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Pelz, Merlin

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.

Text

2024-07-25

Attribution-NonCommercial-NoDerivatives 4.0 International

http://hdl.handle.net/2429/88694

Mathematics

University of British Columbia

UBCV

http://creativecommons.org/licenses/by-nc-nd/4.0/