary condition given in (A.15) of Appendix A.8. . . . . . 183Figure 4.13 Comparison of the numerical solutions computed from theintegro-differential system (4.32) using the algorithm in §4.5.4with that computed from the bulk-cell model (4.5) usingFlexPDE [39] with the artificial boundary condition given in(A.15) of Appendix A.8. . . . . . . . . . . . . . . . . . . . . 184Figure 4.14 Comparison of the amplitudes and periods of the numericalsolutions computed from the integro-differential system (4.32)using the algorithm in §4.5.4 with that computed from thebulk-cell model (4.5) using FlexPDE [39] with the artificialboundary condition given in (A.15) of Appendix A.8. . . . . . 185Figure 4.15 Hopf bubbles for system with two diffusion-coupled cells . . . 186xxFigure 4.16 Intracellular dynamics u2 j(t) between 600 < t < 800 at thelabeled points along the vertical and horizontal slices inFig. 4.15a. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 187Figure 4.17 Same caption as in Fig. 4.9 except that parameters are nowD = 3, σ = 1/2. . . . . . . . . . . . . . . . . . . . . . . . . 188Figure 4.18 Hopf bubbles for system with two diffusion-coupled cells. . . 189Figure 4.19 Intracellular dynamics for system with two diffusion-coupledcells. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 190Figure 4.20 Intracellular dynamics for system with four diffusion-coupledcells. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 192Figure 4.21 Intracellular dynamics for system with four diffusion-coupledcells. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 193Figure 4.22 Hopf bubbles for system with four diffusion-coupled cells and,among them, two active ones . . . . . . . . . . . . . . . . . . 194Figure 4.23 Intracellular dynamics for system with four diffusion-coupledcells and, among them, two active ones . . . . . . . . . . . . 195Figure 4.24 Hopf bubbles for four diffusion-coupled cells with modifiedinflux rates for two cells . . . . . . . . . . . . . . . . . . . . 196Figure 4.25 Intracellular dynamics for system with four diffusion-coupledcells with larger influx rates for two cells . . . . . . . . . . . 197Figure 4.26 Hopf bubbles for system with seven diffusion-coupled cells. . 200Figure 4.27 Intracellular dynamics u2 j(t) for the ring and center cellconfiguration with rc = 2 computed using the algorithm of§4.5.4 with ∆t = 0.005 at the points indicated in Fig. 4.26 usingfor uniformly random perturbations about the steady-state asthe initial condition. The center cell is labeled as C7. . . . . . 202Figure 4.28 Averaged order parameter (4.124) for the ring and center cellconfiguration of identical cells in Fig. 4.5c along the red-dashed path in the scatter plot of Fig. 4.26c with σ = 4/7 and0.25 < D < 1.25. . . . . . . . . . . . . . . . . . . . . . . . . 204Figure 4.29 Intracellular dynamics for system with seven diffusion-coupled cells. . . . . . . . . . . . . . . . . . . . . . . . . . . 206Figure 4.30 Eight- and nineteen-cell arrangements. . . . . . . . . . . . . . 207xxiFigure 4.31 Intracellular dynamics u2 j(t) for the two-ring configuration ofcells in Fig. 4.30a computed using the algorithm of §4.5.4 with∆t = 0.005 for three values of D. . . . . . . . . . . . . . . . . 218Figure 4.32 Short- (left panels) and long-time (right panels) intracellulardynamics u2 j(t) for the finite lattice configuration in Fig. 4.30bas computed using the algorithm of §4.5.4 with ∆t = 0.005 andfor three values of D. . . . . . . . . . . . . . . . . . . . . . . 220Figure 5.1 Compartmental-reaction diffusion system in R3 . . . . . . . . 222xxiiList of AbbreviationsBE Backward-EulerCN Crank-NicolsonCPU Central processing unit (of a processor in a computer)ETD2 Exponential time differencing of order 2FN FitzHugh-NagumoGCEP Globally coupled eigenvalue problemGM Gierer-MeinhardtHB Hopf bifurcationIC Initial conditionIMEX Implicit-explicitNAS Nonlinear algebraic systemODE Ordinary differential equationPDE Partial differential equationRD Reaction-diffusionRK4 Runge-Kutta 4 (classical)RM Rauch-Millonas1-D One-dimensional space2-D Two-dimensional space3-D Three-dimensional spacexxiiiAcknowledgmentsI am deeply grateful for everything I have learned from my advisor Michael J. Wardand for his relentless support through open-ended and open-minded meetings,discussions about everything, and through pushing me continuously in a healthyand organic way. Michael, look how far we’ve come with your initial idea of themembrane “fluxes”!Further, I am highly appreciating months-long discussions with MirandaHolmes-Cerfon about randomness in compartmental-reaction diffusion systems,before working on our current research project that is independent from thisdissertation. Those discussions, along with stochastic simulations, have givenme solid ideas where to start when venturing into the development of bifurcationtheory for random compartmental-reaction diffusion systems.Of course, I would also like to express my gratitude and love to my parents.Mama and Papa, you are always interested in everything I’m doing and discusswith me about all this, even though sometimes the math is a little deep for you tounderstand immediately — thank you.I am also thankful for the support I received through the Four Year DoctoralFellowship (4YF) of UBC’s Graduate School, the UBC Faculty of ScienceGraduate Award, the UBC International Tuition Award, and the President’sAcademic Excellence Initiative PhD Award of UBC.xxivDedicationThis dissertation is dedicated to the general advancement of mathematical biology.May this dissertation exhibit the power and insight of theoretical mathematicaldevelopment, and may it be used for the good.xxvChapter 1IntroductionA central issue in many chemical and biological systems that involve the couplingof diffusive processes and nonlinear reactions is to determine conditions for whichspatio-temporal patterns can form from either a patternless or a pre-patterned state.In a pioneering theoretical study, Alan Turing [138] established that diffusingmorphogens with different diffusivities can destablilize a spatially uniform andstable steady-state of the nonlinear reaction kinetics. As applied to two-componentactivator-inhibitor reaction-diffusion (RD) systems, this Turing stability analysisshows that a sufficiently large diffusivity ratio is typically needed to obtainspatial pattern formation from the destabilization of a spatially uniform state,unless the nonlinear reaction kinetics are finely tuned (cf. [109], [6], [29]). Forcertain chemical systems, this large diffusivity ratio requirement needed for patternformation may be feasible to achieve in situations where one of the chemicalspecies can bind to a substrate, which has the consequence of reducing theeffective diffusivity of this species (cf. [86], [33]). However, in many cellularprocesses related to developmental biology and morphogenesis, the theoreticallarge diffusivity ratio threshold needed for freely diffusing morphogens to createsymmetry-breaking patterns is often unrealistic as different small moleculestypically have very comparable diffusivities (cf. [99], [116]). In [99], variousmodifications of the simple “freely diffusing” morphogen paradigm such as,facilitated diffusion, transient binding, immobilization and transcytosis, amongothers, have been postulated to play a central role in specific applications of1diffusive transport at the cellular level. Qualitatively, the postulated overall effectof these mechanisms is to modify an effective diffusivity ratio of the morphogens,which can, therefore, lead to the emergence of spatial patterns and symmetry-breaking behavior in cellular processes related to developmental biology and earlymorphogenesis (cf. [123]).As a result, one key long-standing theoretical question in RD theory is howto modify the two-component RD paradigm so as to robustly generate stablespatial patterns from a spatially homogeneous state when the time scales fordiffusion of the interacting species are comparable. By including an additionalnon-diffusible component, which roughly models either membrane-bound proteinsor an immobile chemically active substrate, it has been shown (cf. [108], [72], [75])that this “2+ 1” extension of the two-component RD framework can yield stablespatial patterns even when the two diffusible species have a common diffusivity.In another direction, which is based on graph-theoretic properties associated withnonlinear reactions between multiple species that are either immobile or freelydiffusing, it has been shown that with certain activating and inhibiting feedbackrelations in the chemical kinetics, spatial patterns can form without the largediffusivity ratio requirement (cf. [95], [31], [84]). More recently, the authors in[54] have revealed that in random, multi-component, RD systems the requireddiffusivity threshold for pattern formation typically decreases as the number ofinteracting and diffusing species increases.From a theoretical viewpoint, in specific applications where a large diffusivityratio is a realistic assumption, it has been shown both analytically and fromnumerical simulations (cf. [141], [143], [57], [56]) that two-component RDsystems admit a wide range of spatially localized patterns and instabilities thatoccur in the “far-from-equilibrium” regime, far from where a Turing linear stabilityanalysis will provide any insight into pattern-forming properties.The first goal of this dissertation is to formulate and quantitatively analyzea new theoretical model in a 1-D and 2-D setting that robustly leads to patternformation even when the two diffusing species have a comparable or equaldiffusivity. More specifically, we analyze symmetry-breaking pattern formationfor a 1-D and 2-D PDE-ODE bulk-cell RD model in which spatially segregatedlocalized reaction compartments, referred to as “cells”, are coupled to a two-2component linear bulk diffusion field with constant bulk degradation rates. Inthe cells, which are in the 2-D setting assumed to have a common radius thatis small compared to the domain length-scale and the inter-cell distances, two-component intracellular activator-inhibitor reaction kinetics are specified. In the1-D setting, the cells are modeled by separated singular points. The intracellularspecies undergo an exchange with the two bulk species across the cell boundaries,as mediated by membrane reaction rates in a Robin boundary condition that isspecified on each cell boundary. The two extracellular diffusing bulk species,with comparable diffusivities and degradation rates, provide the mechanism thatcouples the nonlinear intracellular reactions that occur in the union of the spatiallysegregated cells. We refer to this modeling framework as a compartmental-reaction diffusion system. From a mathematical viewpoint, these bulk-cellmodels in 2-D or 3-D are singularly perturbed nonlinear diffusive processes thathave spatially localized “defects”. These “defects” consist of a collection of smallcells in which the nonlinear reactions occur. This type of perturbation, which hasa large magnitude but small spatial extent, is called a strong localized perturbation[144]. It may be contrasted with a weak perturbation, which has small magnitudebut large extent. Strong localized perturbations are singular perturbations in thesense that they produce large, but localized, changes in the solutions. As such, theperturbed solutions can be constructed by a matched asymptotic analysis, whichmatches an inner solution defined near the strong perturbation to an outer solutiondefined in the region away from the defect. This matching procedure is facilitatedusing Dirac masses, and as a result certain Green’s functions feature prominentlyin the analysis.The numerical implementation of our theoretical analysis for this model forvarious specific intracellular reaction kinetics reveals that it is the ratio of thereaction rate of the inhibitor component to that of the activator component onthe compartment boundaries that plays a central role in the initiation of symmetry-breaking bifurcations of a symmetric steady-state. The magnitude of this ratioultimately controls whether linearly stable asymmetric steady-states for the bulk-cell model can occur even when the bulk diffusivities are comparable or equal. Thebifurcation threshold condition for this key membrane reaction rate ratio parameteris distinct from the usual large diffusivity ratio threshold that is required for pattern3formation from a spatially uniform state for typical two-component activator-inhibitor RD systems (cf. [94], [76]). We emphasize that our linear stabilityanalysis predicting symmetry-breaking bifurcations for the bulk-cell model, asregulated by the membrane reaction rate ratio, is significantly more challengingthan performing a simple Turing stability analysis [138] since it is based on thelinearization of the bulk-cell model around a spatially non-uniform symmetricsteady-state. Such a steady-state generally looks spatially hat-shaped around thecells with the same hat-shape and -height at each cell. This makes this non-uniform steady-state the symmetric base state. In our 1-D study, where nonlinearreactions are restricted either to domain boundaries or at lattice sites on a 1-Dperiodic chain, it is shown for some specific nonlinear kinetics that symmetry-breaking bifurcations can occur from a symmetric steady-state when the ratio ofmembrane reaction rates exceeds a threshold. In our 2-D study we are not anymoreassuming the cells to be singular but rather to be disk-shaped with small commonradius. We are deriving similar results in the 2-D case by using strong localizedperturbation theory in this small radius of the cells. This more intricate steady-stateand perturbation analysis involves Green matrices with modified Bessel functions.In the special case of having cells centered on an infinite Bravais lattice embeddedin a coupling diffusion field in whole R2 outside of the cells, the Green matricesreduce to a single scalar Green function regular part that is an infinite sum of acombination of Bessel functions. Dealing with this infinite sum is made tractableby using the summation over the reciprocal lattice to the Bravais lattice.We remark that our 2-D study, and related 1-D analysis, is largely inspiredby the agent-based numerical computations in [116] where it was shown thatnonlinear kinetic reactions restricted to lattice sites on a 2-D lattice can generatestable Turing-type spatial patterns when coupled through a spatially discretizedtwo-component bulk diffusion field in which the two diffusible species have acomparable diffusivity.In a broader context, the study of novel pattern-forming properties associatedwith compartmentalized reactions interacting through a passive bulk diffusionfield originates from the 1-D analysis in [46] for the FitzHugh-Nagumo model andthe bulk-membrane analysis of [87] in disk-shaped domains. In a 1-D context,and with one bulk diffusing species, this compartmental-reaction diffusion system4modeling paradigm has been shown to lead to triggered oscillatory instabilitiesfor various reaction kinetics involving conditional oscillators (cf. [49], [47],[50]). Amplitude equations characterizing the local branching behavior forthese triggered oscillations have been derived in [105] using a weakly nonlinearanalysis. Applications of this framework have been used to model intracellularpolarization and oscillations in fission yeast (cf. [147], [148]). In a 2-D domain,similar bulk-cell models, but with only one diffusing bulk species, have beenformulated and used to model quorum-sensing behavior (cf. [48], [64], [117],[44]). With regards to bulk-membrane RD models in a multi-spatial dimensionalcontext, where nonlinear kinetics are restricted to the membrane, the associatedpattern-forming properties have been studied both theoretically (cf. ([113], [36],[93], [92], [104]), and for some specific biological applications (cf. [21], [114],[115], [125], [106]).Moving away from patterning solely of steady-states and turning towardthe time-dependent analysis of spatio-temporal pattern formation with focus onintracellular oscillations caused and synchronized by a collection of cells is thetheme of the second goal of this dissertation.Over the past four decades, the Kuramoto model [79]θ˙i = ωi+KNN∑j=1sin(θ j−θi) , (1.1)and its variants have provided the primary theoretical modeling framework foranalyzing synchronization properties of a collection of N nonlinear oscillators.In (1.1), θi is the phase of the ith oscillator, ωi is its natural frequency, andK is the coupling strength. Key quantities for analyzing the synchronizationbehaviour for (1.1) are the Kuramoto order parameters r(t) and ψ(t) defined byreiψ = N−1∑Nj=1 eiθ j . In the mean field limit N → ∞, phase transitions to completephase synchronization r= 1 as K is increased have been characterized. For a surveyof results see [126], [2] and [23].One key aspect of Kuramoto-type phase oscillator systems is that they canbe formally derived from nonlinear dynamical systems of coupled oscillators by5using asymptotic phase reduction methods that are valid in the limit of weakcoupling (cf. [101], [83] and [82]). An important current research theme is thestudy of oscillator synchronization on networks, as surveyed in [111], owing totheir many diverse applications such as power grids [52] and cortical brain activity[17], among others.With a rather different focus, in [81] Kuramoto emphasized the need forcharacterizing synchronization behavior for a collection of coupled oscillators or“cells”, each fixed in space, in which an extra substance secreted from each cellcan diffuse over the entire space and effectively mediate the inter-cell interaction.His phenomenological model of this interaction has the formu′j = F(u j)+g(A(x j, t)) , εAt = D∆A−ηA+N∑k=1h(uk)δ (x−xk) , (1.2)where g and h are prescribed coupling functions. In R1 and in the quasi-steadylimit ε → 0, (1.2) reduces tou′j = F(u j)+g(N∑k=1σ(|x j−xk|)h(uk)). (1.3)where σ(r) = Ce−r√η/D is the 1-D quasi-static Green function for some C >0. Further approximating (1.3) near a Hopf bifurcation point of the uncoupleddynamics u′j = F(u j) by using a weakly nonlinear analysis, the normal form ofthe underlying dynamics leads to a nonlocally coupled Complex Ginzburg-Landau(CGL) system. These nonlocal CGLs have been shown to yield a wide varietyof highly intricate spatio-temporal dynamics, commonly classified as chemicalturbulence ([81], [80], [100]). More recently, in a series of articles ([4], [122],and [22]) there has been a renewed focus on analyzing synchronization propertiesfor phase oscillator systems with coupling as in (1.2), with the main emphasis beingfor the quasi-steady limit (1.3).Two primary limitations of the phenomenological model (1.2) is that A(x, t)has a singularity at x = x j in R2 and R3, and that it is not clear how to choosethe coupling functions g and h for a specific application. For many problemsin biology and chemical physics, it is essential to explicitly model the diffusive6coupling of dynamically active units, without incorporating extraneous couplingfunctions. In particular, for certain microbial systems, cell-cell communicationoccurs through the diffusion of extracellular signaling molecules, referred to asautoinducers, that are both produced and absorbed by the entire collection ofcells (cf. [34], [130], [25]). For a colony of Dictyostelium discoideum, it isknown that the autoinducer cyclic adenosine monophosphate (cAMP) triggersintracellular oscillations that initiate the process of the spatial aggregation ofthe colony in low nutrient environments (cf. [51], [102], [43]). In addition,the autoinducer acetaldehyde (Ace) leads to glycolytic oscillations in a colonyof yeast cells (cf. [26], [25], [27], [58]), while acylated homoserine lactones(AHLs) are implicated in triggering bioluminescence behavior associated withthe marine bacterium Aliivibrio fischeri that resides in the light organ of certainspecies of tropical squid (cf. [130]). For these microbial systems, one key aspectis to characterize quorum-sensing behavior, whereby collective dynamics can onlyoccur if the cell population exceeds a threshold (cf. [118], see [110] for a survey).In a chemical physics context, collective oscillatory dynamics of catalyst loadedpellets can occur owing to the diffusive coupling of Belousov-Zhabotinsky (BZ)reagents in a chemical mixture (cf. [121], [131], [133], [136], [135]). In addition,certain microemulsions of dynamically active surface-stabilized BZ droplets thatare immersed in oil lead to synchronized oscillatory dynamics (cf. [137]). Morerecently, chaotic oscillations have been observed for a compartmentalized surfacereaction nanosystem [112].An ideal modeling framework to investigate collective dynamics arising fromthe coupling of a bulk diffusion field are the cell-bulk models originating from[98] and [97] in a 3-D setting and from [48] and [64] in 2-D domains. Froma mathematical viewpoint, cell-bulk models are compartmental-reaction diffusion(RD) systems with rich dynamics that can be used for analyzing how intracellularoscillations associated with spatially segregated dynamically active “units” or“cells” are mediated by one or more diffusing substances that are both producedand absorbed by the entire cell population. For these models, the diffusive couplingis nonlocal in both space and time. This modeling framework is also well-alignedwith Kuramoto’s original aims of investigating oscillator synchronization throughthe effect of diffusive coupling as discussed in [81]. The survey article [121]7has also emphasized the need for theoretical modeling frameworks that couplediscretely interacting dynamical units.Connecting to these modeling frameworks, we are introducing acompartmental-reaction diffusion system with a single diffusing species, that islater extended to have two diffusing species, and derive a reduced integro-ODEsystem for the intracellular dynamics at all times using an extension of stronglocalized perturbation theory to time-dependent systems. The reduction is overthe spatial diffusion field and derives coupling functions that capture the couplinginduced by the diffusion field taking into account the geometry of the wholedomain including the cell locations. Thus, using oscillatory intracellular kinetics,a model similar to the Kuramoto model is derived that includes precise mechanicalcoupling terms and functions. We derive such a model in both R2 and R3, and inR2 we extensively analyze the synchronization behavior of cell systems for variouscell numbers and arrangements choosing, for illustration, intracellular Sel’kovkinetics [119]. A Kuramoto-type order parameter is derived to find transitions tosynchronization in systems of many cells, and we carefully illustrate quorum anddiffusion sensing in the treated cell systems.This dissertation is organized as follows. In Chapter 2 a first model of acompartmental-reaction diffusion system in 1-D is introduced and a correspondingnonlinear algebraic system (NAS) is derived that captures all steady-states. Aglobally coupled eigenvalue problem (GCEP) is derived for the growth rates ofall eigenperturbations that can possibly destabilize the respective steady-statetowards symmetry-broken steady-states with similar geometry as the respectiveeigenperturbation with dominant growth rate. Illustrations of pattern formationare given using FitzHugh-Nagumo, Gierer-Meinhardt, and Rauch-Millonasintracellular reaction kinetics. With Chapter 3 we are venturing into 2-D spaceand construct a NAS for the steady-state and a GCEP for all eigenperturbationsusing strong-localized perturbation theory in the small cell radius, in a finitedisk-domain and whole R2 with cells placed at the nodes of an infinite Bravaislattice. The Kuramoto coupled oscillators viewpoint is taken in Chapter 4 inwhole R2 and in Chapter 5 in whole R3. With an extension of strong-localizedperturbation theory for the fully time-dependent system, not only for the steady-8state system, using Laplace transformation, an integro-ODE system with memoryfor the intracellular chemical concentrations is constructed in the 2-D setting inChapter 4 and in the 3-D setting in Chapter 5. A NAS for the steady-states and itscorresponding GCEP are derived in Chapters 4 and 5. Additionally, in Chapter 4 atime-marching scheme using a sum-of-exponentials method is developed that canefficiently solve the derived integro-ODE system for all possible dynamics, and,with it, the synchronization behavior is extensively illustrated for the case when allcells have intracellular Sel’kov reaction kinetics. Quorum and diffusion sensingare illustrated therein. Finally, the results are summarized and put in context inChapter 6. Various future directions that are tractable to tackle with our theory andcompartmental-reaction diffusion framework are discussed as well.9Chapter 2Compartmental ReactionKinetics Coupled by Two BulkDiffusing Species withComparable Diffusivities in 1-D2.1 IntroductionIn this chapter we analyze symmetry-breaking pattern formation for a 1-D coupledbulk-compartment RD system in which spatially localized reaction compartmentsare coupled through a two-component linear bulk diffusion field with constant bulkdegradation rates. The two-component reaction kinetics inside each compartmentare assumed to be identical and the chemical exchange between the bulk mediumand the spatially localized reaction compartments is modeled by a Robin boundarycondition with certain permeabilities. We refer to this modeling framework asa 1-D compartmental-reaction diffusion system. Although this coupled systemdoes not admit spatially homogeneous steady-states, it does allow for the existenceof steady-state solutions that are symmetric about the domain mid-line when thereaction compartments are identical. This symmetric steady-state solution is thebase state for our analysis. From a linear stability analysis, and as confirmed10by full PDE simulations, we will show for various specific activator-inhibitorintra-compartmental reaction kinetics that this coupled system admits symmetry-breaking pitchfork bifurcations of the symmetric base state, leading to stableasymmetric steady-state patterns, even when the bulk diffusing species havecomparable or even equal diffusivities. We emphasize that the linear stabilityanalysis predicting these symmetry-breaking bifurcations is more intricate than in[138] as it is based on a linearization around a spatially non-uniform symmetricsteady-state, rather than around a spatially uniform steady-state as in [138]. Inour coupled system, a key parameter regulating the emergence of these stableasymmetric steady-states is shown to be a sufficiently large permeability ratio ofthe two species on the compartment boundary. This feature is in contrast to theusual large diffusivity ratio requirement that is a necessary condition for patternformation from a spatially uniform state for typical two-component activator-inhibitor RD systems (cf. [94], [76]).The 2-D extension in Chapter 3 of the theoretical 1-D framework adoptedin this chapter can potentially be applied to characterizing symmetry-breakingpattern-forming properties associated with biological or chemical modelingscenarios that involve small dynamically active, but spatially localized, discretecompartments or “cells” that are coupled through multi-component linear bulkdiffusion fields.The outline of this chapter is as follows. In §2.2 we formulate and analyzea compartmental-reaction diffusion system when the reaction compartments arelocated on the boundaries of a 1-D spatial domain. For this model, symmetricand asymmetric steady-state solutions are determined for a class of reactionkinetics when the bulk diffusing species have comparable or equal diffusivities.This construction involves the solution of a nonlinear algebraic system, whosebifurcation structure can be used to predict pitchfork bifurcation points on thesymmetric solution branch where asymmetric patterns emerge. Moreover, awayfrom bifurcation points, the linear stability problem for the symmetric steady-stateis studied numerically after first deriving a nonlinear matrix eigenvalue problemfor the eigenvalue parameter. In this way, symmetry-breaking bifurcations of thesymmetric steady-state, leading to stable asymmetric patterns, as the permeabilityratio increases past a threshold are illustrated for FitzHugh-Nagumo [46], Gierer-11Meinhardt [42], and Rauch-Millonas [116] reaction kinetics. For FitzHugh-Nagumo kinetics we show that oscillatory instabilities of the symmetric steady-state are also possible in certain parameter regimes. In §2.3 we provide asimilar analysis for a compartmental-reaction diffusion system where the localizedreaction compartments are periodically spaced on a ring. For this periodic problem,symmetric steady-states are constructed analytically and, through a linear stabilityanalysis based on a Floquet-type approach, we derive a nonlinear matrix eigenvalueproblem that can be used to determine parameter values where symmetry-breakingbifurcations occur due to multiple possible modes of instability. The theory isillustrated for the three specific reaction kinetics. The theoretical predictions in§2.2 and §2.3 of symmetry-breaking bifurcations and stable asymmetric steady-states are confirmed from full time-dependent simulations of our PDE-ODEmodel. Owing to the novel form of the bulk-compartment coupling, off-the-shelf PDE software was not available for the PDE numerical simulations. Theimplicit-explicit (IMEX) numerical schemes and discretizations we developed forthe full PDE simulations in §2.2 and §2.3 are derived in Appendix A.1 and A.2,respectively. In §2.4 it is discussed how to introduce randomness meaningfully inour compartmental-reaction diffusion framework, and in §2.5 an overall discussionand prospects are provided.2.2 Two Diffusion-Coupled Reaction CompartmentsIn our modeling framework, we assume that there is a compartment or “cell”located at each boundary of a one-dimensional spatial bulk domain (0,L). Thesecompartments are assumed to be reactive in the sense that two species withconcentrations µ(t) and η(t) are produced according to µ˙(t) = f (µ,η) and η˙(t) =g(µ,η), for some nonlinear reaction kinetics f and g. Here the dot “˙” denotes thetime derivative d/dt. Further, we assume that these two compartmental species arecoupled to two bulk-defined species u(t,x) and v(t,x) on (0,L) through a chemicalexchange across the compartment boundary. The bulk or extracellular species areassumed to undergo a linear diffusion process with constant diffusivities Du and Dvand constant degradation rates σu and σv (see Fig. 2.1 for a schematic plot). This12leads to a coupled PDE-ODE system formulated asbulk∂tu = Du ∂xxu−σuu , x ∈ (0,L)∂tv = Dv ∂xxv−σvv , x ∈ (0,L) (2.1a)reaction fluxesDu∂xu(t,0) = βu (u(t,0)−µ1(t)) (Robin boundary conditions)Dv∂xv(t,0) = βv (v(t,0)−η1(t))−Du∂xu(t,L) = βu (u(t,L)−µ2(t))−Dv∂xv(t,L) = βv (v(t,L)−η2(t))(2.1b)compartmentsµ˙1 = f (µ1,η1)+Du ∂xu(t,0) (reaction kinetics at x = 0)η˙1 = g(µ1,η1)+Dv ∂xv(t,0)µ˙2 = f (µ2,η2)−Du ∂xu(t,L) (reaction kinetics at x = L)η˙2 = g(µ2,η2)−Dv ∂xv(t,L) .(2.1c)To illustrate the analysis, in this section we will initially assume that thecommon reaction kinetics in the two compartments is the FitzHugh-Nagumo (FN)kinetics (with parameters q > 0,z > 0,ε > 0), as considered in [46], given byf (µ j,η j) = µ j−q(µ j−2)3+4−η j, g(µ j,η j) = εµ jz− εη j , (2.2)for j ∈ {1,2}. We refer to the parameters βu and βv in (2.1b) as thepermeabilities, as they model the strength of the exchange between the bulk andintra-compartmental species.Figure 2.1: The domain [0,L] with two diffusion-coupled compartments(cells) and intracellular reaction kinetics on its boundary.132.2.1 Symmetry-breakingWith this formulation, we will calculate a steady-state solution that is symmetricin the sense that the concentrations in the two compartments are identical. Byanalyzing the linear stability of this steady-state, we will show that, dependingon the parameters, it can be destabilized through either an oscillatory instabilitydue to a Hopf bifurcation or from the emergence of a new stable steady-state thatis asymmetric in the two compartments. In contrast to the typical Turing-typeanalysis for two component RD systems, this stable asymmetric steady-state forour model, which arises from a symmetry-breaking pitchfork bifurcation, occurseven when the bulk diffusivities are comparable.2.2.1.1 Uncoupled system equilibriumIn the absence of diffusion, the ODE system for the intra-compartmental species isuncoupled from the bulk medium and reduces toµ˙(t) = f (µ,η) , η˙(t) = g(µ,η) . (2.3)Let µe, ηe be an equilibrium point for (2.3) and label F(µ,η) :=( f (µ,η),g(µ,η)). For a specific parameter set, the linear stability propertyof the equilibrium state is characterized by whether the eigenvalues λ of theJacobian DF(µe,ηe) have positive (unstable, exponentially growing perturbations)or negative (stable, exponentially decaying perturbations) real parts Re(λ ).2.2.1.2 Coupled system equilibriumIn the presence of bulk diffusion, the compartments are coupled to the bulkmedium. For this case, we first construct a steady-state solution for (2.1) withoutany symmetry assumptions.In the bulk region x ∈ (0,L), the steady-state solution for (2.1a) has the formue(x)=A1 sinh(ωu(L−x))+A2 sinh(ωux) , ve(x)=B1 sinh(ωv(L−x))+B2 sinh(ωvx) ,(2.4)where ωu :=√σu/Du and ωv :=√σv/Dv. Here the constants A1,A2,B1 and B2are to be determined. By employing the boundary conditions in (2.1b) for u, we14obtain(A1A2)=βuDuωu(γ2u −1)(γu 11 γu)(µe1µe2)where γu := cosh(ωuL)+βuDuωusinh(ωuL) ,(2.5a)while from the boundary conditions in (2.1b) for v we get(B1B2)=βvDvωv(γ2v −1)(γv 11 γv)(ηe1ηe2)where γv := cosh(ωvL)+βvDvωvsinh(ωvL) .(2.5b)Next, from the compartmental reaction kinetics for the v-species in (2.1c), andfrom use of (2.4), we getβvγ2v −1A˜(ηe1ηe2)=(g(µe1 ,ηe1)g(µe2 ,ηe2))where A˜ :=(γv cosh(ωvL)−1 cosh(ωvL)− γvcosh(ωvL)− γv γv cosh(ωvL)−1).(2.6)For the special case where g(µ,η) is linear in η , i.e., g(µ,η) = g1(µ)− g2η , asoccurs for the FN kinetics in (2.2), we can use (2.6) to obtainA(ηe1ηe2)=(g1(µe1)g1(µe2))with A :=βvγ2v −1A˜+g2I , (2.7)where I is the identity matrix.Next, we use (2.4) in (2.1c) for the u-species. This yieldsβuγ2u −1B(µe1µe2)=(f (µe1 ,ηe1)f (µe2 ,ηe2))where B :=(γu cosh(ωuL)−1 cosh(ωuL)− γucosh(ωuL)− γu γu cosh(ωuL)−1).(2.8)Finally, by solving (2.7) for ηe1 and ηe2 , we substitute the resulting expressions into(2.8) to obtain a nonlinear algebraic system for µe1 and µe2(f (µe1 ,(1,0)(βvγ2v−1 A˜+g2I)−1(g1(µe1),g1(µe2))T )f (µe2 ,(0,1)(βvγ2v−1 A˜+g2I)−1(g1(µe1),g1(µe2))T ))− βuγ2u −1B(µe1µe2)= 0 , (2.9)where T denotes transposition. In terms of a solution to (2.9) we can determine theother steady-state values from (2.8) and (2.5). We emphasize that (2.9) applies to15all steady-states of (2.1) regardless of symmetry.The matrices A and B that effectively couple the two compartments aresymmetric and circulant, and hence have the common eigenspace spanned byq1 := (1,1)T and q2 := (1,−1)T . The corresponding eigenvalues a j and b j forA and B for j = 1,2 are simplyAq1 = (βvγ2v−1(γv(cosh(ωvL)−1)+(cosh(ωvL)−1))+g2) q1 = a1q1 ,Aq2 = (βvγ2v−1(γv(cosh(ωvL)+1)− (cosh(ωvL)+1))+g2) q2 = a2q2 ,Bq1 = (γu(cosh(ωuL)−1)+(cosh(ωuL)−1)) q1 = b1q1 ,Bq2 = (γu(cosh(ωuL)+1)− (cosh(ωuL)+1)) q2 = b2q2 .(2.10)To determine a symmetric solution for (2.9) for which (µe1 ,µe2) = µeq1, whereq1 = (1,1)T , we simply use the eigenvalues a1 and b1 in (2.10) to reduce (2.9) tof (µe,g1(µe)a1)− βuγ2u −1b1µe = 0 . (2.11)This scalar nonlinear algebraic problem for µe applies whenever g(µ,η) has thespecific form g(µ,η) = g1(µ)−g2η , where g2 is a constant.To detect any symmetry-breaking pitchfork bifurcation points of the symmetricsteady-state solution we can perform a linear stability analysis of (2.1) around thesteady-state solution and seek λ = 0 eigenvalue crossings. More conveniently, todetect zero-eigenvalue crossings for the linearization of the full system (2.1) wecan more simply determine bifurcation points associated with the linearization ofthe nonlinear algebraic system (2.9) around a symmetric steady-state. To do so, weintroduce the perturbation (µ1,µ2)T = µeq1+δφ from the symmetric steady-stateinto (2.9) to obtain, for δ ≪ 1, the linearized problem(ddµ1f (µ1,(1,0)A−1(g1(µ1),g1(µ2))T ) ddµ2 f (µ1,(1,0)A−1(g1(µ1),g1(µ2))T )ddµ1f (µ2,(0,1)A−1(g1(µ1),g1(µ2))T ) ddµ2 f (µ2,(0,1)A−1(g1(µ1),g1(µ2))T ))φ− βuγ2u −1Bφ = 0 ,(2.12)where the derivatives are to be evaluated at the symmetric state µ1 = µe and µ2 =µe. In this way, we obtain the linearized problem(J11 J12J21 J22)φ − βuγ2u −1Bφ = 0 , (2.13a)16where the Jacobian matrix coefficients are given, in terms of any solution µe of(2.11), byJ11 := ∂µ f (µe,g1(µe)a1)+∂η f (µe,g1(µe)a1)(1,0)A−1(g′1(µe),0)T ,(2.13b)J12 := ∂η f (µe,g1(µe)a1)(1,0)A−1(0,g′1(µe))T , (2.13c)J21 := ∂η f (µe,g1(µe)a1)(0,1)A−1(g′1(µe),0)T , (2.13d)J22 := ∂µ f (µe,g1(µe)a1)+∂η f (µe,g1(µe)a1)(0,1)A−1(0,g′1(µe))T .(2.13e)From (2.13), a bifurcation with an in-phase perturbation φ = q1, if it exists,must satisfy∂µ f (µe,g1(µe)a1)+∂η f (µe,g1(µe)a1)g′1(µe)a1− βuγ2u −1b1 = 0 . (2.14)Moreover, a bifurcation with an anti-phase perturbation φ = q2, if it exists, has tofulfill∂µ f (µe,g1(µe)a1)+∂η f (µe,g1(µe)a1)g′1(µe)a2− βuγ2u −1b2 = 0 . (2.15)Below in §2.2.2.1 we will show for FN kinetics that the symmetric steady-state is never destabilized through a zero-eigenvalue crossing of the linearizationwith an in-phase perturbation. In contrast, to find a bifurcation point for an anti-phase eigenperturbation, we must solve the following coupled system for µe and abifurcation parameter:f (µe,g1(µe)a1)− βuγ2u −1b1µe = 0 , (2.16a)∂µ f (µe,g1(µe)a1)+∂η f (µe,g1(µe)a1)g′1(µe)a2− βuγ2u −1b2 = 0 . (2.16b)The bifurcation parameter is taken either as a reaction kinetic parameter or asthe ratio ρ := βv/βu of the binding rates. In (2.16) the explicit expressions forthe matrix eigenvalues a1, a2, b1, and b2 are defined in (2.10) in terms of the17parameters.2.2.2 Compartments with FitzHugh-Nagumo kinetics2.2.2.1 Application of the theoryWhen uncoupled from the bulk, the intra-compartmental dynamics with FitzHugh-Nagumo (FN) kinetics is (see [46])µ˙ = f (µ,η) := µ−q(µ−2)3+4−η , η˙ = g(µ,η) := εzµ− εη . (2.17)As shown in §2 of [47], there are up to three equilibria for (2.17) that aredetermined by the cubicµe−q(µe−2)3+4− zµe = 0 .With F(µ,η) := ( f (µ,η),g(µ,η)), and for a specific parameter set, the sign of thereal parts of the eigenvalues of DF(µe,ηe) can be read off the tr-det-plot with, inthis setting, tr(DF(µe,ηe)) = 1−3q(µe−2)2−ε and det(DF(µe,ηe)) = 3εq(µe−2)2+ε(z−1). For z> 1, for which det(DF(µe,ηe))> 0, the linear stability of µe isdetermined by the sign of tr(DF(µe,ηe)). We will choose a parameter set for whichthere is a unique linearly stable steady-state of the intra-compartmental dynamics(2.17) (see the green dot in Fig. 2.2).To apply the steady-state theory of §2.2.1.2 for the bulk-cell coupled systemfor FN kinetics we first identify that g(µ,η) = g1(µ)−g2η , where g1 = εµz andg2 = ε . From (2.7) we getA(ηe1ηe2)= εz(µe1µe2)with A :=βvγ2v −1A˜+ εI , (2.18)where A˜ was defined in (2.6). In terms of A, we obtain from (2.9) that the nonlinearalgebraic system characterizing both symmetric and asymmetric steady-states is(µe1µe2)−q((µe1 −2)3(µe2 −2)3)+4(11)− εzA−1(µe1µe2)− βuγ2u −1B(µe1µe2)= 0 . (2.19)18Figure 2.2: The instability boundary tr(DF(µe,ηe)) = 0 (magenta curve) inthe (q,z)-plane for a unique steady-state of (2.17) with ε = 0.7, anda generic curve (blue curve) where the steady-state is linearly stable.For q = 1 and z = 4.48430 (green dot) there is a unique stable steadystate µe ≈ 1.26293 and ηe = zµe ≈ 5.66336 for which tr(DF(µe,ηe))≈−1.32981 < 0.From (2.11), the symmetric steady-state solution, for which µe1 = µe2 = µe, isobtained from−q(µe−2)3+4+(1− εza1− βuγ2u −1b1)µe = 0 , (2.20)where a1 is an eigenvalue of A as defined in (2.10) with g2 = ε .Bifurcation points (if they exist) for either in-phase and anti-phaseperturbations of the symmetric steady-state, as obtained from (2.16), satisfy−3q(µe−2)2+1− εza1 −βuγ2u −1b1 = 0 , (in-phase) (2.21a)−3q(µe−2)2+1− εza2 −βuγ2u −1b2 = 0 , (anti-phase) (2.21b)where a1,a2,b1,b2 are the eigenvalues of A and B as given in (2.10) with g2 = ε .To show that there can be no bifurcation point for the in-phase perturbation ofthe symmetric steady-state when µe ≥ −1, we use (2.21a) in (2.20) to eliminate19εz/a1+βu/(γ2u −1). This yields0 = q(µe−2)3−4+(εza1+βuγ2u −1b1−1)µe= q(µe−2)2((µe−2)−3µe)−4 =−2q(µe−2)2(µe+1)−4≤ 0 ,which shows that a bifurcation with an in-phase eigenperturbation is impossiblewhen µe ≥−1.To determine a bifurcation point associated with the anti-phase perturbation ofthe symmetric steady-state we must solve the coupled system (2.21b) and (2.20)on the range where εz/a2+βub2/(γ2u −1)≤ 1. By eliminating µe between (2.21b)and (2.20) we obtainq−(1− εza2 −βuγ2u−1 b2)27(6− 2εza1 −2βuγ2u−1 b1)2 (−2+ εz( 3a1 − 1a2)+βuγ2u −1(3b1−b2))2= 0 .(2.22)For a fixed parameter set, we view the parameter constraint (2.22) as a nonlinearalgebraic equation for either z or for the permeability ratio ρ := βv/βu.In terms of z, in Fig. 2.3(left) we plot both symmetric and asymmetric steady-state solution branches, as computed from (2.20) and (2.19) using MatCont [28]for the parameter set given in the figure caption. We observe that asymmetricsteady-state solution branches form a bubble structure between two pitchforkbifurcation points zP,1 and zP,2 associated with anti-phase perturbations of thesymmetric steady-state. For z = zP,1, in Fig. 2.3(right) we plot symmetric andasymmetric solution branches in terms of ρ := βv/βu for the same parameter setin Fig. 2.3(left). This latter figure shows the existence of a symmetry-breakingbifurcation as the permeability ratio ρ increases past a critical value, and wherethe bulk diffusivity ratio is fixed near unity. In order to check the stability ofthe pitchfork bifurcation in ρ , we use the function F− of (2.30) to obtain thegrowth rates for the asymmetric eigenperturbation in direction q1 = (1,−1)T ,sliding along the symmetric equilibrium branch in Figure 2.3(right) through thepitchfork bifurcation point ρp. In Figure it is shown that precisely on ρp there isa zero-growth rate crossing, so that the asymmetric eigenperturbation destabilizes20the symmetric equilibrium branch past the bifurcation point, causing convergenceto an asymmetric steady-state.Figure 2.3: Left: µe1 versus z showing that asymmetric equilibria exist insidea pitchfork bubble delimited by zP,1 ≈ 4.48430 and zP,2 ≈ 5.07294 whenρ = βv/βu = 80. Right: For z = zP,1, there is a symmetry-breakingbifurcation of the symmetric steady-state as ρ increases past the criticalvalue ρp = 80. Parameters: Du = 1,Dv = 3,σu = σv = 1,ε = 0.7,q =1,L = 1, and βu = 0.1.2.2.2.2 Hopf bifurcation bubbleThe analysis in (2.2.2.1) has constructed both symmetric and asymmetricequilibria of the coupled bulk-cell model with FN kinetics and has detectedsymmetry-breaking bifurcations of the symmetric solution branch ue,ve,µe,ηe thatcorrespond to zero-eigenvalue crossings of the linear stability problem.We now formulate and study the linear stability problem for the symmetricsteady-state solution branch to determine whether other instabilities, such asHopf bifurcations, are possible. For the linear stability analysis we perturb thesymmetric steady-state by introducing u(t,x) = ue(x)+ φ(x)eλ t ,v(t,x) = ve(x)+ψ(x)eλ t ,µ j(t) = µe + ξ jeλ t ,η = ηe + ζ jeλ t , where |φ | ≪ 1, |ψ| ≪ 1, |ξ j| ≪ 1and |ζ j| ≪ 1 for j ∈ {1,2}, into (2.1) and linearizing. This yields the linearized21Figure 2.4: Zero-eigenvalue crossing of the eigenvalue λ of the linearizedsystem (2.23) to the asymmetric eigenvector q2 = (1,−1)T . This λis the exponential growth rate of the asymmetric eigenperturbationdestabilizing the symmetric steady-state when Re(λ )> 0 right after thepitchfork bifurcation point ρp. Parameters as for Figure 2.3 and withz = zP,1.eigenvalue problembulk∂xxφ −Ω2uφ = 0 , x ∈ (0,L)∂xxψ−Ω2vψ = 0 , x ∈ (0,L) (2.23a)reaction fluxesDu∂xφ(0) = βu (φ(0)−ξ1) (Robin boundary conditions)Dv∂xψ(0) = βv (ψ(0)−ζ1)−Du∂xφ(L) = βu (φ(L)−ξ2)−Dv∂xψ(L) = βv (ψ(L)−ζ2)(2.23b)compartmentsλξ1 = ∂µ feξ1+∂η feζ1+βu (φ(0)−ξ1) (reaction kinetics at x = 0)λζ1 = ∂µgeξ1+∂ηgeζ1+βv (ψ(0)−ζ1)λξ2 = ∂µ feξ2+∂η feζ2+βu (φ(L)−ξ2) (reaction kinetics at x = L)λζ2 = ∂µgeξ2+∂ηgeζ2+βv (ψ(L)−ζ2).(2.23c)In (2.23c) the partials of f and g evaluated at the symmetric steady-state22Figure 2.5: Symmetry-breaking pitchfork bifurcation thresholds zP,1 and zP,2versus ρ corresponding to zero-eigenvalue crossings for the anti-phasemode where the asymmetric steady-state bifurcates from the symmetricbranch. The anti-phase mode is destabilizing on the range zP,1 < z <zP,2. The symmetric branch is unstable to an oscillatory instability forthe in-phase mode when zH,1 < z < zH,2, where zH,1 and zH,2 are Hopfbifurcation thresholds for the in-phase mode. On the range ρ > 75,we observe the ordering zP,1 < zH,1 < zH,2 < zP,2. For this range of ρ ,where the unstable oscillatory range for the in-phase mode is containedwithin the pitchfork range, the asymmetric branches of equilibria thatbifurcate from the symmetric branch are predicted to be locally linearlystable. Parameters: Du = 1,Dv = 3,σu = σv = 1,ε = 0.7,q = 1,L = 1,βu = 0.1, and βv = ρβu.are denoted by ∂µ fe, ∂η fe, ∂µge, and ∂ηge. In (2.23a) we defined Ωu :=√(λ +σu)/Du and Ωv :=√(λ +σv)/Dv.Our goal is to determine all of the eigenvalues λ that are associated withnontrivial solutions to (2.23). The symmetric steady-state solution is linearlystable if all such eigenvalues satisfy Re(λ ) < 0, and it is unstable if for onesuch eigenvalue we have Re(λ ) > 0. Although our eigenvalue problem is notself-adjoint, and as such there may be a short-term transient growth of initialperturbations (cf. [71]), our numerical simulations below, confirming results fromthe linear stability theory, suggest that transient growth effects are unlikely to play23Figure 2.6: For an initial condition near the unstable symmetric branch, andfor z = 4.52211 and ρ = 80, we predict that the asymmetric solutionbranch is linearly stable since zP,1 < z < zH,1 (see Fig. 2.5). The fullPDE simulation of (2.1) by the CN-RK4 IMEX method of Appendix A.1supports this conjecture and shows the existence of a stable asymmetricsteady-state. For z = 4.52211, the ρ-pitchfork point was ρ ≈ 70 (seeFig. 2.5). Parameters: Du = 1,Dv = 3,σu = σv = 1,ε = 0.7,q = 1,L =1,βu = 0.1,ρ = βv/βu = 80..a key role in the pattern-forming process.Owing to the symmetry of the steady-state solution about the mid-line, theeigen-modes for (2.23) can be decomposed into two distinct classes: in-phaseeigenperturbations (+) or anti-phase eigenperturbations (-), for which φ ′+(L2 ) =0 = ψ ′+(L2 ) or φ−(L2 ) = 0 = ψ−(L2 ), respectively. We now derive the eigenvalueproblem for these two possible eigenmodes.24For the in-phase mode, we calculate from (2.23a) and (2.23b) thatφ+(x) = d+1cosh(Ωu(L2 − x))cosh(ΩuL)with d+1 :=βuβu+DuΩu tanh(Ωu L2 )ξ1,(2.24)ψ+(x) = d+2cosh(Ωv(L2 − x))cosh(ΩvL)with d+2 :=βvβv+DvΩv tanh(Ωv L2 )ζ1.(2.25)For this in-phase mode, we have ξ1 = ξ2 := ξ and ζ1 = ζ2 := ζ . Then, by using(2.23c), we obtain the 2×2 nonlinear matrix eigenvalue problemM+(λ )(ξζ)=(00), (2.26a)whereM+ :=(λ −∂µ fe+βu(1− βuβu+DuΩu tanh(ΩuL/2)) −∂η fe−∂µge λ −∂ηge+βv(1− βvβv+DvΩv tanh(ΩvL/2)) .)(2.26b)Eigenvalues associated with the in-phase eigenperturbation are roots λ ofdet(M+(λ )) = 0.Similarly, for the anti-phase mode, we calculate from (2.23a) and (2.23b) thatφ−(x) = d−1sinh(Ωu(L2 − x))sinh(ΩuL)with d−1 :=βuβu+DuΩu coth(Ωu L2 )ξ1 ,(2.27)ψ−(x) = d−2sinh(Ωv(L2 − x))sinh(ΩvL)with d−2 :=βvβv+DvΩv coth(Ωv L2 )ζ1 ,(2.28)where we now label ξ1 =−ξ2 := ξ and ζ1 =−ζ2 := ζ . From (2.23c), eigenvalues25λ associated with the anti-phase mode are roots of det(M−(λ )) = 0, whereM− :=(λ −∂µ fe+βu(1− βuβu+DuΩu coth(ΩuL/2)) −∂η fe−∂µge λ −∂ηge+βv(1− βvβv+DvΩv coth(ΩvL/2))).(2.29)By a simple scaling, the eigenvalues for the in-phase and anti-phase mode areequivalently written as the roots ofF±(λ ) = 0, whereF±(λ ) :=det(M±(λ ))det(λ I−DFe) with DFe :=(∂µ fe ∂η fe∂µge ∂ηge). (2.30)Roots of (2.30) with Re(λ )> 0 correspond to destabilizing modes, while roots withλ = iλi for λi > 0 yield Hopf bifurcations. To determine the number of zeroes ofF± in the right-half plane of C, denoted byN+(F±), we numerically implementthe Argument Principle given by12πi∫ΓRddλF±(λ )F±(λ )dλ =N+(F±)−P+(F±) , (2.31)whereP+(F±) is the number of poles ofF± in Re(λ )> 0. Here we have labeledthe counterclockwise integration contour as ΓR := {Rexp(iϕ)|ϕ ∈ [−π2 , π2 ]} ∪[−iR, iR], with R sufficiently large so that all zeroesN+(F±) and polesP+(F±)are in int(ΓR).Our numerical results based on (2.31) applied toF−(λ ) show for the parameterset in Fig. 2.3(left) that there is a unique real destabilizing eigenvalue for theanti-phase mode on the range zP,1 < z < zP,2 between the two pitchfork points.Moreover, we numerically verified that the symmetric branch in Fig. 2.3(right) islinearly stable to the anti-phase mode only when ρ is below the pitchfork point.In Fig. 2.5, the pitchfork points zP,1 and zP,2 are plotted versus ρ . Froma numerical root finding of F+(iλi) = 0 in (2.30), together with a numericalimplementation of the Argument Principle (2.31), we identify two Hopf bifurcationthresholds zH,1 and zH,2 for the in-phase mode, with the symmetric branch beingunstable to an oscillatory instability only on the range zH,1 < z < zH,2. Usingthe function F+ to recover as its root the complex eigenvalues to the in-phase26perturbation, we see in Figure that inside the Hopf bubble the exponential growthrate Re(λ ) is positive and so causes the in-phase Hopf bifurcation leading to stablein-phase oscillations of both compartments. In Fig. 2.5 we observe for ρ > 75 thatthe unstable oscillatory range is contained within the pitchfork range, in that theordering zP,1 < zH,1 < zH,2 < zP,2 holds. For this range of ρ , the symmetric branchis linearly stable to both in-phase and anti-phase modes on z< zP,1 and on z> zP,2.As a result, we predict that the symmetry-breaking bifurcation at zP,1 and zP,2 leadsto linearly stable asymmetric solution branches at least locally near either zP,1 orzP,2. As shown in Fig. 2.6, this conjecture is verified in the full PDE simulations of(2.1) by the CN-RK4 IMEX method of Appendix A.1.Figure 2.7 shows that there is a zero-crossing of the real part of the eigenvalueλ , which represents the exponential growth rate of the corresponding in-phaseoscillatory eigenperturbation. The scatter plot was obtained by varying the reactionkinetic parameter z in the range shown in Figure 2.5 for fixed ρ = 80.Figure 2.7: The complex eigenvalues to the symmetric eigenperturbation ofthe linearized system (2.23) have a positive real part inside the Hopfbubble depicted in Figure 2.5. Thus, the symmetric equilibrium branchis destabilized by in-phase oscillations in the corresponding regime.Parameters as in Figure 2.5 with ρ = 80.272.2.3 Compartments with Gierer-Meinhardt kineticsNext, we extend our analysis to a compartmental-reaction variant of the Gierer-Meinhardt (GM) model, where the reaction kinetics are now confined to theboundaries. The original GM model, as introduced in [42] and [41] to modelpattern formation in biological morphogenesis, given by∂tu = ρ0(x)+ cuρu(x)u2v +Du∆u−σuu ,∂tv = cvρv(x)u2+Dv∆v−σvv ,disregards that in biological tissues morphogen-producing reactions mostly occurintracellularly and on the membranes of cells. For simplicity, we illustrate ourcompartmental-reaction diffusion theory for the 1-D case when ρ0 ≡ 0,cuρu(x) ≡1, and cvρv ≡ 1. In our model, we restrict the reaction kinetics to the boundaries sothat for the uncoupled bulk-cell model we haveµ˙(t) = f (µ,η) :=µ2η, η˙(t) = g(µ,η) := µ2 . (2.32)Similar to the case with FN kinetics we observe that µ activates self-productionand the production of η , whereas the species η only inhibits the production ofµ . Hence, the GM model has a more indirect activator-inhibitor structure thanthe FN model, in which η also inhibits directly its own production. The uncoupledequilibrium for (2.32) is non-hyberbolic in all directions, as DF(µe,ηe) = 0, whereµe = 0 and η is a constant.To apply the bulk-cell steady-state analysis of §2.2.1.2 we simply identify thatg(µ,η) = g1(µ)−g2η , where g1 = µ2 and g2 = 0. From (2.7) we getA(ηe1ηe2)=((µe1)2(µe2)2)with A =βvγ2v −1A˜ :=(A11 A12A12 A11), (2.33a)where, by using (2.6) for A˜, we identify thatA11 =βvγ2v −1(γv cosh(ωvL)−1) , A12 = βvγ2v −1(cosh(ωvL)− γv) . (2.33b)In terms of A11 and A12, we obtain from (2.9) after calculating A−1 that the28nonlinear algebraic system for µe1 and µe2 reduces to((µe1)2(µe2)2)− βuγ2u −11(A211−A212) (A11(µe1)2−A12(µe2)2 00 −A12(µe1)2+A11(µe2)2)B(µe1µe2)= 0 .(2.34)From (2.11), or equivalently from (2.34), the symmetric steady-state solution, forwhich µe1 = µe2 = µe, is given explicitly byµe =a1b1(γ2u −1)βu, ηe =µe(γ2u −1)βub1=a1b21(γ2u −1)2β 2u, (2.35)where a1 =A11+A12 and b1 are the in-phase eigenvalues of A and B given in (2.10)with g2 = 0.The linear stability of the symmetric steady-state solution is studiednumerically using the methodology in §2.2.2.2. In contrast to the case with FNkinetics, there are no Hopf bifurcation points associated with the in-phase modefor the GM model. As a result, the symmetric steady-state cannot be destabilizedby the in-phase mode. Moreover, the anti-phase mode for the linearization of thesymmetric steady-state undergoes a zero-eigenvalue crossing only at any pitchforkbifurcation point ρ = ρp that can be calculated from the nonlinear algebraic system(2.34). This symmetric steady-state is unstable to the anti-phase mode only whenρ > ρp (as suggested by the full numerical results shown below in Fig. 2.9 andFig. 2.10).In Fig. 2.8 we plot solution branches of (2.34) for Du =Dv = σu = σv = 1, L=1, and βu = 0.1, as a function of ρ = βv/βu. We observe that the symmetric branchundergoes a subcritical pitchfork bifurcation at ρ = ρp ≈ 13.10. The emergingunstable asymmetric equilibria undergo a secondary fold bifurcation at ρ f ≈ 5.59.The asymmetric branches are found to be linearly stable for ρ > ρ f . As a result,we conclude that linearly stable asymmetric patterns occur even when Du = Dv.From Fig. 2.8, the bifurcation structure of equilibria for (2.34) exhibits ahysteresis structure. As shown in Fig. 2.9 for ρ = 4 and ρ = 8, for an initialcondition near the symmetric branch when ρ ∈ [1,ρp), the full PDE simulations of(2.1) by the CN-RK4 IMEX method of Appendix A.1 verifies the convergenceto the symmetric steady-state branch. In contrast, the full numerical results29Figure 2.8: 3-D bifurcation diagram for µe1 and µe2 with bifurcationparameter ρ = βv/βu for steady-states of the GM compartmental-reaction diffusion system as computed from (2.34) using MatCont[28]. There is a subcritical pitchfork bifurcation from the symmetricsteady-state at ρp ≈ 13.09728, with (secondary) fold point bifurcationsoccurring on the asymmetric branches at ρ f ≈ 5.59447. For ρ < ρpthe symmetric steady-state is linearly stable, while for ρ > ρ f theasymmetric branches are linearly stable. Parameters: Du =Dv = 1,σu =σv = 1,L = 1, and βu = 0.1.shown in Fig. 2.10(left) for ρ > ρ f confirms that for an initial condition nearthe stable asymmetric solution branch the time-dependent solution converges tothe asymmetric branch. Moreover, in Fig. 2.10(right) when ρ = 25 > ρp aninitial condition near the unstable symmetric branch is shown to converge to theasymmetric solution branch.For the classical GM RD model without reaction compartments, it is well-known from a standard Turing analysis that stable spatially non-uniform solutionscan only occur when the diffusivity ratio Dv/Du is sufficiently large. In ourcompartmental-reaction diffusion model, in Table 2.1 we show that asymmetricsolution branches bifurcate from the symmetric branch even when Dv < Du. Asseen from this table, the required permeability ratio threshold ρp increases as30Dv/Du decreases. Furthermore, the extent of the hysteresis, as measured by thevalues of ρp − ρ f and µe1 − µe2 , becomes larger as Dv/Du decreases. Regardingthe dependence of the hysteresis on the permeabilities βu and βv = ρβu, Table 2.2shows that if βu ≥ 0.35 there is now a supercritical pitchfork bifurcation from thesymmetric branch and the hysteresis structure disappears.Dv/Du 0.4 0.5 0.6 0.7 0.8 0.9 1ρp > 100 52.18853 26.86097 19.71589 16.34470 14.38188 13.09728µe 4.83783 4.60295 4.43970 4.31969 4.22776 4.15511ρ f 9.33646 7.71235 6.82347 6.26265 5.87653 5.59447µe1 4.24548 4.03936 3.89609 3.79078 3.71011 3.64635µe2 0.70991 0.67544 0.65148 0.63387 0.620385 0.60972Dv/Du 1.1 1.2 1.3 1.4 1.5 1.6ρp 14.38188 11.51761 10.99734 10.58334 10.24607 ...µe 4.22776 4.04759 4.00670 3.97186 3.94181ρ f 5.87653 5.20997 5.07306 4.96011 4.86535µe1 3.71011 3.55200 3.51612 3.48554 3.45917µe2 0.62038 0.59395 0.58795 0.58283 0.57842Table 2.1: Numerical values (rounded to 5th decimal place) of the subcriticalpitchfork bifurcation point ρp for the symmetric branch and the two foldbifurcation points ρ f along the asymmetric branches. Bifurcation valuesfor the symmetric (µe) and one of the asymmetric (µe1 , µe2) solutionbranches are listed. As the ratio Dv/Du decreases the range of ρ wherehysteresis occurs increases. Parameters: Du = 1,σu = σv = 1,L = 1, andβu = 0.1.2.2.4 Compartments with Rauch-Millonas kineticsWe now briefly show that linearly stable asymmetric patterns can occur for acompartmental-reaction diffusion model using the generic intracellular reactionkinetics proposed in [116]µ˙ = f (µ,η) := cu−quµ+ αu1µγu1+µ− αu2µηγu2+µη˙ = g(µ,η) := cv+wvµ−qvη ,(2.36)31Figure 2.9: Full numerical computations of (2.1) by the CN-RK4 IMEXmethod of Appendix A.1 for GM kinetics.32βu 0.05 0.1 0.15 0.2 0.25 0.3 0.35ρp 20.44213 13.09728 11.42015 11.59667 13.33412 17.91935 33.18867µe 7.05336 4.15511 3.21380 2.76056 2.50167 2.33930 2.23156ρ f 4.69415 5.59447 6.82631 8.63523 11.58010 17.27125 -µe1 3.82650 3.64635 3.45845 3.25882 3.03717 2.76171 -µe2 0.32048 0.60972 0.87895 1.13950 1.40694 1.71692 -Table 2.2: Numerical values (rounded to 5th decimal place) of the subcritical(or supercritical) pitchfork bifurcation point ρp, the two fold bifurcationpoints ρ f , and the corresponding values for the symmetric (µe) and one ofthe asymmetric (µe1 , µe2) solution branches. As βu increases from 0.05,the range of ρ where hysteresis occurs decreases, until a supercriticalpitchfork bifurcation arises when βu ≈ 0.35. Parameters: Du = Dv =1,σu = σv = 1,L = 1.Figure 2.9: Left: Convergence to the symmetric branch for ρ = 4 startingclose to the symmetric branch. Right: Convergence to the symmetricbranch for ρ = 8 and starting near the symmetric branch. The remainingparameter values are as in the caption to Fig. 2.8.where we identify g1(µ) = cv+wvµ and g2 = qv. This kinetic model was proposedin [116] as a simplified universal description for signal transduction kinetics incells.Owing to the special form of g(µ,η), we can readily implement the steady-state theory of §2.2.1.2 when the kinetics (2.36) are restricted to the domainboundaries. Parameters are chosen as in [116] to ensure that the uncoupleddynamics (2.36) has a stable equilibrium point.With ωv or the permeability ratio ρ = βv/βu as bifurcation parameters, we useMatCont [28] to numerically compute steady-states of the bulk-cell model. Withthe other parameters similar to those in [116], in Fig. 2.11(right) we observe thatasymmetric patterns do emerge from the symmetric solution branch as ρ exceedsa threshold. Pitchfork bifurcations in wv also occur (see Fig. 2.11(left)), and theycorrespond as before to a zero-eigenvalue crossing of the anti-phase perturbationof the symmetric steady-state. The zero-eigenvalue crossing can be observed inFigure 2.12. The full-time dependent numerical computations shown in Fig. 2.1333Figure 2.10: Full numerical computations of (2.1) by the CN-RK4 IMEXmethod for GM kinetics.34Figure 2.10: Left: Convergence to the asymmetric branch for ρ = 8 startingnear one of the stable asymmetric branches for ρ > ρ f . Right:Convergence to an asymmetric branch for ρ = 25 starting close to thesymmetric branch with perturbation toward the respective asymmetricbranch. The remaining parameter values are as in the caption toFig. 2.8.confirms the existence of stable asymmetric steady-state patterns.Figure 2.11: 3-D bifurcation diagrams using MatCont [28] for the two-cell compartmental-reaction diffusion system with Rauch-Millonas(RM) kinetics (2.36). Left: Plot of µe1 showing that asymmetricsteady-states occur inside the degenerate pitchfork bubble delimitedby wP,1v ≈ 6.67610 and wP,2v ≈ 7.28516 when ρ = βv/βu = 7. Right:Supercritical pitchfork bifurcation from the symmetric branch occurswhen wP,2v ≈ 7.28516. Linearly stable asymmetric branches occur pastthis threshold in ρ . Parameters: Du = Dv = 1,σu = σv = 0.01,L =1,cu = cv = 1,qu = 1/100,qv = 7,αu1 = 600,αu2 = 6,γu1 = 100,γu2 =1/10, and βu = 0.3.2.3 A Periodic Compartmental-Reaction DiffusionModelWe now extend the two-compartment model to allow for a periodic array ofidentical reaction compartments in 1-D. Our goal is to investigate symmetry-breaking behavior from a symmetric steady-state for this extended model. For aperiodic chain of compartments there can be multiple possible modes of instability35Figure 2.12: Zero-eigenvalue crossing for the growth rate λ of theasymmetric eigenperturbation in direction q2 = (1,−1)T between thetwo supercritical pitchfork points wP,1v and wP,2v , as obtained as rootsof F− of (2.30). This perturbation is exponentially growing fromthe symmetric steady-state and therefore causes convergence to anasymmetric state. Parameters are as for Figure 2.11 with ρ = 7.of the symmetric steady-state.In our formulation, we consider a periodic chain of a fixed length nL,n ∈N\{0,1},L > 0, with compartments centered at x j = (2 j−1)2 L for j ∈ {1, ...,n},so that for x ∈ R\(nL ·Z) we have the following bulk-cell system on the domain0 < x < nL (see Fig. 2.14 for a schematic):bulk∂tu = Du ∂xxu−σuu , t ∈ (0,∞) , x ∈ (0,nL)\⋃nj=1{x j}∂tv = Dv ∂xxv−σvv, t ∈ (0,∞), x ∈ (0,nL)\⋃nj=1{x j} (2.37a)bulk boundaryu(t,0) = u(t,nL) , v(t,0) = v(t,nL) (periodic BC)∂xu(t,0) = ∂xu(t,nL) , ∂xv(t,0) = ∂xv(t,nL) (2.37b)reaction fluxes[Du∂xu]|x=x j = βu (u(t,x j)−µ j(t)) (cell jump conditions)[Dv∂xv]|x=x j = βv (v(t,x j)−η j(t)) (2.37c)compartmentsµ˙ j = f (µ j,η j)+ [Du∂xu]|x=x j (reaction kinetics at x = x j)η˙ j = g(µ j,η j)+ [Dv∂xv]|x=x j , (2.37d)36Figure 2.13: Full numerical computations of (2.1) by the CN-RK4 IMEXmethod of Appendix A.1 for Rauch-Millonas kinetics (2.36). Forρ = 15 and wv =wP,2v ≈ 7.28516, and with an initial condition near theunstable symmetric branch (see Fig. 2.11(right)), the time-dependentsolution converges to a stable asymmetric pattern. Parameters are as inthe caption of Fig. 2.11.for j = {1, . . . ,n}. Here for any functionF we have defined [F ]|x=x˜ :=F (x˜+)−F (x˜−).Figure 2.14: The periodic compartmental-reaction diffusion model on [0,nL]with compartments centered at the blue dots. Two bulk diffusingspecies provide the inter-compartment coupling.372.3.1 Compartments with general reaction kineticsWe will first construct a symmetric steady-state solution for (2.37) and we willanalyze the linear stability of this steady-state by using a Floquet-based approach.Owing to the complexity of (2.37), we will not construct asymmetric steady-statesof (2.37).2.3.1.1 Symmetric steady-state and the linear stability problemTo construct a symmetric steady-state ue, ve for (2.37) we need only consider thefundamental domain [0,L] upon which we impose periodic boundary conditionsue(0) = ue(L) , ve(0) = ve(L) , ∂xue(0) = ∂xue(L), ∂xve(0) = ∂xve(L) .(2.38)The symmetric steady-state is obtained by a periodic extension to the entire chain.To represent the steady-states in the bulk, we introduce the periodic Greenfunction Gω(x).Definition 2.3.1. The periodic Green function on [0,L] with singularity at x = L2 isthe solution of∂xxGω −ω2Gω =−δ (x− L2 ) , x ∈ [0,L] ; Gω(0) = Gω(L) , ∂xGω(0) = ∂xGω(L) ,(2.39a)Gω(L2+) = Gω(L2−) , ∂xGω(L2+)−∂xGω(L2−) =−1 . (2.39b)The solution to (2.39) is readily calculated asGω(x) = 12ω coth(ωL2)cosh(ω(L2 − x))− 12ω sinh(ω(L2 − x) , x ∈ [0, L2 ],12ω coth(ωL2)cosh(ω(L2 − x))+ 12ω sinh(ω(L2 − x)) , x ∈ [L2 ,L] ,(2.40)where we identify that Gω (L/2) = 12ω coth(ωL/2).Upon defining ωu :=√σu/Du and ωv :=√σv/Dv the bulk steady-statesolutions satisfying (2.37a) and (2.37c) on the fundamental domain [0,L] with38periodic boundary conditions (2.38) areue(x) = c1Gωu (x)Gωu (L/2)with c1 :=βuβu+2Duωu tanh(ωuL/2) µe ,ve(x) = c2Gωv (x)Gωv (L/2)with c2 :=βvβv+2Dvωv tanh(ωvL/2) ηe .(2.41)Finally, upon substituting (2.41) into the steady-state of the intra-compartmentdynamics (2.37d) we obtain that µe and ηe satisfy the nonlinear algebraic systemf (µe,ηe)=2Duωuβuβu coth(ωuL/2)+2Duωuµe , g(µe,ηe)=2Dvωvβvβv coth(ωvL/2)+2Dvωvηe .(2.42)In this way, for a given reaction-kinetics, we can determine the symmetricequilibrium (µe,ηe) by a numerical root finding on (2.42).We now formulate and study the linear stability problem for the symmetricsteady-state solution branch. For the linear stability analysis we perturb thesymmetric steady-state by introducing u(t,x) = ue(x)+ φ(x)eλ t ,v(t,x) = ve(x)+ψ(x)eλ t ,µ j(t) = µe+ξ jeλ t ,η = ηe+ζ jeλ t , where |φ | ≪ 1, |ψ| ≪ 1, |ξ j| ≪ 1 and|ζ j| ≪ 1 for j ∈ {1,2}, into (2.37) and linearizing. This yields the linearizedeigenvalue problembulk∂xxφ −Ω2uφ = 0 , x ∈ (0,nL)\⋃nj=1{x j}∂xxψ−Ω2vψ = 0 , x ∈ (0,nL)\⋃nj=1{x j}(2.43a)bulk boundaryφ(0) = φ(nL) , ψ(0) = ψ(nL) (periodic BC)∂xφ(0) = ∂xφ(nL) , ∂xψ(0) = ∂xψ(nL) (2.43b)reaction fluxes[Du∂xφ ]|x=x j = βu (φ(x j)−ξ j) (cell jump conditions)[Dv∂xψ]|x=x j = βv (ψ(x j)−ζ j) (2.43c)compartmentsλξ j = ∂µ feξ j +∂η feζ j +[Du∂xφ ]|x=x j (reaction kinetics at x = x j)λζ j = ∂µgeξ j +∂ηgeζ j +[Dv∂x]|x=x j ,(2.43d)for j ∈ {1, ...,n}. Here we have defined Ωu :=√(λ +σu)/Du and Ωv :=√(λ +σv)/Dv.To study this eigenvalue problem, it is convenient to use a Floquet-based39approach where we consider the fundamental cell problem on [0,L] and imposea Floquet-type boundary condition:bulk∂xxφ −Ω2uφ = 0 , x ∈ (0,L)\{L2}∂xxψ−Ω2vψ = 0 , x ∈ (0,L)\{L2} (2.44a)bulk boundaryφ(0) = Zφ(L), ψ(0) = Zψ(L) (Floquet BC)∂xφ(0) = Z∂xφ(L) , ∂xψ(0) = Z∂xψ(L) (2.44b)reaction fluxes[Du∂xφ ]|x= L2 = βu (φ(L2 )−ξ ) (cell jump conditions)[Dv∂xψ]|x= L2 = βv (ψ(L2 )−ζ ) (2.44c)intracellularλξ = ∂µ feξ +∂η feζ +[Du∂xφ ]|x= L2 (reaction kinetics at x = L2 )λζ = ∂µgeξ +∂ηgeζ +[Dv∂xψ]|x= L2 , (2.44d)for Z ∈ C. Upon using translational invariance, we get the chainφ(0) = Zφ(L) = Z2φ(2L) = ...= Znφ(nL) ,so that in order to satisfy φ(0) = φ(nL), we must have thatZn = 1 ⇔ Zk = e2πik/n , for k ∈ {0, ...,n−1} . (2.45)Such a Floquet-inspired approach to more readily derive the eigenvalueequation in this periodic setting has been used previously in other contexts. In§3 of [74] a similar approach was used to analyze the stability of a periodic patternof localized hot-spots or “spikes” for a reaction-diffusion model of urban crime.It was also used in §5.2 of [50] to analyze the linear stability of a compartment-reaction diffusion model with a periodic array of compartments, but with only one-bulk diffusing species.Remark 2.3.1. Since Z = 1 when k = 0, the mode k = 0 represents in-phaseperturbations of the symmetric steady-state. There are n−1 other possible modesof instability, which we shall refer to as “asymmetric modes”. For the special casen = 2, we have Z1 = −1 and φ(0) = −φ(L), and we refer to this mode as the40anti-phase mode.In order to derive an explicit formula for the eigenvalues λ of (2.44) for a givenZ, we must first introduce the the quasi-periodic Green function GΩ,Z .Definition 2.3.2. The quasi-periodic Green function on the fundamental domain[0,L] with singularity at x = L2 is the solution of∂xxGΩ,Z−Ω2GΩ,Z =−δ (x− L2 ) , x ∈ [0,L] , (2.46a)GΩ,Z(0) = ZGΩ,Z(L) , ∂xGΩ,Z(0) = Z∂xGΩ,Z(L) , (2.46b)GΩ,Z(L2+) = GΩ,Z(L2−) , ∂xGΩ,Z(L2+)−∂xGΩ,Z(L2−) =−1 . (2.46c)Upon satisfying the continuity and jump conditions in (2.46c), the explicitsolution to (2.46a) can be written in terms of two constants AΩ and BΩ asGΩ,Z(x) =AΩ cosh(Ω(L2 − x))+BΩ sinh(Ω(L2 − x)), x ∈ [0, L2 ] ,AΩ cosh(Ω(L2 − x))+( 1Ω +BΩ)sinh(Ω(L2 − x)), x ∈ [L2 ,L] .(2.47)Then, by satisfying the quasi-periodic boundary conditions in (2.46a), we obtain amatrix problemE(AΩBΩ)=ZΩ(−sinh(ΩL2 )cosh(ΩL2) ) , with E :=((1−Z)cosh(ΩL2 ) (1+Z)sinh(ΩL2 )(1+Z)sinh(ΩL2)(1−Z)cosh(ΩL2 )).Next, we invert the matrix E by using det(E) = 1− 2Z cosh(ΩL) + Z2 and Z +1/Z = 2Re(Z) when |Z|2 = 1. This determines AΩ and BΩ as(AΩBΩ)=12Ω(Re(Z)− cosh(ΩL))(−sinh(ΩL)−Z+ cosh(ΩL)). (2.48)The explicit formula for GΩ,Z immediately follows from (2.48) and (2.47). Bysetting x = L/2 in (2.47) and using (2.48), we getGΩ,Z(L2)=sinh(ΩL)2Ω [cosh(ΩL)−Re(Z)] . (2.49)41As a remark, for Z = 1 in (2.47) and (2.48) we recover the periodic Green functionin (2.40).In terms of this Green function, the solution to (2.44a)–(2.44c) isφ(x) = d1GΩu,Z(x)GΩu ,Z(L/2)with d1 =βuGΩu ,Z(L/2)βuGΩu ,Z(L/2)+Duξ ,ψ(x) = d2GΩv,Z(x)GΩv,Z(L/2)with d2 =βvGΩv,Z(L/2)βvGΩv,Z(L/2)+Dvζ .Finally, from (2.44d) we obtain an explicit nonlinear matrix eigenvalue problemfor determining λMZ(λ )(ξζ)=(00), (2.50a)where MZ(λ ) :=(λ −∂µ fe+ βuDuβuGΩu,Z(L/2)+Du −∂η fe−∂µge λ −∂ηge+ βvDvβvGΩv,Z(L/2)+Dv).(2.50b)In calculating GΩv,Z (L/2) and GΩu,Z (L/2) from (2.49) we recall that Ωu =√(λ +σu)/Du and Ωv =√(λ +σv)/Dv.It follows that λ is an eigenvalue of the linearization (2.43) if and onlyif for a given Z = Zk with k ∈ {0, . . . ,n−1}, we have det(MZ(λ )) = 0. Todetermine zero-eigenvalue crossings for a given mode Zk when a parameter, suchas the permeability ratio ρ = βv/βu, is varied, we must simultaneously solvedet(MZk(0)) = 0 together with the nonlinear algebraic system (2.42) for the steady-state to identify the bifurcation point. Such zero-eigenvalue crossings for mode Zkcorrespond to symmetry-breaking bifurcations from the symmetric steady-state.We now clarify how the Floquet modes Zk are related to the form of theperturbation near each cell centered at x = x j = ( j−1/2)L for j ∈ {1, . . . ,n},whenever we have a zero-eigenvalue crossing λ = 0 obtained from settingdet(MZk(0)) = 0 for any k ∈ {0, . . . ,n−1}. Labeling φcell j,k := φ (( j−1/2)L),we have by translation invariance that φcell j,k = Zj−1k φ (L/2) when Z = Zk. Inthis way, we can define a vector characterizing the perturbation of the symmetric42steady-state ue at the cell locations byφcell,k :=φcell1,kφcell2,k...φcelln,k=1Zk...Zn−1kφ (L/2) . (2.51)In a similar way, we can identify for (2.43) in terms of the fundamental cellvariables that ψcell j,k := ψ (( j−1/2)L) = Z j−1k ψ (L/2), ξ jk = Z j−1k ξ , and ζ jk =Z j−1k ζ for j ∈ {1, . . . ,n}.Since GΩ,Zk (L/2) from (2.49) and, consequently, MZk(λ ) in (2.50) are real-valued when λ is real, we conclude that φ (L/2) is real-valued when λ is real. Bytaking the real and imaginary parts of (2.51) we conclude thatφcell,k ∈ span{vck,vsk} , vck :=1...cos(2π( j−1)kn)...cos(2π(n−1)kn), vsk :=0...sin(2π( j−1)kn)...sin(2π(n−1)kn).(2.52)When n = 2, we have either Z0 = 1 or Z0 = −1. For the in-phase modeZ0 = 1, we have vsk = 0, and so (2.52) yields φcell,0 ∈ span{vc0}= span{(1,1)T}.Likewise, for the anti-phase mode Z0 = −1 we have φcell,1 ∈ span{vc1} =span{(1,−1)T}. For any n ≥ 2, we always have the in-phase mode Z0 = 1 forwhich φcell,0 ∈ span{vc0}= span{(1, . . . ,1)T}.Now suppose that n > 2 is even. Then, since Zn/2 = eiπ = −1, wewill have a zero-eigenvalue crossing for an anti-phase perturbation φcell,k/2 ∈span{(1,−1, . . . ,1,−1)T}, whenever there is a bifurcation parameter for whichdet(MZn/2(0)) = 0. Furthermore, since ReZk = ReZn−k and, consequently,MZk(λ ) = MZn−k(λ ) in (2.50) for k ∈ {1, . . . , n2 −1}, we conclude that thebifurcation parameter thresholds for which det(MZk)(0) = 0 and det(MZn−k)(0) = 0are identical. As a result, we have n/2−1 “degenerate” asymmetric modes whenn > 2 is even. Since vck = vc(n−k) and vsk = −vs(n−k) for k ∈ {1, . . . , n2 −1}, we43conclude from (2.52) that φcell,k ∈ span{vck,vsk} for k ∈ {1, . . . , n2 −1}.In contrast, for n ≥ 3 with n odd, there is no anti-phase mode and there areexactly n+12 − 1 possible degenerate asymmetric modes for which the bifurcationparameter thresholds are determined from setting det(MZk)(0) = 0 for k ∈{1, . . . , n+12 −1}. We again conclude from (2.52) that φcell,k ∈ span{vck,vsk}for k ∈ {1, . . . , n+12 −1}. In particular, when n = 3, we conclude that atthe zero-eigenvalue crossing for either Z = Z1 or Z = Z2 there is a two-dimensional eigenspace for the vector φcell,1 of cell perturbations given by theplane (1,1,1)Tφcell,1 = 0.2.3.2 Two compartments with g(µ,η) = g1(µ)−g2ηThe analysis so far for the periodic problem has focused on constructing thesymmetric steady-state and formulating the linear stability problem so as to readilydetect symmetry-breaking bifurcation points for any n ≥ 2 and arbitrary intra-compartment reactions f and g.In this section, we briefly focus on the two-compartment case n = 2when g(µ,η) = g1(µ)− g2η with an aim of constructing both symmetric andasymmetric branches of equilibria. Since the analysis is similar to that in §2.2.1.2we only briefly highlight it here.For n = 2 compartments, the steady-state for ue has the formue(x) =A1 sinh(ωu(L2 − x))+A2 sinh(ωux) , x ∈ [0, L2 ] ,A3 sinh(ωu(3L2 − x))+A4 sinh(ωu(x− L2 )), x ∈ [L2 , 3L2 ] ,A5 sinh(ωu(2L− x))+A6 sinh(ωu(x− 3L2 )), x ∈ [3L2 ,2L] ,where ωu :=√σu/Du. Upon imposing that ue(0) = ue(2L), ∂xue(0) = ∂xue(2L),and ue is continuous at x = L/2 and x = 3L/2, we obtain thatA1 = A6 = A3+A4 , A2 = A3sinh(ωuL)sinh(ωuL/2), A5 = A4sinh(ωuL)sinh(ωuL/2).By enforcing the jump conditions for ue at x = L/2 and x = 3L/2, we determine44A3 and A4 as(A3A4)=βuDuωu(γ2u −4)(γu 22 γu)(µe1µe2), with γu := 2cosh(ωuL)+βuDuωusinh(ωuL) .Likewise, we can calculate ve(x). Then, from the steady-state of the intra-compartment dynamics, we proceed as in §2.2.1.2 to obtain a non-linear algebraicequation for µe1 and µe2 characterizing all steady-states of the two-compartmentsystem. We obtain as in §2.2.1.2 that(f (µe1 ,(1,0)A−1(g1(µe1),g1(µe2))T )f (µe2 ,(0,1)A−1(g1(µe1),g1(µe2))T ))− βuγ2u −4B(µe1µe2)= 0 , (2.53)where A and B are now defined byA :=βvγ2v −4A˜+g2I , A˜ :=(2γv cosh(ωvL)−4 4cosh(ωvL)−2γv4cosh(ωvL)−2γv 2γv cosh(ωvL)−4),(2.54a)B :=(2γu cosh(ωuL)−4 4cosh(ωuL)−2γu4cosh(ωuL)−2γu 2γu cosh(ωuL)−4). (2.54b)Here γv := 2cosh(ωvL)+ βvDvωv sinh(ωvL).In place of (2.10), the eigenvalues a1,b1 and a2,b2 of the matrices A and B forthe in-phase q1 = (1,1)T and anti-phase q2 = (1,−1)T modes area1 =βvγ2v −4[2γv (cosh(ωvL)−1)+4(cosh(ωvL)−1)]+g2 (2.55a)a2 =βvγ2v −4[2γv (cosh(ωvL)+1)−4(cosh(ωvL)+1)]+g2 , (2.55b)b1 = 2γu (cosh(ωuL)−1)+4(cosh(ωuL)−1) , (2.55c)b2 = 2γu (cosh(ωuL)+1)−4(cosh(ωuL)+1) . (2.55d)In terms of a1 and b1, and in analogy with (2.11), the symmetric steady-state45(µe1 ,µe2) = µeqT1 , is obtained from the scalar algebraic equationf (µe,g1(µe)a1)− βuγ2u −4b1µe = 0 . (2.56)As a remark, for the special case where g = g1(µ)− g2η , we can verify after adirect but tedious calculation that the steady-state condition (2.42) is equivalent to(2.56).2.3.3 FitzHugh-Nagumo kineticsTo illustrate the theory, we first choose the FN kinetics as given in (2.2). Forn = 2, for which (2.53), (2.56) and (2.55) apply, the pitchfork points zP,1 and zP,2delimiting the pitchfork bubble in z are recovered from the parameter constraint(2.22) in which a1, a2, b1, and b2 are now given as in (2.55). For n = 2, we can useMatCont [28] on (2.53) to calculate both symmetric and asymmetric steady-statesusing either z or the permeability ratio ρ = βv/βu as parameters. The bifurcationdiagrams are qualitatively similar to those in Fig. 2.3 (not shown).For n = 2, in Fig. 2.15 we show, for an initial condition near the unstablesymmetric branch, that the time-dependent numerical solution computed using theBE-RK4 IMEX scheme of Appendix A.2 tends to a stable asymmetric pattern.For n = 3, and for a parameter set where the symmetric steady-state is unstable,in Fig. 2.16 we show the time-dependent convergence to two different stableasymmetric patterns depending on the specific form of the initial perturbation ofthe symmetric state. For n = 3, we conclude from (2.52) and the discussion in§2.3.1.1 that at the zero-eigenvalue crossing for an asymmetric mode there is atwo-dimensional eigenspace for the vector cell perturbation φcell,1, given by theplane (1,1,1)Tφcell,1 = 0.2.3.4 Gierer-Meinhardt kineticsNext, we consider the periodic problem with GM intra-compartmental kinetics(2.32). By solving (2.53) and (2.56) using MatCont [28], in Fig. 2.17 we showthat the bifurcation diagram of steady-states for n = 2 compartments and withDu =Dv = 1 exhibits a hysteresis structure between the symmetric and asymmetric46Figure 2.15: FN kinetics with n= 2: For an initial condition near the unstablesymmetric branch, and for z = zP,2 ≈ 6.63675 and ρ = βv/βu = 60,the full time-dependent solution computed using the BE-RK4 IMEXscheme of Appendix A.2 converges to a stable asymmetric steady-state. For z = 6.63675, the pitchfork bifurcation point in ρ is ρp = 50.Parameters: Du = 1,Dv = 2,σu = σv = 1,ε = 0.6,q = 1,L = 1, andβu = 0.3.steady-state solution branches.From the numerical results given in Table 2.3, we observe that asymmetricbranches do bifurcate from the symmetric branch even when Dv/Du < 1, and thatthe range of ρ where hysteresis occurs is larger as the ratio Dv/Du decreases. InTable 2.4 we show for Du = Dv = 1 that as βu is increased beyond βu = 0.7 thepitchfork bifurcation switches from sub- to super-critical. The overall trends arequalitatively similar to those given in Tables 2.1 and 2.2 for the case of reactivedomain boundaries.Although we have not provided a linear stability theory for the asymmetric47Figure 2.16: FN kinetics with n = 3: Full-numerical time-dependent resultsfrom the BE-RK4 IMEX scheme of Appendix A.2.48Figure 2.16: Left: Convergence to the asymmetric branch for ρ = 8 startingnear one of the stable asymmetric branches for ρ > ρ f . Right:Convergence to an asymmetric branch for ρ = 25 starting close to thesymmetric branch with perturbation toward the respective asymmetricbranch. The remaining parameter values are as in the caption toFig. 2.8.DvDu0.4 0.5 0.6 0.7 0.8 0.9 1ρp > 100 53.17080 35.32088 28.23715 24.43531 22.06333 20.44213µe 8.21230 7.81358 7.53645 7.33272 7.17669 7.05336ρ f 6.25939 5.66710 5.29081 5.03058 4.83989 4.69415µe1 4.45524 4.23893 4.08858 3.97806 3.89341 3.82650µe2 0.37314 0.35503 0.34243 0.33318 0.32609 0.32048DvDu1.1 1.2 1.3 1.4 1.5 1.6ρp 19.26390 18.36889 17.66592 17.09916 16.63252 ...µe 6.95344 6.87085 6.80144 6.74229 6.69129ρ f 4.57915 4.48608 4.40922 4.34468 4.28971µe1 3.77230 3.72749 3.68983 3.65775 3.63008µe2 0.31594 0.31219 0.30904 0.30635 0.30403Table 2.3: Numerical values (rounded to 5th decimal place) of the subcriticalpitchfork bifurcation point ρp for the symmetric branch and the foldpoints ρ f along the asymmetric branches. Bifurcation values for thesymmetric (µe) and one of the asymmetric (µe1 , µe2) solution branchesare listed. As Dv/Du decreases, the range of ρ where hysteresis occursincreases. Parameters: Du = 1,σu = σv = 1,L = 1, and βu = 0.1.steady-state solution branches in Fig. 2.17, the full PDE numerical results shown inFigs. 2.18–2.19, as computed using the BE-RK4 IMEX scheme of Appendix A.2,confirm the hysteresis structure of Fig. 2.17 and the prediction of stable asymmetricpatterns.2.3.5 Rauch-Millonas kineticsFinally, we briefly show that stable asymmetric patterns can also occur with Rauch-Millonas kinetics (2.36) for n= 2 compartments when Du =Dv = 1. Upon solving(2.53) with (2.36) using MatCont [28], in the left and right panels of Fig. 2.20 we49Figure 2.17: 3-D Bifurcation diagram, exhibiting hysteresis, with bifurcationparameter ρ = βv/βu for steady-states with GM kinetics and n = 2computed from (2.53) using MatCont [28]. The subcritical pitchforkbifurcation point from the symmetric steady-state is at ρ = ρp ≈11.42015 and the fold points on the asymmetric branches are atρ f ≈ 6.82631. The symmetric steady-state is linearly stable only for1 < ρ < ρp, while the asymmetric branches become stable only pastthe fold points. Parameters: Du = Dv = 1,σu = σv = 1,L = 1, andβu = 0.3. Note that the geometry of the asymmetric branches near thefold points is just mirrored and it looks different in the figure becauseof the 3-D plot.plot bifurcation diagrams of µe1 versus either wv or ρ , respectively. In Fig. 2.20(left)we observe that asymmetric steady-states occur in a wv-pitchfork bubble, whilefrom Fig. 2.20(right) we observe that asymmetric steady-states emerge from asupercritical ρ-pitchfork bifurcation that opens as ρ increases above a threshold.The full-numerical simulations shown in Fig. 2.21 for ρ = 15 and wv = wP,2v areconsistent with the prediction of Fig. 2.20(right) of stable asymmetric patterns.50Figure 2.18: GM kinetics with n = 2. Full numerical time-dependent resultsfrom the BE-RK4 IMEX scheme.51Figure 2.18: Left: Convergence to the stable symmetric branch when ρ = 6starting just prior to the fold point where a stable asymmetric steady-state is born. Right: Convergence to the stable symmetric branch forρ = 9 with an initial condition near the symmetric branch. Remainingparameter values are as in the caption to Fig. 2.17.βu 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4ρp 36.28648 20.44213 15.39744 13.09728 11.94362 11.42015 11.32738 11.59667µe 12.90881 7.05336 5.11479 4.15511 3.58673 3.21380 2.95240 2.76056ρ f 4.32660 4.69415 5.11246 5.59447 6.15768 6.82631 7.63502 8.63523µe1 3.91408 3.82650 3.73728 3.64635 3.55352 3.45845 3.36053 3.25882µe2 0.16492 0.32048 0.46828 0.60972 0.74617 0.87895 1.00951 1.13950βu 0.45 0.5 0.55 0.6 0.65 0.7ρp 12.23745 13.33412 15.08439 17.91935 22.88248 33.18867µe 2.61497 2.50167 2.41175 2.33930 2.28021 2.231556ρ f 9.90668 11.58010 13.88586 17.27125 22.73469 -µe1 3.15185 3.03717 2.91034 2.76171 2.55994 -µe2 1.27099 1.40694 1.55225 1.71692 1.93262 -Table 2.4: Similar caption as in Table 2.3 except that the bifurcationquantities for ρ = βv/βu are tabulated for various values of βu whenDv = Du = 1. The range where hysteresis occurs decreases as βuincreases, and when βu ≈ 0.7 the pitchfork bifurcation switches fromsub- to super critical. Other parameters are as in the caption of Table 2.3.2.4 Extension: Introduction of Randomness into theKineticsThis section provides a discussion of work in progress on random kinetics incompartmental-reaction diffusion systems. Since the behavior of biological cellsis continuously randomly perturbed due to the nature of the chemical reactionsinvolved in the DNA-mRNA-protein production cascade (cf. [129], [145], [60]), itis of importance to extend the deterministic theory by deriving and incorporatingprecise random noise terms into the intracellular reaction kinetics. As evidencethat analyzing the symmetry-breaking behavior of random RD systems is yieldingnovel results different from their PDE system pendants, in [54] it is shown that52Figure 2.19: GM kinetics with n = 2. Full numerical time-dependent resultsfrom the BE-RK4 IMEX scheme.53Figure 2.19: Left: Convergence to a stable asymmetric branch when ρ = 9starting close to the stable asymmetric branch. Right: Convergence toa stable asymmetric branch when ρ = 25 starting close to the unstablesymmetric branch. Remaining parameter values are as in the captionto Fig. 2.17.Figure 2.20: 3-D Bifurcation diagrams for Rauch-Millonas kinetics (2.36)with n = 2 computed from (2.53) using MatCont [28]. Left: Plot ofµe1 showing that asymmetric steady-states occur inside a degeneratepitchfork bubble bounded by wP,1v ≈ 6.34518 and wP,2v ≈ 7.64062 whenρ = βv/βu = 7. Right: Supercritical pitchfork bifurcation in ρ from thesymmetric branch occurs when wv =wP,2v . Stable asymmetric branchesoccur past this threshold in ρ . Parameters: Du = Dv = 1,σu = σv =0.01,cu = cv = 1,qu = 1/100,qv = 7,αu1 = 600,αu2 = 6,γu1 = 100,γu2 =1/10, and βu = 0.3.including stochasticity in RD systems can lower the threshold for the onset ofpattern formation. For an overview of stochastic modeling in RD systems see [38],where the authors illustrate that in random systems switching between stochasticequilibria and stochastic resonance can occur, so that the equilibrium of theexpected value of the stochastic process can be quite different from the equilibriumof the deterministic reaction rate formulation of the same system. However, it mustbe noted that, due to the random fluctuations of a stochastic process, there is nousual equilibrium. Rather, the stationary distribution of an SDE system, if it exists,provides insight into the attractor the system is converging to. The classic book [5]provides a first introduction to stochastic bifurcation theory.Now I would like to show the derivation of the noise term corresponding to the54Figure 2.21: Rauch-Millonas kinetics with n = 2. For an initial conditionnear the unstable symmetric branch, and for ρ = 15 and wv =wP,2v , the full time-dependent solution computed using the BE-RK4-IMEX scheme of Appendix A.2 converges to a stable asymmetricsteady-state. The results are consistent with the prediction from thebifurcation diagram in Fig. 2.20(right). The remaining parametervalues are as in the caption of Fig. 2.20. Observe that ue is nearlypiecewise linear owing to the fact that the bulk degradation σu is verysmall, i.e. σu = 0.01.Michaelis-Menten approximation [124] of a generic enzymatic reaction cascadefrom a substrate S to a product P through forming a complex C with an enzyme Egiven byE+Sc f−−⇀↽−cb Ccp−−→ E+P ,where all capitalized variables are denoting the molecule counts of the respectivechemical species and c f , cb, and cp are the exponential reaction rates. The55deterministic Michaelis-Menten assumptions are that E ≪ S, that C is in quasi-equilibrium, that is,dCdt= c f ES− cbC− cpC != 0(in an appropriate time scale t), and that the enzyme building block count isconserved, i.e., E +C = ET = const. This immediately givesC =c fcb+ cp(ET −C)S ⇔ C = ETKM +SSwith the Michaelis-Menten constant KM :=cb+cpc f. In this way, the reaction cascadeis reduced to the single artificial overall reactionScM(S)−−−→ Pwith the now nonlinearly S-dependent reaction rate cM(S) := ET/(KM + S). Justas described in [60], this reaction can be simulated with the stochastic simulationalgorithm. To derive an SDE formulation of the reaction, also called the chemicalLangevin equation, one proceeds as follows. If the propensity cM(S)S does notchange significantly over time intervals [t, t + τ) with small τ > 0, which is thecase if S≫ KM or if the time increment τ is small enough, the reactions happeningwith exponentially distributed waiting times with their respective reaction rateparameter during the time intervals [t, t + τ) can be combined via a Poissonrandom variable P ∼ Poi(cM(S)Sτ). This then gives S(t + τ) = S(t)−P andP(t + τ) = P(t)+P . If additionally cM(S)Sτ is large, which is the case if S≫ 1τ ,the Poisson random variable can be well-approximated by a, now continuouslydistributed, Gaussian random variable with the same mean cM(S)Sτ and variancecM(S)Sτ . With a standard Gaussian random variable Z∼N (0,1), we hence obtainS(t+ τ)−S(t) = −cM(S)Sτ−√cM(S)S√τZ,P(t+ τ)−P(t) = cM(S)Sτ+√cM(S)S√τZ.56In the limit τ → 0, the equations reduce to the SDE systemdS = −cM(S)S dt−√cM(S)S dW (t),dP = cM(S)S dt+√cM(S)S dW (t).with a standard Wiener process W . All in all, we see that we arrive atdS = − ET SKM +Sdt−√ET SKM +SdW (t),dP =ET SKM +Sdt+√ET SKM +SdW (t),and directly recognize that the nonlinear noise term consists of the square root ofthe propensity multiplying a standard Wiener process differential interpreted in theIto´ sense.Since randomness of reaction kinetics is most apparent when theconcentrations of the involved chemical species are low, it would especially payto incorporate the correct random noise into the intracellular and cell membranekinetics rather than into the bulk kinetics. Then, because only randomness ofstrictly nonlinear reaction kinetics yield different behavior in expectation than theirdeterministic reaction rate formulation, it seems appropriate to disregard randomperturbations in our linear membrane reaction kinetics and to randomly perturbsolely the nonlinear reaction kinetics inside the cells.Referring back to the reaction terms of the intracellular Rauch-Millonas kinetics(2.36), we can identify the corresponding Michaelis-Menten terms, which are alsodiscussed in the original article [116]. Adding the square root of each singlereaction term multiplied by seven independent Brownian motions correspondingto the reactions of the activator-inhibitor system, I solved the compartmental-reaction diffusion system with parameters as in Figure 2.21 but with wv = 7 insidethe deterministically computed pitchfork bubble (delimited by wP,1v ≈ 6.34518 andwP,2v ≈ 7.64062 when ρ = βv/βu = 7) around the deterministic pitchfork point. Forwv = 7, the deterministic theory developed in 2.3 yields the supercritical pitchforkpoint ρb ≈ 4.08982. Figure 2.22 shows 20 realizations of the system as computedby a similar centred IMEX operator splitting scheme to the one in Appendix A.257which now includes integrating the additional random operator just simply with theEuler-Maruyama method, causing a step-wise error of O(√∆t). Here we can seethat still a kind of symmetry-breaking is preserved but also that it may be difficultto quantify it precisely. It seems that stochastic resonance does not take place to alarger extent for random RM kinetics with this parameter set since the fluctuatingtrajectories seem to be approximately centered about the deterministic steady-statesafter a short time has passed. However, the bifurcation point may be substantiallydifferent.The next step would be to derive a Fokker-Planck equation for the stationarydistribution of the dynamics, if it exists, since one can observe an indication ofa stochastic attractor in Figure 2.22 and because two bumps in the stationarydistribution would correspond to two asymptotically stable asymmetric attractors.Now I would like to illustrate how to compute such a probability density forour periodically extended system (2.37) for two cells with the simpler Gierer-Meinhardt intracellular kinetics. The randomly perturbed kinetics of the jthcoupled cell readdµ j(t) =µ2jη j dt+[Du∂xu]|x=x j dt+µ j√η j dW1, j(t) ,dη j(t) = µ2j dt+[Dv∂xv]|x=x j dt+µ j dW2, j(t) ,(2.57)j ∈ {1, ...,n}, with independent standard Wiener processes W1, j,W2, j. Close tothe deterministic version of the equilibrium, one finds from (2.53), lumping allcell concentrations into one vector ν := (µ1, ...,µn,η1, ...,ηn)T , that the stochasticprocess fluctuates according todν ≈F(ν)dt−Cν dt+DdW (t) with combined coupling C :=( βuγ2u−4 B 00 βvγ2v−4 A˜),(2.58)with deterministic reaction terms F(ν) :=( f (µ1,η1), ..., f (µn,ηn),g(µ1,η1), ...,g(µn,ηn))Tand isotropic diffusion matrix D := diag(√f (µ1,η1), ...,√f (µn,ηn)). Thereis a “≈” used above since we have assumed that the bulk medium is in quasi-equilibrium for (2.53). The stationary distribution of this approximate SDE58Figure 2.22: Full stochastic numerical simulation results of (2.37) in Julia[15]. The cells are taken to have RM kinetics (2.36) with additionalsquare rooted versions of each reaction term multiplying one of sevenindependent Wiener processes each. Due to the finite time step forthe numerical IMEX operator splitting method I used to solve thesystem, trajectories could leave the domain getting negative, whichwould not be permissible analytically. Hence, I have added reflectiveboundary conditions to still conserve mass (cf. [61]). The parameterwv = 7 is chosen so that it is inside the deterministically computedpitchfork bubble delimited by wP,1v ≈ 6.34518 and wP,2v ≈ 7.64062when ρ = βv/βu = 7. The other parameters are as in Figure 2.21.The deterministic supercritical pitchfork point is ρp,det ≈ 4.08982. Toprow: a single realization of the system for ρ around the stochasticpitchfork bifurcation point ρp < 7. Remaining rows: 20 realizationsof the system for ρ around the stochastic pitchfork point. I wouldlike to note that the initial condition for the simulation was thedeterministically computed symmetric steady-state perturbed by thequite large asymmetric perturbation 0.3 ·Z1 = (0.3,−0.3) so that initialrandom fluctuations would not regularly kick the solution in a differenteigenperturbation direction.59Figure 2.22: The color magenta is used for the concentration of µ1(t) ofcell 1, which is perturbed upward, and orange for µ2(t) of cell 2,which is perturbed downward. We see that with random perturbationsthe stochastic pitchfork point (P-bifurcation point in [5]) seemsactually larger than its deterministic pendant, since for ρ = 6.5 all 20trajectories appear to show synchronization of the random chemicalconcentration fluctuations in both compartments after about t = 100,just as in the top left plot. However, also an in-phase oscillatoryinstability exists for the system with RM kinetics.system, if it exists has to solve the Fokker-Planck equation [107] at equilibrium0 = ∂t ps(ν) = ∇ · ((−F(ν)+Cν)ps+ 12∇ · (D(ν)ps)= −∑2nl=1 ∂νl (eTl (F(ν)−Cν)ps)+ 12 ∑2nl=1 ∂ 2νl (eTl Del ps).(2.59)supplied with the mass-conserving reflecting boundary condition j ·n = 0 at zeroboundaries for which ∃ l ∈ {1, ...,2n} : νl = 0 (cf. [61], [107]). Here j is the fluxj = (F(ν)−Cν)ps− 12(∂ν1(eT1 De1 ps), ...,∂ν1(eT2nDe2n ps))Tand n is the outward normal vector to the corresponding boundary. Additionally,since concentrations remain bounded, ps(ν) → 0 as |ν | → ∞ is the remainingboundary condition.Solving such a PDE analytically seems involved. However, it would alreadygive insight solving it numerically with an iterative method, like the Gauss-Seidel method (cf. [55]), on a discretization of the corresponding 2n-dimensionalspace, where n denotes the compartment number. In order to compute aprobability density, a simple admissible initial condition would be the constant1/(#grid points) at all grid points, so that ∑xk∈ grid points p˜s(xk) = 1 where p˜srepresents the numerical approximation to p˜s. Here it remains to be analyzed towhich extent the stationary distribution to the approximate SDE system (2.58) isapproximating the stationary distribution of the orginal PDE-SDE system.It would already provide a feeling for the diffusion-coupling effect to computethe covariance and cross-covariance functions of, e.g., µ1(t) and µ2(t) for this60randomly perturbed two ring-cell system.This small illustration shows that it is both interesting and importantmathematically and biologically to consider the random extension of acompartmental-reaction diffusion system. Further more precise investigation isneeded here.Connecting back to graph theory, a Geometric Inhomogeneous Random Graph(cf. [68]) could be a fruitful graph-theoretic model to address stochastic patternformation with, on geometric graphs with diffusion on the edges.2.5 DiscussionWe have formulated and analyzed a new class of 1-D compartmental-reactiondiffusion system in which a two-component nonlinear reaction kinetics is restrictedto the domain boundaries and where two bulk diffusing species effectivelycouple the spatially localized reactions. For the case of a common reactionkinetics, and when one of the kinetic species has a linear dependence, we havederived a nonlinear algebraic system characterizing the steady-states of the model.Through a bifurcation analysis and from path-following numerical methods wehave shown that symmetry-breaking pitchfork bifurcations can occur even whenthe two bulk diffusing species have identical or very comparable diffusivities.The linear stability properties of the symmetric steady-state was investigatednumerically by deriving a nonlinear matrix eigenvalue problem characterizingboth in-phase and anti-phase perturbations of a symmetric steady-state. Throughnumerical computations of branches of steady-state solutions together with thenumerical solution of the nonlinear matrix eigenvalue problem, the theory wasillustrated for either FitzHugh-Nagumo, Gierer-Meinhardt, or Rauch-Millonasreaction kinetics. The theoretical prediction of symmetry-breaking leading tostable asymmetric patterns in these models was confirmed through full time-dependent PDE computations. Overall, it was shown that it is the ratio of thetwo permeabilities to the compartments that controls whether stable asymmetricpatterns can occur. This is in contrast to the case of a traditional activator-inhibitorRD system where stable spatial patterns exist only for a sufficiently large, buttypically biologically unrealistic, diffusivity ratio of the two diffusing species. For61FitzHugh-Nagumo and Rauch-Millonas kinetics we have also shown that stableasymmetric patterns can emerge from a symmetric pattern at a fixed permeabilityratio when a control parameter in the intracellular kinetics is varied. Extensions ofthe analytical theory to treat a periodic chain of reaction compartments was made,and the results confirmed through full PDE simulations.We remark that our findings are qualitatively similar to those associatedwith a postulated diffusive transport mechanism for biological morphogens,referred to as a binding-mediated hindrance diffusion process [99], in whicha differential permeability ratio on the cell boundaries for two morphogenspecies with comparable diffusivities was suggested to provide the large effectivediffusivity ratio that is required for pattern formation and symmetry-breaking intissues. However, since our condition (2.16) for the existence of symmetry-breaking bifurcations is highly implicit in the bulk diffusivities, it seems ratherintractable analytically to isolate via a simple scaling analysis an effectivediffusivity bifurcation parameter for the bulk species that incorporates themembrane permeability ratio and/or the ratio of bulk degradation rates. Aneffective diffusivity ratio condition was also derived analytically in [75] for a 2+1RD system with an immobile substrate. It is an open problem to isolate an effectivediffusivity ratio parameter for our compartmental-reaction diffusion system thatencapsulates analytically how the symmetry-breaking bifurcation point dependson the membrane permeability ratio and the ratio of bulk degradation rates.In the context of our 1-D framework, there are four main additional openproblems. Firstly, it would be worthwhile to formulate a linear stability problemfor asymmetric patterns of the 1-D compartmental-reaction diffusion system, whileallowing for arbitrary reaction kinetics. Secondly, it would be interesting toextend the weakly nonlinear analysis of [105], which dealt with one-bulk diffusingspecies, to our compartmental-reaction diffusion setting. With this extension,amplitude equations near the symmetry-breaking bifurcation would be valuable soas to classify whether the bifurcation is sub- or super-critical, and thereby predictwhether the emerging asymmetric patterns are linearly stable or unstable. Thirdly,it would be insightful to analytically represent the time-dependent bulk diffusionfields in terms of a time-dependent Green function so as to reduce the bulk-cellsystem to an integro-differential system of ODEs with memory. In this setting,62large amplitude solutions, including time-periodic solutions, would represent a newfrontier to be investigated. This is done in Chapter 4 in the infinite 2-D setting andin Chapter 5 in the infinite 3-D setting. A further extension of our 1-D model wouldbe to investigate how the symmetry-breaking bifurcations are influenced by randomnoise arising from the compartmental reaction kinetics. In [54] it was shown thatincluding stochasticity in RD systems can lower the threshold for the onset ofpattern formation (see [38] for an overview of stochastic modeling in RD systems).In §2.4 we have illustrated how randomness can be introduced meaningfully intoour compartmental-reaction diffusion framework.An important open direction is to analyze symmetry-breaking behaviorassociated with a similar modeling framework in a 2-D spatial setting whereintra-compartmental reactions occur only within a disjoint collection of smallcircular disks within a bounded 2-D domain. In this context, two bulk diffusingspecies with comparable diffusivities can effectively be transported across thecompartment boundaries as regulated by certain permeabilities. This exchangebetween the compartments and the bulk medium can either trigger or modulatethe intra-compartmental reactions. Similar bulk-cell models in 2-D, but with onlyone diffusing bulk species, have been analyzed recently to model quorum-sensingbehavior (cf. [48], [64], [117], [44]). We are tackling this problem in the nextChapters 3 and 4.Another possible extension of our modeling framework is to consider acompartmental-reaction diffusion system on a network. In this graph-theoreticsetting, the reaction kinetics are restricted to the nodes of a graph, while two bulkspecies are taken to diffuse along the edges. Related network models of this typehave been analyzed in [95], [31] and [84] for general multi-component systems,and in [9] for a spatially-extended SIR epidemic model.63Chapter 3Compartmental ReactionKinetics Coupled by Two BulkDiffusing Species withComparable Diffusivities in 2-D3.1 IntroductionIn this chapter we formulate and quantitatively analyze a new theoretical model ina 2-D setting that robustly leads to pattern formation even when the two diffusingspecies have a comparable or equal diffusivity. More specifically, we analyzesymmetry-breaking pattern formation for a 2-D PDE-ODE bulk-cell RD modelin which well-separated localized reaction compartments, referred to as “cells”,are coupled to a two-component linear bulk diffusion field with constant bulkdegradation rates. In the cells, which are assumed to have a common radius thatis small compared to the domain length-scale and the inter-cell distances, two-component intracellular activator-inhibitor reaction kinetics are specified. Theintracellular species undergo an exchange with the two bulk species across the cellboundaries, as mediated by membrane reaction rates in a Robin boundary conditionthat is specified on each cell boundary. The two extracellular diffusing bulk species,64with comparable diffusivities and degradation rates, provide the mechanism thatcouples the nonlinear intracellular reactions that occur in the union of the spatiallysegregated cells. We refer to this modeling framework as a 2-D compartmental-reaction diffusion system.The numerical implementation of our theoretical analysis for this model forvarious specific intracellular reaction kinetics reveals that it is the ratio of thereaction rate of the inhibitor component to that of the activator component on thecompartment boundaries that plays a central role in the initiation of symmetry-breaking bifurcations of a symmetric steady-state. The magnitude of this ratioultimately controls whether linearly stable asymmetric steady-states for the bulk-cell model can occur even when the bulk diffusivities are comparable or equal. Thebifurcation threshold condition for this key membrane reaction rate ratio parameteris distinct from the usual large diffusivity ratio threshold that is required for patternformation from a spatially uniform state for typical two-component activator-inhibitor RD systems (cf. [94], [76]). We emphasize that our linear stabilityanalysis predicting symmetry-breaking bifurcations for the bulk-cell model, asregulated by the membrane reaction rate ratio, is significantly more challengingthan performing a simple Turing stability analysis [138] since it is based on thelinearization of the bulk-cell model around a spatially non-uniform symmetricsteady-state. In our 1-D study in Chapter 2, where nonlinear reactions wererestricted either to domain boundaries or at lattice site on a 1-D periodic chain,it has been shown for some specific nonlinear kinetics that symmetry-breakingbifurcations can occur from a symmetric steady-state when the ratio of membranereaction rates exceeds a threshold. It should be stressed that in the present 2-Dsetting the cells are not assumed to be singular but have rather the geometry of adisk with small radius ε . Leveraging this small parameter, both the steady-statesare constructed and the perturbation theory is being performed using asymptoticanalysis (strong localized perturbation theory). This procedure is more intricatethan in the 1-D setting and leads to Green matrices with modified Bessel functions.The outline of this chapter is as follows. In §3.2 we formulate our bulk-cellmodel and use a singular perturbation approach in the limit of a small commoncell radius to derive a nonlinear algebraic system characterizing all steady-statesolutions of the model. In §3.3 we show that the discrete eigenvalues of the65linearization of the bulk-cell model around a steady-state solution are determinedby a root-finding condition on a nonlinear matrix eigenvalue problem. For a certaintype of spatial configuration of the cells, the bulk-cell model is shown to admita symmetric steady-state solution in which the steady-states of the intracellularreactions are identical. The possibility of symmetry-breaking bifurcations alongthis symmetric steady-state solution branch, leading to the existence of linearlystable asymmetric patterns, are analyzed by applying solution path continuationsoftware to our bifurcation-theoretic analytical results. For a certain two-cellconfiguration in the unit disk, and for either Gierer-Meinhardt [42], Rauch-Millonas [116], or FitzHugh-Nagumo [46] intracellular reactions, we show in§3.4 that it is the magnitude of the ratio of the reaction rates for the two bulkspecies on the cell membranes that controls whether linearly stable asymmetricpatterns can bifurcate from the symmetric steady-state. Our theoretical predictionsof symmetry-breaking behavior, leading to stable asymmetric steady-states evenwhen the two bulk species have comparable or equal diffusivities, are confirmedfrom full PDE numerical simulations. For a closely-spaced arrangement of cells asis typical in biological tissues, and where our asymptotic theory no longer applies,the PDE numerical simulations shown in §3.4.4 illustrate that symmetry-breakingbifurcations can still be controlled by the reaction rate ratio on the cell boundaries.In particular, our numerical results suggest that such bifurcations occur with asmaller membrane reaction-rate ratio than for the situation where the cells are morespatially segregated. In §3.6 we discuss our theoretical results in a wider context,and suggest a few open directions.3.2 Compartmental-Reaction Diffusion System in 2-D3.2.1 Model formulationWe consider a bounded 2-D domain with length scale L, denoted by ΩL ⊂R2, thatcontains m disconnected circular compartments ΩLj , for j ∈ {1, ...,m}, referred toas “cells”. We will assume that these cells have a common radius that is smallin comparison with the length scale L of the domain. The bulk or extracellularmedium is the region ΩL\⋃mj=1ΩLj .66In the bulk we assume that there are two extracellularly diffusing anddegrading chemical species with concentrations U and V . These messengermolecules are synthesized on the “cell” membranes through the interaction withtwo corresponding intracellular species M j and H j. With the molecule countsU,V,M j and H j corresponding to respectively U,V,M j and H j, the chemicalequations areUβU−−⇀↽−βUM j , VβV−−⇀↽−βVH j . (3.1)Here we made the assumption that the exponential forward reaction rates equal thebackward reaction rates and that all compartments are identical in that they havecommon membrane reaction rates. The intra-compartmental species, in turn, areproduced by certain reaction kinetics, denoted by f (M,H) and g(M,H), that areassumed to be identical in each compartment.More precisely, in dimensional variables, our bulk-cell coupled model isbulk∂TU = DU ∆XU−κU U , X ∈ΩL\⋃mj=1ΩLj ,∂TV = DV ∆XV −κV V , X ∈ΩL\⋃mj=1ΩLj ,∂n˜XU = ∂n˜XV = 0 , X ∈ ∂ΩL , (Neumann condition)(3.2a)reaction fluxesDU ∂n j,XU = βU,1U−βU,2 M j , X ∈ ∂ΩLj , (Robin condition)DV ∂n j,XV = βV,1V −βV,2 H j , X ∈ ∂ΩLj , (3.2b)compartmentsddT M j = κR µc f(1µc M j,1µc H j)+∫∂ΩLj(βU,1U−βU,2 M j) dSX ,ddT H j = κR µc g(1µc M j,1µc H j)+∫∂ΩLj(βV,1V −βV,2 H j) dSX (reaction kinetics) ,(3.2c)with j ∈ {1, ...,m} and where n j,X is the outward unit normal vector to ΩLjwhile n˜X is the outward unit normal vector to ΩL. The diffusivities (diffusioncoefficients) for U and V are DU and DV , and U and V are degrading in thebulk with exponential rate constants κU and κV , respectively. The exponentialreaction rates on the compartment boundaries are βU and βV with correspondingrates βU,1 and βV,1 in length per time and βU,2 and βV,2 in per length times timeunits, and µc is a normalizing constant for the intracellular species. Lastly, κRis a dimensional reaction rate for the intracellular reactions. The macroscopic67Robin boundary conditions (3.2b) can be derived from the microscopic stochasticformulation of biochemical particle trajectories at a, for example, stochasticallyswitching boundary (controlling the influx and efflux) along the lines of [85] and[18].In Appendix A.3 we non-dimensionalize (3.2) to obtain the dimensionlessPDE-ODE modelbulk∂tu = Du∆u−σuu , x ∈Ω\⋃mj=1 Bε(x j) ,∂tv = Dv∆v−σvv , x ∈Ω\⋃mj=1 Bε(x j) ,∂n˜u = ∂n˜v = 0 , x ∈ ∂Ω ,(3.3a)reaction fluxesεDu∂n j u = du1u−du2µ j , x ∈ ∂Bε(x j) ,εDv∂n j v = dv1v−dv2η j , x ∈ ∂Bε(x j) , (3.3b)compartmentsdµ jdt = f (µ j,η j)+1ε∫∂Bε (x j)(du1u−du2µ j) dS ,dη jdt = g(µ j,η j)+1ε∫∂Bε (x j)(dv1v−dv2η j) dS ,(3.3c)for j ∈ {1, . . . ,m}. Here n and n˜ are the outward unit normal vectors to Bε(x j)and Ω, respectively, and we have dropped the label “x” for ∆ and the outward unitnormal vectors. In (3.3), the compartments are disks of a common radius ε ≪ 1centered at x j ∈ Ω, i.e. Bε(x j) := {x | |x−x j| ≤ ε}. We will refer to du1 , dv1, du2 ,and dv2 as dimensionless membrane reaction rates. An illustration of the bulk-cellmodel is shown in Figure 3.1.We will use strong localized perturbation theory [143] to construct the steady-state solutions of (3.3) and to analyze their linear stability properties in theasymptotic limit ε ≪ 1 and under the assumption that m circular cells are well-separated in the sense that the cell centers satisfy |xi − x j| = O(1), for i, j ∈{1, ...,m} and i ̸= j.3.2.2 Asymptotic construction of the steady-statesOur main goal is to construct a symmetric steady-state solution for (3.3) in whichthe concentration of each species is the same inside and in the local vicinity ofeach compartment. We will show below that even when the bulk diffusing species68Figure 3.1: A 2-D bounded domain with four diffusion-coupled circular cellsof a common radius. In the jth cell, activator-inhibitor reaction kineticsoccur for the activator µ j and inhibitor η j. Across the cell membrane,there is an exchange between the intracellular and bulk species. In thebulk, u and v diffuse and undergo degradation. The cells are assumed tobe small but are drawn larger here for illustration only.have comparable diffusivities this symmetric steady-state is unstable to symmetry-breaking perturbations that occur beyond a pitchfork bifurcation point associatedwith the membrane reaction rate ratio ρ := dv1/du1 = dv2/du2 . This leads to theexistence of linearly stable asymmetric steady-state solutions to (3.3).In the absence of diffusion, the ODE system for the intra-compartmentalspecies is decoupled from the bulk medium and reduces toµ˙(t) = f (µ,η) , η˙(t) = g(µ,η) . (3.4)Let (µe,ηe) be an equilibrium point for (3.4) and label F(µ,η) :=( f (µ,η),g(µ,η)). For a specific parameter set, the linear stability property ofthe equilibrium state is characterized by whether the eigenvalues λ of the Jacobianmatrix DF(µe,ηe) have positive (unstable, exponentially growing perturbations) ornegative (stable, exponentially decaying perturbations) real parts Re(λ ). However,when there is bulk diffusion and the compartments are coupled through the bulk,69the steady-state solution in the compartments depends on the bulk diffusivities, themembrane reaction rates, and the spatial configuration of the cells.We now use the method of matched asymptotic expansions to constructsteady-state solutions for (3.3). In the jth inner region, defined within an O(ε)neighborhood of the boundary of the jth cell, we introduce the local variablesy j = ε−1(x−x j), u j(x) = u(εy j +x j), and v j(x) = v(εy j +x j), where p j := |y j|.Upon writing the steady-state of (3.3a) in terms of the inner variables, for ε → 0the steady-state problem in the jth inner region is ∆u j = 0 and ∆v j = 0, for p j ≥ 1,subject to Du ∂p j u j = du1u j − du2µ j and Dv ∂p j v j = dv1v j − dv2η j on p j = 1. Theradially symmetric solutions to these problems areu j(p j)=Auj log p j+1du1(Du Auj +du2µ j), v j(p j)=Avj log p j+1dv1(Dv Avj +dv2η j),(3.5)for j ∈ {1, . . . ,m}, where Auj and Avj for j ∈ {1, . . . ,m} are constants to bedetermined. Upon substituting (3.5) into the steady-state problem of (3.3c), weobtain for the jth cell thatf (µ j,η j)+2πDu Auj = 0 , g(µ j,η j)+2πDv Avj = 0 , j ∈ {1, . . . ,m} .(3.6)Next, we must determine Auj and Avj by matching the far-field behavior of the innersolutions (3.5) to the outer solutions defined in the bulk region.Now we halt for a bit and discuss what changes if one had nonlinear boundaryconditions Du ∂p j u j = h(u j,µ j) with some nonlinear reaction kinetic function hinstead of the linear Robin boundary conditions above. In this case, the constantBuj in u j(p j) = Auj log p j +Buj needs to be determined numerically through DuAuj =h(Buj ,µ j). If this gives multiple solutions for Buj , one has to select one of them toproceed with matching to the global solution.In the limit ε → 0, in the bulk region the compartments formally shrink topoints and from the far-field behavior of (3.5), when written in outer variables, weobtain that the steady-state bulk species U satisfies∆U−ω2u U = 0 , x ∈Ω\{x1, . . . ,xm} ; ∂nU = 0 , x ∈ ∂Ω ; (3.7a)U ∼ Auj log |x−x j|+Aujν+1du1(DuAuj +du2µ j) , as x→ x j , j ∈ {1, . . . ,m} ,(3.7b)70where ν := −1/ logε ≪ 1 and ωu :=√σu/Du. Similarly, with ωv :=√σv/Dv,for the bulk species V we have that∆V −ω2v V = 0 , x ∈Ω\{x1, . . . ,xm} ; ∂nV = 0 , x ∈ ∂Ω ; (3.8a)V ∼ Avj log |x−x j|+Avjν+1dv1(DvAvj +dv2η j) , as x→ x j , j ∈ {1, . . . ,m} .(3.8b)To represent solutions to (3.7) and (3.8), we introduce the reduced-wave Greenfunction Gω(x,x j) that satisfies∆Gω − ω2Gω =−δ (x−x j) , x ∈Ω ; ∂nGω = 0 , x ∈ ∂Ω ;(3.9a)Gω ∼− 12π log |x−x j|+Rω(x j)+o(1) , as x→ x j . (3.9b)Here Rω(x j) is the regular, or non-singular, part of the singularity at x = x j. Thesolutions to (3.7) and (3.8) are represented asU(x) =−2πm∑i=1Aui Gωu(x;xi) , V (x) =−2πm∑i=1Avi Gωv(x;xi) . (3.10)The pre-specification of the regular part of each singularity condition in (3.7)and (3.8) yields a constraint. These constraints provide algebraic systems for Aujand Avj for j ∈ {1, . . . ,m}. By expanding (3.10) as x→ x j, we enforce that that thenon-singular terms in the resulting expression agree with the conditions that arerequired in (3.7b) and (3.8b) for each j ∈ {1, . . . ,m}. This leads to linear algebraicsystems for A u := (Au1, . . . ,Aum)T and A v := (Av1, . . . ,Avm)T , given in matrix formby((1+νDudu1)I+2πνGωu)A u =−νdu2du1µ ,((1+νDvdv1)I+2πνGωv)A v =−νdv2dv1η ,(3.11)where µ := (µ1, . . . ,µm)T and η := (η1, . . . ,ηm)T . In (3.11), Gω with either ω =ωu71or ω = ωv is the symmetric reduced-wave Greens’ interaction matrix defined byGω :=Rω1 Gω12 . . . Gω1mGω21 Rω2 . . . Gω2m....... . ....Gωm1 Gωm2 . . . Rωm . (3.12)Here Gω ji = Gωi j := Gω(x j;xi) for i ̸= j, and Rω j := Rω(x j) for j ∈ {1, . . . ,m},are obtained from the solution to (3.9).To determine a nonlinear algebraic system that characterizes our steady-statesolution, we solve (3.11) for A v and A u, and substitute the resulting expressionsinto (3.6). In this way, we obtain a 2m dimensional nonlinear algebraic system forµ j and η j, for j = 1, . . . ,m, given byf (µ j,η j)− eTj Θuµ = 0 , g(µ j,η j)− eTj Θvη = 0 , for j ∈ {1, . . . ,m} ,(3.13a)where e j := (0, . . . ,0,1,0, . . . ,0)T is the unit vector in the jth direction. In (3.13a),Θu and Θv are defined byΘu := 2πνDudu2du1[(1+νDudu1)I+2πνGωu]−1, Θv := 2πνDvdv2dv1[(1+νDvdv1)I+2πνGωv]−1.(3.13b)We can simplify our steady-state analysis for the special case where g(µ,η) islinear and inhibiting in η , with the formg(µ,η) = g1(µ)−g2η , (3.14)where g2 ≥ 0 is a constant. This specific form applies to Gierer-Meinhardt[42], Rauch-Millonas [116], and FitzHugh-Nagumo [46] reaction kinetics, and isrelevant for the illustrations of the theory given in §3.4. In this case, we obtainfrom the second equation in (3.13a) thatη = [g2I+Θv]−1 g1 where g1 := (g1(µ1), . . . ,g1(µm))T . (3.15)Then, from the first equation in (3.13a) we obtain an m dimensional nonlinear72algebraic system for µ = (µ1, . . . ,µm)T given byf(µ j,eTj (g2I+Θv)−1g1)− eTj Θuµ = 0 , j ∈ {1, . . . ,m} . (3.16)Next, we define a symmetric cell arrangement for which the steady-state analysiscan be further simplified.Definition 3.2.1. A symmetric cell arrangement is defined by the condition thatthe symmetric Green matrix Gω satisfies the following two properties:• Property 1: e := (1, . . . ,1)T is an eigenvector of Gω for all ω > 0:• Property 2: The eigenspace of Gω orthogonal to e is independent of ω .These two properties certainly hold when Gω is a circulant matrix. In particular,Gω is a circulant matrix when m small cells are equidistantly spaced on a ring thatis concentric within a circular domain Ω. Such an arrangement of cells is called aring pattern.For a symmetric cell arrangement, Gωu and Gωv have a common eigenspace,and so we can seek a symmetric solution to (3.13) of the formµ = µce , η = ηce , A u = Auce , Av = Avce , (3.17)where the scalars µc, ηc, Auc , and Avc are to be found. Upon substituting (3.17) into(3.13), we obtain that µc and ηc satisfy the nonlinear algebraic systemf (µc,ηc)−αuµc = 0 , g(µc,ηc)−αvηc = 0 , (3.18)where αu and αv, denoting the eigenvalues of Θu and Θv for the eigenvector e,respectively, are defined byαu :=2πνDudu2/du11+νDu/du1 +2πνκu, αv :=2πνDvdv2/dv11+νDv/dv1 +2πνκv. (3.19a)Here κu and κv are the eigenvalues of the Green matrices for the eigenvector e,given byGωue = κue , Gωve = κve . (3.19b)73Moreover, if g(µ,η) has the specific form in (3.14), we obtain from (3.16) that fora symmetric pattern of cells, there is a symmetric steady-state solution wheneverthere is a root µc to the scalar nonlinear algebraic equationf(µc,g1(µc)g2+αv)−αuµc = 0 . (3.20)In summary, for a symmetric pattern of cells, the asymptotic construction of asymmetric steady-state solution for (3.3) is reduced to the much simpler problem ofdetermining a solution to the two-dimensional nonlinear algebraic problem (3.18)for general reaction kinetics, or to (3.20) when g has the specific form in (3.14).In these algebraic problems, the eigenvalues κu and κv, as needed in (3.19a), arethe constant row sums of the Green matrices for the two bulk species. The bulkdiffusivities, the membrane reaction rates, and the spatial configuration of the cellsall influence αu and αv.3.2.3 Symmetry-breaking bifurcationsTo detect any symmetry-breaking pitchfork bifurcation points along the symmetricsteady-state solution branch we can perform a linear stability analysis of(3.3) around the steady-state solution and seek λ = 0 eigenvalue crossings.An equivalent, but simpler, approach to detect zero-eigenvalue crossings forthe linearized problem is to determine bifurcation points associated with thelinearization of the nonlinear algebraic system (3.13) around a symmetric steady-state.To do so, we introduce the perturbationsµ = µce+ µ˜ , η = ηce+ η˜ , A u = Auce+ ˜Au , A v = Avce+ ˜Av ,(3.21)into (3.13) and linearize the resulting system. In this way, we obtain that asymmetry-breaking bifurcation occurs if and only if there is a non-trivial solutionto the 2m×2m homogeneous linear system(f cµ I−Θu f cη Igcµ I gcη I−Θv)(µ˜η˜)=(00), (3.22)74at some point along the symmetric solution branch given by (3.18). In (3.22)we have labeled f cµ by fcµ := ∂µ f (µ,η) when evaluated at µ = µc and η = ηc,while I is the m×m identity matrix. For the special case where g has the specificform in (3.14), we can solve (3.22) for η˜ and reduce (3.22) to the m-dimensionalhomogeneous linear system(f cµ I+ fcηg′1(µc)(g2I+Θv)−1−Θu)µ˜ = 0 . (3.23)Next, by Property 2 for a symmetric cell arrangement, it follows that Gωu andGωv have a common orthogonal eigenspaceQ⊥ := span{q2, . . . ,qm}, where qTj e =0 for j ∈ {2, . . . ,m} and qTi q j = 0 for i ̸= j. The eigenvalues of Gωu and Gωv in thiscommon eigenspace are labeled byGωuq j = κ⊥u, jq j , Gωvq j = κ⊥v, jq j , j ∈ {2, . . . ,m} , (3.24)so thatΘuq j = α⊥u, jq j , Θvq j = α⊥v, jq j , j ∈ {2, . . . ,m} , (3.25)withα⊥u, j :=2πνDudu2/du11+νDu/du1 +2πνκ⊥u, j, α⊥v, j :=2πνDvdv2/dv11+νDv/dv1 +2πνκ⊥v, j. (3.26)By setting µ˜ = µ˜cq j and η˜ = η˜cq j in (3.22), we conclude that a symmetry-breaking bifurcation occurs for the jth mode with j ∈ {2, . . . ,m} whenever(f cµ −α⊥u, j f cηgcµ gcη −α⊥v, j)(µ˜cη˜c)=(00), (3.27)has a nontrivial solution. This is equivalent to the condition that(f cµ −α⊥u, j)(gcη −α⊥v, j)− f cηgcµ = 0 , j ∈ {2, . . . ,m} , (3.28)is satisfied at some point along the symmetric solution branch defined by thesolution to (3.18).Finally, for the special case where g has the specific form in (3.14), we obtain75that there is a symmetry-breaking bifurcation for the jth mode, with j ∈ {2, . . . ,m},when there is a root to the scalar problemf cµ +f cηg′1(µc)g2+α⊥v, j−α⊥u, j = 0 , (3.29)whenever ηc = g1(µc)/(g2+αv) where µc satisfies (3.20).In the examples shown in §3.4 we will use ρ := dv1/du1 = dv2/du2 as thebifurcation parameter to detect whether symmetry-breaking bifurcations can occuralong the symmetric solution branch.3.2.4 A symmetric cell arrangement with two cellsConsider a symmetric cell arrangement with two cells, i.e., m = 2, for the specialcase where g has the form in (3.14). Then, to determine all steady-state solutionswe need only solve the nonlinear algebraic system (3.16) for µ1 and µ2. Thesymmetric steady-state solution, for which µc := µ1 = µ2, is obtained by solvingthe scalar problem (3.20). To detect whether symmetry-breaking bifurcations canoccur, we note that q2 = (1,−1)T spans the common eigenspace of Gωu andGωv orthogonal to e, and that κu = Rωu1 −Gωu12 and κv = Rωv1 −Gωv12 are theassociated eigenvalues for q2. This yields that the root-finding condition (3.29)becomesf cµ +f cηg′1(µc)g2+α⊥v,2−α⊥u,2 = 0 , (3.30)where in terms of the entries of the Green matrices we haveα⊥u,2 :=2πνDudu2/du11+νDu/du1 +2πν [Rωu1−Gωu12], α⊥v,2 :=2πνDvdv2/dv11+νDv/dv1 +2πν [Rωv1−Gωv12].(3.31)To detect any pitchfork bifurcation points on the symmetric steady-state branchparameterized by ρ = dv1/du1 = dv2/du2 we numerically solve (3.20) together with(3.30). In §3.4 we illustrate this approach for certain reaction kinetics when Ωis the unit disk. The advantage of considering a disk-shaped confining domain isthat the reduced-wave Green function is known analytically by using separation ofvariables (see Appendix A.4). We remark that it would also be readily feasible to76illustrate our asymptotic theory for a rectangular-shaped confining domain, sincethe reduced-wave Green function is also available analytically for such a domain.3.3 The Linear Stability AnalysisIn this section, we formulate the linear stability problem for the steady-statesolutions constructed in §3.2.2. We denote the bulk steady-state solutions of §3.2.2by ue(x) and ve(x), and the steady-state vector of intracellular steady-states byµe = (µe1, . . . ,µem)T and ηe = (ηe1, . . . ,ηem)T .To formulate the linear stability problem, we first introduce the perturbationsu(t,x) = ue(x)+ eλ tφ(x) , v(t,x) = ve(x)+ eλ tψ(x) ,µ j(t) = µe j + eλ tξ j , η j(t) = ηe j + eλ tζ j , for j ∈ {1, . . . ,m} ,into (3.3) and linearize the resulting system. This yields the eigenvalue problembulk∆φ −Ω2uφ = 0 , x ∈Ω\⋃mj=1 Bε(x j) ,∆ψ−Ω2vψ = 0 , x ∈Ω\⋃mj=1 Bε(x j) ,∂n˜φ = ∂n˜ψ = 0 , x ∈ ∂Ω ,(3.32a)reaction fluxesεDu∂n jφ = du1φ −du2ξ j , x ∈ ∂Bε(x j) ,εDv∂n jψ = dv1ψ−dv2ζ j, x ∈ ∂Bε(x j) , (3.32b)compartments(λ I− J j)ξ jζ j= ε−1∫∂Bε (x j)(du1φ −du2ξ j) dS∫∂Bε (x j)(dv1ψ−dv2ζ j) dS ,(3.32c)j ∈ {1, . . . ,m}. Here the Jacobian matrix J j of the intracellular kinetics, as well asΩu and Ωv are defined byJ j :=(∂µ f (µ,η) ∂η f (µ,η)∂µg(µ,η) ∂ηg(µ,η))∣∣∣µ=µe j,η=ηe j, Ωu :=√λ +σuDu, Ωv :=√λ +σvDv.(3.33)We now use strong localized perturbation theory [143] to analyze (3.32) in thelimit ε → 0. In this way we will derive a nonlinear matrix eigenvalue problem,77referred to as the globally coupled eigenvalue problem (GCEP), for the discreteeigenvalues λ of the linearization. This GCEP will be used to investigate variousinstabilities of the steady-state solutions constructed in §3.2.2.In the O(ε) inner region near the jth cell we introduce the local variables y j =ε−1(x−x j), φ j(x) := φ(x j +εy j) and ψ j(x) := ψ(x j +εy j), with p j = |y j|. Uponwriting (3.32a) in terms of the inner variables, for ε → 0 we obtain in the jth innerregion that ∆φ j = 0 and ∆ψ j = 0, for p j ≥ 1, subject to Du ∂p jφ j = du1φ j − du2ξ jand Dv ∂p jψ j = dv1ψ j−dv2ζ j on p j = 1. The radially symmetric solutions to theseproblems areφ j(p j)= cuj log p j+1du1(Du cuj +du2ξ j), ψ j(p j)= cvj log p j+1dv1(Dv cvj +dv2ζ j),(3.34)for j ∈ {1, . . . ,m}, where cuj and cvj for j ∈ {1, . . . ,m} are constants to bedetermined. Upon substituting (3.34) into (3.32c) we obtain, in terms of theJacobian J j of (3.33), that(λ I− J j)(ξ jζ j)=(2πDucuj2πDvcvj), for j ∈ {1, . . . ,m} . (3.35)To determine cuj and cvj we must match the far-field behavior of the innersolutions (3.34) to the outer solutions defined in the bulk region. Similar to theanalysis of the steady-state solution, we obtain that∆φ −Ω2u φ = 0 , x ∈Ω\{x1, . . . ,xm} ; ∂nφ = 0 , x ∈ ∂Ω ; (3.36a)U ∼ cuj log |x−x j|+cujν+1du1(Ducuj +du2ξ j) , as x→ x j , j ∈ {1, . . . ,m} ,(3.36b)where ν :=−1/ logε≪ 1. Similarly, for the perturbation of the other bulk specieswe obtain∆ψ−Ω2v ψ = 0 , x ∈Ω\{x1, . . . ,xm} ; ∂nψ = 0 , x ∈ ∂Ω ; (3.37a)ψ ∼ cvj log |x−x j|+cvjν+1dv1(Dvcvj +dv2ζ j) , as x→ x j , j ∈ {1, . . . ,m} .(3.37b)78The solutions to (3.36) and (3.37) are represented asφ(x) =−2πm∑i=1cui Gu,λ (x;xi) , ψ(x) =−2πm∑i=1cvi Gv,λ (x;xi) , (3.38)where, to simplify the notation and emphasize the dependence on the eigenvalueparameter λ , we have definedGu,λ (x;x j) := GΩu(x;x j) , Gv,λ (x;x j) := GΩv(x;x j) , (3.39)where Gω(x;x j) is defined by the solution to (3.9). Upon letting x→ x j in (3.38)and ensuring that the singularity conditions in (3.36b) and (3.37b) are satisfied,we obtain a linear algebraic system for the vectors cu := (cu1, . . . ,cum)T and cv :=(cv1, . . . ,cvm)T , given in matrix form by((1+νDudu1)I+2πνGu,λ)cu =−νdu2du1ξ ,((1+νDvdv1)I+2πνGv,λ)cv =−νdv2dv1ζ ,(3.40)where ξ := (ξ1, . . . ,ξm)T and ζ := (ζ1, . . . ,ζm)T . In (3.40), Gu,λ and Gv,λ denotethe reduced-wave Green matrix given in (3.12) with either ω = Ωu or ω = Ωv,respectively. Here Ωu and Ωv are defined in terms of λ by (3.33).Assuming that λ is not an eigenvalue of J j for any j ∈ {1, . . . ,m}, we obtainupon inverting (3.35) and writing the system in matrix form thatξ = 2πDuK11cu+2πDvK12cv , ζ = 2πDuK21cu+2πDvK22cv , (3.41)where ξ = (ξ1, . . . ,ξm)T and ζ = (ζ1, . . . ,ζm)T . Here K11, K12, K21, and K22 arethe diagonal matrices defined byK11 := diag(K11 j) , K12 := diag(K12 j) , K21 := diag(K21 j) , K22 := diag(K22 j) ,(3.42a)with diagonal entries given byK11 j := eT1 (λ I− J j)−1e1 , K12 j := eT1 (λ I− J j)−1e2 ,K21 j := eT2 (λ I− J j)−1e1 , K22 j := eT2 (λ I− J j)−1e2 ,(3.42b)79where e1 = (1,0)T and e2 = (0,1)T .Then, upon substituting (3.40) into (3.41), we obtain the 2m × 2mhomogeneous algebraic system, which we write in block matrix form asM (λ )(cucv)=(00), where M (λ ) :=(Mu(λ ) Hu(λ )Mv(λ ) Hv(λ )), (3.43a)withMu(λ ) :=(1+νDudu1)I+2πνDudu2du1K11+2πνGu,λ , Hu(λ ) := 2πνDvdu2du1K12 ,(3.43b)Hv(λ ) := 2πνDudv2dv1K21 , Mv(λ ) :=(1+νDvdv1)I+2πνDvdv2dv1K22+2πνGv,λ .(3.43c)The nonlinear matrix eigenvalue problem (3.43) is referred to as the globallycoupled eigenvalue problem (GCEP). The GCEP has a nontrivial solution(cu,cv)T ̸= (0,0)T , if and only λ satisfies detM (λ ) = 0. The set Λ(M ), definedbyΛ(M ) := {λ | detM (λ ) = 0} , (3.44)is the union of all such roots. Any element λ ∈ Λ(M ) satisfying Re(λ ) > 0provides an approximation, valid as ε → 0, for a destabilizing discrete eigenvalueof the linearized problem (3.32). However, if for all λ ∈Λ(M )we have Re(λ )< 0,then the steady-state solution is linearly stable.When there is a large number of cells m, the determination of the discreteeigenvalues comprising Λ(M ) is in general a very challenging numerical problem.A survey of nonlinear matrix eigenvalue problems and available solution strategiesthat apply to only certain classes of matrices is given in [53] and [10]. Specificapplications of nonlinear matrix problems in simpler contexts where M (λ ) iseither a polynomial or a rational function of λ are discussed in [11]. Since forour problem, M (λ ) is not symmetric and has a complicated dependence on theeigenvalue parameter through the Green matrices and from the diagonal K matricesof (3.42), these previously developed numerical strategies are not applicable forcomputing the set Λ(M ) in (3.44) for a steady-state solution with an arbitrarycollection of cells.80For a symmetric steady-state solution corresponding to a symmetric cellarrangement, we now verify that the condition det(M (0)) = 0 in (3.43) isequivalent to the zero-eigenvalue crossing condition derived in (3.22), which wasbased on a linearization of the nonlinear algebraic system around the symmetricsteady-state. When λ = 0, we have2πDucu =−Θuξ , 2πDvcv =−Θvζ ,where Θu and Θv are defined in (3.13b). Since K11, K12, K22, and K21 areall multiples of the identity for a symmetric steady-state, we obtain from (3.43)together with (3.40) that when λ = 0 we have in block matrix form(ξζ)+(K11I K12IK12I K22I)(Θu 00 Θv)(ξζ)=(00). (3.45)Here we have re-defined the scalars K11, K12, K21, and K22 by K11 = K11I, K12 =K12I, K21 = K21I, and K22 = K22I. When λ = 0, we calculate thatK =(K11I K12IK21I K22I)=−J−1c , where Jc :=(f cµ I fcη Igcµ I gcη I).Finally, upon multiplying (3.45) by Jc, and using JcK =−I, we readily obtain that(f cµ I−Θu f cη Igcµ I gcη I−Θv)(ξζ)=(00), (3.46)which is precisely the same as in (3.22).3.3.1 Re-formulation of the linear stability problemA simpler formulation of the linear stability problem that applies to both symmetricand asymmetric steady-state solutions can be done when g has the specific form in(3.14). In this situation, we can write (3.35) in the formζ j =g′1(µ j)λ +g2ξ j +2πDvλ +g2cvj ,(λ − fµ(µ j,η j))ξ j− fη(µ j,η j)ζ j = 2πDucuj .81Then, upon relating cvj and cuj to ζ j and ξ j by using (3.40), we obtain in matrixform thatζ =1λ +g2Λ2ξ − 1λ +g2Θv,λζ , Λ3ξ −Λ4ζ =−Θu,λξ , (3.47)where Θu,λ , Θv,λ , and the diagonal matrices Λ2, Λ3, and Λ4 are defined byΘu,λ := 2πνDudu2du1[(1+νDudu1)I+2πνGu,λ]−1, Θv,λ := 2πνDvdv2dv1[(1+νDvdv1)I+2πνGv,λ]−1,(3.48a)Λ2 := diag(g′1(µ j)) , Λ3 := diag(λ − fµ(µ j,η j)) , Λ4 := diag( fη(µ j,η j)) .(3.48b)Upon eliminating ζ in (3.47), we obtain the following nonlinear eigenvalueproblem for the case where g has the form in (3.14):N (λ )ξ = 0 , where N (λ ) := Λ3−Λ4[(λ +g2)I+Θv,λ]−1Λ2+Θu,λ .(3.49)Observe that setting det(N (λ )) = 0 involves root-finding on the determinant of amatrix of size m×m rather than that for the larger 2m×2m matrix, as needed for(3.44).The characterization (3.49) is particularly useful for determining the linearstability properties of a symmetric steady-state for a symmetric cell arrangementwhen g has the form in (3.14). For a symmetric steady-state with λ = 0, we obtainfrom (3.48) that Λ2 = g′1(µc)I, Λ3 =− f cµ I, and Λ4 = f cη I. From (3.49), this yieldsthatN (0) =− f cµ I− f cηg′1(µc) [g2I+Θv,0]−1+Θu,0 .Since Θu,λ = Θu and Θv,λ = Θv when λ = 0, where Θu and Θv were definedin (3.13b), we obtain that the condition det(N (0)) = 0 is equivalent to theformulation (3.23) derived in §3.2.3, which was based on linearizing the nonlinearalgebraic system around the symmetric steady-state solution.For a symmetric steady-state solution of a symmetric cell arrangement, onekey advantage of the re-formulation (3.49) is that the discrete eigenvalues of thelinearization (3.32) can be determined by finding the union of the roots of m scalar82problems. This is done by using det(N (λ )) =∏mj=1σ j(λ ), where σ j(λ ) for j ={1, . . . ,m} are the eigenvalues of N (λ ). More specifically, since Gu,λ and Gv,λhave the common eigenspaceGu,λ e= κu,λ e , Gv,λ e= κv,λ e ; Gu,λq j = κ⊥u,λ jq j , Gv,λq j = κ⊥v,λ jq j , j∈{2, . . . ,m} ,(3.50a)the eigenvalue σ1(λ ) corresponding to e and the eigenvalues σ j(λ ) correspondingto q j, for j ∈ {2, . . . ,m} are readily calculated. A simple calculation yields thatσ1(λ ) = λ − f cµ −f cηg′1(µc)λ +g2+αv,λ+αu,λ , (3.51a)σ j(λ ) = λ − f cµ −f cηg′1(µc)λ +g2+α⊥v,λ j+α⊥u,λ j , j ∈ {2, . . . ,m} ,(3.51b)where we have definedαu,λ :=2πνDudu2/du11+νDu/du1 +2πνκu,λ, αv,λ :=2πνDvdv2/dv11+νDv/dv1 +2πνκv,λ, (3.51c)α⊥u,λ j :=2πνDudu2/du11+νDu/du1 +2πνκ⊥u,λ j, α⊥v,λ j :=2πνDvdv2/dv11+νDv/dv1 +2πνκ⊥v,λ j, (3.51d)j ∈ {2, . . . ,m}.With this re-formulation, for a symmetric steady-state of a symmetric cellarrangement, and with g of the form in (3.14), the set of discrete eigenvalues ofthe linearization, Λ(M ), in (3.44) can be written conveniently asΛ(M ) := {λ |σ1(λ ) = 0} ∪m⋃j=2{λ |σ j(λ ) = 0} . (3.52)In summary, to determine the linear stability properties of this symmetric steady-state solution we need only solve m scalar root-finding problems and determinewhether there are any roots in Re(λ ) > 0. This is considerably more tractablenumerically than performing a root-finding based on the determinant of the GCEPin (3.49).833.4 Illustrations of the Theory: A Ring Pattern of CellsIn this section we illustrate the steady-state and linear stability theory developed in§3.2.2 and §3.3 for a ring pattern of cells inside unit disk Ω, for which the Greenfunction is given analytically in Appendix A.4. We will show that symmetry-breaking bifurcations can occur for the Gierer-Meinhardt [42], Rauch-Millonas[116] and FitzHugh-Nagumo [46] reaction kinetics. The theoretical predictionof stable asymmetric patterns will be confirmed through full time-dependentnumerical simulations of (3.3) computed using FlexPDE [39].For a ring pattern of m cells in the unit disk with ring radius r, with 0 < r < 1,the cell centers are located atxk = r(cos(2π(k−1)m), sin(2π(k−1)m)), k ∈ {1, . . . ,m} . (3.53)For a ring pattern of cells, all Green matrices are symmetric and circulant and havethe common eigenspacevk =(1,Zk,Z2k , ...,Zm−1k )T with Zk := exp(2πi(k−1)m)and k∈{1, ...,m} ,which are a basis ofCm. In (A.4) of Appendix A.4 we summarize how to obtain thematrix spectrum of a symmetric and circulant matrix that has a real-valued basisfor the eigenspace.In our illustrations of the theory below, we will assume for simplicity that themembrane reaction rates satisfydu1 = du2 =: du dv1 = dv2 =: dv , with ρ :=dvdu, (3.54)and that g(µ,η) has the specific form in (3.14). We will focus on a two-cell ringpattern in the unit disk, as shown schematically in Figure 3.2, for three specificreaction kinetics.To numerically implement our asymptotic theory, the steady-state solutionbranches are computed from (3.16) with m = 2 using the parameter continuationsoftware MatCont [28], while the symmetric solution branch is obtained from(3.20). Symmetry-breaking bifurcation points in ρ along the symmetric branch are84Figure 3.2: A schematic plot of a ring pattern in the unit disk with twocells. The bifurcation parameter for symmetry-breaking is ρ , while thediffusivities satisfy Du = Dv.identified by numerically solving (3.30) together with (3.20). Finally, to determinethe linear stability properties of the symmetric branch we need only determine ifthere exists a λ with Re(λ ) > 0 in the set Λ(M ) given in (3.52). For m = 2, thisis done by calculating all the roots of σ1(λ ) = 0 and σ2(λ ) = 0 by using (3.51)and the explicit expressions for the eigenvalues of the Green matrices as can beobtained from Appendix A.4.3.4.1 Gierer-Meinhardt reaction kineticsWe consider a prototypical Gierer-Meinhardt model (GM), where the nonlinearreaction kinetics are confined within the compartments. The original GMmodel, introduced in [42] and [41] to model pattern formation in biologicalmorphogenesis, has the form∂tu = Du∆u−σuu+ρ0(x)+ cuρu(x)u2v, ∂tv = Dv∆v−σvv+ cvρv(x)u2 .This model disregards that in biological tissues morphogen-producing reactionsmostly occur intracellularly and on the membranes of cells. For simplicity,we illustrate our compartmental-reaction diffusion theory for the specific casewhere ρ0 ≡ 0,cuρu(x) ≡ 1, and cvρv ≡ 1. When there is no bulk diffusion, thecompartments are uncoupled from the bulk and we impose the intracellular reaction85kineticsµ˙(t) = f (µ,η)≡ µ2η, η˙(t) = g(µ,η)≡ µ2 . (3.55)The uncoupled equilibrium for (3.55) given by µe = 0, and where ηe is an arbitraryconstant, is non-hyberbolic in all directions.To apply the bulk-cell steady-state analysis of §3.2 for a two-cell ring pattern,we first identify that g(µ,η) = g1(µ)−g2η , where g1 = µ2 and g2 = 0. For m= 2,we conclude from (3.16) that all steady-states of the bulk-cell system are associatedwith the nonlinear algebraic problemf (µe1,eT1Θ−1v ((µe1)2,(µe2)2)T )− eT1Θu(µe1,µe2)T = 0f (µe2,eT2Θ−1v ((µe1)2,(µe2)2)T )− eT2Θu(µe1,µe2)T = 0 .(3.56)The symmetric equilibrium (µe,ηe), which satisfies (3.20), is readily calculatedasµe =αvαu, ηe =αvα2u, (3.57)where αu and αv are defined in (3.19). By combining (3.57) with (3.30), weconclude that a symmetry-breaking bifurcation from the symmetric steady-stateoccurs whenever the conditionαvα⊥v,2+α⊥u,22αu−1 = 0 , (3.58)is satisfied at some point along the symmetric solution branch. Here α⊥u,2 and α⊥v,2were defined in (3.31).In the left panel of Figure 3.3 we plot the bifurcation diagram of solutions to(3.56) for a parameter set where Dv =Du and with the other parameter values as inthe figure caption. We observe that a supercritical symmetry-breaking pitchforkbifurcation from the symmetric branch occurs at the critical value ρ = ρp ≈9.79168. In Figure 3.4 we show full PDE results for (3.3) computed with FlexPDE[39] for values of ρ on either side of this theoretically predicted bifurcation value.In the left panels of this figure, we observe that when ρ = 5 < ρp, an initialperturbation of the symmetric steady-state converges back to the symmetric steady-state as time increases. In contrast, when ρ = 15 > ρp, we observe from the right86panels of Figure 3.4 that, for an initial condition near the symmetric steady-state,the time-dependent PDE solution converges as time increases to the asymmetricsteady-state predicted in the left panel of Figure 3.3.In the right panel of Figure 3.3 we show that the pitchfork bifurcation value forthe emergence of asymmetric steady-states increases substantially when the ringradius for the two-cell pattern increases. As a result, we conclude that for cells thatare farther apart, a larger value of ρ is needed to create a stable asymmetric pattern.Figure 3.3: Left: 3-D Bifurcation diagram, computed from (3.56), showingsymmetric and asymmetric steady-states of a two-cell ring pattern withring radius r = 0.5 and GM kinetics (3.55). Asymmetric steady-statesemerge at the supercritical pitchfork bifurcation point ρ = ρp ≈ 9.79168along the symmetric branch. Right: The pitchfork bifurcation value ofρ increases rapidly as the ring radius r, and consequently the distancebetween the cells, increases. The dots are the values computed from(3.58), while the curve is the interpolation by the plotting function inJulia [15]. Parameters: Du = Dv = 5,σu = σv = 0.6,du = 0.09, andε = 0.03.We now show that by varying the membrane reaction rate du, which necessarilyvaries the membrane reaction rate to the v-species according to dv = ρ du, thesteady-state solution branches with GM kinetics (3.55) computed from (3.56) canexhibit a hysteresis structure for low du. The numerical results of Figure 3.5 showsuch a hysteretic bifurcation structure between the asymmetric and symmetricsolution branches for two values of du. We observe that as du decreases theextent of the hysteresis increases. The range where hysteresis occurs is given87Figure 3.4: Full numerical PDE simulation results of (3.3) with FlexPDE [39]for GM kinetics (3.55). The bottom panels show the concentration ofu. Left: Convergence to the symmetric branch for ρ = 5 before thesupercritical pitchfork point ρp = 9.79168, for an initial condition closeto the symmetric branch. Right: Convergence to the asymmetric branchselected by the eigenperturbation direction q2 = (1,−1)T for ρ = 15and starting near the symmetric branch. Parameters: Du =Dv = 5,σu =σv = 0.6,du = 0.09,ε = 0.03, and r = 0.5.88Figure 3.5: 3-D Bifurcation diagram, computed from (3.56), for symmetricand asymmetric steady-states of a two-cell ring pattern with ring radiusr = 0.5 and two different values of du with GM kinetics (3.55). Leftpanel: du = 0.08. Right panel: du = 0.05. For these values ofdu, the steady-states exhibit hysteresis, i.e., a subcritical pitchforkbifurcation occurs from the symmetric equilibrium branch, with theemerging unstable asymmetric equilibrium branches regaining stabilityat a secondary fold point. Observe that the extent of the hysteresisincreases when du decreases. Parameters: Du = Dv = 5,σu = σv =0.6,ε = 0.03, and r = 0.5.by the separation ρp− ρs between the pitchfork point ρp and the secondary foldbifurcation point ρs along each asymmetric branch. Numerical results for this rangefor a parameter set where hysteresis occurs when du < 0.09 is given in Table 3.1.For the parameter set with du = 0.05, which corresponds to the bifurcationdiagram shown in the right panel of Figure 3.5, the full time-dependentcomputations of (3.3) with FlexPDE [39], as shown in Figure 3.8, illustrate thatfor an initial condition near the symmetric steady-state branch, and with ρ eithersatisfying ρ < ρs or ρs < ρ < ρp, the time-dependent solution converges to thestable symmetric steady-state solution. However, as shown in the left panel ofFigure 3.9, for an initial condition near the asymmetric branch when ρ is inthe hysteresis region ρs < ρ < ρp, the time-dependent solution converges to theasymmetric steady-state. Moreover, if ρ > ρp, the right panel of Figure 3.9 showsthat for an initial condition near the unstable symmetric steady-state the time-dependent solution converges to the asymmetric steady-state solution.To determine the linear stability properties of the symmetric steady-state89du 0.05 0.06 0.07 0.08 0.09 0.1 0.11 0.12 0.13 0.135ρp 7.70971 7.66508 11.42015 8.62258 9.79168 11.81838 15.65552 24.82347 70.62460 > 1000 or ∄µe 2.78094 2.54938 3.21380 2.28060 2.19994 2.14061 2.09668 2.06422 2.04050ρs 6.27944 6.93251 6.82631 8.60260 - - - - - -µe1 3.27136 3.09001 3.45845 2.56189 - - - - - -µe2 1.12375 1.34489 0.87895 1.93057 - - - - - -Table 3.1: Numerical values (rounded to 5th decimal place) of the subcritical(or supercritical) pitchfork bifurcation point ρp, the fold bifurcationpoint ρs, and the associated values for the symmetric µe and one of theasymmetric (µe1, µe2) solution branches. As du increases from 0.05,the range of ρ where hysteresis occurs decreases, until a supercriticalpitchfork bifurcation occurs when du ≈ 0.85. Parameters: Du = Dv =5,σu = σv = 0.6,ε = 0.03,r = 0.5.solution branch as ρ is varied in Figures 3.3 and 3.5 we must determine theeigenvalues λ in the set (3.52). This is done by numerically computing the largestroots to σ1(λ ) = 0 and to σ2(λ ) = 0, as defined in (3.51). In Figure 3.6 we plotthese roots versus ρ for two values of du. From this figure, we observe that in-phaseperturbations of the symmetric steady-state solution branch, as determined by theroots of σ1(λ ) = 0, are always linearly stable. In contrast, anti-phase perturbationsof the symmetric steady-state, as characterized by the roots of σ2(λ ) = 0, arelinearly stable only for ρ < ρp, where ρp is the symmetry-breaking threshold. Forρ > ρp, the symmetric steady-state solution branch is unstable to an anti-phaseeigenperturbation q2 = (1,−1)T .Next, to study the linear stability properties of the asymmetric steady-state solution branches we must determine whether the root-finding conditiondet(N (λ )) = 0 in (3.49) yields an eigenvalue with Reλ > 0. The numericalresults shown in Figure 3.7 for du = 0.05 (corresponding to the right panel in Figure3.5), establishes that the asymmetric branch on the subcritical range ρs < ρ < ρp,which emanates from the symmetric steady-state branch, is unstable. However, asobserved from Figure 3.7, the upper portion of the asymmetric branch for ρ > ρsis linearly stable.We now explore how the pitchfork bifurcation point depends on decreasingvalues of the diffusion coefficient ratio Dv/Du when du = 0.09. When this ratiois unity, there was no hysteresis between the symmetric and asymmetric steady-90Figure 3.6: Plots of the numerically computed largest roots of σ1(λ ) = 0and σ2(λ ) = 0 versus ρ , as defined in (3.51), that determine the linearstability properties to either in-phase e = (1,1)T or anti-phase q2 =(1,−1)T eigenperturbations of the symmetric steady-state solution,respectively. Left panel: For du = 0.09 we have ρp ≈ 9.79168.Right panel: For du = 0.05 we have ρp ≈ 7.70971. Observe that in-phase eigenperturbations are always linearly stable, whereas anti-phaseeigenperturbations are linearly stable only on the range ρ < ρp beforethe pitchfork point ρp. Parameters: Du = Dv = 5,σu = σv = 0.6,ε =0.03,r = 0.5.Figure 3.7: Zero-crossings of det(N (λ ))= 0, as defined in (3.49), determinethe linear stability properties of an asymmetric steady-state solutionwith two cells. On the range ρp < ρ < ρs, before the secondaryfold point along the asymmetric branch, we observe that λ > 0. Thisimplies that the subcritical portion of the asymmetric steady-statebranch between the pitchfork point and the fold point is unstable.Further along past the fold point the asymmetric branch regains stability.Parameters: Du = Dv = 5,σu = σv = 0.6,du = 0.05,ε = 0.03,r = 0.5.91Figure 3.8: Full numerical simulation results of (3.3) with FlexPDE [39] forGM kinetics (3.55). Left: For an initial condition near the symmetricbranch we observe convergence to the symmetric branch when ρ =4, which is before the hysteresis region bounded by the fold pointρs ≈ 6.27945 and the subcritical pitchfork point ρp ≈ 7.70971. Right:Convergence to the symmetric branch for ρ = 7.2, which lies on therange ρs < ρ < ρp, when starting near the symmetric branch. Thebottom panels show the concentration of u. Parameters: Du = Dv =5,σu = σv = 0.6,du = 0.05,ε = 0.03,r = 0.5.92Figure 3.9: Full numerical simulation results of (3.3) with FlexPDE [39] forGM kinetics (3.55). Left: Convergence to the asymmetric branch foran initial condition near this branch when ρ = 7.2 lies in the hysteresisregion between the fold point ρs ≈ 6.27945 and the subcritical pitchforkpoint ρp ≈ 7.70971. Right: Convergence to an asymmetric steady-stateas selected by a small initial perturbation of the symmetric solution inthe direction q2 = (1,−1) when ρ = 15 > ρp. The bottom panels showthe concentration of u. Parameters: Du = Dv = 5,σu = σv = 0.6,du =0.05,ε = 0.03, and r = 0.5.93state solution branches (see Table 3.1). A similar numerical experiment wasperformed in Section 2.3.4 for a 1-D compartmental-reaction diffusion model withGM kinetics with dynamically reactive boundaries. In our 2-D setting, we observefrom the numerical results in Table 3.2 that a symmetry-breaking bifurcation canoccur on the range Dv/Du < 1, but only up until some minimum diffusion ratiothreshold at which the pitchfork bifurcation point given by the root of (3.58) nolonger exists. In addition, we observe that reducing the diffusion ratio thresholdDv/Du below unity for fixed du = 0.09 does not introduce new hysteresis behavior,and the symmetry-breaking bifurcation remains supercritical.Next, we set du = 0.08 where hysteresis occurs when Dv/Du = 1, and wevary this diffusion ratio to determine whether hysteresis can be eliminated. Ournumerical results in Figure 3.10 and Table 3.3 indicates that varying Dv/Du doesnot eliminate the hysteresis between the symmetric and asymmetric steady-statebranches. However, the extent of the hysteresis decreases as the ratio Dv/Duincreases.Dv/Du 0.42 0.43 0.5 0.6 0.8 1 1.2 1.4ρp > 1000 or ∄ 573.56743 38.45836 19.56926 12.06861 9.79168 8.69082 8.04185µe 2.32310 2.29271 2.26170 2.22305 2.19994 2.18456 2.17360ρ f - - - - - - - -µe1 - - - - - - - -µe2 - - - - - - - -Table 3.2: Decreasing the ratio Dv/Du does not trigger hysteresis whendu = 0.09, but rather there is a minimum threshold of the diffusivityratio where the symmetry-breaking pitchfork bifurcation point exists.The numerical values for the pitchfork point (ρp,µe) on the symmetricsteady-state branch are again rounded to the 5th decimal place.Parameters: Du = 5,σu = σv = 0.6,du = 0.09,ε = 0.03,r = 0.5.To obtain some analytical insight into the disappearance of the pitchfork pointas shown in Tables 3.2 and 3.3 when the diffusivity ratio Dv/Du decreases belowa threshold, in Figure 3.11 we plot the function Fα(ρ) := αv/α⊥v,2 +α⊥u,2/(2αu)−1 versus ρ , representing the left-hand side of the pitchfork bifurcation condition(3.58), for several values of Dv/Du, and for either du = 0.08 (left panel) or du =0.09 (right panel). From (3.58) a root of Fα(ρ) = 0 corresponds to a symmetry-94Dv/Du 0.37 0.38 0.4 0.6 0.8 1 3 5 8ρp > 1000 or ∄ 197.98732 72.56533 14.30013 10.14347 8.62258 6.13144 5.79198 5.61640µe 2.43797 2.42519 2.34462 2.30457 2.28061 2.21699 2.20432 2.19719ρ f 188.58078 71.25577 14.24676 10.11612 8.60260 6.12105 5.78264 5.60759µe1 2.7386611 2.72431 2.63380 2.58881 2.56189 2.49043 2.47619 2.46819µe2 2.063782 2.05297 1.98476 1.95085 1.93057 1.87672 1.86599 1.85996Table 3.3: Increasing the diffusivity ratio Dv/Du when du = 0.08 does noteliminate hysteresis, as the symmetry-breaking bifurcation point remainssubcritical. The numerical values for the pitchfork point (ρp,µe) on thesymmetric equilibrium branch and for one of the fold points (ρs,µe1,µe2)on the asymmetric branch are again rounded to the 5th decimal place.Parameters: Du = 5,σu = σv = 0.6,du = 0.08,ε = 0.03,r = 0.5.breaking bifurcation point along the symmetric solution branch. From Figure 3.11we observe that the asymptote of Fα(ρ) as ρ→∞ is positive when Dv/Du is belowa threshold, which eliminates the possibility of a pitchfork bifurcation point.Figure 3.10: Effect of the diffusivity ratio Dv/Du of the two bulk specieson the extent of the hysteresis when du = 0.08, as measured bythe distance between the fold bifurcation points and the subcriticalpitchfork bifurcation point (left) and by the distance of the twoasymmetric equilibria µe1 and µe2 from each other (right). Thediffusivity Du = 5 is fixed and the remaining parameters are as in Table3.3. The dots are the numerically computed values using MatCont [28]that are interpolated by the plotting function in Julia [15].95Figure 3.11: Effect of the diffusivity ratio Dv/Du on the existence of thepitchfork point when du = 0.08 (left) and du = 0.09 (right). Thenumerical results show that the asymptote of Fα(ρ) := αv/α⊥v,2 +α⊥u,2/(2αu)− 1 is positive for smaller values of Dv/Du, as suggestedby Tables 3.2 and 3.3. Therefore, when Dv/Du falls below a threshold,the pitchfork bifurcation condition Fα(ρ) = 0, which is equivalent to(3.58), no longer holds for any ρ > 0.3.4.2 Rauch-Millonas reaction kineticsNext, we consider the activator-inhibitor system proposed in [116] to universallymodel two-species signal transduction reaction kinetics between cells. This Rauch-Millonas (RM) intracellular kinetics of [116] is given byµ˙ = cu−quµ+ au1µbu1+µ− au2µηbu2+µ ≡ f (µ,η)η˙ = cv+wvµ−qvη ≡ g(µ,η) .(3.59)Since g has the form in (3.14), we identify that g1(µ) = cv+wvµ and g2 = qv. Wewill choose a parameter set for which the reaction kinetics when uncoupled fromthe bulk has a unique linearly stable steady-state.From (3.16), all steady-states of the bulk-cell model for a two-cell ring patternare associated with the nonlinear algebraic systemf (µe1,eT1 (qvI+Θv)−1 (cv+wvµe1,cv+wvµe2)T )− eT1Θu(µe1,µe2)T = 0f (µe2,eT2 (qvI+Θv)−1 (cv+wvµe1,cv+wvµe2)T )− eT2Θu(µe1,µe2)T = 0 .(3.60)By using (3.20), the symmetric steady-state solution branch is obtained from the96solution µe tocu−quµe+ au1µebu1+µe− au2µebu2+µe(cv+wvµe)qv+αv−αuµe = 0 , (3.61)where αu and αv are given in (3.19). Symmetry-breaking bifurcation points areobtained by solving the zero-eigenvalue crossing condition (3.30) together with(3.61). By solving for wv = wv(µe) in (3.30), we calculatewv(µe) =−qu+ au1bu1+µe− au1µe(bu1+µe)2− au2bu2+µecvqv+αv +au2µe(bu2+µe)2cvqv+αv −α⊥u,2au2bu2+µeµeqv+αv −au2µe(bu2+µe)2µeqv+αv +au2µebu2+µe1qv+α⊥v,2, (3.62)where α⊥u,2 and α⊥v,2 are defined in (3.31). By using (3.62) to eliminate ωv in (3.61)we obtain a nonlinear algebraic equation that determines any symmetry-breakingbifurcation value for µe along the symmetric solution branch. The correspondingbifurcation value for wv is obtained from (3.62).For the parameter set given in the figure caption, we show in the left panel ofFigure 2.11 that, for the fixed value ρ = 15, there is a degenerate wv-pitchforkbubble, which is characterized by the emergence of asymmetric steady-statesolutions at two values of wv. From the right panel of Figure 2.11 we observe that interms of ρ , and at a fixed wv, the symmetry-breaking bifurcation is supercritical inρ . For the parameter set in the right panel of Figure 2.11, we observe from Figure3.13 that the eigenvalue λ determined by the root-finding condition σ2(λ ) = 0,with σ2 given in (3.51), crosses through zero at the ρ-pitchfork bifurcation pointalong the symmetric steady-state branch. As a result, when wv = wP,2v ≈ 7.08723,the symmetric steady-state solution is linearly stable for ρ < ρp = 15, and isunstable on the range ρ > ρp = 15 to eigenperturbations in the direction ofq2 = (1,−1)T .3.4.3 FitzHugh-Nagumo reaction kineticsFinally, we consider a ring pattern for the bulk-cell system (3.3) with two cells andwith FitzHugh-Nagumo (FN) intracellular reaction kinetics [46]. The uncoupled97Figure 3.12: 3-D Bifurcation diagram, computed from (3.60) using MatCont[28], for symmetric and asymmetric steady-states of a two-cell ringpattern with ring radius r = 0.5 and with RM kinetics (3.59). Left:3-D Plot of (µe1,µe2) versus the kinetic parameter wv in (3.59) at afixed ρ = dv/du = 15, showing that asymmetric steady-states occurinside the degenerate pitchfork bubble delimited by wP,1v ≈ 6.88285and wP,2v ≈ 7.08723. Note that the bubble lobes are stretching intodecreasing wv and that there exists hysteresis at wP,2v . Right: Interms of ρ , a supercritical pitchfork bifurcation from the symmetricbranch occurs when wv = wP,2v ≈ 7.08723. The asymmetric branchesare linearly stable past this bifurcation threshold in ρ . Parameters:Du = Dv = 1,σu = σv = 0.1,ε = 0.03,cu = cv = 1,qu = 1/100,qv =7,au1 = 600,au2 = 6,bu1 = 100,bu2 = 1/10, and du = 0.14.intracellular kinetics areµ˙(t) = µ−q(µ−2)3+4−η ≡ f (µ,η) , η˙(t) = δ zµ−δη ≡ g(µ,η) ,(3.63)with q > 0, δ > 0 and z > 0. Since g has the form in (3.14), we identify thatg1(µ) = δ z and g2 = δ .We will choose a parameter set for which there is a unique linearly stablesteady-state of the intra-compartmental dynamics (3.63). From (3.16), all steady-states of the bulk-cell model for a two-cell ring pattern are obtained from thenonlinear algebraic problemf (µe1,δ zeT1 (δ I+Θv)−1(µe1,µe2)T )− eT1Θu(µe1,µe2)T = 0f (µe2,δ zeT2 (δ I+Θv)−1(µe1,µe2)T )− eT2Θu(µe1,µe2)T = 0 .(3.64)98Figure 3.13: For the two-cell system with RM kinetics (3.59) and parametersas in the caption of Figure 3.12 with wv = wP,2v , we plot theeigenvalue λ , satisfying σ2(λ ) = 0 in (3.51), versus ρ that determinesthe linear stability of the symmetric steady-state solution branch toeigenperturbations of the form q2 = (1,−1)T . We observe that thesymmetric steady-state branch is unstable only for ρ > ρp = 15. Thereis no root to σ1(λ ) = 0 on this range.The symmetric steady-state solution branch, as characterized by (3.20), is obtainedfrom the root µe of the cubic equationµe−q(µe−2)3+4− δ zµeδ +αv −αuµe = 0 , (3.65)where αu and αv are given in (3.19). The symmetry-breaking bifurcation condition(3.30) along the symmetric steady-state solution branch yields that1−3q(µe−2)2− δ zδ +α⊥v,2−α⊥u,2 = 0 ⇔ z(µe)=δ +α⊥v,2δ(1−3q(µe−2)2−α⊥u,2),where α⊥u,2 and α⊥v,2 are defined in (3.31). We substitute z(µe) into (3.65), and solvethe resulting equation numerically for µe. For ρ = 150, and with the parameters asin the caption of Figure 3.14, we obtain that there are two supercritical pitchforkbifurcation points zP,1 and zP,2 on the symmetric steady-state branch. The linearlystable asymmetric steady-state branches that exist on the range zP,1 < z < zP,2between the two pitchfork points are shown in the left panel of Figure 3.14. Whenz = zP,2, we observe from the bifurcation diagram in the right panel of Figure993.14, together with the eigenvalue computations in Figure 3.15, that the symmetry-breaking bifurcation is supercritical in terms of ρ .Next, we illustrate that the bulk-cell model with FN kinetics can also exhibitoscillatory instabilities for in-phase perturbations of the symmetric steady-state. InFigure 3.16 we plot the bifurcation diagram of µe1 versus z for the same parameterset as in the caption of Figure 3.14 except that the bulk degradation rates havebeen decreased slightly to σu = σv = 0.9. We observe that there are now two Hopfbifurcation values zH,1 and zH,2 of z along the symmetric steady-state branch for thein-phase mode that lie within the interval delimited by the two pitchfork bifurcationpoints. In the right panel of Figure 3.16, we plot the real and imaginary parts ofthe complex-valued root of σ1(λ ) = 0, as computed from (3.51), which shows thatRe(λ )> 0 and Im(λ ) ̸= 0 when zH,1 < z < zH,2. This leads to the possibility of asynchronous oscillatory instability. As a result, on the range zH,1 < z < zH,2, thesymmetric steady-state solution branch is unstable to both anti-phase and in-phaseperturbations. However, as seen from the right-panel of Figure 3.16, where wealso plot the growth rate λ for the anti-phase mode as obtained by setting σ2(λ ) =0 in (3.51), the anti-phase instability has a larger growth rate than the in-phaseinstability.100Figure 3.14: 3-D Bifurcation diagram, computed from (3.64) using MatCont[28], for symmetric and asymmetric steady-states of a two-cell ringpattern with ring radius r = 0.5 and with FN kinetics (3.63). Left: 3-DPlot of (µe1,µe2) showing that asymmetric steady-states occur insidethe supercritical pitchfork bubble delimited by zP,1 ≈ 36.75458 andzP,2 ≈ 41.26889 when ρ = dv/du = 150. Right: Supercritical pitchforkbifurcation from the symmetric branch occurs at ρp = 150 when z =zP,2. Linearly stable asymmetric branches exist past this threshold in ρ .Parameters: Du = 1,Dv = 4,σu = σv = 1,ε = 0.03,r = 0.5,q = 1,δ =0.1, and du = 0.04.3.4.4 Numerical experiments with closely-spaced cells: GM kineticsWe now briefly explore, from full PDE simulations of (3.3), symmetry-breakingbehavior leading to stable asymmetric patterns that can occur for closely spacedcells when the ratio ρ = dv/du is increased. For realistic modeling of patternformation properties of biological tissues one needs to consider the situation wherecells are closely spaced in the sense that the cell radii are either comparable tothe distance between the cells, or that there are only narrow gaps between cells.Although the asymptotic theory of §3.2 and §3.3 is no longer valid for such closelyspaced cell arrangements, the FlexPDE simulations of (3.3) shown below reveal asimilar qualitative solution behavior as we have analyzed for spatially segregatedcells. More specifically, although we no longer have an analytical theory to predicta bifurcation diagram of all steady-state solutions, our full PDE numerical resultssuggest that stable symmetric steady-states occur only when ρ is below somethreshold. When ρ exceeds some symmetry-breaking threshold, stable asymmetric101Figure 3.15: For the two-cell system with FN kinetics (3.63) and parametersas in the caption of Figure 3.14 with z = zP,2, we plot the numericallycomputed eigenvalue λ , satisfying σ2(λ )= 0 in (3.51), versus ρ . Sinceλ > 0 only on the range ρ > ρp = 150, we conclude that the symmetricsteady-state solution is linearly stable to anti-phase eigenperturbationsof the form q2 = (1,−1)T only when ρ < ρp = 150. Moreover,since the root to σ1(λ ) = 0 satisfies λ < 0, we conclude that thesymmetric steady-state branch is always linearly stable to in-phaseeigenperturbations of the symmetric steady-state.steady-states will be the preferred state. Our numerical results suggest that thecritical threshold of ρ that is needed to establish this symmetry-breaking behaviorfor closely spaced cells is smaller than that needed for spatially segregated cells, ifin fact such a threshold exists.To illustrate this, in Figure 3.17 we take two closely spaced cells centered nearthe origin that have a minimum separation of 0.002. The degradation rates, cellradius, and the value of du used for Figure 3.17 are the same as in Table 3.3, wherebifurcation values were given for the two-cell arrangement at different ratios ofDv/Du with Du = 5 and for a ring radius r = 0.5. In the cell arrangement in Figure3.17, the only difference is that the two cells are now much more closely spacedthan in Table 3.3 and we fix Dv/Du = 0.3 and Du = 5. For these parameter values,we observe from Table 3.3 that no symmetry-breaking bifurcations are possiblefor this diffusivity ratio when the ring radius is r = 0.5. However, as suggestedfrom the results shown in Figure 3.17, when the cells are closely spaced there is a102Figure 3.16: Left panel: Bifurcation diagram of µe1 versus z, computed from(3.64) using MatCont [28], for symmetric and asymmetric steady-states of a two-cell ring pattern with the same parameters as in theleft panel of Figure 3.14 except that now the degradation rates aredecreased slightly to σu = σv = 0.9. For this parameter set, thereare Hopf bifurcation points associated with in-phase perturbations ofthe symmetric steady-state that emerge at z = zH,1 ≈ 34.65328 andz = zH,2 ≈ 38.02834 between the two symmetry-breaking pitchforkbifurcation points located at z = zP,1 ≈ 33.41022 and z = zP,2 ≈38.80742. Right panel: The root λ of σ2(λ ) = 0 versus z (pinkcurve), as computed from (3.51), shows that the symmetric steady-state solution branch is unstable to anti-phase perturbations on therange zP,1 < z < zP,2. The plotted real and imaginary parts of thecomplex-valued root λh := λr + iλi to σ1(λh) = 0, from (3.51), showsthat Re(λh) > 0 on the range zH,1 < z < zH,2. On this range ofz, a synchronous oscillatory instability of the symmetric steady-statesolution can also occur, but it has a smaller growth rate than that forthe anti-phase mode.symmetry-breaking pitchfork bifurcation point that occurs on the range 3< ρ < 8.We remark that, rather surprisingly, if we use the symmetry-breaking bifurcationcondition (3.58) from the asymptotic theory for this case of two-closely spacedcells it predicts that ρp ≈ 6.53, which lies within the range 3 < ρ < 8. However,we emphasize that the asymptotic theory is not valid for closely-spaced cells.Finally, in Figures 3.18 and 3.19 we show that the stable asymmetric steady-state patterns can also occur for respectively three and four closely-spaced cellswhen ρ exceeds some threshold.103Figure 3.17: Full numerical simulation results of (3.3) with FlexPDE [39] forGM kinetics (3.55) with two closely spaced cells centered on a ring ofradius r = 0.031 and with minimum cell separation of 0.002. Theother parameters are the same as in Table 3.3. The only differencehere is that the cells are now much more closely spaced. The bottomtwo panels show the concentration of u. Left: Convergence to a stablesymmetric steady-state solution when ρ = 3. Right: Convergence toa stable asymmetric steady-state solution for ρ = 8 when starting witha symmetric initial condition. Parameters: Du = 5, Dv = 1.5, σu =σv = 0.6, du = 0.08, and ε = 0.03. It is difficult to observe symmetry-breaking in the unfiltered extracellular species plots, but it is apparentin the plots of the intracellular species concentrations.104Figure 3.18: Full numerical simulation results of (3.3) with FlexPDE [39]for GM kinetics (3.55) with three closely spaced cells with cell centerslocated on the vertices of an equilateral triangle centered at the origin.The ring radius is r = 0.2√3/6. Left: Convergence to a symmetricsteady-state when ρ = 5. Right: Convergence to an asymmetricsteady-state when ρ = 10. Parameters: Du = Dv = 5,σu = σv =0.6,du = 0.05,ε = 0.099. The cells have a minimum separation of0.002. The bottom two panels show the concentration of v. By plottingv rather than u, the bottom right panel clearly shows the asymmetry inthe three cells.105Figure 3.19: Full numerical simulation results of (3.3) with FlexPDE [39]for GM kinetics (3.55) in four close cells. Left: Convergence tothe symmetric branch starting close to for ρ = 5 before the nearlysupercritical pitchfork point ρp ≈ 8.28593. Right: Convergence tothe asymmetric branch selected by a smaller magnitude version ofeigenperturbation v3 = (1,−1,1,−1) for ρ = 20 and starting close tothe symmetric branch. The true pitchfork point value is rather closeto 16 because of the slow convergence rate we saw when starting withρ = 16. Parameters: Du =Dv = 5,σu =σv = 0.6,du = 0.05,ε = 0.099.106Figure 3.19: The cells have a minimum separation of 0.002. We note that wehad to stop the simulation for the right-hand side plots due to numericaloscillations after t = 150.3.5 Compartments Centered on a Bravais LatticeUp to now we have only considered systems with a relatively small number ofcells in 2-D. In 1-D we have already analyzed the periodic domain with infinitelymany cells, which can be viewed as a circular 1-D domain with finitely manycells. In the present section we regard a similar but still infinite case of a periodicplacement of cells in R2. Here it has to be defined what we mean by periodic,and we will introduce a natural extension of the 1-D periodic domain which isnot the torus: compartments centered on an infinite Bravais lattice. This 2-Dsetting can be understood to be a first step towards describing pattern formationin tissues with compartmental-reaction diffusion systems. Already in previoussections Gierer-Meinhardt tissue reaction kinetics have been used for illustration,which would now be especially interesting to include inside the cells in an infinite2-D cell system. Surprisingly, in such a 2-D system there is a continuous,not discrete, set of eigenperturbations that can perturb the symmetric steady-states, in which all cells have the same equilibrium. We illustrate with Gierer-Meinhardt, FitzHugh-Nagumo, and Rauch-Millonas intracellular reaction kineticswhich of those eigenperturbations can destabilize the symmetric steady-state forchosen parameter sets. It seems that, upon sliding the parameter values of thesystem closer to a bifurcation point, eigenperturbations which are correspondingto non-oscillatory and asymmetric steady-states cause the symmetric steady-stateto become unstable in case of dealing with Gierer-Meinhardt and Rauch-Millonaskinetics. We stress that, to prove this statement, all possible growth rates wouldhave to be computed, so that we know that the ones shown in Figures 3.21, 3.23,and 3.24 are indeed representing the dominant growth rates, i.e., the ones of biggestmagnitude. This remains to be done in future work, but we conjecture at this pointthat it is seems to hold.After the steady-state and perturbation analysis of the Bravais cell system, weconnect back to a system with finitely many cells by taking an excerpt of the107Bravais system, still embedded in R2, in order to discuss its relation to the fullBravais cell system. For the finite system, the symmetric equilibrium actually doesnot exist, and we offer a conjecture on which steady-state would correspond to thebase state for a perturbation analysis that would converge to the symmetric steady-state for cell numbers growing to infinity.3.5.1 Construction of the theory3.5.1.1 Model formulationWe consider disk-shaped compartments with small radius ε centered well-separated (O(ε0)) on a Bravais lattice which is generated by the vectors l1 ∈ R2and l2 ∈R2, l1 ∦ l2. We denote the Bravais lattice by Λ= {ml1+nl2 |m,n∈Z} andthe compartments by Bε(q) for q ∈ Λ. The extra-compartmental space diffusivelycouples with two scalar fields (u and v) all compartments through their reactiveboundary conditions to corresponding two intracellular scalar fields (µq and ηqinside compartment q), evolving through nonlinear intra-compartmental dynamics.Analogously to Appendix A.3 for the finite 2-D domain, we nondimensionalize thesystem, so that the dimensionless PDE-ODE model readsbulk∂tu = Du∆u−σuu , t ∈ (0,∞), x ∈ R2\⋃mq∈ΛBε(q) ,∂tv = Dv∆v−σvv , t ∈ (0,∞), x ∈ R2\⋃mq∈ΛBε(q) ,(3.66a)reaction fluxesεDu∂nqu = du1u−du2µq , t ∈ (0,∞), x ∈ ∂Bε(q) ,εDv∂nqv = dv1v−dv2ηq , t ∈ (0,∞), x ∈ ∂Bε(q) , (3.66b)compartmentsdµqdt = f (µq,ηq)+1ε∫∂Bε (q)(du1u−du2µq) dS , t ∈ (0,∞)dηqdt = g(µq,ηq) +1ε∫∂Bε (q)(dv1v−dv2ηq) dS , t ∈ (0,∞)(3.66c)for q ∈ Λ. Here nq is the outward unit normal vector to the compartment Bε(q).We will refer to du1 , dv1, du2 and dv2 as dimensionless membrane reaction rates. Foran illustration of the Bravais cell system please refer to Figure 3.20.108Figure 3.20: A disk-shaped excerpt of the considered lattice cell system inwhich cells of small radius ε are centered in a well-separated manneron a Bravais lattice with generators l1 and l2. The Voronoi cellsassociated with the compartment centers are shown with grey dashed-dotted lines. Intracellular nonlinear reaction kinetics are producingand consuming µq and ηq in the qth cell, which are reacting to theextracellularly diffusing species u and v through reactive membraneboundary conditions. The cells drawn in are rather large in thisschematic for illustration purposes only, and the cell at the origin ismarked with a dark-blue membrane.3.5.1.2 Symmetric steady-state constructionWe now asymptotically construct the symmetric steady-state, for which allcompartments have the same intra-compartmental equilibrium, using the smallradius of the compartments. In the qth local region, i.e., in an O(ε)-neighborhoodabout compartment Bε(q), introduce the local coordinates y := 1/ε · (x−q) andp := |yq| and the local scalar fields for the steady-state ueq(x) := ue(q+ εyq) andveq(x) := ve(q+εyq). In this local neighborhood the steady-state is recovered from,109in local coordinates,∆queq = 0, yq ∈ Bδ (q)\B1(q), 1 < δ ≪ 1ε ,∆qveq = 0,Du∂nueq = du1ueq−du2µeq, yq ∈ ∂B1(q),Dv∂nveq = dv1veq−dv2ηeqwhere Ba(c) always denotes the two-dimensional closed ball of radius a centeredat c ∈ R2. Due to the radial symmetry of the local Laplacian setting, the equationsyieldueq(pq) = Auq ln(pq)+ 1du1 (DuAuq+du2µq),veq(pq) = Avq ln(pq)+ 1dv1 (DvAvq+dv2ηq)with constants Auq and Avq for q ∈ Λ, which still remain to be determined.Inserted into the intracellular equilibrium equations, we obtainf (µq,ηq)+2πDuAuq = 0,g(µq,ηq)+2πDvAvq = 0.To determine Auq and Avq by matching to a global solution, we construct the globalsolution switching back to the original (global) coordinate x from the global steady-state problem∆Ue−ω2uUe = 0, x ∈ R2\Λ,Ue ∼ Auq ln |x−q|+ Auqν +1du1(DuAuq−du2µq) as x→ qwhere ωu :=√σu/Du and ν :=−1/ ln |ε| is a gauge function. The global steady-state problem for Ve is analogous but with ωv :=√σv/Dv etc. Now follows the keystep in the global solution construction. We focus on the region of points whichare closer to the center of the cell at the origin than to all other cell centers. Thisregion is called the Wigner-Seitz cell or Voronoi cell of the origin in the lattice (cf.[70]; see also [14]), and we denote it by Ω. The central building block for Ue and110Ve is the periodic reduced-wave Green function Gω which solves in Ω∆Gω −ω2Gω = −δ (x−0), x ∈Ω,Gω(x) = Gω(x−q), q ∈ Λ,Gω ∼ − 12π ln |x−q|+Rω(q)+o(1) as x→ q.(3.67)Here the singular behavior in the third line is not what we enforce but is a well-known property of the solution Gω of the first two lines (cf. [73]). Here Rω(q) isthe regular, or nonsingular and nondecaying, part as x→ q.Now we can represent, informed by matching to the singular behavior of the localsolution,Ue(x) =−2πAus Gωu(x;q), Ve(x) =−2πAvsGωv(x;q).where we have used the notation Aus for all Auq etc. since all cells have the sameequilibrium in the symmetric steady-state.One last step remains for determining the constants Aus and Avs . Matching now alsothe regular parts of the global and local solutions, we see that−2πRωu(0)Aus = 1ν Aus +du1(DuAus +du2µes ),⇔ (1+2πνRωu(0)+ν Dudu1 )Aus = −ν du2du1µes .This determines Aus and Avs is determined analogously. With these constants athand, the equations for the intracellular equilibrium readf (µes ,ηes )−2πνDudu2du1(1+2πνRωu(0)+νDudu1)−1µes = 0, (3.68a)g(µes ,ηes )−2πνDvdv2dv1(1+2πνRωv(0)+νDvdv1)−1ηes = 0, (3.68b)from which the steady-state solution (µes ,ηes ) can be extracted by using a rootfinding routine. This constitutes the symmetric steady-state in the Bravais latticecell system, where each Voronoi cell has the same equilibrium structure. As theinverted scalar above is positive, no problems arise during inversion.Remark 3.5.1. Comparing with the nonlinear algebraic system for the symmetric111steady-state for the finite no-flux-disk domain of Section 3.2.1, the scalar2πνDudu2du1(1+ 2πνRωu(0)+ νDudu1)−1 can be interpreted as both the eigenvalue tothe one-vector as eigenvector of an infinite version of a coupling matrix (Θu in(3.13)) which only has diagonal entries with this scalar standing repeatedly alongits diagonal. The scalar of the second equation can be interpreted similarly.3.5.1.3 Finding asymmetric states branching off the symmetric stateWe now perturb the symmetric steady-state to recover its stability propertiesand to find bifurcation points from its symmetric solution branch to asymmetricsteady-state solutions, in which compartments have different intra-compartmentalequilibria. For this, we introduce small O(δ )-sized space- and time-dependentperturbations, δ ≪ 1, and use the fact that δ is small to obtain a linearizedsystem about the symmetric steady-state. The eigenvalues of the right-hand sideJacobian matrix of the PDE system then give the growth rate λ to the correspondingeigenperturbation eλ tφ(x) perturbing the steady-state Ues (x), and similarly for theother variables. Hence, we writeu(t,x) = Ues (x)+ eλ tφ(x), v(t,x) = V es (x)+ eλ tψ(x),µ(t) = µes + eλ tξ , η(t) = ηes + eλ tζwhere the perturbations have to satisfy, using the notation ∂µ fs := ∂µ f (µes ,ηes ) etc.,∆φ −Ω2uφ = 0, x ∈ R2\⋃q∈ΛBε(q),∆ψ−Ω2vψ = 0,εDu∂nφ = du1φ −du2ξq, x ∈ ∂Bε(q),εDv∂nψ = dv1ψ−dv2ζq,λξq = ∂µ fsξq+∂η fsζq+ 1ε∫∂Bε (q)(du1φ −du2ξq) dS,λζq = ∂µgsξq+∂ηgsζq+ 1ε∫∂Bε (q)(du1ψ−du2ζq) dSwith Ωu :=√λ+σuDuand Ωv :=√λ+σvDv.Infinitely many perturbation shapes can be possible here. To make this linearstability analysis more tractable, in the following we only consider the subsetof such existent perturbations which satisfy a quasi-periodic Floquet-Bloch type112condition on the Voronoi cell boundary of the Bravais lattice (cf. [77], [78]). This issimilar to the Floquet-type boundary conditions on the fundamental domain in the1-D cell system linear stability analysis of Chapter 2. The corresponding systemof equations reads∆φ −Ω2uφ = 0, x ∈Ω\Bε(0),∆ψ−Ω2vψ = 0,εDu∂nφ = du1φ −du2ξq, x ∈ ∂Bε(0),εDv∂nψ = dv1ψ−dv2ζq,φ(x+q) = e−ik·qφ(x), ψ(x+q) = e−ik·qψ(x), x ∈Ω\Bε(0), q ∈ Λ,λξ0 = ∂µ fsξ0+∂η fsζ0+1ε∫∂Bε (0)(du1φ −du2ξ0) dS,λζ0 = ∂µgsξ0+∂ηgsζ0+1ε∫∂Bε (0)(du1ψ−du2ζ0) dSNotice that this system is restricted to the fundamental Wigner-Seitz cell Ω andextended to R2 through the Floquet-Bloch boundary condition for a q-shift forq ∈ Λ from any x ∈ Ω. The wave-vector k ∈ R2 selects the specific geometry ofthe perturbation (the perturbation mode).Proceeding in the same way as before, we first observe that in the localcoordinates y := 1ε · x and p := |y| and with the local functions φ0(p) := φ(εx)and ψ0(p) := ψ(εx), the lowest order system∆pφ0 = 0, y ∈ Bδ (0)\B1(0), 0 < δ ≪ 1ε ,∆pψ0 = 0,Du∂nφ0 = du1φ0−du2ξ0, y ∈ ∂B1(0),Dv∂nψ0 = ds1ψ0−ds2ζ0is solved byφ0(p) = cu1 ln |p|+1du1(Ducu1+du2ξ0),ψ0(p) = cv1 ln |p|+1dv1(Dvcv1+dv2ζ0)113where the constant terms were again determined by inserting cu1 ln |p|+C etc. intothe boundary condition. Thus, the intracellular equilibrium equations become, withF(µ,η) := ( f (µ,η),g(µ,η))T ,(λ I−DF(µes ,ηes ))(ξ0ζ0)= 2π(Ducu1Dvcv1).To find the perturbation growth rates λ , it remains to determine the constants cu1and cv1 by matching to the global perturbation solution satisfying in the originalx-coordinate, for ε → 0,∆Φ−Ω2uΦ = 0, x ∈Ω\{0},∆Ψ−Ω2vΨ = 0,Φ(x+q) = e−ik·qΦ(x), Ψ(x+q) = e−ik·qΨ(x), x ∈Ω\{0}, q ∈ Λ,Φ(x) ∼ cu1 ln |x|+ cu1ν +1du1(Ducu1+du2ξ0) as x→ 0,Ψ(x) ∼ cv1 ln |x|+ cv1ν +1dv1(Dvcv1+dv2ζ0)The fundamental building block of the solution for Φ and Ψ is the quasi-periodic reduced-wave Green function Gk,Ξ satisfying∆Gk,Ξ−Ξ2Gk,Ξ = −δ (x), x ∈Ω,Gk,Ξ(x+q) = e−ik·qGk,Ξ(x), q ∈ Λ,Gk,Ξ(x) ∼ − 12π ln |x|+Rk,Ξ(x)+o(1) as x→ 0.(3.69)Matching the singular behaviors givesΦ(x) =−2πcu1Gk,Ωu(x), Ψ(x) =−2πcv1Gk,Ωv(x),and matching the regular parts gives−2πRk,Ωu(0)cu1 = cu1ν +1du1(Ducu1+du2ξ0)⇔ (1+2πνRk,Ωu(0)+ν Dudu1 )cu1 = −ν du2du1ξ0and, for cv1,cv1 =−νdv2dv1ξ0(1+2πνRk,Ωv(0)+νDvdv1)−1.114With these expressions, the matrix equation for the perturbations to the intracellularequilibrium becomes0 =Mk(λ )(ξ0ζ0)(3.70a)where the linear stability matrix to a eigenperturbation with wave-vector k readsMk := λ I−DF(µes ,ηes )+(2πνDudu2du1(1+2πνRk,Ωu(0)+νDudu1)−1 00 2πνDvdv2dv1(1+2πνRk,Ωv(0)+νDvdv1)−1).(3.70b)with the identity matrix I. Hence, there only exists a nontrivial eigenperturbation tothe symmetric steady-state branch and to wave-vector (-mode) k if det(Mk(λ )) =0. The exponential growth rate of such an existent perturbation is then givenby λ , which can also be strictly complex. Thus, for any parameter set, only ifRe(λ ) > 0, the symmetric equilibrium can be destabilized by the correspondingk-eigenperturbation, causing convergence to an asymmetric oscillatory or steadystate with similar geometry as the eigenperturbation.Similarly to our treatment of the periodic 1-D cell system in Section 2.3.1.1,we now consider the geometry of the eigenperturbations in more detail and startwith focusing on the perturbation φ from the symmetric steady-state Ues . Bytranslational invariance, the perturbation at cell q(i, j) = il1 + jl2 or cell (i, j) issimply given by φi j,k := φ(il1 + jl2) = e−ik·(il1+ jl2)φ(0) with the generators of theBravais lattice, l1 and l2. Thus, if we were to write the k-eigenperturbation as aninfinite vector associated with the infinite lattice, it satisfiesφk :=φ00,kφ01,kφ10,kφ11,kφ02,k...=1e−ik·l2e−ik·l1e−ik·(l1+l2)e−i2k·l2....115Taking the real and imaginary parts we see that φk ∈ span{vck,vsk) wherevck =1cos(k · l2)cos(k · l1)cos(k · (l1+ l2))cos(2k · l2)...=(cos(k·q))q∈Λ and vsk =0sin(k · l2)sin(k · l1)sin(k · (l1+ l2)sin(2k · l2)...=(sin(k ·q))q∈Λ.Assuming that the intra-compartmental inhibitory kinetics depend linearly on theinhibitory species η , i.e., g(µ,η) = g1(µ)− g2η , the matrix equation (3.70) canbe rewritten into(λ +g2+2πνDvdv2dv1(1+2πνRk,Ωv(0)+νDvdv1)−1)ζ0 = g′1(µes )ξ0,(λ −∂µ fs)ξ0−∂η fsζ0 = 2πDucu1.The first equation can be used to eliminate ζ0, so that we obtainξ0(λ −∂µ fs− ∂η fsg′(µes )λ +g2+ϑv(k,λ ))+ϑu(k,λ ))where we have defined ϑu(k,λ ) := 2πνDudu2du1(1+ 2πνRk,Ωu(0)+ νDudu1)−1 and ϑvanalogously. This, in turn gives the functionσk(λ ) := λ −∂µ fs− ∂η fsg′(µes )λ +g2+ϑv(kλ )+ϑu(k,λ ) (3.71)which has to satisfy σk(λ ) = 0 for nontrivial perturbations to exist. Thus, thereexists a bifurcation point for a k-eigenperturbation with k ∈ Ω⋆ if and only ifσk(0) = 0.The attentive reader may have noticed that we have not written k∈R2 anymorebut k ∈ Ω⋆. The reason behind this is that it turns out that the perturbationwave-vectors k are distributed for the Bravais lattice cell system continuously, notdiscretely, in a finite domain Ω⋆, which is the Wigner-Seitz cell of the reciprocallattice to the Bravais lattice Λ. More on this stands in the next section.1163.5.2 Illustrations of the theoryIn this section we are presenting bifurcation theoretic results to three differentintra-compartmental reaction kinetics: the Gierer-Meinhardt, FitzHugh-Nagumoand Rauch-Millonas reaction kinetics. Crucial for finding bifurcation points fromthe nonlinear algebraic equations involving the regular part of the quasi-periodicGreen function, Rk,Ξ, is to find a way to efficiently compute the evaluations of thisregular part. We present the method of splitting the Green function into a partsummed with respect to the original Bravais lattice Λ and a second part summedwith respect to the reciprocal lattice Λ⋆ to Λ. Once this numerical evaluationis discussed, we mention the (first) Brillouin zone Ω⋆, which is the continuousdomain for all possible eigenperturbation wave-vectors k and the Wigner-Seitzcell of the reciprocal lattice Λ⋆. After that, we illustrate our theory using thementioned reaction kinetics by showing which eigenperturbations with whichwave-vectors can destabilize the symmetric steady-state — depending on thebifurcation parameter, the membrane reaction rate ratio ρ = dv/du.3.5.2.1 Summing the quasi-periodic reduced-wave Green functionThe quasi-periodic reduced-wave Green function is the solution to∆Gk,Ξ−Ξ2Gk,Ξ = −δ (x), x ∈Ω,Gk,Ξ(x+q) = e−ik·qGk,Ξ(x), q ∈ Λ,(3.72)whereΛ= {ml1+nl2 |m,n∈Z} is the Bravais lattice generated by l1 ∈R2 and l2 ∈R2 and Ω is the Wigner-Seitz cell of Λ. Just like the reduced-wave Green function,the quasi-periodic reduced-wave Green function has the singularity behaviorGk,Ξ(x)∼− 12π ln |x|+Rk,Ξ(x)+o(1) as x→ 0.Notice the Floquet-Bloch quasi-periodic boundary condition giving the quasi-periodic reduced-wave Green function its name, along with the decay terminvolving Ξ. WritingGk,Ξ(x) = ∑q∈Λeik·qGfreek,Ξ(x+q), (3.73)117we see that the q-shift boundary condition is satisfied and that the PDE is solved ifthe, then so-called free-space reduced-wave, Green function Gfreek,Ξ(x) satisfies∆Gfreek,Ξ −Ξ2Gfreek,Ξ =−δ (x), x ∈ R2.Fourier transforming the PDE with respect to the vector κ and using a hat to denotethe transformed function,(∥κ∥2+Ξ2) Gˆfreek,Ξ(κ ) = 1 ⇔ Gˆfreek,Ξ(κ ) =1∥κ∥2+Ξ2 , κ ∈ R2. (3.74)Using the Poisson summation formula to combine (3.73) with (3.74), the analyticalexpression for the quasi-periodic reduced-wave Green function isGk,Ξ(x) =1|Ω| ∑d∈Λ⋆eix·(2πd−k)∥2πd−k∥2+Ξ2with the reciprocal lattice Λ⋆ := {md1 + nd2 | m,n ∈ Z} generated by d1 and d2that satisfy di · l j = δi j with the discrete Dirac delta function δi j. We normalize theWigner-Seitz cell Ω so that its area is |Ω|= 1.An extended version of the Ewald summation technique of [14] was used to obtaina rapidly converging series for Rk,Ξ. This series is derived in [65] and readsRk,Ξ = 12π(ln(2)− ln(Ξ)− γe)− 14πE1(Ξ2η2)+∑l∈Λ,l ̸=0 eik·lFsing(∥l∥)+∑d∈Λ⋆ exp(−∥2πd−k∥2+Ξ2η2)1∥2πd−k∥2+Ξ2(3.75a)where γe is Euler’s constant, E1(z) :=∫ ∞1 x−1e−zx dx is the exponential integral,defined for Re(z)> 0 andFsing(r) :=12πK0 (rΞ)− J(r;Ξ)4π with J(r;Ξ) :=∫ ∞1x−1e−xΞ2/η2−rη2/(4x) dx.(3.75b)The parameter η > 0 above is the Ewald cutoff parameter, chosen so that theinfinite sums in (3.75) converge rapidly and so that approximately similarly manyterms are needed for each sum for a chosen accuracy.Now we can efficiently evaluate the regular part Rb f k,Ξ using (3.75). In the118following subsections we are using it for three different intracellular reactionkinetics: Gierer-Meinhardt, FitzHugh-Nagumo, and Rauch-Millonas reactionkinetics.3.5.2.2 Gierer-Meinhardt kineticsFirst we take for illustration the nonlinear reaction kinetics of a prototypical Gierer-Meinhardt model (GM) inside the compartments, as it has already been done inSection 3.4.1 in the finite domain case. When there is no bulk diffusion, thecompartments are uncoupled from the bulk and we impose the intracellular reactionkineticsµ˙(t) = f (µ,η) :=µ2η, η˙(t) = g(µ,η) := µ2 . (3.76)The uncoupled equilibrium for (3.76) given by µe = 0, and where ηe is an arbitraryconstant, is non-hyberbolic in all directions.With these kinetics we proceed as follows. As a space-filling geometry whichallows to find regions with positive perturbation growth rates Re(λ )> 0 at as manyeigenperturbation wave-vectors k as possible in the continuous Brillouin zone Ω⋆,we take the Fibonacci spiral [142] and overlay it with the Brillouin zone. At each kof the spiral we compute the corresponding perturbation growth rate λ via σk(λ ) =0 of (3.71). Here we find that the first k ∈ Ω⋆ that are corresponding to growthrates λ with Re(λ ) > 0, so perturbation growth rates that are destabilizing thesymmetric steady-state, are the ones at the corners of the Brillouin zone Ω⋆ asthe parameters are varied toward bifurcation bubbles found with σk(0) = 0 (seeFigure 3.21). Geometrically, the reason behind this is that there exists a bifurcationcircle in k-space centered at the Brillouin zone origin, not a point, which is gettingsmaller the closer the parameters of the system (ρ , du, D=Du =Dv, σ = σu = σv,ε) approach a bifurcation point from one side.From Figure 3.22 it can be seen that the growth rates computed for k on eachline from the origin to any of the six vertices of the Brillouin zone hexagon Ω⋆are the same. This suggests that Ω⋆ with its corresponding overlaid growth rateshas dihedral group D6 symmetry. Additionally, in the right panel the growth ratevalues are shown when walking along the full hexagonal boundary.119Figure 3.21: Left: A Fibonacci spiral [142] is used to compute the growthrates λ ∈ R to as evenly distributed eigenperturbation wave-vectorsk ∈ Ω⋆ as possible. Additionally, we go along the boundary of theBrillouin zone Ω⋆ (the hexagon). Red means that the correspondinggrowth rate is positive (exponentially growing perturbation), blue thatit is negative (exponentially decaying perturbation) and white or closeto white means that the growth rate is close to zero. All dots which aresurrounded by orange circles correspond to purely imaginary growthrates, and all other dots are real. Right-top: Zero-transition of thegrowth rate λ (k(s)) ∈ R when following a line that starts from theorigin toward one of the vertices, linearly parametrized by s ∈ [0,1].Right-bottom: One can see that the growth rate vanishes from the realline and appears in the purely complex plane at about 75% of the linelength from any vertex toward to origin. All such lines give equalgrowth rate transitions due to the lattice symmetries. Parameters: Du =Dv = 9,σu = σv = 0.75,du = 0.03,ε = 0.03, and ρ = 20.3.5.2.3 FitzHugh-Nagumo kineticsWe now consider cells centered on the Bravais lattice which have FitzHugh-Nagumo (FN) intracellular reaction kinetics [46] as in Section 3.4.3. Thesekinetics have a cubic nonlinearity for the activator µ and are inspired by neuronalaction potential dynamics that can exhibit stable oscillations. For convenience, the120Figure 3.22: Top: The discrepancy between computed eigenperturbationgrowth rates λ ∈ R along all six lines to the Brillouin zone verticescorresponding to the top-right plot of Figure 3.21. One can see thatthe discrepancy is very low and we interpret it as simply a numericalerror, so that the Brillouin zone hexagonΩ⋆ with its growth rate valuesat each k ∈Ω⋆ has dihedral group D6 symmetry. Bottom: The growthrates λ (k(s)) ∈ R when walking over the full hexagonal boundary ofΩ⋆, parametrized by s∈ [0,1]. The dots are coloured in the same colouras the vertices of the full Brillouin zone plot in Figure 3.21.uncoupled intracellular kinetics are again given here:µ˙(t) = µ−q(µ−2)3+4−η ≡ f (µ,η) , η˙(t) = δ zµ−δη ≡ g(µ,η) ,(3.77)121with q > 0, δ > 0 and z > 0. We choose a parameter set out of the wide parameterrange for which the uncoupled FN kinetics exhibit a unique stable steady-state, i.e.,oscillatory behavior is unstable.The symmetric equilibrium for which all cells have the same intracellularequilibrium (µes ,ηes ) can easily be found as a root via (3.68) again. Especiallydue to g being linear in the inhibitor component η with the form g(µ,η) =g1(µ)−g2η , equation (3.68b) can be rearranged to yield ηes which in turn can beplugged into equation (3.68b). With this single equation the root finding procedureis simplified.As in the previous section we now use (3.71) to find the growth rate to anyeigenperturbation wave-vector k sampled from a Fibonacci spiral (cf. [142] inand around the Brillouin zone Ω⋆ of the reciprocal lattice Λ⋆. In Figure 3.23 it canbe seen that there exists a continuum of eigenperturbations which destabilize thesymmetric steady-state at each corner of the Brillouin zone. Additionally, stableoscillations can be obtained with small wave-vectors just near the origin. Red,blue and white dots represent growth rates λ ∈ R which are respectively positive,negative and zero. Orange encircled dots represent growth rates λ ∈ C\R withRe(λ ) either being positive, negative, or zero, and the two plots to the right showboth Re(λ ) and Im(λ ) to growth rate λ to eigenperturbations with wave-vector kgoing on a straight line from the origin to any of the Brillouin zone vertices.3.5.2.4 Rauch-Millonas kineticsLastly, as has been done for the finite domain in Section 3.4.2, we show that thereis the emergence of patterns from the symmetric steady-state of the Bravais latticecell system with our usual diffusive signal transduction kinetics between cellsintroduced by Rauch and Millonas [116]. The two-species intracellular kineticsread for single uncoupled cellsµ˙ = cu−quµ+ au1µbu1+µ− au2µηbu2+µ ≡ f (µ,η)η˙ = cv+wvµ−qvη ≡ g(µ,η) .(3.78)where all parameters therein are positive real numbers. The authors claim that, withadded diffusive coupling, these kinetics quite universally describe inter-cellular122Figure 3.23: Left: A Fibonacci spiral [142] is used to compute the growthrates λ ∈ R to as evenly distributed eigenperturbation wave-vectorsk ∈ Ω⋆ as possible. Red, blue and white dots represent growth ratesthat are respectively positive (exponentially growing perturbation),negative (exponentially decaying perturbation) and close to zero. Dotsencircled by orange represent purely complex growth rates. Right:Real and imaginary parts of growth rates λ (k(s)) ∈R when followinga line that starts from the origin going toward any of the vertices,linearly parametrized using s ∈ [0,1]. All such lines give equal growthrate transitions due to the lattice symmetries. Here it can be seenthat eigenperturbations with wave-vectors k close to the Brillouinzone corners can destabilize the symmetric equilibrium and can leadto steady asymmetric patterns. However, there also exist oscillatoryeigenperturbations with growth rates λ ∈ C\R and Re(λ ) > 0 withwave-vectors close to the origin that can destabilize the symmetricsteady-state and can lead to stable oscillations of cell concentrations inthe Bravais lattice cell system. Parameters: Du =Dv = 5, σu =σv = 107 ,du = 0.03, ε = 0.03, q = 1, δ = 0.1,z = 40, and ρ = 150.signal transduction mechanisms. We are again choosing a parameter set for whichthe reaction kinetics, when uncoupled from the bulk, have a unique linearly stablesteady-state. Due to the linear inhibitor dependence of the inhibitory kineticsg(µ,η) = g1(µ)− g2η , the symmetric steady-state (µes ,ηes ) can easily be foundby plugging (3.68b) into (3.68b) and obtaining the root µes . We then perturb from123this steady-state with eigenperturbations with wave-vector k, whose growth ratesare determined by (3.71). First we locate parameter ranges for Du, Dv/Du, σu, σv,du, ρ = dv/du and wv for which some wave-vectors — as it turns out, especially theones near the corners of the Brilouin zone — correspond to positive perturbationgrowth rates. Then, with such a close to optimal parameter set, we sample k froma Fibonacci spiral in and around the Brillouin zone and compute the correspondinggrowth rates λ ∈ R and λ ∈ R\C. As for the Gierer-Meinhardt and FitzHugh-Nagumo kinetics of the previous subsections, the eigenperturbation growth ratesare strictly complex in the middle of the Brillouin zone but still have negative realpart, meaning that induced perturbative oscillations are quickly decaying back tothe symmetric steady-state, when the growth rates corresponding to perturbationswith wave-vector k close to the Brillouin zone boundary have already becomepositive. Thus, such perturbations close to the boundary can destabilize thesymmetric steady-state and push the cell system into a steady asymmetric statewith similar spatial shape as the corresponding eigenperturbation. Note that k = 0simply gives the unique symmetric eigenperturbation. Please see Figure 3.24showing plots of these results for one parameter set.3.5.3 The finite lattice systemFor comparison to our infinite Bravais lattice results and to make numericalsimulations tractable, we now consider a finite disk excerpt around thecompartment at the origin still in R2 (see Figure 3.25). The compartmentsare numbered starting with compartment 1 at the origin and proceeding in acounterclockwise manner around each closest compartment-ring, with the firstcompartment per ring on or just above the horizontal axis. The compartment-ringsare populated with compartments that have the same distance from the origin.As for the 2-D single-ring system of Section 3.4, all equilibria (symmetric andasymmetric) solvef (µ j,η j)− eTj Θuµ = 0, g(µ j,η j)− eTj Θvη = 0 for j ∈ {1, ...,m} (3.79a)where m is the total number of compartments and the coupling matrices coupling124Figure 3.24: Left: A Fibonacci spiral [142] is used to compute the growthrates λ ∈ R to as evenly distributed eigenperturbation wave-vectorsk ∈ Ω⋆ as possible. Red, blue, and close to white (only very few)dots represent growth rates that are respectively positive (exponentiallygrowing perturbation), negative (exponentially decaying perturbation)and close to zero. Orange-encircled dots represent purely complexgrowth rates with either positive, negative or zero real part. Right: Realand imaginary parts of growth rates λ (k(s)) ∈ R when following aline that starts from the origin toward any of the vertices, parametrizedby s ∈ [0,1]. All such lines give equal growth rate transitions due tothe lattice symmetries. Here it can be seen that eigenperturbationswith wave-vectors k close to the Brillouin zone corners can destabilizethe symmetric equilibrium and can lead to steady asymmetric patterns.There also exist oscillatory eigenperturbations with growth rates λ ∈C\R but with Re(λ ) < 0 for wave-vectors close to the origin thatcannot destabilize the symmetric steady-state toward stable oscillatorybehavior of cell concentrations for the chosen parameter set. Wehave not found parameter ranges for which there are oscillatoryperturbations with Re(λ ) > 0 yet but they may exist. Parameters:Du =Dv = 5, σu = σv = 0.5, du = 0.12, ε = 0.03, au1 = 600, bu1 = 100,au2 = 6, bu2 = 1/10, cu = 1= cv, qu = 1/100, qv = 7, wv = 7, and ρ = 15.all compartments with one another areΘu := 2πνDudu2du1((1+νDudu1)I+2πνGωu)−1, Θv := 2πνDvdv2dv1((1+νDvdv1)I+2πνGωv)−1(3.79b)125Figure 3.25: A finite version of the Bravais lattice compartment system.There are three fully populated compartment-rings (compartmentswith the same distance to the origin) about the compartment centeredat the origin.with ωu =√σu/Du and ωv =√σv/Dv. The reduced-wave Green matrix Gω hasthe following entries of the free-space reduced-wave Green function Gω(x;y) (cf.[73]) that is the solution to∆Gω −ω2Gω = −δ (x−y), x,y ∈ R2Gω → 0 as ∥x−y∥→ ∞.Along the diagonal of Gω there stand the regular parts of Gω(xi;xi) to eachcompartment i ∈ {1, ...,m} with center xi, namely Rω = 12π (ln(2)− γe− ln(ω)),and, at position (i, j), i ̸= j, in the matrix, there stands Gω(xi;x j) = 12π K0(ω∥xi−x j∥) with the modified Bessel function of the second kind of order 0, K0. Noteagain that, due to the symmetry of the Green function Gω , the Green matrix Gω is126symmetric.In this finite lattice system there cannot exist a symmetric equilibrium, in whichall compartments have the same intracellular equilibrium. This is due to the factthat the compartment at the origin is encircled by all compartment-rings, whichare directly interacting with the origin compartment, whereas, for example oneof its direct neighbouring compartments is encircled only by a smaller number of(translated) compartment-rings.Theorem 3.5.1. The equilibrium for which all compartments on the same ringabout the origin have the same concentration exists. This is due to the Greenmatrix in (3.79) consisting of blocks of circulant matrices. In what follows we arecalling it the multi-ring equilibrium or multi-ring steady-state. Furthermore, withthe above cell numbering, this equilibrium corresponds to the eigenvector of theGreen matrix that has an increasing or decreasing step-function like shape.Proof. First, as an illustration, consider only two compartment-rings about theorigin. Denoting by Gk =Gω(xi;x j) the value of the Green function for ∥xi−x j∥=rk where rk is the kth closest compartment-distance from the compartment i, theGreen matrix in this setting has the structureGω =R G1 G1 G1 G1 G1 G1 G2 G2 G2 G2 G2 G2G1 R G1 G2 G3 G2 G1 G1 G1 G3 G4 G4 G3G1 R G1 G2 G3 G2 G3 G1 G1 G3 G4 G4G1 R G1 G2 G3 G4 G3 G1 G1 G3 G4G1 R G1 G2 G4 G4 G3 G1 G1 G3G1 R G1 G3 G4 G4 G3 G1 G1G1 R G1 G3 G4 G4 G3 G1G2 R G2 G5 G6 G5 G2G2 R G2 G5 G6 G5G2 R G2 G5 G6G2 R G2 G5G2 R G2G2 RSince Gω is symmetric and has the same regular part repeatedly on its diagonal,127the Green matrix is symmetric as well and it is enough to consider the entriesabove the diagonal. No matter from which compartment center xi one observes,the distances to the other same-ring-as-ith-compartments are arising in the sameordering as the x j we are observing from. Hence, the corresponding matrixblock in Gω is circulant. The same is coming up when observing from x j allcompartment distances to compartments on a specific bigger compartment-ringabout the origin. Hence, the corresponding matrix block is again circulant withdimensions #cells on ring 1×#cells on ring 2 etc.As all circulant matrices have the eigenvectors v j = (1,Z j,Z2j , ...,Zmj ) with Z j :=exp(2πi( j−1)m)for j ∈ {1, ...,m}, the full set of eigenvectors of Gω is known (cf.[134]; also see Appendix A.4). Note that the first eigenvectors with this choiceof numbering of a circulant matrix is the one-vector v1 = 1. Concatenating theseone-vectors weighted by a, generally different, scalar for each ring gives the firsteigenvector of Gω , which has the multi-ring structure.Now note that the nonlinear algebraic system for the steady-states (3.79) includesthe two coupling matrices Θu and Θv, which have the same set of eigenvectorsas the Green matrix. However, f and g are nonlinear functions and (3.79) wouldonly give a single nonlinear algebraic equation for a then existent symmetric basestate equilibrium if f and g are bilinear. Thus, even though the multi-ring steady-state will still satisfy (3.79) the nonlinearities in f and g will generally modify theseparation of the values of µe and ηe between compartment-rings.This insight leads us to the following conjecture. Future work will prove ordisprove its statement. The cell ring number k should not be confused with theeigenperturbation wave-vector k.Conjecture 3.5.1. The multi-ring steady-state of Theorem 3.5.1 converges to thesymmetric steady-state for cell ring numbers k → ∞. This means that, for thefinite lattice cell system, the multi-ring steady-state is the base state from whichasymmetric steady-states are bifurcating off. This also shows that for k → ∞ thefinite lattice cell system is converging to the infinite Bravais lattice cell system. Inwhich sense this convergence occurs has to be determined.1283.6 DiscussionWe have analyzed symmetry-breaking behavior associated with the PDE-ODEbulk-cell model (3.66) where identical two-component intra-compartmentalreactions occur only within a disjoint collection of small circular compartments,or “cells”, of a common radius within a bounded 2-D domain. In the bulk, orextra-cellular, medium two bulk species with comparable diffusivities and bulkdegradation rates diffuse and globally couple the spatially segregated intracellularreactions. The bulk species are coupled to the intracellular species through anexchange across the compartment boundaries, as modeled by a Robin boundarycondition that depends on certain membrane reaction rates. In the limit of a smallcell radius, we have used a singular perturbation methodology to derive a nonlinearalgebraic system (3.13) characterizing all the steady-states for the bulk-cell model(3.66). Moreover, the linear stability properties of the steady-state solutions of thebulk-cell model were shown to be determined by the nonlinear matrix eigenvalueproblem (3.43) of size 2m×2m, where m is the number of compartments. A root-finding condition on the determinant of this matrix yields the discrete eigenvaluesof the linearization (3.32) around an arbitrary steady-state solution, as defined bythe set (3.44).We have shown that the steady-state and linear stability theory simplifiesconsiderably for a symmetric cell arrangement, as characterized by Definition3.2.1, and when one of the intracellular species has a linear dependence of theform (3.14). In this more restricted scenario, we have shown that a symmetricsteady-state solution, in which the steady-states of the intracellular species are thesame for each cell, will exist if the scalar nonlinear algebraic equation (3.20) hasa solution. We emphasize that since our bulk-cell model does not admit spatiallyhomogeneous steady-state solutions that can be analyzed by a simple Turing-typelinear stability approach [138], this symmetric steady-state solution of the bulk-cell model (3.66) represents the base state in our construction. Instabilities andbifurcations associated with this base state are challenging to analyze owing to thefact that the base state is not spatially uniform. Asymmetric steady-state solutions,as determined from (3.16), were shown to bifurcate from the symmetric steady-state solution branch whenever the algebraic criterion (3.29) is satisfied at some129point on the symmetric branch. For a symmetric cell arrangement, the linearstability properties of the symmetric and asymmetric steady-state solution branchesare characterized by (3.52) and the roots of the nonlinear matrix eigenvalueproblem (3.49), respectively.We have implemented our steady-state and linear stability theory for a specificsymmetric cell arrangement in which two cells are equally spaced on a ringconcentric within a unit disk, and where we have specified either Gierer-Meinhardt,Rauch-Millonas, or FitzHugh-Nagumo intracellular reactions, which all have thesimplified form in (3.14). By using parameter continuation numerical software [28]to implement the asymptotic theory, we have shown that the symmetric steady-statesolution branch can undergo symmetry-breaking pitchfork bifurcations, leadingto linearly stable asymmetric patterns, even when the two bulk diffusing specieshave identical diffusivities and degradation rates. Overall, we have shown that it isthe magnitude of the ratio of the reaction rates for the two bulk species to the cellmembranes that determines whether stable asymmetric patterns can occur. Thismembrane reaction rate ratio threshold condition for the emergence of symmetry-breaking bifurcations is in marked contrast to the well-known large diffusivityratio threshold condition for pattern formation from a spatially uniform state thatis typically derived by a Turing stability analysis for two-component activator-inhibitor RD systems. For FitzHugh-Nagumo and Rauch-Millonas kinetics wehave also shown that stable asymmetric patterns can also emerge from a symmetricsteady-state pattern at a fixed, but large, membrane reaction rate ratio when acontrol parameter in the intracellular kinetics is varied. Our theoretical predictionsof symmetry-breaking behavior leading to linearly stable asymmetric patterns for asymmetric two-cell arrangement were confirmed through full time-dependent PDEcomputations of (3.66).Further, we have regarded a system of diffusively coupled cells centered on aBravais lattice, which is infinite and periodic in multiple directions. Key was inthe steady-state solution to use the periodic reduced-wave Green function (3.67)and in the eigenperturbation construction to use the quasi-periodic reduced-waveGreen function with Floquet-Bloch type boundary conditions (3.69). The searchfor quasi-periodic eigenperturbations led to a relatively simple expression for theperturbation growth rates given in (3.71), which can directly be used to find when130the real part of the growth rate to an eigenperturbation becomes positive and so candestabilize the respective steady-state it perturbs from. A Fibonacci spiral was usedto sample in a space-filling manner from the central Voronoi cell of the reciprocallattice: the continuous Brillouin zone, in which the wave-vectors of possiblequasi-periodic eigenperturbations are “living”. This facilitates finding parametersets with which a growth rate zero-transition (and so a bifurcation) occurs. ForGierer-Meinhardt, FitzHugh-Nagumo, and Rauch-Millonas reaction kinetics thishas been successfully illustrated, and it seems that at least for the Gierer-Meinhardtand Rauch-Millonas kinetics the symmetric steady-state loses its stability due tonon-oscillatory and asymmetric eigenperturbations, kicking the system towardsasymmetric solution branches with similar geometry as the respective dominatingeigenperturbation. A theoretical excursion to a system in R2 with a finite excerptof the Bravais lattice with a multi-ring structure about a cell in the center was done.In such a system the symmetric steady-state does not exist but the steady-statein which cells on the same ring about the center cell have the same intracellularequilibrium concentrations may be the base state that converges to the symmetricsteady center for cell numbers converging to infinity.We now briefly relate our theoretical results to some qualitative behaviorthat has been suggested in chemical and biological applications. Firstly,compartmental-reaction diffusion models of the form (3.3) could potentiallybe useful for theoretically modeling the collective behavior that occurs fora microemulsion consisting of Belousov-Zhabotinsky (BZ) chemical reactantsthat are confined within small aqueous droplets that is dispersed in oil [137](see also [37], [19]). In this experimental set-up, polar BZ reactants anda catalyst are confined within small immobile droplets, while two non-polarintermediate species generated during the reaction can be transported acrossthe droplet boundaries. These intermediate species diffuse across the domain,with comparable diffusivities, and provide the mechanism for inter-drop coupling[137]. The recent experimental study of [19] has suggested that it is the relativemagnitude of the membrane reaction rates of these intermediates on the dropletboundaries that plays a key role for determining pattern-forming properties for BZmicroemulsions. Secondly, with regards to the transport of biological morphogens,it has been suggested in [99] that a differential reaction rate ratio on the cell131boundaries for two morphogen species with comparable diffusivities can yieldthe large effective diffusivity ratio that is needed for pattern formation andsymmetry-breaking in tissues. This membrane attachment mechanism, whichreduces the effective diffusivity of one of the morphogens and is referred to in[99] as a binding-mediated hindrance diffusion process, may be relevant in manybiological applications. Moreover, detailed intracellular mechanisms in biologicalcells, affecting signaling pathways and gene expression rates, play a pivotal role inlarge-scale pattern-forming properties of biological tissues [99]. Such mechanismswould need to be included into the nonlinear intracellular kinetics and the boundaryconditions if they modify the relevant dynamics, giving novel kinetic modelsdifferent from the FN, GM, and RM kinetics combined with linear Robin reactiveboundary conditions. By way of qualitative comparison, our theoretical analysisof the 2-D bulk-cell model (3.3) for a very simple 2-cell pattern has revealed that alarge membrane reaction rate ratio, together possibly with changes in a parameterin the intracellular kinetics, can trigger the emergence of stable asymmetric steady-state patterns that bifurcate from a symmetry steady-state. However, owing tothe complexity of the analysis needed for (3.3), where certain Green’s matriceswere found to be central to the analysis, it appears rather intractable analyticallyto isolate via a simple scaling analysis an effective diffusivity for the bulk speciesthat incorporates the membrane reaction rates.Although we have only applied our theoretical framework to a simple two-cellarrangement, it is rather straightforward to numerically implement the steady-stateand linear stability theory for a symmetric cell arrangement with a much largernumber of cells. For this scenario, the symmetric steady-state solutions are againdetermined by the scalar nonlinear algebraic equation (3.20) and the linear stabilityproperties of this steady-state are readily studied by computing the union of all theroots of the scalar problems σ j(λ ) = 0, for j ∈ {1, . . . ,m}, in (3.51) that comprisethe set (3.52) that approximates the discrete eigenvalues of the linearization (3.32)of (3.3) around the steady-state. However, for an arbitrary (not only on a Bravaislattice) spatial arrangement of a large number of cells, one key impediment forimplementing the linear stability theory for steady-state solutions is with regardsto numerically computing the eigenvalues λ from a root-finding condition on thedeterminant of the full 2m×2m GCEP matrixM (λ ) in (3.43). This matrix is non-132Hermitian, is not sparse, and has an intricate dependence on λ through the Greenmatrices. In contrast to the availability of efficient numerical solution strategies fornonlinear matrix eigenvalue problems with special structure, as was discussed in[53], [10] and [11], it appears to a significant open challenge to develop efficientnumerical methods to determine all such eigenvalues λ for which M (λ ) is asingular matrix when m is large. Recall that if there are any such eigenvaluesin Re(λ )> 0, the steady-state for (3.3) is unstable.A few other open problems related to our analysis are as follows. Firstly,it would be interesting to develop an extension of our asymptotic approach totreat closely-spaced cell configurations that are more relevant to modeling pattern-forming properties in biological tissues. Our numerical results shown in §3.4.4have suggested that only a smaller membrane reaction rate ratio is needed toinitiate symmetry-breaking behavior for closely-spaced cells than for arrangementswith more spatially segregated cells. To theoretically analyze pattern-formingproperties of the bulk-cell model with closely-spaced cells, an extension of theapproach developed in [63] to analyze the mean first passage time for a clusterof small traps may be fruitful. Secondly, it would be interesting to formulateand analyze a related bulk-cell model where the chemical reactions occur on theboundaries of a collection of small compartments, rather than in the interior ofthe compartments. In this scenario, chemical species produced on the membranecan then detach and diffuse in the bulk medium. Such an extension is relevant foranalyzing collective behavior that occurs for dynamically reactive solid pellets thatare chemically coated and are coupled through a bulk diffusion field (cf. [132],[131], [136], [135]). Finally, it would be worthwhile to extend our 2-D analysisto a 3-D setting. For a 3-D bounded domain that contains a collection of smallspherical compartments, the analysis would be rather different than in 2-D sincethe free space Green function has a rapid decay at infinity instead of a logarithmicgrowth. This would suggest that the inter-cell coupling effect would be, in general,much weaker in 3-D than in 2-D. This is analyzed in Chapter 5 in regard of the fulltime-dependent dynamics.133Chapter 4SynchronizedMemory-Dependent IntracellularOscillations for a Bulk-CellODE-PDE Model in R24.1 IntroductionThe introduction of this dissertation mentions a switch in mindset betweenChapters 2 and 3 and the Chapters 4 and 5. After having analyzed the patternforming behavior of the steady-state of compartmental-reaction diffusion systemswith two extracellularly diffusing species in 1-D and 2-D, we now view eachcompartment or “cell” as an oscillator, in which chemical concentrations areoscillating in time. Further, the extracellular diffusion field couples all suchoscillators with a diffusive time lag in information transmission between them.This setting — coupled oscillators — reminds us of the Kuramoto model.Over the past four decades, the Kuramoto model [79]θ˙i = ωi+KNN∑j=1sin(θ j−θi) , (4.1)134and its variants have provided the primary theoretical modeling framework foranalyzing synchronization properties of a collection of N nonlinear oscillators.In (4.1), θi is the phase of the ith oscillator, ωi is its natural frequency, andK is the coupling strength. Key quantities for analyzing the synchronizationbehaviour for (4.1) are the Kuramoto order parameters r(t) and ψ(t) defined byreiψ = N−1∑Nj=1 eiθ j . In the mean field limit N → ∞, phase transitions to completephase synchronization r= 1 as K is increased have been characterized. For a surveyof results see [126], [2] and [23].One key aspect of Kuramato-type phase oscillator systems is that they canbe formally derived from nonlinear dynamical systems of coupled oscillators byusing asymptotic phase reduction methods that are valid in the limit of weakcoupling (cf. [101], [83] and [82]). An important current research theme is thestudy of oscillator synchronization on networks, as surveyed in [111], owing totheir many diverse applications such as power grids [52] and cortical brain activity[17], among others.With a rather different focus, in [81] Kuramoto emphasized the need forcharacterizing synchronization behavior for a collection of coupled oscillators or“cells”, each fixed in space, in which an extra substance secreted from each cellcan diffuse over the entire space and effectively mediate the inter-cell interaction.More precisely, Kuramoto wrote:Imagine a population of N identical cells each exhibiting oscillatoryactivity. Let the intrinsic dynamics of the individual cell activity bedescribed by the equation X˙ j = F(X j) ( j = 1,2, ...,N) which isassumed to admit a stable limit-cycle solution. The cells are fixed inspace, the j-th cell sitting at r j. There exists no direct cell-cellinteraction. Instead, the system involves extra substance whichmediates cellular interaction. This substance, denoted by A, issecreted from each cell, and can diffuse over the entire space. Therate of local secretion of A is a function of the local cell activity, thelatter being in turn influenced by the local concentration A of A. Withpossible variations from case to case, similar situations seem rathercommon in biological systems. — Yoshiki Kuramoto (1995)135His phenomenological model of this interaction has the formu′j = F(u j)+g(A(x j, t)) , εAt = D∆A−ηA+N∑k=1h(uk)δ (x−xk) , (4.2)where g and h are prescribed coupling functions. In R1 and in the quasi-steadylimit ε → 0, (4.2) reduces tou′j = F(u j)+g(N∑k=1σ(|x j− xk|)h(uk)). (4.3)where σ(r) = Ce−r√η/D is the 1-D quasi-static Green function for some C > 0.More recently, in a series of articles ([4], [122], and [22]) there has been a renewedfocus on analyzing synchronization properties for phase oscillator systems withcoupling as in (4.2), with the main emphasis being on the quasi-steady limit (4.3).One important drawback of (4.3) is that it includes cells as singular objects thatdo not have a spatial extent. Even though in the previous chapters we have used thelimit ε→ 0 as well to construct a global steady-state solution, we have done this inthe framework of strong localized perturbation theory in the asymptotic matchingprocedure of the local, or inner, solution with the global, or outer, solution. Thisgives an asymptotic solution that describes the system asymptotically over allspatial scales while still letting cells to have a finite and non-singular extent.A major drawback of Kuramoto’s model (4.2) and of the original Kuramotomodel (4.1) is that it is not clear how to rigorously derive the coupling functionsKN sin, g, and h for applications at hand.In order to tackle both of these issues, we now formulate the coupleddimensional bulk-cell ODE-PDE model of [48] and [64], as inspired by [98] and[97], with one bulk species in R2. We assume that there are N dynamically activecells, modeled by a collection of disks of a common radius R0, denoted by Ω jand centered at x j ∈ R2 for j ∈ {1, . . . ,N}. In the bulk, or extracellular, regionR2 \∪Nj=1Ω j, the concentration U (X,T ) of the autoinducer or bulk signal, with136diffusivity DB and bulk degradation kB, is assumed to satisfy∂TU = DB∆U − kBU , T > 0 , X ∈ R2 \∪Nj=1Ω j , (4.4a)DB ∂nXU = β1 jU −β2 j µ1 j , x ∈ ∂Ω j , j ∈ {1, . . . ,N} . (4.4b)The permeabilities β1 j > 0 and β2 j > 0 control the influx and efflux into and out ofthe jth cell, while ∂nX is the outer normal derivative pointing into the bulk region.(a) Cells coupled by bulk diffusion (b) Exchange between a cell and the bulkFigure 4.1: Left: Dynamically active circular signaling compartments (incyan) coupled through a bulk diffusion field (red dots) in R2. Right:Zoom of the chemical exchange between the bulk and any given cellwhen there are two intracellular species (green and red dots). Only thered species is secreted into the extracellular bulk region, while the bulkspecies can bind to the cell membrane. A Robin boundary conditionmodels this chemical exchange.Inside each cell we assume that there are m species µ j := (µ1 j, . . . ,µm j)T thatundergo interactions with the local reaction kinetics F j, and that one species,labeled by µ1 j, can permeate across the cell boundary. The cells are assumed to besufficiently small so that we can neglect any spatial gradients in the intracellularspecies within each cell. The intracellular dynamics in the jth cell are coupled tothe bulk diffusion field (4.4) via an integration of the diffusive flux across the cell137membrane asdµ jdT= kR µc F j (µ j/µc)+ e1∫∂Ω j(β1 jU −β2 j µ1 j) dSX , j ∈ {1, . . . ,N} .(4.4c)In (4.4c), e1 := (1,0, . . . ,0)T , kR > 0 is the dimensional intracellular reaction rateand µc > 0 is a typical magnitude of the vector µ j. In this formulation, µ1 j canpermeate the cell membrane with an efflux rate β2 j per unit length. The influx rateβ1 j controls the feedback into the jth cell from the global bulk diffusion field thatis produced by the entire collection of spatially segregated cells.For our asymptotic limit, we will assume that the common radius R0 of thesignaling compartments is small relative to the minimum inter-cell separationdistance L, and so we introduce a small parameter ε := R0/L≪ 1. From the non-dimensionalization of the bulk-cell model (4.4) done analogously to Appendix A.3we obtain that the dimensionless bulk field U(x, t) satisfies∂tU =D∆U−σU , t > 0 , x ∈ R2 \∪Nj=1 Bε(x j) ; U(x,0) = 0 , (4.5a)εD∂nU = d1 j U−d2 j u1 j , x ∈ ∂Bε(x j) , j ∈ {1, . . . ,N} , (4.5b)which is coupled to the dimensionless intracellular dynamics within the jth cell bydu jdt= F j (u j)+e1ε∫∂Bε (x j)(d1 j U−d2 j u1 j) dSx , j ∈ {1, . . . ,N} . (4.5c)Here u j = (u1 j, . . . ,um j)T is the dimensionless vector of intracellular species in thejth cell, labeled by Bε(x j) := {x | |x−x j| ≤ ε}. We assume that the centers ofthe cells are well-separated in the sense that dist(x j,xk) = O(1) for j ̸= k. In thisdimensionless formulation (4.5), the key dimensionless parameters areD :=DBkRL2, d1 j := εβ1 jkRL=O(1) , d2 j := εβ2 jLkR=O(1) , σ :=kBkR. (4.6)We refer to D and σ as the effective bulk diffusivity and effective bulk degradationrate, respectively. In (4.6), the permeability parameters β1 j and β2 j are chosen asO(ε−1) to ensure that there is an O(1) transport across each cell membrane. The138bulk-cell coupling for the dimensionless model is illustrated in Fig. 4.1.Our goal for the conceptual bulk-cell model (4.5) is to develop a hybridasymptotic-numerical approach to study how intracellular oscillations are bothinitiated and synchronized by the bulk diffusion field that is created by the entirecollection of cells. In contrast to the study in [64] that focused only on the steady-state problem and the spectral properties of the linearization of the steady-state, themain goal in this paper is to use a hybrid approach to study large-scale intracellulardynamics for (4.5).In §4.3 we will extend the strong localized perturbation theory, as surveyed in[144], to the time-dependent setting of (4.5) in order to derive, in the limit ε → 0of small cell radii, a new integro-differential ODE system that characterizes howintracellular dynamics are coupled to the bulk diffusion field that is produced bythe entire cell population. This reduced integro-differential system, as describedin Proposition 4.3.2, is asymptotically accurate to all orders in ν = −1/ logε .As discussed in Remark 4.3.1 the “outer problem” in our analysis shares somesimilarity with Kuramoto’s original model (4.2).Assuming that U(x,0) = 0, in §4.2 we analyze the short-time behavior of thebulk-cell system on the time-scale τ = ε2t within an O(ε) neighborhood of eachcell. This transient solution is essential for providing the initial behavior for theintegro-differential ODE system in Proposition 4.3.2 that is valid on the O(1)time-scale. In §4.4, we show that the steady-states of the bulk-cell model (4.5),as obtained from strong localized theory, coincide with the steady-state limitingvalues of the integro-differential ODE system. In addition, the linear stabilityproperties of the steady-states are shown to be characterized by the roots of anonlinear matrix eigenvalue problem, referred to as the GCEP.We emphasize that a direct numerical study of the integro-differential systemin Proposition 4.3.2 is challenging owing to the fact that N(N+1)/2 memory-dependent convolution integrals, each with integrable singularities, would have tobe computed numerically up to time t in order to advance the solution one time stepto time t+∆t. To overcome this difficulty, in §4.5 we develop a fast time-marchingnumerical scheme, which relies significantly on the sum-of-exponentials method of[66] and [12] (see also [13]) together with Duhamel’s principle, that allows us toboth rapidly and accurately compute solutions to the integro-differential system for139arbitrary reaction kinetics over long time intervals. The fast marching algorithm issummarized in §4.5.4.In §4.6 we illustrate our hybrid asymptotic-numerical theory for the choice ofSel’kov reaction dynamics, which have been used previously in simple models ofglycolysis oscillations (cf. [102], [119]). With Sel’kov kinetics the steady-statesolution branches of (4.5) have no transcritical or fold bifurcations and, as a result,steady-states are destabilized only via Hopf bifurcations. For various specificspatial configurations of cells, phase diagrams in the 1/σ versus D parameterplane, as computed from the winding-number of the determinant of the nonlinearmatrix eigenvalue problem, are provided to identify the number of destabilizingmodes of the linearization of the steady-state. The corresponding eigenvector of theGCEP is shown to encode both the relative magnitude of the signaling gradient neareach cell, as well as the phase shift in small amplitude oscillations between cellsthat occur near an unstable steady-state. Quorum-sensing and diffusion-sensingbehavior, as well as the key effects on intracellular oscillations of varying the influxand efflux permeabilities, will be examined for various spatial configurations ofcells. In particular, we show that a single oscillating cell can trigger significantintracellular oscillations in an entire collection of cells. Moroever, by computingthe Kuramoto order parameter, we show an apparent phase transition to completephase coherence as the bulk diffusivity is increased for a one-shell hexagonalarrangement of identical cells. These cells would be in a quiescent state withoutany bulk-cell coupling. Finally, in §4.8 we briefly summarize our findings and wediscuss several related bulk-cell problems that are now tractable to study with ourhybrid asymptotic-numerical approach.4.2 Transient Analysis Near the CellsWe first analyze the transient solution near the jth cell, which is valid for t =O(ε2)when U(x,0) = 0 and for arbitrary initial values u j(0) for j ∈ {1, . . . ,N}. Thisshort-time analysis will provide the initial conditions for the long-time dynamicsstudied in §4.3.We first assume that u1 j(0) ̸= 0. In the jth inner region we let y = ε−1(x−x j)with ρ = |y| and we introduce the short time-scale τ , defined by t = ε2τ . From140(4.5), we obtain that V (ρ,τ) =U(x j+εy,ε2τ) satisfies, to leading-order, the localproblem∂τV = D(∂ρρV +1ρ∂ρV), ρ > 1 , τ ≥ 0; V (ρ,0) = 0 ,D∂ρV = d1 jV −d2 ju1 j(0) , on ρ = 1 .(4.7)By taking the Laplace transform of (4.7), we determine Vˆ (ρ,s) =L [V (ρ,τ)] asVˆ (ρ,s) = u1 j(0)d2 js K0(ρ√s/D)d1 jK0(√s/D)+√sDK1(√s/D) , (4.8)where the branch cut is taken along Re(s) ≤ 0 with Im(s) = 0. Here K0(z) andK1(z) are modified Bessel functions of the second kind.Since the singularity of (4.8) with the largest real part occurs at s = 0, todetermine the large τ behavior of V for fixed ρ > 1 we must first determine thebehavior of (4.8) as s → 0. Upon using K0(z) ∼ − logz− γe + log2+ o(1) andK1(z)∼ z−1+o(1) as z→ 0, where γe is Euler’s constant, we calculate thatVˆ (ρ,s)∼ d2 ju1 j(0)d1 js+2d2 ju1 j(0)d1 j(logρ+Dd1 j)1s log(κ0 js), as s→ 0 ,(4.9a)where κ0 j is defined byκ0 j :=14De2(γe−D/d1 j) . (4.9b)To determine V (ρ,τ) for τ ≫ 1 and ρ = O(1), we must determine a functionwhose Laplace transform is analytic in Re(s)> 0 and where its singularity with thelargest real part occurs at s= 0 with the leading-order local behavior [s log(κ0 js)]−1as s→ 0. We emphasize that Vˆ (ρ,s) in (4.8) is analytic in Re(s)> 0, and that thepole at s = 1/κ0 j in the local behavior as s→ 0 in (4.9a) is spurious. As shown in[88], and summarized in Appendix A.6, for any κ0 j > 0 we have the exact relationL [N (τ/κ0 j)] =1s−1/κ0 j −1s log(sκ0 j), (4.10)141where N(x) for x > 0 is Ramanujan’s integral defined byN(x) :=∫ ∞0e−xξξ [π2+(logξ )2]dξ . (4.11)It is readily shown that s = 1/κ0 j is a removable singularity for (4.10) and thatL [N (τ/κ0 j)] is analytic in Re(s)> 0 and the singularity with the largest real partsatisfies L [N (τ/κ0 j)] ∼ −[s log(sκ0 j)−1]+O(1) as s → 0. By using the well-known integral asymptotics (cf. [146], [16], [88])N(x)∼ 1logx− γe(logx)2+O((logx)−3) , as x→+∞ , (4.12)where γe is Euler’s constant, we absorb the second term into the leading-orderapproximation to obtain the renormalized expressionN(x)∼ 1log(xeγe)−O([log(xeγe)]−3) , as x→+∞ . (4.13)By using this result and (4.9b), we readily obtain for τ ≫ 1 and ρ > 1 fixed thatV (ρ,τ)∼ u1 j(0)d2 jd1 j +B j2πD(logρ+Dd1 j), (4.14a)where B j satisfiesB j ∼−u1 j(0) 4πDd2 jd1 j log(τ/ [κ0 je−γe ]) +O([log(τ/(κ0 je−γe))]−2), for τ ≫ 1 .(4.14b)A similar analysis can be done for the case where u1 j(0) = 0. By evaluating(4.5c) at t = 0, and using U(x,0) = 0 and u1 j(0) = 0, we conclude for τ = O(1)thatu1 j ∼ ε2τu′1 j(0) , where u′1 j(0) = eT1 F j(u j(0)) . (4.15)As a result, we replace u1 j(0) in the boundary condition on ρ = 1 in (4.7) with142ε2τu′1 j(0) to derive, in place of (4.8), thatVˆ (ρ,s) = d2 jε2u′1 j(0) K0(ρ√s/D)s2[d1 jK0(√s/D)+√sDK1(√s/D)] . (4.16)To determine the large τ behavior for ρ fixed, we let s → 0 in (4.16) to concludethatV (ρ,τ)∼ ε2τu′1 j(0)d2 jd1 j++2ε2u′1 j(0)d2 jd1 j(logρ+Dd1 j)L −1[1s2 log(κ0 js)],(4.17)where κ0 j is given in (4.9b). By convolving L −1[(s log(κ0 js)−1]= −N (τ/κ0 j)andL −1[s−1]= 1, we obtain from (4.17) for τ ≫ 1 and ρ = O(1) thatV (ρ,τ)∼ ε2τu′1 j(0)d2 jd1 j+B j2πD(logρ+Dd1 j), (4.18a)where B j satisfiesB j ∼−ε2u′1 j(0)4πDd2 jd1 j∫ τ0N (ξ/κ0 j) dξ , for τ ≫ 1 . (4.18b)Here N(x) is Ramanujan’s integral (4.11) and u′1 j(0) = eT1 F j(u j(0)).Finally, to estimate B j for the range 1 ≪ τ ≪ ε−2, we observe that since∫ ∞0 N (ξ/κ0 j) dξ is not integrable, the dominant contribution to the integral in(4.18b) arises from the upper endpoint ξ = τ≫ 1 where we can use the asymptotics(4.13). Upon integrating by parts, and using E1(z) ∼ e−z/z for z → +∞, wecalculate that∫ τ01log(ξ/(κ0 je−γe))dξ =−κ0 jE1[− log(τ/(κ0 je−γe))] ,∼ τlog(τ/(κ0 je−γe)), as τ →+∞ .(4.19)143In this way, we obtain from (4.13), (4.18b) and (4.19) thatB j ∼−ε2∫ τ0u′1 j(0)γ jlog(ξ/(κ0 je−γe))dξ =−ε2 u′1 j(0)γ jτlog(τ/(κ0 je−γe)), 1≪ τ ≪ ε−2 .(4.20)The results (4.14) and (4.20), valid for 1≪ τ≪ ε−2 and ρ =O(1), are key forasymptotically matching the transient solution to the long-time dynamics in §4.3.4.3 Derivation of ODE System for Intracellular KineticsWe now derive a reduced ODE system with memory for the intracellular kineticsvalid for t ≫ O(ε2).For t = O(1), in the jth inner region we let y = ε−1(x− x j) with ρ = |y|, toobtain that V (y, t) =U(x j +εy, t) satisfies the leading-order quasi-steady problem∆yV = 0 , for ρ > 1; D∂ρV = d1 jV −d2 ju1 j , on ρ = 1 , (4.21)which has the radially symmetric solutionV (y, t) =B j(t)2πDlog |y|+ B j(t)2πd1 j+d2 jd1 ju1 j(t) , (4.22)for some B j(t) to be found. In terms of B j(t), we calculate the diffusive flux asε−1∫Ω j (d1 jU−d2 ju1 j) dSx = B j(t), so that from (4.5c)du jdt= F j(u j)+B j(t)e1 , j ∈ {1, . . . ,N} . (4.23)By asymptotically matching (4.22) to the outer solution U , we conclude that Usatisfies∂tU = D∆U−σU , x ∈ R2\{x1, . . . ,xN} ; U(x,0) = 0 ,U ∼ B j2πDlog |x−x j|+ B j2πDν +B j2πd1 j+d2 jd1 ju1 j , as x→ x j , j ∈ {1, . . . ,N} ,(4.24)with U(x, t)→ 0 as |x| → ∞ for each t > 0. Here ν := −1/ logε , B j = B j(t), and144u1 j = u1 j(t).Remark 4.3.1. The limiting “inner” problem (4.23) coupled to the “outer” bulkdiffusion field via (4.24) will self-consistently lead to an integro-differential ODEsystem for B j(t) and u j(t) for j ∈ {1, . . . ,N}. For the bulk-cell model, (4.23) and(4.24) replace the phenomenological diffusive coupling model of [81] as given in(4.2).To solve (4.24) we consider an auxiliary problem for vk(x, t) given by∂tvk = D∆vk−σvk−Bk(t)δ (x−xk) ; vk → 0 as |x| → ∞ ; vk(x,0) = 0 ,vk(x, t)∼ Bk(t)2πD log |x−xk|+Rk(t) , as x→ xk ,(4.25)where Rk(t) is to be determined in terms of Bk(t). As shown in our analysis of thetransient solution in §4.2, we must impose that Bk(t)→ 0 as t → 0+.Denoting the Laplace transforms vˆk(x,s) =L [vk(x, t)] and Bˆk(s) =L [Bk(t)],we obtain from (4.25) that∆vˆk− (σ + s)D vˆk =BˆkDδ (x−xk) ; vˆk → 0 as |x| → ∞ ,vˆk ∼ Bˆk2πD log |x−xk|+ Rˆk , as x→ xk .(4.26)Then, upon decomposing vˆk =−BˆkGˆk, we obtain that Gˆk(x,s) satisfies∆Gˆk− (σ + s)D Gˆk =−1Dδ (x−xk) ; Gˆk → 0 as |x| → ∞ , (4.27)which has the solutionGˆk(x,s) =12πDK0(√s+σD|x−xk|), (4.28)where K0(z) is the modified Bessel function of the second kind of order zero. In(4.28) we specify the principal branch of the square root, so that Gˆk is analytic inthe complex s plane except along the branch cut where Re(s)≤−σ and Im(s) = 0.145Upon using K0(z)∼− logz− γe+ log2 as z→ 0, where γe is Euler’s constant,we can calculate vˆk(x,s) as x → xk and enforce that this limiting behavior agreeswith that in (4.26). By specifying the principal branch of log(s + σ), thisdetermines thatRˆk(s) =14πDBˆk(s) log(s+σ)+Bˆk(s)2πD(γe− log(2√D)). (4.29)Then, by using the well-known inverse Laplace transform (cf. [1])L −1[K0(a√s)]=12te−a2/(4t) for a > 0 ,together with the shift property of the Laplace transform, we conclude from (4.27)thatGk(x, t) =L −1[Gˆk(x,s)]=14πDte−σte−|x−xk|2/(4Dt) . (4.30)This is simply the fundamental solution of the heat equation with bulk degradation.By using the convolution property we calculate vk(x, t) =−L −1[BˆkGˆk]asvk(x, t) =−∫ t0Bk(τ)Gk(x, t− τ)dτ =−∫ t0Bk(τ)e−σ(t−τ)4πD(t− τ) e−|x−xk|2/(4D(t−τ)) dτ .(4.31)Next, we invert (4.29) under the assumption that Bk(0) = 0 to derive thatRk(t) =14πDL −1[sBˆk(s)(log(s+σ)s)]+Bk(t)2πD(γe− log(2√D)). (4.32)By using the inverse transforms B′k(t) =L−1 [sBˆk(s)], and L [s−1 log(s+σ)] =E1(σt) + logσ for σ > 0 (cf. [1]), where E1(z) is the exponential integral, wecalculate from the convolution property and Bk(0) = 0 thatL −1[Bˆk log(s+σ)]=∫ t0B′k(τ)E1(σ(t− τ))dτ+Bk(t) logσ . (4.33)In this way, we obtain from (4.32) thatRk(t) =Bk(t)2πD[γe− log(2√Dσ)]+14πD∫ t0B′k(τ)E1(σ(t− τ))dτ . (4.34)146We represent the solution to (4.24) as the superposition U(x, t) =∑Nk=1 vk(x, t).By letting x → x j, and enforcing that the limiting behavior of U(x, t) agrees withthat required in (4.24), we conclude thatR j(t)+N∑k=1k ̸= jvk(x j, t) =B j(t)2πD(1ν+Dd1 j)+d2 jd1 ju1 j(t) , j ∈ {1, . . . ,N} . (4.35)Upon using (4.34) for R j(t), (4.35) determines B j(t) in terms of u1 j(t).Finally, upon combining (4.35) with the intracellular dynamics (4.23) weobtain an integro-differential system for the intracellular concentrations u j(t) thatis coupled to the time-history of the source strengths B j(t). We summarize ourresult in the following formal proposition.Proposition 4.3.2 (The asymptotically spatially reduced integro-ODE system).For ε→ 0, and with the initial condition U(x,0) = 0, the solution U(x, t) and u j(t),for j ∈ {1, . . . ,N}, to the bulk-cell model (4.5) is approximated for t ≫ O(ε2) bythe integro-differential systemdu jdt= F j(u j)+ e1B j(t) , (4.36a)∫ t0B′j(τ)E1(σ(t− τ))dτ = η jB j(t)+ γ ju1 j(t)+N∑k=1k ̸= j∫ t0Bk(τ)e−σ(t−τ)(t− τ) e−|x j−xk|2/(4D(t−τ)) dτ ,(4.36b)for j ∈ {1, . . . ,N}. Here η j and γ j are defined byη j := 2(1ν+Dd1 j+ log(2√Dσ)− γe)=− log(ε2κ0 jσ) , γ j := 4πDd2 jd1 j ,(4.36c)where κ0 j is defined in (4.9b). In terms of B j(t), the approximate solution in the147bulk region is the superpositionU(x, t)∼− 14πDN∑j=1∫ t0B j(τ)e−σ(t−τ)(t− τ) e−|x−x j|2/(4D(t−τ)) dτ , (4.37)while in the vicinity of the jth cell we have for ρ = ε−1|x−x j|= O(1) thatU ∼ B j(t)2πDlogρ+B j(t)2πd1 j+d2 jd1 ju1 j(t) . (4.38)4.3.1 Matching to the transient solutionWe now study the limiting behavior as t → 0+, but with t ≫ O(ε2), for B j(t) in(4.36), which satisfies B j → 0 as t → 0. Our small time analysis is based on takingthe Laplace transform of (4.36b) and letting s → ∞. More specifically, we willderive the following precise asymptoticsB j(t)∼ −u′1 j(0)γ jt/ log(t/(κ je−γe)) , as t → 0+ if u1 j(0) = 0 ,−u1 j(0)γ j/ log(t/(κ je−γe)) , as t → 0+ if u1 j(0) ̸= 0 ,(4.39)where κ j = ε2κ0 j with κ0 j defined in (4.9b). These limiting results agree with thosederived in §4.2 from the far-field behavior of the transient solution. Moreover, wehaveB′j(t)∼− u′1 j(0)γ jlog(t/(κ je−γe ))(1− 1log(t/(κ je−γe ))), as t → 0+ if u1 j(0) = 0 ,u1 j(0)γ j/(t [log(t/(κ je−γe))]2), as t → 0+ if u1 j(0) ̸= 0 .(4.40)Therefore, B′j(t)→ 0 as t → 0+ if u1 j(0) = 0, while |B′j(t)| → ∞ as t → 0+ ifu1 j(0) ̸= 0.To derive (4.39), we will first assume that u1 j(0) = 0. Setting B j(0) = 0, weobtain from (4.36a) that u1 j(t) ∼ tu′1 j(0) as t → 0+, where u′1 j(0) = eT1 F(u j(0)).By taking the Laplace transform of (4.36b) and using L[B′j(t)]= sBˆ j(s) and148L [E1(σt)] = log(1+ s/σ)/s we readily calculate thatBˆ j [log(1+ s/σ)−η j]−2N∑k=1k ̸= jBˆkK0(√s+σD|x j−xk|)= uˆ1 jγ j . (4.41)By using the exponential decay of K0 and uˆ1 j ∼ u′1 j(0)/s2 for s → ∞, (4.41)becomesBˆ j[log(se−η jσ)+O(s−1)]∼ γ js2u′1 j(0) , as s→ ∞ . (4.42)By neglecting the O(s−1) term, we use (4.36c) for η j to calculateκ j :=e−η jσ= ε2κ0 j , where κ0 j =14De2(γe−D/d1 j) . (4.43)In this way, (4.42) becomesBˆ j ∼u′1 j(0)γ js2 log(κ js), as s→ ∞ . (4.44)As a result, for t → 0+, but with t ≫ O(ε2), we use the convolution property andκ j = ε2κ0 j to obtainB j(t)∼−u′1 j(0)γ j∫ t0N(ξε2κ0 j)dξ , for ε2 ≪ t ≪ 1 , (4.45)where N(x) is Ramanujan’s integral (4.11), κ0 j is defined in (4.44) and u′1 j(0) =eT1 F j(u j(0)). This limiting result matches identically with the result derived in(4.18b) of §4.2, and by setting t = ε2τ in (4.20) we obtain the first result in (4.39).A very similar short-time analysis can be done for the case where u1 j(0) ̸= 0.In place of (4.44), we obtain thatBˆ j ∼ u1 j(0)γ js log(κ js) , as s→ ∞ . (4.46)By inverting (4.46), and recalling that γ j = 4πDd2 j/d1 j, we obtain the second resultin (4.39). This result matches identically with the far-field behavior (4.14b) as149obtained from our transient analysis in §4.2.4.4 Steady-State and Linear Stability AnalysisIn this section, we first classify the long-time dynamics of (4.36) for solutionsthat tend to limiting values as t → ∞. We show that this limiting dynamicscoincides with the steady-state solution, which is constructed using stronglocalized perturbation theory from the steady-state of the bulk-cell PDE system(4.5). We also formulate the linear stability problem for steady-state solutions. Webegin with the following lemma:Lemma 4.4.1. Suppose that B j(0) = 0, B j(t) is bounded for t > 0 and that B j(t)→B j∞ as t → ∞ for j ∈ {1, . . . ,N}. Then,limt→∞C jk(t) = 2B j∞K0(√σD|x j−xk|), limt→∞D j(t) = 0 , (4.47a)where C jk(t) :=∫ t0B j(τ)e−σ(t−τ)t− τ e−|x j−xk|2/(4D(t−τ)) dτ , (4.47b)and D j(t) :=∫ t0B′j(τ)E1 (σ(t− τ)) dτ . (4.47c)Proof. To establish the first limit in (4.47a), we take the Laplace transform of(4.47b) to obtainCˆ jk(s) =L[e−σtte−|x j−xk|2/(4Dt)]Bˆ j(s) = 2K0(√s+σD|x j−xk|)Bˆ j(s) .(4.48)To determine the long-time behavior of C jk(t), we will use the Tauberian theorem.Since B j(t) is bounded for t > 0 and satisfies B j(t)→ B j∞ as t →∞, it follows thatBˆ j(s) is analytic for Re(s)> 0 and that the singularity with the largest real part is asimple pole at s = 0 for which Bˆ j(s)∼ B j∞/s as s→ 0. As a result, the singularityof Cˆ jk(s) with the largest real part is a simple pole at s = 0 and we havelimt→∞C jk(t) = lims→0sCˆ jk(s) = 2K0(√σD|x j−xk|)lims→0sBˆ j(s) , (4.49)where lims→0 sBˆ j(s) = B j∞. This proves the first result in (4.47a).150The second result in (4.47a) is established in a similar way. Upon taking theLaplace transform of D j(t) in (4.47c), we use B j(0) = 0 to obtain thatDˆ j(s) =L[B′j(t)]L [E1(σt)] = sBˆ j(s)log(1+ s/σ)s= Bˆ j(s) log(1+ s/σ) ,(4.50)where the branch cut for log(1+ s/σ) occurs for Re(s)≤−σ < 0 and Im(s) = 0.To establish that D j(t)→ 0 as t → ∞ it suffices to show that Dˆ j(s) is analytic inRe(s)≥ 0. This follows since from (4.50) we observe that, although Bˆ j ∼ B j∞/s ass→ 0, the point s = 0 is a removable singularity for Dˆ j(s).With this lemma, we can readily characterize those solutions to (4.36) that tendto limiting values as t → ∞. We summarize this result as follows:Proposition 4.4.2. For (4.36), suppose that B j(0) = 0 and that B j(t) and u j(t)are bounded for t ≥ 0 with limiting values B j(t)→ B j∞ and u j → u j∞ as t →∞ foreach j ∈ {1, . . . ,N}. Then, B j∞ and u j∞ satisfy the N(m+1) dimensional nonlinearalgebraic system (NAS)F j(u j∞)+ e1B j∞ = 0 , j ∈ {1, . . . ,N} , (4.51a)η jB j∞+ γ ju1 j∞+2N∑k=1k ̸= jBk∞K0(√σD|x j−xk|)= 0 , j ∈ {1, . . . ,N} , (4.51b)where u1 j∞ = eT1 u j∞ is the first component of the intracellular species. In (4.51),η j and γ j are defined in (4.36c).We now show that the limiting NAS (4.51) is precisely the same system as canbe derived from a strong localized perturbation analysis for steady-state solutionsof (4.5). In the limit ε → 0, the steady-state problem for the outer bulk solution isD∆Us−σUs = 0 , x ∈ R2\{x1, . . . ,xN} ,Us ∼ B js2πD log |x−x j|+B js2πDν+B js2πd1 j+d2 jd1 ju1 js , as x→ x j , j ∈ {1, . . . ,N} ,(4.52)where ν :=−1/ logε . This system is coupled to the steady-state of the intracellular151kinetics, given byF j (u js)+ e1B js = 0 , j ∈ {1, . . . ,N} . (4.53)The solution to (4.52) isUs(x) =−N∑k=1Bks2πDK0(√σD|x−xk|). (4.54)To determine the constants B js, for j ∈ {1, . . . ,N}, we enforce that the localbehavior of Us(x) as x → x j agrees with that required in (4.52). Upon usingK0(z)∼− logz+ log2− γe+o(1) as z→ 0, this matching condition yields thatη jB js+ γ ju1 js+2N∑k=1k ̸= jBksK0(√σD|x j−xk|)= 0 , j ∈ {1, . . . ,N} . (4.55)We observe that the coupled system (4.55) and (4.53) characterizing the steady-state solution is the same as that in (4.51).4.4.1 Linear stability analysisWe now derive a globally coupled eigenvalue problem (GCEP) for the linearizationof the bulk-cell model (4.5) around the steady-state solution.To derive the GCEP we first perturb around the steady-state solution byintroducing the eigen-perturbationU =Us(x)+ eλ tΦ(x) , u j = u js+ eλ tζ j , j ∈ {1, . . . ,N} , (4.56)into (4.5). Upon linearizing, we obtain the eigenvalue problem∆Φ− (λ +σ)DΦ= 0 , x ∈ R2 \∪Nj=1 Bε(x j) , (4.57a)εD∂nΦ= d1 jΦ−d2 j ζ1 j , x ∈ ∂Bε(x j) , j ∈ {1, . . . ,N} , (4.57b)152which is coupled to the linearized intracellular dynamics within the jth cell byλζ j = J jζ j +e1ε∫∂Bε (x j)(d1 jΦ−d2 j ζ1 j)dSx , j ∈ {1, . . . ,N} . (4.57c)Here J j := Du j F j is the Jacobian of the intracellular kinetics evaluated at u j = u js.For ε → 0, we now analyze (4.57) using strong localized perturbation theory.In the inner region near the jth cell, we have to leading-order that∆yΦ= 0 , for ρ > 1; DΦρ = d1 jΦ−d2 jζ1 j , on ρ = 1 , (4.58)which has the radially symmetric solutionΦ=c j2πDlog |y|+ c j2πd1 j+d2 jd1 jζ1 j , (4.59)where c j is to be found. In terms of c j, we calculate thatε−1∫Ω j (d1 jΦ−d2 jζ1 j) dSx = c j, so that (4.57c) becomes(λ I− J j)ζ j = e1c j , j ∈ {1, . . . ,N} , (4.60)where I is the m×m identity matrix. Upon assuming that λ I− J j is invertible, wecalculate ζ1 j asζ1 j = K jc j , where K j := eT1 (λ I− J j)−1 e1 . (4.61)By writing (4.59) in terms of the outer variable, the asymptotic matchingcondition shows that in the outer region Φ satisfies∆Φ− (λ +σ)DΦ= 0 , x ∈ R2 \∪Nj=1 Bε(x j) , (4.62a)Φ∼ c j2πDlog |x−x j|+ c j2πDν +c j2πd1 j+d2 jd1 jζ1 j , as x→ x j , (4.62b)for j ∈ {1, . . . ,N} where ν :=−1/ logε . The solution to (4.62) isΦ(x) =−N∑k=1c j2πDK0(√σ +λD|x−xk|). (4.63)153Upon enforcing that the limiting behavior of Φ(x) as x → x j agree with thatspecified in (4.62b), and where ζ1 j is given in terms of c j by (4.61), we concludethat2(1ν+Dd1 j− γe+ log(2√Dσ +λ))c j +4πDd2 jd1 jζ1 j+2N∑k=1k ̸= jckK0(√σ +λD|x j−xk|)= 0 , j ∈ {1, . . . ,N} .(4.64)Finally, upon writing (4.64) and (4.61) in matrix form, we obtain that thediscrete eigenvalues λ of the linearization around a steady-state solution of (4.5)are obtained from a nonlinear matrix eigenvalue problem, which we refer to as theGCEP. Our result is summarized as follows:Proposition 4.4.3. For ε → 0, the discrete eigenvalues λ associated with thelinearization around a steady-state solution to (4.5), for which det(λ I− J j) ̸= 0for any j ∈ {1, . . . ,N}, are the set of valuesΛ(M ) := {λ | detM (λ ) = 0} , (4.65a)where the N×N dimensional matrixM (λ ) is defined byM (λ ) := I+2πνGλ +νDP1+2πνDP2K (λ ) , (4.65b)with ν =−1/ logε . In (4.65b), P1, P2 andK are the diagonal matricesP1 := diag( 1d11, . . . ,1d1N), P2 := diag(d21d11, . . . ,d2Nd1N), (4.66a)K (λ ) := diag(K1, . . . ,KN) , where K j := eT1 (λ I− J j)−1 e1 . (4.66b)154In addition, Gλ is the eigenvalue-dependent Green matrix with matrix entries(Gλ )i j = (Gλ ) ji :=12πK0(√σ +λD|x j−xk|), i ̸= j , (4.67)(Gλ ) j j = Rλ j :=12π(log(2√Dσ +λ)− γe). (4.68)For any specific λ0 ∈ Λ(M ), we have detM (λ0) = 0, and so M (λ0)c = 0 has anontrivial solution c=(c1, . . . ,cN)T that can be normalized as |c|= 1. We concludethat the steady-state is linearly stable if, whenever λ ∈Λ(M ), we have Re(λ )< 0.We remark that the normalized nullvector c= (c1, . . . ,cN)T encodes the relativemagnitude of the perturbation of the spatial gradient of the bulk signal near thecell boundaries. It also can be used to predict the relative magnitude and phaseshift of intracellular oscillations for the permeable species u1 j that can arise frombifurcations of the steady-state. To see this, we use the steady-state solution Us(x)and the perturbation (4.56) to calculate for the jth cell thatD∂ρU |ρ=1 ∼ 12π(B js+ ∑λ0∈Λ(M )c jeλ0t), j ∈ {1, . . . ,N} , (4.69a)u1 j ∼ u1 js+ ∑λ0∈Λ(M )K j(λ0)c jeλ0t , j ∈ {1, . . . ,N} , (4.69b)where K j(λ0) = eT1 (λ0I− J j)−1 e1. From (4.69b) we observe that if λ0 is complex-valued, the real and imaginary parts of the jth component of the complex-valuedmatrix-eigenvector productK c encode both the relative magnitude and phase shiftof oscillations for the permeable species u1 j in the cell population. Moreover, from(4.69a), the components of the eigenvector c determine the strength and phase shiftof the eigen-perturbation of the signaling gradients near the cells.4.5 Time-Marching Scheme for the Integro-DifferentialSystemA direct numerical approach to solve (4.36) would require at each time step anumerical quadrature of O(N2) memory-dependent convolution integrals. This155naive approach would be prohibitively expensive for large N and would also requirestoring the full time history of each B j(t) in order to advance one time-step.As such, in this section we develop a time-marching algorithm to computesolutions to (4.36). This approach is based on an accurate approximation of thekernels in the nonlocal terms of (4.36b) by a sum of exponentials, which leadsnaturally to an exponential time differencing marching scheme. Rigorous resultsfor the sum-of-exponentials approximation, together with the development of time-marching methods for convolution integrals in other contexts are given in [12],[13], [91], [89] and [66].With an exponential kernel, the development of a time-marching scheme relieson the following simple lemma:Lemma 4.5.1. Let f (t) be a continuous function and define the convolutionintegral F (t) :=∫ t0 eω(t−τ) f (τ)dτ . Then, we have F ′(t) = ωF (t) + f (t) withF (0) = 0. Moreover, we have the marching schemeF (t+∆t) =F (t)eω∆t +U (t,∆t) , with U (t,∆t) := eω∆t∫ ∆t0e−ωz f (t+ z)dz .(4.70)An exponential time differencing ETD2 scheme (cf. [91]), ensuring that the updateintegral U is exact for linear functions f (t), yields, with an error O((∆t)3), theapproximationF (t+∆t)≈F (t)eω∆t+ f (t)(eω∆t −1ω)+[ f (t+∆t)− f (t)](eω∆t −1−ω∆tω2∆t).(4.71)Proof. The proof of (4.70) is immediate. To derive (4.71) we substitutef (t+ z) = f (t)+z∆t[ f (t+∆t)− f (t)]+O ((∆t)2) ,into the update integral U and integrate the resulting expression explicitly.We now develop a time-marching scheme for (4.36b), which we writecompactly asD j(t) = η jB j(t)+ γ ju1 j(t)+N∑k=1k ̸= jC jk(t) , (4.72)156where C jk(t) and D j(t) are defined in (4.47b) and (4.47c), respectively. We observethat both memory integrals C jk(t) and D j(t) are improper, as their kernels eachhave an integrable singularity at t = τ .4.5.1 Sum-of-exponentials approximation for D j(t) and C jk(t)We first develop a sum-of-exponentials approximation for D j(t). To do so, weuse the sector analyticity of the Laplace transform of the exponential integral todeform the initial vertical Bromwich line ΓB to the curve Γ, defined by (4.74), withendpoints at infinity in the left-half plane Re(s) < 0 (see Fig. 4.2 below). Thisyields thatE1(σt) =− 12πi∫ΓE (s)est ds , where E (s) =log(1+ s/σ)s. (4.73)Observe that E (s) is analytic except along the branch cut Re(s)≤−σ with Im(s) =0.The sum-of-exponentials method following [12] (see also [66]) establishesrigorous results for the quadrature of (4.73) along a family of hyperbolic shapedcurvesΓ := {s = χP(x) , x ∈ R} , where P(x) := 1− sin(α+ i x) , (4.74)with 0 < α < π/2 and χ > 0. We observe that s = χ(1− sinα) > 0 at x = 0 andthat Γ has the limiting behaviorIm(s)→−∞ , Re(s)→−∞ , Im(s)Re(s)→ χ cotα , as x→ ∞ ,Im(s)→ ∞ , Re(s)→−∞ , Im(s)Re(s)→−χ cotα , as x→ ∞ .To evaluate (4.73) on Γ we use ds = χP′(x)dx and (4.74) for P(x) to obtainthatE1(σt) =χ2π∫ ∞−∞eχP(x)tE [χP(x)] cos(α+ i x)dx . (4.75)157By discretizing (4.75) uniformly in x, with xℓ = ℓh for |ℓ| ≤ n, we getE1(σt)≈ En(t) :=n∑ℓ=−neℓ esℓt , (4.76a)where the coefficients areeℓ =χh2πcos(α+ iℓh)log(1+ sℓ/σ)sℓ, (4.76b)sℓ = χP(xℓ) = χ [1− sin(α)cosh(ℓh)]− iχ cos(α)sinh(ℓh) . (4.76c)Since |E (s)| ≤ 1/(2|s|1/2) in the cut plane C\(−∞,0) as |s| →∞, the rigorousresult of Lemma 1 and Corollary 1 of [66], as adapted from [90], provides theestimate for the difference |E1(σt)−En(t)|. The result is as follows.Lemma 4.5.2. (cf. [66]) Consider the time interval 0 < δ ≤ t ≤ Tf , with Tf ≥1000δ and let ε f with 0 < ε f < 0.1 be a prescribed error-tolerance. Then, for thechoice of parameters h and χ defined byh =a(θ)n, χ =2πβn(1−θ)Tf a(θ), where a(θ) := cosh−1(2Tfδ (1−θ)sinα),(4.77)where 0 < α−β < α+β < π/2 and 0 < θ < 1, we have the uniform estimate∥E1(σt)−En(t)∥ ≤ ε f√t on δ ≤ t ≤ Tf , (4.78)when n is sufficiently large of the ordern =O ((− logε f + log log(Tf /δ )) log(Tf /δ )) . (4.79)In (4.77), the parameters α , β and θ , satisfying the constraint in Lemma 4.5.2,can be optimized so as to minimize the number of terms needed to achieve aprescribed accuracy. As in [66], we chose α = 0.8 and β = 0.7 in all the resultsbelow with θ in the range [0.90,0.95] (see below). The key conclusion is that thenumber of terms in the sum grows very slowly as either the tolerance ε f decreasesor as Tf /δ increases.In a similar way, we can develop a sum-of-exponentials approximation for158C jk(t) defined in (4.47b). From the Laplace-transform pairL[G(a jk, t)]= 2K0(2a jk√s+σ),where G(a jk, t) :=e−σtte−a2jk/t , with a jk :=|x j−xk|√4D,(4.80)we observe that K0(2a jk√s+σ)is analytic in the complex plane except across thebranch cut Re(s)≤−σ with Im(s) = 0. As such, we can approximate G(a jk, t) bya sum of exponentials, similar to that done in (4.76), to obtainG(a jk, t)≈ Gn(a jk, t) :=n∑ℓ=−nω jkℓ esℓt , (4.81a)where the coefficients areω jkℓ =χh2πcos(α+ iℓh)2K0(2a jk√sℓ+σ), (4.81b)sℓ = χP(xℓ) = χ [1− sin(α)cosh(ℓh)]− iχ cos(α)sinh(αℓ) . (4.81c)Owing to the same branch cut structure as for approximating E1(σt), for simplicityin our approximation (4.81a) we choose the same values of the parameters α = 0.8and β = 0.7 used in (4.76), which provides a comparable accuracy. We emphasizethat the coefficients ω jkℓ will depend on j and k owing to the inter-cell distances|x j−xk|. For N cells, there are N(N−1)/2 sets {ω jkℓ | −n≤ ℓ≤ n} of coefficientsthat need to be calculated once and stored for the algorithm developed below in§4.5.4.We now provide some numerical results for three sum-of-exponentialsapproximations. Under the conditions of Lemma 4.5.2 the difference betweena function f (x, t), with Laplace transform fˆ (x,s), and its sum-of-exponentialsapproximation fa(x, t) = ∑n−n wkeskt with an error magnitude ε f is∥ f (x, t)− fa(x, t)∥ ≤ ε f√t , on t ∈ [δ ,T ] . (4.82)In the tables below f (x, t) is taken either as the 1-D heat-kernel G1D(x, t)considered in [66], the 2-D heat-kernel with bulk degradation G2D(x, t) :=159(4πt)−1e−x2/(4t)−σt = (4π)−1G(x, t), where G(x, t) is given in (4.80), or theexponential integral E1(σt). Recall that the discretization points aresk = χ(1− sin(α+ ikh)), wk = χh2π cos(α+ ikh) fˆ (x,sk).In order to estimate√t ∥ f (x, t)− fa(x, t)∥ depending on the number 2n+1 of termsin the sum-of-exponentials approximation we proceed as in [66]. We take a 50×1000 grid (x j, tk) where x0 = 0, x j = 2−16+ j for j ∈ {1, ...,49} (not applicable forE1(σt)), and tk = δek▲ with ▲ := log(T/δ )/999 for k ∈ {0, . . . ,999} (equi-spacedon a logarithmic scale). We then compute max j,k√t | f (x j, tk)− fa(x j, tk)|, whichprovides the error results given in the tables below on the three different intervalsI1 := [δ ,T ] = [10−3,1], I2 := [δ ,T ] = [10−3,103] and I3 := [δ ,T ] = [10−5,104].In Table 4.1 we first reproduce the numerical error table of [66] for the sum-of-exponentials approximation of the 1-D heat kernel G1D(x, t) := (4πt)−1/2 e−x2/(4t).ε f n(I1) n(I2) n(I3)10−3 15 (6.598 ·10−4) 23 (9.768 ·10−4) 32 (9.095 ·10−4)10−6 31 (7.795 ·10−7) 50 (8.709 ·10−7) 68 (9.643 ·10−7)10−9 47 (9.199 ·10−10) 77 (9.107 ·10−10) 105 (8.817 ·10−10)Table 4.1: The number 2n + 1 of terms needed to approximate the 1-D heat kernel G1D(x, t) := (4πt)−1/2e−x2/(4t) with Laplace transformL [G1(x, t)] = 12√s e−√s|x| to a precision ε f corresponding to (4.82) for thethree time intervals I1 = [10−3,1], I2 = [10−3,103] and I3 = [10−5,104].This table essentially reproduces Table 1 in [66] with a few additionalexplicit error values.In Table 4.2 and Table 4.3 we provide similar numerical error tables for the 2-D heat kernel with degradation G2D(x, t) = (4πt)−1e−x2/(4t)−σt with σ = 1 andfor the exponential integral E1(σt) =∫ ∞σt η−1e−η dη with σ = 1, respectively.For the 2-D heat kernel, we observe that we require a slightly higher number ofterms in the sum-of-exponentials approximation than for the 1-D heat kernel or theexponential integral. However, this causes essentially no time constraint for ouroverall numerical scheme in §4.5.4 since the exponential sum representation hasto be created only once at the beginning of the time marching stepping for a givenspatial arrangement of cells. Creating these exponential sum approximations for160both the 2-D heat kernel and the exponential integral takes less than 2s on a laptopfor n(I2) = 77.ε f n(I1) n(I2) n(I3)10−3 31 (8.085 ·10−4) 49 (8.663 ·10−4) 91 (8.089 ·10−5)10−6 45 (6.690 ·10−7) 75 (9.272 ·10−7) 114 (9.799 ·10−7)10−9 64 (7.028 ·10−10) 110 (8.417 ·10−11) 150 (9.343 ·10−10)Table 4.2: Number 2n + 1 of terms needed to approximate the 2-D heatkernel with degradation G2D(x, t) = (4πt)−1e−x2/(4t)−σt with LaplacetransformL [G2D(x, t)] = (2π)−1K0(x√s+σ) to a precision ε f for σ =1 corresponding to (4.82) for the three time intervals I1 = [10−3,1],I2 = [10−3,103] and I3 = [10−5,104].ε f n(I1) n(I2) n(I3)10−3 31 (8.824 ·10−5) 49 (9.457 ·10−5) 91 (8.424 ·10−5)10−6 45 (8.362 ·10−7) 75 (7.902 ·10−7) 114 (9.532 ·10−7)10−9 64 (9.620 ·10−10) 110 (9.229 ·10−10) 150 (9.525 ·10−10)Table 4.3: Number 2n+ 1 of terms needed to approximate the exponentialintegral E1(σt) for σ = 1 with Laplace transform L [E1(σt)] =log(1+ s/σ)/s to a precision ε f corresponding to (4.82) for the threetime intervals I1 = [10−3,1], I2 = [10−3,103] and I3 = [10−5,104].We remark that θ = 0.95 was chosen for Table 4.2 while θ = 0.90 was chosenfor Table 4.3. For Table 4.1, θ = 0.9 was chosen only for I1, with θ = 0.95otherwise. In our time-stepping algorithm of developed below in §4.5.4 we usedθ = 0.95 and n = 75 for the sum-of-exponentials approximation for E1(σt) andG2D(x, t), so that the discretization points sℓ were common to both approximationsOverall, this achieved an accuracy of roughly 10−9 for E1(σt) on the time interval[10−3,103].4.5.2 Time-marching scheme for D j(t)To derive a time-marching scheme for D j(t), we first decompose D j(t) into thesum of a local term DL j(t) near the singularity of the kernel and a history termDH j(t) asD j(t) = DH j(t)+DL j(t) , (4.83a)161Figure 4.2: Left: Plot of the hyperbolic-shaped contour Γ in the Laplacetransform plane defined by (4.74) with θ = 0.95, α = 0.8 and β = 0.7in (4.77). The discretization points {sℓ} for the sum-of-exponentialsapproximation of E1(σt) with σ = 1 and n= 75 are the red dots. Right:A zoomed plot showing the discretization near the origin.where for some ∆t, with 0 < ∆t ≪ 1, we defineDH j(t) :=∫ t−∆t0B′j(τ)E1 [σ(t− τ)] dτ , (4.83b)DL j(t) :=∫ tt−∆tB′j(τ)E1 [σ(t− τ)] dτ =∫ ∆t0B′j(t− z)E1(σz)dz . (4.83c)The approximation given in the next lemma for the local contribution DL j(t)ensures that the quadrature is exact for linear functions on ranges where B′j(τ) iscontinuous. However, the estimate for DL j(∆t) is more delicate since, owing to thesingular behavior for B′j(τ) given in (4.39) when u1 j(0) ̸= 0, we cannot in generalassume that B′j(τ) is approximately linear on 0 < τ < ∆t.Lemma 4.5.3. For 0 < ∆t ≪ 1 and for t ≥ 2∆t, we haveDL j(t) = B′j(t)b1+B′j(t−∆t)b2+O((∆t)3), (4.84a)162where b2 and b1 are defined byb2 =∆t2E1(σ∆t)− e−σ∆t2σ+(1− e−σ∆t)2σ2∆t, b1 = ∆tE1(σ∆t)+(1− e−σ∆t)σ−b2 .(4.84b)Moreover, for t = ∆t, and with B j satisfying the limiting behavior in (4.39), wehave the exact relationDL j(∆t) = E1(σ∆t)B j(∆t)+∫ ∆t0(B j(∆t)−B j(∆t− z)) e−σzzdz . (4.85)Proof. On 0≤ z≤ ∆t, and assuming t ≥ 2∆t, we can approximateB′j(t− z) = B′j(t)+z∆t[B′j(t−∆t)−B′j(t)]+O((∆t)2),in DL j(t) of (4.83c). By integrating the resulting expression explicitly we getDL j(t)=B′j(t)∆t∫ σ∆t0E1(η)dη+[B′j(t−∆t)−B′j(t)]σ2∆t∫ σ∆t0ηE1(η)dη+O((∆t)3).By using the two integrals∫ x0 E1(η)dη = xE1(x)+ 1− e−x and∫ x0 ηE1(η)dη =[x2E1(x)− (x+1)e−x+1]/2, we readily obtain (4.84).To derive (4.85), we integrate DL j(∆t) by parts using B j(0) = 0 to obtainDL j(∆t) =− limz→0E1(σz) [B j(∆t)−B j(∆t− z)]+E1(σ∆t)B j(∆t)+∫ ∆t0(B j(∆t)−B j(∆t− z)) e−σzzdz .(4.86)Upon using the local behavior (4.39) for B j, together with E1(σz) ∼ − log(σz)+O(1) as z→ 0, we readily obtain that the first term on the right-hand side of (4.86)vanishes when either u1 j(0) = 0 or u1 j(0) ̸= 0. This yields (4.85). In particular, foru1 j(0) ̸= 0 we have the estimatelimz→0E1(σz) [B j(∆t)−B j(∆t− z)] =− u1 j(0)γ j∆t log(∆t/(κ je−γe)) limz→0 [−z log(σz)] = 0 .163For the history term DH j(t), we substitute (4.76a) into (4.83b) to obtainDH j(t)≈n∑ℓ=−neℓHD jℓ(t) where HD jℓ(t) :=∫ t−∆t0B′j(τ)esℓ(t−τ) dτ . (4.87)For each history-mode HD jℓ(t) we adapt Lemma 4.5.1 to readily obtain for t ≥ ∆tthatHD jℓ(t+∆t) = HD jℓ(t)esℓ∆t +UD jℓ(t,∆t) , with HD jℓ(∆t) = 0 ,where UD jℓ(t,∆t) := e2sℓ∆t∫ ∆t0e−sℓzB′j(t−∆t+ z)dz .(4.88a)To approximate (4.88a) for t ≥ 2∆t, we use the ETD2 scheme that ensures that theupdate integral UDℓ j is exact for linear functions. This yields thatUD jℓ(t,∆t)≈ B′j(t)b3ℓ+B′j(t−∆t)b4ℓ , for t ≥ 2∆t , (4.88b)where b3ℓ and b4ℓ are defined byb3ℓ =esℓ∆ts2ℓ∆t(esℓ∆t −1− sℓ∆t), b4ℓ = esℓ∆t(esℓ∆t −1)sℓ−b3ℓ . (4.88c)For t = ∆t, we integrate UD jℓ(∆t,∆t) from (4.88a) by parts and use B j(0) = 0 togetUD jℓ(∆t,∆t) = esℓ∆tB j(∆t)+ sℓe2sℓ∆t∫ ∆t0e−sℓzB j(z)dz . (4.88d)By using an ETD1 scheme to estimate∫ ∆t0 e−sℓzB j(z)dz ≈ B j(∆t)∫ ∆t0 e−sℓz dz, wereadily obtain after calculating the integral∫ ∆t0 e−sℓz dz explicitly thatUD jℓ(∆t,∆t)≈ B j(∆t)e2sℓ∆t . (4.88e)In this way, by combining (4.87)–(4.88b), we obtain for t ≥ 2∆t thatDH j(t+∆t)≈n∑ℓ=−neℓ esℓ∆tHD jl(t)+(n∑ℓ=−neℓ b3ℓ)B′j(t)+(n∑ℓ=−neℓ b4ℓ)B′j(t−∆t) .(4.89)164Moreover, we haveDH j(2∆t)≈(n∑ℓ=−neℓ e2sℓ∆t)B j(∆t) , DH j(∆t) = 0 . (4.90)Finally, by using (4.84), (4.89) and (4.90) we readily obtain the marchingschemeD j(t+∆t)≈ B′j(t+∆t)b1+B′j(t)(b2+n∑ℓ=−neℓb3ℓ)+(n∑ℓ=−neℓ b4ℓ)B′j(t−∆t)+n∑ℓ=−neℓ esℓ∆tHD jl(t) , t ≥ 2∆t ,D j(2∆t)≈ B′j(2∆t)b1+B′j(∆t)b2+(n∑ℓ=−neℓe2sℓ∆t)B j(∆t) ,D j(∆t) = E1(σ∆t)B j(∆t)+∫ ∆t0(B j(∆t)−B j(∆t− z)) e−σzzdz .(4.91a)From (4.88) we have the following update rule for the next time-step:HD jℓ(t+∆t) = HD jℓ(t)esℓ∆t +B′j(t)b3ℓ+B′j(t−∆t)b4ℓ , t ≥ 2∆t ,B j(∆t)e2sℓ∆t , t = ∆t . (4.91b)In (4.91), b1, b2, b3ℓ and b4ℓ are defined in (4.84b) and (4.88c).4.5.3 Time-marching scheme for C jk(t)A similar approach can be used to derive a time-marching scheme for C jk(t)defined in (4.47b).As in (4.83), we decompose C jk(t) into local and history-dependent terms asC jk(t) =CH jk(t)+CL jk(t) , (4.92a)165where for some ∆t, with 0 < ∆t ≪ 1, we defineCH jk(t) :=∫ t−∆t0Bk(τ)G(a jk, t− τ)dτ , (4.92b)CL jk(t) :=∫ tt−∆tBk(τ)G(a jk, t− τ)dτ =∫ ∆t0Bk(t− z)G(a jk,z)dz . (4.92c)The marching scheme for the history term is derived in the same way as in§4.5.2. We haveCH jk(t)≈n∑ℓ=−nω jkℓHCkℓ(t) where HCkℓ(t) :=∫ t−∆t0Bk(τ)esℓ(t−τ) dτ . (4.93)For each history-mode HCkℓ(t) we use Lemma 4.5.1 to obtain the update schemeHCkℓ(t+∆t) = HCkℓ(t)esℓ∆t +UCkℓ(t,∆t) , withHCkℓ(∆t) = 0 ,where UCkℓ(t,∆t) := e2sℓ∆t∫ ∆t0e−sℓzBk(t−∆t+ z)dz .(4.94)To approximate (4.94) for t ≥ 2∆t, we use the ETD2 scheme to obtainUCkℓ(t,∆t)≈ Bk(t)b3ℓ+Bk(t−∆t)b4ℓ , for t ≥ 2∆t , (4.95)where b3ℓ and b4ℓ are defined in (4.88c). For t = ∆t, we use an ETD1 scheme togetUCkℓ(∆t,∆t)≈ Bk(∆t)b40ℓ , where b40ℓ := esℓ∆t(esℓ∆t −1sℓ). (4.96)The approximation of the local contribution CL jk(t) is simpler than for DL jkowing to the exponential decay of G(a jk,z) for z > 0. We use an ETD1 schemeand estimateCL jk(t)≈ Bk(t)∫ ∆t0G(a jk,z)dz≈ Bk(t)E1(a2jk∆t). (4.97)By using E1(z)∼ e−z/z as z→ ∞, we conclude that CL jk(t) is exponentially small166when a2jk/∆t ≫ 1. However, if the diffusivity D is large so that a2jk/∆t is not solarge, then we may need to use the higher order approximation for CL jk(t) given byCL jk(t)≈ Bk(t)E1(a2jk∆t)−(∆te−a2jk/∆t −a2jkE1(a2jk∆t))(σBk(t)+Bk(t)−Bk(t−∆t)∆t).(4.98)In this way, by combining (4.93)–(4.96) for the history modes and (4.97) forthe local term, and using only the leading-order result for CL jk(t), we obtain themarching scheme for C jk(t) for t ≥ ∆t:C jk(t+∆t)≈ Bk(t+∆t)E1(a2jk∆t)+(n∑ℓ=−nω jkℓb3ℓ)Bk(t)+(n∑ℓ=−nω jkℓ b4ℓ)Bk(t−∆t)+n∑ℓ=−nω jkℓ esℓ∆tHCkℓ(t) , for t ≥ 2∆t ,C jk(2∆t)≈ Bk(2∆t)E1(a2jk∆t)+(n∑ℓ=−nω jkℓb40ℓ)Bk(∆t) ,C jk(∆t)≈ Bk(∆t)E1(a2jk∆t),(4.99a)where, from (4.95) and (4.96), we have the update ruleHCkℓ(t+∆t) =HCkℓ(t)esℓ∆t +Bk(t)b3ℓ+Bk(t−∆t)b4ℓ , t ≥ 2∆t ,Bk(∆t)b40ℓ , t = ∆t . (4.99b)Here b3ℓ, b4ℓ and b40ℓ are defined in (4.84b), (4.88c) and (4.96), respectively.4.5.4 Marching scheme for integro-differential systemBy substituting (4.99) and (4.91) into (4.72), and recalling (4.36a) for theODE intracellular dynamics, we now develop a time-marching algorithm for167approximating solutions to (4.36). Let {t0, ..., tq} be a discretization of the timedomain [0,T ] where T is some final time of choice. Then, t0 = 0, tq = T ,ti+1− ti = ∆t for i ∈ {0, ...,q−1}, where ∆t = T/q and ti = i∆t. In the formulationof our algorithm we will use B′j(ti) ≈ (B j(ti)−B j(ti−1))/∆t, for i ≥ 2 in (4.91).Below, we denote the numerical approximation to u1 j(ti) by u(i)1 j and the vector ofall u(i)1 j for j ∈ {1, . . . ,N} by u(i)1 . We use a similar notation for the reaction-kineticfunctions.Step 1; 0↷ ∆t: For this first time-step we discretize the reaction kinetics Fj ineach cell by the explicit Runge-Kutta RK4 method [128] and useu(1)1 = u(0)1 +∆t6(k(0)1 +2k(0)2 +2k(0)3 + k(0)4 )+B(0)∆t ,u(1)2 = u(0)2 +∆t6(k˜(0)1 +2k˜(0)2 +2k˜(0)3 + k˜(0)4 ) ,(4.100)where the RK4 weights arek(0)1 = F1(u(0)1 ,u(0)2 ) , k˜(0)1 = F2(u(0)1 ,u(0)2 ) ,k(0)2 = F1(u(0)1 +∆t2k(0)1 ,u(0)2 +∆t2k˜(0)1 ) , k˜(0)2 = F2(u(0)1 +∆t2k(0)1 ,u(0)2 +∆t2k˜(0)1 ) ,k(0)3 = F1(u(0)1 +∆t2k(0)2 ,u(0)2 +∆t2k˜(0)2 ) , k˜(0)3 = F2(u(0)1 +∆t2k(0)2 ,u(0)2 +∆t2k˜(0)2 ) ,k(0)4 = F1(u(0)1 +∆tk(0)3 ,u(0)2 +∆tk˜(0)3 ) , k˜(0)4 = F2(u(0)1 +∆tk(0)3 ,u(0)2 +∆tk˜(0)3 ) .In (4.100) we imposed the explicit short-time behavior for B(0) given in (4.39),which is valid since ε2 ≪ ∆t ≪ 1. The truncation error for this approximation isO(∆t), whereas RK4 gives a truncation error of O(∆t5) for the reaction kinetics.In terms of the computed u(1)1 ≈ u1(t1), in Appendix A.7 we derive an improvedapproximation for B(1) = (B(1)1 , . . . ,B(1)N )T , in which the components areB(1)j =−u(1)1 j γ jlog(∆t/(κ je−γe))(1− π26 [log(∆t/(κ je−γe))]2). (4.101)Step 2; ∆t ↷ 2∆t: We use an RK4 scheme for the reaction kinetics, with RK4168weights as given in the first time step, and with a lagged B(1), so thatu(2)1 = u(1)1 +∆t6(k(1)1 +2k(1)2 +2k(1)3 + k(1)4 )+B(1)∆t,u(2)2 = u(1)2 +∆t6(k˜(1)1 +2k˜(1)2 +2k˜(1)3 + k˜(1)4 ) .(4.102)In terms of the computed u(2)1 , we determine B2 from the linear systemB(2) = A−1(M1B(1)+∆tΓu(2)1 −b2∆tB′(∆t)), (4.103)where B′(∆t) is given in (4.40). In (4.103), Γ := diag(γ1, . . . ,γN), while the matricesA and M1 have entriesA j j = b1−η j∆t , A jk = −∆tE1(a2jk/∆t),M1, j j = b1−∆t(∑nℓ=−n eℓe2sℓ∆t), M1, jk = ∆t∑nℓ=−nω jkℓb40ℓ ,(4.104)for k ̸= j ∈ {1, ...,N}.Recursive step; ti ↷ ti+1, for i ≥ 2: We use the RK4 method with a laggedB(i), so thatu(i+1)1 = u(i)1 +∆t6(k(i)1 +2k(i)2 +2k(i)3 + k(i)4 )+B(i)∆t ,u(i+1)2 = u(i)2 +∆t6(k˜(i)1 +2k˜(i)2 +2k˜(i)3 + k˜(i)4 ) .(4.105)In terms of the computed u(i+1)1 we calculate B(i+1) asB(i+1) = A−1(MB(i)+N B(i−1)+(n∑ℓ=−neℓb4ℓ)B(i−2)+∆tΓu(i+1)1 −∆tH(i)),(4.106)where the matrices M andN have entriesM j j = b1−b2−∑nℓ=−n eℓb3ℓ , M jk = ∆t∑nℓ=−nω jkℓb3ℓ ,N j j = b2+∑nℓ=−n eℓ(b3ℓ−b4ℓ) , N jk = ∆t∑nℓ=−nω jkℓb4ℓ,(4.107)for k ̸= j ∈ {1, ...,N}. In (4.106) the history vector H(i) := (H(i)1 , . . . ,H(i)N )T has169entriesH(i)j =n∑ℓ=−n(eℓesℓ∆tH(i)D jℓ−N∑k=1,k ̸= jω jkℓesℓ∆tH(i)Ckℓ), (4.108a)and is updated with the schemeH(i)Dℓ = H(i−1)Dℓ esℓ∆t +b3ℓ(B(i−1)−B(i−2))∆t+b4ℓ(B(i−2)−B(i−3))∆t , if i≥ 4 ,B′(∆t)b4ℓ if i = 3 ,H(2)Dℓ = e2sℓ∆tB(∆t) ,(4.108b)together withH(i)Cℓ = H(i−1)Cℓ esℓ∆t +b3ℓB(i−1)+b4ℓB(i−2) , if i≥ 3 ,H(2)Cℓ = b40ℓB(∆t) .(4.108c)Overall, our formulation is an an operator-splitting scheme of a semi-implicitkind in the sense that the right-hand sides of the reaction kinetic vector fields aretreated explicitly using the RK4 method with a lagged B(i−1), while in (4.106) u(i)1appears implicitly in the update to B(i).4.6 Numerical Results: Sel’kov Intracellular DynamicsTo illustrate our analysis of (4.5), we will consider the two-componentSel’kov kinetics. For this choice, the intracellular kinetics F j(u1 j,u2 j) :=(F1 j(u1 j,u2 j),Fj2(u1 j,u2 j))T are given byF1 j = α ju2 j +u2 ju21 j−u1 j , F2 j = ζ j[µ j−(α ju2 j +u2 ju21 j)]. (4.109)We refer to α j, ζ j and µ j as the reaction-kinetic parameters for the jth cell.1704.6.1 An isolated cellFor an isolated cell uncoupled from the bulk, the unique steady-state for du j/dt =F j is u1 j = µ j and u2 j = µ j/(α j +µ2j). At this steady-state we calculate thedeterminant and trace of the Jacobian asdet(J j) = ζ j(α j +µ2j), tr(J j) =1α j +µ2j(2µ jµ j− (α j +µ2j )−ζ j(α j +µ2j )2).(4.110)Since det(J j) > 0, the unique steady-state for an isolated cell is linearly stable ifand only if tr(J j)< 0. The Hopf bifurcation boundary in the α j versus µ j parameterplane occurs when tr(J j) = 0. From the Poincare´-Bendixson theorem, we concludethat whenever the steady-state is unstable the isolated cell will have limit cycleoscillations. For ζ j = 0.15, in Fig. 4.3a we show the region in the α j versus µ jparameter plane where limit cycle oscillations occur, and in Fig. 4.3b we plot thelimit cycle in the u2 j versus u1 j plane for α j = 0.4 and µ j = 2.0 (black star inFig. 4.3a).0 0.5 1 1.5 2 2.5 300.20.40.60.81(a) Instability region: isolated cell0 1 2 3 4 5 600.20.40.60.811.2(b) Limit cycle in phase planeFigure 4.3: Left: Blue-shaded region of instability in the α j versus µ j planefor the steady-state of an isolated cell when ζ j = 0.15. In this region,since tr(J j)> 0 the unique steady-state is unstable, and a time-periodic(limit cycle) solution occurs for an isolated cell. In the unshaded regionthe steady-state is linearly stable for an isolated cell. Right: For α j = 0.4and µ j = 2.0 (black star in left panel), there is a limit cycle in the u2 jversus u1 j plane.Next, we determine how the instability region for an isolated cell changes171when we include the effect of efflux across the cell membrane, but neglect anyinflux from the bulk medium (see the schematic in Fig. 4.1b). Setting U = 0in (4.5c), for an isolated cell with boundary efflux the unique steady-state fordu j/dt = F j−2πd2 ju1 je1 is now u1 j = µ j/(1+2πd2 j) and u2 j = µ j/(α j +u21 j).The Hopf bifurcation boundary in the α j versus d2 j plane occurs when tr(J j) = 0,which yieldsα j =−µ2j(1+2πd2 j)2+12ζ j−(1+2πd2 j)+√(1+2πd2 j)2+8µ2j ζ j1+2πd2 j .(4.111)For ζ j = 0.15 and µ j = 2.0, in Fig. 4.4 we show that when d2 j increases pasta threshold, the range of α j where intracellular oscillations occur decreasessignificantly.0 0.2 0.4 0.6 0.8 100.20.40.60.81Figure 4.4: Blue-shaded region of instability in the α j versus d2 j plane forthe steady-state of an isolated cell with boundary efflux when ζ j = 0.15and µ j = 2.0. As d2 j increases past a threshold, the range in α j wherelimit cycle oscillations will occur decreases.4.6.2 Steady-states and their stability with bulk-cell couplingWith bulk-cell coupling, the steady-state solutions of (4.5) are obtained from thesolution to the NAS (4.53) and (4.55) where F j is given in (4.109). We readilyobtain thatu1 js = µ j +B js , u2 js =µ jα j +u21 js, (4.112a)172where the steady-state source strengths B js satisfy(γ j +η j)B js+2N∑k=1k ̸= jBksK0(√σD|x j−xk|)=−γ jµ j , j ∈ {1, . . . ,N} . (4.112b)Here η j and γ j are defined in (4.36c). We remark that although (4.55) and (4.53)generally yields a nonlinear algebraic system characterizing steady-state solutions,with Sel’kov kinetics one must only solve the linear algebraic system (4.112b).To analyze the linear stability of this steady-state solution, we must solve thenonlinear eigenvalue problem (4.65a) of Proposition 4.4.3. In calculating M (λ )from (4.65b), the only term that depends on the choice of the intracellular kineticsis the diagonal matrix K (λ ) defined in (4.66b). For the Sel’kov kinetics (4.109),we ready calculateK (λ ) := diag(K1, . . . ,KN) , where K j :=λ +det(J j)λ 2− tr(J j)λ +det(J j) , (4.113)where det(J j) and tr(J j) are now given bydet(J j) = ζ j(α j +u21 js)> 0 ,tr(J j) =1α j +u21 js(2µ ju1 js−(α j +u21 js)−ζ j (α j +u21 js)2) . (4.114)Here u1 js = µ j +B js, where B js for j ∈ {1, . . . ,N} are obtained from (4.112b).Since the linear system (4.112b) is always solvable, there are no transcriticalor fold bifurcation points along the steady-state solution branch as parameters arevaried for Sel’kov kinetics. Therefore, since λ = 0 is not a root of detM (λ ) = 0,the steady-state solution is never destabilized by a zero-eigenvalue crossing (seeProposition 1 of [64]).In this way, for reaction-kinetic parameters for which each isolated cell hasa stable steady-state, any instability that arises from the bulk-cell coupling mustoccur through a Hopf bifurcation. As a result, we will identify stability boundariesin the 1/σ versus D parameter plane for steady-state solutions with Sel’kovkinetics by seeking Hopf bifurcation (HB) thresholds for which λ = iλI ∈ Λ(M )with λI ∈ R. For arbitrary locations x j, with j ∈ {1, . . . ,N}, of a collection173of non-identical cells, with possibly cell-dependent permeability and reaction-kinetic parameters, we must compute all paths in the 1/σ versus D parameterspace where detM (iλI) = 0 for some λI > 0. Previously, HB boundaries inparameter space were obtained numerically for a ring arrangement of identicalcells, with and without a defective center cell, for which the eigenvalues ofM (λ ) are known analytically (see [64]). When explicit analytical formulaeare known for the eigenvalues of M (λ ), simple scalar root-finding algorithmscan be used to determine the HB thresholds in parameter space using a pseudo-arclength continuation scheme in D (see [64] for details). However, for a generalspatial arrangement of non-identical cells, the numerical solution of the nonlinearmatrix eigenvalue problem detM (iλI) = 0 in the 1/σ versus D parameter space ishighly challenging. Nonlinear matrix eigenvalue problems and effective solutionstrategies for various classes of matrices are discussed in [10], [11], [53].To determine regions of instability in open sets of the 1/σ versus D parameterplane off of the HB boundaries and to count the number, Z , of destabilizingeigenvalues of the linearization of the bulk-cell model (4.5) around the steady-state,as defined by the number of λ ∈Λ(M ) with Re(λ )> 0 (counting multiplicity), weuse the argument principle of complex analysis applied to F (λ ) := det(M (λ )),where M (λ ) is the complex symmetric GCEP matrix (4.65b) in which K (λ ) isgiven in (4.113). In the right-half plane, we take the contour ΓR as the unionof the imaginary axis ΓI = iλI , for |λI| ≤ R, and the semi-circle CR , defined by|λ | = R > 0 with |arg(λ )| ≤ π/2. Provided that there are no zeroes or poles onΓR for F (λ ), the argument principle yields that the number ZR of zeroes ofF (λ ) = 0 inside ΓR isZR =12π[argF (λ )]ΓR+PR . (4.115)HerePR is the number of poles ofF (λ ) inside ΓR , while[argF (λ )]ΓRdenotesthe change in the argument of F (λ ) over the counter-clockwise oriented contourΓR . We pass to the limit R → ∞ and use (4.113) to obtain that K (λ )→ 0 asR → ∞ on the semi-circle CR . Moreover, for R ≫ 1, (4.67) and (4.65b) yieldsthat M (λ ) ≈ −(ν/2) log(R) I +O(1) on CR . As a result, we conclude thatlimR→∞[argF (λ )]CR= 0. Finally, to evaluate the change of argument on the174imaginary axis we use F (λ ) =F (λ ) with λ = iλI to reduce the computation tothe positive imaginary axis. In this way, by letting R→ ∞ we get that the numberof zeroes Z ofF (λ ) in Re(λ )> 0 isZ =P− 1π[argF (iλI)]ΓI+, (4.116)whereP denotes the number of poles (counting multiplicity) ofF (λ ) in Re(λ )>0, and ΓI+ denotes the entire positive imaginary axis now directed upwards startingfrom λI = 0. To determine P , we observe, since Gλ is analytic in Re(λ ) > 0,that any singularity of F (λ ) must arise from the poles of the diagonal matrixK (λ ) with entries given in (4.113). Since det(J j)> 0, we conclude thatP = 2pwhere p is the total number of integers j ∈ {1, . . . ,N} for which tr(J j) > 0. Fora given parameter set, we will use (4.116) to numerically calculate Z in a finediscretization of the 1/σ versus D parameter plane. Below, we refer to this “phasediagram” as a scatter plot. For cell configurations where the matrix spectrum ofM (λ ) is known, the determinant is readily evaluated at a fine discretization alongthe imaginary axis. The axis-crossing algorithm of [3] can be used to compute thewinding number.(a) Two-cell configuration (b) Four-cell configuration (c) Ring with center cellFigure 4.5: Left: A two-cell configuration. Middle: four equally-spaced cellson a ring. Right: A hexagonal ring arrangement of identical cells wherethe center cell is a pacemaker or “defective cell”. Equally-colored pairsof cells are taken to have identical permeabilities and reaction-kineticparameters.In the results shown in §4.6.3–4.6.6 we have fixed the cell radius as ε = 0.03and the Sel’kov kinetic parameters as µ j = 2 and ζ j = 0.15 for j ∈ {1, . . . ,N}.We will explore the effect of changing the kinetic parameter α j and the influx and175efflux parameters d1 j and d2 j, respectively. More specifically, we will study theeffect of choosing pairs (d2 j,α j) either inside or outside the blue-shaded region ofinstability, as shown in Fig. 4.4, for an isolated cell with boundary efflux.4.6.3 Two-cell configurations0.5 1 1.5 2 2.502468(a) d1 j = 0.40 1 2 3 4 5 60123456(b) d1 j = 0.80 2 4 6 8 100246(c) d1 j = 1.5Figure 4.6: Left: HB boundaries in the 1/σ versus D parameter planecorresponding to the linearization of the steady-state for the two-cellconfiguration of Fig. 4.5a for identical cells with permeabilities d1 j =0.4, d2 j = 0.2 and kinetic parameter α j = 0.9. The HB boundariesfor the in-phase and anti-phase modes are the solid and dashed curves,respectively. Only inside the union of the two lobes is the steady-stateunstable. Middle: same plot but now d1 j = 0.8. Right: d1 j = 1.5. Therange in D of the lobes of instability increases as the influx parameterd2 j increases.For the linearization of the steady-state for a two-cell configuration such asshown in Fig. 4.5a, in Fig. 4.6 we plot the HB boundaries for the in-phase mode(solid curve) and the anti-phase mode in the 1/σ versus D parameter plane for thecase of two identical cells that are initially in a quiescent state when uncoupledfrom the bulk. The plots are shown for three values of the influx parameter d1 j.The other parameters are given in the figure caption. The steady-state is unstable tothe in-phase and the anti-phase perturbations only inside the corresponding lobes,and this steady-state is linearly stable outside the union of the two lobes. FromFig. 4.6 we observe the possibility of either a purely anti-phase instability or apurely in-phase instability for some parameter pairs (1/σ ,D). Upon comparingFigs. 4.6b–4.6c, we observe that the range in D where the lobes of instability occurare larger when the influx parameter is increased.1760.5 1 1.5 2 2.502468(a) d1 j = 0.40 1 2 3 4 5 60123456(b) d1 j = 0.80 5 100123456(c) d1 j = 1.5Figure 4.7: Scatter plot of the number Z of destabilizing eigenvaluessatisfying Re(λ ) > 0 in the 1/σ versus D parameter planecorresponding to the linearization of the steady-state for the two-cellconfiguration of Fig. 4.5a for identical cells with parameter valuesd1 j = 0.4 (left), d1 j = 0.8 (middle), d1 j = 1.5 (right). Here, Z = 0is white,Z = 2 is blue, andZ = 4 is magenta. The HB boundaries aresuperimposed. Remaining parameters are as for Fig. 4.6.For Fig. 4.7 we numerically implemented the winding number criterion (6.3)to provide a scatter plot in the D versus 1/σ plane that indicates the number ofdestabilizing eigenvalues in Re(λ ) > 0 associated with the linearization of thesteady-state. In Fig. 4.8 we plot the path of both the real and imaginary partsof the two dominant eigenvalues λ ∈C along the path indicated by the vertical andhorizontal dotted lines in Fig. 4.7a and Fig. 4.7c, respectively. Along the verticalpath in Fig. 4.7a, we observe from Fig. 4.8a that the in-phase and anti-phase modesof instability have comparable growth rates when D = 0.75 and σ = 2.0 and thatthe in-phase mode becomes stable first as 1/σ is increased. The imaginary partIm(λ ) is roughly the same for both modes and shows little variation with σ . Alongthe horizontal path in Fig. 4.7c, we observe from Fig. 4.8c that the growth rate forthe in-phase mode is larger than that for the anti-phase mode as D is increased.We now compare our numerical solution of the reduced integro-differentialsystem (4.32), as computed with our algorithm of §4.5.4, with that of thefull bulk-cell system (4.5), as computed with the commercial PDE solverFlexPDE Professional 6.50/L64 [39] using the domain-truncation approach given1772 4 6-0.0100.010.020.030.04(a) Re(λ ) vertical slice in Fig. 4.7a2 4 60.770.780.790.8(b) Im(λ ) vertical slice in Fig. 4.7a2 4 6 8 10-0.06-0.04-0.0200.020.04(c) Re(λ ) horizontal slice in Fig. 4.7c2 4 6 8 100.740.760.780.80.82(d) Im(λ ) horizontal slice in Fig. 4.7cFigure 4.8: Top row: two dominant eigenvalues, Re(λ ) (left) and Im(λ )(right), computed by solving detM (λ ) = 0 along the vertical dottedpath with D = 0.75 indicated in Fig. 4.7a. Bottom row: two dominanteigenvalues, Re(λ ) (left) and Im(λ ) (right), computed by solvingdetM (λ ) = 0 along the horizontal dotted path with σ = 0.5 indicatedin Fig. 4.7c. The horizontal red line is the marginal stability thresholdRe(λ ) = 0.in Appendix A.8. Our comparison of results is for the indicated point shown inFig. 4.7a, where rather intricate long-time dynamics occur. Our goal with thiscomparison is to validate two key features with our overall hybrid asymptotic-numerical approach. More specifically, our results below suggest that both theintegro-differential system, as derived in (4.32) under the asymptotic assumptionε≪ 1, and the mixed-order numerical time-marching scheme formulated in §4.5.4,are able to accurately replicate detailed fine features in intracellular oscillations for178the full bulk-cell model (4.5) over long time intervals. For the algorithm in §4.5.4,we used a time step size ∆t = 0.01 for the first two time steps for the B-part ofthe right-hand side with a more refined time-step size ∆t f = ∆t/5 = 0.002 for thereaction kinetics with the RK4. After the first two time steps we used ∆t f for boththe reaction kinetics and the B part. The same discretization of the Laplace spacecontour for the sum-of-exponentials approximation was taken as in Fig. 4.2. Weemphasize that for the FlexPDE solution, the time-integration to t ≈ 669 took manyhours of CPU time. In contrast, our algorithm, implemented in Fortran on a DellPrecision laptop with an Intel Core I7 processor, completed in roughly one minute.In Fig. 4.9 we show a very close agreement between our hybrid asymptotic-numerical results and the full numerical results. We observe that our hybridapproach is able to capture a transiently decaying oscillation, which transitionsto a mixed-mode oscillation on some intermediate time scale, and that ultimatelytends to a steady-state solution as characterized by (4.112b). Our final steady-state values correctly approximates the FlexPDE computed steady-state to severaldecimal places of accuracy. We remark that this intricate mixed-mode behaviorover a rather long time scale seen in Fig. 4.9 stems from the proximity to the anti-phase Hopf bubble in parameter space as shown in Fig. 4.7a, together with thetwo dominant, but rather small magnitude, closely-spaced eigenvalues shown inFig. 4.8a with σ = 1/7. For Fig. 4.10 we additionally extracted the amplitude andperiod of both our hybrid marching scheme solution to the integro-ODE systemand the FlexPDE solution to the full PDE system of Fig. 4.9 and plotted themfor comparison in one plot per intracellular chemical species (shown only foru1 j). While the FlexPDE solution’s amplitude and period look more rugged, theycompare well with our hybrid marching scheme solution’s amplitude and period.A parameter set along the dashed line in Fig. 4.6a just outside the synchronousbubble and still inside the asynchronous bubble is interesting to test out. Weexpect that in-phase oscillations are unstable for D = 0.75 and σ = 1/5.8 whileanti-phase oscillations of the cell chemical concentrations are stable, and we alsoexpect initial mixed-mode oscillations until a stable attractor is reached, due tothe proximity to both bubbles leading to competition between the oscillatorymodes. In Fig. 4.11 one can observe clearly that starting with a combination ofthe anti-phase perturbation (1,−1)T with the in-phase perturbation (1,1)T from179(a) u1 j hybrid scheme (b) u1 j FlexPDE(c) u2 j hybrid scheme (d) u2 j FlexPDEFigure 4.9: Comparison of the numerical solutions computed from theintegro-differential system (4.32) using the algorithm in §4.5.4 withthat computed from the bulk-cell model (4.5) using FlexPDE [39]with the artificial boundary condition given in (A.15) of AppendixA.8. Parameters: D = 0.75, σ = 1/7 with d1 j = 0.4, d2 j = 0.2 andα j = 0.9 corresponding to the labeled point in Fig. 4.7a. The initialcondition imposed was the steady-state with an anti-phase perturbation:u(0)1 = (u11s,u21s)T + 0.01 · (1,−1)T , and similarly for u(0)2 . The initialbulk solution for (4.5) was U(x,0) = 0. The steady-state is the blackhorizontal line.180(a) Comparison of u11 amplitudes (b) Comparison of u11 periods(c) Comparison of u12 amplitudes (d) Comparison of u12 periodsFigure 4.10: Comparison of the amplitude and period of the numericalsolutions to u1 j computed from the integro-differential system (4.32)using the algorithm in §4.5.4 with that computed from the bulk-cell model (4.5) using FlexPDE [39] with the artificial boundarycondition given in (A.15) of Appendix A.8. This figure adds additionalinformation to Figure 4.9.the steady-state leads to stable anti-phase oscillations. However, starting with anin-phase perturbation from the steady-state as was done for Fig. 4.13, the in-phaseoscillations are exponentially decaying to the steady-state obtained from (4.112b).Again, there is very good agreement of our solution to the integro-ODE system(4.32) computed using the algorithm in §4.5.4 with the FlexPDE solution to thebulk-cell model (4.5). Figures 4.12 and 4.14 show how the extracted amplitudesand periods of the hybrid marching scheme solution and the FlexPDE solutioncompare. While they compare well besides the ruggedness of the extracted periodsin time, one can see that the amplitudes of the FlexPDE solution are slightly higher181at later times.(a) u1 j hybrid scheme (b) u1 j FlexPDE(c) u2 j hybrid scheme (d) u2 j FlexPDEFigure 4.11: Comparison of the numerical solutions computed from theintegro-differential system (4.32) using the algorithm in §4.5.4 withthat computed from the bulk-cell model (4.5) using FlexPDE [39]with the artificial boundary condition given in (A.15) of AppendixA.8. Parameters: D = 0.75, σ = 1/5.8 with d1 j = 0.4, d2 j = 0.2and α j = 0.9 corresponding to the labeled point in Fig. 4.7a. Theinitial condition imposed was the steady-state with a combination of ananti-phase with an in-phase perturbation: u(0)1 = (u11s,u21s)T + 0.05 ·(1,−0.1)T , and similarly for u(0)2 . The initial bulk solution for (4.5)was U(x,0) = 0. The steady-state is the black horizontal line.Next, we consider the same parameters as in Figs. 4.6c and 4.7c except thatwe now modify the kinetic parameter to α1 = 0.4 for the first cell centered at x1 =(1,0). For α1 = 0.4, an isolated cell with no boundary efflux would have limit cycleoscillations (see the blue star in Fig. 4.3a). Since d21 = 0.2, we further observe fromFig. 4.4 that the cell centered at x1 would have limit cycle oscillations even with182(a) Comparison of u11 amplitudes (b) Comparison of u11 periods(c) Comparison of u12 amplitudes (d) Comparison of u12 periodsFigure 4.12: Comparison of the amplitude and period of the numericalsolutions to u1 j computed from the integro-differential system (4.32)using the algorithm in §4.5.4 with that computed from the bulk-cell model (4.5) using FlexPDE [39] with the artificial boundarycondition given in (A.15) of Appendix A.8. This figure adds additionalinformation to Figure 4.11.boundary efflux when it is uncoupled from the bulk. As a result, we refer to thiscell as the signaling cell. In the scatter plot shown in Fig. 4.15a we observe that thesteady-state is always unstable in the 1/σ versus D parameter plane. In the blue-shaded region, the unique destabilizing mode is the one for which the amplitudeof intracellular oscillations for the signaling cell centered at x1 is large, and wherethe second cell centered at x2 has very small oscillations, and so is effectivelysilent. In Fig. 4.15a there is only one HB boundary and it corresponds to a marginalmode where intracellular oscillations emerge in the otherwise silent cell centeredat x2 = (−1,0) and where the signaling cell centered at x1 is effectively silent.183(a) u1 j hybrid scheme (b) u1 j FlexPDE(c) u2 j hybrid scheme (d) u2 j FlexPDEFigure 4.13: Same conditions as for Figure 4.11, but now the initial conditionimposed was the steady-state with an in-phase perturbation: u(0)1 =(u11s,u21s)T +0.01 · (1,1)T , and similarly for u(0)2 . One can see nicelythat in-phase oscillations are unstable while anti-phase oscillations arestable as seen from Figure 4.11.However, by plotting the path of the two dominant eigenvalues in Fig. 4.15b andFig. 4.15c along the horizontal and vertical slices shown in Fig. 4.15a, respectively,we observe that the mode with the largest growth rate is always the one for whichthe cell centered at x2 has only very small amplitude oscillations in comparisonto that for the signaling cell. By computing the eigenvector c for the two modesat some points along the horizontal slice in Fig. 4.15a, we use the criterion in(4.69b) to determine the relative magnitude of the intracellular oscillations for thelinearized problem in the two cells. As shown in Table 4.4, the mode with thelargest growth rate is the one for which the second cell is effectively quiet.By using our algorithm in §4.5.4 with ∆t = 0.002, and with a uniformly random184(a) Comparison of u11 amplitudes (b) Comparison of u11 periods(c) Comparison of u12 amplitudes (d) Comparison of u12 periodsFigure 4.14: Comparison of the amplitude and period of the numericalsolutions to u1 j computed from the integro-differential system (4.32)using the algorithm in §4.5.4 with that computed from the bulk-cell model (4.5) using FlexPDE [39] with the artificial boundarycondition given in (A.15) of Appendix A.8. This figure adds additionalinformation to Figure 4.13.initial perturbation (Uni f ([−0.01,0.01])) about the steady-state, in Fig. 4.16 weplot u2 j versus t on the time window 600 < t < 800 at the labeled points alongthe horizontal and vertical slices in Fig. 4.15a. We conjecture that when both thein-phase and anti-phase modes are destabilizing for the linearization of the steady-state solutions, such as for D = 2, σ = 1/2 in Fig. 4.16a and for D = 4, σ = 1/2in Fig. 4.16b, the oscillations for the second cell centered at x2 are of wave-packettype, as is characteristic of mixed-mode oscillations.For the scatter plot of Fig. 4.15a with D = 3 and σ = 1/2, in Fig. 4.17we show that our hybrid algorithm is able to reproduce to high accuracy the1850 2 4 6 8 1012345(a) one non-quiescent cell2 4 6 8 10-0.100.10.20.3(b) Re(λ ) horizontal slice2 4 600.050.10.150.2(c) Re(λ ) vertical sliceFigure 4.15: Left: Scatter plot of the numberZ of destabilizing eigenvalues,with Z = 2 (blue) and Z = 4 (magenta), for the linearization of thesteady-state in the 1/σ versus D plane for two non-identical cells withdifferent kinetics parameter α1 = 0.4 and α2 = 0.9. The remainingparameters are d1 j = 1.5 and d2 j = 0.2 for j ∈ {1,2}. Only thecell with α1 = 0.4 (signaling cell) would have limit cycle oscillationswith boundary efflux when it is uncoupled to the bulk. The sole HBboundary is superimposed. The steady-state is now always unstable.Middle: Re(λ ) versus D along the horizontal slice. Right: Re(λ )versus 1/σ along the vertical slice. The dominant mode of instabilityis for intracellular oscillations to be concentrated to the first cell, whilethe second cell is essentially in a quiescent state.delicate mixed-mode wave-packet type oscillations for cell 2 over long time scalesthat occurs in the FlexPDE numerical solution of the full bulk-cell model (4.5).The FlexPDE numerical results required hours of CPU time, whereas the hybridalgorithm completed in less than a minute on a laptop.Next, we consider the effect on intracellular oscillations of increasing the effluxpermeability for the cell centered at x1 to d21 = 0.5. From Fig. 4.4 we observe thatlimit cycle oscillations no longer occur for this cell when it is uncoupled from thebulk. Moreover, we decrease the influx parameter for this cell to d11 = 0.4, so thatthere is less influx from the bulk medium as compared to the cell at x2. In Fig. 4.18awe show the resulting scatter plot in the 1/σ versus D plane. We now observe thatthere is a parameter range (i.e., the white region) where the steady-state is linearlystable. Along the dotted parameter path shown in Fig. 4.18a, in Figs. 4.18b-4.18c186600 650 700 750 80000.511.5cell 1cell 2(a) D = 2, σ = 1/2.600 650 700 750 80000.511.5cell 1cell 2(b) D = 4, σ = 1/2.600 650 700 750 80000.511.5cell 1cell 2(c) D = 2, σ = 1/6.Figure 4.16: Intracellular dynamics u2 j(t) between 600 < t < 800 at thelabeled points along the vertical and horizontal slices in Fig. 4.15a.Left: D = 2, σ = 1/2: wave-packet solution for second cell.Middle: D = 4, σ = 1/2: wave-packet solution transitions to regularoscillations. Right: D = 2, σ = 1/6: regular oscillations for secondcell but with smaller amplitude. The dashed horizontal lines are theunstable steady-state values for u2 j. Parameters: α1 = 0.4, α2 = 0.9,with d1 j = 1.5 and d2 j = 0.2 for j ∈ {1,2}.D |c1| |c2| ∆θ(rad) |(Kc)1| |(Kc)2| ∆θ(rad) Re(λ ) Quiet2 1.0000 0.0078 2.71 0.9989 0.0467 0.78 0.215 cell 22 0.1018 0.9948 2.31 0.0468 0.9989 2.25 0.017 cell 14 0.9999 0.0127 1.26 0.9973 0.0736 0.59 0.252 cell 24 0.1422 0.9898 3.77 0.0731 0.9973 2.46 0.0135 cell 16 0.9998 0.0182 0.93 0.9965 0.0838 0.49 0.269 cell 26 0.1556 0.9878 3.69 0.0834 0.9965 2.56 0.0010 cell 18 0.9998 0.0210 0.80 0.9962 0.0872 0.45 0.278 cell 28 0.1588 0.9873 3.64 0.0804 0.9963 2.60 −0.013 cell 1Table 4.4: Moduli of the components c j of the eigenvector c = (c1,c2),normalized by cHc= 1, and of the normalized components of Kc at somespecific points along the horizontal slice through the phase diagram inFig. 4.15a for the two roots of detM (λ ) = 0 with the largest real part.∆θ indicates the phase shift for the eigenvector c and for Kc betweenthe cells. We observe that the dominant mode of instability alwayscorresponds to the cell at x2 having very small amplitude oscillationsas compared to that for the signaling cell.187(a) u1 j hybrid scheme (b) u1 j FlexPDE(c) u2 j hybrid scheme (d) u2 j FlexPDEFigure 4.17: Same caption as in Fig. 4.9 except that parameters are nowD = 3, σ = 1/2. α1 = 0.4, α2 = 0.9 and d1 j = 1.5 and d2 j = 0.2for j ∈ {1,2} corresponding to the scatter plot in Fig. 4.15a. Theinitial condition imposed was the steady-state with an anti-phaseperturbation: u(0)1 = (u11s,u21s)T + 0.1 · (1,−1)T , and similarly foru(0)2 . The initial bulk solution for (4.5) was U(x,0) = 0. The hybridalgorithm accurately reproduces the wave-packet oscillations of cell 2over long time intervals. The steady-states for cell 1 and cell 2 are theupper and lower black horizontal lines, respectively188we plot Re(λ ) on the left vertical axis for each of the two dominant modes togetherwith our criterion in (4.69b) on the right vertical axes that predicts which cell willhave larger amplitude oscillations near the steady-state. In contrast to the behaviorin Fig. 4.15 where the permeabilities were identical for the two cells, we observefrom the right vertical axes in Figs. 4.18b-4.18c that, as D is increased with σ = 2,the dominant mode of instability is the one for which the cell with α1 = 0.4 nowhas much smaller intracellular oscillations than the cell with α2 = 0.9.2 4 6 80.511.52(a) d11 = 0.4 and d21 = 0.52 4 6 8-0.2-0.100.10.20.960.970.980.991(b) Re(λ ) for one mode2 4 6 8-0.0100.010.020.030.0401234510 -4(c) Re(λ ) for second modeFigure 4.18: Left: Scatter plot after decreasing the influx parameter tod11 = 0.4 and increasing the efflux parameter to d21 = 0.5 for thecell centered at x1 with α1 = 0.4. The number Z of destabilizingeigenvalues for the linearization of the steady-state is: Z = 0 (white),Z = 2 (blue) and Z = 4 (magenta). Other parameters are as forFig. 4.15. Middle: Left vertical axis is Re(λ ) for one mode along thedotted line in the left panel. The right vertical axis is |(Kc)1|2, whichmeasures the relative amplitude of the oscillations near the steady-statefor the signaling cell. Here we normalized ∑2j=1 |(Kc) j|2 = 1. Right:Same as middle plot but for the other dominant mode. Red horizontallines are the stability threshold Re(λ ) = 0.In Fig. 4.19 we plot u2 j versus t at three points with increasing D along theparameter path in Fig. 4.18a. As D increases, the amplitude of the intracellularoscillations in the signaling cell decreases as suggested by Figs. 4.18b-4.18c.However, when D = 1 and σ = 1/2, the signaling cell has much larger amplitudeoscillations than does the other cell. The results in Fig. 4.19 where computed fromour algorithm in §4.5.4 with ∆t = 0.005 for uniformly random perturbations of189Uni f ([−0.1,0.1]) about the steady-state as the initial condition.0 100 200 300 40000.511.522.5cell 1cell 2(a) D = 1, σ = 2.0 100 200 300 40000.511.522.5cell 1cell 2(b) D = 2.5, σ = 2.0 100 200 300 4000.511.522.53cell 1cell 2(c) D = 4, σ = 2.0 100 200 300 40000.511.52cell 1cell 2(d) D = 1, σ = 1/2.Figure 4.19: Top left, top right, bottom left: Intracellular dynamics u2 j(t) atthe three labeled points along the path in Fig. 4.18a. As D increases,the amplitude of oscillations in the signaling cell decreases. Bottomright: u2 j(t) when D = 1 and σ = 1/2 (see Fig. 4.18a), where largeoscillations in the signaling cell occur. Dashed horizontal lines arethe steady-states for u2 j. Parameters: α1 = 0.4, α2 = 0.9, d11 = 0.4,d21 = 0.5, d12 = 1.5, d22 = 0.2.4.6.4 A ring configuration of cellsNext, we consider the four-cell pattern shown in Fig. 4.5b with rc = 1 where thereare two pairs of cells with common kinetic parameters and permeabilities. For thiscell configuration, and where a, b, h, and d can be identified from (4.65b), the190GCEP matrixM in (4.65b) has the formM =a h d hh b h dd h a hh d h b . (4.117)The matrix spectrum ofM is readily obtained as follows:Lemma 4.6.1. For the matrix in (4.117), we havedetM = (a−d)(b−d)[(a+d)(b+d)−4h2] . (4.118)The matrix spectrum forM c = χc isc1 = (1,0,−1,0)T , χ1 = a−d ; c2 = (0,1,0,−1)T , χ2 = b−d ,c± = (1, f±,1, f±)T , χ± = (a+d)+2h f± ; f± :=(b−a)4h±√(b−a)2(4h)2+1 .(4.119a)The proof of this result is straightforward and is omitted.To determine the eigenvalues λ of the linearization of the steady-state, whichsatisfy detM (λ ) = 0, we need only find the union of the roots of the scalar root-finding problems a= d, b= d and 4h2 = (a+d)(b+d). The HB boundaries in the1/σ versus D plane are obtained by setting λ = iλI . For identical cells where a= b,the modes c1 and c2 are degenerate since a= b. For identical cells, where f±=±1,the in-phase and anti-phase modes are c+ = (1,1,1,1)T and c− = (1,−1,1,−1)T ,respectively.For identical cells with d1 j = 0.4, d2 j = 0.2 and α j = 0.9 for j ∈ {1, . . . ,4}, inFig. 4.20a we show the scatter plot, while in Fig. 4.20b we plot the real parts ofthe three dominant eigenvalues of detM (λ ) = 0 along the horizontal slice shownin Fig. 4.20a. In the blue-shaded region, only the in-phase mode is destabilizing,while in the green-shaded region all modes are destabilizing. Moreover, for D =0.4, all the destabilizing modes have comparable growth rates. With this parameterset, Fig. 4.4 shows that no intracellular oscillations would occur when there is no191coupling to the bulk.0.5 1 1.5 2 2.50.511.522.53(a) Four identical cells on a ring0.5 1 1.5 2-0.06-0.04-0.0200.020.04(b) Re(λ ) on horizontal sliceFigure 4.20: Left: Scatter plot of the numberZ of destabilizing eigenvalues,withZ = 0 (white),Z = 2 (blue),Z = 6 (rust) andZ = 8 (green) forthe linearization of the steady-state for four identical cells with a ringradius rc = 1, with d1 j = 0.4 and d2 j = 0.2 for j ∈ {1 . . . ,4}. The HBboundaries (solid black curves) are superimposed. Right: Real part ofthe dominant eigenvalues of detM (λ ) = 0 along the horizontal slicewith σ = 1 in the left panel. Owing to the circulant matrix structurethere is mode degeneracy. In the blue region only the in-phase mode isdestabilizing. For D= 0.4, three destabilizing modes have comparablegrowth rates.In Fig. 4.21 we plot u2 j versus t at the three indicated points shown alongthe parameter path in Fig. 4.20a as computed using the algorithm of §4.5.4 with∆t = 0.005. The choice of initial conditions imposed near the steady-state valuesare indicated in the figure subcaptions. As predicted by the large growth rate ofthe in-phase mode in Fig. 4.20b, the intracellular oscillations become synchronousas time increases for D = 2.4, D = 1.7 and D = 1.2 as shown in the top row inFig. 4.21. Furthermore, for D = 0.4, where the different modes of instability havecomparable growth rates, the bottom row of Fig. 4.21 shows that the long-termdynamics can be established by the precise form of the initial conditions imposed.Next, we consider two pairs of identical cells with α2 = α4 = 0.5, α1 = α3 =0.9, and with the same permeabilities d1 j = 0.4 and d2 j = 0.2 for j ∈ {1, . . . ,4} asin Fig. 4.20a. For this parameter set, from Fig. 4.4 we observe that cells 2 and 4would exhibit limit-cycle oscillations when uncoupled from the bulk. We refer to1920 200 4000.750.80.850.90.95cell 1cell 2cell 3cell 4(a) D = 2.4 (random)0 200 4000.50.60.70.80.9cell 1cell 2cell 3cell 4(b) D = 1.7 (random)0 100 200 300 4000.50.60.70.80.9cell 1cell 2cell 3cell 4(c) D = 1.2 (random)400 450 5000.40.50.60.70.8cell 1cell 2cell 3cell 4(d) D = 0.4 (random)400 450 5000.40.50.60.70.8cell 1cell 2cell 3cell 4(e) D = 0.4 (in-phase)400 450 5000.40.50.60.70.8cell 1cell 2cell 3cell 4(f) D = 0.4 (anti-phase)Figure 4.21: Intracellular dynamics u2 j(t) at the indicated points along thehorizontal slice with σ = 1 in Fig. 4.20a computed using the algorithmin §4.5.4 with ∆t = 0.005. The type of initial condition (IC) isindicated in the captions: Random: perturbing the steady-state valuesfor u1 j and u2 j by 0.1U , where U is uniformly distributed on [−1,1].In-phase: perturbing steady-states by 0.1(1,1,1,1)T . Anti-phase:perturbing steady-states by 0.1(1,−1,1,−1)T . Top left: For D = 2.4,random IC lead to synchronous oscillations that slowly decay to thesteady-state limit. Top middle and top right: For D = 1.7 and D = 1.2,random IC lead to persistent in-phase oscillations. Bottom row: ForD = 0.4, the choice of IC imposed leads to distinctly different long-time dynamics as suggested by Fig. 4.20b. Parameters: α j = 0.9,d1 j = 0.4 and d2 j = 0.2 for j ∈ {1, . . . ,4}.193these two cells as signaling cells. In Fig. 4.22a we show the scatter plot, while inFig. 4.22b we plot the real parts of the four dominant eigenvalues of detM (λ ) = 0along the parameter path shown in Fig. 4.22a. In the magenta-shaded region thedominant instability is where the signaling cells are activated. For D = 0.4, thereare four destabilizing modes. In Fig. 4.23 we plot u2 j versus t at the two indicatedpoints along the parameter path in Fig. 4.22a as computed using the algorithmof §4.5.4 with ∆t = 0.005. For D = 1.75, the non-signaling cells have in-phasesmall amplitude oscillations. However, for D = 0.5, as a result of the number ofdestabilizing modes shown in Fig. 4.22b, the behavior of the oscillation that occursin the non-signaling cells depends on the choice of initial condition near the steady-state that is imposed (see Fig. 4.23b–4.23c).0.5 1 1.5 20.511.522.53(a) α2 = α4 = 0.50.5 1 1.5-0.0500.050.10.150.2(b) Re(λ ) on horizontal sliceFigure 4.22: Left: Scatter plot of the numberZ of destabilizing eigenvalues,with Z = 4 (magenta), Z = 6 (rust) and Z = 8 (green) for thelinearization of the steady-state for two pairs of identical cells withrc = 1 where α2 =α4 = 0.5 and α1 =α3 = 0.9. The HB boundaries aresuperimposed. Same permeabilities d1 j and d2 j as in Fig. 4.20a. Cells2 and 4 admit limit cycle oscillations when uncoupled from the bulk.Right: Real part of the four dominant eigenvalues of detM (λ ) = 0on the horizontal slice with σ = 1 in the left panel. The labels forthe eigenmodes from (4.119a) are c1 = (1,0,−1,0)T (cells 2, 4 silent),c2 =(0,1,0,−1)T (cells 1, 3 silent), and c±=(1, f±,1, f±)T (all cells).There are two destabilizing modes when D = 1.75 and four whenD = 0.5.Finally, we explore the key effect of increasing the efflux to d22 = d24 = 0.5194400 500 60000.511.5cell 1cell 2cell 3cell 4(a) D = 1.75 (random)450 500 550 60000.511.5cell 1cell 2cell 3cell 4(b) D = 0.5 (random)400 450 500 550 60000.511.5cell 1cell 2cell 3cell 4(c) D = 0.5 (anti-phase)Figure 4.23: Intracellular dynamics u2 j(t) at the two indicated points alongthe horizontal slice with σ = 1 in Fig. 4.22a computed using thealgorithm in §4.5.4 with ∆t = 0.005. The choice of initial condition(IC) is indicated (see caption of Fig. 4.23 for details). Left: ForD = 1.75 cells 2 and 4 trigger small amplitude oscillations in cells1 and 3. Middle: For D = 0.5 with a uniformly random IC, thereis a phase shift in the oscillations of the signaling cells. Incoherentoscillations occur in the other cells. Right: For D = 0.5 with anti-phase IC, the signaling cells oscillate in-phase. Cells 1 and 3 exhibitmixed-mode type oscillations.for the signaling cells with α2 = α4 = 0.5, while also increasing the influx tod11 = d13 = 2.0 for the non-signaling cells where α1 = α3 = 0.9. For this modifiedparameter set, from Fig. 4.4 we observe that cells 2 and 4 would no longer exhibitlimit cycle oscillations when uncoupled from the bulk. As a result, we refer tothese cells as de-activated signaling cells. In Fig. 4.24a we show how the scatterplot of Fig. 4.22a is modified, while in Fig. 4.24b we plot the real parts of the fourdominant eigenvalues of detM (λ ) = 0. In contrast to Fig. 4.22b we now observethat the dominant instability at the right-end of the horizontal parameter path whereD = 2 and σ = 4 in Fig. 4.24a is where the de-activated signaling cells 2 and 4are now effectively quiet. In Fig. 4.25 we plot u2 j versus t at the four indicatedpoints in Fig. 4.24a, as computed using the algorithm of §4.5.4 with ∆t = 0.005.For D = 2 and σ = 4, in Fig. 4.25a we observe that the non-signaling cells, cells1 and 3 (which would not oscillate with bulk-cell coupling), have much largeramplitude oscillations than do the de-activated signaling cells, as is consistent with195the prediction from Fig. 4.22b. Moreover, when both D and the degradation σare smaller, i.e., D = σ = 1, Fig. 4.25c shows that the non-signaling cells havea much small amplitude oscillation in comparison with the de-activated signalingcells. For smaller bulk diffusivity, and with a lower degradation rate, the spatialgradient in (4.69a) that occurs in the vicinity of the de-activated cells can be large.As a result of the influx permeability, this signaling gradient can effectively re-activate these cells when D is small. We suggest that this behavior is related to thequalitative mechanism of diffusion-sensing (cf. [59]). By increasing D to D= 1.65,Fig. 4.25d shows that both pairs of cells exhibit only small amplitude oscillations.These distinct time-dependent behaviors clearly illustrate that it is the componentsof the eigenvector c of the GCEP matrix that encodes key predictive informationon which cells will have larger oscillations near an unstable steady-state.0.5 1 1.5 2 2.50.511.5(a) d11 = d13 = 2.0, d22 = d24 = 0.50.5 1 1.5 2-0.2-0.100.10.20.3(b) Re(λ ) on horizontal sliceFigure 4.24: Left: Scatter plot with rc = 1, α2 = α4 = 0.5 and α1 = α3 = 0.9,but now d11 = d13 = 2.0 and d22 = d24 = 0.5. Remaining parametersare d12 = d14 = 0.4 and d21 = d23 = 0.2. The de-activated signalingcells 2 and 4, now have an increased efflux and no longer admitlimit cycle oscillations when uncoupled from the bulk. The influxpermeability for the other two cells is much larger than in Fig. 4.22a.Right: Real part of the four dominant eigenvalues of detM (λ ) = 0 onthe horizontal slice with σ = 0.25 in the left panel.1960 100 200 3000.511.52cell 1cell 2cell 3cell 4(a) D = 2, σ = 4 (random)0 100 200 3000.511.52cell 1cell 2cell 3cell 4(b) D = 1, σ = 4 (random)0 100 200 3000.511.52cell 1cell 2cell 3cell 4(c) D = 1, σ = 1 (random)0 100 200 3000.511.52cell 1cell 2cell 3cell 4(d) D = 1.65, σ = 1 (random)Figure 4.25: Intracellular dynamics u2 j(t) at the four indicated pointsin Fig. 4.24a with uniformly random initial perturbations(Uni f ([−0.1,0.1])). Top left: Synchronized oscillations occurin cells 1 and 3, while the de-activated signaling cells, cells 2 and 4,are quiet. Top right: The de-activated signaling cells oscillate out ofphase, and mixed-mode oscillations occur in the other cells. Bottomleft: Large amplitude in-phase oscillations occur in the de-activatedsignaling cells, with very small in-phase oscillations in the other cells.Bottom right: Both pairs of cells have only small amplitude in-phaseoscillations. Parameters as in Fig. 4.24.1974.6.5 A ring configuration with a center cellNext, we consider the centered-hexagonal cell configuration shown in Fig. 4.5c.We label the ring cells by x j = rc (cos(π( j−1)/3),sin(π( j−1)/3))T for j ∈{1, . . . ,6}, and the center-cell at the origin by x7 = 0. The ring cells are assumedto be identical with d1 j = d1r, d2 j = d2r and α j = αr for j ∈ {1, . . . ,6}. The centercell can have different permeabilities d17 and d27 and a distinct kinetic parameterα7.For this cell configuration, the GCEP matrix M in (4.65b) can be partitionedasM (λ ) =aM6...aa . . . a b , (4.120a)whereM6 ∈ C6,6 is both circulant and symmetric. Here a and b are defined bya := νK0(√σ +λD|x1−x7|),b := 1+ν(log(2√Dσ +λ)− γe)+νDd17(1+2πd27K7) ,(4.120b)with ν = −1/ logε and K7 as given in (4.113). In (4.120a), M6 is written asa cyclic permutation of its first row, labeled by (m1, . . . ,m6), in which m2 = m6and m3 = m5 since M6 is symmetric. The matrix spectrum of M6 consists ofan in-phase eigenvector e6 = (1, . . . ,1)T ∈ R6 and five anti-phase modes, two ofwhich are degenerate. The corresponding anti-phase eigenvectors of M6 are allorthogonal to e6. ForM , these anti-phase modes are lifted to R7 by requiring thatthe center cell is silent. In contrast, the in-phase mode for M6 will lead to twoeigenmodes in R7 for M where the center cell is active. The matrix spectrum ofM is characterized as follows:Lemma 4.6.2. The matrix eigenvalaues forM c = χc, withM as in (4.120a), are198χ j =6∑k=1cos(π j(k−1)3)mk , j ∈ {1, . . . ,5} ,χ± = ω6+a f± , f± :=(b−ω6)2a±√(b−ω6)2(2a)2+6 ,detM = (χ+χ−)(5∏j=1χ j)=(bω6−6a2)( 5∏j=1χ j).(4.121a)Here ω6 is defined by M6e6 = ω6e6 and (m1, . . . ,m6) is the first row of M6. Thecorresponding eigenvectors arec j =(1 ,cos(π j3), . . . ,cos(5π j3),0)T, j ∈ {1,2} ,c3 = (1 ,−1 ,1 ,−1 ,1 ,−1 ,0)T ,c6− j =(0 ,sin(π j3), . . . ,sin(5π j3),0)T, j ∈ {1,2} ,c± = (1, . . . ,1, f±)T ,(4.121b)where c3 is referred to as the sign-alternating anti-phase mode. Since χ j = χ6− jfor j ∈ {1,2}, two pairs of anti-phase modes for which the center cell is silent aredegenerate. The roots λ to detM (λ ) = 0 are obtained from the union of the fivescalar root-finding problems χ1 = 0, χ2 = 0, χ3 = 0 and χ± = 0. Setting χ± = 0,yields the parameter constraint 6a2 = bω6 and f± =∓√6ω6/b.Proof. The proof of this result for χ j and c j for j ∈ {1, . . . ,5} is immediate sinceM6 is a circulant matrix and eT c j = 0 for j ∈ {1, . . . ,5}, where e = (e6,1)T ∈ R7.For the remaining two eigenmodes, we let c = (e6, f )T and calculateM c =(M6e6+a f e66a+b f)= χ(e6f). (4.122)WithM6e6 = ω6e6, we obtain that ω6+a f = χ and 6a+b f = χ f , which leads toa quadratic equation for f . The solution yields (4.121a) and (4.121b) for χ± andf±. Setting χ = 0 in (4.122) we readily obtain the constraint 6a2 = bω6.1991 2 3 4 50.511.522.5(a) signaling cell on1 2 3 40.511.52(b) signaling cell de-activated0.5 1 1.5 2123(c) identical cellsFigure 4.26: Scatter plots for the ring and center cell configuration ofFig. 4.5c with rc = 2. The number of destabilizing eigenvaluesZ for the linearization of (4.5) around the steady-state solution isadditionally coded asZ = 10 (orange),Z = 12 (purple) andZ = 14(gray). The HB boundaries are shown: dashed black curves aredegenerate anti-phase modes, dot-dashed black curve is the sign-alternating anti-phase mode, and solid black curves are the HB modeswhere the center cell is active. Left: Identical ring cells have d1r = 0.8,d2r = 0.2 and αr = 0.9. Center signaling cell is on with d17 = 0.4,d27 = 0.2 and α7 = 0.5. Middle: Same parameters as in the left panelexcept that the efflux from the center cell is increased to d27 = 0.5.The center cell is now de-activated. Right: All cells identical withd1r = d17 = 0.4, d2r = d27 = 0.2 and αr = α7 = 0.9. Each cell wouldbe in a quiescent state without any cell-cell coupling. The Kuramotoorder parameter is computed on the red-dashed line.Fixing d1r = 0.8, d2r = 0.9 and αr = 0.9 for each of the six ring cells, and withinflux permeability d17 = 0.4 and kinetic parameter α7 = 0.5 for the center cell, inFig. 4.26a and Fig. 4.26b we show the scatter plots when the efflux permeabilityis either d27 = 0.2 or d27 = 0.5, respectively. When d27 = 0.2 the center cellis a signaling cell (see Fig. 4.4) and would have intracellular oscillations whenuncoupled from the bulk. However, when d27 = 0.5 this signaling cell is de-activated. In Fig. 4.26c we show the scatter plot for seven identical cells withparameter values d1 j = 0.4, d2 j = 0.2 and α j = 0.9 for j ∈ {1, . . . ,7}. These cellswould all be in a quiescent state without any cell-cell coupling. In generating thescatter plots in Fig. 4.26 we used the analytical expression in (4.121a) for detM to200calculate Z from (6.3). In Fig. 4.26 the HB boundaries for the various modes areshown. Setting λ = iλI , the degenerate anti-phase HB modes occur on the blackdashed curves, for which χ1 = 0 or χ3 = 0, the black dot-dashed curve correspondsto the sign-alternating anti-phase HB mode, for which χ3 = 0, while the solid blackcurves are the HB boundaries where the center cell is active, as obtained fromsetting χ± = 0. The slight raggedness in Fig. 4.26 shows that our winding numberalgorithm has some challenges in correctly calculating Z very close to the HBboundaries where the eigenvalues associated with the degenerate anti-phase modessimultaneously cross the imaginary axis of the spectral plane.In Fig. 4.27 we plot u2 j versus t at the indicated points in the scatter plots ofFig. 4.26, as computed using the algorithm of §4.5.4 with ∆t = 0.005 and withuniformly random perturbations of Uni f ([−0.1,0.1]) about the steady-state as theinitial condition. For D = 2.5 and σ = 2 in the scatter plot of Fig. 4.26a whereall the modes are destabilizing, in Figs. 4.27a–4.27b we show that the ring cellssynchronize on a wave packet solution and that large oscillations occur in the centersignaling cell. By increasing D to D = 4.5 the ring cells now exhibit regular in-phase oscillations but with rather small amplitude. In contrast, when the centersignaling cell has been de-activated, for the values D= 2.5 and σ = 2 in the scatterplot of Fig. 4.26b the results in Figs. 4.27d–4.27e show that the ring cells eventuallysynchronize to a rather large in-phase oscillation in comparison to that of the de-activated center cell. Observe from the HB boundaries in Fig. 4.26b that whenD = 2.5 and σ = 2 only one of the two modes where the center cell is active isdestabilizing. This is different than in Fig. 4.26a where, for D = 2.5 and σ = 2,both modes where the center cell are active are linearly destabilizing. For thepair D = 1 and σ = 5/9 in Fig. 4.26b all anti-phase modes where the center cellis silent are linearly stable, while exactly one mode for which the center cell isunstable and the ring cells are in-phase is unstable. As expected, Fig. 4.27f showsthat the center cell has much larger amplitude oscillations than do the ring cells,which oscillate in-phase. Finally, for identical cells that are all quiescent withoutbulk-cell coupling, the last row of Fig. 4.27 shows the intracellular dynamics at thethree indicated points in Fig. 4.26c. For D= 1 and σ = 4/7 (purple-shaded region),Fig. 4.27h shows that the ring cells have incoherent oscillations while the centercell has a small amplitude regular oscillation. When σ = 1, Fig. 4.27g shows that2010 100 200 30000.511.5c1c2c3c4c5c6c7(a) D = 2.5, σ = 2 (short)550 600 650 70000.511.5c1c2c3c4c5c6c7(b) D = 2.5, σ = 2 (long)550 600 650 70000.511.5c1c2c3c4c5c6c7(c) D = 4.5, σ = 2.0 100 200 3000.511.52 c1c2c3c4c5c6c7(d) D = 2.5, σ = 2 (short)550 600 650 7000.511.52 c1c2c3c4c5c6c7(e) D = 2.5, σ = 2 (long)550 600 650 7000.511.52 c1c2c3c4c5c6c7(f) D = 1, σ = 5/9650 660 670 680 690 7000.50.60.70.80.9c1c2c3c4c5c6c7(g) D = 1, σ = 1650 660 670 680 690 7000.50.60.70.80.9c1c2c3c4c5c6c7(h) D = 1, σ = 4/70 100 200 3000.50.60.70.80.9c1c2c3c4c5c6c7(i) D = 1, σ = 5/14Figure 4.27: Intracellular dynamics u2 j(t) for the ring and center cellconfiguration with rc = 2 computed using the algorithm of §4.5.4 with∆t = 0.005 at the points indicated in Fig. 4.26 using uniformly randomperturbations of Uni f ([−0.1,0.1]) about the steady-state as the initialcondition. The center cell is labeled as C7. Top row: The short andlong-time dynamics for D = 2.5 and σ = 1 in Fig. 4.26a is shown.A synchronous wave-packet solution finally emerges for the ring cellsas t increases. For D = 4.5, the ring cells synchronize to a regularoscillation with a small amplitude.202Figure 4.27: Middle row: For D = 2.5 and σ = 2 (left and middle) inFig. 4.26b the multiple destabilizing anti-phase modes of the ring cellseventually synchronize to a periodic oscillation that is larger than thatof the center cell. For D = 1 and σ = 5/9, the center cell now hasmuch larger amplitude oscillations than do the ring cells. Bottom row:For the identical cells in Fig. 4.26c all cells exhibit roughly in-phaseoscillations with a similar amplitude when D = σ = 1. For D = 1,but with a smaller bulk degradation σ = 4/7, the ring cells show largeamplitude incoherent mixed-mode oscillations (middle). The centercell has a small amplitude regular oscillation. For the even smallervalue σ = 5/14, the mixed-mode oscillations in the ring cells and theregular oscillation of the center cell tend to their linearly stable steady-state values.all cells eventually oscillate in-phase and with a comparable amplitude. When thedegradation is decreased to σ = 5/14, Fig. 4.27i shows that the amplitudes of themixed-mode oscillations for the ring cells decays to the common steady-state limit.4.6.6 Measuring phase synchronizationFor a few cell configurations we now compute the phase synchronization orcoherence of intracellular oscillations based on the Kuramoto order parameterQ(t) :=1NN∑j=1eiθ j , where θ j := arctan(u2 j(t)−u2 jsu1 j(t)−u1 js)∈ (0,2π) . (4.123)Here u2 js and u1 js are the steady-state values from (4.112). To safely disregard theeffect of transients, we compute a time-averaged order parameter Qave defined byQave =1tup− tlow∫ tuptlowQ(t)dt , (4.124)where we chose tlow = 1300 and tup = 1500 in the results below. To use ourmarching algorithm in §4.5.4 with ∆t = 0.005 on the longer time interval t ≤ 1500,we used 114 quadrature points to discretize the Laplace space contour (see Tables4.2 and 4.3).For the ring with center cell configuration of identical cells, with scatter plot as203in Fig. 4.26c, in Fig. 4.28 we plot Qave versus D along the parameter path 0.25 ≤D ≤ 1.25 for σ = 4/7 (red dashed line in Fig. 4.26c), for two choices of initialcondition. For either choice of initial condition, we observe an apparent abrupttransition to complete phase coherence when D is near unity. To illustrate thistransition to phase coherence, for D= 0.75, D= 1 and D= 1.25 in Fig. 4.29 we plotthe trajectories u2 j(t) versus t, as computed from our algorithm with ∆t = 0.005,on both short 700 < t < 750 and long 1450 < t < 1500 time intervals.0.4 0.6 0.8 1 1.200.20.40.60.81(a) random IC0.4 0.6 0.8 1 1.200.20.40.60.81(b) sign-alternating ICFigure 4.28: Averaged order parameter (4.124) for the ring and center cellconfiguration of identical cells in Fig. 4.5c along the red-dashed pathin the scatter plot of Fig. 4.26c with σ = 4/7 and 0.25 < D < 1.25.Black (left panel) and red (right panel) marked points are for eithera uniformly random initial perturbation or a sign-alternating initialperturbation of the steady-state of magnitude 0.1. A phase transitionto phase coherence occurs for D near unity for either choice of IC.Parameters: rc = 2, d1 j = 0.4, d2 j = 0.2 and α j = 0.9 for j∈{1, . . . ,7}.Next, we consider the two-ring configuration of cells as shown in Fig. 4.30a.For the rightmost ring, where the cells are identical with d1 j = 1.5, d2 j = 0.2and α j = 0.9 for j ∈ {1, . . . ,4}, the cell centers are x1 = (4.5,0), x2 = (2.5,2),x3 = (0.5,0) and x4 = (2.5,−2). These cells are in a quiescent state whenuncoupled from the bulk. For the leftmost ring with cell centers x5 = (−0.5,0.0),x6 = (−1.5,1), x7 = (−2.5,0) and x8 = (−1.5,−1), the cells have identicalpermeabilities d1 j = 0.4 and d2 j = 0.2 for j ∈ {5, . . . ,8}, but have the distinctkinetic parameters α5 = 0.6, α6 = 0.45, α7 = 0.4, and α8 = 0.5. As a result,204when decoupled from the bulk, these cells are activated and would exhibit differentfrequencies of intracellular oscillations.From a numerical computation of Qave in (4.124), in Table 4.5 and fromthe intracellular dynamics plotted in Fig. 4.31 we show that an increase in thebulk diffusivity D has the dual effect of increasing both the phase coherence ofthe activated cells in the leftmost ring as well as triggering larger intracellularoscillations for the otherwise quiescent cells in the rightmost ring.DQaveσ = 1 σ = 0.50.5 0.329 0.4271.0 0.419 0.5402.0 0.496 0.6245.0 0.596 0.82610.0 0.763 0.875Table 4.5: Averaged Kuramoto order parameter Qave from (4.124) for thetwo-ring configuration of Fig. 4.30a for various bulk diffusivities D wheneither σ = 1 or σ = 0.5. The IC were the steady-state values. Phasecoherence increases as D increases for fixed σ . Moreover, when the bulkdegradation rate σ is smaller, the bulk signal has a weaker spatial decayand so is more able to promote phase synchronization.Our last numerical experiment is for the centered hexagonal arrangement ofcells shown in Fig. 4.30b with two shells. The generators for the finite lattice aretaken to be l1 =((43)1/4,0)Tand l2 =(43)1/4(12 ,√32)T, so that the area |l1× l2|of the primitive cell is unity, just as for the Bravais lattice in §3.5.We fix d1 j = 0.8 and d2 j = 0.2 for all cells, i.e., for j ∈ {1, . . . ,19} and wechoose the bulk degradation as σ = 1. We will show that a single activated cell atthe center can trigger oscillations in the otherwise quiescent cells that form theinner and outer shells of the truncated lattice, and that the coherence of theseoscillations increases strongly with increasing D. For the center cell we takeα1 = 0.5, so that from Fig. 4.4 it is activated when uncoupled from the bulk. Forthe inactive cells on the inner and outer shells we take α j = 0.7 for j ∈ {2, . . . ,7}and α j = 0.8 for j ∈ {8, . . . ,19}, respectively.In Fig. 4.31 we plot u2 j(t) versus t for three values of the bulk diffusivity D205700 720 7400.50.60.70.8c1c2c3c4c5c6c7(a) D = 0.75 (short)1450 1460 1470 1480 1490 15000.50.60.70.8c1c2c3c4c5c6c7(b) D = 0.75 (long)700 720 7400.550.60.650.70.750.8c1c2c3c4c5c6c7(c) D = 1 (short)1460 1480 15000.550.60.650.70.750.8c1c2c3c4c5c6c7(d) D = 1 (long)700 720 7400.680.70.720.740.760.78c1c2c3c4c5c6c7(e) D = 1.25 (short)1460 1480 15000.680.70.720.740.76c1c2c3c4c5c6c7(f) D = 1.25 (long)Figure 4.29: Intracellular dynamics u2 j(t) for the ring and center cellconfiguration with rc = 2 computed using the algorithm of §4.5.4 with∆t = 0.005 at three values of D along the red-dashed parameter path inFig. 4.5c. Uniformly random initial conditions of magnitude 0.1 nearthe steady-state were used. The center cell is C7. Short- and long-timedynamics are shown. There is a transition to phase coherence nearD = 1.206(a) Eight-cell configuration (b) Centered hexagon with two shellsFigure 4.30: Left: Two non-concentric rings of cells. One ring has identicalcells, while the other ring has cell-dependent parameters. Right: Acentered hexagonal configuration of cells with two shells. Each shellhas identical cells, while the center cell is the pacemaker cell.using our algorithm in §4.5.4 with ∆t = 0.005 and uniformly random perturbationsof Uni f ([−0.1,0.1]) about the steady-state as the initial condition. From this figureand Table 4.6 we observe that there are intracellular oscillations for the cells on theshells that become increasingly coherent as D increases.D Qave0.2 0.1710.3 0.2370.4 0.4980.5 0.6640.6 0.7980.7 0.9350.8 0.9580.9 0.973Table 4.6: Averaged Kuramoto order parameter Qave from (4.124) for thefinite hexagonal lattice configuration of Fig. 4.30b for various bulkdiffusivities D. Uniformly random IC of magnitude 0.1 were used. Phasecoherence increases significantly with D. Parameter values as in thecaption of Fig. 4.32.2074.7 Two Diffusing SpeciesNow suppose both intra-compartmental chemical species can react on or leakthrough the compartment boundary to corresponding passive extracellularlydiffusing and degrading species. The system then becomesbulk∂tU = Du∆U−σuU , t ∈ (0,∞), x ∈ R2\⋃Nj=1 Bε(x j) ,∂tV = Dv∆V −σvV , t ∈ (0,∞), x ∈ R2\⋃Nj=1 Bε(x j) ,reaction fluxesεDu∂n jU = du1 jU−du2 ju1 j , t ∈ (0,∞), x ∈ ∂Bε(x j) ,εDv∂n jV = dv1 jV −dv2 ju2 j , t ∈ (0,∞), x ∈ ∂Bε(x j) ,compartments{du jdt = F j(u j)+e1ε∫∂Bε (x j)(du1 jU−du2 ju1 j) dS+ e2ε∫∂Bε (x j)(dv1 jV −dv2 ju2 j) dS(4.125)for j ∈ {1, ...,N}. Here n j is the outward unit normal vector to the compartmentBε(x j). We will refer to du1 j, dv1 j, du2 j, and dv2 j as dimensionless membrane reactionrates. Both U and V diffusively couple all cells with one another.To obtain an asymptotic ODE system for the intracellular concentrations atall times, we proceed in the same way as in §4.2 and §4.3. Since the interactionbetween the chemical species only takes place inside each cell, one directly obtainsfrom (4.14a) and (4.18a) that the behavior of the transient solution of U locallyaround each compartment with local coordinate ρ := |x−x j|ε just after t = 0 is givenfor the short time scale τ := t/ε2 but for long 1≪ τ ≪ ε−2 byU˜(ρ,τ)∼ u1 j(0)du2 jdu1 j+Buj2πDu(logρ+Dudu1 j),with the transient coupling function Buj given byBuj ∼−u1 j(0)4πDudu2 jdu1 j log(τ/[κu0 je−γe]) +O ([log(τ/(κu0 je−γe))]−2)208in the case u1 j(0) ̸= 0 and byU˜(ρ,τ)∼ ε2τu′1 j(0)du2 jdu1 j+Buj2πDu(logρ+Dudu1 j),where Buj satisfiesBuj ∼−ε2∫ τ04πDudu2 ju′1 j(0)du1 j log(ξ/(κu0 je−γe)) dξ =−ε2 4πDdu2 ju′1 j(0)τdu1 j log(τ/(κu0 je−γe))in the case u1 j(0) = 0. Recall thatκu0 j :=14Due2(γe−Du/du1 j). (4.127)The transient extracellular solution of the second species V is given analogouslywith the obvious relabeling u↔ v.Already at this point we notice that nothing changes in the derivation becausethe intracellular reaction kinetics F j do not need to be used — only the initialintracellular concentrations u1 j(0) and u2 j(0)were needed to obtain the asymptotictransient behavior.Proceeding just as in Section 4.3, we solve, using an extension of strong-localized perturbation theory to a time-dependent system, for the extracellularsolution for U and V that arbitrarily depends on u1 j and u2 j and on thecorresponding coupling terms Buj and Bvj, and we add Buj and Bvj to the unmodifiedreaction kinetics F j. In this way, we can directly write down our resulting integro-ODE system with two diffusing species.Proposition 4.7.1. For ε → 0, and with the initial condition U(x,0) = 0 andV (x,0) = 0, the solution U(x, t), V (x, t) and u j(t), for j ∈ {1, . . . ,N}, to the bulk-cell model (4.125) is approximated for t ≫O(ε2) by the integro-differential system209du jdt= F j(u j)+ e1Buj(t)+ e2Bvj(t) , (4.128a)∫ t0(Buj)′(τ)E1(σu(t− τ))dτ = ηuj Buj(t)+ γuj u1 j(t)+N∑k=1k ̸= j∫ t0Buk(τ)e−σu(t−τ)t− τ e−|x j−xk|2/(4Du(t−τ)) dτ ,(4.128b)∫ t0(Bvj)′(τ)E1(σv(t− τ))dτ = ηvj Bvj(t)+ γvj u2 j(t)+N∑k=1k ̸= j∫ t0Bvk(τ)e−σv(t−τ)t− τ e−|x j−xk|2/(4Dv(t−τ)) dτ ,(4.128c)for j ∈ {1, . . . ,N}. Here ηuj ,ηvj ,γuj , and γvj are defined byηuj := 2(1ν+Dudu1 j+ log(2√Duσu)− γe)=− log(ε2κu0 jσu) , γuj :=4πDudu2 jdu1 j,(4.128d)ηvj := 2(1ν+Dvdv1 j+ log(2√Dvσv)− γe)=− log(ε2κv0 jσv) , γvj :=4πDvdv2 jdv1 j,(4.128e)where κu0 j is defined in (4.127) and similarly κv0 j. In terms of Buj(t) and Bvj(t), theapproximate solution in the bulk region is the superpositionU(x, t)∼− 14πDuN∑j=1∫ t0Buj(τ)e−σu(t−τ)t− τ e−|x−x j|2/(4Du(t−τ)) dτ , (4.129a)V (x, t)∼− 14πDvN∑j=1∫ t0Bvj(τ)e−σv(t−τ)t− τ e−|x−x j|2/(4Dv(t−τ)) dτ , (4.129b)210while in the vicinity of the jth cell we have for ρ = ε−1|x−x j|= O(1) thatU ∼ Buj(t)2πDulogρ+Buj(t)2πdu1 j+du2 jdu1 ju1 j(t) , V ∼Bvj(t)2πDvlogρ+Bvj(t)2πdv1 j+dv2 jdv1 ju2 j(t) .(4.130a)Similarly, for the construction of the steady-state solution, the reaction kineticsare left untouched, so that we arrive at the following proposition.Proposition 4.7.2. For (4.36), suppose that Buj(0) = 0 = Bvj(0) and that Buj(t),Bvj(t), and u j(t) are bounded for t ≥ 0 with limiting values Buj(t)→ Buj∞, Bvj(t)→Bvj∞, and u j → u j∞ as t → ∞ for each j ∈ {1, . . . ,N}. Then, Buj∞, Bvj∞ and u j∞satisfy the N(m+1) dimensional nonlinear algebraic system (NAS)F j(u j∞)+ e1Buj∞+ e2Bvj∞ = 0 , j ∈ {1, . . . ,N} , (4.131a)ηuj Buj∞+ γuj u1 j∞+2N∑k=1k ̸= jBuk∞K0(√σuDu|x j−xk|)= 0 , j ∈ {1, . . . ,N} ,(4.131b)ηvj Bvj∞+ γvj u2 j∞+2N∑k=1k ̸= jBvk∞K0(√σvDv|x j−xk|)= 0 , j ∈ {1, . . . ,N} ,(4.131c)where u1 j∞ = eT1 u1 j∞ and u2 j∞ = eT2 u2 j∞ are the first and second component ofthe intracellular species. In (4.131), ηuj ,ηvj ,γuj , and γvj are defined in (4.128d) and(4.128e).Next, to recover the stability properties of the steady-states and to findbifurcation branches to other symmetry-broken steady-states or to oscillatorystates, introduce a generic eigenperturbation to eigenvalue λ about the steady-state(Us,Vs,u1s, ...,uNs) asU =Us(x)+eλ tΦ(x), V =Vs(x)+eλ tΨ(x), u j = u js+eλ tζ j, j ∈ {1, ...,N}.Plugging this into (4.7.2) just as in §4.4, we notice that the prefactor c j of Φ in(4.59) is the coupling factor for the eigenperturbation solution, just as B j(t) is211the coupling function for the general dynamics in (4.36). Thus, Proposition 4.4.3only needs to be slightly modified, to account for the possibly different diffusing,degrading and on the cell-membranes reacting species U and V . We obtain theeigenvalue problem∆Φ− (λ +σu)DuΦ= 0 , x ∈ R2 \∪Nj=1 Bε(x j) ,∆Ψ− (λ +σv)DvΨ= 0 , x ∈ R2 \∪Nj=1 Bε(x j) ,εDu ∂nΦ= du1 jΦ−du2 j ζ1 j , x ∈ ∂Bε(x j) , j ∈ {1, . . . ,N} ,εDv ∂nΨ= dv1 jΨ−dv2 j ζ2 j , x ∈ ∂Bε(x j) , j ∈ {1, . . . ,N} ,which is coupled to the linearized intracellular dynamics within the jth cell byλζ j = J jζ j +e1ε∫∂Bε (x j)(du1 jΦ−du2 j ζ1 j)dSx+e2ε∫∂Bε (x j)(dv1 jΨ−dv2 j ζ2 j)dSx , j ∈ {1, . . . ,N} .Here J j := Du j F j is the Jacobian of the intracellular kinetics evaluated at u j = u js.The outer solutionΦ(x)=−N∑k=1cuj2πDuK0√σu+λDu|x−xk| , Ψ(x)=− N∑k=1cvj2πDvK0√σv+λDv|x−xk|(4.133)is easily obtained, along with(λ I− J j)ζ j = e1cuj + e2cvj, j ∈ {1, ...,N}, (4.134)so that the intracellular eigenperturbation magnitudes are given byK jc j with K j := (λ I− J j)−1 and c j := (cuj ,cvj).These encode much information about the specific eigenperturbation and can beused for recovering the relative magnitude and phase shifts to the oscillationscaused by the eigenperturbation (causing in-phase, anti-phase, etc. oscillations)in a certain scenario determined by the parameter set (see the explanation after212Proposition 4.4.3).The perturbation asymptotic behavior obtained from the local solution as eachcell is approached isΦ∼ cuj2πDulog |x−x j|+cuj2πDuν+cuj2πdu1 j+du2 jdu1 jζ1 j , as x→ x j ,Ψ∼ cvj2πDvlog |x−x j|+cvj2πDvν+cvj2πdv1 j+dv2 jdv1 jζ1 j , as x→ x j .Matching this behavior to the behavior of the global solution (4.133) gives2(1ν+Dudu1 j− γe+ log(2√Duσu+λ))cuj +4πDudu2 jdu1 jζ1 j+2N∑k=1k ̸= jcukK0√σu+λDu|x j−xk|= 0 , j ∈ {1, . . . ,N} , (4.135a)and2(1ν+Dvdv1 j− γe+ log(2√Dvσv+λ))cvj +4πDvdv2 jdv1 jζ2 j+2N∑k=1k ̸= jcvkK0√σv+λDv|x j−xk|= 0 , j ∈ {1, . . . ,N} . (4.135b)Putting both equations (4.135a) and (4.135b) together in matrix form, we obtainthe following proposition.Proposition 4.7.3. For ε → 0, the discrete eigenvalues λ associated with thelinearization around a steady-state solution to (4.125), for which det(λ I− J j) ̸= 0for any j ∈ {1, . . . ,N}, are the set of valuesΛ(M ) := {λ | detM (λ ) = 0} , (4.136a)213where the N×N dimensional matrixM (λ ) is defined byM (λ ) :=(Mu(λ ) Hu(λ )Mv(λ ) Hv(λ ))(4.136b)withMu(λ ) := I+2πνG uλ +νDu Pu1 +2πνDuPu2 K11(λ ) , (4.136c)Mv(λ ) := I+2πνG vλ +νDv Pv1 +2πνDvPv2 K22(λ ) , (4.136d)Hu(λ ) := 2πνDuPu2 K12(λ ) , (4.136e)Hv(λ ) := 2πνDvPv2 K21(λ ) , (4.136f)(4.136g)with ν =−1/ logε . In (4.136c), Pu1 , Pu2 and K12 are the diagonal matricesPu1 := diag( 1du11, . . . ,1du1N), Pu2 := diag(du21du11, . . . ,du2Ndu1N), (4.137a)K12(λ ) := diag(K1,12, . . . ,KN,12) , where K j,12 := eT1 (λ I− J j)−1 e2 ,(4.137b)and Pv1 , Pv2 , K11, K21, and K22 are defined analogously. In addition, Guλ is theeigenvalue-dependent Green matrix with matrix entries(G uλ )i j = (Guλ ) ji :=12πK0√σu+λDu|x j−xk| , i ̸= j , (4.138)(G uλ ) j j = Ruλ j :=12π(log(2√Duσu+λ)− γe), (4.139)and G vλ is analogously defined by relabeling u↔ v. For any specific λ0 ∈ Λ(M ),we have detM (λ0) = 0, and so M (λ0)c = 0 has a nontrivial solution c =(cu,cv)T = (cu1, . . . ,cuN ,cv1, . . . ,cvN)T that can be normalized as |cu| = 1 = |cv|. Weconclude that the steady-state is linearly stable if, whenever λ ∈ Λ(M ), we haveRe(λ )< 0.It is curious that the matrix blocks ofM (λ ) in this proposition are differently214defined to the ones obtained for the finite 2-D disk domain with Neumann boundaryconditions in (3.43). However, besides the different Green matrices, there is notreally a difference between the matrix equationM (λ )c = 0 of the last Proposition4.7.3 and the one in (3.43). The illusion of a difference between the equationsstems from the different scaling of cuj and cvj in (4.134) compared to the one withDu and Dv in (3.35).The above Propositions 4.7.1, 4.7.2 and 4.7.3 can now readily be used to findsymmetry-breaking bifurcations from (not necessarily only symmetric) steady-states and to find transitions to synchronization behaviors just as in Chapter 3 andin this chapter before this section. It would be highly interesting to find whetherand how transitions from cell-group oscillations to a patterned steady-state and viceversa can be induced by reaction kinetic parameter variation of a single (defectingor activating) cell in the cell colony.4.8 DiscussionThe hybrid asymptotic-numerical theory developed herein has provided newtheoretical and computationally efficient approaches for studying how oscillatorydynamics associated with spatially segregated dynamically active “units” or “cells”are regulated by a PDE bulk diffusion field that is produced and absorbed bythe entire population of cells. For the bulk-cell ODE-PDE system (4.5) weextended the strong localized perturbation theory, surveyed in [144], to a time-dependent setting in order to derive a new integro-differential ODE system thatcharacterizes intracellular dynamics in a memory-dependent bulk-diffusion field.For this nonlocal system, a novel fast time-marching scheme, which relies onthe sum-of-exponentials method of [66] and [12], was developed to numericallycompute solutions over long time intervals.For the special case of Sel’kov reaction kinetics, we have used our hybridapproach to investigate the effect that the influx and efflux permeability parameters,the bulk degradation rate and bulk diffusivity, and the reaction kinetic parametershave in both triggering intracellular oscillations and in mediating oscillatorsynchronization for the entire collection of cells. Quorum and diffusion-sensingbehavior were illustrated for various cell configurations. Comparisons of numerical215results from our fast time-marching algorithm for the integro-differential ODEsystem with the numerical results for the full bulk-cell system (4.5) computedusing the commercial solver FlexPDE [39] have shown that our hybrid asymptotic-numerical theory accurately reproduces, within roughly a minute of CPU timeon a laptop, the intricate mixed-mode oscillatory dynamics that can occur for thefull system (4.5) over long-time intervals. From a computation of the Kuramotoorder parameter, we have exhibited an apparent phase transition to complete phasecoherence for a hexagonal arrangement of identical cells (see Fig. 4.28). Thesecells would not exhibit oscillations without any inter-cell coupling.Although we have only illustrated and carefully benchmarked our theory for asmall or moderate number of cells, with our fast time-marching scheme, possiblyapplied to Fitzhugh-Nagumo kinetics, it is viable numerically to compute targetor spiral wave patterns of oscillatory dynamics, or to seek to identify chimerastates, associated with a large N ≥ 100 collection of cells. We emphasize thatour integro-differential ODE system can also be used for specific multi-componentreaction kinetics such as those modeling the initiation and synchronization ofglycolysis oscillations (see [58] for a recent survey) or for those modeling quorum-sensing transitions between bistable states that are believed to be responsible forbioluminscence phenomena for the marine bacterium Aliivibrio fischeri (cf. [130],[96], [117] see also the references therein).Moreover, our new theoretical and computational approaches for first derivingand then computing solutions to integro-differential systems of ODE’s for bulk-cellmodels with one diffusing species can readily be extended to R3, as is done in thenext Chapter 5, and to the case of finite domains Ω ∈Rd with d ∈ {1,2,3}. In a 1-D setting, steady-states and their linear stability properties for bulk-cell modelsconsisting of discrete oscillators located at either the domain boundaries or atinterior points in the domain, and which are coupled by a bulk diffusion field, havebeen studied in [49], [50] and [104] (see also the references therein). Our approachis also readily extended to bulk-cell models with two bulk diffusing species, suchas in Chapters 2 and 3, where the focus was mainly on describing bifurcationand pattern-formation properties of the model. This corresponding extension ofthe theory has been done in §4.7 of this chapter. Bulk-cell models, allowingfor a spatially uniform bulk drift velocity as in [140], should also be tractable216to analyze with our approach. The main features that we require to derive thefast time-marching algorithm is an exact analytical representation of the Laplace-transformed bulk diffusion field and that this representation has singularities onlyalong the negative real axis in the Laplace transform space. These properties areinherent for Green’s functions for heat-type equations with bulk degradation ineither bounded or unbounded domains.Finally, we briefly remark on two numerical challenges that warrant furtherinvestigation. Firstly, the primary numerical errors in our time-marching schemeare due to the ETD2 discretization of the Duhamel integrals and the Forward-Euler discretization in (4.105) for the dynamics. By increasing the truncationorder of this aspect of our algorithm it should be possible to take larger time stepsthan we have done while still preserving the same level of accuracy. Secondly,the computation of roots of the GCEP that determines the HB boundaries andoscillation frequencies near the steady-state becomes highly challenging when thenumber of cells becomes large. As a result, efficient numerical methods to computesolutions to nonlinear matrix eigenvalue problems that are complex-symmetric (butnot Hermitian) are required.2171350 1400 1450 150000.511.52(a) D = 0.5 (all cells)1350 1400 1450 15000.40.450.50.550.6(b) D = 0.5 (rightmost ring cells)1350 1400 1450 150000.511.52(c) D = 1 (all cells)1350 1400 1450 15000.30.40.50.60.70.8(d) D = 1 (rightmost ring cells)1350 1400 1450 150000.511.52(e) D = 5 (all cells)1350 1400 1450 15000.40.50.60.70.80.9(f) D = 5 (rightmost ring cells)Figure 4.31: Intracellular dynamics u2 j(t) for the two-ring configuration ofcells in Fig. 4.30a computed using the algorithm of §4.5.4 with ∆t =0.005 for three values of D. The left panels show the dynamics in allthe cells, identified according to the colors in Fig. 4.30a. The rightpanels show the intracellular dynamics for the cells in the rightmostring, where the solid cyan curve is for the rightmost cell x1. With acounterclockwise orientation, solid-dashed, dashed-dotted and dottedcurves identifies the dynamics in the cells x2, x3 and x4, respectively.218Figure 4.31: Top row: For D = 0.05, the leftmost ring cells show little phasecoherence (left panel), and the only cell in the rightmost ring thatexhibits oscillations is x3 (which is the cell closest to the leftmostring). Middle row: For D = 1 there is enhanced phase coherence inthe left ring cells, and all cells in the rightmost ring have significantoscillations. Bottom row: For D = 5, the cells on the left ring showsignificant phase coherence, while the rightmost ring cells exhibit in-phase mixed-mode oscillations. Parameter values are given in the text.219(a) D = 0.3 (short time) (b) D = 0.3 (long time)(c) D = 0.5 (short time) (d) D = 0.5 (long time)(e) D = 0.7 (short time) (f) D = 0.7 (long time)Figure 4.32: Short- (left panels) and long-time (right panels) intracellular dynamicsu2 j(t) for the finite lattice configuration in Fig. 4.30b as computed using thealgorithm of §4.5.4 with ∆t = 0.005 and for three values of D. The cell at thecenter and on the two shells are identified by the colours in Fig. 4.30b. Onlythe center cell is activated when uncoupled from the bulk. Uniformly randomIC of magnitude 0.1 around the steady-state intracellular values were given.Phase coherence increases substantially near D = 0.5. Parameters: σ = 1,d1 j = 0.8, d2 j = 0.2 for j ∈ {1, . . . ,19}, α1 = 0.5, α j = 0.7 for j ∈ {2, . . . ,7}and α j = 0.8 for j ∈ {8, . . . ,19}.220Chapter 5Extensions: Theory for R35.1 IntroductionAssuming a 3-D spherical shape of the cell may be more realistic in many casesthan assuming a 2-D disk-shape that can be viewed as a cell at fixed positionwithout any protrusions flat in a petri dish. While we still assume that the separatedcells of small radius ε are immobile and are perfectly round, the 3-D shape causesa change in the relevant scaling of the kinetics in the compartmental-reactiondiffusion system. It turns out that the efflux of the intracellular species that causesthe diffusive coupling has to be much larger (O(ε−1)) than in the 1-D and 2-D case(O(1)) in order for the cell-coupling to come in at leading order in the asymptoticsolution construction. This is due to the shape of the local (inner) solution in 3-D. Thus, the O(ε−1)-scaling of the efflux is the realistic scaling in the 3-D setting.Further, the 3-D Green function that appears in the coupling has a faster time-decaythan the 2-D one. Hence, we expect that the cells are more weakly coupled in the3-D setting.Pattern forming properties of reaction-diffusion (RD) systems in 3-D have beenmuch less theoretically analyzed than the 1-D and 2-D counterparts, and mainlynumerical studies have been undertaken in 3-D (cf. [120], [103]). More theoreticalstudies of the symmetry breaking properties are [139] for the 3-D Schnakenbergmodel and [45] for the 3-D Gierer-Meinhardt model. Besides those, carefullyengineered chemical microemulsion experiments led to 3-D pattern formation in221capillaries (cf. [7], [40]).In this chapter we are deriving a 3-D compartmental-reaction diffusion systemwhose finite domain version can be used to analyze pattern formation of thesteady-state along the lines of our 2-D analysis in Chapter 3. However, insteadof following the steps of this steady-state analysis, we are directly going overto the construction of the time-dependent asymptotic integro-differential systemfor the coupled intracellular chemical concentrations at all times for arbitraryintracellular reaction kinetics, as was done in Chapter 4 in 2-D. Thus, we againtake the Kuramoto oscillator viewpoint and regard our system as a system withdiffusively coupled synchronizing oscillators. Besides the still relevant referenceson quorum and diffusion sensing already mentioned in the previous Chapter 4, thework on cluster synchronization in 3-D lattices of (discretely) diffusively coupledoscillators in [8] should be mentioned and on 3-D chimera patterns in networksof spiking neuron oscillators in [67], in which a coupling matrix with nonlocaltopology is derived.Figure 5.1: Left: A schematic diagram showing dynamically active signalingcompartments (in cyan) in 3-D. Right: A cross section of one 3-Dcompartment with illustration of the secretion of signaling moleculesinto the bulk region and the feedback of chemical into the cells. Thegreen and red dots represent the signaling chemicals in the cells, whereonly the red is secreted into the extracellular bulk region.We now formulate the bulk-cell model with one bulk species in R3 withN dynamically active cells. The cells are assumed to be a collection ofnon-overlapping spheres of a common radius R0 and are labeled as Ω j :={X | |X−X j| ≤ R0} for j ∈ {1, ...,N}. In the bulk, or extracellular, region R3 \222∪Nj=1Ω j, the bulk concentration U (X,T ) is assumed to satisfyUT =DB∆U − kBU , T > 0 , X ∈ R3 \∪Nj=1Ω j , (5.1a)DB ∂nXU = β1 jU −β2 j µ1 j , x ∈ ∂Ω j , j ∈ {1, ...,N} , (5.1b)where DB and kB are the bulk diffusivity and bulk degradation, respectively. Theinflux and efflux into and out of the jth cell are regulated by the permeabilitiesβ1 j > 0 and β2 j > 0. In (5.1b), ∂nX denotes the outer normal derivative pointinginto the bulk region.Within each cell we assume that m intracellular species µ j = (µ1 j, . . . ,µm j)Tinteract with reaction kinetics F j, and we neglect any spatial gradients within thecells. Moreover, we assume that the cell boundary (membrane) is either permeableto one species, labeled by µ1 j, or that this species reacts to an extracellularlydiffusing and degrading species on the membrane which can react on the membraneback to the intracellular species µ1 j as well. The intracellular dynamics in the jthcell are coupled to the bulk diffusion field (5.1) via an integration of the diffusiveflux across the boundary of the cell asdµ jdT= kR µc F j (µ j/µc)+ e1∫∂Ω j(β1 jU −β2 j µ1 j) dSX , j ∈ {1, ...,N} ,(5.1c)where e1 := (1,0, . . . ,0)T . In this model, kR > 0 is the dimensional intracellularreaction rate, µc > 0 is a typical magnitude of the vector µ j, β2 j is the efflux rateof µ1 j per unit area of the cell surface, and the influx parameter β1 j regulates thefeedback into the jth cell from the bulk diffusion field that is generated by the entirecollection of cells. A schematic plot of the bulk-cell model is shown in Fig. 5.1 forthe case of two intracellular species.We will analyze the bulk-cell model in the asymptotic limit where the commonradius R0 of the signaling compartments is small relative to the minimum inter-cellseparation distance L = min |Xi−X j| for i ̸= j. As such, we introduce a smallparameter by ε := R0/L ≪ 1. In Appendix A.5 we non-dimensionalize the bulk-223cell model (5.1) to obtain that the dimensionless bulk field U(x, t) satisfies∂tU =D∆U−σU , t > 0 , x ∈ R2 \∪Nj=1Ωε j ; U(x,0) = 0 , (5.2a)εD∂nU = d1 j U− d2 jε u1 j , x ∈ ∂Ωε j , j ∈ {1, ...,N} . (5.2b)This bulk diffusion field is coupled to the dimensionless intracellular species u j =(u1 j, . . . ,um j)T within the jth cell bydu jdt= F j (u j)+e1ε∫∂Ωε j(d1 j U− d2 jε u1 j)dSx , j ∈ {1, ...,N} . (5.2c)In (5.2), the small signaling cells are labeled by Ωε j := {x | |x−x j| ≤ ε}, and weassume that they are spatially segregated in the sense that dist(x j,xk) = O(1) forj ̸= k. The dimensionless parameters in the bulk-cell model (5.2) areD :=DBkRL2, d1 j := εβ1 jkRL=O(1) , d2 j := ε2β2 jL2kR=O(1) , σ :=kBkR, (5.3)where D and σ are referred to the effective bulk diffusivity and effective bulkdegradation parameter, respectively. In (5.3), the permeability parameters β1 j andβ2 j are assumed to have a different scaling in ε in order to ensure that the smallcells contribute an O(1) diffusive flux into the bulk medium.In §5.2 we derive the behavior of the transient solution of (5.2), i.e., thesolution shortly after t = 0, whose long-time behavior will be matched with thesolution in the standard (O(1)) time-scale. The integro-ODE system for the lattersolution is derived in §5.3, and its steady-state and linear stability analysis is donein §5.4, in which a perturbation theory of the linear stability analysis leads tofurther insight into to which order terms of our original system (5.2) influencethe stability properties of the steady-state and thus the transition to oscillatorysynchronizing states. More specifically, this analysis shows that the effect of thebulk-cell coupling in 3-D is much weaker than the O(1)-effect in the 2-D case.Finally, a short discussion of the results is provided in §5.5.2245.2 Transient Analysis Near the CellsWe begin by studying the transient solution near the jth cell on the time-scalet = O(ε2) when U(x,0) = 0. As such, we introduce τ by t = ε2τ . The far-fieldbehavior as τ → ∞ of this transient solution provides the initial conditions for thelong-time dynamics analyzed below §5.3.We first consider the case where u1 j(0) ̸= 0. Near the jth cell, we introducethe local coordinates y = ε−1(x− x j) with ρ = |y|. From (5.2), we obtain thatV (ρ,τ) =U(x j + εy,ε2τ) satisfies the leading-order local problemVτ = D(Vρρ +2ρVρ), ρ > 1 , τ ≥ 0; V (ρ,0) = 0 ,DVρ = d1 jV − d2 jε u1 j(0) , on ρ = 1 .(5.4)To determine V (ρ,τ) we take the Laplace transform of (5.4) in τ and solve theresulting boundary value problem. This yields that Vˆ (ρ,s) =L (V (ρ,τ)) satifiesVˆ (ρ,s) = u1 j(0)d2 jρεe−√s/D(ρ−1)s(d1 j +D+√sD) , (5.5)where the branch cut is taken along the negative real axis {s ∈ C : Re(s) ≤ 0 andIm(s) = 0}.To determine the far-field behavior of V (ρ,τ) for τ ≫ 1 and ρ fixed, we mustcalculate Vˆ (ρ,s) near s= 0, which corresponds to the rightmost edge of the branchcut. By letting s→ 0, we readily calculate from a Taylor series approximation thatVˆ (ρ,s)∼ d2 ju1 j(0)ερ(d1 j +D)(1ρ[1s+1√sD(d1 jd1 j +D)]− 1√sD), as s→ 0 .(5.6a)Upon recalling that L −1(s−1/2)= 1/√πτ , we invert (5.6a) to obtain for ρ > 1fixed and τ → ∞ thatV (ρ,τ)∼ d2 ju1 j(0)ε(d1 j +D)(1ρ[1+d1 j(d1 j +D)√πDτ]− 1√πDτ). (5.6b)Proceeding in a similar way for the case where u1 j(0) = 0, we evaluate (5.2c)225at t = 0 and use U(x,0) = 0 and u1 j(0) = 0 to conclude for τ = O(1) thatu1 j ∼ ε2τu′1 j(0) , where u′1 j(0) = eT1 F j(u j(0)) . (5.7)Upon replacing u1 j(0) in the boundary condition on ρ = 1 in (5.4) with ε2τu′1 j(0)we derive, in place of (5.5), thatVˆ (ρ,s) = u′1 j(0)d2 jρεe−√s/D(ρ−1)s2(d1 j +D+√sD) , (5.8)By expanding (5.8) as s→ 0, and usingL −1(s−2) = τ andL −1(s−3/2) = 2√τ/π ,we obtain for ρ =O(1) and τ → ∞ thatV (ρ,τ)∼ εd2 ju′1 j(0)τ(d1 j +D)(1ρ[1+2√πDτd1 j(d1 j +D)]− 2√πDτ), (5.9)where u′1 j(0) = eT1 F j(u j(0)).The far-field behaviors (5.6b) and (5.9) of the transient solution near thejth cell, which are valid for 1 ≪ τ ≪ ε−2 and ρ = O(1), will be used forasymptotically matching the transient solution to the long-time dynamics studiedin §5.3.5.3 Derivation of ODE System for Intracellular KineticsWe now derive an integro-ODE system for the intracellular species that is valid onlong time scales t ≫ O(ε2).For t = O(1), in the jth inner region we introduce the local coordinatesy = ε−1(x− x j), ρ = |y|, and V (y, t) = U(x j + εy, t). From (5.2a), we obtainthe leading-order quasi-steady state problem∆yV = 0 , for ρ > 1; DVρ = d1 jV − d2 jε u1 j , on ρ = 1 , (5.10)which has the radially symmetric solutionV (y, t) =A j(t,ε)ρ+B j(t,ε) , (5.11)226where A j and B j are to be determined. We expand A j and B j asA j(t,ε) = ε−1A0 j(t)+A1 j(t)+O(ε) , B j(t,ε) = B0 j(t)+O(ε) . (5.12)We substitute (5.11) and (5.12) into the boundary condition on ρ = 1 in (5.10) andcollect powers of ε to obtain−DA0 j = d1 jA0 j−d2 ju1 j , −DA1 j = d1 j(A1 j +B0 j) . (5.13)Upon solving for A0 j and A1 j we obtain the quasi-steady inner solutionV ∼ 1ερ(A0 j + εA1 j)+B0 j +O(ε) , (5.14a)whereA0 j =d2 ju1 jd1 j +D, A1 j =− d1 jB0 jd1 j +D . (5.14b)Next, we substitute (5.14a) into the intracellular dyamics (5.2c) to obtaindu jdt−F j (u j) = e1ε∫∂Ωε jDVρ |ρ=1 dSx = e1ε∫∂Ωε jD(−A j0ε−A1 j)dSx , (5.15)for each j ∈ {1, ...,N}. Then, by using |∂Ωε j | = 4πε2 and (5.14b), we concludethat the intracellular dynamics is determined in terms of the unknown B0 j(t) bydu jdt= F j (u j)−κ ju1 je1+ εd1 jd2 j κ jB0 je1 , where κ j :=4πDd2 jd1 j +D. (5.16)We now determine B0 j(t) by matching the inner solutions near the cells to anouter bulk solution. Upon writing (5.14a) in outer variables, by using ερ = |x−x j|,we obtain that the bulk solution U(x, t) satisfies∂tU = D∆U−σU , x ∈ R3\{x1, . . . ,xN} ; U(x;0) = 0 ,U ∼ b j4πD1|x−x j| +B0 j +O(ε) , as x→ x j , j ∈ {1, ...,N} ,(5.17a)227where b j(t) is defined byb j(t) := κ ju1 j(t) . (5.17b)We impose that U(x, t)→ 0 as |x| → ∞ for each t > 0.To determine B0 j(t), we take the Laplace transform of (5.17) to obtain forUˆ(x,s) =L (U(x, t) that∆Uˆ− (σ + s)DUˆ =−N∑k=1bˆkDδ (x−xk) ; Uˆ → 0 as |x| → ∞ ,Uˆ ∼ bˆ j4πD1|x−x j| + Bˆ0 j , as x→ x j , j ∈ {1, ...,N} .(5.18)The solution to (5.18) isUˆ =N∑k=1bˆk4πD|x−xk|e−|x−xk|√(σ+s)/D . (5.19)By expanding Uˆ as x → x j, and enforcing that the limiting behavior agrees withthat required in (5.18), we determine Bˆ0 j asBˆ0 j =− bˆ j√σ + s4πD3/2+N∑k=1k ̸= jbˆk4πD|x j−xk|e−|x−xk|√(σ+s)/D . (5.20)To determine B0 j(t) =L −1(Bˆ0 j), we first use the inverse transformL −1[e−|x j−xk|√(σ+s)/D4πD|x j−xk|]=G(a jk, t) :=e−σte−a2jk/(4t)(4πDt)3/2where a jk :=|x j−xk|√D.(5.21)Then, from the convolution property of the Laplace transform, we getB0 j =− 14πD3/2L−1 [bˆ j√s+σ]+ N∑k=1k ̸= j∫ t0bk(τ)G(a jk, t− τ)dτ . (5.22)Finally, we must invert the remaining term in (5.22). By adding and subtracting228a term, and recalling (5.17b), we obtainL −1[bˆ j√s+σ]= κ jL −1[(suˆ1 j−u1 j(0))√s+σs]+κ ju1 j(0)L −1(√s+σs).(5.23)By using the two inverse transformsL −1(√s+σs)= g(t) :=e−σt√πt+√σ erf(√σt), L −1 (suˆ1 j−u1 j(0))= u′1 j(0) ,(5.24)(5.23) can be calculated explicitly using the convolution formula. Uponsubstituting the resulting expression into (5.22) we obtain B0 j(t) as needed in(5.16). Overall, this leads to an integro-differential system for the intracellularconcentrations u j(t) that is coupled to the time-history of the source strengthsB j(t). We summarize our result in the following formal proposition.Proposition 5.3.1. For ε → 0, and with the initial condition U(x;0) = 0, thesolution U(x, t) and u j(t), for j ∈ {1, ...,N}, to the bulk-cell model (5.2) isapproximated for t ≫ O(ε2) by the integro-differential systemdu jdt= F j(u j)−κ ju1 je1+ εd1 jd2 j κ jB0 j(t)e1 , (5.25a)B0 j(t) =−κ ju1 j(0)4πD3/2 g(t)−κ j4πD3/2∫ t0u′1 j(τ)g(t− τ)dτ+N∑k=1k ̸= jκk∫ t0u1k(τ)G(a jk, t− τ)dτ , (5.25b)whereg(t) :=e−σt√πt+√σerf(√σt), G(a jk, t) :=e−σte−a2jk/(4t)(4πDt)3/2. (5.25c)Here erf(z) is the error function, while κ j and a jk are defined bya jk :=|x j−xk|√D, κ j :=4πDd2 jd1 j +D. (5.25d)229In terms of B0 j(t), the asymptotic solution in the vicinity of the jth cell, with ρ =ε−1|x−x j|= O(1), isU ∼ d2 jερ (d1 j +D)(u1 j(t)− εd1 jd2 j B0 j(t))+B0 j(t)+O(ε) . (5.26)Moreover, the asymptotic solution in the bulk region isU(x, t)∼N∑k=1κk∫ t0u1k(τ)G(a jk, t− τ)dτ . (5.27)5.3.1 Matching to the transient solutionWe now show that the limiting behavior as t → 0+, but with t ≫ O(ε2), for B0 j(t)in (5.25), agrees with the far-field behavior of the transient solution as derivedin (5.6b) and (5.9) of §5.2. The analysis below is based on taking the Laplacetransform of (5.25b) and letting s→ ∞.We will first analyze the case where u1 j(0) ̸= 0. From (5.25b), we obtain using(5.25d) for κ j thatB0 j(t)∼− d2 ju1 j(0)(d1 j +D)√πDt, as t → 0 . (5.28)Upon substituting (5.28) into (5.26), we conclude for t → 0+ thatU ∼ d2 ju1 j(0)ερ (d1 j +D)(1+εd1 j(d1 j +D)√πDt)− d2 ju1 j(0)(d1 j +D)√πDt. (5.29)Recalling that t = ε2τ , we observe that (5.29) matches identically with the far-fieldbehavior of the transient solution given in (5.6b).Next, we consider the case where u1 j(0) = 0. By setting u1 j(0) = 0 in (5.25b)and taking the Laplace transform we obtainBˆ0 j(s) =− κ j4πD3/2 suˆ1 j√s+σs+N∑k=1k ̸= jκkuˆ1k4πD|x j−xk|e−|x j−xk|√(σ+s)/D . (5.30)230Since u1 j(t) ∼ u′1 j(0)t as t → 0+, we have uˆ1 j ∼ u′1 j(0)/s2 as s → ∞. Therefore,by letting s→ ∞ in (5.30), and using (5.25d) for κ j, we conclude thatBˆ0 j(s)∼− d2 j(d1 j +D)√Du′1 j(0)s3/2, as s→ ∞ . (5.31)From the inverse Laplace transform of (5.31) we getB0 j(t)∼− 2d2 j(d1 j +D)u′1 j(0)√tπD, as t → 0+ . (5.32)We substitute (5.32) into (5.26) where we use u1 j(t)→ tu′1 j(0). This yields nearthe jth cell thatU ∼ td2 ju′1 j(0)ερ (d1 j +D)(1+2ε√πDtd1 jd1 j +D)− 2d2 ju′1 j(0)(d1 j +D)√tπD, as t → 0+ ,(5.33)where u′1 j(0) = eT1 F(u j(0)). Recalling t = ε2τ , we observe that (5.33) matchesidentically with the far-field behavior of the transient solution given in (5.9).In our numerical solution of (5.25) along the lines of §4.5, (5.28) and (5.32) willprovide the initial values for B0 j(t) when u1 j(0) ̸= 0 and u1 j(0) = 0, respectively.5.4 Steady-State and Linear Stability AnalysisIn this section, we construct steady-state solutions to (5.2) and we derive a globallycoupled eigenvalue problem (GCEP) that determines the linear stability of thesteady-states.We first study the long-time dynamics of solutions to (5.25) that tend to steady-state values as t → ∞. To do so, we need the following lemma:Lemma 5.4.1. Suppose that u1 j(t) is bounded for all t > 0 and that limt→∞ u1 j(t)=u1 j∞ for j ∈ {1, ...,N}. Then, with κ j as given in (5.25d), we havelimt→∞B0 j(t) = B0 j∞ :=−κ j√σ4πD3/2u1 j∞+N∑k=1k ̸= jκku1k∞4πD|x j−xk|e−|x j−xk|√σ/D . (5.34)Proof. To determine the long-time behavior of B0 j(t), we use the Tauberian231theorem. From (5.21) and (5.24) the Laplace transform of (5.25b) isBˆ0 j(s) =−κ ju1 j(0)4πD3/2√s+σs− κ j4πD3/2(suˆ1 j(s)−u1 j(0))√s+σs+N∑k=1k ̸= jκkuˆ1k(s)4πD|x j−xk|e−|x j−xk|√(s+σ)/D ,=−κ juˆ1 j(s)4πD3/2√s+σ +N∑k=1k ̸= jκkuˆ1k(s)4πD|x j−xk|e−|x j−xk|√(s+σ)/D , (5.35)where the branch cut is chosen along {s ∈ C : Re(s) ≤ −σ < 0 and Im(s) = 0}.Since u1 j(t) is bounded for t > 0 and has the limiting behavior u1 j(t)→ u1 j∞ ast → ∞, it follows that uˆ1 j(s) is analytic in Re(s)> 0 and has the singular behavioruˆ1 j(s)∼ u1 j∞/s as s→ 0. As a result, the singularity of Bˆ0 j(s) with the largest realpart is at s = 0. To determine the long-time behavior of B0 j(t), we expand (5.35)near this dominant singularity to obtainBˆ0 j(s)∼− κ j√σ4πD3/2u1 j∞s+N∑k=1k ̸= jκku1k∞4πsD|x j−xk|e−|x j−xk|√σ/D , as s→ 0 . (5.36)The Tauberian theorem B0 j(∞) = lims→0 sBˆ0 j(s) yields (5.34).From this lemma, we obtain the following result that characterizes thosesolutions to (5.25) tending to limiting values as t → ∞.Proposition 5.4.2. For (5.25), suppose that u j(t) is bounded for t > 0 and satisfiesu j → u j∞ as t →∞ for each j ∈ {1, ...,N}. Then, limt→∞B0 j =B0 j∞ and u j∞ satisfythe nonlinear algebraic system (NAS)F j(u j∞)−κ ju1 j∞e1+ ε d2 jd1 j κ jB0 j∞e1 = 0 , j ∈ {1, ...,N} , (5.37a)B0 j∞ =−κ jD Re u1 j∞−N∑k=1k ̸= jκkDu1k∞Ge jk , j ∈ {1, ...,N} , (5.37b)232where u1 j∞ = eT1 u j∞. In (5.37), Ge jk and Re are defined byRe :=14π√σD, Ge jk :=− 14π|x j−xk|e−|x j−xk|√σ/D . (5.38)We identify Ge jk :=Ge(x j,xk), where Ge(x,xk) is the steady-state free-space Greenfunction for x ∈ R3 satisfying∆Ge− σDGe = δ (x−xk) ; Ge ∼−14π|x−xk| +Re as x→ xk . (5.39)We now establish that the limiting NAS (5.37) is the steady-state system thatresults from an asymptotic analysis of (5.2) for the time-independent case. To doso, we use a strong localized perturbation analysis to construct steady-states of(5.2).In the limit ε → 0, the steady-state problem for the outer bulk solution Us isD∆Us−σUs = 0 , x ∈ R3\{x1, . . . ,xN} ,Us ∼ κ ju1 js4πD|x−x j| +B0 js , as x→ x j , j ∈ {1, ...,N} ,(5.40)where κ j is given in (5.25d). From the steady-state of the intracellular kinetics in(5.2) we obtain thatF j (u js)−κ ju1 jse1+ ε d1 jd2 j κ jB0 jse1 = 0 , j ∈ {1, ...,N} . (5.41)We represent the solution to (5.40) in terms of the Green function of (5.39) asUs(x) =−N∑k=1κku1ksDGe(x,xk) . (5.42)By enforcing that the local behavior of Us(x) as x→ x j agrees with that specifiedin (5.40) we readily determine B0 js asB0 js =−κ jD Reu1 js−N∑k=1k ̸= jκkDu1ks Ge jk , j ∈ {1, ...,N} , (5.43)233where Ge jk and Re are given in (5.38). We conclude that the coupled system (5.41)and (5.43) is the limiting system (5.37) obtained from the long-time dynamics of(5.25).Finally, we combine (5.41) and (5.43) to obtain a NAS for the intracellularsteady-states. To so so, we let B0s = (B01s, . . . ,B0Ns)T and u1s = (u11s, . . . ,u1Ns)T ,and write (5.43) in matrix form asB0s =− 1DGeK u1s , (5.44a)where the steady-state Green matrix and the diagonal matrixK are defined by(Ge)i j = (Ge) ji = Gei j , i ̸= j , (Ge) j j = Re ; K := diag(κ1, . . . ,κN) .(5.44b)Since B0 js = eTj B0s, where e j is the jth unit basis vector, we substitute (5.44a) into(5.41) to obtain an mN dimensional NAS for the steady-state intracellular speciesF j (u js)−κ ju1 jse1− ε d1 jd2 j κ je1(eTj GeK u1s)= 0 , j ∈ {1, ...,N} . (5.45)5.4.1 Linear stability analysisNext, we derive a globally coupled eigenvalue problem (GCEP) for thelinearization around the steady-state solution of (5.2).To determine the linear stability of the steady-states we introduce theperturbationU =Us(x)+ eλ tΦ(x) , u j = u js+ eλ tζ j , j ∈ {1, ...,N} , (5.46)where u js = (u1 js, . . . ,u1Ns)T into (5.2). By linearizing the resulting system weobtain the eigenvalue problem∆Φ− (λ +σ)DΦ= 0 , x ∈ R3 \∪Nj=1Ωε j , (5.47a)εD∂nΦ= d1 jΦ− d2 jε ζ j1 , x ∈ ∂Ωε j , j ∈ {1, ...,N} , (5.47b)234which is coupled to the linearized intracellular dynamics within the jth cell by(λ I− J j)ζ j = e1ε∫∂Ωε j(d1 jΦ− d2 jε ζ j1)dSx , j ∈ {1, ...,N} . (5.47c)Here I is the m×m identity and J j := ∂F j/∂u j is the Jacobian of the intracellularkinetics evaluated at the steady-state u j = u js.For ε → 0, we now analyze (5.47) using strong localized perturbation theory.In the inner region near the jth cell, we have to leading-order that∆yΦ= 0 , for ρ > 1; DΦρ = d1 jΦ− d2 jε ζ j1 , on ρ = 1 , (5.48)which has the radially symmetric solutionΦ=c jερ− c jε(1+Dd1 j)+d2 jεd1 jζ j1 , (5.49)where c j is to be found. In terms of c j, we get ε−1∫Ω j(d1 jΦ−d2 jε−1ζ j1)dSx =−4πDc j so that from (5.47c) we conclude that(λ I− J j)ζ j =−4πDc je1 , j ∈ {1, ...,N} . (5.50)Next, we write (5.49) in terms of the outer variable, which yields the asymptoticmatching condition for the perturbation Φ in the outer bulk region. In this way, weobtain that Φ satisfies∆Φ− (λ +σ)DΦ= 0 , x ∈ R3 \∪Nj=1Ωε j , (5.51a)Φ∼ c j|x−x j| −c jε(1+Dd1 j)+d2 jεd1 jζ j1 , as x→ x j , j ∈ {1, ...,N} .(5.51b)The pre-specification of the regular part of this singularity condition (5.51b) willprovide a linear homogeneous system for c1, . . . ,cN .235We represent the solution to (5.51) asΦ(x) =−4πN∑k=1ckGλ (x,xk) , (5.52)where Gλ (x,xk) is the eigenvalue-dependent Green function for x ∈ R3 satisfying∆Gλ −(σ +λ )DGλ = δ (x−xk) ; Gλ ∼−14π|x−xk| +Rλ as x→ xk .(5.53)The solution to (5.53) is given explicitly byGλ (x,xk) =−e−|x−xk|√(σ+λ )/D4π|x−xk| , Rλ =14π√σ +λD. (5.54)Upon enforcing that the limiting behavior as x→ x j forΦ in (5.52) agrees withthat required in the matching condition (5.51b), we obtain the linear systemc j(1+Dd1 j)−4πεc jRλ −4πεN∑k=1k ̸= jckGλ (x j;xk) =d2 jd1 jζ j1 , j ∈ {1, ...,N} ,(5.55)where ζ j1 is given in terms of c j by (5.50). When λ is not an eigenvalue of J j, weuse (5.50) to calculate ζ j1 =−4πc jeT1 (λ I− J j)−1 e1.Finally, by using this expression in (5.55) and writing the resultinghomogeneous linear system for c = (c1, . . . ,cN)T in matrix form, we conclude thatthe discrete eigenvalues λ of the linearization around a steady-state solution of(5.2) are obtained from a nonlinear matrix eigenvalue problem, which we refer toas the GCEP. Our result is summarized as follows:Proposition 5.4.3. For ε → 0, the discrete eigenvalues λ associated with thelinearization around a steady-state solution to (5.2), for which det(λ I− J j) ̸= 0for all j ∈ {1, ...,N}, are collected inΛ(M ) := {λ | detM (λ ) = 0} . (5.56a)236Here the N×N dimensional matrixM (λ ) is defined byM (λ ) := I+DP1−4πεGλ +4πP2H (λ ) . (5.56b)In (5.56b), P1, P2 andK are the diagonal matricesP1 := diag( 1d11, . . . ,1d1N), P2 := diag(d21d11, . . . ,d2Nd1N), (5.57a)H (λ ) := diag(H1, . . . ,HN) , where H j(λ ) := eT1 (λ I− J j)−1 e1 . (5.57b)In addition, Gλ is the eigenvalue-dependent Green matrix with matrix entries(Gλ )i j = (Gλ ) ji :=−e−|xi−x j|√(σ+λ )/D4π|xi−x j| , i ̸= j ; (Gλ ) j j = Rλ =14π√σ +λD.(5.58)The steady-state is linearly stable if, whenever λ ∈Λ(M ), we have Re(λ )< 0. Thevector c=(c1, . . . ,cN)T , satisfyingM (λ )c= 0, is nontrivial only when λ ∈Λ(M ).To interpret the information encoded by the nullvector c = (c1, . . . ,cN)T , weuse the steady-state solution Us(x) and the perturbation (5.46) to calculate boththe diffusive flux into the jth cell and the corresponding permeable intracellularspecies asD∂ρU |ρ=1 ∼−Dε(d2 jd1 j +Du1 js+ ∑λ∈Λ(M )c jeλ t), j ∈ {1, ...,N} , (5.59a)u1 j ∼ u1 js−4πD ∑λ∈Λ(M )H j(λ )c jeλ t , j ∈ {1, ...,N} , (5.59b)where H j(λ ) = eT1 (I−λJ j)−1 e1. From (5.59) we conclude that the componentsof the normalized eigenvector c determines the relative magnitude of the spatialgradient of the bulk signal near the cell boundaries. If λ ∈ Λ(M ) is complex-valued, the nullvector c is also complex-valued, and the real and imaginary partsof c j in (5.59b) encode both the relative magnitude and phase shift of intracellularoscillations within the cell population.2375.4.2 Perturbation theory to the linear stability analysisIn contrast to the 2D case, for our 3D bulk-cell model we observe from the mainresults (5.45) and (5.56) of our steady-state and linear stability analyses, thatthere is only an O(ε) effect due to cell-cell coupling, as mediated by the Greenmatrices that depend on the cell locations. As such, it is worthwhile to developa perturbation analysis to extract information from the GCEP (5.56), under theassumption that the steady-state NAS (5.45) can readily be solved numerically forany ε .We first consider the ε = 0 problem for the GCEP (5.56). For this limitingproblem, we seek all values λ0 for which det(M0) = 0 where M0 := I +DP1 +4πP2H (λ0). SinceM0 is a diagonal matrix, the spectrum of the limiting GCEP isΛ0(M0) = {λ0 | eT1 (λ0I− J j)−1e1 =d1 j +D4πd2 j, j ∈ {1, ...,N}} . (5.60)An equivalent formulation of the condition in (5.60) can be obtained by thefollowing well-known matrix determinant lemma (see Lemma 1.1 [32]).Lemma 5.4.4. Let A be an invertible m×m matrix and let a ∈ Rm and b ∈ Rm betwo column vectors. Then,det(A+abT)=(1+bT A−1a)det(A) . (5.61)As such, (A+abT )φ = 0 has a nontrivial solution if and only if bT A−1a =−1.By applying this lemma to (5.60) we obtain the following equivalent re-formulation of the set Λ0(M0).Lemma 5.4.5. Suppose that det(λ0I− J j) ̸= 0 for each j ∈ {1, ...,N}. Then, wehavedet(I−λ0J j + 4πd2 jd1 j +De1eT1)= 0 , (5.62)if and only ifeT1 (λ0I− J j)−1e1 =(d1 j +D)4πd2 j. (5.63)238Therefore, the set in (5.60) is equivalent toΛ0(M0) ={λ0 | det(I−λJ j + 4πd2 jd1 j+De1eT1)= 0 , j ∈ {1, ...,N}}. (5.64)Proof. The result readily follows by setting A = λ0I − J j, a = e1 and b =e14πd2 j/(d1 j +D) in the matrix determinant lemma (5.61).Next, for any λ0 ∈Λ0(M0)we will determine a two-term asymptotic expansionof the form λ = λ0 + ελ1 + · · · , for which we can ensure that λ ∈ Λ(M ). To doso, we first write the NLEPM (λ )c = 0 in (5.56b) as(M0(λ )−4πεGλ )c = 0 , with M0(λ ) = diag(m1(λ ), . . . ,mN(λ )) , (5.65)wherem j(λ ) := 1+Dd1 j+4πd2 jd1 jeT1 (λ I− J j)−1e1 . (5.66)The perturbation expansion that we will now develop depends on the dimension ofthe null space ofM (λ0).We first suppose that mJ(λ0) = 0 and m′J(λ0) ̸= 0 for some J ∈ {1, . . . ,N}, andthat m j(λ0) ̸= 0 for j ̸= J. This implies that λ = λ0 is a simple root of det(λ ) = 0and that M (λ0)c0 = 0 with c0 = eJ , where eJ is the unit basis vector in the Jthdirection. In (5.65) we expandλ = λ0+ ελ1+ . . . , c = eJ + εc1+ . . . , (5.67)and collect powers of ε to obtainM0c1 = r , where r :=(4πGλ0 −M ′0(λ0)λ1)eJ . (5.68)SinceM0 =M T0 , the solvability condition for (5.68) is that eTJ r = 0, which yieldsthatλ1 =4πeTJ Gλ0eJeTJM ′0(λ0)eJ=4πRλ0m′J(λ0), where m′J(λ0) =−4πd2 jd1 jeT1 (λ0I− J j)−2 e1 .(5.69)Next, we consider the case of identical cells where d1 j = d1, d2 j = d2 and where239J j = J for all j. For this case, we haveM0 = m(λ )I , where m(λ ) := 1+Dd1+4πd2d1eT1 (λ I− J)−1e1 . (5.70)We assume that m(λ0) = 0 and that m′(λ0) ̸= 0. For this case we have c0 =span(e1, . . . ,eN), and readily conclude that λ1 and c1 satisfydet(4πGλ0 −m′(λ0)λ1I)= 0 ,(4πGλ0 −m′(λ0)λ1)c1 = 0 . (5.71)Our key conclusion is that the spectrum of the coupled bulk-cell problemaround the steady-state are O(ε) close to the leading-order intracellular problemobtained by setting ε = 0 in (5.25a),du jdt= F j(u j)−κ ju1 je1 , j ∈ {1, ...,N} . (5.72)For the steady-state of (5.72), the perturbation results in (5.69) and (5.71) showhow the eigenvalue of the linearization is altered by O(ε) owing to the bulk-cellcoupling. In particular, if the kinetic parameters are O(ε) close to an instabilitythreshold of (5.72), then the bulk-cell coupling may effectively destabilize thesteady-state. This behavior is in contrast to the 2-D case where the bulk-cellcoupling has a much stronger O(1) effect on coupling the intracellular dynamicsto the bulk.5.5 DiscussionWe have set up a compartmental-reaction diffusion system in R3 (see (5.2)). Herethe efflux (or reaction rate into the bulk) of the diffusively coupling chemicalspecies has to be of order O(ε−1) for the cell coupling to come in at leadingorder in the asymptotic construction of the solution. This is to be compared tothe O(1)-efflux in the 2-D case. We derived the integro-ODE system (5.25) for thefull time-dependent asymptotic solution for the intracellular concentrations of allcells, whose steady-states coincide with the steady-states of the full system (5.2),and matched its solution with the long-time behavior of the constructed transientsolution. Compared to the reduced system (4.3.2) in the 2-D setting in Chapter2404, the 3-D integro-ODE system (5.25) is much simpler since it does not involvean integro-differential equation for the coupling function B0 j anymore but simplya direct integral-equation. This is again due to the different scaling in the full 3-D system (5.2). Further, in the linear stability analysis we have derived a GCEP(5.56) yielding the set of eigenvalues λ for which nontrivial eigenperturbationsexist that can possibly destabilize the steady-states if their growth rate Re(λ ) ispositive. Zero-transitions of these growth rates correspond to bifurcation pointsfrom the steady state branch the perturbations are perturbing about. Lastly, anasymptotic perturbation theory was developed that is sensible since the 3-D GCEP(5.56) includes the small parameter ε , the cell radius. This gave the key insightthat the spectrum of the linear stability analysis, i.e., the set of eigenvalues λ ofthe GCEP, are O(ε)-close to the leading-order intracellular problem obtained bysetting ε = 0 in (5.25a) of the integro-ODE system. Thus, the diffusive couplingcan only change the eigenvalues with a change of magnitude O(ε) and possiblycause a bifurcation in this way, unless the coupling functions B0 j are of largemagnitude in certain cases. This is to be seen in contrast to the 2-D setting, wherethe coupling function comes in at O(1).It remains to apply our 3-D theory by choosing specific intracellular kineticslike the oscillatory Sel’kov kinetics (4.109) used in the previous Chapter 4. Anumerical treatment using the sum-of-exponentials approximation, together withthe development of time-marching methods for the convolution integrals alongthe lines of [12], [13], [91], [89], and [66] should be straightforward to developand similar to §4.5. It would be highly interesting to see and analyze thesynchronization behavior for certain 3-D cell arrangements and to compare themwith corresponding 2-D cell arrangements.241Chapter 6Discussion & ProspectsIn the chapters of this dissertation we have defined compartmental-reactiondiffusion systems in 1-D (Chapter 2), in 2-D (Chapters 3 and 4), and in 3-D(Chapter 5). We have developed asymptotic theory using the small commoncompartment radius for analyzing both the symmetry-breaking, so pattern-forming,properties of the steady-states of the systems as well as the synchronizationproperties in the case of compartments with oscillatory intra-compartmentalkinetics. The common theme was to first find the systems’ extra-compartmentalsolution asymptotically in the small compartment radius to arrive at a system ofequations for either the diffusively coupled intracellular steady-state (nonlinearalgebraic system; Chapters 2, 3, 4, and 5) or the diffusively-coupled dynamics ofthe intracellular concentrations at all times (time-dependent integro-ODE system;Chapters 4 and 5). Both systems turn out to have a linearly added coupling term orfunction in the differential equation for each intracellular chemical species thathas direct cell membrane interaction with the bulk. The key building blocksin this asymptotic system construction are the respective Green functions thatcapture the geometry of the domain and the cell arrangement. In 1-D theseGreen functions were easily found to be a combination of hyperbolic functionsfor both the finite domain with cells at the boundary (Robin boundary conditions;(2.4)) and the infinite (or finite circular) domain with cells inside the domain(periodic boundary conditions; (2.40)). In 2-D we needed to solve for thereduced-wave Green function in a finite disk with no-flux (Neumann) boundary242condition (3.9), in an infinite Bravais lattice with periodic boundary conditions(3.67), and in Laplace-space in full R2 (4.27). After a system for the steady-states was obtained in each case, a linear stability analysis led us to solve for theeigenperturbations that exist and can perturb the respective steady-state. Againthe key building block for the eigenperturbations in the extracellular space was aGreen function that was a combination of hyperbolic functions in 1-D for the in-phase and anti-phase eigenperturbation ((2.24) and (2.27)), a reduced-wave Greenfunction with Neumann boundary conditions in the 2-D disk (3.9), a reduced-wave Green function quasi-periodic with respect to each edge of the fundamentalVoronoi/Wigner Seitz domain centered at the origin in the R2 Bravais lattice cellsystem (3.72), and a reduced-wave free-space Green function in R2 (4.62) and inR3 (5.53). Subsequently, this led to a globally coupled eigenvalue problem in eachcase, det(M (λ ))= 0 whereM (λ ) is the linear stability matrix, for the eigenvaluesto all possible eigenperturbations. Bifurcation points from the steady-state branchto any eigenperturbation can directly be found if there is a crossing of the real part(= growth rate of the perturbation) of the respective eigenvalue λ from negativeto positive upon parameter variation of any parameter in the system. As soon asa single eigenperturbation with a positive growth rate Re(λ ) is found, such aneigenperturbation destabilizes the respective steady-state and kicks the dynamicstowards a steady or oscillatory behavior with similar shape as the eigenperturbationif no other eigenperturbation has a bigger growth rate and thus would dominate it.In Chapters 2 and 3 steady-state symmetry-breaking and so cell specializationwas illustrated with Gierer-Meinhardt (generic tissue kinetics), FitzHugh-Nagumo(neuronal kinetics), and Rauch-Millonas (generic cell signal transduction kinetics)intracellular kinetics. It should be stressed that any intracellular kinetics caneasily be used in our theoretical framework. In Chapters 4 and 5 Sel’kovintracellular kinetics were used for illustration, which have the property thatthey are, like FitzHugh-Nagumo and Rauch-Millonas kinetics, oscillatory forcertain parameter regimes and, in addition, only have a unique steady-state. Thus,only Hopf bifurcations can possibly destabilize this steady-state. The oscillatorycell-synchronization properties were analyzed and illustrated with plots producedby a newly developed and efficient numerical scheme based on the sum-of-exponentials method combined with a marching scheme in §4.5. Both quorum243and diffusion sensing were illustrated with bifurcation bubble and time-dynamicsfigures. Further, we introduced Kuramoto order parameters in (4.123) and (4.124)that show that there seems to be a phase transition to synchronization as thediffusivity in the extracellular space is increased past a threshold, exemplifiedwith a certain parameter set (see Figure 4.28). Additional perturbation theory wasdeveloped in Chapter 5 in the R3-setting, in which the small cell radius ε entersthe linear stability problem. This is due to the different scaling of the efflux ratein 3-D that is guided by letting the cell-bulk coupling come in to leading orderin the asymptotic solution construction. Thus, the efflux, or reaction rate into theextracellular space, on the cell membranes has to be much larger (O(ε−1)) in 3-Dthan in the 1-D and 2-D setting (O(1)). Additionally, in §2.4 it is discussed howto include stochasticity meaningfully in our 1-D compartmental-reaction diffusionframework.Prospects of the present work include the following ones. In the context of our1-D framework, there are five main additional open problems. Firstly, it wouldbe worthwhile to formulate a compact linear stability problem for asymmetricpatterns of the 1-D compartmental-reaction diffusion system, while allowing forarbitrary reaction kinetics. Secondly, it would be interesting to extend the weaklynonlinear analysis of [105], which dealt with one-bulk diffusing species, to ourcompartmental-reaction diffusion setting. With this extension, amplitude equationsnear the symmetry-breaking bifurcation would be valuable so as to classify whetherthe bifurcation is sub- or super-critical, and thereby predict whether the emergingasymmetric patterns are linearly stable or unstable. Thirdly, it would be interestingto analytically represent the time-dependent bulk diffusion fields in terms of atime-dependent Green function so as to reduce the bulk-cell system to an integro-differential system of ODEs with memory, as has been done in Chapters 4 and 5.In this setting, large amplitude solutions, including time-periodic solutions, wouldrepresent a new frontier to be investigated in the 1-D context. A further extensionof our 1-D model would be to investigate how the symmetry-breaking bifurcationsare influenced by random noise arising from the compartmental reaction kinetics.In [54] it was shown that including stochasticity in RD systems can lower thethreshold for the onset of pattern formation (see [38] for an overview of stochastic244modeling in RD systems). Finally, another possible extension of our modelingframework is to consider a compartmental-reaction diffusion system on a network.In this graph-theoretic setting, the reaction kinetics are restricted to the nodes ofa graph, while two bulk species are taken to diffuse along the edges. Relatednetwork models of this type have been analyzed in [95], [31] and [84] for generalmulti-component systems, and in [9] for a spatially-extended SIR epidemic model.It would be fruitful to combine the network setting with the third prospect, theasymptotic reduction to an integro-ODE system for the intracellular concentrationsat all times.Going over to our 2-D steady-state symmetry-breaking analysis, we nowbriefly relate our theoretical results to some qualitative behavior that has beensuggested in chemical and biological applications. Firstly, compartmental-reactiondiffusion models of the form (3.3) could potentially be useful for theoreticallymodeling the collective behavior that occurs for a microemulsion consisting ofBelousov-Zhabotinsky (BZ) chemical reactants that are confined within smallaqueous droplets that is dispersed in oil [137] (see also [37], [19]). In thisexperimental set-up, polar BZ reactants and a catalyst are confined within smallimmobile droplets, while two non-polar intermediate species generated during thereaction can be transported across the droplet boundaries. These intermediatespecies diffuse across the domain, with comparable diffusivities, and providethe mechanism for inter-drop coupling [137]. The recent experimental study of[19] has suggested that it is the relative magnitude of the membrane reactionrates of these intermediates on the droplet boundaries that plays a key role fordetermining pattern-forming properties for BZ microemulsions. Secondly, withregards to the transport of biological morphogens, it has been suggested in [99] thata differential reaction rate ratio on the cell boundaries for two morphogen specieswith comparable diffusivities can yield the large effective diffusivity ratio that isneeded for pattern formation and symmetry-breaking in tissues. This membraneattachment mechanism, which reduces the effective diffusivity of one of themorphogens and is referred to in [99] as a binding-mediated hindrance diffusionprocess, may be relevant in many biological applications. Moreover, detailedintracellular mechanisms in biological cells, affecting signaling pathways and geneexpression rates, play a pivotal role in large-scale pattern-forming properties of245biological tissues [99]. Such mechanisms would need to be included into thenonlinear intracellular kinetics and the boundary conditions if they modify therelevant dynamics, giving novel kinetic models different from the FN, GM, andRM kinetics combined with linear Robin reactive boundary conditions. By wayof qualitative comparison, our theoretical analysis of the 2-D bulk-cell model (3.3)for a very simple 2-cell pattern has revealed that a large membrane reaction rateratio, together possibly with changes in a parameter in the intracellular kinetics, cantrigger the emergence of stable asymmetric steady-state patterns that bifurcate froma symmetry steady-state. However, owing to the complexity of the analysis neededfor (3.3), where certain Green’s matrices were found to be central to the analysis,it appears rather intractable analytically to isolate via a simple scaling analysis aneffective diffusivity for the bulk species that incorporates the membrane reactionrates.Although we have only applied our theoretical framework to a simple two-cellarrangement, it is rather straightforward to numerically implement the steady-stateand linear stability theory for a symmetric cell arrangement with a much largernumber of cells.A few other open problems related to our 2-D analysis are as follows. Firstly,it would be interesting to develop an extension of our asymptotic approach totreat closely-spaced cell configurations that are more relevant to modeling pattern-forming properties in biological tissues. Our numerical results shown in §3.4.4have suggested that only a smaller membrane reaction rate ratio is needed to initiatesymmetry-breaking behavior for closely-spaced cells than for arrangements withmore spatially segregated cells. To theoretically analyze pattern-forming propertiesof the bulk-cell model with closely-spaced cells, an extension of the approachdeveloped in [63] to analyze the mean first passage time for a cluster of smalltraps may be fruitful. Secondly, it would be interesting to formulate and analyze arelated bulk-cell model where the chemical reactions occur on the boundaries of acollection of small compartments, rather than in the interior of the compartments.In this scenario, chemical species produced on the membrane can then detach anddiffuse in the bulk medium. Such an extension is relevant for analyzing collectivebehavior that occurs for dynamically reactive solid pellets that are chemicallycoated and are coupled through a bulk diffusion field (cf. [132], [131], [136],246[135]).With regard of our 2-D full time-dependent analysis, although we have onlyillustrated and carefully benchmarked our theory for a small or moderate number ofcells, with our fast time-marching scheme, possibly applied to Fitzhugh-Nagumokinetics, it is viable numerically to compute target or spiral wave patterns ofoscillatory dynamics, or to seek to identify chimera states, associated with alarge N ≥ 100 collection of cells. We emphasize that our integro-differentialODE system can also be used for specific multi-component reaction kinetics suchas those modeling the initiation and synchronization of glycolysis oscillations(see [58] for a recent survey) or for those modeling quorum-sensing transitionsbetween bistable states that are believed to be responsible for bioluminscencephenomena for the marine bacterium Aliivibrio fischeri (cf. [130], [96], [117]see also the references therein). We would like to stress at this point thatmany research articles have been published in recent years that report onsynthetically engineering microbial populations so as to induce division of laborand synchronization. Synthetic compartmentalized systems with lipid bilayermembranes have been shown to produce signal propagation in form of travelingpulses in [35]. Microfluidic devices tailored to microbial populations have beenused to investigate spatiotemporal waves and even collective synchronizationproperties in [24]. The authors use the expressions genetic clocks and geneticoscillators for what we model at high-level with our compartments with oscillatoryintra-compartmental kinetics. In [69], diffusive coupling is described to causelong-range temporal coordination (synchronization) in microbial consortia, andthe authors show that positive feedback in the intracellular kinetics is essentialfor global synchrony in the microbial colonies.Moreover, cell-bulk models, allowing for a spatially uniform bulk drift velocityas in [140], should also be tractable to analyze with our approach. The mainfeatures that we require to derive the fast time-marching algorithm is an exactanalytical representation of the Laplace-transformed bulk diffusion field and thatthis representation has singularities only along the negative real axis in the Laplacetransform space. These properties are inherent for Green’s functions for heat-typeequations with bulk degradation in either bounded or unbounded domains.Finally, we briefly remark on two numerical challenges that warrant further247investigation. Firstly, the primary numerical errors in our time-marching schemeare due to the ETD2 discretization of the Duhamel integrals and the Forward-Eulerdiscretization in (4.105) for the dynamics. By increasing the truncation order of thisaspect of our algorithm it should be possible to take larger time steps than we havedone while still preserving the same level of accuracy. Secondly, the computationof roots of the GCEP (det(M (λ )) = 0 whereM (λ ) is the linear stability matrix)that determines the HB boundaries and oscillation frequencies near the steady-state becomes highly challenging when the number of cells becomes large. Asa result, efficient numerical methods to compute solutions to nonlinear matrixeigenvalue problems that are complex-symmetric (but not Hermitian) are required.One approach to tackle this problem is the following one. Since the linear stabilitymatrix M (λ ) of Chapters 3, 4, and 5 is a complex symmetric matrix, there is analternative approach to perform the computation, e.g., in (4.116)Z =P− 1π[argF (iλI)]ΓI+,whereP denotes the number of poles (counting multiplicity) ofF (λ ) in Re(λ )>0, and ΓI+ denotes the entire positive imaginary axis now directed upwards startingfrom λI = 0. The approach is to introduce the Takagi decomposition [62] of Malong the path λ = iλI for λI > 0, which we write asM (iλI) =U (λI)S (λI)U T (λI) . (6.1)Here T denotes transpose and S is the diagonal matrix of singular values of Malong the path, i.e.,S = diag(s1, . . . ,sN) with si ≥ 0 for i= 1, . . . ,N, andU (λI) isa unitary matrix with U U † = I where U † denotes the usual conjugate transpose.Since U is unitary, we have |det(U )|= 1 so that det(U ) = eiΦ(λI), for some real-valued Φ(λI). From (6.1), this yields thatF (iλI) = det(M (iλI)) = e2iΦ(N∏i=1si). (6.2)It follows that det(M (iλI)) = 0 if and only if a singular value vanishes.The expression (6.2) can be used to calculate the number Z of destabilizing248eigenvalues in Re(λ ) > 0 in open regions of the 1/σ versus D parameter space.Assuming there are no eigenvalues on the positive imaginary axis, by using (6.2)in (4.116) we getZ =P− 2π[Φ(λI)]ΓI+. (6.3)If the singular values ofM (iλI) remain distinct on the positive imaginary axis, thenU (λI) is a smooth function for λI > 0 [30]. Since the coalescence of two singularvalues of M (iλI) is a codimension-two phenomenon (cf. [30]), we can expect onthe one-parameter path λ = iλI with λI ≥ 0 that we have distinct singular values anda smooth Takagi factorization U (λI), and correspondingly a smoothly determinedphaseΦ(λI). On this path, we initialize the phase asΦ= 0 at λI = 0. The algorithmin Theorem 3.1 of [30], which can be implemented to provide differential equationsfor U and S on the path λ = iλI for λI > 0, was used to numerically compute asmoothly depending U and phase Φ along this path. This allows us to calculateZ from (6.3).It remains to apply our 3-D time-dependent and steady-state theory by choosingspecific intracellular kinetics like the oscillatory Sel’kov kinetics (4.109) used inChapter 4. A numerical treatment using the sum-of-exponentials approximation,together with the development of time-marching methods for the convolutionintegrals along the lines of [12], [13], [91], [89], and [66] should be straightforwardto develop and similar to §4.5. It would be highly interesting to see and analyze thesynchronization behavior for certain 3-D cell arrangements and to compare themwith corresponding 2-D cell arrangements.249Bibliography[1] M. Abramowitz and I. Stegun. Handbook of mathematical functions, withformulas, graphs and mathematical tables. 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Biophysical J., 114(3):711–722, 2018. → page 5263Appendix ASupporting MaterialsA.1 CN-RK4 IMEX method on a non-centered gridIn this appendix we provide details on the numerical method used to solve the 2-cell system of §2.2. For illustration, we focus on the FN kinetics as in §2.2.2. LetM ∈ N\{0,1},N ∈ N\{0,1}, and let ∆x := LN−1 and ∆t :=t fM−1 be, respectively,the spatial step and the time step for some final time t f . Let then {xn}n with xn :=(n−1)∆x, n ∈ {1, ...,N}, be the spatial grid and {tm}m with tm := (m−1)∆t, m ∈{1, ...,M}, be the temporal grid.We discretize the 1-D Laplacian by the centered scheme∂xxu(t,xn) =u(t,xn+1)−2u(t,xn)+u(t,xn−1)(∆x)2+O(∆x2) ,for all points away from the boundary, i.e., for n ∈ {2,3, ...,N−1}. For n = 1, usethe Robin boundary conditions of our system, so that∂xxu(t,0)=u(t,∆x)−u(t,0)∆x −∂xu(t,0)12∆x+O(∆x)=u(t,∆x)−u(t,0)12(∆x)2− βuDuu(t,0)−µ1(t)12∆x+O(∆x) .264Similarly, for x = N, we have∂xxu(t,L)=∂xu(t,L)− u(t,L)−u(t,L−∆x)∆x12∆x+O(∆x)=βuDuµ2(t)−u(L, t)12∆x− u(t,L)−u(t,L−∆x)12(∆x)2+O(∆x) .This yields the discretization matrix defined byLu∆x :=−2−2 βuDu∆x 2 0 ... 01 −2 1 0 ... 0. . . . . . . . .1 −2 1. . . . . . . . .0 ... 0 2 −2− βuDu∆x,so that to O(∆x) we haveddtu(t,x1)u(t,x2)...u(t,xN−1)u(t,xN)=Du(∆x)2Lu∆xu(t,x1)u(t,x2)...u(t,xN−1)u(t,xN)+1∆x2βuµ1(t)0...02βuµ2(t)−σuu(t,x1)u(t,x2)...u(t,xN−1)u(t,xN).The discretization matrix Lv∆x can be obtained analogously from the equations for v.Then, upon including the intracellular FN kinetics of (2.2), we obtain thelumped systemddtw =Lw+N [w]+ϑ , (A.1)265wherew :=u(t,x1)...u(t,xN)v(t,x1)...v(t,xN)µ1(t)η1(t)µ2(t)η2(t)N [w] :=0...00...0−q(µ1−2)30−q(µ2−2)30, ϑ :=0...00...04040,and where the linear matrix operatorL is defined asL :=2 βu∆x 0 0 0Du(∆x)2 Lu∆x−σuI......0 0 2 βu∆x 00 2 βv∆x 0 0Dv(∆x)2 Lv∆x−σvI......0 0 0 2 βv∆xβu 0 0 0 1−βu −1 0 00 0 βv 0 εz −ε−βv 0 00 βu 0 0 0 0 1−βu −10 0 0 βv 0 0 εz −ε−βv.As inspired by the operator-splitting approach of [20], we consider the splitequationsddtwL =Lw+ϑ ,ddtwN =N [w] ,for a single time step of length ∆t, and note that w(t) = wL(t)+wN(t). The affineequation can easily be integrated implicitly with the Crank-Nicholson scheme (CN)yielding an O(∆t2)-error and the nonlinear equation can be approximated with the266classical Runge-Kutta scheme (RK4) yielding an O(∆t5) error. All combined, thisgives the scheme for the full numerical solution at time step m+ 1. Denoting thenumerical solution at time step tm by wm, we havewm+1−wm∆t =12(Lwm+1+Lwm)+16(k1,m+2k2,m+2k3,m+ k4,m)+ϑ⇔ wm+1 = (I− ∆t2 L )−1((I+ ∆t2 L )wm+ ∆t6 (k1,m+2k2,m+2k3,m+ k4,m)+∆tϑ)where the weights for RK4 are given by k1,m :=N [wm], k2,m :=N [wm+ ∆t2 k1,m],k3,m :=N [wm+ ∆t2 k2,m], and k4,m :=N [wm+∆tk3,m].267A.2 BE-RK4 IMEX method on a centered gridIn this appendix we provide details on the numerical method used to solve thering-cell system of §2.3. For illustration, we again use the FN kinetics (2.2) as in§2.3.3.Let N ∈N\{0,1} be even and denote the number of grid points per fundamentaldomain with centered cell. Let M ∈ N\{0,1} and label ∆x := LN and ∆t :=t fM−1 ,respectively, as the spatial step and the time step for some final time t f . We let{xk}k with xk := (k− 1)∆x, k ∈ {1, ...,N}, be the spatial grid while {tm}m withtm := (m− 1)∆t, m ∈ {1, ...,M} is the temporal grid. We note that xnN = nL− ∆x2and that the j-th cell is centered exactly in between x 2 j−12 Nand x 2 j−12 N+1for j ∈{1, ...,n}.We discretize the 1-D Laplacian by the centered scheme∆u(t,xk) =u(t,xk+1)−2u(t,xk)+u(t,xk−1)(∆x)2+O(∆x2) ,for all points away from the cells, i.e., for k ∈ {1, ...,nN}\⋃{2 j−12 N, 2 j−12 N+1}.For k j :=2 j−12 N, which labels the index of the grid point to the left of the j-th cell,we discretize the 1-D Laplacian by∂xxu(t,xk j) =u˜k j+1(t)−2u(t,xk j)+u(t,xk j−1)(∆x)2+O((∆x)2) ,∂xxu(t,xk j+1) =u(t,xk j+2)−2u(t,xk j+1)+ u˜k j(t)(∆x)2+O((∆x)2) ,where the coloured points are values at the ghost points depicted in the schematicfigure above and are obtained fromu(t,xk j+1)+ u˜k j(t)2=u˜k j+1(t)+u(t,xk j)2, (continuity condition)u(t,xk j+1)− u˜k j(t)∆x− u˜k j+1(t)−u(t,xk j)∆x=βuDu(u(t,xk j+1)+u(t,xk j)2−µ j(t)), (jump condition) .268Upon solving for the ghost point values we getu˜k j(t) =12(2− βu∆x2Du)u(t,xk j)+βu∆x2Duµ j(t)− βu∆x4Du u(t,xk j+1) ,u˜k j+1(t) =−βu∆x4Duu(t,xk j)+βu∆x2Duµ j(t)+12(2− βu∆x2Du)u(t,xk j+1) .This yields the discretization matrix defined byLu∆x :=−2 1 0 ... 0 11 −2 1 0 ... 0. . .1 −2− βu∆x4Du 1−βu∆x4Du1− βu∆x4Du −2−βu∆x4Du1. . .1 0 ... 0 −2 1satisfying, to O(∆x),ddtu(t,x1)u(t,x2)...u(t,xk j)u(t,xk j+1)...u(t,xN−1)u(t,xN)=Du(∆x)2Lu∆xu(t,x1)u(t,x2)...u(t,xk j)u(t,xk j+1)...u(t,xN−1)u(t,xN)+1∆x0...0βu2 µ j(t)βu2 µ j(t)0−σuu(t,x1)u(t,x2)...u(t,xk j)u(t,xk j+1)...u(t,xN−1)u(t,xN).Note the abuse of notation for the entries indexed by j. Read here that suchentries are in Lu∆x for each j ∈ {1, ...,n}. The discretization matrix Lv∆x can beobtained analogously from the equations for v.269Next, the intracellular FN reaction kinetics (2.2) are approximated byddtµ j(t) = (1−βu)µ j(t)−q(µ j(t)−2)3+4−η j(t)+βu(u(t,xk j)+u(t,xk j+1)2),ddtη j(t) = εzµ j(t)+(−ε−βv)η j(t)+βv(v(t,xk j)+ v(t,xk j+1)2).Then, upon including the intracellular FN kinetics, we obtain the lumpedsystemddtw =Lw+N [w]+ϑ , (A.2)where now we definew :=u(t,x1)...u(t,xk j)u(t,xk j+1)...u(t,xN)v(t,x1)...v(t,xk j)v(t,xk j+1)...v(t,xN)µ j(t)η j(t), N [w] :=0...00...00...00...0−q(µ j−2)30, ϑ :=0...00...00...00...040,270and where the linear matrix operatorL is now defined byL :=0 0...Du(∆x)2 Lu∆x−σuI 0 βu2∆x 0βu2∆x 0...0 00 0...0 Dv(∆x)2 Lv∆x−σvI βv2∆x0 βv2∆x...0 00 βu2βu2 0 0 0 1−βu −10 0 0 βv2βv2 0 εz −ε−βv.Again note that there is an abuse of notation for the j-indexed rows, including thereaction kinetics rows, as it refers to all j ∈ {1, ...,n}.With the operator-splitting approach inspired by [20], we again consider thesplit equationsddtwL =Lw+ϑ ,ddtwN =N [w]for a single time step of length ∆t, and note that w(t) = wL(t) + wN(t). Theaffine equation is approximated implicitly with the Backward-Euler scheme (BE)yielding an O(∆t)-error, while the nonlinear equation is integrated explicitly usingthe classical Runge-Kutta scheme (RK4) yielding an O(∆t5) error. All combined,this gives the scheme for the full numerical solution at time step m+1. Denotingthe numerical solution at time step tm by wm, we havewm+1−wm∆t = Lwm+1+16 (k1,m+2k2,m+2k3,m+ k4,m)+ϑ⇔ wm+1 = (I−∆tL )−1(wm+ ∆t6 (k1,m+2k2,m+2k3,m+ k4,m)+∆tϑ) ,271where the weights for RK4 are given by k1,m :=N [wm], k2,m :=N [wm+ ∆t2 k1,m],k3,m :=N [wm+ ∆t2 k2,m], and k4,m :=N [wm+∆tk3,m].We remark that this scheme is of orderO((∆x)2) in contrast to the orderO(∆x)scheme of Appendix A.1. This is due to using a centered grid with two ghost pointsper cell. Also note that we have used the implicit O(∆t)-scheme BE instead ofthe implicit O((∆t)2)-scheme CN since the latter can possibly generate numericalinstabilities. As a result, the scheme presented in this appendix is more somewhatrobust than that given in Appendix A.1.272A.3 Non-dimensionalization of the 2-D bulk-cell modelHere we non-dimensionalize (3.2) to obtain the PDE-ODE system (3.3). Let [z]denote the unit of some variable z. In the SI unit system, we have[U ] = molesm2 , [DU ] =m2s , [κU ] =1s , [T ] = s , [X ] = m ,[M j] = moles , [κR] = 1s , [µc] = moles , [βU,1] =ms , [βU,2] =1ms .Letting L denote the length-scale of the domain, we introduce the dimensionlessvariables u := L2U/µc, v := L2V/µc, t := κRT , x := X/L, µ j := M j/µc, and η j :=H j/µc. Since we assume that the common radius, denoted by Lc, which we havenormalized to Lc = 1 in (3.2), of the cells is much smaller than the domain length-scale L, we introduce ε ≪ 1 by ε = Lc/L = 1/L≪ 1. Then, we obtain thatκR ∂tU = DUL2 ∆xU− κUU ,κR ∂tV = DVL2 ∆xV − κVV ,⇔ ∂tu = Du∆xu−σuu, x ∈Ω\⋃mj=1 Bε(x j) ,∂tv = Dv∆xv−σvv, x ∈Ω\⋃mj=1 Bε(x j) ,where we have defined the dimensionless effective diffusivities Du and Dv anddegradation rates σu and σv byDu :=DUL2κR, σu :=κUκR, Dv :=DVL2κR, σv :=κVκR.The 2-D ball, or disk, of radius ε centered at x j is denoted by Bε(x j).Next, by non-dimensionalizing the Robin boundary conditions in (3.2), weobtainDULµcL2 ∂nxu = βU,1µcL2 u−βU,2 µcµ j ,DVLµcL2 ∂nxv = βV,1µcL2 v−βV,2 µcη j ,⇔ εDu∂nxu = du1u−du2µ j, x ∈ ∂Bε(x j) ,εDv∂nxv = dv1v−dv2η j, x ∈ ∂Bε(x j) ,where we have defined du1 , du2 , dv1, and dv2 bydu1 :=βU,1κRLε , du2 :=βU,2LκRε , dv2 :=βV,1κRLε , dv2 :=βV,2LκRε .Here, in order that there is an O(1) exchange across the cell membranes we haveassumed that βU,1κRL ,βU,2LκR ,βV,1κRL andβV,2LκR are all O(ε−1).273Lastly, we non-dimensionalize the intracellular reaction kinetics in (3.2) byκRµc ddt µ j = κRµc f (µ j,η j)+∫∂Bε (x j)(βU,1 µcLL2 u−βU,2µcLµ j)dSx ,κRµc ddtη j = κRµc g(µ j,η j)+∫∂Bε (x j)(βV,1 µcLL2 v−βV,2µcLη j)dSx ,which yields the dimensionless intracellular reactionsdµ jdt= f (µ j,η j)+1ε∫∂Bε (x j)(du1 u−du2 µ j) dSx ,dη jdt= g(µ j,η j)+1ε∫∂Bε (x j)(dv1 v−dv2η j) dSx ,for each j ∈ {1, ...,m}. This completes the derivation of (3.3).274A.4 Reduced-wave Green’s function for the unit diskWhen Ω is the unit disk, the reduced-wave Green function Gω(x;ξ ) and its regularpart, satisfying (3.9) can be determined analytically using separation of variablesas (see equations (6.10) and (6.11) of [48])Gω (x;ξ ) =12πK0 (ω|x−ξ |)− 12π∞∑n=0βn cos(n(ψ−ψ0)) K′n(ω)I′n(ω)In (ω|x|) In (ω|ξ |) ,(A.3a)Rω (ξ ) =12π(log2− γe− logω)− 12π∞∑n=0βnK′n(ω)I′n(ω)[In (ω|ξ |)]2 , (A.3b)where γe ≈ 0.5772 is Euler’s constant, and In(z) and Kn(z) are the modified Besselfunctions of the first and second kind of order n, respectively. In (A.3), β0 ≡ 1,βn ≡ 2 for n≥ 1, while x≡ |x|(cosψ,sinψ)T , and ξ ≡ |ξ |(cosψ0,sinψ0)T .For a ring pattern where the cell centers xk for k ∈ {1, . . . ,m}, are equidistantlyspaced on a ring of radius r concentric within the unit disk, as in (3.53), all theGreen matrices used in the steady-state and linear stability analysis are circulantand symmetric matrices. As a result, each such matrix spectrum is availableanalytically.Following Appendix A of [117], for an m×m circulant matrixA , with possiblycomplex-valued matrix entries, its complex-valued eigenvectors v˜ j and eigenvaluesα j are α j =m∑k=1A1kωk−1j and v˜ j =(1,Z j, . . . ,Zm−1j)T, for j ∈ {1, . . . ,m}. HereZ j := exp(2πi( j−1)m)and A1k, for k ∈ {1, . . . ,m}, are the elements of the firstrow of A . Since A is also a symmetric matrix, we have A1, j = A1,m+2− j, forj ∈ {2, . . . ,⌈m/2⌉}, where the ceiling function ⌈x⌉ is defined as the smallest integernot less than x. Therefore, α j = αm+2− j, for j ∈ {2, . . . ,⌈m/2⌉}, so that thereare m− 1 eigenvalues with a multiplicity of two when m is odd, and m− 2 sucheigenvalues when m is even. As a result, we conclude that 12 [v˜ j + v˜m+2− j] and12i [v˜ j− v˜m+2− j] are two independent real-valued eigenvectors ofA , correspondingto the eigenvalues of multiplicity two. In summary, the matrix spectrum of acirculant and symmetric matrix A , where the eigenvectors have been normalizedby vTj v j = 1, isα j =m∑k=1A1k cos(θ j(k−1)), j ∈ {1, . . . ,m} ; θ j := 2π( j−1)m ; (A.4a)v1 =1√me , v j =√2m(1,cos(θ j), . . . ,cos(θ j(m−1)))T, (A.4b)vm+2− j =√2m(0,sin(θ j), . . . ,sin(θ j(m−1)))T, (A.4c)275for j ∈ {2, . . . ,⌈m/2⌉}, where θ j := 2π( j−1)/m. When m is even, thereis an additional normalized eigenvector of multiplicity one given by vm/2+1 =m−1/2(1,−1,1, . . . ,−1)T .276A.5 Non-dimensionalization of the 3-D bulk-cell modelIn this appendix we non-dimensionalize (5.1) to obtain (5.2). The dimensions,labeled by [. . .], of the various terms in (5.1) are[U ] =moles(length)3, [µ j] = moles , [µc] = moles , [DB] =(length)2time,[kB] = [kR] =1time, [β1] =lengthtime, [β2] =1length2× time .(A.5)We introduce the dimensionless variables t, x, U and u ast = kRT , x = X/L , U =L3µcU , u j =µ jµc, ε =R0L≪ 1 , (A.6)where L=min|xi−x j| for i ̸= j and R0 is the common cell radius scaled to R0 = 1.In this way, the dimensionless bulk diffusion field satisfies (5.2a), while (5.1b) onthe jth cell boundary becomesεD∂nx U =εβ1 jLkRU− εβ2 jL2kRu1 j , x ∈ ∂Bε(x j) . (A.7)By noting that dSX = L2dSx, the intracellular kinetics (5.1c) becomesdu jdt= F j(u j)+e1ε∫∂Bε (x j)(εβ1 jLkRU− εL2β2 jkRu1 j)dSX . (A.8)Noting that the surface area has the scaling |∂Bε(x j)| = O(ε2), in order to ensurethat there is O(1) efflux out of the jth cell into the bulk medium, we must considerthe limit β2 jL2kR= O(ε−2). The feedback into the jth cell from the bulk mediumis O(ε) when the influx parameter satisfies β1 jLkR = O(ε−1). These considerationsmotivate the ε-dependent scalings in (5.3) for the permeabilities.277A.6 Long-time dynamics of the transient solutionWe follow [88] and summarize some results needed for inverting the Laplacetransform in (4.9a).To do so, we define Hˆ(s) := 1/ [s logs], which has a branch cut along thenegative real axis. To determine H(τ) =L −1[Hˆ]we integrate over a BromwichcontourH(τ) =12πi∫ c+i∞c−i∞esτs logsds .where c> 1. Upon calculating the residue at the simple pole s= 1, we obtain froma routine path deformation argument thatH(τ) = eτ +12πi∫ ∞0(e−ξτξ (logξ + iπ)− e−ξτξ (logξ − iπ))dξ ,where the two integrand terms represent contributions from the top and the bottomof the branch cut along the negative real axis. In this way, we obtain the identityN(τ) :=∫ ∞0e−τξξ [π2+(logξ )2]dξ = eτ −H(τ) ,where N(τ) is Ramanujan’s integral. This yields the identityNˆ(s) =1s−1 −1s logs.As s → 1 we calculate using logs ∼ (s− 1)− (s−1)2/2 that Nˆ(s) ∼ 1/2 ass → 1, so that s = 1 is a removable singularity for Nˆ. Therefore, Nˆ is analytic inRe(s)> 0 and satisfies Nˆ(s)∼− [s logs]−1+O(1) as s→ 0. By the scaling relationL [N(τ/c)] = cNˆ(cs) for the Laplace transform with c= κ0 j > 0, we obtain (4.10).278A.7 Improved approximation for B j(∆t)For u1 j(0) ̸= 0, we derive the approximation B j(∆t) in (4.101) that is animprovement of that given in (4.39).By substituting D j(∆t) and C jk(∆t) from (4.91a) and (4.99a) into (4.72), weobtain for j ∈ {1, . . . ,N} that[E1(σ∆t)−η j]B j(∆t) = γ ju1 j(∆t)− I+N∑k ̸= jBk(∆t)E1(a2jk/∆t), (A.9)whereI :=∫ ∆t0(B j(∆t)−B j(∆t− z)) e−σzzdz . (A.10)We let ∆t ≪ 1, and use E1(σ∆t) = − log(σ∆t)− γe +O(∆t), and neglect the lastterm on the right-hand side of (A.9) since E1(z) is exponentially small for z ≫ 1.This yields thatB j(∆t) =− u1 j(∆t)γ jlog(∆t/(κ je−γe)) +Ilog(∆t/(κ je−γe)), (A.11)where we used σeη j = 1/κ j. Upon substituting the two-term expansion for B j(∆t),given byB j(∆t) =c jlog(∆t/(κ je−γe))+d j[log(∆t/(κ je−γe))]2, (A.12)with c j =−u1 j(∆t)γ j into (A.11) and (A.10), we determine d j asd j = c j∫ ∆t0log(1− z/∆t)e−σzz log((∆t− z)/(κ je−γe)) dz = c j∫ 10log(1−ξ )e−σξ∆tξ log(∆tκ je−γe (1−ξ )) dξ .(A.13)By using e−σξ∆t ≈ 1 and | log(∆t/(κ je−γe)) | ≫ 1, we obtain for ∆t ≪ 1 thatd j ∼ c jlog(∆tκ je−γe) ∫ 10log(1−ξ )ξdξ . (A.14)Upon using∫ 10 ξ−1 log(1 − ξ )dξ = −π2/6, we substitute (A.14) and c j =−u1 j(∆t)γ j into (A.12), which yields (4.101).279A.8 An artificial boundary condition for full PDEsimulationsTo numerically solve the bulk-cell system (5.2) using FlexPDE 6.50 [39], we musttruncate R2 by both choosing and formulating a boundary condition on a confiningboundary ∂Ω that encloses all of the cells.For the full PDE computations shown in Fig. 4.9, we chose the artificialboundary ∂Ω to have the same shape as the asymptotic levels sets of the steady-state solution away from the two cells. Assuming that the solution is approximatelyradially symmetric due to the cell arrangement with center of mass at xc = 0, wetook Ω = {x ∈ R2 | |x| ≤ Ra} with Ra ≫ 1. The steady-state far away from theasymptotic support of the two cells satisfies ∆Us−ω2Us ≈ C1δ (x), for some C1where ω :=√σ/D. This yields the asymptotic behavior Us =C2K0(ω|x|) whereC2 = −C1/(2π). Upon using K0(z) ∼√π/2z−1/2e−z(1+O(z−1))for z≫ 1, welet r = |x|, and readily calculateddrUs ∼−(ω+12r)Us+O(1r2)Us , for r ≫ 1 .Neglecting the O(r−2) term, the artificial boundary condition that we imposed inour FlexPDE simulations, which asymptotically approximates the steady-state onR2, was∂rU(r, t) =−(ω+12r)U(r, t) on r = Ra , (A.15)where ω =√σ/D. We chose Ra = 6 for the results shown in Fig. 4.9.We remark that more sophisticated, memory-dependent, artificial boundaryconditions for time-dependent parabolic problems have been formulated in [127].However, owing to the very close agreement between our asymptotic and FlexPDEresults in Fig. 4.9 using (A.15), we did not find it essential to implement theapproach in [127].280
Thesis/Dissertation
eng
Mathematics
Vancouver : University of British Columbia Library
Attribution-NonCommercial-NoDerivatives 4.0 International
Graduate
Ward, Michael J. (Michael Jeffrey)
Coombs, Daniel
Symmetry-breaking bifurcations in compartmental-reaction diffusion systems with comparable diffusivities
Text