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Modelling the dynamics of actin in cells Civelekoglu, Esma Gul 1995

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MODELLING THE DYNAMICS OF ACTIN IN CELLSbyESMA GUL CIVELEKOLUB.Sc. Bosphorus University, Turkey, 1987M.Sc. Simon Fraser University, 1989A THESIS SUBMITTED IN PARTIAL FULFILLMENT OFTHE REQUIREMENTS FOR THE DEGREE OFDOCTOR OF PHILOSOPHYinTHE FACULTY OF GRADUATE STUDIES(Department of Mathematics)We accept this thesis as conformingto the required standardTHE UNIVERSITY OF BRITISH COLUMBIADecember 1994©Esma Gui Cive1eko1uIn presenting this thesis in partial fulfilment of the requirements for an advanceddegree at the University of British Columbia, I agree that the Library shall make itfreely available for reference and study. I further agree that permission for extensivecopying of this thesis for scholarly purposes may be granted by the head of mydepartment or by his or her representatives. It is understood that copying orpublication of this thesis for financial gain shall not be allowed without my writtenpermission.Department of________________The University of British ColumbiaVancouver, CanadaDate T4 3, ‘75DE-6 (2)88)AbstractThe cytoskeleton is a macromolecular scaffold which gives the cell its shape and controlscellular motion. Actin is the most abundant proteins in the cytoskeleton and an important determinant of its structure and mechanical properties. Actin monomers polymerizeinto filaments that are then linked to one another by a variety of binding proteins. Filaments can organize into unipolar and bipolar bundles as well as orthogonal networks.The formation of these structures and the transitions between them depend on the types,quantities, and properties of the binding proteins.The problem addressed in this thesis concerns interactions of actin filaments withactin binding proteins. I investigate the main mechanisms governing the formation of avariety of cytoskeletal actin structures as well as transitions between them. In particularI discuss how the type of binding protein and its binding kinetics affects the structuresformed. I further investigate the influence of the geometry of the molecules and thedimensionality of the environment (for example the presence of a surface near which thestructures form).Dynamic continuum models analogous to the mean field approximation in physicsare used to study the time evolution of angular distributions of actin filaments. Integropartial differential equations are derived for two types of events: (a) rapid binding offilaments, and (b) gradual turning and alignment of filaments. Linear stability analysisis applied to 2D and 3D versions of such models. Numerical analysis and explicit solutionsare discussed in special cases.It is found that as the actin filament density increases in the cell, a spontaneous11tendency to organize into bundles or networks occurs. Both the linear stability analysis and the nnmerical results indicate that the structures formed are highly sensitiveto changes in the parameters including the total mass of actin filaments, the rotationaldiffusion coefficient and rate constants representing binding and unbinding. Criteria (involving combinations of these parameters) are obtained for instability of the homogeneoussteady state and appearance of order. Similar results are obtained for both rapid andgradual alignment models, suggesting robustness of the modelling approach.111Table of ContentsAbstract iiTable of Contents ivList of Figures viiiList of Tables xAcknowledgements xi1 Introduction 11.1 The cellular cytoskeletou 11.2 Actin structure and function 21.3 Actin associated proteins 51.4 Review of previous work 71.5 Goals and Objectives 92 Models for rapid alignment of actin filaments 122.1 Introduction 122.2 The molecular approach: a new model based on molecular interactions 132.3 Model I: one type of binding protein interacting with F-actin (2 dimensional model) 142.4 Analysis of Model I 222.4.1 Linear stability analysis 24iv2.4.2 Numerical analysis 272.5 Model II: two types of binding protein interacting with F-actin (2 dimensional model) 302.6 Analysis of Model II 332.6.1 Linear Stability Analysis . 332.6.2 Numerical Analysis 362.7 Generalization of Models I and II to 3 dimensions 392.7.1 Linear Stability Analysis . 432.8 Nonlinear analysis in 2D and 3D. 472.9 Discussion and conclusions 483 Models for gradual alignment of actin filaments 523.1 Introduction 523.2 Angular drift model 543.3 An overview of binding proteins 563.3.1 Unipolar bundling proteins (I) 573.3.2 Bipolar bundling proteins (II) 573.3.3 Orthogonal networking proteins (III) 583.3.4 Myosin (IV) 583.4 Classification of Kernels 593.4.1 Kernels for unipolar bundling (I) 613.4.2 Kernels for bipolar bundling (II) and myosin (IV) 623.4.3 Kernels for orthogonal binding (III) 643.4.4 Competition between two types of binding proteins (V) 643.5 Linear stability analysis 653.5.1 Dispersion relation for unipolar bundling (I) 69v3.5.2 Dispersion relation for bipolar bundling (II) and myosin (IV)3.5.3 Dispersion relation for orthogonal binding (III)3.5.4 Dispersion relation for competition of binding proteins (V) .3.6 Steady state equation and explicit solutions in three special cases .3.6.1 Steady state solutions for a special class of kernels3.6.2 Relations satisfied by the constants c and D3.6.3 Investigating the transcendental equation for c3.6.4 The explicit steady state solutions3.6.5 Examples of several steady state solutions for specific kernels3.7 Discussion4 Models for actin filament alignment associated with4.1 Actin arrangement in specialized structures4.2 Modelling actin associated with a surface4.2.1 The kernel for binding near a surface4.3 The analysis of the models4.4 First steps towards a more realistic model4.5 Discussiona membrane 9192971011061131165 DiscussionBibliographyA Non-dimensionalizationB Linearization and Linear Stability AnalysisC The properties of SSH11912312913113471737375777981838789viD SSH on the surface of the unit hemi-sphere 137viiList of Figures1.1 The structure of an actin filament 31.2 Actin filaments in orthogonal network and bundles 52.3 Orthogonal and parallel binding kernels 202.4 The dispersion relation for orthogonal and parallel binding kernels . . . 262.5 Numerical results for orthogonal binding of microfilaments 282.6 Numerical results for parallel binding of microfilaments 292.7 Kernels representing the angle dependence of binding for various mixturesof orthogonal and parallel binding proteins 342.8 The dispersion relation for mixed kernels 352.9 Numerical results for mixed kernels 372.10 The angular coordinates (, 0) 402.11 Kernels representing the angle dependence of binding in 3D 422.12 The dispersion relation in 3D 463.13 Actin binding proteins 563.14 Kernels representing interactions of various actin binding proteins with actin 603.15 Direction of the angular drift in case of ‘attraction’ 613.16 Direction of the angular drift in case of ‘repulsion’ 633.17 Direction of the angular drift in case of orthogonal binding 643.18 Visualization of the dispersion relation for various kernels 703.19 Visualization of the solution of equation (3.78) 82yin4.20 Actin filaments in the Contractile Ring 944.21 Actin filaments in the Adhesion Belt 964.22 Typical shape of Listeria and its actin tail 984.23 Angles on the surface of the unit hemi-sphere 994.24 The position of the bacteria relative to the coordinate system 1054.25 The slab of material representing the filaments oriented at ç5’ 1144.26 The shape of a kernel including the effect of the presence of a surface 115ixList of Tables2.1 The parameter representing the relative binding rate and concentrationof orthogonal and parallel binding proteins 333.2 Binding rate kernels representing different actin binding proteins 663.3 Outcome of the dispersion relation 764.4 Thickness of the cell cortex in various cells 93xAcknowledgementsFirst and foremost I would like to express my gratitude to my research supervisor,Prof. Leah Edeistein-Keshet, for providing continuous advice, support and guidance, andfor her sincerity and understanding which led to the completion of this work.I would like to thank Prof. W. Alt and E. Geigant for discussions leading to preliminary ideas for the model in Chapter 3. I also would like to thank Alex Mogilner for hisvaluable input in Chapters 3 and 4 of this thesis, and in particular for his continuousfriendship, support and encouragement.I owe a very special gratitude to my parents who have done tremendous amount forme, and many many thanks to my friends without whom graduate school would havebeen no fun at all.I am grateful to Profs. Lionel Harrison and Don Brunette for serving on my advisorycommittee and for many helpful comments during various stages of this work. I am alsograteful to Profs. Lionel Harrison and Birger Bergersen, and Carol Oakley for carefullyreading drafts of this thesis.My work was supported by the Mathematics Department at UBC, an NSERC operating grant and a NATO collaborative research grant to L. Edelstein-Keshet.xiChapter 1Introduction1.1 The cellular cytoskeletonA molecular scaffold, referred to as the cytoskeleton, euables a cell to adopt a variety ofshapes and to carry out various functions including motility. This highly complex networkconsists of numerous protein filaments and may exhibit physical properties similar to aliquid or a solid at different times and at different locations in the intracellular mediumof the cell, the cytoplasm. The cytoskeleton is distinct for two main evolutionary classesof living organisms: the eucaryotic cells (e.g. animal cells) which have a cytoskeleton andthe procaryotes (e.g. bacteria), which do not.The three principal types of protein filaments forming the cytoskeleton are: actin filaments, microtubules and intermediate filaments. Even though these three major proteinfilaments all participate in the cytoskeletal structure in animal cells, they have differentfunctions. For example, the intermediate filaments are found in a basket-weave of fiberswhich extend from the nucleus to the membrane and provide mechanical integrity. Microtubules are the thickest of the filaments. They are long cylindrical rods which radiatefrom one site (the microtubule organizing center or centrosome) to the cell periphery, andplay a key role in over-all organization of cell movement (for example, by transportingorganelles or vesicles, thereby facilitating communication between different parts of thecell). Actin filaments are also vitally important in the mechanical structure and motility of the cell. Actin filaments are rarely found solitary, but associate into networks,1Chapter 1. Introduction 2bundles, or various complex structures which undergo dynamic rearrangement duringcell motion, cell division, and other functions of the cell. Both actin and microtubulesare dynamic structures that polymerize and depolymerize, associate and dissociate (intobigger structres) on a ps time scale.Recent studies suggest that all of the above classes of filaments are implicated in themotility and alignment of cells along certain preferred axes of orientation (Oakley andBrunette, 1993), particularly when the cells are grown on grooved surfaces. Abundantinformation is available on actin dynamics and strnctures and this will form the maintopic of this thesis.A variety of smaller proteins are associated with actin and are believed to take aleading role in these structural transitions. Indeed, actin filaments by themselves havelittle mechanical strength. However, in the presence of these auxiliary proteins, whichconnect actin filaments to one another in various configurations, stronger well-definedstructures are formed. The types, the amounts, and the affinities of the attachmentproteins determine the type of structure that forms, and are thus of great importanceto cellular function. The actin structures are not static, but constantly changing as anintegral part of cell movement and cell function. In this thesis we concentrate exclusivelyon the dynamics of actin structures, and on transitions that take place under the influenceof the actin-related proteins.1.2 Actin structure and functionActin has become one of the most carefully characterized proteins in cell biology sinceits discovery in muscle in the 1940s, and non-mnscle cells in the 1960s. It is an abundantprotein in cells and an important determinant of the structure and mechanical propertiesof the cytoplasmic matrix.C1liapter 1. 1nrodiiction 3a ACTIN b ACTINSUBUNIT NUCLEATION FILAMENTMYOSINFILAMENTQ__BARBEDBARBED END POINTED END ENDELONGATES FAST ELONGATES SLOWLYFigure 1.1: A schematic representation of the helical structure of an actin filament (a),and attached myosin filaments and heads which determine its pointed amidbarbed end (b). Taken from Stossel (1994).The actin molecule exists both as a monomer (globular or 0-actin), and as a polymer(fliamentous actin, F-actin, or microfiiainents) in cells. Each globular actin has bindingsites on jt,s surface that allow it to associate with two other monomers in a helical arrangement (see, e.g. Stossel, 1994; Alberts et al., 1989; Bray, 1990). The orientation ofthe actin monomers in this double helical structure provides it with a unique polarity,see Fig. 1.1. This polarity is most easily detected by allowing the filaments to react witha protein molecule, myosin, which binds to each actin subunit in a filament in a veryprecise fashion giving the appearance of a series of arrowheads. The direction in whichthe arrowheads point defines the pointed end, and the other end is called the barbed end,see Fig. 1.1. Actin filaments grow bidirectionally by addition of monomers with differentrates of assembly (i.e. different growth rates) at opposite ends. Thus F-actin may heviewed as a polar macromolecule with a pointed (slowly growing) and a barbed (fastgrowing) end.MYOSINHEADChapter 1. Introduction 4The slow step of assembly of a few actin molecules provides a nucleus for the formationof an actin filament. An equilibrium is quickly reached whereby the so-called ‘criticalconcentration’ of monomer is in apparent equilibrium with F-actin filaments.Actin filaments are essential for many forms of cellular motility. In non-muscle cells,actin filaments are highly dynamic on a second to minute time scale. Since the 1970’s ithas been generally recognized that in cultured motile cells, polymerized actin occurs in atleast two distinguishable states of structural organization: in linear fibrillar bundles -commonly referred to as stress fibres- and in isotropic meshworks or networks confinedto the motile lamella zones and ruffling membranes, see Fig. 1.2a-b, (Small et al., 1982;Stossel et al., 1985; Stossel, 1984; Weeds, 1982). The transitions between bundles andmeshworks are vital to cell motion and organization.In some actin structures the filaments display locally uniform polarity whereas in others they display opposite polarity or no polarity. The bundling proteins such as fascin,fimbrin and villin create polarized bundles (Pollard and Cooper, 1986). Unidirectionally polarized microfilament structures are found in microvilli of epithelial cells, and instreocilia of cochlear hair cells. F-actin structures which do not display any polarityare observed in the cell cortex, and in the periphery of various cells including amoebas,macrophages, leukocytes and blood platelets. In these latter cases the filaments intersect in a perpendicular fashion. In both stress fibres in fibroblasts and epithelial cells inculture the filaments are organized into bundles without being polarized (Stossel, 1984).Populations of actin filaments have been observed to rearrange in a variety of cells,for example during differentiation of embryonic carcinoma cells, during locomotion offibroblasts or during development of yeast cells. See (Way and Weeds, 1990; Meulemansand De Loof, 1992). It has been revealed that the structural organization of actin filaments can also form and disappear rapidly in various cellular phenomena such as mitosisChapter 1. Introduction 5Figure 1.2: Network of actin ifiaments (a), joined and held nearly perpendicular by thecross-linking proteins (e.g. ABP or fllamin), and bundles of actin filaments(b), joined and held nearly parallel by the bundling proteins (e.g. vil]in,fascin). Note that the helical arrangement of actin subunits in the ifiamentsis not reflected in this figure.and fertilization (Pollard, 1990). The rearrangement of actin cytoskeleton in a cell is nowknown to affect many functions of the cell. It also plays a dominant role in various phenomena, such as the motility of a parasitic bacterium inside a host cell. The bacteriumListeria monocytogenes propels itself through the host cell cytoplasm using a tail-likeactin meshwork that it assembles on its posterior end (Theriot and Mitchison, 1992).1.3 Actin associated proteinsThe microfilaments interact with various other proteins in the cell: anneal end to end,fragment, become capped at the ends, crosslink and organize into diverse structures. Theself-assembly of actin structures is regulated in a remarkably precise manner; it occursat particular times and in discrete places within the cell. It is now recognized that thiscontrol is conferred by actin binding proteins. After the discovery of the major classes(a)Chapter 1. Introduction 6of actin binding proteins in the 1980’s it seemed possible that the assembly and functionof actin in cells might be explained by relatively simple mechanisms involving a smallhandful of proteins (Cooper, 1991; Pollard and Cooper, 1986; Pollard et al., 1990). Someof these actin binding proteins lead to interactions between filaments by linking themtogether in various ways. The cross linking proteins promote the formation of orthogonalmeshworks and the bundling proteins promote the alignment of filaments in bundles. Therole and function of each binding protein is determined by observing how it interacts withmicrofilaments. However, the mechanism by which a variety of filament structures formor switch from one to another in a cell when all actin binding proteins are acting inconcert is unclear.The actin binding proteins are classified in different groups according to how theyinteract with actin. Actin monomer binding proteins, capping proteins, severing proteins,proteins that bind to the sides of actin filaments and membrane attachment proteinsare the major different groups in eucaryotic cells. The proteins that bind to the sidesof actin filaments are generally considered in three different subgroups based on theirfunctional properties. Among those, except for the group including tropomyosin whichbind to only one filament at a time, the cross linking proteins and the bundling proteinspromote interactions between filaments. In this thesis we focus attention to proteins inthese two classes. Cross-linking proteins include ABP and filamin, and the filamentsform networks or meshworks where they are joined approximately at 90° angles via theseproteins (Stossel, 1990; Tilney et al., 1992a-b; Hartwig, 1992; Hartwig et aL, 1980;Weeds, 1982). The bundling proteins include villin and fascin, and the filaments producestructures where actin filaments are aligned parallel to each other (Cooper, 1991; Pollardand Cooper, 1986; Weeds, 1982). Further details about the binding proteins are given insection 3.3.Chapter 1. Introduction 71.4 Review of previous workThere is a wealth of literature which focuses ou actiu and its dynamics. Several branchesof science have led to recent progress in this subject, including cell biology, biochemistry,biomathematics, biophysics, colloid and polymer chemistry, rheology and thermodynamics. As a result of interdisciplinary efforts, several important symposia and conferenceshave served to summarize the progress in this field. The effect has been to stimulateresearchers to explore the molecular and mechanical basis of important phenomena suchas cell motility or mitosis. Numerous studies have attempted to unravel the mechanismof motility at the cytoskeletal level. Although knowledge of the constituents and theorganization of components has been elucidated, the details of mechanisms are largelyunknown.Experimental and theoretical studies of the formation of different cytoskeletal structures and the properties of the resulting structures have been previously considered.Recent work by Dufort and Lumsden (l993a,b) has provided dynamic visual images ofthe actin cytoskeleton and its interactions with many binding proteins. The cellular automaton simulation that they have produced allows an exploration of the interactionsof a small population of actin molecules. The three dimensional spatial positions, binding and unbinding, and the spatial and rotational diffusions of individual molecules isshown. The results reveal the dramatic transitions that these molecules undergo. (Seevideo supplement to l993a.) Their papers also contain detailed values of parametersassociated with actin kinetics. Their model is a complex and realistic simulation, withmany parameters. The realism of the simulation makes it hard to dissect the essentialeffects from the many competing influences.In other previous theoretical considerations the approach is a mechanical one, considering the effects and the balance of the forces in and outside of the cell and neglectingChapter 1. Introduction 8the microscopic interactions and their influences on the mechanical properties of the cytoskeleton. In a recent publication, Sherratt and Lewis (1993) consider the alignment ofintracellular actin filaments as a response to external forces (stress and strains) or to ananisotropy in the stress field of the filaments themselves. Their approach is a mechanicalone, based on a balance of forces in the system. In their model, the interactions betweenthe filaments, as well as the turnover rate and the strength of the bonds between themis reflected in a single parameter: the sensitivity parameter.Oster, Murray and Odell (1985) present a model accounting for the formation of regular hexagonal patterns in microvilli solely as a consequence of the mechanical instabilityof the contractile acto-myosin gel. In (Oster and Odell, 1984), the actin-myosin meshwork is considered, and the dynamic contractile behavior of the cytogel is captured in amodel based on the mechanical properties of the gel which, in turn, are regulated by achemical trigger. In these models, the cross-links between actin filaments are assumed tobe permanent, and the cytogel is viewed as an elastic material. However, according to(Sato et al., 1987), the mechanical properties of the cytoskeleton also depend on (or areinfluenced by) the dynamics of the rapid rearrangement of these bonds. Thus, there is aproblem with the above approach, namely, on the time scales of interest, the cytoskeletalnetwork behaves as a viscous fluid with negligible elasticity. Oster (1989) gives a reviewof the role of the mechanical aspects in cell motility and morphogenesis.Other mechanical models of the contractile behavior of the actin-myosin meshworkappear in (Alt, 1987; Pohl, 1990). In (Alt, 1987) the actin-myosin meshwork is viewedas a creeping viscous fluid with negligible elasticity. Thus, in this model the filamentcross-links are not assumed to be permanent. Pohl (1990) models iv. vitro experimentsof actin-myosin based contraction waves, stimulated by external forces, regarding thecytoplasmic matrix as a mixture of a fibroid network and an aqueous solution. ApplyingChapter 1. Introduction 9the laws of fluid mechanics to this mixture, he describes the dynamic behavior of thecytogel. His model is based on the Reactive Flow Model of the cytoplasm reviewed in(Dembo, 1989). Dembo (1989) reviews the mechanical theory of the dynamics of thecontractile cycle of actin cytoskeleton, considering a dynamic F-actin network. In thismodel, the network is assumed to be isotropic and the network synthesis and breakdown,as well as the formation of cross-links between the filaments are described by single termsin the equations.The importance of the key structural elements in these phenomena, the actin bindingproteins, has been noted in the above papers. However the interactions between the actinfilaments and the binding proteins and the consequences of these interactions have notbeen included in most of these models.1.5 Goals and ObjectivesIn this thesis we investigate the hypothesis that molecular interactions betweenactin filaments and the actin associated proteins lead to the formation oforder and the transitions between different structures formed by the actincytoskeleton. We further propose that these transitions occur even in the absence ofexternal mechanical forces.To investigate this hypothesis we must derive models for the dynamic changes inactin. (The problem cannot be studied as an equilibrium phenomenon, but rather as akinetic one.) These models will permit characterization of the essential aspects of molecular interactions promoting order of several types: bundles, networks, versus isotropicarrangements of microfilaments. We focus on the orientational distribution of filamentsin these structures, not on the spatial density. (But see Mogilner and Edelstein-Keshet1994c for a general framework of spatio-angular models.)Chapter 1. Introduction 10Biological experiments have produced a wealth of iuformation about details of thechemistry, biochemistry, and molecular biology of the actin structures. However, much ofthis knowledge is focused either on individual molecules or on macromolecular assemblies,but not on the relationship of one level of complexity to the next. The goal of this thesisis to link the properties at the level of individual molecules with the behavior of theensemble. The mathematical model provides this linkage.An additional objective of the modelling approach is to determine the sensitivity ofactin dynamics to parameters such as the kinetic rate constants, the concentrations, andthe affinities of the various intermediates. The importance of this issue has been recentlyproposed by Wachsstock et al (1994). Recent interest has arisen in the comparison ofcytoskeletal structures found in different species of organisms whose actiu associatedproteins are related, but have slightly different kinetic rates.The models discussed here describe the angular density of the actin structures, such asmight occur in a particular location inside the cell. I will study two structurally distinctclasses of models here. One class (Chapter 2) accounts for rapid (one-step) actin filamentalignment in response to interactions with other filaments mediated by specific types ofbinding-proteins. A second class (Chapter 3) deals with more gradual drift-like turningand alignment. In chapter 4, the basic model of the first sort is modified to allow for thepresence of a surface, such as the cell membrane.The challenge of deriving a suitable model is that it is desirable to capture the essentialaspects of the phenomenon, whilst keeping the model simple enough to analyse. This isa rather difficult task considering the overwhelming quantity of biological, bio-physicaland bio-chemical information on the components of the cell. Further, it is of interest toinvestigate the robustness of the predictions to the formulation of the model. For thispurpose, comparisons were made between the two distinct ways of modelling analogousChapter 1. Introduction 11situations; that is, I investigated a given class of binding proteins in each modellingscenario, and found essentially similar conclusions.Chapter 2Models for rapid alignment of actin filaments2.1 IntroductionIn this chapter I develop a model to study the formation of parallel and orthogonal actinfilament structures as well as the transitions between these. We first ask several questionsabout the formation of such actin filament structures. We ask which type of molecularinteractions and properties observed biologically can account for the observed dynamicsof actin in the cell. We also consider how properties at the molecular level (for example,affinities of binding proteins) can affect the macro-molecular structure and organization,and how changes in the details of the interactions can affect the outcome of the structuresthat form. Towards this goal we will reformulate, in mathematical terms, the dynamicsof the actin filaments in the cell based on the elements and properties reported in thebiological literature (Stossel, 1990). Second, we address the question of a spontaneousswitch between the orthogonal and parallel structure and the sharpness of this transitionin a model that accounts for the presence of two types of actin binding proteins.The model(s) will allow us to reach the following conclusions:(1) When the density of aetin reaches a critical level, a spontaneous tendency to organize into an orthogonal or parallel structure occurs.(2) The structure depends on the concentrations of active cross-linking or parallel binding proteins, e.g. filamin and ABP-50 or fibrillin and villin.12Chapter 2. Models for rapid alignment of actin filaments 13(3)Furthermore, the switch between the orthogonal and the parallel aligned structurescan occur as a result of a change in the relative binding rates and concentration of thetwo types of actin binding proteins (cross linking and bundling).The results presented in this chapter have appeared in the paper Civelekoglu andEdelstein-Keshet (1994).2.2 The molecular approach: a new model based on molecular interactionsExperimental evidence indicates that forces are not essential for the cytoskeletal rearrangement and the rapid changes in the cytoskeletal structure can be mediated by theactin binding proteins. Actin interacts with several different proteins at once depending on the relative binding affinities, concentrations of different proteins, and regulatoryfactors (Way and Weeds, 1990). A new set of actin binding proteins may be responsiblefor a change in the cytoskeletal organization of a cell (Vandekerckhove, 1990). Also, theactin binding proteins may act differently under different conditions. For example, someproteins act as cross-linking proteins in the absence of Ca++, and as capping proteinsin the presence of Ca. The sol-gel transformation can therefore be regulated by theresponse of a single molecule to changes in Ca concentrations (Korn, 1982; Hartwig,1992). Thus, there exists biological evidence that the changes in the molecular propertiesof these elements affect the resulting structure, and changes from one structure to theother occurs also in the absence of external forces, via activation or inactivation of theactin binding proteins. Based on the above evidence, we view the cell as a pool of interacting molecules. In the following section we present a model based on the geometry ofthe molecular interactions, and on the differences between binding proteins that promotea variety of actin structures that form.Chapter 2. Models for rapid alignment of actin filaments 14According to Stossel (1990) and Pollard and Cooper (1986) the steps in the formation of the actin meshwork are as follows: First, needlelike actin filaments are createdby polymerization of individual actin monomers. This process has two stages: the singlemolecules aggregate to form small groups of three or four molecules -nucleation-, andthen the nuclei elongate, eventually generating long, stiff rods of actin. When the lengthand mass of these filaments reach a certain level, the filaments start to join under theinfluence of cross-linking proteins in orthogonal networks or bundles. As seen under theelectron microscope, some cross-linking proteins join the actin filaments at approximatelyright angles (Hartwig, 1992; Hartwig et al., 1992; Hartwig et al., 1980; Stossel, 1984; Stossel, 1990; Tilney et al., 1992a-b; Weeds, 1982), whereas the bundling proteins promotebinding in parallel. The ways in which the cross-linking and the bundling proteins bringabout the high angle branching or the parallel alignment of actin is a function of theirstructure (Stossel, 1984; Stossel, 1990; Hartwig and Stossel, 1981; Pollard and Cooper,1986).The models we formulate in the following sections account for the formation of structure in a pool of actin filaments in the cell and focuses on orientation rather than spatialdistributions. We assume the existence of short (ready to bind) actin filaments ratherthan explicitly modeling the nucleation of filaments.2.3 Model I: one type of binding protein interacting with F-actin (2 dimensional model)In this section we consider a two dimensional analogue of a truly three dimensionalmolecular system. The model corresponds to a mean-field approximation of the molecularsystem. A similar simplification was made in (Sherratt and Lewis, 1993). The modelChapter 2. Models for rapid alignment of actin filaments 15here closely resembles a model for orientations of interacting cells described in (EdelsteinKeshet and Ermentrout, 1990).We distinguish between filaments which are bound to other filaments, referred toas Bound filaments, and those which are not, referred to as Free filaments. Thefollowing simplifications have been made in deriving this model:(a) We consider only angular distribntions of filaments, not spatial distributions.(Thus the model applies to a small part of the cell.) A fuller model which includesspatial variations is discussed in Mogiliier and Edelstein-Keshet (1994b).(b) Binding and unbinding of filaments is similar at all stages of the process. (Actuallyonce a dense network forms unbinding will be restricted to its exposed surface.)(c) Monomers are added to the filaments at a rate proportional to the total length offilaments. (Actually, monomers are only added at the ends of an actin filament but weassume that filaments have some fixed average length so that the total number of endsis proportional to the total length of filaments.)(d) We assume that only free filaments can rotate freely. (In reality small clusters ofbound filaments will also undergo rotational diffusion but we do not distinguish betweensmall and large clusters.)The model is based on the following variables:t = time,U = an angle, 0 U 27r, with respect to some arbitrary fixed direction,L(O, 1) = the concentration (total length) of free actin filaments at orientation U at time1,B(O, t) = the concentration (total length) of bound actin filaments at orientation 0 attime 1,= the rate constant for binding of filaments via actin binding proteins,Chapter 2. Models for rapid alignment of actin filaments 16K(O) = the kernel representing the angnlar dependence of the rate constant forbinding,p = the concentration of free actin binding protein,6 = the dissociation rate of the actin binding proteins,g = the concentration of actin monomers,xi = the rate of elongation of filaments by addition of monomers at the ends,= the rate of shortening of filaments by loss of monomers from the ends.The concentrations of L and B are the total length of filaments (in terms of monomersubunits) inside a unit element of the region, for example, length per unit area in a twodimensional model, or length per nnit volume in a three dimensional version. These areanalogous to the density function F(, p) defined by Sherratt and Lewis (1993), who alsoneglect the spatial dependence of F. Note, however that L and B in our model are timedependent, as we explore a fully dynamic model.The quantity /3K(O) is the rate constant for binding of one filament to another filament at a relative angle 0 in the presence of actin binding proteins. 5 is the magnitudeof the rate constant. The kernel K is a normalized function which represents the effective interaction of molecules at various relative configurations. It is known in manychemical reactions that molecules must first come into the correct relative configurationsbefore they can react. The nature of the kernel K, discussed below, is deduced fromseveral remarks in the literature (Stossel, 1990; Hartwig and Stossel, 1981; Hartwig etal., 1980; Tilney et al., 1992a-b) taking into consideration the molecular properties andthe structure of the actin binding proteins. For example, the orthogonal binding protein,Actin Binding Protein, promotes binding of filaments at right angles, see the histogramin (Hartwig et al., 1980) or (Stossel, 1994). Filament densities L(0,t) and B(0,t) arefunctions of time and of 0. Since 0 is an angle of orientation, all functions of 0 areChapter 2. Models for rapid alignment of actin filaments 17assumed to be periodic, i.e. L(O, t) = L(2r, t) for all t.In deriving the equations of the model we start by considering the behavior of anindividual filament. The repertoire of a single filament consists of:(a) rotational diffusion which results in tumbling and thus random reorientation ofthe molecules (frictional forces in the cytoplasm will limit this effect for larger molecules),(b) binding upon contact with another filament and an actin binding protein (thisbinding is angle dependent)The rotational diffusion of actin filaments in the cytoplasm can be depicted as arandom walk in 0. The associated diffusion coefficient, ji, has been determined in theliterature for biopolymers, see (Mossakowska et al., 1988; Phillips et al., 1991; Sawyer etal., 1988; Thomas et al., 1979).Next, we consider how the free actin filaments binding to others can affect the freefilament density at a given orientation 0, namely L(0, t). To this end, we first considerthe rate that a single free filament at orientation 0 attaches to another free filament, sayat orientation 0’, in presence of actin binding protein. This rate depends on the densityof free filaments oriented at 0’, i.e. on L(0’, t), and on their relative orientation, i.e. on(0— 0’). Ler r be the effect of filaments at angle 0’ on the rate of realignment of a filamentat angle 0. Then,r = p/3K(0 — 0’)L(O’,t), (2.1)where /3 is the binding rate and p is the concentration of actin binding protein. Summingover the density of free filaments at all possible orientations results in R, the cumulativeeffect of all filaments oriented at 0’ on the rate of realignment of a filament at angle 0.Chapter 2. Models for rapid alignment of actin filaments 18Then,p2wR=p/3 J K(O — O’)L(O’, t)dO’. (2.2)0Finally, we consider the effect of such binding on the total density of free filamentsoriented at 0, which is:OL(0, t)= —p/3L(0, t) j K(0 — 0’)L(O’, t)dO’. (2.3)For notational simplicity, we adopt the * notation for the above convolution integral, i.e.p2wK * L= J K(0 — 0’)L(O’,t)dO’. (2.4)0As mentioned above, actin filaments bind to each other via auxiliary protein moleculesof different structures. With cross-binding proteins, e.g. ABP or filamin, F-actinforms networks or meshworks joined approximately at 90° angles (Stossel, 1990; Tilneyet al., 1992a-b; Hartwig, 1992; Hartwig et al., 1980; Weeds, 1982), whereas the bundllngproteins, e.g. villin or fascin, produce parallel actin filaments (Cooper, 1991; Pollardand Cooper, 1986; Weeds, 1982). ABP and filamin are long flexible hinge-like moleculeswhereas villin and fascin are short rod-like molecules (Stossel, 1990; Pollard and Cooper,1986). Differences in the structures of these binding proteins implies differences in thegeometry of binding. In the presence of a binding protein, two filaments bind uponcontact depending on (a) the kinetic rate constant of the binding protein, and (b) theproper configuration being attained by the filaments at the binding site. The criticalangular range for successful binding depends on the molecular structure of the givenbinding protein. In the model, the relative angle formed by the actin filaments, 0, mustbe within some critical range for binding to occur in each case. This is depicted byChapter 2. Models for rapid alignment of actin filaments 19the function K(O). In the thesis we will consider several types of kernels associatedwith binding proteins but in this chapter we restrict attention to two types: one whichaccounts for orthogonal cross-linking of F-actin, and a second one for the bundling ofF-actin. The critical angles a and b in the equations below reflect this range for theorthogonal binding and bundling proteins. Thus, modelling the orthogonal binding ofF-actin we consider kernels of the following form (see Fig. 2.3a):If(O) for IO—<a orK1(O) = 2 2, (2.5)I. 0 otherwiseand modelling parallel binding we consider the following type of kernels (see Fig. 2.3b):(h(O) forO<b or or 2ir—O<b1(2(0) =. (2.6)0 otherwiseIt was argued by Edelstein-Keshet and Ermentrout (1990) that the specific form of thesefunctions is of no consequence for the conclusions of the model as long as they satisfycertain symmetry properties. The critical angles a and b, beyond which the binding doesnot take place represent a range of angular attraction (a = 20° and b = 30° in Fig. 2.3a-b,respectively). We also normalize K by requiring,JK(0)dO = 1. (2.7)This means that the angle dependence, summed over all possible angles of interactionis set to 1. The following fnnctional differences are assumed between L and B typefilaments:(1) Free filaments reorient randomly but bound filaments do not.Chapter 2. Models for rapid alignment of actin filaments 200.6C0.40.2Figure 2.3: Shapes of angle dependent kernels representing the angle dependence of binding of two actin filaments via (a) orthogonal actin binding proteins and (b)bnndling proteins. We assume a uniform concentration of actin bindingproteins in the cell. The vertical axes represent the kernel and the horizontal axes represent the angle between two contacting ifiaments. The criticalangles are as follows: a = 200 in (a) and b = 30° in (b).(2.8)a)0.8b) °C00 80 120 180 240B300(2) Binding of two filaments occur if two filaments contact in presence of actin bindingproteins.(3) All bound filaments can become free by dissociation of proteins at some fixedunbinding rate 6. (Actually, 6 would probably be density dependent as bound filamentson the inside of a large network would have very low rate of dissociation. We do notinclude this effect in the model.)(4) Filaments can elongate by addition of actin monomers, g, at the constant rate i’.(5) Filaments can shorten (loss of actin monomers from ends) at a constant rate 7.(Recall the assumption that the number of free ends is proportional to the total lengthof the filaments.)The following set of equations depict the interactions described above:f(O, 1)=— 7L + vgL + 6B — /3pL(K * B)— /3pL(K * L)%f(o, t) = —7B + vgB — 6B + 3pB(K * L) + /3pL(K * L)Chapter 2. Models for rapid alignment of actin filaments 21The terms in the equations (2.8) have the following meanings: L(K*B) represents the rateat which free filaments, oriented at 0, bind to bound filaments at arbitrary orientations,L(K * L) denotes the rate at which they bind to free filaments at arbitrary orientations,and B(K * L) denotes the rate at which free filaments, at arbitrary orientation, bind tobound filaments oriented at 0. p denotes the rotational diffusion constant of F-actin,and the first terms in the right hand side of the equation for free filaments representsthe angnlar diffusion of filaments freely rotating in the cytoplasm. p denotes the actinbinding protein concentration. 3 denotes the rate constant for binding of filaments by anactin binding protein and denotes the dissociation rate of the actin binding proteins.The 0-independent steady state of these equations, (L, B), corresponds to the casein which the total addition of actin monomers to filaments equals the total loss of actinmonomers from filaments. (This equilibrium state is referred to as the treadmilhingstate in Stossel (1990).) Thus the second and third term in the equation for free filamentdensity and the first two terms in the equation for bound filament density cancel eachother and (2.8) reduces to the following system of equations, similar to equations (17) in(Edelstein-Keshet and Ermentrout, 1990):t)= p + SB — /3pL(K * B) — /3pL(K * L)oo. (2.9)-1(0, t) = —SB + /3pB(K * L) + /3pL(K * L)Also, the total mass density of actin filaments in the system is conserved, i.e.M=L2r{L(0,t) + B(0,t)}dO (2.10)is constant. The quantity M will be treated as a constant throughout the present analysis.Later we will be interested in the situation in which M is allowed to vary slowly.The equations (2.9) can be written in the following dimensionless form:Chapter 2. Models for rapid alignment of actin filaments 22(O, t) = + B — L(K * B) — L(K * L),(2.11)(O,t) = —B + B(K * L) + L(K * L),where = 6/j3pM and e = p/B pM are dimensionless parameters. Also, the densities Land B are dimensionless quantities. The details of this non-dimensionalization is givenin the Appendix A.2.4 Analysis of Model IThe analysis of the model is similar to the analysis of the model in (Edelstein-Keshetand Ermentrout, 1990). The homogeneous steady state (L, B) of the system (2.8), orequivalently (2.9) is found by setting:DL— 0 — DLIat ao1 DB0O ,at ao (2.12)and satisfies.B/3pMl bL 6This represents a time independent population in which all orientations are equally represented. If this steady state is stable, the population will persist in the ratio (2.12b) andno angle or orientation will be favored. However, if noise can disrupt this steady state,i.e. if it is unstable, the situation might change to one where some angles are favored.XVe investigate this possibility by considering small perturbations from the steady state.Equations (2.9) can be linearized about the homogeneous steady state by substitutingL(O, t) = L-i- L0(O, t) and B(O, t) = B + B0(O, t) into (2.9) and retaining linear terms:Chapter 2. Models for rapid alignment of actin filaments 23%(O, t) = + 6B0 — /3p(L(K * B0) + L0B) — /3p(L(K * L0) + L0)2 13-(O,t) —6B0 + /3p(B(K * L0) + B0L) + 73p(L(K * L0) + L0)When L0 and B0 are sufficiently small to render the linear approximation (2.13) a validrepresentation of the full equations (2.9), the linear stability theory is an adequate methodfor the analysis of the states near the steady state value (L, B). The details of thelinear stability analysis are given in Appendix B. These equations are now linear integro—02partial differential equations containing the Laplacian operator () and the linearoperator (K* ), and they describe the evolution of a small perturbation from steadystate. As discussed in Mogilner and Edelstein-Keshet (1994a), both these operatorsshare a common set of eigenfunctions, namely:lv C R. (2.14)The fact that the domain 0 0 2ir is periodic (i.e. all functions H(0) = H(0 + 2vrn))will restrict permissible values of lv to the integers, lv = 0, 1, 2,.. . , n. The terms 6jk8form an orthonormal basis for periodic functions which satisfy Dirichiet ‘s Conditions,i.e. functions with finite number of finite discontinuities and finite number of turningpoints. Moreover, any such periodic function f(0), with period 2K can be expressedas a convergent sum of terms ake, referred to as its Fourier series expansion. Thus,considering perturbations where the dependence on 0 is of the form e18 is sufficient toinvestigate the stability of this steady state. We thus consider perturbations of the form:L(0,t) L= + eJVOe)t. (2.15)B(0,t) B B0Chapter 2. Models for rapid alignment of actin filaments 24where L0,B0 are small amplitudes, k is the wavenumber (the number of peaks or thenumber of dominant orientations in [0, 27rJ) and A is the growth rate of the perturbation.As the domain is periodic, with period 27r, the wavenumber k must be an integer. Weseek conditions for which such small perturbations from the steady state are amplifiedwith time, i.e. for which A > 0 for some wavenumber lv.2.4.1 Linear stability analysisUsing the form (2.15) for the perturbations and substituting into (2.13) the equationscan be written in the following matrix form:L0A=J , (2.16)B0wherefan a12\ f(izk2+/3pL(1+k)+/3pB) 6—/3pLk \J=I 1=1— 1. (2.17)a21 a22) \ $pL(1 + K) + /3pBK —(6 — /3pL))Here k is the Fourier transform of the kernel K, lv is the wavenumber, and A is thegrowth rate of the perturbation as above. See Appendix B for the description of theFourier transform of K. For stability to uniform perturbations and instability to 0-dependent perturbations it is necessary that the determinant of the Jacobian,det J =a1122—a1221, (2.18)be non-negative for lv = 0 and negative for some integer wavenumber lv. Such an integeris then a destabilizing wave number or mode. Thus the stability condition is determinedChapter 2. Models for rapid alignment of actin filaments 25by an algebraic equation, referred to as the dispersion relation, obtained by setting thedeterminant of the Jacobian matrix to be negative. It is a condition on the type ofperiodicity that leads to instability. The determinant of the Jacobian in (2.17), withslight rearrangement, leads to the following condition:The Dispersion Relation:Ck2 <k(i - k) (2.19)where,C= ((L ) = (2.20)is a combination of the parameters in (2.9) or the dimensionless parameters in (2.11).The steady-state (IL, B) of (2.9) can be destabilized by pertnrbations of the form (2.15),provided that the wave nnmber Ic satisfies (2.19). Only wavennmbers satisfying thisinequality will give rise to growing strnctnres. Thns (2.19) mnst be satisfied for eitherbundles or networks of actin to form. We can visualize (2.19) graphically as done in(Edelstein-Keshet and Ermentrout, 1990) by plotting the right hand side and the lefthand side of it on a common set of axes. This has been done in Fig. 2.4 for varionssettings of the parameters. The expression on the right hand side of (2.19) as a fnnctionof Ic (the wavy curve in Fig. 2.4a-b) is fundamentally different for the two types of kernelsin Fig. 2.3a-b and is scaled differently for different choices of critical angles a and b. Theleft hand side of (2.19) is a parabola in Ic with coefficient C, as shown superimposed inFig. 2.4a-b. The inequality (2.19) depends on the shape of K(k)(1— k(k)) and onthe value of C. Though only integral Ic values are relevant (dne to periodic boundaryconditions), we plot this expression as a continuous function of Ic for easier visualization.Chapter 2. I1ode1s for rapid alignmcnt of actin filaments 26—0The expression K(k)(1 — K(k)), the wavy curve in Fig. 2.4a-h. is shown as afunction of the wave number k for k, the Fourier transform of the kernels inFig. 2.3a.-b. Superimposed is a set of parabolas y = Ck2. (The parabola andthe function K(k)(i —K(k)) are plotted as continuous functions ofk, howeveronly integer Ii values arc of interest.) The uniform steady state of (2.9) canhe (list urhed and pattern formation can be initialed only by pertH rhat ions(2.15) whose wave number k is a integer satisfying Ck2 < k(1 — K). whereC depends on biological parameters. The sequence of parabolas in (a) and(b) can be generated by varying the total mass of F—actin, Al = (L + B).given that the other parameters are constant. Parameters are as follows: in(a.) the critical angle is a = 20° and the coefficient C is 0.04 and 0.01, in (h)the critical angle is b = 30° and the coefficient C is 0.06 and 0.02.1r1 other words the parabola Ck2 must be lower than the function of K displayed on theright hand side of the inequality at some integer value k for instability at that wavenumber. In the case where we have a kernel accounting for the orthogonal binding ofF—actin, as in Fig. 2.3a, the first i ritegral wave number at winch the mequality (2. I 9) issatisfied is k = 4, (see Fig. 2.4a). This means that a perturbation of the form 40 grows,the steady state loses stability and four orientations, 90° apart, become accentuatedamong all possible orientations from 0 to 2ir. As a result, the filaments are mostlyorthogonal to each other. In the case where we have a kernel accounting for the bundlingof F-actin, as in Fig. 2.3b, the first such wave number is k = 2, as shown in Fig. 2.4b.A perturbation of the form e20 grows and results in two accentuated orientations 180°a) C = 0.04 b)C = 0.01(1C=0.06 C=0.0200.I0.Figure 2.4:-0.Chapter 2. Models for rapid alignment of actin filaments 27apart. In this case most filaments lie parallel to each other. In both cases the positionsof the accentuated orientations are determined by the initial disturbance that disruptsthe steady state. However the spacing between them is determined by a wave numberlv, which satisfies (2.19). The instability discussed here is somewhat analogous to theisotropic-nematic transition in liquid crystals.2.4.2 Numerical analysisThe equations of the model were simulated numerically by using an explicit finite difference scheme. To insure stability of the numerical scheme a small value of zit = 0.01 andforward differencing for 15,000-100,000 iterations were used. The numerical code usedto obtain these results is written in Fortran. Numerical solutions to (2.9) in the case oforthogonal or parallel binding kernels is given in Fig. 2.5a-b and Fig. 2.6a-b. A varietyof initial densities were nsed, including random (in Fig. 2.5a-b and Fig. 2.6a-b) or sinusoidal deviations from the homogeneous steady state densities 1 and B (obtained fromthe steady state equation 2.12b), and from other homogeneous densities for L and B.The magnitude of these deviations was roughly 10% of the initial homogeneous densities.(Smaller deviations also cause instability but the time evolution is much slower.) Thevariables were discretized typically on a grid of 30 to 36 points (AU = = 12° and360° . . .AU =-- = 10 ). For numerical stability I made sure that the values of AU, At satisfiedthe Courant-Friedrichs-Lewy condition for a given value of II:[LAt 1(AU)2 (2.21)See Press et al. (1988). The kernel in Fig. 2.3a was used for Fig. 2.5a-b and the kernelin Fig. 2.3b was used for Fig. 2.6a-b. In the results shown in Fig. 2.5a-b the criticalChapter 2. Models for rapid alignment of actin filaments 28Formation of orthogonal network of F-actin in a pooi of initially randomlydistribnted bonnd (a) and free (b) actin filaments. Shown are numericalresults to (2.9) where K is as in Fig. 2.3a. The horizontal axis is orientationand the vertical axis is the density of bound F-actin at a given orientationin (a) and of free F-actin in (b). Initial densities (not shown) are L =L +L0(U,t), and B = B + B0(8,t) where L = 0.8, B = 9.2, L0 and B0 are10% random noise. Other parameters are 6 = = 0.5, p = 5, p = 0.4 and M10. The grid size is AU = 36° and At = 0.01. The solutions were foundfor 16,000 iterations, with plots shown at 3,200, 6,400 and 16,000 iterations.Note the scale on free and bound F-actiu indicating that most filaments arebound. In (a) and (b) four orientations 90° apart have been accentuated.angle is a = 20° and in Fig. 2.6a-b the critical angle is b = 30°.In Fig. 2.5 and Fig. 2.6 we present the evolution of bound and free actin filamentdensities over time. Fig. 2.5 shows formation of parallel filament structures (two preferredorientations) whereas Fig. 2.6 shows orthogonal meshworks of filaments (four preferredorientations), as anticipated from our assumptions about the kernels in each case. It canbe seen that structures that develop in the bound population are similar to those thatarise in the free actin density. Pattern formation occurred either in both populations or inneither. The number of preferred orientations and their location was identical for boundand free actin filaments. However, pattern formation appeared sooner in one populationthan in the other for certain choices of parameters. For example if 6 << p, which meansa)0tFigure 2.5:b) 0.50, 60 120 180 240 300 36C0Chapter 2. Models for rapid alignment of actin filaments 29a) 2.6: As in Fig. 2.5a-b, but showing the formation of parallel networks of F-actin.Shown are numerical results to (2.9) where K is as in Fig. 2.3b, (a) boundF-actin and (b) free F-actin. Initial densities as in Fig. 2.5 where 1 = 0.5,B = 4.5. Other parameters are 6 = 0.6,,8=O.S,p = 4,/A = 1.2 and M5. The grid size is z\O = 100 and zXi = 0.01. The solutions were foundfor 30,000 iterations, with plots shown at 6,000, 18,000, 24,000 and 30,000iterations. In (a) and (b) two orientations 1800 apart have been accentuated.biologically that the rotational diffusion of filaments is considerably higher than thedissociation rate of the actiu binding proteins with filaments, pattern formation in freeactin filaments took considerably longer than in the bound actin filaments. Also, in allsimulations, the free actin filament density level was considerably lower than the boundactin filament density level at the final stable configuration. In the following section wewill only present the evolution of the bound filament density since the evolution of thetwo populations is essentially the same.The results of the numerical simulations matched the predictions of the analysis andpattern formation in networks (Fig. 2.5a-b) or in bundles (Fig. 2.6a-b) was obtained fora choice of parameter values which satisfied (2.19). Changing any of the parameters M,t, 6, /3 or p affects the value of the dimensionless constant C that appears in (2.19) andthus the stability of the system. For example, when monomers assemble into filaments,0•01155 10b)2’0.3ILI.. 0.200 60 120 180 240 300-80 120 180 240 300 3616the total mass of actin filaments, M, increases. Therefore C decreases and this leads toChapter 2. Models for rapid alignment of actin filaments 30the formation of a meshwork or bundles. Similarly, increasing the binding rate of thebinding protein, 3, increasing the actin binding protein concentration, p, or decreasingthe dissociation rate of the actin binding protein, 6, in the cell results in formation ofmeshworks or bundles.We have also observed that, in the case where the critical angle a, or b was either toosmall, a, b 50, or too large, a, b 40°, no pattern formed (for 200, 000 iterations) forany choice of the other parameter values. This means that when the range of angularattraction is too small, very few filaments become bonnd and they are released beforegetting a chance to form big groups. Most filaments remain free, and thus the directionalhomogeneity is preserved. In the latter case, i.e. when the range of angular attractionis too wide, the filaments bind to each other at nearly every possible relative angle.Most filaments become bound with no apparent structure, and hence, the directionalhomogeneity is preserved in this case too.To summarize, both numerical and analytical results of the model show that theorganization of F-actin into orthogonal networks or bundles depends on the biologicaland chemical properties of the molecules, the parameters in the system. Typical valuesof parameters taken from biological literature are given in the discussion.2.5 Model II: two types of binding protein interacting with F-actin (2 dimensional model)In this section we consider the case where both orthogonal and parallel binding canoccur. The question addressed is under what circumstances will one of the two forms ofstructure dominate. To this end we extend the model in section 2.3 to account for theexistence of two types of actin binding proteins simultaneously: the cross linking and thebundling proteins. We now allow the actin filaments to bind orthogonally and in parallel.Chapter 2. Models for rapid alignment of actin filaments 31We also investigate the transition from the network structure to the bundles and viceversa. K1 and K2 denote the orthogonal and the parallel binding kernels as in section2.3. Also p1, /31 and P2, /32 will denote the concentrations and the binding rate constantsof orthogonal cross-linking (1) and parallel bundling proteins (2), respectively.The equations depicting the effect of the two types of binding simultaneously can bewritten as follows:j(o, t) =— 7L + vgL + 6B — /31pL(Ki * B) — /31pL(Ki * L)—/32pL(K * B)—52pL(K * L)-rno’ t) = —“yB + vgB — 6B + ,3ipiB(K1* L) + /3ipiL(K1 * L)+* L) + /32pL(K * L)In the above equations we have assumed for simplicity that 6 = 62 = 6, i.e. thatdissociation rates for both types of proteins are approximately equal. In order to reduceto the previous method of analysis, we now define:K = (1— i/)K1 + ?J’K2, (2.23)where/ !32P222.4and= /3iPi +/32P2= /32P2 (2.25)Chapter 2. Models for rapid alignment of actin filaments 32Here K is a combined binding kernel and /3p is a combined binding rate and bindingprotein density. Note that /32 = 0 (or P2 = 0) results in all orthogonal binding and= 0 (or pi = 0) results in all parallel binding as in section 2.4. For example, p2 = 0stands for the situation in which the parallel binding protein, villin, is absent. /32 = 0represents the case of binding protein that has no affinity to actin, similar conclusionshold for P1 = 0, /3 = 0 with respect to the orthogonal binding protein, see Table 2.1.The parameter i/’ represents the ratio of parallel binding to orthogonal binding, and issummarized in Table 2.1. For purposes of the analysis, it is convenient to vary the singleparameter t,b. As discussed later, in numerical investigations results are calculated forvarious values of the parameters 5i, Pi, /32 and P2. After slight rearrangement of terms,equations (2.22) can be reduced to the previous system, (2.8), but with the new kerneldefined above, in (2.23).In this section we study both extremes as well as intermediate situations, i.e. weare interested in all values of t/’ in 0 < /‘ 1. Also note that since K1 and K2 werenormalized, so is K, and further,(2.26)The shape of the kernel K (see Fig. 2.7a-b) in this case depends not only on the twocritical angles but also on the parameter & representing the ratio of the concentrationsand the binding rates of the two types of auxiliary proteins. Here we assume thatthe effect of two different binding proteins is simply linear in their concentrations. This,together with the linear property of the operator K* allows us to reduce the new problemto the old one via (2.23).Chapter 2. Models for rapid alignment of actin filaments 33b=0 b=0.5actin /32=0 13i=Obinding or !3iPi = /32P2 orproteins P2 = 0 = 0kernel = = K1 + K2 =type of binding orthogonal both kinds of parallelbinding only binding binding onlyTable 2.1: The proportion of parallel and orthogonal binding rate and binding protein concentration can be represented by a single parameter /‘ definedby (2.25).2.6 Analysis of Model II2.6.1 Linear Stability AnalysisThe analysis is identical to the previous section, and the stability condition is exactlyas given in (2.19), but with the new interpretations of /3p and K as above in (2.25) and(2.23). The left hand side of (2.19) in this case, too, is a parabola as a function of thewave number k. and its coefficient depends on the parameters in the system. The righthand side is a function of the Fourier transform of the combined kernel, K, as in (2.26).The inequality (2.19) can be rearranged to obtain:+ B))k <(/31pk+/32pk)(/ ipi + /32p2 — /31pk — /32pk) . (2.27)Chapter 2. Models for rapid alignment of actin filaments 34H60 120 180 240 300 36C8Shapes of the kernels K representing the combined angle dependence of bothorthogonal and parallel binding. The values of the critical angles are a = 20°for K1 and b = 20° for K2, note that K is also dependent on the parameterb. (a) b = 0.3 (for example /3 = 0.7, /32 = 0.3 and Pi = p2). (b) 4 = 0.7(for example /3 = 0.3, /32 = 0.7 and P1 = P2).In order to study the transition from the extreme case where the bundling proteins areinactive or absent, i/’ = 0, to the other extreme case where the cross-linking proteinsare inactive or absent, = 1, we vary j3 (or equivalently p) from 1 to 0 and /32 (orequivalently p2) from 0 to 1 simultaneously. The reason for this is that we wish toinvestigate only the effect of the changes of the binding rates or binding protein ratiowhile all other conditions remain the same. See Fig. 2.8a-e for plots of the function onthe right hand side of (2.27) for various values of the parameters as varies from 0 to1. We also display the parabola on the left hand side of (2.27) on these figures. Asin the previous section, instability at integer wavenumbers k occurs if the parabola onthe left hand side of (2.27) is lower than the function on the right hand side of (2.27),i.e. the uniform steady state of (2.22) is disrupted and pattern formation is initiated byperturbations of the form (2.15) whose wavenumbers satisfy (2.19) or equivalently (2.27).The first integer wavenumber for which (2.27) can be satisfied depends on the value of5, and for the choice of critical angles a = b = 20°, k changes from 4 to 2 as ,L’ changesa) 60 120 180 240 300 36(b) 010.50.400.30.20.10Figure 2.7:(hapier 2. Models for rapid alignment of act in filaments 35C = 0.04 0.12 C = 0.04C = 0.12 C = 0.04 C = 0.3 C 0.12Figure 2.8: The expression on the right hand side of (2.27) is shown as a function of thewavenumber k. K is as in Fig. 2.7a and 2.7b for (b) and (d), respectively.The critical angles are a = 20° for K1 and b = 20° for K2 in all cases. Alsom = P2 = 2 and hence /3p 2 in all cases. The superimposed parabolasfrom left to right can be obtained by increasing the ‘total mass’ of F-actin inthe system. In (a) /3 = I and /32 0 and the coefficient C of the parabolasare 0.12 and 0.04, in (b) i3 0.7 and /32 = 0.3 and C 0.12 and 0.04, in(c) /3 = 0.5= /2 and C = 0.12 and 0.04, in (d) /9 = 0.3 and /32 = 0.7 andC = 0.3 and 0.12, and in (e) /3 = 0 and /32 1 and C = 0.3 and 0.065. Thefirst wavenumber for which the uniform steady state is disturbed is k = 4 in(a)-(c) i.e. perturbations of the form eh grow resulting in four accentuatedorientations 90° apart, a network structure. For (d) and (e) the first suchwavenumber is k = 2 i.e. perturbations of the form e2tO grow resulting intwo accentuated orientations 180° apart, bundles.C = 0.12I .50.5C = 0.065Chapter 2. Models for rapid alignment of actin filaments 36from zero to one (or equivalently 5 from one to zero and 52 from zero to one), (see Fig.2.8a-e). The transition from k = 4 to k = 2 is sharp, as predicted by the analysis, andwill be discussed in the subsection below.2.6.2 Numerical AnalysisThe numerical solutions of (2.22) are in agreement with the results of the analysis. Themethods of the numerical computations are identical to those of the previous section. InFig. 2.9a-e the numerical solutions to (2.22) corresponding to the kernels used in Fig.2.8a-e are shown. We note that the number of peaks that arise correspond to the integerfor which the parabolas in Fig. 2.8a-e first cross below the curve on the right handside of (2.27). For example, this occurs at k = 4 in Fig. 2.8a-c, whereas at k = 2 inFig. 2.8d-e. Initial densities were random deviations from the uniform steady state. Theresults of cases where deviations were sinusoidal were similar and we do not present themhere. We first summarize the results of the simulations in which the initial densitieswere uniform with small deviations. In the cases where the quantity ib was smaller than0.5 (and even when it was equal to 0.5 in some cases), indicating a higher binding rateor a higher concentration of the orthogonal cross linking proteins, pattern formation oforthogonal networks resulted for the choice of parameter values which satisfied (2.27),see Fig. 2.9a-c. For values of iJ’ closer to i/’ = 0.5, in some cases, two peaks appeared firstand later divided into four peaks. However whether this occurs depends on the valuesof the critical angles, a and b, and the parameter 6 which represents the dissociationrate of the binding proteins. Also for the choice of parameter values for which b = 0.5,i.e. equal binding rates and/or equal concentrations for both types of binding proteins,the resulting structure is dependent on the values of the critical angles a and b, andcould be either orthogonal networks or bundles. For the values a = 20° and b = 20° theChapter 2. Models for rapid alignment of actin filaments 37a)t18c)Figure 2.9: Formation of the network or bundles of F-actin in a pool of initially randomlydistributed bound filaments and two type of binding proteins: orthogonaland parallel. K(k) is identical to the ones used for Fig. 2.8a-e correspondingto 2.9a-e in order. Initial densities (not shown) are 10% random noise on theuniform steady state (L, B), L = 0.25 and B = 4.75, and other parametersare @p = P1 = P2 = 2, i = 1.84,6 = 0.5,1W 5,At = 0.01 and thegrid size is AU = 12°. The solutions were found for: 70,000 iterations,with plots shown at 1 , 42,000 and 70,000 iterations in (a) and (b), 100,000iterations, with plots shown at 1 , 60,000 and 100,000 iterations in (c),50,000 iterations, with plots shown at 1, 10,000 and 50,000 iterations in(d), and 130,000 iterations, with plots shown at 1 , 104,000 and 130,000iterations in (e). In (a)-(c) four orientations 90° apart have been accentuated(network structure), and in (d) and (e) two orientations 1800 apart have beenaccentuated (bundles).12•0ise)Chapter 2. Models for rapid alignment of actin filaments 38filaments organize into a network when 7/’ = 1, see Fig. 2.9c. In the cases where theqnantity 7/’ was larger than 0.5 (higher binding rate or a higher concentration of bundlingproteins) pattern formation in the form of bundles resulted for the choice of parametervalues which satisfied (2.27), see Fig. 2.9d-e. The transition from one type of structureto the other was very sharp, as predicted by the analysis. We have also simulated caseswith pre-structnred initial densities to analyze how stable these structures are to snddenchanges in their environment. For example, we started with a pool of filaments organizedmostly parallel to each other as in Fig. 2.9d, and let the parameter 7/ be very close to0 (a sudden change from high parallel binding rate to high orthogonal binding rate), orwe started with a network of filaments as in Fig. 2.9b, and let the parameter 7/’ be veryclose to 1 (a sudden change from high orthogonal binding rate to high parallel bindingrate). Through these simulations we have found that for the same parameter values, thesame type of structure results regardless of the choice of initial densities, i.e. whetheruniform or pre-structnred. However, in the case of pre-structured initial densities theorientations that appeared were usually determined by the initial ones, with either twonew peaks appearing in between the existing ones (change from bundles to networks) ortwo alternating peaks disappearing (change from networks to bundles). This transitiondoes not require the complete break up of the existing structure; rather the new structureforms on the remnants of the old one. Thus in our model the cell is capable of switchingits cytoskeletal structure while preserving its polarity, rather than choosing a randomnew direction after every switch. This might be compared to the situation where cellsmoving in a particular direction tend to continue in that direction even in the absence ofexternal stimuli.Chapter 2. Models for rapid alignment of actin filaments 392.7 Generalization of Models I and II to 3 dimensionsThe models in sections 2.3 and 2.5 are two dimensional analogues of a truly three dimensional structure. In the 2D models, I and II, the functions L(O) and B(Q) have as thierdomain an interval of length 2K with periodic boundaries. This domain is formally equivalent to a unit circle. Thus the problem of pattern formation in angle 0 can be thoughtas a pattern formation on a unit circle. Similarly, in 3D, a given orientation can be put incorrespondence with a unit vector, and thus with a point on the surface of a unit sphere.Thus to extend the analysis to 3D it is necessary to generalize the domain from a unitcircle to a unit sphere. We represent the points on the surface of a unit sphere by (ci, 0),where 4) is in [O,ir] and 0 is in [0, 2K]. The equations of the model in three dimensions arelargely analogous to (Mogilner and Edelsteiu-Keshet, l994a). One studies perturbationsof the uniform steady state that are spherical harmonics, i.e. Legeudre polynomials. Thedispersion relation analogous to (2.19) or (2.27) then involves the inner product of Kwith these spherical eigenfunctions, rather than the Fourier transform K.Directions in 3-D can be represented by unit vectors on the surface of a unit sphere.A spherical coordinate transformation leads to the representation of the vectors in thecartesian coordinate system (x, y, z) in terms of angular coordinates (r, 4), 0). The angularcoordinates Q= (, 0) will be used to describe an orientation. (See Figure 2.10). Thisis commonly referred to as the surface spherical coordinates. A unit vector in cartesiancoordinates has the corresponding representation:u = (x,y,z) = (cos0sin4),sinOsin4),cos4)). (2.28)The problem in 2-D, described by the equations (2.9), can then be generalized to 3-Dby converting the convolution terms and the rotational diffusion term to account for theChapter 2. Models for rapid alignment of actin filaments 40Figure 2.10: The angular coordinates (, 8) shown in 3D. (k, 8) represent the angles onthe unit sphere.3-D dynamic.The rotational diffusion in 3-D corresponds to a random walk on the surface of a unitsphere. Hence this can be described by the angular part of the Laplacian operator in3-D. In surface spherical coordinates this is:1 8 8 1 82A= sino sin +sin28O(2.29)Similarly the convolution terms which account for the interaction of filaments on the rimof a unit circle in 2-D can be generalized to account for interactions with filaments overthe unit sphere. These interactions depend on the relative orientation of the filaments asexplained in 2.4. The angle between two filaments oriented at 0 and 0’, in 2-D, is simply0— 0’. In 3-D the angle between two filaments oriented at Q and Q’, say 7, can also beexpressed in terms of the angles th, 0, 4/ and 0’, however it is a complicated expression.Conveniently, the cosine of the angle 7, cos 7, can be calculated simply by consideringzxyChapter 2. Models for rapid alignment of actin filaments 41two unit vectors, n and n’, oriented at !2 and IV, and taking their scalar product n n’.n = (cosOsin,sinOsinç&,cosç&),= (cos 0’ sin ‘, sin 0’ sin qS’, cos‘), . (2.30)cos 7 = n n’ = cos cos ç’ + sin sin ‘ cos(O — 0’)Since we assume that filaments interact in 3-D depending on the relative angle, 7, betweenthem, the kernel K is a function of this variable: K(7). In the light of the abovecalculations it is convenient to express this kernel as a function of the cosine of the angle.Defining ri = cos 7 we denote this new function with the same symbol, K(r1). Weconsider three types of interactions:1- Orthogonal binding or binding at close to right angles, K1(i7), Fig. 2.lla,2- Parallel binding or binding at acute angles, K2(), Fig. 2.llb,3- Both orthogonal and parallel binding, with a large critical angle value for orthogonalbinding and a small critical angle value for parallel binding K3(i1), Fig. 2.Ilc.4- Both orthogonal and parallel binding, with a small critical angle value for orthogonal binding and a large critical angle value for parallel binding rate K4(i7), Fig. 2.lld.Also, the line integral in 2-D will be replaced by a surface integral (a double integral)in 3-D. The surface differential dS in spherical coordinates being:dS = sin d0d, (2.31)the convolution terms take the following form:Chapter 2. Models for rapid alignment of actin filaments0.30.142Figure 2.11: The kernels K.j(r1) for i = 1, 2, 3 and 4 representing the four types of interactions listed above. Note that the argument of K, the cosine of an angle:= cos’y, is varying from -1 to 1. The kernels in Fig. 2.lla and b correspond to the ones shown in Fig. 2.3a and b. Only here, the argument is irather than 0. The kernels shown in Fig. 2.llc and d are of the form (2.23)where K1 and K2 as in Fig. 2.lla and b, and = 0.33 and = 1 for c andd respectively.K * L= Is K(, !Y)L(’, t)dSçir r2Tr= Jo Jo K(, 0, gY, 0’)L(çb’, 0’, t) sin qYd0’dY= ff1 K()L(qY, 0’, t)d(cos çY)dO’(2.32)where S is the surface of the unit sphere. Hence the equations analogous to (2.9) in 3-Dcan be written as follows using the new definitions of the convolution and the diffusiona) b)0.4d).80.6r0.20 I I —-0.8 -0.4 0 0.4 0.8110.2c) -0.4 0 04 0.8TI-0.8 -04 0 0.4 0.811terms:Chapter 2. Models for rapid alignment of actin filaments 43= pAL + SB — /3pL(K * B) — /3pL(K * L),233V(Q,t)= —SB+pB(K*L)+pL(K*L).2.7.1 Linear Stability AnalysisAnalogously to the 2-D case, we look for the homogeneous steady state of the equations(2.33) where filaments are equally distributed in every direction !l= (‘, 0). We wish todetermine whether this steady state can be destabilized leading to patterns in 1! signifyinga transition from isotropic to nonisotropic networks of filaments.The linearized set of equations which govern perturbations, are, by direct analogywith equations (2.13):(O, t) = pAL0 + SB0 — flp(L(K * B0) + L0B) — p(L(K * L0) + L0)2 34—rn11°’ t) = —SB0 + /3p(B(K * L0) + B0L) + ,Øp(L(K * L0) + L0)•The operators A and K* in these linear equations now have as their domain the set offunctions on the unit sphere. This surface spherical geometry restricts eigenfunctions ofthese operators to the set of surface spherical harmonics (SSH) Y(Q),Y(Q) = Y(,U) = AF(cos) + E(A cosmO + BsinmO)F’(cos), (2.35)m=1where F are the Legendre polynomials of degree n and F, are the associated Legendrefunctions of degree n and order in. Indeed, both operators share this set of eigenfunctions,andChapter 2. Models for rapid alignment of actin filaments 44= —n(n+1)Y(236)K*Y = k(n)Ywhere the multipliers of Y, on the right hand side are the corresponding eigenvalues and— p1K(n) = 2rrj K(7)F(9i)d97. (2.37)—1(See Appendix C for details.) It is a well-known result that the set of SSH forms anorthonormal basis for functions satisfying Dirichlet ‘.s Conditions on the unit sphere(MacRobert, 1927; Arsenin, 1968; Hobson, 1931), so that such a function f(’,O) foro <q r, 0 <0 2r can be expressed as:f(, 0)=Y@, 0). (2.38)Thus, to investigate the stability of the steady state we consider perturbations of thefollowing form:L(fl,t) L=— + Y(Q)e. (2.39)B(fl,t) B B0As in the 2-D case here too L and B are steady state values, L0 and B0 are smallamplitudes. A is the growth rate and n is the mode number of the perturbation.Substituting these into (2.34) and retaining only the linear contributions, we arriveat the following inequality, the dispersion relation, which is analogous to equation (2.19)in 2-D.Cn(n +1) <f?(n)(i - k(n)) (2.40)Chapter 2. Models for rapid alignment of actin filaments 45where,C= ((L ) (2.41)The details of these calculations is given in Appendix C and follow closely the 2-D case.The dispersion relation (2.40) differs from the dispersion relation (2.19) in its dependenceon the mode number n: note that in the 2-D case the dependence on the mode numberk is of the form k2 whereas here the dependence on the mode number n is of the formn(n + 1). We conclude that the homogeneous steady state is destabilized and patternformation is initiated by perturbations of the form (2.39) provided that the mode numbern satisfies (2.40).An integral form of the kernel K, K, as in (2.37), appears in this equation. This isanalogous to K, the Fourier transform of K, appearing in the dispersion relation (2.19)in the 2D case. Similarly to the 2-D case we visualize (2.40) graphically in Fig. 2.12a-dfor the four types of kernels given in Fig. 2.lla-d. In Fig. 2.12 the expressions on bothsides of the inequality (2.40) are plotted superimposed as functions of the mode numbern. The coefficient C of the quadratic function on the left hand side of (2.40) is positivesmall in all biologically relevant cases.The inequality (2.40) depends on the shape of K(n)(l— [((n)) and on the constantC. For this case, since the eigenfnnctions are 5511 and hence the function k(n) involvesLegendre polynomials of degree n, this expression can only be evaluated for integer n. Itsdiscrete values have been connected by line segments in Fig. 2.12 for ease of visualization.For instability the quadratic function of n must be lower than the function on the righthand side for some integer mode number n. In Fig. 2.12a and c, n = 4 is the first suchmode number. Thus the harmonic Y4 breaks the stability and becomes the leading mode.a)Chapter 2. Models for rapid alignment of actin filaments 460.0.C=0.015 C=0.005 b) C = 0.05 C = 0.015-o-0.C = 0.0150.0.0.0.-0..d)0.4C = 0.015C = 0.0050.30.20.1n-0.-0.C = 0.052 4 6-0.2nFigure 2.12: The ftmct.ion on the right hand side of (2.40) is shown superimposed withthe quadratic function of n on the left hand side for various values of thecoefficient C. The plots a-d correspond to the four different type of kernelshown in Fig. 2.ila-d. Iii cases a and c the first mode causing instability is4, while in b and d it is 2. The quadratic function on the left hand side isshown for the following values for the coefficient C: in a and c C = 0.015and 0.005 and in b and d C = 0.05 and 0.015.This mode is rotationally and pattern-wise highly degenerate. However, it is shown in(Busse, 1987) that the mode competition removes the pattern degeneracy. The resultingpattern has the form of six mutually perpendicular smooth peaks on the surface of theunit sphere. (Two of them at the north and the south pole and four of them mutuallyperpendicular on the equator.) Most filaments are oriented in a mutually orthogonalmanner: the 3D orthogonal network architecture of microfilaments observed in the cortex.In Fig. 2.12b and d the first mode for which (2.40) is satisfied is ii = 2, thus the secondharmonic Y is the leading mode. This mode is rotationally degenerate. It is shown in(Mogilner and Edelstein-Keshet, 1994) that the harmonics P2’ and P die out and theChapter 2. Models for rapid alignment of actin filaments 47leading mode is P’. The pattern evolved is axisymmetric and the angular distributionof filaments looks like two smooth peaks at the north and south pole. The filamentdensity is increased on the poles and diminished on the equator suggesting the alignmentof F-actin in a parallel manner. This type of parallel alignment of microfilaments in 3Dis commonly observed in stress fibers. The competition of different modes, F,, for thecase of the second and the fourth harmonics, Y2 and Y4, is discussed in greater detailin (Mogilner and Edelstein-Keshet, 1994). We conclude that the results of the linearstability analysis of the model in 3D are highly analogous to the results of the linear andnumerical analysis in 2D case, one important difference being the value of the constantC for which the stability breaks. This constant is a combination of the parameters in themodel such as the total mass of actin, M, the kinetic rates of the binding proteins, fi, andthe amount of available binding protein, p. In the 3D case C = 0.005 and C = 0.015, seeFig. 2.12a-d. In the 2D case, C = 0.065 and C = 0.12 (one order of magnitude higher)(see Fig. 2.8a,b,d and e for comparable kernels). This indicates that the steady state ismore stable in 3D than it is in 2D. That is, stronger interactions or higher total mass ofactin are necessary for the organization of filaments in ordered structures. (See Mogilnerand Edelstein-Keshet, 1994-a.)2.8 Nonlinear analysis in 2D and 3DThe instability discussed in sections 2.4, 2.6.1 and 2.7.1 is analogous to the isotropicnematic transition in liquid crystals. In that case, linear stability analysis is sufficient todescribe pattern formation in 2D but not in 3D. A complete bifurcation analysis of ModelI in both 2D and 3D has been carried out in Mogilner and Edelstein-Keshet (1994a). Theyapplied analysis following the synergetics approach (Haken, 1977; Friedrich and Haken,Chapter 2. Models for rapid alignment of actixi filaments 481989) in which an assumption is made that the fastest growing mode controls the amplitudes of all other (slave) modes. A full mode expansion is substituted into the nonlinearmodel and terms up to third power in the leading mode and up to second power in theother modes are kept. This leads to a system of equations for the mode amplitudes. In2D, from these equations, it can be deduced that the bifurcation is supercritical (implyinga non-equilibrium phase transition of second order). This implies that as a governing parameter increases past its bifurcation value the amplitude of the inhomogeneity increasesgradually. In 3D a similar analysis can be applied but the modes are described by theSSH and the calculations are more complex. In contrast to the 2D case, the transitionin 3D case is a transcritical bifurcation implying a non-equilibrium phase transition offirst order. This means that the amplitude of the pattern jumps abruptly from zero to ahigher value. Physically this means that the stable inhomogeneous pattern can co-existwith the homogeneous distribution.2.9 Discussion and conclusionsThe main points and results of this chapter can be summarized as follows:(1)- The model presented here accounts for directional distribution of F-actin withoutconsidering its spatial distribution.(2)- The observed dynamics of assembly and disassembly of F-actin structures in thecell can be explained by relatively simple interactions of the molecules in the cell.(3)- The switch between an orthogonal network and bundles of F-actin may resultsimply from a change in the binding rates or in the concentrations of actin bindingproteins, see Fig. 2.8 and 2.9 for 2D results and Fig. 2.12 for 3D case. These in turn,could be governed by messages received by the cell and expression of the genes codingfor these actin binding proteins.Chapter 2. Models for rapid alignment of actin filaments 49Recent experimental evidence also suggest the key importance of the kinetic rates ofthe binding proteins in the formation of actin cytoskeleton. Wachsstock et al. (1993)and (1994) showed that the structure of actin filament gels depends strongly on affinities(kinetic rates) of the binding proteins.As previously mentioned actin filaments are polar structures with two structurallydifferent ends. The polarity of filaments has not been explicitly included as yet in theabove models, but, in cases where it is important, it can readily be accommodatedby a slight change. In actin structures the filaments can display any of the followingconfigurations: (a) bundles with locally uniform polarity, (b) bundles where filaments arearranged in anti-parallel fashion (opposite polarity) and (c) networks of perpendicularfilaments. The polarized binding of filaments can be accommodated in the model simplyby changing the kernel, K2(O), in section 2.3 to allow binding only in the case of acutecontact angle. An example of this sort would be a kernel as in Fig. 2.3a, but withoutthe hump in the middle. Our conclusions, and the results of the linear analysis and thenumerical computations remain valid also with this type of kernel.Examples of actin structures considered in this chapter include orthogonal networksof filaments observed in the periphery or cortical cytoplasm of motile cells, for examplepseudopods, lamellipodia and membrane ruffles of moving or spreading cells and bundlesof actin filaments observed in stress fibres, microvilli (column-like structures) of epithelialcells and filopodia (finger-like projections) of blood cells (Hartwig, 1980; Hartwig, 1992;Stossel, 1984; Way and Weeds, 1990; Weeds, 1982).We base all interactions and physical and molecular properties on the biological data.Most of the parameters in the model appear in biological literature, but not in formscorresponding exactly to the parameters in the equations. In (Sato et al., 1987), thedissociation constant for the complex Acanthamoeba a-actinin (a cross-linking proteinChapter 2. Models for rapid alignment of actin filaments 50found in amoeba as wefl as in many other organisms (Pollard and Cooper, 1986; Sato etal., 1987; Stossel et al., 1985)) with actin ifiaments has been measured in sedimentationbinding experiments as 26,uM. From this value, they also give estimates of the associationand dissociation rate constants of the a-actinin with F-actin as io — 107M’s and2-200.s—’, respectively. These correspond to our model parameters /3 and.The valuesof these rate constants are known for various other actin binding proteins too (Pollardet al., 1990).The rotational motion of F-actin has been extensively studied (Mossakowska et al.,1988; Phillips et al., 1991; Sawyer et al., 1988; Thomas et al., 1979). A typical value forthe rotational correlation time of actin filaments of average length 1im is (10-lOOjzs),from various cells (for example rabbit skeletal muscle or chicken gizzard smooth muscleactin) have been measured using various techniques, for example by solid-state nuclearmagnetic resonance (NMR) spectroscopy. Note that these are the results of in vitrostudies, and the average length of actin filaments in vitro and in vivo differ significantly(e.g. 1zm and 0.ljtm). The results show that the time scale of filament motion isof the order of microseconds. The rotational diffusion coefficient, z, of F-actiu can becalculated from the rotational correlation time viewing the actin filaments as a rigid bodydiffusing about its long axis. The rotatioual correlation time given above corresponds toa rotational diffusion coefficient (10— 104r).The time scale of dissociation and association rates of the actin binding proteins arecomparable to the time scale of the rotational diffusion rate of F-actiu. Many of the otherparameters in our model, such as the elongation rate constant, 7, or the total filamentconcentration. M, are provided in (Cooper et al., 1983; Cooper, 1991). Typical valuesare: M = 300-400,uM (local concentration in lamellae), and 7 = 107Mr’. We havenot yet gathered a complete set of biological parameters for our equations, but this isChapter 2. Models for rapid alignment of actin filaments 51an important future goal. This is a rather difficult task since the parameters appearingin literature are being measured under different circumstances (some in vitro and someothers in vivo), from various species, and under various chemical conditions.Chapter 3Models for gradual alignment of actin filaments3.1 IntroductionAfter the investigation of the models described in Chapter 2, certain drawbacks of theseequations appeared: while the models were a reasonable description of a process of rapid(nearly instantaneous) alignment mediated by certain binding proteins, they could notdescribe cases in which the actin filaments are gradually pulled into alignment. Theinteractions of myosin with actin are of this type: myosin binds to two actin filamentsand gradually pulls them into an aligned (anti-parallel) configuration. This is referredto as the actin-myosin sliding filament mechanism (Alt, 1987; Mabuchi, 1986). Thisimportant class of proteins could not be omitted from consideration in models of actindynamics, and motivated a new model.As will be shown, a model which accounts for gradual turning differs from the previousmodels in having drift terms (in angle) where these were absent before. In Chapter 2,models consisted of two coupled equations for the bound and free actin filaments, but aswe will show below, the model for gradual alignment can be based on a single equationwhich, by itself, reproduces the phenomena of alignment. We still view the process ofalignment as a phenomenon mediated by a variety of actin-binding proteins, and weconsider not only myosin, but also proteins which would have effects similar to thosediscussed in Chapter 2.Aside from being the simplest type of model for gradual alignment, the model we will52Chapter 3. Models for gradual alignment of actin filaments 53describe here has several convenient featnres: first, it can be analysed for linear stabilityin a straight-forward way. Second, it allows some explicit steady state solutions to befound. These allow us to study several classes of interactions. Some of these are essentiallysimilar to cases discussed in Chapter 2, but some new cases, including that of myosinand unipolar bundling proteins are also included. The results of the model are given forinteractions in two dimensions. (The analysis leading to explicit solutions necessitates2D, but the linear stability analysis can be generalized to 3D in a straightforward way,as before.)In the gradual alignment model, the turning rate of an actin filament is assumed tobe influenced by all other actin filaments, which interact with it in an angle-dependentway. The kernels, in this case, represent the turning rate of one filament towards another.(The convolution K * A is the cumulative turning rate, or drift velocity, induced by theactin density.) Note that this is in contrast to the meaning of convolutions in Chapter 2,where they represented probabilities of binding.As before, the details of the functions taken to represent the turning rates do notheavily influence the predicted behavior. Only some general symmetry properties ofthese kernels are essential. Therefore, in this chapter, we approximate the various classesof interactions (corresponding to various binding proteins) by using two sorts of convenient forms of the kernels. We investigate the linear stability analysis of the modelusing piecewise linear kernels (for which it is easy to compute the fourier transform),and we discuss explicit steady-state solutions using trigonometric kernels (for which theappropriate integrals are easily evaluated.) The symmetry properties of correspondingcases are the same, even though their functional form is distinct. As we will see, thesetwo distinct functional approximations lead to essentially similar qualitative results.Parallel to chapter 2, we extend the model to account for the competition of bindingChapter 3. Pviodels for gradual alignment of actin filaments 54proteins when more than one type is present. The results obtained for cases correspondingto those discussed in chapter 2 are in agreement. This leads to the conclusion that thepredictions are robust, i.e. are not highly sensitive to the precise details of the models.3.2 Angular drift modelWe consider one type of actin density, A, as it is mathematically simpler to deal with asingle equation. Also this model leads to pattern formation with a single equation whichwas not possible in the case of previous models.The two-dimensional model is based on the following variables:0 = an angle, —7r 0 ir, with respect to some arbitrary fixed direction,A(0, t) = the concentration of actin filaments at orientation 0 at time t,K(0) the rate of turning of filaments meeting at a relative angle 0.The rate of turning of a given filament, K, depends on its interaction with all otherneighboring filaments; moreover it depends on the type of binding protein that mediatesthese interactions. The following equation describes how the density of filaments in agiven direction changes through filament-filament interaction mediated by actin bindingproteins:ÔA(0 t) U 8A2Ut = oV(0t)A(0t) + (3.42)V(0, t) = K * A= L K(0 — 0’)A(O’, t)d0’. (3.43)Chapter 3. Models for gradual alignment of actin filaments 55Equation (3.42) is a standard conservation equation:= —V. (Flux), (3.44)FMwhere the Flux term includes both a convective (VA) and a diffusive term (—p--). Thefirst term in equation (3.42) represents a continuous drift with angular velocity V(O, t).This velocity, which is given by the convolution integral, represents the turning of a singlefilament under the cumulative influence of interactions with all other filaments. We assume implicitly that the total effect of the other filaments is a simple superposition. Thisassumption leads to the convolution shown in (3.43) and to the quadratic nature of thenonlinearity. The second term in (3.42), which is the rotational diffusion, represents therandom turning of filaments as before. The drift term represents deterministic dynamicswhile the laplacian accounts for stochastic effects. As before, we do not represent thespatial distribution of filaments. We consider A(O, t), K(O) as smooth periodic functionsof 0 on (—ir, ir). From equation (3.42) it can be shown that the total mass of of thesystem is conserved, so that:M= £ A(0)dO (3.45)is constant.The equation (3.42) is a phenomenological description. However it can be derivedrigorously as a Fokker-Planck approximation from assumptions about the underlyingstochastic turning process (see Segel and Jaeger, 1992).Details of the interactions will depend heavily on the shape of the kernel K which hasdifferent forms to account for different kinds of actin binding proteins. Actin filamentsin the cell orient at various angles to one another depending on the type of actin bindingChapter 3. Models for gradual alignment of actin filaments 56Figure 3.13: Schematic diagram summarizing some actin binding proteins, their shapeand interaction with actin filaments.protein mediating their interaction. We discuss four classes of actin binding proteins inthe section below.3.3 An overview of binding proteinsPollard and Cooper (1986) give an excellent review of actin binding proteins. (See Fig.3.13 for a summary.) Here we restrict our attention to the proteins that bind to the sidesof actin filaments, linking them together. Briefly we can consider four major categories,nnipolar and bipolar bundling, orthogonal networking and myosin. We have alreadydescribed the first three classes in Chapter 2 under the assumption of rapid turning andalignment. We now reconsider them with the gradual turning model. Further, we cannow treat myosin, which is known to mediate slow turning and sliding of actin filaments.sliding cappingbundling actin ifiaments/orthogonal networkingstiffening fragmentingChapter 3. Models for gradual alignment of actin filaments 573.3.1 Unipolar bundling proteins (I)This group includes fimbrin, fasciu, and villin. These proteins promote the formationof unidirectional F-actin bundles in which the filaments have the same polarity. Suchbundles are observed in numerous cell types including the brush border microvifli ofepithelial cells, the cortical microvilli of fertilized egg oocytes, the streocilia of cochlearhair cells, the processes extending from the surface of blood platelets and others. Theactin binding proteins in these structures hold the filaments tightly together in bundles.The physiological purpose of this type of actin structure is to stabilize cell protrusions, e.g.for the purpose of increasing surface area for exchange of material with the surroundings(Pollard and Cooper, 1986; Pollard, 1990; Stossel, 1984). While we only have informationabout the final configuration of the actin bundles we can speculate about the dynamicsthat lead to its formation. In particular, it is evident that the filaments have a tendencyto converge to a parallel orientation in the presence of these proteins, with the pointedends all converging to the same direction.3.3.2 Bipolar bundling proteins (II)Bipolar bundling proteins attach filaments to each other both in parallel and antiparallel.These binding proteins are generally observed in stress fibres on the ventral surface ofmammalian fibroblasts, in epithelial cells in culture, in endothelial cells in vivo, and inthe cytoplasm of amoebas. Annular rings seen around the whole cell consist of suchbidirectional arrays of fibers, closely resembling stress fibers. Similar filament bundlesare observed at the peripheral margin or the leading edge of blood platelets and of tissueculture cells. Spectrin, tropomyosin and a-actinin are from this group and most of themcan bind the sides of two different actin filaments (Pollard and Cooper, 1986; Pollard,1990). It is evident that filaments converge to either parallel or anti-parallel orientationChapter 3. Models for gradual alignment of actin filaments 58and we will assume this occurs with equal probability.3.3.3 Orthogonal networking proteins (III)There is a class of binding proteins that promote orthogonal networks. Orthogonalnetworks of actin filaments are often observed in the periphery of motile cells such asamoebas, macrophages, leukocytes and some blood platelets. In these structures, longactin fibers are linked together in oblique or right angle relationships. A substantialmajority of filaments in the periphery of the cell cortex consist of short fibers and areoften found in T- and X-shaped junctions. Such binding proteins include ABP andfilamin (Stossel, 1994; Stossel, 1990; Pollard and Cooper, 1986). In particular, filamin isbelieved to be a floppy hinge with two nearly perpendicular arms. Stossel suggests thatthe role of the orthogonal networking proteins is to preserve the isotropic 3D structureof the cytoskeleton (without filamin, bundling and cross-linking proteins would collapsethe network into linear structnres, changing the mechanical properties of the cytogeldrastically).3.3.4 Myosin (IV)The small bipolar myosin molecules arrange actin filaments in bidirectional bundles bypulling adjacent randomly oriented fibers against each other. Such bundles have a role incontractile events associated with cytokinesis and motility such as endocytosis, exocytosisand membrane ruffling. A functional myosin unit consists of a complex of two myosinmolecules. Each molecule has an active head capable of binding to and walking towardsthe plus end (the pointed end) of an actin filament. If two actin filaments are spatiallyfixed at their ends and connected by the myosin complex then the gliding of the myosinheads along the filaments leads to the gradual turning of the filaments to an anti-parallelChapter 3. Models for gradual alignment of actin filaments 59configuration. The velocity of gliding of myosin is about liz/s (see Peskin et al., 1994).This means that, in this case, the process of alignment of the attached microfilamentsis not rapid as in Chapter 2. The gradual turning modeled by equation (3.42) is moreappropriate for this case.3.4 Classification of KernelsWe assume that the binding protein causes the gradual alignment of filaments to whichit is attached. As there is no quantitative data on the dynamics of this process, wewill consider several reasonable scenarios. In this section we introduce a few functionalrepresentations of the alignment rates. That is, we suggest possible forms for the kernelswhich would be representative of the classes of actin binding proteins introduced in section3.3. We will assume symmetry of turning towards positive and negative directions (weneglect any possible chirality of the molecular interactions). It then follows that allkernels K(O) are odd functions, so that:L K(O)dO = 0. (3.46)Two basic types of functions (leading to four distinct kernels) will be investigated:( L3+aO forO0,(a) K(O) =—3+aO forO<0.(3.47)a(O—7r) for—7rO<—,(b) K(O)= riO for —<O<a(O+vr) for<O<irIn this model the kernel is not normalized (i.e. the integral of )K) is not set equal to1). This means that the rate constants are included as part of the expressions for K.Chapter 3. Models for gradual alignment of actin filaments 60Figure 3.14: Shapes of the kernels K representing various types of interactions betweenactin binding proteins and actin. The values (in particular the signs) of theparameters a and j3 determine the type of interaction the kernel represents.In (a) the kernel is of type 3.47a with = —land a = —1, unipolar bundlingproteins, in (b) the kernel is of type 3.47a with 3 = 1 and a = 1, bipolarbundling proteins, in (c) the kernel is of type 3.47a with = 4 and a = —1,bipolar bundling proteins, and in (d) the kernel is of type 3.47b with a = 1,orthogonal binding proteins.rfhe parameters a and ,8 determine the type of interaction represented. a is the slopeof the function K(O), and is the y-intercept of K, the magnitude of the velocity ato = 0. (See Figure 3.14 for the graph of kernels of type a and b above for some choices ofthe parameters a and.) The biological and mathematical meaning of these parametersis discussed in greater detail below for each of the cases. We now describe the specificforms these kernels take for different choices of a and , and the binding protein theyrepresent.b)—. thetaz. theta/theta—ir/1Chapter 3. Models for gradual alignment of actin filaments 61I’direction of V-Ic Itrelative angleFigure 3.15: Shown is the direction of the angular drift due to interactions of ifiaments(mediated by unipolar bundling proteins) meeting at a relative angle 9. Thiscase is referred to as ‘attraction’ since filaments converge to a configurationwhere they have the the same orientation. Note that the direction of movement is counterclockwise when the kernel is negative and clockwise when thekernel is positive.3.4.1 Kernels for unipolar bundling (I)We first look at some kernels representative of interaction between filaments mediatedby unipolar bundling proteins. In these cases, the kernel is of the form (3.47a) wherej3 0 (see 3.2). Thus the rate of change of the relative angle between filaments K(9) isnegative for positive 0 and positive for negative 0, that is motion is towards 0 = 0 so thefilaments converge towards each other, a parallel configuration (see Fig. 3.15). We referto this convergence as ‘attraction’.We further consider the following specific cases:(1) 5 = 0, a < 0.The filaments oriented at acute relative angles are attracted weakly to zero while thoseat obtuse relative angles are attracted strongly. Note that the angular velocity is zero ifChapter 3. Models for gradual alignment of actin filaments 62the relative angle is zero.(2) < 0, a = 0.In this case, the angular velocity is independent of the relative angle. All filaments areattracted to a relative orientation U = 0 at the same rate.(3) < 0, a < 0.This case is same as case (1), however the angular velocity is non-zero for all angles. (SeeFig. 3.14a for the graph of a kernel of this type.)(4) 3 < 0, a> 0 and fi < —ar.This case is similar to case (3) with the exception that here filaments oriented at acuterelative angles are attracted to a zero relative angle strongly while those oriented atobtuse relative angles are attracted weakly.3.4.2 Kernels for bipolar bundling (II) and myosin (IV)In these cases, the kernel is either of the form (3.47a) where j3 0, or (3.47b) wherea < 0 (see 3.2). In case (3.47a), K(O) is positive for positive 0 and negative for negative 0so motion is away from 0 = 0 and filaments diverge towards an anti-parallel configuration(see Fig. 3.16). We refer to this as ‘repulsion’. In case (3.47b) K(0) is negative for acuterelative angles and positive for obtuse relative angles implying the tendency for parallelalignment at acute angles and for anti-parallel alignment at obtuse angles.We consider the following specific cases for the kernel (3.47a):(1) /3 = 0, a> 0.The filaments oriented at acute relative angles are repulsed weakly from zero while thoseat obtuse relative angles are repulsed strongly. At 0 = 0 the angular velocity is zero.Chapter 3. Models for gradual alignment of actin filaments 630relative angleFigure 3.16: Shown is the direction of the angular drift due to interactions of filaments(mediated by bipolar bundling proteins) meeting at a relative angle 0. Thiscase is referred to as ‘repulsion’ since filaments converge to a configurationwhere they have anti-parallel orientations.(2) 41> 0, a = 0.In this case the angular velocity is again independent of the relative angle. Filaments arerepulsed from 0 = 0 equally regardless of their relative orientation.(3) 3> 0, a> 0.This case is same as case (1). However the angular velocity is non-zero for all angles.(See Fig. 3.14b for the graph of a kernel of this type.)(4) 3> 0, a < 0 and /3> —air.This case is similar to case (2) with the exception that filaments oriented at acute relativeangles are repulsed from 0 = 0 strongly while those oriented at obtuse relative angles arerepulsed weakly. (See Fig. 3.14c for the graph of a kernel of this type.)-Tvdirection of VICChapter 3. Models for gradual alignment of actin filaments 64/direction of V-It -It/2 It/2 It-—--—Urelative angleFigure 3.17: Shown is the direction of the angular drift due to interactions of filaments(mediated by orthogonal binding proteins) meeting at a relative angle Kernels for orthogonal binding (III)In view of the structure of filamin, we assume that filaments converge to an orthogonalconfiguration from both acute and obtuse relative angles. This suggests kernels of theform (3.47b) where a > 0 (see 3.2). (See Fig. 3.14d for the graph of a kernel of thistype.) The rate of motion in this case is illustrated in Fig. Competition between two types of binding proteins (V)We now consider the case of competition between two types of actin binding proteins whenthey are simultaneously present in some proportions. We consider two cases, namely (1)myosin and orthogonal binding proteins and (2) the bipolar bundling and the orthogonalbinding proteins. We represent the interaction of F-actin with two types of biudingproteins by simply taking a linear superposition of the kernels of type (a) and (b) in(3.47). (We are making a simplifying assumption, i.e. that the binding proteins do notinterfere with one another, as this would create nonlinear effects that are not describedChapter 3. Models for gradual alignment of actin filaments 65here.)(1) Orthogonal binding proteins (III) and Myosin (IV)Both kernels for type III and IV interactions are given by the basic form (3.47b) (see3.2). However, the coefficient a is positive in the case of orthogonal binding protein(III) and negative for myosin (IV). We will define K1(O),K2(O) to be kernels of the form(3.47b) with a1 > 0 (III) and a2 < 0 (IV) in place of a, respectively. The coefficientsa1 and a2 are called the ‘effectiveness’ parameters, and represent the combined effectsof binding rate constant and concentration of the binding protein. (Note the similarityto the combined effects of binding proteins described in section 2.5 where jpj representthe binding rate and concentration of binding protein i.)(2) Orthogonal binding (III) and Bipolar bundling proteins (II)The kernel for orthogonal binding kernel is (3.47b) with coefficient a1 > 0, whereas thebipolar bundling is simply given by (3.47a), with /3 > 0 and a2 > 0 (see 3.2). As in (1)above, the coefficients can be viewed as the ‘effectiveness’ parameters of the two bindingproteins.3.5 Linear stability analysisWe now examine steady states of equation (3.42) and their stability. Later on we will focuson the specific forms that the stability criterion takes for different kernels. The analysisof the model is similar to the analysis of the model in Chapter 2. The homogeneoussteady state A of the equation (3.42) satisfies:aA DA-= 0 = -a-, and A = constant. (3.48)Chapter 3. Models for gradual alignment of actin filaments 66Kernel type parameters type of proteinType (a) /3 0, a e J? Tinipolar bundling (I)Type (a) /3 0, a e Bipolar bundling (II)or orType (b) a < 0 Myosin (IV)Type (b) a > 0 Orthogonal binding (III)Type (b) a1 < 0 Orthogonal binding+ +Type (b) a2 > 0 Myosin (V) (1)Type (b) a1 < 0 Orthogonal binding+ +Type (a) /3> 0, a2 > 0 Bipolar bundling (V) (2)Table 3.2: Table summarizing the types of kernels and the parameters a and /3,representing different actin binding proteins. The forms of the kernelof types (a) and (b) are given in 3.47.Chapter 3. Models for gradual alignment of actin filaments 67Note that any constant level of the variable is a steady state, but the value of theconstant is determined by the total mass which is conserved by equation (3.45). We wishto examine the stability of this uniform steady state. As for Model I in Chapter 2 ,terms are the eigenfunctions of the two operators appearing in the equation, namely theLaplacian in 1D, and the integral operator. This fact greatly simplifies the linear analysisof equation (3.42). Thus, we consider perturbations of the form:A(O, t) = A +A0e°J’. (3.49)where A0 is a small amplitude, lv is the wave number and A is the growth rate of theperturbation. We seek conditions for which such small perturbations from the steadystate are amplified with time, i.e. for which A > 0 for some nontrivial wave number lv.Substituting (3.49) into (3.42) and retaining the linear contributions we find that:AAoe3CSe)\t = _ikAA0euJOe)tk —1uk2A0e”°e’t, (3.50)where ft is the Fourier transform of the kernel K, namely:k(k)= £ K(O)ethO dO. (3.51)By cancellation of common factors in (3.50) we find that the growth rate of the perturbations, A, is:A = —ikAf(— 1k2. (3.52)Instability occurs when A is positive. This leads to the following dispersion relation:Chapter 3. Models for gradual alignment of act in filaments 68Ck2 < —ikk (3.53)where,C = = 2ir-j. (3.54)Note that the coefficient C is inversely proportional to the total mass M. We nowexamine in detail what the dispersion relation implies in each type of interaction andkernel ontlined in Sections 3.3 and 3.4.Since the inequality (3.53) depends on the Fourier transform, K, of the kernel K,we prepare the way by computing K, the Fourier transforms of the two basic kernels(3.47a-b) considered in the previous sections. The exact forms of the Fourier transformsare:(a) K(k)= _t(afr cos k7r — sin kK) + 3(cos kK — 1)),(3.55)(b) k(lv) = —(Kcosf_ sinkK).We note that for Ic = 0, k = 0 since K is an odd function for all cases consideredhere. This implies the neutral stability of the homogeneous distribution caused by theconservation of mass. Substituting (3.55a-b) into (3.53) we find the two basic forms ofthe dispersion relation:(a) Ck2 <2(wr cos kir + fl(cos kK — 1)),(3.56)(h) Ck2c(2a7rcosk.Chapter 3. Models for gradual alignment of actin filaments 69We look for the smallest integer wavennmber k which satisfies (3.56). The homogeneonsdistribution destabilizes in favor of the first wave number (the smallest one) which wouldthen govern the growing pattern until the system is drawn far from equilibrium wherenonlinear effects of competing wave numbers dominate. By previous remarks about C.we observe that as the total mass, M, increases, the left hand side of (3.56a-b) decreasesso that the inequality can be satisfied. The first value of M for which these inequalitiesare satisfied (for some integer k) will be called the ‘critical mass’. We now comment onthe specific cases 1-4 in section 3.4. The outcome of the dispersion relation, i.e. the firstmode number causing instability, for the cases considered below is summarized in Table3.3. In Fig. 3.18a-d the left and right hand sides of the dispersion relations (3.56a orb) corresponding to the kernels of type (3.47a or b) shown in Fig. 3.14a-d are plotted.Notice the first integer wave number k causing instability (i.e. the first integer Ic forwhich the parabola is below the curve) in each case.3.5.1 Dispersion relation for unipolar bundling (I)In all cases listed below the kernel is of the form (3.47a) with corresponding Fouriertransform (3.55a), and dispersion relation (3.56a).(1)j3=O,a<O. Ck2<2cv’ircoslcvrFor even wave numbers, the right hand side of the inequality is negative. For odd wavenumbers, Ic = 1, 3, 5,... the right hand side is 2air. Thus, the first mode which breaks thestability is Ic = 1. This means that perturbations of the form e0 will grow. Hence a singledirection in [—ir, ir] is accentuated. Most filaments align along this favored direction.(2)5<0, a=0. Ck2<25(coskw—1)In this case, the right hand side of the dispersion relation is zero for even wave numbersChapter 3. Models for gradual alignment of actin filaments 70\kFigure 3.18: The expression on the right hand side of (3.56a or b) is shown as a functionof the wavenumber k. K is as in Fig. 3.13a-d respectively, with a and 3same as in Fig. 3.13a-d. Superimposed in each graph is a parabola for whichthe coefficient C is chosen to satisfy the inequality. In (a), (b) and (c) thecoefficient C of the parabola is 1, and in (d) C = 0.3. The first wavenumber for which the uniform steady state is destabilized is k = 1 in (a) i.e.perturbations of the form e8 grow, resulting in one accentuated orientation(a unidirectional structure). For (b) the first such wavenumber is k = 2i.e. perturbations of the form e2’9 grow, resulting in two accentuated orientations 180° apart (bundles). For (c) the wavy function assumes negativevalues for all k. Thus the inequality cannot be satisfied for any value of k, sothe homogeneous distribution is stable. For (d) the first such wavenumber isk = 4 i.e. perturbations of the form e46 grow, resulting in four accentuatedorientations 90° apart (orthogonal networks).Chapter 3. Models for gradual alignment of actin filaments 71and positive (and equal to —43) for all odd wave numbers, k = 1, 3, 5... . Thus, the firstdestabilizing wave number is k = 1. This leads to unidirectional alignment of filamentsas in case (1).(3) /3 < 0, a < 0. C1c2 < 2(cnr cos kir + /3(cos kc’r — 1))In this case, for even Ic, the right hand side of the inequality is negative. For oddwave numbers Ic = 1, 3,5,... it is positive and equal to —2Qira + 2/3). Thus, the firstdestabilizing mode is Ic = 1. Filaments align in a unidirectional fashion as in the previouscases. (See Fig. 3.18a for visualization of the dispersion relation corresponding to a kernelof this type, shown in Fig. 3.14a.)(4) /3<0, a>0 and —/3> air. Ck2 <2(aircoskir+/3(coskir— 1))For even wave numbers, Ic = 2, 4, 6,... the right hand side is positive and equal to 2air.For odd wave numbers it is also positive and equal to —2air— 4/3> 2air. Thus, the firstdestabilizing mode is Ic = 1 as before.3.5.2 Dispersion relation for bipolar bundling (II) and myosin (TV)We first consider kernels of the form (3.47b) with a < 0. As shown above, this leads tothe dispersion relation (3.56b). The sign of the expression on the right hand side of theinequality (3.56b), namely 2air cos Icir/2, determines the outcome of stability. For oddwave numbers, Ic = 1, 3, 5... this expression is zero. For wave numbers which are evenmultiples of 2, Ic = 4, 8, 12... it is negative, and for odd multiples of 2, Ic = 2, 6, 10... itis equal to —2air > 0. Thus, the first wave number which breaks the stability is Ic = 2.This means that perturbations of the form e129 will grow and thus two orientations 180°apart become accentuated. Hence filaments bundle in parallel and antiparallel fashionequally.Chapter 3. Models for gradual alignment of actin filaments 72Next, we consider four cases of the dispersion relation of type (3.56a) correspondingto four kernels of the form (3.47a), with fi 0, given in section 3.4.2 (1), (2), (3) and(4).(1) /3=0 and a>0. Ok2 <2aKcosklrClearly, the smallest k for which the inequality holds is k = 2. Thus, two directions, rdegrees apart, will grow in [—K, K]. The uniform steady state will he broken with theappearance of two peaks 1800 apart again.(2) 3>0 and a = 0. Ok2 <2/3(coskr— 1)In this case, the right hand side of the inequality is negative for odd wave numbers,k = 1, 3, 5, 7, ... and is equal to zero for even wave numbers. The homogeneous steadystate is stable to all perturbatious and no pattern will form. We conclude that somedzfference in the interaction (turning rate) at different angles is essential for disruptionof homogeneity.(3) 3> 0 and a> 0. Ok2 <2(aK cos kir + 5(cos kK — 1))The first wave number satisfying this inequality is again k = 2. (See Fig. 3.18b forvisualization of the dispersion relation corresponding to a kernel of this type, shown inFig. 3.14b.) This can be easily seen from the dispersion relation (3.56a) since for oddwave numbers, k = 1, 3, 5... the right hand side is negative and for even wave numbersit is equal to 2aw.(4) /3>0, a< 0 and /3> —a. Ok2 < 2(aKcoskK+/3(coskK— 1))In this case the uniform steady state is stable to all perturbations. This follows from thefact that for odd wave numbers, the right hand side is equal to —2(aK + 2/3) which isnegative, and for even wave numbers it is equal to 2aK, which is also negative. (See Fig.Chapter 3. Models for gradual alignment of actin filaments 733.18c for visualization of the dispersion relation corresponding to a kernel of this type,shown in Fig. 3.14c.) Thus, these types of interactions do not lead to pattern formation.From the results of (2) and (4) we conclude that in the case of bipolar bundling (II)and myosin (IV), i.e. when the interactions are ‘repulsive’, the necessary condition forpattern formation is that the rate of repulsion at acute angles is smaller than the rate ofrepulsion at obtuse angles. The biological implication of this result is as follows: if thecombined effects of the substances present in the cell cause greater repulsive interactionsat acute angles than at obtuse angles, then bundles will not form, even in the presenceof bundling proteins or myosin.3.5.3 Dispersion relation for orthogonal binding (III)The interactions between filaments mediated by orthogonal binding proteins lead to thedispersion relation (3.56b) where a is positive, namely Ck2 < 2air cos kir/2. For oddwave numbers, k = 1, 3,5... the right hand side of the ineqnality is zero, for evenwave numbers which are odd multiples of 2, lv = 2, 6, 10... it is negative, and for evenmultiples of 2, lv = 4, 8, 12... it is positive. (See Fig. 3.18d for visualization of thedispersion relation corresponding to a kernel of this type, shown in Fig. 3.14d.) Thus,the first wave number which breaks the stability is lv = 4, meaning that four directions90° apart will be accentuated, and four peaks will appear. In the presence of filamin, forexample, orthogonal networks of actin filaments would therefore be promoted.3.5.4 Dispersion relation for competition of binding proteins (V)Since the kernel for this case is given by a linear superposition, K(O) = K1(O) + K2(O)(see section 3.4.4 (1)), the Fourier transform will also be a simple linear superposition,i.e. K= Ki + K2 with K1 and K2 as in (3.55a-b).Chapter 3. Models for gradual alignment of actin filaments 74(1) Dispersion relation for myosin (IV) and orthogonal binding proteins (III)In this case, K-i and k2 are of the form (3.55b). Therefore, the dispersion relation, whichis of the form (3.56b) is given by:Ck2 <2(ai +a2)fr cos ). (3.57)Since a1 is positive, and a2 is negative, if jail < 1a2 then (ai —I- a2) is negative. In thiscase, the first mode to satisfy (3.57), and thus break stability is lv = 2. This implies thatfilaments align in a bidirectional fashion. The effect of myosin dominates over orthogonalbinding proteins, so that bundling takes place. If, on the other hand, jail > a2j thenthe coefficient (ai + a2) is positive, and the first mode to break stability is lv = 4. Thus,in this case, the effect of the orthogonal binding proteins dominates and filaments forman orthogonal network.Consider the effect of the presence of a ‘passive’ binding protein, i.e one with smallerrate constant, or with lower concentration (implying a lower ‘effectiveness’ parameter).The critical mass at which instability occurs in this case would be larger than in the caseof a single binding protein because, from previous remarks (see section 3.5), the mass forwhich instability can occur is of the order M a1 +a2jt This means that decreasingthe value of the coefficient ai + a2j hinders the appearance of order. If both types ofbinding proteins are equally effective, aj = a2j, the homogeneous solution is stable.(2) Dispersion relation for bipolar (II) and orthogonal binding (III) proteinsIn this case, the Fourier transform of the kernel representing bipolar binding is of theform (3.55a) with /3 > 0 and a2 > 0, and the Fourier transform of the kernel representingorthogonal binding is of the form (3.55b) with a > 0 as in case (1). Thus the dispersionrelation is:Chapter 3. Models for gradual alignment of actin filaments 75Ck2 < 2(aiKcosk7+(cosk— 1)+a2rcos). (3.58)The results in this case are exactly the same as those of case (1) above, and are independent of the parameter 3. (Note that the term containing fi in (3.58) is either negative orzero for any integer k.) If a1 < a2 then k = 2 is the first unstable wavenumber, and ifa1 > a then k = 4 is the first unstable wavenumber. If a1 = a2, the homogeneons stateis stable.The above results are valid only near bifurcation and will not predict the type oforder away from bifurcation where non-linear effects may dominate. For further analysisand an explicit solution in this case, refer to section Steady state equation and explicit solutions in three special casesFor kernels of the special form described below, it is possible to obtain an explicit formulafor the steady state of equation (3.42). We first note that the steady state of (3.42)satisfies A = 0, that is:02A a— A(K * A) = 0. (3.59)Integrating once over —K U we obtain:= f(O)A(O) + C, (3.60)where C is the constant of integration and f(O) = (K * A)(O).The steady state equation (3.60) is a first order linear ODE, and its general solutionis:Chapter 3. Models for gradual alignment of actin filaments 76Kernel Group First Modeand Type causing instabilityUnipolar bundling (I) k 1Bipolar bundling (II)or k=z2Myosin (IV)Orthogonal binding (III) k = 4Myosin (IV) if a1 < a2 then k = 2+Orthogonal binding (III) if Iai > a21 then k = 4Bipolar bundling (II) if ai < a2 then k = 2+Orthogonal binding (III) if a > a2 then k 2Table 3.3: Table summarizing the outcome of the dispersion relation for differenttypes of kernels. The first mode causing instability is k = 1,2 or4 depending on the type of interaction the kernel represents. Thiswave number k is the number of accentuated orientations breaking thehomogeneous distribution.Chapter 3. Models for gradual alignment of actin filaments 77A(O) = (D + G(O)) exp(F(O)), (3.61)where,F(O) = I f(O)dO,1 (3.62)G(O) = fexp(——F(O))dO,Note that the general solution (3.61) contains two arbitrary constants, C and D. Weneed further conditions to uniquely determine the values of these constants; in this casewe use periodicity of the boundaries (BC) and the normalization condition (3.45) to solvefor these constants. We first note that C = 0 by observing that it represents the total fluxof material (see equation (3.60)). Thus, if we wish to avoid travelling waves of densitycirculating through the periodic domain, we must set C = 0. Therefore, the solution isof the form:A(O) = Dexp(F(O)). (3.63)We now solve for the constant D and the function F. For some special choices of thekernel, K, it is possible to determine the function F. In the next section we present theanalytical solutions for a class of kernels of a special form and compare the results withthose obtained by linear analysis.3.6.1 Steady state solutions for a special class of kernelsWe now consider the class of kernels represented by:K(O) = BsinnO, ii = 1,2,4. (3.64)Chapter 3. Models for gradual alignment of actin filaments 78Note that B is a constant (unrelated to a similar symbol in Chapter 2) and that B> 0corresponds to repulsion, whereas B < 0 corresponds to attraction. A remarkable featureof such integral kernels is that they are degenerate: meaning that they can be representedas a finite sum of the products of the eigenfunctions of the linear integral operator (SeeYoshida, 1960). In particular, the kernel (3.64) has the following finite representation interms of the eigenfunctions:K(O — 0’) = B(sin nO cos nO’— cos nO sin nO’). (3.65)This simplifies the integral term in equation (3.42) greatly. Indeed, with this representation the velocity convolution term (3.43) can be written as:f(O) = (K * A)(0) = c1 cos nO + c2 sin nO (3.66)where,= —B £ sin nOA(O)dO , c2 = —B £ cos nOA(O)dO. (3.67)We now wish to determine the constant D and the function F. To do so, first observethat for the particular choice of kernel given by equation (3.64), the function F(O) isgiven by:F(O)= f f(O)dO = sin nO + cos nO. (3.68)We rewrite F(O) as a phase-shifted cosine:F(O) = cos(nO— w), (3.69)Chapter 3. Models for gradual alignment of actin filaments 79where w is some phase-shift (w is easily calculated from simple trigonometric identities).This function has a peak at w/n and a minimum at (w + ir)/n. To simplify calculationswe bring (3.63) to the following symmetric form by redefining the variable 0 so that0 = w/n corresponds to the origin. (This is equivalent to measuring all angles relativeto the angle at which F has a peak.) Thus, with this new definition of 0, the equation(3.63) can now be written as:A(0) = Dexp(_-ZcosnO) (3.70)where.c = Bj cos nOA(0)dO (3.71)3.6.2 Relations satisfied by the constants c and DNote that the value c is determined by the angular distribution of A(0). To determinethe constant D in (3.70) we use the normalization condition (3.45) on A(0):M= £400 = Djexp(—-_cosn0)d0. (3.72)We observe that the integral in (3.72) involves the modified Bessel function of order zero:10(z) = ifrec050do. (3.73)Thus, M can be written in the following form:Chapter 3. Models for gradual alignment of actin filaments 80M = 2irDI0(——). (3.74)‘an(See Abramowitz and Stegun (1970).) So the constant D can now be written as:MD= (3.75)22r10( —)‘anThe total mass, M, is in principle fixed in the system and known. However c depends onthe angular distribution of A, and is not known or predetermined. Thus to understandequation (3.75) we need to investigate c.Writing the constant c in (3.71) with the expression of A(O) given in (3.70) we findthat:= BDjexp(—.S_cosnO)cosnOdO. (3.76)We observe that the modified Bessel function of order one:Ii(z) = jezc050cosodo. (3.77)appears in this equation, so that:Ii(----)c = B1i/I ‘an, (3.78)Io(--)‘anNote that (3.78) is a transcendental equation for the parameter c, which we now investigate.Chapter 3. Models for gradual alignment of actin filaments 813.6.3 Investigating the transcendental equation for cLet us define the new variable X, byX = —s—. (3.79)‘InFor ease of notation define:L(X) = R(X)=(3.80)(Note that here L and B have meanings which are unrelated to similar symbols in Chapter2.) Then (3.78) is equivalent to:L(X) = R(X). (3.81)Note that 10(X) is an even function, and 1(X) is an odd function so that R(X), theright hand side of (3.81) is odd. We consider p, B and n as fixed parameters and Mas the gradually varying parameter of the model. The solution to this equation can bevisualized as the intersection points of L(X) and R(X) where their graphs are plottedsuperimposed: see Fig. 3.19. L(X) is plotted for various values of the coefficient involvingthe parameter M in Fig. 3.19 to illustrate cases having a single or three solutions. Therecan be solutions for both positive and negative coefficients of L(X). Note that theparameters p, n and M are positive for all biologically relevant cases. Thus, it sufficesto consider the following two cases: B is positive (i), and B is negative (ii). It is easilyseen that in all cases X = 0 is a trivial solution of the equation (3.81). Other solutionsdepend on the sign of B and we consider the following cases:(i) B > 0: This corresponds to the case of repulsion at 0 < rr. The trivial solutionX = 0 is the only solution in this case (see Fig. 3.19 and note the intersection of y = L(X)Chapter 3. Models for gradual alignment of actin filaments 82Figure 3.19: Solution of the equation (3.81) visualized as the intersection of L(X) (shownfor various values of the slope), and R(X) involving modified Besselfurtctiorzsof order one and two. Note that there is a unique solution (intersection) atX = 0 for large negative slopes or all positive slopes and two additionalsymmetric solutions X1 = —X2 appear for negative slopes smaller than acertain value, referred to as the ‘critical slope’.A(O) = D = (3.82)27rI(0)As all quantities in (3.82) are fixed so that the steady state solution is uniform in 0.Thus, in this case, the only steady state solution is one in which the density is constant(Tith positive slope) with y = R(X)). This means that c = 0 (see (3.79)), which in turnimplies that F(0) = 0 (see (3.69), and:Mfor all angles.Chapter 3. Models for gradual alignment of actin filaments 83(ii) B < 0: This corresponds to the case for which filaments with relative angle 0,where 0 < r are attracted. We observe from Fig. 3.19 that the number of intersectionsof L(X) and R(X) depends on the slope of L(X), and therefore on the value of theparameter M. (Note the inverse relationship of slope and M in L(X) in (3.80).) Inparticular, for large negative slope, there is only one intersection, at the origin. (In thiscase, the steady state A(0), given by (3.82), will be uniform as argued above.)For small negative slopes (or equivalently for large M), we will have three intersections: one at the origin, and two symmetric intersections at X = +X. (Note that theseintersections depend on M, i.e. XE = X(M), and furthermore X is an increasing functionof Al.) Now define M to be the value of M for which the two additional intersections justappear. For M> M the slope is small and negative and thus, two non-trivial solutions,X and —X occur. This means that there are two values of c, but as the function Iis even, this will correspond to a single value of D in equation (3.75). Thus, we haveessentially shown the existence of a steady state (3.70) which is non-homogeneous.3.6.4 The explicit steady state solutionsOur next goal is to actually characterize the shape of this non-homogeneous solution. Todo this, we need to take the following steps: (a) Use asymptotic approximations of themodified Bessel functions in equation (3.80) to determine both M and X. (b) Use thesevalues to determine D and F, and eventually, the steady state A(0).First note that the critical mass M corresponds to the case when the slope of L(X)in (3.81) is negative and large enough so that the straight line is tangent to the curveR(X). Also note that since the slope and the mass M are inversely proportional, a largeslope corresponds to a small M = M, and hence a small X value. In order to calculatethe value of the critical mass, M, we use asymptotic expansions of the modified BesselChapter 3. Models for gradual alignment of actin filaments 84functions. We separate this into two cases, namely small and large values of X. (It isevident from Fig. 3.19 that the behavior of R(X) in (3.81) is quite different close to andfar from the origin: for large X this function is nearly constant, and for small X it canbe approximated by a line of slope -1/2.)We first calculate the value of M using the first terms in the asymptotic expansionsof the Modified Bessel functions in (3.81) for small X. For X near 0, retaining up toquadratic terms in the polynomial approximation we have:10(X) 1 and, 1(X) for X ‘P0. (3.83)(See Abramowitz and Stegun (1970).) Substituting these values for J and I in (3.81)we obtain:L(X) = —AX. (3.84)Equating the slopes of the functions on the left and right hand side of (3.84) and solvingfor M leads to:M= 2jtn (3.85)So for M > M, the non-homogeneous solution of (3.70) are = X(M). SinceX(M) is a monotonously increasing function of M the inhomogeneity in the system ismonotonously increasing with growing mass.We now calculate the solution X to (3.81), then obtain c using (3.79) and finally Dand A(O) using (3.75) and (3.70) for two limiting cases: close to bifurcation (a): Mand far from bifurcation (b): M>> M.Chapter 3. Models for gradual alignment of actin filaments 85(a) In this case, the mass is near criticality, that is M = M + AM and AM << M,where M0 is the small quantity given in (3.85). Since X decreases monotonously withit/I we are looking for a small solution X to (3.81) in this case. Taking up to cubic termsin the polynomial approximations of J and I, for small X we have:10(X) —‘ 1 + 0.32X2 and, 1(X) X/2 for X—, 0 (3.86)(See Abramowitz and Stegun (1970).) Substituting these values in R(X) in (3.80) and(3.81), expanding the terms in Maclaurin series and retaining up to cubic terms we obtain:- 1—BAMX c 1.4sf. (3.87)V inhence, using (3.79)/-BAMc 1.4jmj/. (3.88)V tnThis value for c, together with the Maclaurin series expansion of the exponential termleads to the following form of (3.70) for the angular distribution of the density A(O):A(O) = (1 + acosnO) + O((7)2), (3.89)where,!—BAMa = 1.4/ (3.90)V 1unNote that the coefficient of the cosine term a is small since AM is a small deviationfrom the critical mass. Thus, equation (3.89) describes ri small peaks superimposed onChapter 3. Models for gradual alignment of actin filaments 86a larger constant distribution. In the next section we discuss this solution for specificchoices of n.(b) In this case the actin filament density is high, that is M>> M. Here we are usingthe first terms in the asymptotic expansions of I and I for large X values. FollowingAbramowitz and Stegun (1970) we have:110(X), 1(x) xTheX for X (3.91)Substituting these values in (3.81) we obtain:- BMX—. (3.92)/-tnSimilarly to the case (a) we calculate the constant c from (3.79) then the constant Dfrom (3.75) and finally we obtain the following expression for the angular distributionA(O) iu (3.70):A(O) = /3e7@05 nO — 1), (3.93)where,=-(1 + O()), 7 = M(l + Q(MC)) (3.94)Note that, in this case, the solution has the form of n-peaks located at the points:i=0,1,..(n—1) (3.95)Chapter 3. Models for gradual alignment of actin filaments 87In the vicinity of each of these points, O,, A(O)has the following form obtained by nsingthe first two terms of a Taylor series for cos riO in (3.93):A(O) = exp(-(O - O)2) (396)Hence, the width of each peak has order of magnitnde:d= VBZn (3.97)Observe that the constant B which affects the value of the angular drift velocity inequation (3.42) (note that B is a coefficient of the kernel K in (3.64) and the kernelappears in (3.43)) also affects the width of the peak. High values of B represent highangular drift, meaning strong binding and alignment, and this corresponds to sharp peaks(small d in 3.97) in the steady state solution. Also note that as the total mass of actinfilaments, M, increases the peak becomes sharper and the inhomogeneity in the angulardistribution increases.3.6.5 Examples of several steady state solutions for specific kernelsWe now discuss briefly the implication of the steady state solutions obtained in theprevious section in biologically relevant cases, namely for n = 1, 2 and 4 in (3.64).Unipolar bundling (I). n = 1, K(O) = BsinO, (B <0).This kernel represents the attraction filaments at all relative angles corresponding to thecase of unipolar bundling. Indeed the solutions (3.89) and (3.93) have a single peak: theactin filaments orient along one direction. In the case of a very dense filament populationthe amount of alignment is very high.Chapter 3. Models for gradual alignment of actin filaments 88Bipolar bundling (H) and myosin (IV). ii = 2, K(O) = Bsin2O, (B <0).This kernel represents attraction at acute angles and repulsion at obtuse angles corresponding to the case of bipolar bundling. It is seen from the solutions (3.89) and (3.93)that there are two peaks: the population splits in two equal subgroups aligned in paralleland anti-parallel fashion (hi directional bundles).Orthogonal binding (III). ii = 4, K(O) = Bsiu4O, (B <0).This kernel represents the convergence of the filaments to relative orientations 0, ir/2 andit. This corresponds to orthogonal networking alternative to the one considered in theChapter 2. There are four peaks in the explicit solutions (3.89) and (3.93), meaning thatthe filaments organize in an orthogonal network as total mass of filaments increases.Competition of two types of binding proteins (V).The following kernel is a linear superposition of kernels representing bipolar bundling (II)(or myosin (IV)) for which n = 2,and orthogonal bundling (III) for which n = 4. Thisdepicts the combined effect of two proteins, as discussed previously.K(O) = B2 sin 20 + B4 sin 40. (3.98)Here, the first term accounts for bipolar bundling, and the second term accounts for theeffect of orthogonal binding. The ‘effectiveness’ parameters, which we previously calledi, a2 are now replaced by B2 and B4 < 0.Observe that in the steady state solution (3.63), the kernel appears explicitly in thefunction F(0), which is in the exponent. However, this will also affect the calculatedvalue of D. Superposition of two kernels will therefore lead to:Chapter 3. Models for gradual alignment of actin filaments 89A(0) = D exp(—(ci cos 20+ c2 cos 40)), (3.99)where,1D = M( / exp(—---(ci cos20 +c2os40))d0)1, (3.100)J-ir /1and ci and c2 are constant which can be found form transcendental equations as insection 3.6.3. The equation (3.99) represents four narrow peaks located at the angles0, r/2, r and 3ir/2 and the magnitude of the peaks at 0 and r are equal and differentfrom the ones at ir/2 and ‘r/2. This indicates a mixture of actin filaments organizedin orthogonal network and bidirectional bundles. All filaments are positioned along twomutually normal directions if we neglect the polarity of the filaments.3.7 DiscussionThe results this chapter are in qualitative agreement with those of Chapter 2. In Chapter 2 we considered two main types of actin-binding proteins, namely bipolar parallelbundling and orthogonal networking. These correspond to the types (II) and (III) of thischapter, respectively. In both chapters we have found that actin microfilaments wouldorganize either in orthogonal networks or bundles depending on the relative “effectiveness” of the orthogonal and parallel actin binding proteins. (In both cases “effectiveness”represents both relative binding rate constants and the relative concentrations of bindingproteins when they co-exist in a mixture.) The linear stability analysis in both chaptersreveals that the transition between these two structures is sharp: a minute change in therelative effectiveness ( in Chapter 2, and aj here) can result in a switch from one typeof structure to the other.Chapter 3. Models for gradual alignment of actin filaments 90The case of myosin, which causes gradual alignment of filaments could only be described in the context of Chapter 3, in which the gradual turning of filaments is explicitlymodeled by a drift term. (In Chapter 2 filaments align rapidly). We find, from remarksin section 3.5, that the rate of alignment induced by myosin binding affects the formation of order: For faster rate of alignment, a smaller mass can already promote bundleformation.The model in this chapter is of a sufficiently simple mathematical form, that explicitsteady state solutions could be determined. These agree precisely with the predictionsof the linear analysis: i.e. the integer wave number that destabilizes the homogeneoussteady state coincides with the number of peaks appearing in the nonhomogeneous steadystate solution. Although we have not proved so, we expect that these explicit solutionsrepresent stable steady states. These predictions are in qualitative agreement with thenumerical results of section 2.6 (see Fig. 2.8 and 2.9). The agreement of results supportsthe hypothesis that the phenomena modelled here are robust in the sense that manydetails of the type of model used (i.e. rapid or gradual alignment) and of specific formsof kernels (i.e. piecewise linear or trigonometric) do not affect the general conclusions.Chapter 4Models for actin filament alignment associated with a membraneIn this chapter, we focus on several specific examples where actin plays a major role.In these examples, we must incorporate certain geometrical features which have not yetbeen included in our general models. The main structure discussed in this chapter is theassociation of actin with a surface, such as the cell membrane or the surface of a parasiticbacterium which assembles an actin tail. In Chapters 2 and 3 the models described actinfilaments that had freedom of movement in space (whether 2 or 3 dimensional), but thepresence of a surface near the growing and aligning actin structures would significantlyinfluence the types of order that form. In this chapter we discuss the type of modificationsthat have to be made to the models to incorporate this feature, and investigate the results.We are concerned here only with the animal cell environment. Our examples include:(1) the cellular cortex, a relatively dense network of actin that defines the mechanicalproperties and shape of the cell, (2) the contractile ring, a circular ribbon of actinmyosin complex that acts in the last stage of cell-division, (3) adhesion belts, densebands of aligned actin in the apical ends of the side surfaces of epithelial cells, and (4)the actin tail-like structure of the intracellular bacterium Listeria rnonocytogenes.The one common feature shared by all these examples is the fact that actin filamentsare closely associated with a surface. In some cases, actin is actually attached to themembrane by various proteins. In other cases, while no hard attachment is known, actinnucleation and polymerization is restricted to a zone close to a surface.91Chapter 4. Models for actin filament alignment associated with a membrane 924.1 Actin arrangement in specialized structures(1) Actin in the cellular cortexActin in the cytoplasm of non-muscle cells is not evenly distributed over space. It ismost abundant in the cellular cortex, a thin layer adjacent to the plasma membrane.The membrane itself is highly flexible. The cell cortex gives mechanical strength to thecell surface and helps to control the shape, and the contractile properties of the cell. Italso mediates the formation of cellular extensions, and thus, ultimately, the movementof the cell. In the cellular cortex, unlike other regions of the cytoplasm, actiu is attachedto the membrane by specific sequences of proteins such as ponticulin, vinculin, integrin,and talin (Bray, 1992; Alberts et al., 1990).The cellular cortex of some cells is an isotropic meshwork about 5tm thick, i.e. relatively thick compared to the dimension of actin filaments. (See Table 4.4.) It is believedthat the actin filaments in the cortex of these cells do not exhibit any spatial or angularorder (Alberts et aL, 1990). In other types of cells (see Table 4.4), the cortex is muchthinner, about 0.01 jim thick. Since actin filaments can have lengths ranging between0.1-ljzm, the thickness of the cortex can be smaller than or comparable to the lengthof individual filaments. In this case, most actiu filaments tend to ‘lie’ fiat on the innersurface of the membrane and their 3D freedom of motion would be restricted, so that thegeometry is approximately that of 2D. This geometry would promote alignment of thefilaments. This partial alignment (from 3D to 2D) is probably enhanced by interactionsbetween filaments and the cross linking proteins.Chapter 4. Models for actin filament alignment associated with a membrane 93Type of cell Thickness of the cortexCultured fibroblast 0.2 pmlymphocyte 0.1-0.2 pmsea urchin egg 3-5 pmred blood cell 0.01-0.02 pmTable 4.4: Representative ‘order-of-magnitude’ values for the thickness of the cellcortex in some cells. (Modified from Bray (1992)).(2) The contractile ringThe contractile ring (CR) is a filamentous structure formed during mitosis on the equatorial plane of dividing cells (See Fig. 4.20). It is a beltlike bundle of actin and myosinfilaments. It appears beneath the plasma membrane during the initial stages of cytokinesis (cell division), and disappears once the cleavage of the cell is complete. Thecontraction of this ring is believed to be responsible for generating the forces that constrict the mother cell along its equatorial plane, producing two daughter cells. Thisstructure is known to be essential for cell division in animal cells. Once the division iscomplete, actin and myosin filaments in the CR dissolve and disperse.How the assembly and disassembly of this structure are controlled and details ofthe organization of actin and myosin is still poorly understood. Little is known aboutChapter 4. Models for actin filament alignment associated with a membrane 94DIVI DING CELLFigure 4.20: A schematic representation of a.ctin filament bundles in the Contractile Ringformed during mitosis. Taken from Alberts et al. (.1989).other components of the contractile ring, but they may include proteins regulating actinmyosin interactions. The contractile actin-myosin complex must be physically attachedto the plama membrane to achieve effective cleavage.The chain of proteins involved inthis attachment is still uncharacterized. However membrane associated proteins such asspectrin, ankyrin, vincuhin, ta.lin, fibronectin receptors and transmembrane proteins suchas band3 may play a role (Bray, 1992; Schoroeder, 1973; Cao and Wang, 1990; Mahuchi,1986).Various mechanisms have been suggested for the assembly of the contractile ring.rflme chemical signals and the molecular interactions involved in time OCCSS are reviewedin Harris arid Cewalt (1989) and Cao and Wang (1990). lmnmnunofiuorescent, studiesindicate that time primary mechanism underlying the formation of t,he contractile ring istime spatial and angular reorganization of existing actinfilaments in. the cell. It is foundthat both myosin and actin filaments are recruited into thecontractile ring by directionalcontractile ringChapter 4. Models for actin filament alignment associated with a membrane 95movement along the cortex towards the equator of the cell. There they organize parallelto the membrane, and alongthe equator (See Fig. 4.20). We canthink of this alignmentalong the equator as a process that leads to loss of 1, andthen 2 dimensional degrees offreedom, as the final structure is essentially 1 dimensional(Cao and Wang, 1990; Pollardet al., 1990; Mabuchi, 1986).(3) Adhesion beltsSome contractile assembliesof actin are more long-lasting than the contractile ring. Oneexample includes the circularstructures which form the beltdesmosomes. (In the literature there are various definitions of this term, but for thepurposes of this chapter, thesubtle differences between thebelt desmosomes and the adhesion belts are unimportant.)These structures occur close to the apical surfaces of epithelial cells, and are known tomediate shape changes duringdevelopment and differentiation, 4.21. (See Odell et al.(1981) for a model of gastrulation which incorporates the contraction of these bands.)During embryonic development, many events that necessitate the folding of epithelialsheets are ultimately dependent on the contraction of these structures.One fundamental difference between the adhesion belts and the contractile ring, isthat the former are permanent or at least very long-lasting,whereas the latter is transient (Alberts et al., 1990; Bray, 1992). Another difference is that belt-desmosomes areattached to the membrane of not one but two adjacent cells, via proteinswhich perforatethe two neighboring plasma membranes. Proteins such as vinculin are implicated in thisattachment. A similarity shared with the contractile ring is the fact that contraction isprobably due to the sliding ofactin filaments mediated by myosin.(4) Actin in the tail of Listeria monocytogenesListeria monocytogenes is an intracellular parasite which can cause serious, sometimesChapter 4. Models for actin filament alignment associated with a membrane 96EPITHELIAL CELLadhesion beltFigure 4.21: A schematic representation of actin filament bundles iii the Adhesion belts.Taken from Alberts et al. (1989).fatal, infections in pregnant women and newborns (Dabiri arid Sanger, 1990). The bacterium exists inside the host cell, where it uses the host cell actin to create astructure ofits own: an actin tail. Liscrie invades a wide variety of cell types including rnacrophages,fibroblasts, epithelial cells and enterocytes. This rod-shaped, gram-positive, intracellularbacterium spreads from cell to cell by moving to the peripheral membraneof the hostcell and inducing filopodia-like projections on itssurface that are subsequently internalized by the adjacent cell (Tilney and Portnoy, 1989; Dabiri arid Sanger, [990; Mounicr,et al., 1990). Listeria. becomes coated with acloud consisting of a large populationofac.tin filaments after entering the cytoplasm. At later stages, the actin cloudrearrangesto form a tail-like structure extendingoutward from one end of the bacterium (Dabiriand Sanger, 1990; Tilney et al., 1992a; Tilney et al., 1992b). This transition in actinarchitecture is reqvired for bacterial motility (Kuhn et al., 1990).Listeria monocytogenes synthesizes arid secretes an actin filament nucleator on itsChapter 4. Models for actin filament alignment associated with a membrane 97surface. The synthesis of this snrface protein, encoded by the tzctA gene, is necessaryfor bacterially indnced actin assembly (Kocks et al., 1992). Filaments are nucleated atthe bacterial surface with their barbed end(the high affinity end) pointing towards thesurface of Listeria (Tilney et aL, l992a). It is suggested that the nucleatiou processinvolves several actin-binding proteins, including profilin, talin, and vinculin (Kocks,1994; Nanavati et al., 1994; Dold et aL, 1994). They areassembled at a large number ofspecific spots on the bacterial surface (Dabiri and Sanger, 1990). Listeria assembles actinfilaments only on half of its surface, the ‘rear end’ (Tilney, 1990). As the filaments reacha characteristic length, approximately 0.2zm, they become capped and cross-linked viathe actin bundling protein a-actinin (Sanger et al., 1992).This may be a crucial step inthe transition of the 3D actin cloud surroundingthe bacterium into a more ordered (1D)structure.Fluorescence staining studies indicate only thepresence of the bundling protein, aactinin, profilin and tropomyosin throughout thetail and localized around the bacteria(Dabiri and Sanger, 1990). Actin filaments surrounding the bacterium are highly alignedparallel to the surface with their pointed end oriented towards the rear end of Listeria.The proportions of the shape of Listeria resemblea long cylindrical body capped withtwo hemi-ellipsoids (See Fig. 4.22). The typical length of a filament in the tail is of order0.2jzm, and it contains about 70 subunits.4.2 Modelling actin associated with a surfaceBy our remarks above, we restrict attention to actin structures that are associated witha surface. The surface might be the cell membrane (in cases (1),(2),(3)), or the surface ofthe invading bacterium (case(4)). As in Chapter 2, we will consider two populations ofactin filaments, those that are free and those that arebound to other filaments. However,Chapter 4. Models for actin filament alignment associated with a membrane 98Figure 4.22: A hypothetical simplified representation of Listeria and its actin tail in thehost cell.unlike the models of Chapter 2, we assume that filaments polymerize only at the endwhich is adjacent to the surface. We focus on a narrow region near the surface. Actinfilaments may leave or be transported to this region from other parts of the cell, but weconsider this to be a process in dynamic equilibrium.For the purpose of this section we interpret the free filaments as those filamentswhich have one end (the barbed end which is favored for polymerization) either attachedto or near the surface and the other end free to rotate. Bound filaments are those thatare attached to other filaments via actin binding proteins, and thus have no rotationaldegrees of freedom. As in the previous chapters we describe the angular distribution ofthe filaments only, not the spatial distribution. We consider the actin binding proteins asshort rigid rods with two actin binding sites at the ends. Two filaments become boundwhen an actin binding protein links them.We describe a three dimensional version of the model for which the independentvariables are time and angle on the unit hemi-sphere: the presence of an impermeablesurface, e.g. the inner surface of the plasma membrane, restricts the orientations ofthe filaments attached to (or near) the surface to angles on a unit hemi-sphere. (InChapter 4. Models for actin filament alignment associated with a membrane 99Figure 4.23: Shown are angles on the surface of the unit hemi-sphere represented in spherical coordinates. is a latitudinal angle from 0 to K/2, and S is a longitudinalangle from 0 to 27r.Chapter 2 filaments can assume all angles on the unit sphere.) We use a local sphericalcoordinate system. The latitudinal angle = 0 corresponds to the direction normal tothe surface and the angle = lies on the surface (See Fig. 4.23). We assume that theaverage distance between the points of attachment to the surface is much smaller thanthe average length of the filaments (See Mabuchi, 1986). We consider filaments as rigidrods of average length 1. We assume that the surface to which the filaments are attachedis fiat.The variables in the model have identical descriptions to ones in section 2.7. L(Q, t)and B(Q, t) represent concentrations of filaments, oriented at fl at time t. The kernelK(Q, Q’) represents the effective interaction between two filaments, oriented at angles Qand Q’. The interaction between filaments depends on their relative configuration. (AszxyChapter 4. Models for actin filament alignment associated with a membrane 100before, we assume that actin binding proteins mediate this, but we no longer explicitlyrepresent these by individual kernels.) For clarity, we concentrate all relevant definitionsbelow:12= (, 0) = an angle on the unit hemi-sphere, 0 ‘ w/2, 0 0 < 2w,L(12, t) = the concentration of free actin filaments at orientation 12 at time 1,B(Q, 1) = the concentration of bound actin filaments at orientation 12 at time t,/3 = the rate constant for binding of filaments via actin binding proteins,K(Q) the kernel representing the angular dependence of the rate constant forbinding,p = the concentration of free actin binding protein,6 = the dissociation rate of the actin binding proteins,1 = the average length of an actin filament.The dynamic behavior of the densities L and B is given by the following system ofequations:(Q, t) = 1AL + SB — /3pL(K * L) — /3pL(K * B),(4.101)1) = —SB + /3pL(K * L) + j3pB(K * L).These equations are identical to (2.33) with the exception that the integral is taken overthe surface of the unit hemi-sphere, S1:K * L=K(1Z, Q’)L(11’, t)dQ’. (4.102)In (4.101) terms such as fipL(K*L) represent the rate at which free filaments oriented at12 bind to other free filaments. /3 is the binding rate constant of the actin binding protein.SL denotes the rate at which the cross-links are dissolved. 6 represents the dissociationChapter 4. Models for actin filament alignment associated with a membrane 101rate of the binding protein, pAL represents the random reorientation of free filaments,A being the Laplacian in spherical coordinates and t, the rotational diffusion coefficient.It can be easily verified that the total mass in the system:M= LuQt) + B(c2,t))dQ (4.103)is conserved.Although the equations of this model are identical to those of Model I in Chapter 2,the difference is that here interactions occur in a region adjacent to a surface, so thatfunctions are restricted to the hemi-sphere. A second important difference is that here weconsider filaments to be fixed, or approximately stationary at one end, the end adjacentto the membrane. Here, when two filaments contact we consider them to bind rapidly atthe new angle (as in Chapter 2) rather than turning gradually towards it (as in Chapter3).4.2.1 The kernel for binding near a surfaceWe now derive an approximate representation for the angular dependence for binding oftwo filaments close to a surface. As we have discussed in previous chapters, the representation of the binding kernels in terms of eigenfunctions appropriate for the geometry,and shared by the Laplacian and the integral operators is an essential part of the analysisof the models. In the situation we are now considering, the presence of a surface breaksrotational symmetry, so that spherical harmonics are no longer suitable as a class ofeigenfunctions, strictly speaking. (The surface uniquely defines some direction in space,namely its normal vector, so that there is cylindrical rather than spherical symmetry.) Inorder to accommodate this geometry accurately and with due generality we would haveChapter 4. Models for actin filament alignment associated with a membrane 102to treat the problem with more advanced methods. In section 4.4 we outline what thefirst steps in such an approach would be, but here we discuss a simplified special case.In principle, the binding kernel of actin filaments should be represented by a product of two functions, one reflecting the dependence of interactions on the presence of asurface, and the other representing the dependence of interactions on the relative anglesbetween filaments. The latter function should be rotationally invariant, so that sphericalharmonics are appropriate eigenfunctions. The former component is not spherically symmetric, so that its eigenfunctions are different. The special case where the dependenceon the surface is a constant can be treated with spherical harmonics, and we shall hererestrict attention to this simple case. This does not mean that the surface is ignored, asit appears through boundary conditions. However, it means that filaments interact in thesame way given some relative angle between them, independent of their orientation relative to the surface. Further it is equivalent to assuming a form of density-independencewhich will be described in more detail in section 4.4.(1) Binding kernel for actin filaments in the cell cortex.We first derive the kernel, K1 for binding in the cell cortex, as it has the most generalform. From this kernel we later derive the more specific cases of kernels for the contractilering and adhesion belts, and the kernel for actin filaments in the tail of List eria. Wewant this kernel to reflect the angular dependence of the interactions between filamentsin the cell cortex. Namely we want the angular dependence of binding to be zero whenfilaments are perpendicular to the surface, higher if filaments are closer to the surface andcloser to each other, and maximal when filaments are lying flat on the surface. Further,interactions should be symmetric about the long axis of a filament.In arriving at a plausible form for the binding kernel in the cell cortex let us firstChapter 4. Models for actin filament alignment associated with a membrane 103consider an auxiliary function from which we will eventually derive K1. Define:f0(Q, Q’) = sin24sin)’ cos 2(0— 0’). (4.104)This function has the following properties:(i) f° = 0 for = 0 or 4)’ = 0,(ii) f is monotonic in 4), 4/ and continuous for 0 < 4), 4)’ < 42, 0 < 0 — 0’ < 42,(iii) f° is maximal for 4) = 42, &‘ = ir/2, 0 — 0’ = 0, K.(iv) f° is symmetric in 4) and 4)’.(Note that f° itself is not yet a suitable function representing the angular dependenceof binding as it is not strictly positive nor is it normalized.) We choose the positivenormalized kernel of the form:K1(Q, Q’) = A1 + A2f= A1 +A2((sin4)cos 20)(sin4 ’ cos 20’) + (sin2glsin 20)(sin4 ’ sin 20’)),(4.105)where A1 is the normalization constant and A2 is the amplitude. Normalization andpositivity of K1 imply:1 (4.106)As before, the form of the kernel K1(Q, Q’) is of vital importance in the model. The factthat the surface is impermeable to filaments implies a zero flux or Neumann boundarycondition at 4) = 42, 4)’ = 42. This boundary condition is thus the aspect of themodel which incorporates the effect of the surface, as mentioned above. We also assumeperiodicity in the variable 0.Chapter 4. Models for actin filament alignment associated with a membrane 104(2) Binding kernel for actin filaments in the Contractile ring.In order to explain 1D type of alignment in the cases of contractile ring (a narrow strip)we have to take into consideration the fact that high actin density is restricted to a narrowregion on the membrane. If the filaments are oriented along that narrow strip they are ina high density region. Thns, they have a higher probability of forming contacts, and hencebinding, than when they are oriented in directions normal to that strip. Mathematically,this implies an anisotropy of the interactions in 0. Let 0 = 0 be an angle in the plane ofthe surface and parallel to the strip. To reflect this anisotropy we now define:K2(Q, Q’) = A1 +A2(sin cos 20)(sin4/cos 20’). (4.107)Note that there is now a nnique favored direction for interaction, 0 = 0, which is distingnished from other directions (unlike the case of K1.)(3) Binding kernel for actin filaments in the Adhesion belt.The geometry of the adhesion belts is the same as the contractile ring (a narrow 1Dstrip). Thus, we nse the kernel derived for the binding of actin filaments in contractilerings, namely K2, to represent the binding in this case as well. We will treat the analysisof this case together with the contractile ring, case (2), in section 4.3.(4) Binding kernel for actin filaments in Listeria tail.In the case of actin snrronnding the surface of Listeria we do not assnme that the filamentsare attached to the bacterial snrface. (Evidence to this effect is controversial.) Howeverfilaments polymerize only at their barbed ends provided these ends are close to thebacterial surface. We chose 0 = 0 to be parallel to the major axis of Listeria and pointingtowards the rear end (See Fig. 4.24). The fact that Listeria assembles actin filaments onlyChapter 4. Models for actin filament alignment associated with a membrane 105direction of movement ofLicterksFigure 4.24: The longitudinal angle 0 = 0 is chosen to be the direction along the majoraxis of Listeria pointing towards its rear end.on half of its surface, the ‘rear end’ (Tilney, 1990) leads to the conclnsion that filamentscan interact with other filaments most effectively if they are oriented towards that end ofthe bacterium. In this case effective binding occurs for filaments in the direction 0 = 0.This suggests replacing the function in (4.104) by the following:fi (Q, 12’) = sin 4) sin ci’ cos 0 cos 0’, (4.108)Note that the dependence on 0 is no longer of the form cos 20 but rather cos 0. Thisreflects the polarity in the interactions, binding occurs only if filaments are oriented(nearly) at 0 = 0. We then define:K3(2,fl’) =A1+2f(4.109)= A1+A(sin4)cosO)(sinçó’cosO’).For this kernel only filaments with (nearly) the same polarity interact (0 = 0). TheListeria tail has a unipolar, rather than bipolar actin filament distribution.0 0=0Chapter 4. Models for actin filament alignment associated with a membrane 1064.3 The analysis of the models.In order to analyse the dynamic behavior of the equations(4.101) we first bring them tothe following dimensionless form:L = EAL + B — L(K * L) — L(K * B),(4.110)B=—B+B(K*L)+L(K*L),6 iiwhere c = , e = and ill is the total mass in the system. (A similar non/3pM pMdimensionalization is done for Model 1 in section 2.4 and the details are given in AppendixA. Note that this version applies to 3D, angles on the surface of the unit hemi-spherewhereas equations (2.11) in Chapter 2 are in 2D.) As in chapter 2, we expect that atlarge enough rotational diffusion coefficient p the stationary densities are homogeneous.These homogeneous concentrations, L and B, satisfy:B/9pMTm’(4.111)M=L+&(See Appendix A for details in 2D case.) The ratio B/L is the proportion of bound tofree filaments. The product fi represents the strength of interaction and 6 representsthe decay rate of the cross-link. Note that if 6 is small, or equivalently /3p is large, thehomogeneous equilibrium ratio B/L is large. Also note that as the number of filamentsou the surface, iW, increases, the proportion B/L increases.The linear stability analysis of the equations (4.110) together with the boundary conditions lead to an eigenvalue problem closely resembling the one addressed and discussedin section 2.7. In this case, the eigenfunctions of both the integral operator, K*, andthe Laplacian operator, A, on the unit hemi-sphere are a restricted subset of the surfaceChapter 4. Models for actin filament alignment associated with a membrane 107spherical harmonics. In Chapter 2, we used the notation Y to denote the SSH. In thiscase, the elements of this restricted subset are best expressed in the fundamental formgiven below:sinO(4.112)cos 0where n—rn is even and sin 0 term corresponds to harmonics, Ym(Q), for positive in valuesand cos 0 terms corresponds to harmonics, 1çm(Q), for negative ri-i values (MacRobert,1927; Arsenin, 1968; Hobson, 1931). (See Appendix D for the expression of the harmonicsY in terms of the fundamental harmonics }%Th.) The condition on the mode number, thefact that n — i-n should be even stems from the geometry of the problem: namely that ofa herni-sphere with zero flux boundary conditions. (Only the SSH with n — in even areeven functions of ql about ir/2, thus have zero derivative at = ir/2.) The eigenvaluescorresponding to Ym for the integral operator, K*, and the Laplaciau operator, A are—n(n + 1) and k, respectively (See Appendix D). As in section 2.7, by completeness,the densities L(Q, t) and B(IZ, t) can be expanded in series of fundamental sphericalharmonics as follows:L(Q,t) cc n lnm(t)= * Ym(Q) (4.113)B(Q,t) n=ü m=—n bnm(t)where tnm(t), bnm(t) are coefficients depending on time only and Z * means that thesummation is taken over harmonics for which n — in is even (MacRobert, 1927; Arsenin,1968; Hobson, 1931).To determine the stability of the homogeneous steady state, as in section 2.7 weconsider perturbations of the form:Chapter 4. Models for actin filament alignment associated with a membrane 108L(Q,t) L=- + ym(Q)eAt (4.114)B(Q,t) B B0Substituting this expansion into (4.101) with fri — in) even, and taking the linear approximation in small amplitudes L0 and B0, we find that instability of the homogeneousdistribution occurs at any harmonic Ym for which the following inequality is satisfied:Cn(n + 1) <k(i — kr), (4.115)where.6 N2 4116In (4.115) k are coefficients in the expansion of the kernel over the surface harmonicsyin (See Appendix D).We now consider the individual kernels described in section 4.2.1, and summarizeresults in each case. The structure that forms at instability will be determined by thefirst mode ri that satisfies the inequality (4.115). This will depend on the values of thecoefficients isf appearing in (4.115).(1) Structures that form in the cell cortex (1)First note that the coefficients k in this case are as follows:1(22 = f<-2 = A2, (4.117)= 0 otherwise.Chapter 4. Models for actin filament alignment associated with a membrane 109(See Appendix ID for details.) Thus in the case of the cell cortex the inequality (4.115)is first satisfied for ii = in = 2 and for n = 2, in = —2, that is when the right hand sideis positive and maximal. The harmonics Y2 = 3 sin2 çó sin 20 and Yj2 = 3 sin2 ç& cos 20destabilize the homogeneous distribution and initiate pattern formation in this case. Bothharmonics Y2 and p—2 have a peak at 42. However, the positions of the two peaksalong the latitudinal direction 0 is arbitrary.We define Ccr to be the critical value of C at which instability occurs. (The valueof Ccr is calculated by substituting the values of ii, in and k causing instability in(4.115).) In this case we obtain:Ccr = A2(l — 1/3A2). (4.118)Assuming all the parameters are fixed in the system, we treat Al, the total mass, asthe bifurcation parameter. From equation (4.116) we find that the critical value for Ccorresponds to the following critical value for the total mass Al:Mcr=2pA(1— 1/3A2) (4.119)If M > Mcr the stability of homogeneous distribution is broken and a pattern evolvesin which most filaments lie fiat (g = ir/2) on the surface of the membrane. Note thatin (4.119) M depends inversely on /3, the binding rate constant, p, the binding proteinconcentration, and A2 the amplitude of interactions and is proportional to ,u, the rotational diffusion rate, and 6, the dissociation rate of the binding protein. So, as /3, p, or A2increase, the critical mass Mcr decreases, i.e. the homogeneous distributiou becomes unstable at low filament concentrations, and as ii, or 6 increase, the critical mass increases,i.e. the homogeneous distribution becomes unstable at high filament concentrations.Chapter 4. Models for actin filament alignment associated with a membrane 110This analysis shows that the cell cortex loses its 3D isotropic structnre and adoptsa 2D one where most filaments are lying fiat on the snrface parallel to each other. Thestrncture corresponds to a ‘hump’ in the angular distribution of free and bound filamentscentered at = ir/2. The preferred direction of alignment in this 2D structure (the planeof the cell surface) is arbitrary in this case.(2) Structures that form in the contractile ring.In the case of contractile ring the coefficients K are as follows:1’o— Ino— Ill,= A2, (4.120)= 0 otherwise.(See Appendix D for details.) Thus the inequality (4.115) is first satisfied for n = in = 2.The harmonic Y2 = 3 sin2 4 sin 20 destabilizes the homogeneous distribution and initiatespattern formation. }‘2 has peaks at 4’ = ir/2, and 0 = 0, r. Thus the growing patternis one in which filaments are lying flat on the surface (4’ = ir/2), and along the narrowstrip of high density region (0 = 0, ir).The values of Ce,. and M7 are the same as in the previous case:= A2(1— l/3A2), (4.121)and,=2pA(1— 1/3A2) ( .1 )Thus, if iW > 11/Icr the stability of homogeneous distribution is broken and a patternChapter 4. Models for actin filament alignment associated with a membrane 111evolves in which most filaments lie flat on the snrface of the membrane again, but theyare also oriented along the equator. The pattern corresponds to a ‘hnmp’ in the angulardistribution of free and bound filaments centered at = ir/2, U = 0 and at = 7r/2, 0 = r.(3) Structures that form in adhesion belts.The kernel, and hence the results are same as in case (2) above. A pattern in whichmost filaments lie flat and parallel (or anti-parallel) to one another on the surface of themembrane evolves.(4) Structure of the Listeria tail.For the case of actin filaments in the tail of Listeria the coefficients k are as follows:(4.123)= 0 otherwise.(See Appendix D for details.) Thus the inequality (4.115) is first satisfied for n = in = 1.The harmonic Y = sin sin 0 destabilizes the homogeneous distribution and initiatespattern formation in this case. Yj’ has a peak at = 7r/2, and 0 = 0, i.e. the growingpattern is one in which filaments are perpendicular to the bacterial surface (çi =and along the major axis of the bacterium (0 = 0, or).The values of Cer and Me,- are as follows:= A2(1— A2), (4.124)and,Chapter 4. Models for actin filament alignment associated with a membrane 112Mcr= /32pA(1 —A2) (4.125)Thus, if M > Mer the stability of homogeneous distribution is broken and a patternin which most filaments are oriented along the axis of Listeria evolves. This patterncorresponds to a ‘hnmp’ in the angular distribution of free and bound filaments centeredat qS=ir/2,O=O.In all cases, if the mass M is large and the system is drawn far from criticality, sharpnarrow peaks in the angular distribution of the filaments evolves from the mild ‘humps’described above. Results similar to those of section 4.3 were obtained for a general formof the 3D model in Mogilner and Edelstein-Keshet (1994a-b). It is found that for a largeclass of kernels, as the total mass in the system exceeds some critical value (determinedby parameters of the system), a spontaneous pattern formation occurs. The leadingmode having the largest amplitude is determined by the SSH with the largest coefficientappearing in the expansion of the kernel. According to their results, the bifurcation issupercritical and corresponds to a second order non-equilibrium phase transition. As thetotal mass, M, increases the mild ‘hump’ described by the SSH which breaks the stabilityof the homogeneous distribution transforms into a sharp narrow peak (See Mogilner andEdelstein-Keshet (1994a)). The total amount of free and bound filaments in the peakare:B SB= >--,(4.126)where K > 1 is the value of K1 (for i =1,2,3) in the peak (See Appendix D). Hence asthe density of filaments grows, the ratio of the bound to free filaments increases.Chapter 4. Models for actin filament alignment associated with a membrane 1134.4 First steps towards a more realistic modelIn this section we snggest a modified approach to the problem which takes into accountthe effect of the surface on the actual interaction between filaments at various orientatious. (Recall that in section 4.2 the surface appeared only through a boundary conditionfor the problem.)We first derive the angular dependence of binding of a filament at some orientationto any one of the other filaments. We consider only the angle made with the normal ofthe surface and focus attention on a single actin filament (call it A) whose orientationis . As before, we will assume that the average length of the filaments is 1. Considerthe interactions of A with other filaments, for example those at angle ‘. In principle,there could be numerous such filaments in the neighborhood of A. We will treat theseas a continuous slab of material, and assume that interactions with A occur only overthe length of A which is actually embedded in the slab (see Fig. 4.25). Moreover, thestrength of interactions will be proportional to the density of filaments in the slab.The thickness of the slab, r will depend on the angle 4/, i.e r = r(th’) as follows:= lcosgS’, (4.127)(See Fig. 4.25.) If there is a uniform distribution of filament sites along the surface, (forexample a sites per unit area) then the density of filaments in the slab (per unit volume),p(4/), will be:= lcZ4/’(4.128)If > 4/the entire filament A is embedded in the slab and interacts all along its fullChapter 4. Models for actin filament alignment associated with a membrane 114“W)Figure 4.25: Shown are the slab of material representing the filaments oriented at ‘, thefilament A oriented at t, the thickness of the slab r(’), the average lengthof a filament 1, and the portion of the filament A embedded in the slab 1’.length, whereas if < 4/ only a part of the filament A will interact. The effective lengthof interactions, 1’, i.e., the portion of A which is embedded in this slab will be:= 1cos4/ (4.129)cos(See Fig. 4.25.) To avoid singularities in the case of , 4/ c K/2 (when the filaments arenearly lying flat on the surface) it is necessary to assume that the strength of interactionscannot exceed some maximal value. We call this value Cr, the interaction strengthattained if the filaments are within a small angle Ic of r/2Cr = , (4.130)cosQ- — ic)AChapter 4. Models for actin filament alignment associated with a membrane 115ythetaFigure 4.26: The kernel K plotted for various values of ‘, shown for = 0.5, 0.8 and 1radian. The value of the small angle ic is 0.15 radian.We build up the interaction kernel using the effective length of interaction and the effective density of filaments in the slab, obtaining K pi’. Using the above assumptionsabout p and 1’ we find that:cos 4/1cosCrNotice that the kernel is symmetric in 4, 4/. So far this kernel does not yet includedependence on 0 and 0’: the final kernel should be a product of one of the 0, 0’-dependentversions given in section 4.2.1 and the one in (4.131).The graphical representation of this kernel as a function of for a number of different4/ values is plotted in Figure 4.26. It is seen from this figure that the higher the densityof filaments, the higher is the strength of interaction between filaments.The kernel obtained by combining the angle dependence of equations (4.131), and theones in section 4.2.1 (4.105), (4.107) or (4.109), now incorporates an explicit dependenceon the orientation with respect to the surface. The problem thus formulated no longerhas spherical symmetry, and its linear stability can not be investigated with sphericalfor 4/ <for ci < ci’for 4, ç&’> —(4.131)Chapter 4. Models for actin filament alignment associated with a membrane 116harmonics. We do not here treat this more complicated problem. Some steps in itstreatment would include: simplification of the expression using products of trigonometricfunctions to approximate the terms, and explicit solution along the lines of chapter 3.4.5 DiscussionIn this chapter we presented a simple model accounting for the essential features ofassociation of actin filament structures with a surface, such as the cell membrane or theouter surface of the bacterium, Listeria moriocytogenes.We study the alignment of actin filaments into 1 dimensional structures such as thecontractile ring and adhesion belts. We had to modify the previous model (Model I ofChapter 2) to include the effects of a surface. As discussed in detail, the surface presenteda constraint (boundary conditions) which restricted the full 3D freedom of movement ofactin, though the surface did not directly influence the way that two filaments interacted. (This was a simplification, made for mathematical convenience. In section 4.4 wesuggested an approach in which the surface also impacts the filament interactions.)The four cases discussed here included (1) the cellular cortex, (2) the contractile ring,and (3) the adhesion belts, (4) the actin tail of Listeria. These cases were characterizedby different binding kernels (4.105), (4.107) and (4.109). As discussed in section 4.3, ineach case once a critical mass of actin was exceeded, alignment would occur. However,the critical mass was different in each case (see equations (4.119), (4.122) and (4.125)).Situations for which 6 and jz are small (meaning low actin binding protein dissociationand rotational diffusion rate) or p and 5 are large (high binding protein concentrationand rate constant) have a low critical mass value, meaning that spontaneous alignmentoccurs more readily (at smaller densities). The value of amplitude of the binding kernel,A2, which corresponds to the minimal Mer is different in different cases, namely:Chapter 4. Models for actin filament alignment associated with a membrane 117A2 = 3/2 in (1),(2) and (3),(4132)A2 = 1/2 in (4),with,=(4.133)in all four cases. Since the critical mass for bifurcation is the same in all cases, in the caseof Listeria tail formation, case (4), where the interactions are unipolar, the amplitude ofthe binding kernel corresponding to initiation of pattern formation is not as high as in thecases where the interactions are bipolar, cases (1), (2) and (3) (minimal critical mass forbifurcation corresponds to a lower A2 value in case (4)). This makes intuitive sense sincein the former case a single ‘hump’ in the angular distribution of actin filaments grows asa result of binding, whereas in the latter two ‘humps’ grow simultaneously and a singlehump will contain more filaments then two humps. Thus a higher amplitude, i.e. a largerdifference in strength of binding at different angles is required to accentuate the subtleinhomogeneity at two orientations in which filaments may diffuse into other directionsfaster and/or more easily than a single hump containing more filaments. Note that A2 issimilar to the ‘effectiveness’ parameters a and 3 in section 3.5, or B in section 3.6 whichplays a part in the magnitude of the drift velocity. From that point of view also, only ahigher drift velocity will allow the break of homogeneity into two groups compared to asingle group if the same amount of actin filaments are present in the system. In otherwords the drift velocity should be high enough to override the diffusion and dissociationrates of clusters smaller in size.Moreover, the type of alignment was distinct. We found that in cases (2), (3) and (4),Chapter 4. Models for actin filament alignment associated with a membrane 118actin filaments formed structures along specific directions (a ‘ring’ along the equator, a‘belt’ along the apical surface , or a ‘tail’ behind the bacteria), whereas in case (1) actinorganized into structures where filaments were oriented along an arbitrary direction.(This is a direct consequence of assumptions that were made about the kernels in eachcase.)Chapter 5DiscussionIn this thesis we have focussed ou the dyuamics of actiu structures, and on transitions thattake place under the influence of the actin-related proteins. The importance of actin inthe cell stems from the fact that the structure, mechanical properties, and many cellularfunctions are intimately related to the actin cytoskeleton. The rearrangement of actin inthe cytoskeleton determines aspects of cellular motility and many other properties of thecell. As discussed in chapter 4, an actin structure (a “tail”) also plays a dominant rolein the motility of an intra-cellular bacterium Listeria monocytogeries.We have investigated several types of actin binding proteins including those thatmediate unipolar bundles, bipolar bundles, and orthogonal networks. Our models givecontinuum descriptions of angular actin distribution in such structures, and of the temporal behavior of this distribution. This is analogous to the mean field approximation inphysics. The scope of this work is different from recent modelling by Dufort and Lnmsden(1993a,b) in which three dimensional spatial positions, binding and unbinding, and thespatial and rotational diffusions of individual molecules is taken into account. The latteris a complex simulation, whereas our models are aimed at analytic tractability. UnlikeSherratt and Lewis (1993) who consider how actin responds to external forces (stress andstrains), our model is time dependent, and emphasizes the role of the binding proteins.The philosophy of the modelling approach can be summarized as follows. The modelsare based on the following assumptions: (1) The geometry of the binding proteins can119Chapter 5. Discussion 120be represented by a function that depends on the relative orientations of actin filaments.This function is the kernel K. (2) The binding is either rapid (Chapter 2 and 4) or gradual(Chapter 3). (3) Actin filaments can undergo random rotational diffusion governed by theparameter i’ Futhermore in Chapters 2 and 4 the following specific simplifications weremade: (4) Actin filaments exchange between a bound and a free state. Only one typeof bound state is considered. (5) Binding and unbinding of filaments is the same at allstages of the process. No distinction is made between small and large bound clusters. (6)Monomers are added to the filaments at a rate proportional to the total concentration offilaments. Thus the model only includes a limited number of features of a highly complexsystem.The model predicts the following results which were not given initially but whichfollow from the analysis: (1) The above minimal assumptions are already sufficient toobtain the observed pattern formation. (2) The formation of the final structure doesnot depend on whether the molecules bind in a single step (rapidly as in Chapter 2)or whether they pull each other gradually into the right configuration (as in Chapter3). (3) Transitions from one structure to another can take place without completelydisassembling the original structure (see section 2.6). (4) The formation of structuresdepends on combinations of the parameters. If such combinations do not satisfy certaincriteria (dispersion relations), no structures will form.The models have included the following important parameters: p the rotational diffusion coefficient, M the total mass of actin, S the dissociation rate (Chapter 2 and 4), /3the binding rate (Chapter 2 and 4). Pattern formation occurs for small p, 6 and/or largeM, /3. Short actin filaments have rotational diffusion, p, which is orders of magnitudelarger than that of long filaments. (For example, actin monomers rotate much morequickly than F-actin.) This means that polymerization into long filaments must be theChapter 5. Discussion 121first and most important step in initiation of strncture: actin networks or bundles canappear only in a late stage of polymerization according to the model. The model alsopredicts that the total mass, M, must be large enough relative to other parameters forstructures to form. This can be explained by reasoning that a large total mass providesopportunities for contact and binding. (This kind of dependence of self-organization ontotal mass is found in other theories for self-organization in both physics and biology.)The other parameters which reflect binding and unbinding rates also influence the abilityof structures to form. Similar predictions were made in Chapter 4 where the structuresare associated with a surface.Our original hypothesis, stated in the introduction, was that molecular interactionsbetween actin filaments and the actin associated proteins lead to the formation of orderand the transitions between different structures formed by the actin cytoskeleton, even inthe absence of external mechanical forces. The results of Chapters 2, 3, and 4 confirm thatthis hypothesis is correct, subject to the assumptions of the models. Further, the modelshave allowed statements to be made about how the properties of individual moleculesaffect the properties of the macromolecular structures, linking one level of complexity tothe next higher level.Recent experimental work by Wachsstock et al (1994) reveals that actin associatedproteins from different species of organisms may have slight differences in affinities andrate constants. Their work gives evidence to the changes in actin network structurethat stem from these differences in binding proteins. The models investigated in thisthesis predict that specific values of the parameter combinations lead to specific typesof actin alignment, and that minute changes in these parameters (close to bifurcation)can lead to large changes in the structures that form. The comparison between proteinsderived from different species may have some implications about the molecular evolutionChapter 5. Discussion 122of actin-associated proteins.The two distinct types of models discussed were: (a) a model for rapid turning andalignment of actin filaments (Chapter 2), and (b) a model for gradual drift-like turningand alignment of actin filaments (Chapter 3). Comparisons between the results of theseseparate models applied to a given class of binding proteins resulted in similar predictions.These similarities are evidence that the phenomena are robust to changes in the structureof the models.Future areas of extension of this work might proceed in several directions:(1) To determine a complete set of biologically realistic parameter values, and assesswhether these values agree with the predictions corresponding to the given structures,(2) To generalize the models to a full spatio-angular treatment, and investigate bothspatial and angular distributions of actin in the cell, and(3) to include the effects of mechanical properties of the cytoskeleton, and the presenceof external forces.Bibliography[1] ABRAM0wITz, M., STEGUN, l.A. 1970. Handbook of Mathematical functions withformnlas, graphs, and mathematical tables. National Bureau of Standards, AppliedMathematical Series. 55.[2] ALBERTS, B., BRAY, D., LEwIS, J., RAFF, M., RoBERTs, K. 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InterseiencePubl. Inc., NY.Appendix ANon-dimensionalizationIn order to non-dimensionalize the equations (2.9) we define the following dimensionlessquantities:B’— -rLt7-Here B*, L” and t represent scalar dimensionless quantities, whereas B, 1, and r arequantities with dimensions. Substituting the values of B, L and t in the equations (2.9)by the corresponding values in terms of the variables defined above, and rearrangingthem slightly we arrive at the following form:= + 6r!B* — pEL*(K * B*) — prLL*(K * Lj(A.134)= 6rB* + /3prLB*(K * L*) + /3pr4L*(K * L*)Choosing,and L=E=M, (AJ35)129Appendix A. Non- dimensionalization 130and substituting these in (A.134) , and dropping the *‘s in the equations we arrive atthe dimensionless form (2.11) with the dimensionless parameters and 5 as follows:= pM = ppM’ (A.136)The stationary density distribution is expected to be a homogeneous one for largevalues of the diffusion coefficient p or equivalently e. The values of these densities homogeneous in 0 and time t, (IL, B), can be found by setting the time and 0 derivativesequal to zero in (2.9). Thus we have:0= 6B—,dpL(K*B)—/3pL(K*L),——— (A.137)0 = —6B + /3pB(K * L) + 3pL(K * L).We observe that K * B = B and K * L = L. This leads to:= (A.138)Appendix BLinearization and Linear Stability AnalysisThe linearized equations (2.13) constitute an eigenvalue problem. In vector notation,these equations could be written as:a L0 L0—= A . (B.139)at ]3 130where A is a linear operator which contains the Laplacian and K* and constants. Weassume solutions of the form:L0(O,t) 1(0)= e = v(0)et. (B.140)B0(0,t) b(0)Thus, writing the solutions as a product of a time independent and a direction independent part, and substituting them in (B.139) yields:(J — AI)v = 0. (B.141)For non trivial solutions we must havedet (J — XE) = 0. (B.142)Each solution of (B.142), an eigenvalue will correspond to au eigenvector v1. Thegeneral solution of the linearized equations (B.139) can then be written as:131Appendix B. Linearization and Linear Stability Analysis 132L0(O. t)[B0:t]= Zcivi(0)et. (B.143)If one or more of the eigenvalues A is positive (or have positive real parts) the perturbations (L0,B0) will grow with time, i.e. the steady state will be destabilized.As discussed, the eigenfunctions of the linear operator A are the functions ekG. Thus,substituting the perturbations (2.15) in the equations (2.9) we obtain the following system:AL0etkGet = ,tk2L0eCOe)t + 6B0eJVe.Xt — pLB0e1k9eft —AB0ehIOet = _6B0eIkGe + pBLeic8eAtft + pLLeC6eAtft(B.144)Here, ft is the Fourier transform of the kernel K, namely:ft(k) = JK(O)eikOdO, (B.145)and it appears in the linearized equations since:ft(k)eIkj2WK(O — O)eIkO’dO. (B.146)The steady state values (L, B) satisfy:0 = —oB + j3pL2 + /3pBL. (B.147)Substituting this in (B.144) and eliminating eikSeAt terms from the equation we arrive atthe matrix form (2.16). In order to determine the sign of the eigenvalues we examine theAppendix B. Linearization and Linear Stability Analysis 133trace and the determinant of the Jacobian J in (2.17). For .1 a 2 x 2 matrix, the twoeigenvalues are:A1,2 + V — 4d (B.14)where t is thetrace of J and d is the determinant of J:t = tr J = a + a, (B.149)d = det J =a1122 —a1221. (B.150)We observe that a11 and a are always negative since B/L > 0 for all biologicallyrelevant cases and since K < 1. Thus, the trace of the Jacobian is negative in cases ofinterest. In this case, for one of the eigenvalues to be negative, that is for instability toperturbations of the form (L0,B0), the determinant of J must be negative. This stabilitycriterion is equivalent to the dispersion relation (2.19).Appendix CThe properties of SSHThe surface spherical harmonics Y,’s are special cases of solid spherical harmonics V,-’swhich are solutions of Laplace ‘s equation in spherical coordinates. The SSH which arealso solutions of Laplace’s equation are functions q and 0 only, thus independent of r,they are obtained by dividing V1, by rTh and they satisfy the following equation:n(n + 1)Y + -7(sinq)+=0. (C.151)Thus the SSH of degree n, Y,, is the eigenfunction of the Laplacian operator in and 0with corresponding eigenvalue —n(n + 1):AY = —n(n + 1)Y. (C.152)The SSH of degree n can be written as a linear combination of the Legendre polynomialsof degree n, F, (also denoted simply as F), and the associated Legendre functions ofdegree n and of order m, P’:Y. 0) = AF(cos ) + (A cos in0 + B sin m0)F(cos ). (C.153)m1Some of the first few Legendre polynomials are as follows:134Appendix C. The properties of SSJI 135Ff(cos ) = cosF(cos ) = (3 cos2 — 1), (C.154)F3°(cos4) = (5cos3— 3costh).See MacRobert (1927), Arsenin (1968), or Hobson (1931) for more details.The Kernel K in the convolntion in (2.32), can be written as a function of j = cos 7,where 7 is the angle between directions Q and 12’ as in (2.30). In this form, K can beexpressed as a linear combination of Legendre polynomials, F,(cos 7) (MacRobert, 1927;Arsenin, 1968; Hobson, 1931):K(i7) = > K’(n)P,(i7), (C.155)whereK’(n) = 2n+ 1 K(q)F)d. (C.156)The integral of the product of the SSH and the Legendre polynomials has the followingproperty:0 for n#mJ P(cos7)Ym(’,O’)d’dO’= { 4 (C.157)S Y(q,O) for n=m.2n + 1(See MacRobert (1927) or Hobson (1931).) This property of the SSH and the Legendrepolynomials is of extreme significance in the analysis of the linearized equations. Indeed,the convolution of the kernel K and the SSH of degree n can be expressed as a productof the SSH and k(n) as follows:Appendix C. The properties of SSH 136K *=jr K(Q, fl’)Y(Q’)dS= Is K(cos7)Y(qS’, O’)d(cos cY)dO’=I5(Z=1K’(m)F(cos ‘y))Y (4/ , O’)d(cos q&’)dO’=f:Tr A K’(n)F(cos -y)Y(gY, U’)d(cos q5’)dO’ (C.158)= K’(n) f f. P,(cos7)Y(qY, O’)d(cos th’)dO’= K’(n)241Y(,6)=wherek(n)= 2n+1K’(n) = 2K f K(ij)P()d. (C.159)Hence the SSH are also the eigenfunctions of the Convolution operator K* with corresponding eigenvalnes k(n):K * = k(n)Y. (C.160)Appendix DSSH on the surface of the unit hemi-sphereFor notational simplicity in this section we use the fundamental form of the SSH of degree12, yin as in (4.112). These harmonics relate to the basic SSH of degree n, Y,, in thefollowing way:Yn(,O)= k=-n(D.161)Some fundamental SSHs are as follows:= sin 4sinO,Y1’(ç&,O) = sincosO,Y2’ = 3sin4cos4isinO,(D.162)= 3 sin ci cos cos 0,= 3sin2glsin20,= 3sin2qcos20.See MacRobert (1927), Arsenin (1968), or Hobson (1931) for more details.For m — n even, the fundamental SSHs are the eigenfunctions of both the Laplacianand the Convolution operator on the surface of the hemi-sphere with zero flux BC:AY’ = —n(n + 1)g’,- (D.163)137Appendix D. SSH on the surface of the unit hemi-sphere 138where k are the coefficients in the expansion of the kernel K in fundamental SSH:K(Q,f) = z km(Q)gm(c), (D.164)n=O m=—nwhere Z * means that the summation is taken over harmonics for which n — in is even.The coefficients K in this expansion can be easily calculated for the kernels givenin section 4.2.1. Comparing the terms in the above summation with the terms in theexpression of the kernels in (4.105), (4.107) and (4.109) we obtain the coefficients Kfl ineach case.(1) Coefficients for the kernel in the case of the cell cortexThe kernel in this case is Ic’, given in (4.105). The sum in (D.164) corresponds to:K1(Q,fl’) = A1 +kA2(2(Q)2(Q!) +2(Q)’) , (D.165)hence,I?: =A1,= = A2, (D.166)= 0 otherwise.(2) and (3) Coefficients for the kernel in the case of Contractile ring andadhesion beltsThe kernels in these cases are K2, given in (4.107) and the sum in (D.164) correspondsto:Appendix D. SSH on the surface of the unit herni-sphere 139K2(Q, Q’) = A1 +2Y2(Q)}2(ST). (D.167)hence,= A2, (D.168)= 0 otherwise.(4) Coefficients for the kernel in the case of the actin tail of ListeriaThe kernel is, K3, given in (4.109) and the sum in (D.164) corresponds to:K3(Q, Q’) = A1 +A2Y’(Q)Y’(Q’). (D.169)hence,A1,A2, (D.170)0 otherwise.Stability analysis of equations (4.110) to perturbations of the form (4.114) leads tothe dispersion relation (4.115) where C is an algebraic combination of the parameters inthe system:0= jS2pM=(D.171)(This analysis is similar to the one carried out in Appendix B for the Model I or the 3Dmodel in section 2.8.1.)Appendix D. SSH on the surface of the unit hemi—sphere 140In all cases the total density of free and bound filaments in the peak can be foundfrom (4.110):KL + KLB — = 0. (D.172)where K is the value of K(Q, Ii’) within the peak. (For example, for the case of theactin tail of Listeria we have:= = (ct= O = 0),(D.173)K3((, 0), (, 0)) = = !(A1 + A2) = (1 + !A2)> 1.So that, the distributions satisfy:L- KM2=+KM’ (D.174)B M SBF=K=Kt>T.


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