UBC Theses and Dissertations

UBC Theses Logo

UBC Theses and Dissertations

Investigating models for cross-linker mediated actin filament dynamics Spiros, Athan Andrew 1998

Your browser doesn't seem to have a PDF viewer, please download the PDF to view this item.

Item Metadata

Download

Media
831-ubc_1998-346277.pdf [ 12.41MB ]
Metadata
JSON: 831-1.0080002.json
JSON-LD: 831-1.0080002-ld.json
RDF/XML (Pretty): 831-1.0080002-rdf.xml
RDF/JSON: 831-1.0080002-rdf.json
Turtle: 831-1.0080002-turtle.txt
N-Triples: 831-1.0080002-rdf-ntriples.txt
Original Record: 831-1.0080002-source.json
Full Text
831-1.0080002-fulltext.txt
Citation
831-1.0080002.ris

Full Text

I N V E S T I G A T I N G M O D E L S F O R C R O S S - L I N K E R M E D I A T E D A C T I N F I L A M E N T D Y N A M I C S by A T H A N A N D R E W SPIROS B.Sc. (Applied Mathematics), California Polytechnic State University (San Luis Obispo), 1990 M . A . (Mathematics), State University of New York at Stony Brook, 1992 A THESIS S U B M I T T E D IN P A R T I A L F U L F I L L M E N T OF T H E REQUIREMENTS F O R T H E D E G R E E OF D O C T O R OF PHILOSOPHY in T H E F A C U L T Y OF G R A D U A T E STUDIES Department of Mathematics Institute of Applied Mathematics We accept this thesis as conforming to the required standard T H E U N I V E R S I T Y OF BRITISH C O L U M B I A August 1998 © Athan Andrew Spiros, 1998 In presenting this thesis in partial fulfillment of the requirements for an advanced degree at the University of British Columbia, I agree that the Library shall make it freely available for refer-ence and study. I further agree that permission for extensive copying of this thesis for scholarly purposes may be granted by the head of my department or by his or her representatives. It is understood that copying or publication of this thesis for financial gain shall not be allowed without my written permission. Department of Mathematics The University of British Columbia Vancouver, Canada A b s t r a c t Actin, a major component of the cytoskeleton, is responsible for the shape and structural properties of many eucaryotic cells. Actin filaments are made of monomers which bind in a single strand. Cross-linkers bind two filaments together and influence the relative orientation of the filaments. Cross-linkers such as a-actinin favor parallel alignment of filaments, while others such as actin-binding protein favor orthogonal alignment. The fil-ament length, concentration of cross-linkers and cross-linker association-dissociation rates all affect the types of network that form. The resulting network dictates the structural properties of the cell. In this thesis I study how filament length, concentration of cross-linkers and cross-linker association-dissociation rates influence actin network development. I use existing models and create new models to explore these interactions. Integro-partial differential equations and other techniques are used to model the system. Using experimentally determined biological parameters, I predict the geometry and distribution of the resulting network. Computer simulations verify the model predictions. I compare these predictions with experimental results. Increasing the cross-linker concentration first strengthens the isotropic network, but then, beyond a transition, forces the network to be inhomogeneous and more fluid-like. As the cross-linker concentration increases even further, more filaments bind to the network, resulting in a stronger, more solid actin solution. Decreasing the cross-linker dissociation rate constant has the same effects as increasing the cross-linker concentration. Finally, I find that changing the filament length greatly affects rates of diffusion, influencing instabilities. Increasing the filament length, favors alignment and clustering, as well as formation of bundles. Filament length influences the spacing between clusters. ii Increasing the length, forces clusters to be spaced further apart until they eventually disappear. I also find that there is an optimal length for bundle formation. When considering a distribution of filament lengths, I expect to see a wider dispersal of filaments near the bundle transition point. iii T a b l e o f C o n t e n t s Abstract i i Table of Contents iv List of Tables v i i List of Figures vi i i Acknowledgements x i Chapter 1. Introduction and Biological Background 1 1 The Cytoskeleton 1 2 Actin Filaments 2 3 Actin-Binding Proteins 4 4 Networks and Bundles of Actin 5 5 a-Actinin 8 6 The Factors Influencing Actin Networks and Current Problems 9 7 Outline 10 Chapter 2. Effects of Cross-Linker Concentrations on Degree of Cross-Linking of Ac t in Filaments 13 1 The Four Structure Model 13 1.1 Isoclines and Graphical Analysis 17 1.2 Parameter Values 22 1.3 Equilibria 23 1.4 Conclusions of the Four Structure Model 27 2 The Seven Structure Model 30 2.1 Finding Equilibria 34 2.2 Conclusions of the Seven Structure Model '. 35 3 Conclusions and Discussion 36 4 Glossary of Parameters 38 Chapter 3. Modeling the Spatial Clustering and Alignment of Ac t in Fila-ments with Integro-PDE's 39 1 One Dimensional Model -44 1.1 Model Development 44 1.2 Model Analysis 46 1.3 Filament Length Affects Clustering 53 2 Angular and Spatial Model Development 64 3 Model Interpretation 66 i v 4 Glossary of Parameters 69 Chapter 4. Parameter Estimation and Model Refinement 70 1 Estimating the Filament Binding Rate, 72 2 Estimating the Rate of Filament-Network Unbinding, 7 76 2.1 The Chemical Reactions 77 2.2 The Birth and Death Process 78 2.3 Finding 7 83 3 Estimating Filament Rates of Diffusion 84 4 Spatial and Angular Ranges of Interaction 91 4.1 General Remarks 91 4.2 Spatial Part of the Kernel 92 4.3 Angular Part of the Kernel 94 4.4 Angular Kernel for Specific a-actinin Types 96 5 Summary of Parameter Values 99 6 Glossary of Parameters 102 Chapter 5. Numerical Simulations of Actin Filaments in Space and Orien-tation 104 1 Computing the Convolutions 104 1.1 The Fast Fourier Transform 105 1.2 Comparison of Convolution Methods 109 2 Selecting a Finite Difference Scheme I l l 3 Simulations and Results 112 3.1 Effect of Filament Length 113 3.2 Effect of Viscosity 122 3.3 Effect of a-actinin Concentration 123 3.4 Effect of Cross-Linker Affinity 124 3.5 Effect of Kernel 129 4 Conclusions 130 Chapter 6. Interacting Filaments of Different Lengths 133 1 Model for Two Lengths 134 1.1 Preliminary Analysis 136 1.2 Parameter Evaluation 140 1.3 Results 145 1.4 Conclusions 157 2 Fixed Filament Model 160 3 Length Distribution Model 164 Chapter 7. Actin Network Experiments 167 1 Rheology of Polymer Networks : .' 167 2 Results of Experiments 174 v 2.1 Janmey et al. (1994) 176 2.2 Sato et al. (1987) 178 2.3 Wachsstock et al. (1993) and (1994) 180 2.4 Goldman et al. (1997) 184 2.5 Tempel et al. (1996) 186 2.6 Furukawa and Fechheimer (1996) 189 3 Polymer Physics 190 3.1 Regimes for Solutions of Rod-like Polymers 191 3.2 Percolation Theory for Network Elasticity 192 3.3 Percolation Theory for Phase Transitions 194 3.4 Model Comparisons 195 Bibliography 197 vi L i s t o f T a b l e s 1.1 Table of common actin-binding proteins 6 1.2 Association-dissociation rate constants for cv-actinin cross-linker and actin 9 4.1 Parameter values needed for computing diffusivities 90 4.2 Parameter values in units consistent with the model 99 4.3 The model parameters and ranges of values 100 4.4 Effects of biological parameters on model parameters 101 5.1 Comparison of computation times for convolution methods 110 6.1 Special eigenvalues for instabilities in two length model 148 6.2 Average length for two filament solution based on p 154 6.3 The rates of diffusion for the average length solutions 157 vii L i s t o f F i g u r e s 1.1 Actin monomers and filament growth 2 1.2 Two binding sites on monomers 3 1.3 Actin cross-linkers 5 1.4 Actin networks in motile cells 7 1.5 Structure of a-actinin 8 1.6 The simple chemical kinetics of actin and ct-actinin. . : 8 1.7 Outline of thesis 11 2.1 Three states of cv-actinin 14 2.2 Two reactions of actin with a-actinin 14 2.3 Four actin/a-actinin structures 15 2.4 d-isocline 18 2.5 a-isocline 21 2.6 Possible equilibria defined by isoclines 24 2.7 Effect of increased a-actinin on equilibria (isoclines) 25 2.8 Effect of increased a-actinin on equilibria (numerical) 28 2.9 Effect of binding constant on equilibria 29 2.10 Seven structures of actin/a-actinin 30 2.11 Seven reactions of actin with a-actinin 31 2.12 Effects of cross-linkers on equilibria (more detailed) 36 3.1 Different 2-dimensional actin filament network structures 41 3.2 Diagram for a motile cell 42 3.3 2-Dimensional actin network structures in one dimension 42 3.4 Resulting structures as defined by nonuniform distributions 43 3.5 Plot ofk(l-k) 54 3.6 Graphical interpretation of the instability condition 55 3.7 Unstable wavenumbers for L = 0.25/x 57 3.8 Unstable wavenumbers for L = 0.5/i 57 3.9 Unstable wavenumbers for L = l.O/i 58 3.10 Unstable wavenumbers for L = 2.0/i 58 3.11 Unstable wavenumbers for L = 4.0/x 59 3.12 Unstable wavenumbers for L = 8.0// 59 3.13 Simulations in space for L = 0.25 61 3.14 Simulations in space for L = 0.5 61 3.15 Simulations in space for L — 1.0 62 3.16 Simulations in space for L = 2.0 62 3.17 Simulations in space for L = 4.0 63 3.18 Simulations in space for L = 8.0 63 3.19 The position, x, and orientation, 6, of a single filament 65 viii 4.1 Reactions necessary for binding filaments 72 4.2 Binding rate, ft, as a function of cross-linker concentration and 74 4.3 Binding rate, ft, as a function of cross-linker concentration and filament length . 75 4.4 Rates of unbinding based on number of cross-links 79 4.5 Number of cross-links viewed as a birth-death process 80 4.6 Dependence of unbinding rate, 7, on cross-linker concentration and 83 4.7 Restricted volume of filaments 85 4.8 Types of translational diffusion for filaments 86 4.9 Imaginary tube restricting filament motion 87 4.10 Spatial and angular representation of a filament 87 4.11 Restricted rotational diffusion 88 4.12 Length effects on diffusion rates 91 4.13 Spatial aspects for filament bindings 93 4.14 Spatial component of the kernel 94 4.15 Angular aspects for filament bindings 95 4.16 Angular component of kernel for anti-parallel cross-linking 97 4.17 Angular component of kernel for parallel cross-linking 98 5.1 Optimizing the Fast Fourier Transform 108 5.2 Legend for interpreting simulation results 114 5.3 2-D simulations of short lengths at 5 minutes 115 5.4 2-D simulations of short lengths at 30 minutes 116 5.5 2-D simulations of short lengths at 90 minutes 117 5.6 2-D simulations of short lengths at 150 minutes 118 5.7 2-D simulations of long lengths at 15 minutes 119 5.8 2-D simulations of long lengths at 30 minutes 120 5.9 2-D simulations of long lengths at 60 minutes 121 5.10 2-D simulations with 5 cP viscosity at 30 minutes 123 5.11 2-D simulations with 5 cP viscosity at 60 minutes 124 5.12 Effects of changing cross-linker concentration in 2-D simulations 125 5.13 2-D simulation for 1 [iM chicken a-actinin at 60 minutes 126 5.14 2-D simulation for 5 \±M chicken a-actinin at 60 minutes 126 5.15 2-D simulation for 10 \ i M chicken a-actinin at 60 minutes 127 5.16 2-D simulation for 1 fiM amoeba a-actinin at 60 minutes 127 5.17 2-D simulation for 5 \ i M amoeba a-actinin at 60 minutes 128 5.18 2-D simulation for 10 \ i M amoeba a-actinin at 60 minutes 128 6.1 Angular distribution of short filaments 144 6.2 Angular distribution of long filaments 144 6.3 Angular instability patterns for two lengths 146 6.4 Bifurcation diagram for two length model 147 6.5 Simulation of two filament lengths with p = 0.01 149 i x 6.6 Simulation of two filament lengths with p = 0.1 150 6.7 Simulation of two filament lengths with p = 0.22 151 6.8 Simulation of two filament lengths with p — 0.5 152 6.9 Simulation for average filament length when p = 0.01 155 6.10 Simulation for average filament length when p = 0.1 155 6.11 Simulation for average filament length when p = 0.22 156 6.12 Simulation for average filament length when p = 0.01 156 6.13 Hypothesis of how short filaments affect long filaments 159 6.14 Simulation with fixed long filaments when p = 0.5 161 6.15 Simulation with fixed long filaments when p = 1 162 6.16 Simulation with fixed long filaments when p = 10 162 6.17 Simulation with fixed long filaments when p — 100 163 7.1 A solid with shear stress, a, will deform with some relative displacement, 7. 167 7.2 Result of applying a sudden strain 169 7.3 Result of applying a steady strain 170 7.4 Result of applying a sudden stress 171 7.5 Resultant strain for a viscoelastic solid 171 7.6 Resultant strain for a viscoelastic liquid 172 7.7 A strain, 7, may be applied to a substance sinusoidally 172 7.8 Response to sinusoidal strain 174 7.9 Diagram of the complex modulus, G* 175 7.10 Filament length affects the storage modulus of a pure actin solution 177 7.11 Dynamic response for two different lengths of pure actin 178 7.12 Cross-linkers change dynamic response of actin solutions 179 7.13 Effect of temperature on actin filament solutions with or without cross-linkers .. 180 7.14 Dynamic response of different types of «-actinin 181 7.15 Effects of changing cross-linker concentration for two types of a-actinin 182 7.16 Dynamic response for different concentrations of two types of a-actinin 183 7.17 Dynamic response for different concentrations of filamin 185 7.18 The creep for an actin-a-actinin solution at two temperatures 187 7.19 Viscoelastic behavior as a function of temperature 188 7.20 The gel-point temperature for different a-actinin concentrations 189 7.21 Degree of bundling as a function of filament length 190 7.22 A n ideal polymer network 193 7.23 Theoretical result of cross-linker concentration influencing the shear modulus 194 7.24 Phase diagram for actin-a-actinin solutions 195 x A c k n o w l e d g e m e n t s This thesis would not have been possible if not for the help and encouragement of my supervisor, Leah Keshet. Leah has taught me much about mathematical modeling and writing. I cannot believe the amount of patience and guidance she has shown me through-out my studies at U B C . I also must thank Alex Mogilner who got me started on this problem. His assistance and insights have been invaluable. Thanks also go to my advisory committee, Lionel Harrison, Robert Israel, Wayne Vogl and Brian Wetton. Their diverse backgrounds have proven useful in this interdisciplinary field. I really appreciate the time taken by Lionel Harrison and Robert Israel for their reading of this thesis and their suggestions. I am grateful to the Institute of Applied Mathematics for providing such resources as the lab and the lounge. And of course to all those people that make these places happy ones to be, in particular, those veterans Ray, John and Pete for darts and cards during lunch, and Chen, Dave, Greg, Jenn, Judy, Marty and Michele for pretending to listen to me for countless number of hours. Finally, special thanks go to my immediate family, Mom, Dad, Rebecca and Jonathan, and to my second family, the ultimate community of Vancouver and the Northwest. x i Chapter 1 I n t r o d u c t i o n a n d B i o l o g i c a l B a c k g r o u n d 1 The Cytoskeleton Actin filaments are an essential part of the cytoskeleton. The cytoskeleton is a cohesive meshwork of protein filaments inside the cell. Just as our skeleton gives us our overall shape and structural properties, the cytoskeleton gives these and more to eucaryotic cells. The cytoskeleton is also responsible for internal movements of fluids, particles and organelles (Parfenov et al., 1995; Giuliano and Taylor, 1995) as well as the movement of the cell itself (Bray, 1992; Janmey et a l , 1994; Luby-Phelps, 1994; Wachsstock et al., 1993; Wachsstock et al., 1994; Zaner, 1995). The network of filaments that make up the cytoskeleton is dynamic, allowing the cell to coordinate all these functions. Although the protein filaments differ in diameter and function, they are similar in that they each have a basic protein unit associated with them that enables them to grow or shrink with the addition or deletion of this unit. There are three different kinds of protein structures: microtubules, intermediate filaments, and actin filaments. Each of these has its own associated protein unit that is added or deleted to change its length. The protein unit for microtubules is tubulin. The protein unit for intermediate filaments varies depending upon the cell. In general, there are four distinct classes based on systematic differences. The protein unit for 1 Actin Filaments actin filaments is actin. Microtubules are the biggest components of the cytoskeleton. They form the roadways of the cell, providing paths along which various particles move with the help of motor proteins. Intermediate filaments are the next biggest filaments. Some of these form the nuclear lamina which surrounds the cell's nucleus. Others are found throughout the cell and help to distribute various mechanical stresses. Actin filaments are the smallest protein filaments. They are most highly concentrated around the cell's periphery and are key for many movements of the cell. A l l the filaments of the cytoskeleton help to determine the cell's shape and mechanical properties. This account focuses on animal cells which are different than plant cells where transport, cell division and shape are governed by different mechanisms. 2 A c t i n Filaments Actin filaments are formed from actin monomers. Actin is a globular protein with a molecular weight of 42 kDa and a size of roughly 5.5 nanometers (nm). This monomeric protein, also called G-actin, can polymerize into filaments, called F-actin. In other words, actin molecules can join together to form strings of actin called actin filaments. This is much like a Lego™ assembly. Figure 1.1 shows a simplified view of this process. Actin Actin Monomer Filament (G-Actin) (F-Actin) Figure 1.1: Actin monomers are not symmetric. Because of this, they can only join with one another in a certain orientation. This gives the actin filaments polarity, a barbed or plus end, and a pointed or minus end. Figure 1.1 is an idealization of filament growth/decay. The monomers have distinct three dimensional configurations. They have two points where they can bind to other 2 Actin Filaments monomers and one point where they can be bound. This leads to a helical assembly of monomers that form the filaments as shown in figure 1.2. Each actin monomer adds 2.73 Figure 1.2: The figure on the left shows the two binding sites of a single actin monomer. The figure on the right gives a more realistic representation of how actin monomers form actin filaments. The light and dark gray monomers each form a helix so that the filament is actually a double helix. nm to the length of the actin filament. The three dimensional way that they are pieced together forms a double helix. Thus, approximately 370 monomers make a filament one micron (fi) in length. Because the monomers are not joined from end to end, the diameter of an actin filament, 7 to 8 nm, is wider than the width of one monomer. If one were to follow one side of the double helix, one would find that the helix repeats itself every 37 nm. Actin filaments are constantly changing with the addition (polymerization) and dele-tion (depolymerization) of G-actin. The number of monomers required to initiate the formation of a filament (nucleation) is still under debate. Some people believe it takes three monomers (Korn et al., 1987; Frieden, 1983; Alberts et a l , 1989) while others believe it takes four monomers (Fesce et al., 1992; Tobacman and Korn, 1983) for nu-cleation to occur. Once the filament has nucleated it may grow or shrink at either end. The rates of polymerization and depolymerization are different for the different ends and depend upon the concentration of available G-actin (Zigmond, 1993). The barbed or plus end of the filament usually favors polymerization, while the pointed or minus end favors depolymerization. As the length of the filament changes, so does its rigidity. For lengths 3 Actin-Bmding Proteins between 100 and 1000 nm (the lengths commonly found in vivo), the filaments can be viewed as rigid rods (Janmey et a l , 1986). For lengths greater than these (as seen in most in vitro studies) the filament loses its rigidity and can turn back on itself (Janmey et al., 1986). Thus, actin is very dynamic on its own. 3 A c t in -Bind ing Proteins The addition of various substances helps to both control and diversify actin interactions. Those that control actin interactions do so by influencing the rates of polymerization and depolymerization. Cytochalasms, a group of fungal metabolites, bind to the plus end of actin filaments and prevent polymerization. Phallotoxins, a group of proteins from poisonous mushrooms, encourage polymerization. They tend to bind more readily with F-actin rather than G-actin which, in essence, increases the amount of readily available G-actin and thus increases the polymerization rate. These actin-binding chemicals influence polymerization and hence control the inherent actin dynamics. Other actin-binding proteins enrich the functions of actin. Besides changing the rate at which actin polymerizes, they can also fragment, cap, bundle, gelate, contract, stiffen, and anchor filaments as well as bind with monomers (see figure 1.3). Many different actin binding proteins exist and cause different actin structures to form (Otto, 1994). Table 1.1 gives a list of some of the best characterized actin-binding proteins. With this plethora of actin-binding proteins, the cell is able to utilize actin in many different ways. However, this also makes it difficult to perform in vivo experiments, since all of the actin-binding proteins are acting at once. The gelating and bundling proteins (also called cross-linkers because they link two filaments together) enhance the role of F-actin. They add another dimension to the actin interactions so that besides G-actin polymerizing to make F-actin, entire filaments can interact with others. Thus actin filaments can be connected to others so that they build 4 Networks and Bundles of Actin. stiffening contracting monomer binding monomers fragmenting gelating bundling capping Figure 1.3: The different ways actin cross-linkers can influence actin filaments. (Bray, 1992, page 87) an entire structure (also called an actin network) with the facilitation of cross-linkers. The network helps define the cell's shape and structural properties. Thus, cross-linkers are a very important component of the cytoskeleton. 4 Networks and Bundles of A c t i n Actin filaments are abundant in the cell and are extremely versatile. They have been found in every plant and animal cell studied. They are responsible for movements such as phagocytosis (the engulfing of particulate matter by phagocytes), cytokinesis (cell division), cell crawling, and muscle contraction. They also give the cell structure and mechanical stability. To accomplish all this, the cytoplasm contains actin in a variety of networks: linear bundles, two-dimensional networks, and three-dimensional gels (Otto, 1994). Linear bundles are networks where the filaments line up in parallel in some area of 5 Networks and Bundles of Actin Subunit Protein Molecular Weight (kDa) capping/fragmenting gelsolin 90 villin 95 fragmin/severin 42 capping protein 30 gelating spectrin 240 A B P (actin-binding protein) 250 filamin 250 a-actinin 100 bundling a-actinin 100 fimbrin 68 fascin 58 calpactin 36 caldesmon 87 contracting myosin-II 190 myosin-I/minimyosin 120 stiffening tropomyosin 35 cofilin 21 membrane association ponticulin 17 spectrin 460 monomer binding profilin 14 actophorin/depactin 17 Table 1.1: Actin-binding proteins in vertebrates and lower eucaryotes. Many others exist. (Bray, 1992, page 88) 6 Networks and Bundles of Actin the cell. They have been found in many different cases. Linear bundles are observed in contractile rings as the filaments form a band around the middle of the cell and contract to divide the cell. They are observed in stress fibers where they form parallel arrays that connect adhesion points in fibroblasts or moving cells. They also form microvilli where they are periodically aligned to produce many small protusions on the surface of the cell to increase the cell's surface area. Bundles serve other important purposes as seen in cell locomotion and muscles. When filaments link crosswise to one another they form two and three dimensional networks. These networks are important for keeping a cell's shape. The filaments link crosswise to one another and form gels which are semi-solid. These structures are the typical networks found in most cells. A cell that is moving might exhibit all three different types of networks. At the leading edge of the cell, microspikes which are linear bundles may protrude from the cell. Just behind, the cell is usually very thin so that only two-dimensional networks form. Near the nucleus the cell is quite thick where three-dimensional gels form. Figure 1.4 shows an example of a moving cell with the three different types of networks present. Side View Top View Figure 1.4: A crawling cell has all three types of networks. It has linear bundles at the leading edge (1), two dimensional arrays at its tail and close to the leading edge (2), and three dimensional arrays near its center (3). 7 a-Actinin 5 a -Act in in a-Actinin is a very important cross-linker because it has been shown that it can both bundle and gelate actin filaments (Dufort and Lumsden, 1993; Wachsstock et al., 1993; Wachsstock et al., 1994). It has also been very well studied so that some of its properties are known (Burridge and Feramisco, 1981; Colombo et al., 1993; Jockusch and Isenberg, 1981; Maciver et al., 1991; Meyer and Aebi, 1990; Milzani et al., 1995; Taylor and Taylor, 1994). It is a dimer composed of two polypeptide chains, each 100 kDa in molecular weight. It appears as a rod, 30-40 nm in length, with an actin binding site at each end (see figure 1.5). It reacts with actin as shown in figure 1.6. Numerical values for the Figure 1.5: a-actinin is a dimer constructed of two identical subunits. At each end there is a site that can bind actin. association rate constant, k+, and dissociation rate constant, have been found for various types of cv-actinin (Wachsstock et al., 1993; Wachsstock et al., 1994) (see table 1.2). These rates are sensitive to calcium (Burridge and Feramisco, 1981) allowing the cell to change the characteristics of the a-actinin by modulating the concentration of internal calcium. In vitro experiments with actin and a-actinin have helped to determine the effects of these proteins in isolation. Increasing the concentration of a-actinin causes the actin network to change from a gel to a network of bundles (Dufort and Lumsden, 1993; + Figure 1.6: The simple chemical kinetics of actin and a-actinin. 8 The Factors Influencing Actin Networks and Current Problems Parameter Value Source k+ 1 nM^s'1 1 fiM^s-1 chicken smooth muscle a-actinin (Wachsstock et al., 1994) Acanthamoeba a-actinin k_ 0.67 s - 1 5.2 s"1 chicken smooth muscle a-actinin Acanthamoeba a-actinin k+ k_ 3 ^ M _ 1 s _ 1 3 s-1 "generic" (Dufort and Lumsden, 1993) Kd = k_/k+ 0.4 ixM 2.7 iiM 2.7 jiM chicken gizzard a-actinin 2 2 ° C (Meyer and Aebi, 1990) Acanthamoeba Dictyostelium Table 1.2: Association-dissociation rate constants for a-actinin cross-linker and actin. Wachsstock et al., 1994). Increasing the binding constant of a-actinin to actin (i.e. decreasing relative to k+) favors the formation of bundles as well (Wachsstock et a l , 1993; Wachsstock et al., 1994). Bundles give the solution a more fluid-like behavior while gels make the solution behave more like a solid (Wachsstock et al., 1994). Thus a-actinin can alter both the network structures and the mechanical properties of the cell drastically. 6 The Factors Influencing A c t i n Networks and Cur-rent Problems I believe Alberts et al. (1989) stated the importance and problem of actin structures best when they said the following: Actin is involved in a remarkably wide range of structures, from stiff and relatively permanent extensions of the cell surface to the dynamic networks at the leading edge of a migrating cell. Very different structures based on actin 9 Outline coexist in every living cell. In every case the fundamental structure of the actin filament is the same. It is the length of these filaments, their stability, and the number and geometry of their attachments (both to one another and to other components of the cell) that varies in different cytoskeletal assemblies. The challenge is to determine how and under what circumstances the different actin networks are formed. The experimental results point to three main factors that influence the formation of actin filament networks: • concentration of a-actinin, • binding constant of a-actinin to actin, • actin filament length. By developing models that depend on these factors in a reasonable way, I hope to discover how they affect network development. 7 Outline In this thesis, I use several approaches to explore the formation of structures and patterns in actin filament networks. In Chapter 2, I ask how the degree of binding of filaments (in very simplified structures) is affected by the availability of cross-linker. This is the only chapter in which I discuss detailed structures which consist of one or two filaments with up to two cross-links. Only time behavior of the various populations is followed. In Chapter 3, I summarize and adapt previous mathematical models (integro-partial differential equations) to explore the distribution of actin filaments over space and over angles of alignment. Here, the details of the bound filaments (such as number of cross-linkers attached) are sacrificed in favor of simplicity. I extend previous work by Mogilner and Edelstein-Keshet (1996) to the case that is explored in detail with numerical methods. 10 Outline C h a p t e r 1 Biological Background C h a p t e r 2 ] [ C h a p t e r 3 Cross-Linker Dynamics I 2-D Model Development I and Spatial (Cluster) Formation / / C h a p t e r 5 1 ^£ , - Numerical Considerations C h a p t e r 6 U ' " ' ' [ 3 1 1 ( 3  2 ' D Simulations Multiple Length Populations 7 [ Parameter Estimation [ C h a p t e r 4 i C h a p t e r 7 Viscoelastic Measurements and Comparison with Experiments Figure 1.7: F l o w of thesis showing major dependences (solid lines) and minor dependences (dashed lines). In Chapter 4, I tackle the main problem of this thesis - the correspondence between parameters in the model and experimentally measurable quantities. This process results in a number of modeling problems, one for each type of parameter concerned. I show how rate constants, molecular sizes and masses, and typical concentrations as well as other measured values combine to give approximate values for the model parameters. These values are then used in numerical simulations, whose detailed development is described in Chapter 5. Also in Chapter 5, I give results of numerical experiments showing that average filament length can influence the types of actin structures that form. This is one of the most significant results of my thesis, because it leads one to consider the previously unexplored idea that the cell can control cytoskeletal structures such as bundles and networks by adjusting conditions that affect filament length. 11 Outline In Chapter 6,1 explore the effects of unequal lengths of filaments (previously assumed to be of uniform average length). This leads to some surprising results which suggest that a distribution of lengths may lead to behavior that deviates from that of the average. This suggest future modeling work to explore such phenomena. In Chapter 7, I aim to make contact between experimental observations and model predictions. With this in mind, I explain the mechanical basis for some of the experiments found in the literature and then discuss how certain predictions of the model fare when compared to the observations. 12 Chapter 2 E f f e c t s o f C r o s s - L i n k e r C o n c e n t r a t i o n s o n D e g r e e o f C r o s s - L i n k i n g o f A c t i n F i l a m e n t s The first question I study is how the addition of cross-linkers leads to the formation of bundles. I first thought that cross-linkers would act like glue, making filament networks more stable and solid when more were added. However, experiments by Wachsstock et al. (1993; 1994) show that when a-actinin is added to a network of filaments, the filaments tend to align with one another (i.e. bundle) and the resulting network behaves more like a liquid than a solid. Further, a result of Dufort and Lumsden (1993) shows that the number of cross-linkers actually forming cross-links eventually decreases when the total number of cross-linkers increases. This motivates the models in this chapter. To discover why this might be, I studied two simplified models of cross-linker-filament interactions. In particular, I looked at the equilibrium concentrations of the various states of the cross-linker and their dependence on the total concentration of cross-linker. 1 The Four Structure Model Cross-linkers such as a-actinin, only have two available sites, called binding domains, to bind actin filaments. Such cross-linkers may be in one of three different states: free (not bound to any filaments), singly bound (only one site is bound to a filament), and 13 The Four Structure Model O-O free s ing ly b o u n d d o u b l y b o u n d Figure 2.1: Cross-Linkers such as a-actinin exist in three different states: free, singly bound, and doubly bound. The open circle denotes a free binding domain and the closed circle denotes an attached binding domain. doubly bound (both sites are bound to filaments) as shown in Figure 2.1. In order for a free cross-linker to become doubly bound it must go through at least two reactions as shown in Figure 2.2. The goal of this chapter is to investigate how an increase in total cross-linker concentration affects the concentration of the various states of the cross-linker. Figure 2.2: A set of reactions that includes all three states of the cross-linker. The reactions in Figure 2.2 are derived from the cross-linker kinetics as described in Section 5 in Chapter 1. Reaction (i) describes how a free cross-linker may bind with an actin filament forming a singly bound cross-linker (Figure 1.6). The singly bound cross-linker may then bind with another actin filament to form a doubly bound cross-linker as shown in Reaction (ii). The rate constants of association, k+ (in units of p , M - 1 s _ 1 ) , and of dissociation, (in units of s _ 1 ) , in Reaction (i) are well known (Table 1.2). The rates in Reaction (ii) are not well known, but can be readily deduced when compared to O-O + (i) 14 The Four Structure Model Reaction (i) assuming that no conformational changes take place. The dissociation rate constant is 2k-, because one of two binding domains may detach, and the association rate constant is some fraction of k+, u, because this reaction takes more time to happen. The r h a b c d Figure 2.3: There are four different structures in the simplest formulation. Each structure is denoted by the letter labeling it. four different structures shown in Figure 2.2. are simply labeled a, b, c, and d as in Figure 2.3: a(t) is the concentration of free cross-linkers at time t, b(t) is the concentration of actin with no cross-linkers attached at time t, c(t) is the concentration of singly bound cross-linker structures at time t, and d(t) is the concentration of doubly bound cross-linker structures at time t. (Note that the plain type denotes the structure and that the italic type denotes the concentration of that structure.) The reactions in Figure 2.2 can now be recast as k+ a + b ^ c, (i) fc_ c + b ^ d . (ii) 2fc_ The concentrations of the structures obey the Law of Mass Action if the solution of actin and cross-linkers is well mixed. Thus, Reactions (i) and (ii) give rise to the following set of ordinary differential equations for the concentrations a, b, c, and d, at = k-C — k+ab, (2.1a) bt = k-c — k+ab + 2k_d — vk+bc, (2.1b) ct = —k^c + k+ab + 2k-d — vk+bc, (2.1c) 15 O - O The Four Structure Model dt = -2k_d + vk+bc. (2.Id) This system can be reduced through conservation of mass. Because the total concen-tration of actin (denoted by A) and cross-linkers (denoted by a) are fixed (for any one experiment), the concentrations must satisfy: b + c + 2d = A, (2.2a) a + c + d = a. (2.2b) Using these equations, two of the dependent variables may be eliminated. For example, b and c, may be written in terms of a and d, b = (A-a) + a-d, (2.3a) c = a — a — d . (2.3b) System 2.1 is now reduced to two dependent variables, a and d, that satisfy at = k_a — [k- + k+(A — a)} a — k-d — k+a2 + k+ad , (2.4a) dt = vk+(A - a)a + vk+(2a - A)a - (2/c_ + vk+A) d - uk+a2 + uk+d2. (2.4b) The system can be simplified further by dividing by.fc+. The new system (in units of fiM2 as opposed to iiMs"1) is aT = Ka-(K + A-a)a-Kd-a2 + ad, (2.5a) dT = v(A - a)a + v(2a - A)a - (2K + vA) d-va2 + ud2 = v (A - a)a + (2a - A)a - (2— + A^j d - a2 + d2 (2.5b) where r = ^— and k+ K is the cross-linker-to-actin dissociation equilibrium constant (commonly denoted by Kd). 16 The Four Structure Model 1.1 Isoclines and Graphical Analysis Having reduced the system to its simplest form, I can now analyze the steady state solu-tions to discover what happens as more cross-linkers are added, or the binding constant of the cross-linker to actin is changed. The equilibria are found by setting the left hand sides in System 2.5 to zero, Ka - (K + A - a) a - Kd - a2 + ad = 0 , (2.6a) v (A-a)a + {2a-A)a- (2^-+ A^jd-a2+ d2 = 0, (2.6b) and solving for a and d. This is equivalent to finding the points of intersection in the ad-plane of the two curves (called isoclines) defined by System 2.6. Since A, K, a and v are constants, the isoclines described by System 2.6 are conic sections, -a2 + ad + 0d2 — (K + A — a) a — Kd + Ka = 0 , (2.7a) -a2 + 0ad + d2 + 2 [ a - - ) a - 2 ( - + -)d + (A-a)a = 0. (2.7b) Studying these conic sections leads to an understanding of the system equilibria, in particular the dependence of d, the concentration of doubly bound cross-linkers on a, the total concentration of cross-linkers. The isocline defined by Equation 2.7b is a hyperbola opening in the direction of the d-axis as can be seen by rewriting it in the following equivalent form A^2 a — [a 2 + (d-Pl)2=p2 (2.8) where and K A v 2 K (K ; P2 = — ( — + A 17 The Four Structure Model The constant a occurs only inside the first quadratic term in Equation 2.8. Thus, in-creasing a is the same as shifting the hyperbola to the right in the ad-plane as shown in Figure 2.4. increasing a Figure 2.4: Increasing a in Equation 2.7b moves the hyperbola to the right in the ati-plane. The hyperbola on the left was produced for a = 5 and the one on the right for a = 15. The other parameters were fixed at A = 15, K = 1, and v = 1/4. Equation 2.7b represents a conic section with axes rotated rotated away from the co-ordinate axes (see box). The equation of the conic section in the rotated a'd'-coordinates is A / 2 +1 / , Qi — wa\ 2 y/2 — 1 / . o2 — za\ 2 < . . 18 The Four Structure Model where qi = (K + A)w + Kz and q2 = (K + A)z - Kw . Rotating this hyperbola clockwise by 22.5° yields the curve in ad-coordinates. Increasing a in Equation 2.9 has two effects. The first effect is to move the center of the hyperbola in the direction of the a'd'-plane. Rotating this vector by 22.5° (i.e. multiplying the vector by the matrix in the box), transforms it to (0,-1) in the ad-plane. Thus, one effect of increasing a is to move the isocline downwards (see Figure 2.5). The second effect of increasing a is found by studying the right hand side of Equation 2.9. As a increases from zero, the right hand side first decreases, causing the hyperbola to narrow. When a = K + A/2, the right hand side equals zero and the hyperbola is degenerate (it has the form of two intersecting lines). As a increases further, the right hand side becomes increasingly negative; thus, flipping the principal axes and causing the hyperbola to widen. Figure 2.5 shows how increasing a moves the hyperbola downwards and changes its asymptotes. Conic sections with an ad-term are rotated by some angle 0. Recall from linear algebra that the matrix cos 0 — sin 0 sin 9 cos 9 rotates the vector (x, y) clockwise by the angle 6. Thus, a new reference frame (a', d') can be introduced to eliminate the ad term from Equation 2.7a. 19 The Four Structure Model The new coordinates are obtained from the relations a = a' cos 0 — d! sin 6 , d = a! sin 9 + d! cos 9. Substituting these equations into Equation 2.7a, yields another second order equation in terms of a' and d'. The coefficient of the a'd' term is ((2)(0) - (2)(- l ) )cos0sin0 + (1) (cos2 9 - sin 2 9) . By setting the a'd1 coefficient to zero and using the double angle trigonometric identities, one finds that the angle that eliminates the a'd'-term satisfies t a n ( 2 0 ) = _ i _ ^ e=Zl, Thus, by working in the a'd'-plane, the conic section can easily be determined and then rotated clockwise by TT/8 radians (or 22.5°) to give the graph in the ad-plane. Since the cos(—7r/8) and the sin(—7r/8) are needed often in the rotated system, let us define w and z to be W = C O S ( - T T / 8 ) = \\J2 + V2, z = sin(-7r/8) = --^2 - y/2 . 2 Then Equation 2.7a becomes the following in the new coordinate system, _^l±ia'2 + ^ l i d ' 2 -[(K + A-a)w + Kz]a' + [(K + A- a)z- Kw]d' +Ka = 0.(2.10) Completing the square in the above equation yields the canonical form for the conic. (For more information on rotated conic sections, see Appendix II in Adams (1991).) 20 The Four Structure Model Figure 2.5: The top row shows that for small a, increasing a narrows the hy-perbola (a = 3 on left and a = 8 on right). The bottom row shows that as a continues to increase, the hyperbola interchanges its transverse and conjugate axes and begins to widen (a = 10 on left and a = 15 on right). The parameters A, K, and v have been fixed at 15, 1, and 0.25, respectively. 21 The Four Structure Model Before considering the possible equilibria of the system (at intersections of the isoclines), I discuss the biologically relevant values of the parameters A, K and v. 1.2 Parameter Values By choosing realistic parameter values for A, K, and v, one can study the effects of total cross-linker concentration, a, on the equilibria. In experiments by Wachsstock et al. (1993; 1994) and simulations by Lumsden and Dufort (1993), the concentration of actin was fixed at 15 u-M. Thus, I let A = 15 for comparison with their results. Values for K are determined from the kinetics of the cross-linker as seen in the definition of K, K = k-/k+. Table 1.2 lists the kinetic rates for various a-actinin. For chicken smooth muscle a-actinin and the "generic" a-actinin, K 1/J.M. Thus, I usually let K = 1. When I do change K to study its effects, I let K = 5, because K « 5(iM for the other a-actinin in Table 1.2. Unlike A and K, u, the fraction of the dissociation rate constant of Reaction (i) in Reaction (ii), is not explicitly given in experiments, but must be deduced by comparing Reaction (ii) to Reaction (i). In (i), a cross-linker that has two binding domains available binds with an actin filament at the rate k+. In Reaction (ii), only one binding domain is available. Thus, a first approximation to the association rate constant in Reaction (ii) is half the association rate constant of Reaction (i), implying that u = 1/2. However, actin filaments are less mobile than free cross-linkers so that structure c diffuses more slowly than structure a. Thus, the reactants in Reaction (ii) take longer to meet than the reactants in Reaction (i), further reducing the rate constant of association. Therefore, an estimate for v is 1/4, while most certainly 0 < v < 1/2. 22 The Four Structure Model A = 15 (11M) K = 1 (11M) v = - (dimensionless) 1.3 Equilibria The derived parameter values yield the isoclines shown in Figure 2.6 for a = 5. Because the hyperbolas intersect at three points (as one would expect from Equation 2.12), one might think that the system has three equilibria. However, only one of the points of intersection is a feasible steady state. The intersection near (-17,-6) would indicate that the concentrations of free cross-linkers, a, and doubly bound cross-linkers, d, are negative at steady state which is impossible. The intersection near (2,22) would indicate that the concentration of doubly bound cross-linkers, d, exceeds the total cross-linker concentration of a = 5. Therefore, only the intersection near (0.5,2.5) represents an equilibrium of the biological system. By restricting attention to the feasible region, the equilibrium may be tracked for various values of a. Figure 2.7 shows how the equilibrium changes as the concentration of cross-linkers is increased. The a-coordinate of the point of intersection shifts to the right, meaning that the free cross-linker concentration at equilibrium increases with the addition of cross-linkers. The d-coordinate does not change monotonically, though. It is increasing in the top row of Figure 2.7 and decreasing in the bottom row. The doubly bound cross-linker concentration, d, at equilibrium can be found alge-braically through Equation 2.7a: d = a2 + (K + A - a)a - Ka a — K This value of d may then be substituted into Equation 2.7b, reducing the problem from 23 The Four Structure Model 1 5 d 10 Figure 2.6: The isoclines for the system when A = 15, K = 1, v = 0.25, and a = 5. Where the isoclines intersect are the possible equilibria of the system. finding the intersection of two curves to finding the root of (2v - 1) o? - [(1 - 2v) A + (Au - 1) a] a2 (A - 2a) 2 + K[K + A) a - K2a = 0 (2.12) The root is the a value of the equilibrium. The d value of the equilibrium is found by substituting a into Equation 2.11. Further, the concentration of singly bound cross-linkers, c, may also be determined by substituting both a and d into Equation 2.3b. Finding the root(s) of Equation 2.12 may be done in several ways. One method is to 24 The Four Structure Model 2 4 6 8 10 12 14 0 2 4 6 8 10 12 14 a a Figure 2.7: The top row shows the intersection of the hyperbolas in the feasible region for a = 5 (left) and a — 10 (right). The bottom row shows the intersection for a = 15 (left) and a = 20 (right). The equilibrium is the point of intersection. (A = 15, K = 1, v = 0.25 as in box) 25 The Four Structure Model fix the parameter values, A, K, and v and then systematically use Newton's method for various values of a. Caution must be used because the cubic polynomial has three roots (one for each point of intersection in Figure 2.6), but only one of them is of interest. The feasible values of a are between 0 and a because the concentration of free cross-linker cannot exceed the total concentration of cross-linker. Thus, any negative roots are automatically eliminated. Further care must be taken since a may still be in the proper range, but the corresponding value of d may not be feasible. (An example of this is the point of intersection near (2,22) in Figure 2.6: a is between 0 and a = 5, but d exceeds a.) Thus, a further check is to substitute a into Equation 2.11, and test whether d is between 0 and the minimum of a and A. (d cannot exceed the total concentration of cross-linker nor can it exceed the total concentration of actin.) Thus, an algorithm for finding equilibrium values is as follows: 1. Calculate Newton's method to solve Equation 2.12 for A = 15, K — 1, and v = 0.25. 2. Initialize a = 0 and a. = 0. 3. Increase a by some specified step size. 4. Substitute current value of a into Newton's method with the last solution of a as the initial guess and solve for the new a. (a) If a is negative, change initial guess and repeat step 4. (b) If a is positive, calculate d with Equation 2.11. i . If d is not feasible, change initial guess and repeat step 4. i i . If d is feasible calculate c with Equation 2.3b and store a, a, d, and c. 5. If a has not exceeded set limit, repeat step 3. I implemented this algorithm with various values of a. The results in Figure 2.8 show how increasing the total concentration of cross-linker, a, changes the steady state 26 The Four Structure Model concentrations of free, singly bound, and doubly bound cross-linkers when A = 15, K = 1, and v = 0.25. In particular, the concentration of doubly.bound cross-linkers reaches a maximum. This figure also shows that the concentration of singly bound cross-linkers is monotonically increasing, something that could not be determined from the isoclines. Figure 2.9 shows how the results change when K — 5 (as opposed to 1). In this case, cross-linkers dissociate from actin more quickly, and not surprisingly, there are more free cross-linkers and less doubly bound cross-linkers. Figure 2.9 also shows that the maximum concentration of doubly bound cross-linkers is smaller (approximately, 1 u.M as opposed to 3 u,M) and occurs at an increased level of total cross-linker concentration (approximately, a = 12 as opposed to a = 8). Figure 2.8 bears a striking resemblance to Figure 1 in Lumsden and Dufort (1993) as discussed in Section 3. 1.4 Conclusions of the Four Structure Mode l Because the concentration of double bound cross-linkers reaches a maximum, the addition of cross-linkers will eventually lead to a more liquid-like state. Cross-linkers do behave like a glue, but only when they are binding two filaments together (i.e. they are in the doubly bound state). As Figure 2.8 shows, the concentration of doubly bound cross-linkers increases up to some maximum and then decreases. As the concentration of doubly bound cross-linkers approaches its maximum, the concentration of singly bound cross-linkers becomes favored. These results can be explained by considering the number of binding domains (free ends of the cross-linker) vying for the number of available binding sites (places to bind on actin filaments). At low total concentrations of cross-linker, there are many available binding sites for few binding domains. This favors doubly bound cross-linkers. As more cross-linkers are added to the solution, a point will eventually be reached where binding sites become saturated. At this point, singly bound cross-linkers are favored over doubly bound cross-linkers. As even more cross-linkers are added to the solution, most of the cross-linkers will be free because there are not 27 The Four Structure Model 12 10 Concentration of Crosslinker States for K=1 free singly bound doubly bound 8 10 12 14 total concentration of crosslinker 16 18 20 Figure 2.8: As the total concentration of cross-linker increases, the equilibrium states of the cross-linker changes. The concentration of free cross-linkers always increases. The concentration of singly bound cross-linkers increases in the range shown. The concentration of doubly bound cross-linkers reaches a maximum and then begins to decrease. (A = 15, K = 1 and v — 0.25 as in box.) enough sites available for all of the binding domains. Thus, the equilibrium is affected by the competition between binding sites and binding domains. (This can be shown by studying equilibrium equations as well; at equilibrium d is proportional to c2/a. As the concentration of cross-linkers increases d must go to zero. However, when there are few cross-linkers d must also be small. Thus, d must have a maximum at some intermediate concentration of cross-linkers.) A n analogy for this competition is a group of children fighting over a table full of candy. The children can be likened to the cross-linkers and the table of candy likened 28 The Four Structure Model 12 10 Concentration of Crosslinker States for K=5 n 1 1 1 r 2h free singly bound doubly bound 6 8 10 12 14 total concentration of crosslinker, alpha 16 20 Figure 2.9: Changing the binding constant, K, from K = 1 (shown in Figure 2.8) to K = 5 (shown above) has two effects on the doubly bound cross-linkers. It decreases the concentration of doubly bound cross-linkers, but increases the point where the maximum occurs. (As in Figure 2.8, A = 15, K = 1 and v = 0.25.) to the actin. The amount of candy on the table is fixed as is the concentration of actin. When there are not many children present, they can attack the table with both hands. However, as more children arrive, there will be less space at the table. Now rather than being able to use both hands, some children are forced to only use one. This is similar to what Figure 2.8 describes. 29 The Seven Structure Model 2 The Seven Structure Model Although the previous model was insightful, it was not very realistic in its restriction of the possible structures that can form. In this section I explore a more realistic model that allows up to two cross-linkers to bind filaments together. The model is developed by extending the previous model to include three more structures as shown in Figure 2.10. I predict that the addition of these structures will shift the maximum of Figure 2.8 to the right because more binding sites are available. Also, by analyzing this model one can determine whether future extensions should be made (and at what cost) as well as how drastically the effects in the first model change in this more realistic setting (if at all). M L- U U d e f g Figure 2.10: There are now seven different structures under study as labeled above. The letter labeling denotes the concentration of that structure in \xM. By increasing the number of structures from four to seven, the number of reactions increases from two to seven. This is quite a significant jump. Figure 2.11 shows all the possible reactions with their rate constants of association and dissociation labeled. As before, the only reaction with known rates is Reaction (i). A l l other reactions are studied in comparison with Reaction (i), assuming that there is no conformational change in a-actinin, by comparing • the number of binding domains in the reaction (one binding domain detaches at the rate of fc_ and two binding domains attach at the rate of k+), and • the mobility of the reactants (cross-linker meets actin filament as in Reaction (i) or actin filament meets actin filament as in Reaction (ii)). 30 The Seven Structure Model 2vk. + I ^ 1 1 (iv) 2vfe+ + I ^ I I (V) O - O + | | I I (vi) T\k+ 4k (Vii) Figure 2.11: Al l the possible reactions that can occur in order for two filaments to be bound by two cross-linkers are shown above. 31 The Seven Structure Model For example, the mobility of the reactants in Reaction (iii) is the same as Reaction (i), and the number of binding domains is also the same. Therefore, the association rate constant should be the same, k+. The dissociation rate constant is 2fc_ because either one of two binding domains can detach as opposed to one. Reactions (iv), (v), and (vi) are very similar in that they all form the structure, f. The mobilities of the reactants in Reactions (iv) and (v) are the same as Reaction (ii). However, because there are two binding domains as opposed to one, the association rate constant, 2vk+, is twice as fast as Reaction (ii). The reactants in Reaction (vi) are similar to the reactants in Reaction (i), the only difference being the bigger structure d. Because structure d is most likely less mobile than structure b, the association rate constant of Reaction (vi), fik+, is probably smaller than k+. Thus, I estimate that a is close to but smaller than one (0.9 for example). The dissociation rate constants for Reactions (iv)-(vi) are the same. To obtain the reactants in Reactions (iv), (v), or (vi) one specific binding domain must detach in each case. By focusing on only one binding domain for each reaction, one discovers that the dissociation rate constant is similar to k_, implying that £ 1. Unlike the other reactions, Reaction (vii) does not combine two structures, but in-stead changes formation. When comparing the rate constant of association in Reaction (vii) with Reaction (i), one would expect little time to be wasted for the free binding domain to meet with the other filament. Therefore, even though only one binding do-main is available, I'd expect this reaction to be faster than the association rate constant in Reaction (i), letting 77 ~ 2 . The rate constant of dissociation is calculated as before. Because any one of four binding domains can unbind, the dissociation rate constant is Ak_. 32 The Seven Structure Model The reactions in Figure 2.11 can also be written in terms of the structure labels: a + b k+ k-c, 0) b + c 2k-d , (") a + c k+ e, (iii) b + e 2vk+ i k -f, (iv) c + c 2uk+ f, (v) a + d l i k + f, (vi) rjk+ f ^ g 4fc_ (vii) As before, the Law of Mass Action leads to the set of ordinary differential equations for the concentrations of the intermediates shown in Figure 2.11: at = k-C + 2A;_e + — k+ab — k+ac — fik+ad , (2.13a) bt = k„c + 2k^d +£k_f - k+ab - uk+bc - 2uk+be, (2.13b) ct = k+ab + 2k_e + 2/c_d + 2£k-f — A;_c — fc+ac — vh+bc — Avk+c2, (2.13c) dt = £k_f + vk+bc - fik+ad - 2fc_d, • (2.13d) et = k+ac + - 2/c_e - 2vk+be , (2.13e) ft = /ik+ad + 2vk+be + 2vk+c2 + 4h_g - (3£fc_ + vK) f , (2.13f) & = Vk+f-Ak-g. (2.13g) 33 The Seven Structure Model The conservation equations for actin and cross-linker are now A = b + c + 2d + e + 2f + 2g , (2.14a) a = a + c + d + 2e + 2f + 2g . (2.14b) Two variables may be eliminated, reducing System 2.13 to a system of five (long) ordinary differential equations. The system may be simplified further by letting r = and k K — —1 as before. k+ 2.1 Finding Equilibria To find equilibria, one sets the left hand side of System 2.13 to zero and tries to solve for a, b, c, d, e, / and g. Solving for g in Equation 2.13g is trivial, Using this result along with the conservation laws (System 2.14), the problem of finding the equilibria has been reduced to solving four equations for four unknowns. The reduced system of equations is quite cumbersome and is not displayed here. Even if the equations were linear (they are not), finding the equilibria analytically in terms of all the parameters would be difficult. Also, because there are now four equations and four unknowns, the graphical approach cannot be applied since it would involve looking for intersections in four dimensional space. However, since I have estimates for all the parameters used in the model (a. ~ 0.9, £ « 1, v « 1/4, 77 K, 2, A = 15 u.M, and K = 1 (J.M), numerical methods (found in software like "XPP-Aut" ) can be used to find the equilibria. The numerical package "XPP-Aut" was written by Bard Ermentrout. It is free and easily downloaded from his public ftp site. This package allows one to study systems of ordinary differential equations. It not only numerically solves systems of ordinary differential equations, but also finds equilibria and their stability. In addition, it also 34 The Seven Structure Model acts as an interface to the program "Auto" that allows one to study the parametric dependence of differential equations. Using this package, I was able to determine that there is only one equilibrium for System 2.13, and it is stable. I was also able to use the "Auto" portion to quickly determine how the equilibrium changed as a was increased. Thus, " X P P - A u t " was very useful in analyzing this problem. Having found the equilibrium values for a, b, c, d, e, /and g, I was able to determine the concentrations of free cross-linker, singly bound cross-linker and doubly bound cross-linker. The concentration of free cross-linkers is simply the value of a. Singly bound cross-linkers appear once in the c-structure and /-structure, and twice in the e-structure. Thus, the concentration of singly bound cross-linkers is c + 2e + / . Doubly bound cross-linkers appear once in the d-structure and /-structure, and twice in the g-structure. Thus, the concentration of doubly bound cross-linkers is d + / + 2g. The resulting change of the cross-linker states as the total concentration of cross-linker, a, is increased is shown in Figure 2.12. Again, one sees that the concentration of doubly bound cross-linkers eventually reaches a maximum as more cross-linkers are added. 2.2 Conclusions of the Seven Structure Mode l Like the four structure model, the seven structure model shows that the concentration of doubly bound cross-linkers eventually decreases as more cross-linkers are added to the system. As predicted, the maximum occurs at a higher total concentration of cross-linker (around 15 in Figure 2.12 as compared to 8 in Figure 2.8). This gives credence to the reasoning from the previous section about competition for binding sites. Although, this model is more realistic then the previous one, it points to the same mechanism for more fluid-like networks to exist as the total concentration of cross-linker is increased. 35 Conclusions and Discussion 20 18 16 14 12 c o 03 I 1 0 O c o o 8 free single bound double bound 10 15 20 total crosslinker concentration 25 30 Figure 2.12: As the total concentration of cross-linker increases, the equilibrium states of the cross-linker change, (/z = 0.9, £ = 1, v = 1/4, n = 2, A = 15, and K = l ) . 3 Conclusions and Discussion Although the models in this chapter showed how the addition of cross-linkers could lead to a less solid actin network, the current methodology is restricted. Increasing the number of structures included in the model greatly increases the difficulty of solving the ordinary differential equations and of estimating the different rates in the reactions. Also, it is not feasible to describe all the possible structures in this manner. A better approach for the more general case was taken by Lumsden and Dufort (1993). They used a cellular automaton model to simulate the states of the cross-linkers in a solution with actin. My model was motivated by trying to understand their results, and thus a simplification was made. My model was also able to show how a change in the binding constant could affect 36 Conclusions and Discussion the doubly bound cross-linkers. Unfortunately, my model suggests that the addition of any cross-linker might even-tually lead to a less solid actin network. However, the geometry of the actin filaments is not considered and the cross-linker kinetic rates are not fully explored. Both of these factors could influence the mechanical properties of actin networks. It has been shown that cross-linkers such as a-actinin can align the actin filaments into bundles which may impart more fluid-like properties to the solution. Cross-linkers like A B P , however, tend to keep actin filaments orthogonal, which would keep a more solid-like structure intact. Thus a next step would be to consider the geometry of the filaments so that one can study the formation of clusters or bundles. Even though the current model has many shortcomings, it does help to answer how the addition of cross-linkers could lead to a less solid actin network. As the concentra-tion of cross-linkers is increased, the number of binding sites becomes'a limiting factor. Eventually, the singly bound state of a cross-linker will be favored over the doubly bound state. Because cross-linkers stabilize filaments when they are in the doubly bound state, the decrease in doubly bound cross-linkers will cause the actin network to behave less solidly. 37 Glossary of Parameters 4 Glossary of Parameters a the concentration of free cross-linkers. b the concentration of pure actin. c the concentration of actin with one cross-linker. d the concentration of one cross-linker attaching two filaments. e the concentration of actin with two cross-linkers. f the concentration of two cross-linker and two filaments such that one cross-linker is doubly bound and the other is singly bound. 9 the concentration of two cross-linkers attaching two filaments. A the total concentration of actin. a the total concentration of cross-linker. k+ the association rate constant of the cross-linker. the dissociation rate constant of the cross-linker. K the dissociation equilibrium constant of the cross-linker, k^/k+. T the scaled unit of time, t/k+. V the fraction of the association rate in (ii) compared to (i) as shown in Figure 2.2 fi (see Figure 2.11). £ (see Figure 2.11). j] (see Figure 2.11). a! the rotated a-axis. d! the rotated d-axis. 0 the angle of rotation from (a', d') to (a, d). 38 Chapter 3 M o d e l i n g t h e S p a t i a l C l u s t e r i n g a n d A l i g n m e n t o f A c t i n F i l a m e n t s w i t h I n t e g r o - P D E ' s In this chapter I explore the spatial distribution of actin filaments and the possibility of clustering. I show that the length of the filaments influences whether clusters form, and how closely spaced they are. The model I use is an integro-PDE. Integro partial differential equations (integro-PDE's) have been used to model many different phenomena in biology: pattern formation (Levin and Segel, 1985; Murray, 1989; Mogilner and Edelstein-Keshet, 1995), predator-prey systems (Lewis, 1994; Kot, 1992), fibroblast alignment (Edelstein-Keshet and Ermentrout, 1990), actin filament orientation (Civelekoglu and Edelstein-Keshet, 1994), and actin filament interactions (Mogilner and Edelstein-Keshet, 1996; Geigant and Stoll, 1996). In this chapter I will explain how the integro-PDE model for actin is developed, the meaning of the parameters in the model, and how the condition for pattern formation is derived. Whereas the previous chapter focused on cross-linker equilibria, now I wish to study the actin filament dynamics. Because cross-linkers are usually much smaller than actin filaments, they diffuse throughout the solution more quickly. Thus, the assumption in the previous chapter that the solution was homogeneous is reasonable for the study of cross-linker effects, allowing one to consider only one independent variable, time. Because 39 Modeling the Spatial Clustering and Alignment of Actm Filaments with Integro-PDE's actin filaments are long, they interact over an extended region and their mobility is smaller than cross-linkers, leading to the possibility of nonuniform spatial distributions. Modeling this gives rise to partial differential equations rather than ordinary differential equations. In addition, because filaments are elongated and interact with one another along their entire length, nonlocal interactions are possible. These give rise to integral terms in the mathematical model. Thus, a full mathematical model that describes actin filament dynamics is an integro-PDE. The basic ingredients in the analysis are not new, but have been developed in a set of previous papers (Civelekoglu and Edelstein-Keshet, 1994; Mogilner and Edelstein-Keshet, 1996) which concentrated mainly on angular rather than spatial filament distributions. I include a review of this material for completeness, as well as to indicate a number of details and modifications that I made. My study of actin filament networks focuses on the difference between four basic structures as shown in Figure 3.1. I use the word networks to refer to filaments bound together with some actin binding agent such as a-actinin. If these network filaments neither align nor clump, I say that they form a gel (Figure 3.1a). Clusters denote filaments that are clumped spatially (Figure 3.1b). A further categorization depends on filament alignment (Figure 3.1c). I call networks where filaments both cluster and align bundles (Figure 3.Id). This terminology is similar to the one currently used by biologists. In motile cells such as keratocytes and fibroblasts, a lamellipodium forms that fa-cilitates cell movement (see Figure 3.2). The thickness of the lamellipodium, 0.2 /J, is much smaller than the cell diameter, 10-20 \x (Alberts et al., 1989) so that a good ap-proximation is that the distribution of actin is two dimensional in the lamellipodium. In fibroblasts, the interesting dynamics occur within 4 u- of the leading edge of the cell 40 Modeling the Spatial Clustering and Alignment of Actin Filaments with Integro-PDE's (c) (d) Figure 3.1: Actin filament networks may be classified into four different patterns: (a) gel, (b) clusters, (c) alignment, (d) bundles. (Theriot, 1994). By considering a small strip within this band (the shaded area in Figure 3.2), we can reduce one more spatial dimension. Thus, the actin filament distribution will be modeled as though dynamics occur essentially in one spatial dimension, allowing simulations to be completed in a reasonable amount of time. However, the distinction be-41 Modeling the Spatial Clustering and Alignment of Actin Filaments with Integro-PDE's Top View Side View Cortex Lamellipodium Cortex Lamellipodium Figure 3.2: Fibroblasts have a lamellipodium that extends from the cell cortex. Most of the interesting filament dynamics occur near the leading edge, in the thin strip shown shaded above. The one spatial dimension of this strip corresponds to the one-dimensional model described in this thesis. tween the types of structures shown in Figure 3.1, i.e. clusters versus bundles versus gels, will be accommodated by following the angular distribution of the filaments throughout space as in Figure 3.3. The two independent variables x (position) and 6 (angle) allow each type of structure to be distinguished. Figure 3.4 shows how distributions that are non uniform in x and 9 correspond to the various network structures. (a) (b) //K\\^\\\||///IIM^\I 1 w 1 (c) (d) Figure 3.3: The networks as seen in a "one" spatial dimension: (a) gel, (b) clusters, (c) alignment, (d) bundles. The length of the bar corresponds to the length of the shaded strip in Figure 3.2. 42 Modeling the Spatial Clustering and Alignment of Actin Filaments with Integro-PDE's (a) (b) • e MM^ \WII///llM^ \l (c) 3 0 (d) Figure 3.4: Above each network structure is a graph corresponding to the den-sity of filaments in x and 6. (a) A gel is uniform in space and angle, (b) Clusters have a spatial nonuniformity, but are homogeneous in the angular variable, (c) If filaments are aligned, peaks occur in 8 but not necessarily in x. (d) In bundles, filaments are both aligned and clustered corresponding to the density distribu-tion shown. 43 One Dimensional Model 1 One Dimensional Model 1.1 Model Development To get a flavor for the analysis of integro-PDE's, I first develop a model that describes actin filament interactions in one spatial dimension. (This is analogous to the work done by Civelekoglu (1994) on filament orientations.) It is assumed that filaments can be in one of two states: free, meaning that they are not attached to any other filaments, or bound, meaning that they are attached to at least one other filament so that they are part of an actin filament network. Having defined the two types of filaments, the following features can then be modeled: 1. A free filament may bind with another free filament to become a network filament if the filaments are close enough (free-free filament binding). 2. A free filament may bind with a network filament to become a network filament, as well (free-network filament binding). 3. A network filament may unbind from the network to become a free filament (fila-ment unbinding). 4. The free filaments may diffuse in space (filament diffusion). To study the interactions of these filaments, I once again look at concentrations, specifically at the number of filaments per unit volume. The filaments are assumed to have some constant average length, L . Thus, the two independent variables (time, i , and space, x) affect two dependent variables, F(x,t), the concentration of free filaments at time t whose center is at a, and N(x, t), the concentration of network filaments at time t whose center is at x. Features 1, 2 and 3 above can be described using standard chemical kinetics, 44 One Dimensional Model Pi F + F ^ N 2 7 ' F + N ^ N 2 . 7 If the reactants are point masses, then the reactions would lead to the following equations through the law of mass action, Nt(x,t) = 01F(x)F(x) + p2N(x)F(x)-^N(x), (3.1a) Ft(x,t) = -p1F(x)F{x)- (32F{x)N{x)+1N(x). (3.1b) However, this formulation assumes that filaments only interact at a single point. It fails to take into account that filaments can interact along their entire lengths. Allowing filaments at different positions to interact is a central part of these models. This is accomplished with the introduction of a probability density function, K. (This is not to be confused with the dissociation equilibrium constant which will be denoted by Kd)- K{x — x') gives the probability that a filament whose center is at position x interacts with a filament whose center is at position x'. (Technically, K has units of per unit length because it is a probability density function that ranges over the spatial domain). Therefore, the expected concentration of free filaments that can interact with filaments at position x is f K(x-x')F(x')dx'= (K*F)(x) (3.2) where O is the spatial domain. (The above expression is a convolution; hence the common notation, K * F.) Now, rather than having the free filaments at x interacting only with other free filaments at x, we can consider the expected number of free filaments from all locations that can interact with free filaments at x, changing the term 0iF(x)F(x) in System 3.1 to (3iF(x)(K * F)(x). Applying this to all the filament-filament interactions 45 One Dimensional Model changes System 3.1 to Nt(x,t) = PlF(x)(K*F)(x) + p2N(x)(K*F)(x)-1N(x), (3.3a) Ft(x,t) = -p1F(x)(K*F)(x)-02F(x)(K*N)(x) + 'yN(x). (3.3b) Only the fourth criterion, free filament diffusion, remains to be modeled. If \i is the rate of diffusion for a free filament, then Ft(x,t) = u.Fxx{x,t) (3.4) describes the diffusion of free filaments. Including this in System 3.3 gives the complete model of Nt(x,t) = p1F(x)(K*F)(x) + p2N{x)(K*F)(x)-'YN(x), (3.5a) Ft(x,t) = u.Fxx{x)-(3lF{x){K*F){x)-(32F{x)(K*N){x) + 1N{x). (3.5b) As explained in the introduction to this chapter, the spatial domain is a thin strip approximated as a one-dimensional region, 0 < x < R (R 10/j.). Neumann (no flux) boundary conditions are most reasonable. However, these are difficult to implement in the numerical scheme. While the analysis is carried out in the general case, I restrict attention to periodic boundary conditions in my simulations. 1.2 M o d e l Analysis System 3.5 is now analyzed (with Pi ^ /32, unlike previous analyses) to determine under what conditions patterns can form. First, I will show that the system conserves mass. I will then determine what the homogeneous steady state of the system is. Finally, I will linearize the system about the homogeneous steady state and analyze the resulting linear system for stability. 46 One Dimensional Model Conservation of Mass The total number of filaments per unit volume does not change over time, so that mass is conserved. If M is the total number of filaments per unit volume (i.e. concentration in number of filaments), then / (Fix, t) + N(x, t)) dx — M . Jn Differentiating both sides of this equation with respect to time leads to / (Ft(x,t) + Nt(x,t)) dx = 0, Jn (3.6) (3.7) if F and N are smooth functions (as one might expect). System 3.5 gives values for Ft and Nt. Substituting these into the left hand side of Equation 3.7 produces the following: (Ft(x,t) + Nt(x,t)) dx n = / {p2 [N(x)(K * F)(x) - F(x)(K * N)(x)} + /iFxx(x)} dx = p2 [ \N(X) f K(x - x')F(x') dx' - F(x) I K(x - x')N(x') dx'} dx Jn I Jn Jn + / (nFxx(x)) dx Jn = p2\[ f K(x' - x)N(x')F(x) dx1 dx - f I K(x - x')N(x')F(x) dx' dx i + Lt Fx(x) The Fx term is zero if there are either no-flux boundary conditions, as seen in biological experiments, or periodic boundary conditions typical of computer simulations. The P2 term is zero if K is an even function (i.e. K(—x) = K(x)). Because K was defined to be a probability based only on the filaments' relative position, K(x — x') = K(x' — x), so that K is even. Thus, the p2 term is also zero and Equation 3.7 is satisfied. Therefore, the model conserves mass. 47 One Dimensional Model Steady State The homogeneous steady state is found by assuming that F and N are constant, say F(x, t) = F and N(x, t) = N, and solving for F and N when the right hand side of System 3.5 is set to zero. This yields the following result: PiF2 + p2FN = -fN. (3.8) Conservation of mass means that F + N = M, (3.9) where M is a constant determined by the total concentration of actin in the system. By solving for either F or in Equation 3.9 and substituting this into Equation 3.8, one can find the specific values of F and N with the quadratic formula: - (p2M + 7 ) ± J(02M + 7 ) 2 + 4 / 3 l 7 M F=- , (3.10) and N = M — F . (3.11) In the degenerate case when Pi = /32, as noted previously (Civelekoglu and Edelstein-Keshet, 1994; Mogilner and Edelstein-Keshet, 1996). Because the square-root term in Equation 3.10 is always larger than P2M + 7, there will be one positive and one negative root. Since F cannot be negative, System 3.5 has only one homogeneous steady state. If Pi < p2, then the square-root term should be subtracted. If P2 < Pi (as will be shown in Chapter 4), then the square-root term should be added. 48 One Dimensional Model Perturbations from the Steady State Having found the homogeneous steady state of System 3.5, I now consider perturbations away from the steady state. This is accomplished by substituting N = N + n and F = F + / into System 3.5: N + n)t = p1(F + f)[K*(F + f)] +P2 {N + n) [K * (F + /)] - 7 ( i V + n) , (3.13a) -Pi (F + f)[K*(F + f)] -P2(F + f) [K*(N + n)] +7 (N + n) . (3.13b) Multiplying the terms out yields nt = p1F(K*F)+01F{K*f) + p1f[K*F)+p1f(K*f) +P2N (K * F) + p2N(K * /) + p2n (K * F) + P2n(K * /) _ 7 7 V - 7 ? i , (3.14a) ft — l^fxx -PXF (K*F)- P,F(K * f) - pxf (K*F)- pJ(K * /) -P2F (K*N)- p2F{K * n) - p2f (K*N)- P2f(K * n) +jN + ~fn. (3.14b) The term (K * F) may be simplified as follows: (K*F)= [ K{x - x')F(x') dx' = F [ K(x - x) dx' = (F) • (1), (3.15) Jn Jn 49 One Dimensional Model because the probability over the entire domain must be one. Similarly, (K*N) = N. (3.16) Using these two relations, summing the first terms of each line in System 3.14 simplifies with Equation 3.8. Therefore, the new system in terms of / and n is nt = (P2F-y)n + P1Ff+(p1F + /32N)(K*f) + 0ifK*f + /32n(K*f), (3.17a) ft = fifxx + in- p2F(K * n) - ( & F + 02N) f - p1F(K * /) - PJ(K * /) + p2fK * n . (3.17b) I consider perturbations of the form n = nQe ikxeXt, (3.18a) / = fQeik*ext. (3.18b) n 0 and / 0 represent the size of the perturbation. e lhx represents spatial periodicity, k is the wavenumber and is related to the number of peaks (or troughs). Since the domain has length R, the permissible values of k are k = 2nir/R in the case of periodic boundary conditions. (Thus, if R = 2-7T, k can only take on integer values, k = n.) In the Neumann (no flux) boundary condition case, the permissible values of k are k = rnr/R. (In this case if R = 27r, then k can take on the values 1/2, 1, 3/2, ....) Thus, the results shown in Figures 3.6 - 3.12 can be interpreted in either context. A is the growth rate of the system (also called the eigenvalue). If A < 0, perturbations will decay and the homogeneous steady state will be stable. If A > 0, perturbations will grow and the homogeneous steady state will be unstable. 50 One Dimensional Model The perturbations shown in System 3.18 greatly simplify the convolution terms and the diffusive term. The convolution, K * n, can be expressed as (K*n)(x) = f K(x' -x)n0elkx'eXtdx' = n0ext f K(u)eiKu+x) du = n0elkxext f K(u)eiku du = nK(k), (3.19) where k is the Fourier transform of the kernel K. Similarly, (K*f)(x) = fK(k). (3.20) The diffusive term can be expressed as fxx = ^ [f0eikxext] = -k2f0eikxext = -k2 f . (3.21) (Note that elkx is an eigenfunction of the convolution operator, K * •, and the Laplacian operator with eigenvalues k(k) and — k2, respectively.) Substituting for the convolution terms and the diffusive term in System 3.17, one obtains ft (P2F - 7) n + faFf + (faF + p2N) Kf +p1kf + p2knf, -u.k2f +-7n - p2FKn - (P,F + p2N) f -p1Fkf - pxk f2 + p2Knf . (3.22a) (3.22b) Linear Analysis Neglecting terms from System 3.22 that are quadratic or higher, specifically the /2-terms and the n/-terms, one obtains the following linearization of System 3.17: 7 - p2FK -iik2 - p2N - pxF ( l + K) 51 (3.23) One Dimensional Model For notational ease, let us call the above matrix B. The eigenvalues of B determine whether the homogeneous steady state is stable or not. (If the real part of an eigenvalue is positive, the steady state will be unstable). Since B is 2 x 2, the characteristic equation (det (B — XI) = 0) that determines the eigenvalues is quite simple. It is based solely on the trace, Tr(-B), and determinant, det(i?) of the matrix, B: Tr(B) ± A / T r ( £ ) 2 - 4det(B) A — Tv(B)X + det(B) = 0 => X = ———*—— — . (3.24) 2 The trace of B is Tr(B) = (P2F - 7 ) - iik2 - p2N - fcF ( l + k) . (3.25) A l l terms except for (/32F—j) are obviously negative because all the variables are positive. However, (P2F — 7) can be shown to be negative as well by using Equation 3.8: N - 87FN - 77V -B^F2 (fhF - 7) = ^ F - 7) = P 2 R 7 = • (3-26) Thus, the trace is always negative so that A can be positive only if the determinant is negative. Thus, the condition for the steady state to be unstable is det(S) = (j^j k 2 - p2lNK ( l - k ) < 0 . (3.27) Using Equation 3.8, one can simplify the condition for instability to ph2 < (^^j M2K ( l -K), (3.28) where M = F + {Pi/Pi)N. (M is a measure of the total filament concentration). Thus, the following actions may lead to instabilities: 1. Li decreases, 2. Pi or P2 increases, 3. 7 decreases, or 4. M increases. (The physical interpretation of these changes is given in section 3.) 52 One Dimensional Model 1.3 Filament Length Affects Clustering For illustrative purposes, I will focus on how changing the filament length affects the final actin network structure. These results are new and show how the length of the filament (assumed to be rigid) influences the spacing of clusters in the unstable case. For simplicity, the domain size R, will be 2 TT microns (a reasonable approximation of the actual length of the strip shown in Figure 3.2), so that all wavenumbers k will be whole numbers. As will be shown in Chapter 4, changing the filament length, L , directly influences fi, M, Pi, and a as follows: LL oc 1/L, (3.29a) M oc 1/L, (3.29b) Pi oc L , (3.29c) a oc L . (3.29d) These changes can be studied both graphically and numerically to determine their effect on the system. Graphical Analysis By choosing a specific function for the kernel, K, its Fourier transform can be found and used to analyze the instability condition (Equation 3.28). A reasonable choice for K is the normal density function with standard deviation, a, This form of K has a maximum at x = 0, so that filaments that overlap exactly have the greatest probability of binding. Also, the probability of interaction smoothly de-creases as filaments get further and further apart. How fast these probabilities decrease is determined by a, so that changing a affects the overall range of interaction. 53 One Dimensional Model 0.25 0.2 0.15 0.1 0.05 I . \;' v' i I ! I : i ; I : \ '• I i I sigma = 0.5 sigma = 1.0 sigma = 2.0 Figure 3.5: A plot of the expression, K (l — K^J which appears in the condition for instability (Equation 3.28, as a function of the wavenumber, k. Increasing the standard deviation, cr, in the normal distribution, K, causes K — to narrow and move to the left. This Gaussian kernel has a closed form Fourier transform, ' a2k2\ K(k) = exp ( 3 . 3 1 ) that can be applied directly to Equation 3.28. By dividing both sides by the coefficient of the right hand side, one obtains the relation Pk2 < K{k) ( l - k ( k ) ) where P = M 7 M2h(52 (3.32) ( 3 . 3 3 ) 54 One Dimensional Model The graph of the expression on the right hand side of Equation 3.32 (as a function of the wave number k) now only depends on the parameter o. Increasing a narrows the hump in the graph of K(k) (l - K(k)^j as shown in Figure 3.5. The function of k on the left hand side of Equation 3.32 is simply a parabola whose width is determined by all the other parameters lumped together in the quantity P as seen in Equation 3.33. Instability of the homogeneous steady state can now be studied by comparing the graph of the left hand side of Equation 3.32 with the right hand side as shown in Figure 3.6. The homogeneous steady state becomes unstable when the parabola is below the hump. (Note that the list of changes to the parameters that influence instability all force P to decrease; thus, forcing the parabola to become wider). This analysis (but with a distinct form of the kernel suitable for the given situation) is generic and has been done previously (Edelstein-Keshet and Ermentrout, 1990; Civelekoglu and Edelstein-Keshet, 1994). 0.25 0.2 0.15 0.1 0.05 • / \ p = 1.0 P = 0.1 P = 0.01 sigma=1 0.5 1.5 2.5 3 3.5 4 4.5 k Figure 3.6: Changing the parameters to favor instability is the same as open-ing up the parabola in the figure. 'Where the parabola is beneath the curve K (l — K^j determines which wavenumbers, k, are unstable. 55 One Dimensional Model System 3.29 shows how increasing the filament length, L , affects the model param-eters. By studying Equation 3.33, one discovers that L serves to decrease P and thus widen the parabola (Figure 3.6) defined by the left-hand side of Equation 3.32. At the same time, the hump defined by the right-hand side of Equation 3.32 narrows and shifts to the left (Figure 3.5). Thus, an increase in filament length has two effects: by decreas-ing P, it increases the chance of an instability occurring; but by narrowing the hump, the range of possible modes that may become unstable decreases. Figures 3.7 - 3.12 show how changing the filament length from 0.25 ji to 8 \i affect the inequality in Equation 3.32. If filament lengths are 0.25 fi (Figure 3.7), then the left-hand side is greater than the right-hand side for all k meaning that the steady state remains stable. If filament lengths are 0.5 \i (Figure 3.8), then the parabola widens, so that the left-hand side is less than the right-hand side for k < 2.4. In this case, the homogeneous steady state should be unstable and either one or two peaks should form. If filament lengths are 1.0 LL (Figure 3.9), then the parabola widens further, so that the left-hand side is less than the right hand side for k < 3.5. In this case, the homogeneous steady state should be unstable and up to three peaks may form. If filament lengths are 2.0 LL (Figure 3.10), then even though the parabola is wider, the hump has noticeably narrowed so that the left-hand side is less than the right-hand side for k < 2.4, so that the results should be similar to filament lengths of 0.5 \x. If filament lengths are 4.0 [i (Figure 3.11), then the hump narrows further and noticeably shifts to the left, so that the left-hand side is less than the right-hand side for k < 1.5. In this case, the homogeneous steady state is unstable and only one peak should form. Finally, if filament lengths are 8.0 / i (Figure 3.12), then even though the parabola is very shallow, the hump has narrowed to such an extent that the left-hand side is never less than the right-hand side for k > 1. In this case, the homogeneous steady state remains stable. 56 One Dimensional Model LHS RHS 0.2 0.4 0.6 Figure 3.7: Comparison of the left-hand side and the right-hand side of Equation 3.32 for an average filament length of 0.25 fi. The left-hand side of is greater than the right-hand side for all k so that the homogeneous steady state will remain stable. Figure 3.8: As in Figure 3.7 for an average filament length of 0.5 a. The homo-geneous steady state is unstable for values of k less than approximately 2.5. 57 One Dimensional Model 0.35 0.3 0.25 0.2 0.1 0 I I i i 1 1 ' 1 LHS RHS -/ / / v. \ \ / \ / \ / \ / / N / / \ / \ / \ / \ / \ - / \ / \ / / / - / / / / )l - ' 1 1 1 1 1 I I I I 0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 k Figure 3.9: As in Figure 3.8 but with filament length 1.0 u. Homogeneous steady state is unstable for k less than approximately 3.5. Figure 3.10: As in Figure 3.8 but with filament length 2.0 fi. Critical k value for instabilities is approximately 2.5. 58 One Dimensional Model _J L_ LHS RHS 0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 k Figure 3.11: As in Figure 3.8 but with filament length 4.0 p. Critical k value for instabilities is approximately 1.5. Figure 3.12: As in Figure 3.8 but with filament length 8.0 fi. Critical k value for instabilities is approximately 0.9 so that homogeneous steady state remains stable for all k in the case of periodic boundary conditions. 59 One Dimensional Model These results depend on the size of the domain R which was assumed to be 2TT for simplicity, n peaks form when k = 2mr/R. If the size of the domain increases, more peaks should form (smaller values of k may now produce peaks) and clusters may form at longer filament lengths. Therefore, the possible filament lengths that cause clustering can be scaled with the size of the domain. Numerical Simulations Studying the instability condition goes a long way towards determining the effects of the parameters on the system, but does not give the complete picture. Since the instability condition was developed through linear analysis, it cannot fully describe the final outcome of the nonlinear system. The final results are seen only by solving the full integro-PDE's. This is nigh impossible to do analytically. For this reason numerical results for specific parameter values are produced. The current system has only one dimension in space so software packages can be readily used to get numerical results. I used X T C , an interactive program written by Bard Ermentrout, to solve System 3.5, The program is free and easily obtained from Dr. Ermentrout's "bardware" ftp site. The program was specifically designed to solve integro-PDE's with two independent variables. It also has a user-friendly interface that allows one to easily change parameter values, step size, simulation time, and other numerical aspects. X T C was not suitable for implementing no flux boundary conditions with this model, and for this reason, I restricted my simulations to periodic boundary conditions. Thus, X T C was particularly useful for the current task. A l l simulations initialized the concentrations of free and network filaments with a random 10% fluctuation about a homogeneous state. The domain size was just over 6 fi long with periodic boundary conditions. The filament lengths I used in the previous section were used again for comparison purposes. (Note that for Figures 3.13 - 3.18, the 60 One Dimensional Model Filament Concentration Figure 3.13: For an average filament length of 0.25 /z, the homogeneous steady state remains stable. Filament Concentration 100 L-50 V Figure 3.14: For an average filament length of 0.5 /z, clusters grow every TT mi-crons. 61 One Dimensional Model Figure 3.15: For an average filament length of 1.0 fi, what is initially a state where clusters are spaced 27r/3 microns changes to a final state where clusters are spaced n microns apart. Thus, the wavenumber k = 3 becomes dominated by k = 2. 150 -Filament Concentration Figure 3.16: For an average filament length of 2.0 //, only one cluster exists every 2TT microns. 62 One Dimensional Model Filament Concentration 15 -10 r Figure 3.17: For an average filament length of 4.0 /j,, only one cluster exists and it takes longer to form than when the filament length is 2.0 /x (Figure 3.16) as seen by comparing the filament densities. Filament Concentration Figure 3.18: For an average filament length of 8.0 u, the homogeneous steady state remains stable. 63 Angular and Spatial Model Development time scale is in minutes and the space scale is in u). When the average filament length is 0.25 ix (Figure 3.13), the system remained homogeneous as predicted. When the average filament length is 0.50 u, (Figure 3.14), after an initial time where modes of k = 4 are seen, the system eventually settles into a two cluster state, meaning that the mode associated with k = 2 is dominant. When the average filament length is 1.0 fx (Figure 3.15), after an initial time where k = 3 is favored, the system adjusts so k = 2 is again dominant. As you might expect, the clusters change positions so that they are equally spaced for each mode. When the average filament length is 2.0 a-, Figure 3.16 shows that the dominant mode is k = 1. In this case, even though the range of wave numbers that are unstable is the same as when the filament length is 0.5 fx, the final outcome is different. When the average filament length is 4.0 fx (Figure 3.17), only one cluster forms as expected. Finally, when the average filament length is 8.0 fx (Figure 3.18), the system remains in a homogeneous state. By using the numerical results along with the graphical analysis, much can be learned about how the filament length affects the clustering of network filaments. The graphical analysis shows that clustering will occur only for a particular range of filament lengths. If filaments are too short or too long, clustering will not occur. The numerical results show how the clusters form and what spacing between the clusters is favored for the particular simulation. The graphical analysis only gives bounds on the spacing of clusters. These results are new and suggest that filament length regulation through increasing the pool of available monomers or severing filaments with actin severing agents such as gelsolin may govern the clustering of filaments in the cell. 2 Angu la r and Spatial M o d e l Development The preceding section showed how filaments could cluster in space. However, in motile cells such as fibroblasts, filaments not only cluster, but they also align with one another 64 Angular and Spatial Model Development to form bundles as shown in Figure 3.3. Thus, not only is the position of the actin filament important, but also its orientation. Mogilner (1996) extended the integro-PDE in the preceding section to study actin filaments in both space and angle. He developed models for filaments in three dimensions, two dimensions and one dimension. In this section filament orientation is included along with filament position. As the previous section showed, the model consists of free-free filament binding, free-network filament binding, filament unbinding, and filament diffusion that includes rotational diffusion as well as translational diffusion. Because the relative orientation of filaments affects the probability that filaments bind (much like the relative distance does), the kernel K is changed to depend on both space and angle (producing two dimensional convolutions). We let x denote the position of the filament and 9 denote the orientation of the filament (with 9 = 0 being the orientation that is parallel to the one-dimensional cut) as shown in Figure 3.19. The variable 9 is an angle, 0 < 9 < 2ir, so periodic boundary Figure 3.19: The position, x, and orientation, 9, of a single filament. conditions in 9 are assumed. The variable x is taken to be 0 < x < R and for simplicity, periodic boundary conditions are assumed. The notation from the previous section can be extended, so that F(x, 9, t) is the concentration (in number of filaments of length L per unit volume) of free filaments whose center is at position x with orientation 9 at time t and N(x,9,t) is the concentration of network filaments whose center is at position x with orientation 9 at time t. The new system of equations that describe all the actions and interactions in two dimensions is 65 Model Interpretation Nt{x,9,t) = p ^ K * F) + P2N(K * F) - 7 i V (3.34a) filament filament binding unbinding /* PiF(K *F)- (32F(K * N) +jN Ft(x, 6, t) XX (3.34b) rotational translational diffusion diffusion where K*F = (3.35) The analysis of these equations has already been done by Mogilner (1996) for the case where Pi = (32. It is analogous to the analysis in the preceding section and the resulting instability criterion is where M — F + (P2/P\)N. K is the Fourier transform of K, and ki and k2 are the wave numbers in orientation and space, respectively. 3 M o d e l Interpretation The model parameters Pi, 7, and \Xi are all constant; yet, they must describe many biological processes and situations. The following assumptions made in this model serve to both restrict and define the parameters: 1. A l l cross-linker interactions are implicitly defined in the model. They are included in the binding rates, Pi, and the unbinding rate, 7. (3.36) 66 Model Interpretation 2. The model does not differentiate between a network filament that is bound by few cross-linkers and one that is bound by many cross-linkers. 3. The model assumes that all filaments are rod-like and have a fixed length, L . 4. There are no restrictions on the density of filaments at a single point or orientation. 5. Crowding effects and entanglement of filaments are not considered explicitly, al-though they do appear implicitly in the length dependent diffusions discussed later. 6. The probability function, K, depends only on the relative differences in position and orientation. One of the main goals of this thesis has been to make a connection between what was previously an abstract mathematical model, and the information in the literature about actin filament dynamics (in particular, the concentration of cross-linkers, the rates of filament-cross-linker interactions and the length of the filaments). Thus, unlike previous work, most of my attention focussed on understanding and interpreting the parameters, finding biologically relevant values for them, and then simulating the resulting actin networks. The detailed work in elucidating parameters will be described in Chapter 4. Here, I briefly summarize their meanings. Pi is the rate that free filaments associate with other free filaments, and P2 is the rate that free filaments associate with network filaments. Both the concentration of cross-linkers in the solution and the filament length affect the binding rate, so that Pi must depend on these biological factors. Also, because no distinction is made between filaments with many cross-linkers and few cross-linkers attached to them, Pi must describe an average overall binding rate. (See Section 1 in Chapter 4). The rate that filaments dissociate from the network should depend on the number of cross-linkers that attach the filament to the network. Thus, the filament unbinding rate, 67 Model Interpretation 7, is an aggregate rate that combines all the stages of cross-linker detachment. Also, because no distinction is made as to the number of cross-linkers that attach a filament to the network, 7 describes an average rate of unbinding that would be expected to depend on the total concentration of cross-linker. (See Section 2 in Chapter 4). Hi is the rate that free filaments diffuse rotationally, and fi2 is the rate that free filaments diffuse translationally. The length of the filament affects these rates of diffusion, so that ^ depends on L. The total concentration of filaments influences how crowded a solution is which in turn also influences diffusivity. (See Section 3 in Chapter 4). The kernel, K, describes the probability that filaments may interact based on their relative position and orientation. When cross-linkers bind to filaments, they usually prefer one alignment over another. Thus, the angular component of K should describe the influences of the particular cross-linker. Also, the filament length dictates a range where filament interaction can occur. Thus, the spatial component of K should depend on the filament length, L. (See Section 4 in Chapter 4). Having all the parameters depend on the biological situation gives the instability condition (Equation 3.36) biological significance. When the system is at a homogeneous steady state, no one direction or position is favored (ki = k2 = 0). This is the case in an isotropic network. When the system becomes unstable, filaments can either favor an orientation, so that there is filament alignment (ki ^ 0, k2 = 0), favor a position, so that there is clustering (ki = 0, k2 ^ 0), or favor both, so that there is bundling (hi 0, k2 ^ 0). Thus, the instability condition can predict the type of actin filament network that will form based on the model parameters. Numerical simulations can then be used to discover how the actin network forms and to determine the final outcome which is influenced by the nonlinear effects. 68 Glossary of Parameters The partial analysis of System 3.34 by Mogilner (1996) indicates that intriguing patterns form and bifurcate. Geigant and Stoll (1996) have studied similar models ana-lytically and with numerical methods. However, this is the first analysis to date that uses realistic parameter values and has a full numerical treatment. Thus, I first develop the model parameter values based on the understood biology (Chapter 4) and then interpret how the biological factors influence the model through direct analysis and interpretation of numerical simulations (Chapter 5). A publication based on this work has recently appeared (Spiros and Edelstein-Keshet, 1998). 4 Glossary of Parameters x Spatial position 9 Orientation L Average length of an actin filament d Diameter of an actin filament N(x, 9, t) Number density of network (i.e. bound) filaments at x, 9 F(x, 9, t) Number density of free filaments at x, 9 T Total concentration of actin (in all forms) in jiM Hi Rotational diffusion coefficient for actin filaments /i2 Translational diffusion coefficient for actin filaments Pi Effective binding rate for two free actin filaments via binding protein Pi Effective binding rate for free and network filaments via binding protein 7 Effective unbinding rate for network filaments K(x, 9) Probability that filaments whose position differ by x and orientation differ by 6 will in interact. 69 Chapter 4 P a r a m e t e r E s t i m a t i o n a n d M o d e l R e f i n e m e n t Values for the parameters in System 3.34 must be deduced from the basic processes that underlie the model and biological rate constants or observations. The literature gives values for cross-linker interactions such as rate constants of association and dissociation of cross-linkers with actin (recall Table 1.2), but no direct measurement for filament to filament interactions. Each parameter in the model describes the effects of several processes. Thus, estimating each of the parameter values is a miniature modeling problem of its own. In this chapter, I describe each of these little models in turn, and finally arrive at an approximate relationship between the known biological quantities and the model parameters. Parameter estimates take into account the more specific physical situation that is being described. Estimating the rates of filament binding and unbinding is accomplished by considering the dynamics of the cross-linker interacting with actin. Polymer physics gives the basis for determining rates of filament diffusion based on the filament's length. Ranges of interaction are based on cross-linker geometries and filament lengths. By modeling these interactions, the biological features of the model may be made explicit. Getting a consistent set of units is important, especially for relating the model pa-rameters to the cited, physical parameters. The length scale that seems most appropriate 70 Parameter Estimation and Model Refinement for this problem is the micron, LL, since filaments up to one micron in length stay stiff, and the average length in in vitro experiments is 5fx. Thus, F and N are to be measured in number of filaments per LL 3. Since actin is normally measured in units of micro-Molars, LLM, it must be converted to the proper units as shown in the box. The time scale is measured in seconds, s, as this is commonly used for cross-linker dynamics and diffusivity. The angular domain is measured in radians. To convert from LLM to number of filaments per LL 3, it is useful to recall that 1 Mole contains 6.02 x 10 2 3 molecules, 1M = 1 Molar = IMole per liter , and lml = lcm3 = 101 2Ai3 . Thus, 1/j.M of actin = 1 x 1 0 ~ 6 M of actin monomers = 6.02 x 101T monomers per liter = 6.02 x 101 4 monomers per ml = 6.02 x 102 monomers per LL 3 = 602 monomers per LL 3 . To complete the conversion, note that 370 monomers make a filament of length 1 LL. Then, for a collection of filaments of identical length, L, ILLM of actin / 602 monomers \ / 1 LL of length \ (l filament \ y 1/i3 J \ 370 monomers / \ LLL J 1-63 P 1 3 = —^- filaments per [i . 71 Estimating the Filament Binding Rate, Pi 1 Es t imat ing the Filament B i n d i n g Rate , fa The parameters Pi represent the rate of filament binding through the influence of cross-linkers: Pi is the rate that two free filaments bind to form two network filaments, and p2 the rate that a free filament binds to part of the network. The Pi are measured in units of LI? per number of filaments per second. In the simplest case, filament binding occurs when one of the binding domains of a cross-linker such as a-actinin attaches to an actin filament and then the second binding domain adheres to a neighboring filament. (This was referred to as free-free filament binding in Chapter 3). Figure 4.1 shows the reactions necessary for this association along with the interaction rates of the cross-linkers. F a Fa 2N Figure 4.1: Two free filaments must first interact with an a-actinin before they can bind to one another. The kinetic rates are explained in Section 1 of Chapter 2. The letters underneath denote the structures. The first step in the above set of reactions can be represented by the differential equation, d[a] ~dT = k+[F][a] - k^Fa] (4.1) [Fa} = ^_[a}[F]. If this reaction is rapid in comparison with the second, I can assume that the concentra-tions [Fa] and [F\ are at quasi-steady state so that roughly, (4.2) The second reaction in Figure 4.1 describes how two free filaments bind, creating two network filaments. Thus, f U | I W (4.3) 72 Estimating the Filament Binding Rate, 3i Substituting the quasi-steady state value of [Fa] into this expression leads to f i ^ l W H p f ^ W . (4-4) The coefficient in front of the F2 term is then the estimate for 6\, i.e. A . (SB). The estimate for (3X is given in terms of free a-actinin concentration. I relate this (generally unknown) parameter to the total concentration of a-actinin, A, and the total concentration of actin F as follows: when Wachsstock (1993) determines the dissociation equilibrium constant for a-actinin, Kd = k_/k+, he notes that bound a-actinin free a-actinin total actin Kd + free a-actinin or in terms of my parameters A - [a] _ [a] T ~ Kd + [a] " Solving for [a] in the above equation yields [a] = 1 \{A - T - Kd) + yJ(A-F- Kd)2 - 4AKd (4.6) (4.7) (4 i I can substitute this value into the expression for f3\ given by Equation 4.5 to express the estimate in terms of the total amount of a-actinin. To apply these results to actual (numerical) parameter values, units conventionally used in rate constants and concentrations, LLM, must be converted to units appropriate for the variables used here, namely number of filaments per unit volume. Using the conversion from LLM in the box and noting that a typical value for k+ is liiM~ls_1, I find that (3\ (in units of per filament density per second) is 0.61L[a] , . Pi ~ k_ • (4-9) 73 Estimating the Filament Binding Rate, Pi Effects of K-MINUS on beta, length = 2 microns 4.51 1 , 4 -3.5 -k-minus = 1 k-minus = 3 — k-minus = 5 3- . -2.5-iS 2-1.5 . „ ' ' ' ' „ - -0.5 - "~ " ' 0- 1 1 : 0 5 10 15 total alpha-actinin Figure 4.2: The dependence of Pi on A, the total a-actinin for various values of k-. Pi is inversely proportional to the cross-linker dissociation rate constant, k-, so that an increase in k- leads to a decrease in Pi. The amount of actin is fixed at lhuM for these results. The factor of 0.61 converts the fiM units for [a] and the ii units for the filament length L to the appropriate units for Pi, namely per filament density per second. Figures 4.2 and 4.3 show how changing the total amount of a-actinin affects the estimate of Pi based on different cross-linker dissociation rate constants and lengths, respectively, while the total actin concentration is fixed at 15/iM and k+ is fixed at \LLM~ 1 S_1 . Both figures show that Pi increases in a nonlinear way as the total a-actinin concentration, A, increases. This means that changes in k_ or L are more pronounced at higher a-actinin concentrations. Although Equation 4.9 might seem to imply that changes in a-actinin should affect Pi linearly, Equation 4.8 shows that [a] does not 74 Estimating the Filament Binding Rate, Pi Effects of LENGTH on beta, k-minus = 3.0 CO a> 3 length=0.5 length=2.0 length=3.5 length=5.0 o' ^- i 1 1 0 5 10 15 total alpha-actinin Figure 4.3: The dependence of $\ on A, the total a-actinin for various values of L. Pi is directly proportional to the filament length, L so that an increase in filament length leads to an increase in Pi. The amount of actin is fixed at 15uM for these results. change linearly as A is increased. Figure 4.2 highlights the way that Pi is inversely proportional to as seen in Equation 4.9, so that cross-linkers that dissociate more readily from actin will decrease the rate that filaments can bind (as expected). Figure 4.3 highlights the fact that Pi is directly proportional to L as seen in Equation 4.9, so that longer filaments will associate at a faster rate than short ones. Equation 4.9 gives only a first approximation to Pi based on the simplest way that two free filaments can be joined together with one cross-linker. If the first reaction shown in Figure 4.1 is not rapid, then my estimate for Pi will be an overestimate since it will take longer for the complex, Fa, to form. However, if a filament has many cross-linkers 75 Estimating the Rate of Filament-Network Unbinding, 7 attached to it, on average, then my estimate for Pi will be an underestimate, because more than just one binding domain could then bind filaments together. Because both of these situations are likely, they may balance, so that the estimate given by Equation 4.9 is a good first approximation. When a free filament binds to a network filament, only one new network filament is formed. Thus, an estimate for P2 might be p2 w Pi/2. However, since the mobility of a network filament should be very small compared to a free filament, the odds of an encounter occurring between these two filaments should be less than two free filaments. Thus, a better estimate for L32 is P2 ~ /?i/4. We ran simulations where j32 was P\/2 and compared these results with simulations where P2 was Pi/4. There were some changes in the amount of clustering at short lengths, but the final outcome for most simulations changed very little; it only took longer for the system to reach its final state. 2 Es t imat ing the Rate of Fi lament-Network U n b i n d -ing, 7 The parameter, 7, represents the rate that filaments dissociate from the network. It is measured in units of per second. In order for a filament to dissociate from the network, all the cross-linkers that bind it must detach. However, the model does not differentiate between filaments with many cross-links binding them to the network and those with few cross-links. The model also does not distinguish between filaments that are relatively isolated so that they are more likely to dissociate from the network, and those that are surrounded by many others so that unbinding is unlikely. (A much more detailed and, therefore, cumbersome model would be needed to follow such fine details.) The estimate of 7 must average all of these situations, to come up with one aggregate unbinding rate. I consider the necessary steps that lead to a filament dissociating from the network. 76 Estimating the Rate of Filament-Network Unbinding, 7 A n upper bound for 7 is k- (in the unlikely event that just one cross-linker binds the filament to the network). However, it is more likely that several cross-linkers will be involved. In this case, one must consider the time it takes for each of the cross-linkers to detach. Unfortunately, at the same time that the cross-linkers are detaching from the filament others may attach. This leads to a complex process that I model below. 2.1 The Chemical Reactions I estimate 7 by studying the set of chemical reactions that lead to a single filament dissociating from the network. Let Xi denote a network filament with i attached a-actinin cross-linkers. I consider the simultaneous association-dissociation steps that can occur: a cross-linker can either bind to or unbind from the filament. Thus Xi can become xi+x at the rate which depends on the availability of free cross-linkers, and Xi goes to Xi-i at the rate ik- which depends only on the number of attached cross-linkers. If up to n cross-linkers can bind to a filament, then the entire process is described by the following reactions nfc_ ( n - l ) f c - i k - 3fc_ 2fc_ Xn  T Xn—\ Xn—2 - ' • T • • • T ^ %1 ^  • [a]fc+ [Q]'c+ [ a]' c+ [Q]k+ [Q]'c+ The above system can be analyzed to determine how quickly, on average, a filament can move through the sequence of steps that allow it to become liberated of all cross-linkers. The analysis begins by representing the chemical kinetics with a system of ordinary differential equations for the x^s as shown below. The system is linear if we assume that the level of free cross-linkers is held fixed (Jacquez, 1972): % ! = - n M * « ] + N * + [ * « - i ] i (4-10) (i + l)/c_[£j+i] — + ik-) [xi] + [a]A;+[xj_i] for n — 1 > i > 2, d[xj] dt 77 Estimating the Rate of Filament-Network Unbinding, 7 * J = -([#+ + L ) N . I assume that the maximum number of cross-links that can bind to a filament is n, so that the above is a system of n linear differential equations. The corresponding n x n matrix is tridiagonal with negative diagonal entries and positive off-diagonal entries: - ([a]k+ + 2k. ... 0 0 0 [a]k+ - ([a]k+ + 2/c_) ... 0 0 0 0 [a]k+ ... 0 0 0 0 0 ... [a]k+ - ([a]k+ + (n - l)fc_) n/c_ 0 0 ... 0 [a]k+ -nk_ The eigenvalues of this matrix describe the "rates of flow" through the system. If all the eigenvalues are negative, an initial group of network filaments will eventually disappear from the system as they are liberated, one by one. In this case, the negative eigenvalue of smallest magnitude, —A m , represents the "rate limiting" decay, i.e.'the slowest rate of decay in the system, which I take to be an estimate for 7. The estimate for 7 might seem to be sensitive to the assumed maximal cross-linker occupancy level, n. However, as shown by Figure 4.4, 7 reaches a limit as n increases (while other parameters are held fixed). Thus, rather than calculate all the eigenvalues for some arbitrarily large n (which may be very time consuming), I can determine a suitable n that sufficiently describes the average cross-linker occupancy rate. 2.2 The B i r t h and Death Process Probability models can be used to determine an appropriate maximum for n. The number of cross-linkers that attach a filament to a network may be viewed as the state of a continuous-time Markov chain. Specifically, this can be likened to a birth and death process (Ross, 1989). In order for the process to be Markovian or memoryless, the 78 Estimating the Rate of Filament-Network Unbinding, 7 matrix size Figure 4.4: The negative eigenvalue of smallest magnitude, — A m , reaches a limit as the maximal number of cross-links per filament, n (and, thus, the size of the n x n matrix) increases. The biological parameters for this figure have been fixed at [a] = luM, k+ = l/j,M~1s~1 and fc_ = I s - 1 . 7 (in units of s _ 1 ) does not change significantly after n = 5 in this case. time between births and deaths must be exponentially distributed with the average time between births being l / A * and between deaths being 1/in- By considering the reciprocal of these average times, birth and death rates may be found. Thus, the number of cross-linkers at state i increases at the "birth" rate A = [a]k+ and is independent of i. The number of cross-linkers at state i decreases at the "death" rate / i ; = ik_, because any one of i cross-linkers may detach. Figure 4.5 shows the birth and death process which is nearly identical to the chemical reactions above. The birth and death process can be studied to determine the limiting probability, Pi, i.e. the probability that the system will be in state i at any time t. In our system, p , 79 Estimating the Rate of Filament-Network Unbinding, 7 Figure 4.5: The number of cross-linkers attached to a filament can be viewed as a birth and death process. Each circle represents one "state" of the filament (i.e. how many cross-linkers are attached). The rate of cross-linker attachment is independent of the number already attached. Thus the birth rate, A = [a]&+, is the same for all states. The death rate, LL1 depends on the number attached to the filament: since any single one can unbind at the rate k-, the rate that state i goes to i — 1 is m = ik-. gives the probability that there will be i cross-linkers attaching a filament to the network. Pi is found from the balance equations that equate the rates of entry and departure from a given state, 2 : (A +/ii) P x = LI2P2 + XP0 (\ + li2)P2 = LI3P3 + XP, 1 ! (A 4- Lti) Pi = iM+iPi+i + APj. By adding successive equations, we obtain a simple recursive relation: AP 0 = MiP APi = H2P2 AP 2 = M3P3 APi — /ii+iPi+i, 80 Estimating the Rate of Filament-Network Unbinding, 7 Each limiting probability can now be expressed in terms of P 0 , P i = - P o , Mi P 2 = - P i = ~ P 0 , Hi HlVl P 3 = — P i — Po , £*3 M1M2M3 Pi = - P - i = Po- (4.11) Mi M1M2 • • • Hi However, an additional constraint is that the limiting probabilities must sum to one. Thus, 0 0 0 0 £ P = 1 P 0 + £ p i = l . (4.12) i=0 i=l By substituting Pj in terms of P 0 ma Equation 4.11, one obtains ( ° ° A* \ i + E = 1 ( 4 - 1 3 ) ~[^2 • • • Hi) Using the facts that A = [a]k+ and Hi = the terms in the above series may be simplified: A1 _ ([a]k+y 1 f[a]k+ ' H1H2 • • • Hi (Ik-) • (2fc_) • (3fc_) • • • il \ k. ' • ^ 4' 14^ [al A; Now, let r = — and substitute the above into Equation 4.13 to obtain The summation in the above equation is the Taylor series of the exponential function. Thus, P0e r = l PQ = e~ r. (4.16) 81 Estimating the Rate of Filament-Network Unbinding, 7 Having found P 0 , Equation 4.11 can be used in conjunction with Equation 4.14 to obtain a closed form for p ; specifically, n r % -r 1 \a]k+ Pi = —e r where r = . (4.17) The way that Pi depends on r is analogous to a result by Edelstein-Keshet and Ermentrout (1998) on actin length distributions for the case of simple polymerization. They focused on the number of monomers in an actin filament and how this number changed. Monomers can attach at the rate k+a (where a is the concentration of available monomers) and detach at the rate k-. This is slightly different than my case that focuses on the number of cross-linkers attached to a filament. The association rate constant is the same, k+[a] ([a] is the concentration of free cross-linkers). The difference is that the dissociation rate constant, i/c_, depends on the number of cross-linkers attached to the filament. Even though the points of view were different, deterministic versus stochastic and equilibrium concentrations versus limiting probabilities, both analyses solved a difference equation and rz appears in both solutions. Having found the limiting probability, p , a bound may be placed on the maximum number of cross-linkers per filament, n. Recall that P denotes the probability that a filament will have i cross-linkers attaching it to the network. By placing a bound on this probability, n can be determined. For example, if states whose probability is less than 0.1% are considered to be insignificant, then the greatest number of cross-linkers attached to a filament is the maximum i such that p > .001. Using [a] = \\iM, k+ = 1LIM" 1S~ 1, and = I s - 1 , the maximum number of cross-linkers per filament at the 0.1% level of significance is n = 5, because P 5 = .003 but P 6 = .0005. This value of n fits Figure 4.4 which uses the same parameter values, indicating that a bound of at most 0.1% on the probability is necessary. 82 Estimating the Rate of Filament-Network Unbinding, 7 2.3 Finding 7 I am now able to find 7 with a simple algorithm. First, take the biological parameter values and calculate n. Next, calculate the eigenvalues from the resulting n by n matrix and take 7 « A m where — Xm is the negative eigenvalue of smallest magnitude. The computer package, Matlab, is able to compute 7 quickly with this algorithm. The results are summarized in Figure 4.6. As the total amount of a-actinin is increased, 7 decreases, meaning that it becomes harder for filaments to dissociate from the network. As the dissociation rate constant of the cross-linker is increased 7 is also increased so that network filaments dissociate more quickly. 0.9 0.8 0.7 0.6 g 0.5 I 0.4 0.3 0.2 0.1 \ Dependence of gamma on alpha-actinin and k-minus 15 20 25 30 35 40 45 total alpha-actinin km = 5 • km = 3 km= 1 5 10 total alpha-actinin Figure 4.6: The dependence of 7 and on total available a-actinin, A (in units of /iM). The figure on the left shows that 7/fc- (a dimensionless quantity) goes to zero as the total concentration of a-actinin is increased. This indicates that 7 itself goes to zero and that affects 7 more than simple scaling. The figure on the right shows a rough linear dependence of 7 on the total concentration of a-actinin, A, when A is in the range of levels used in experimental situations. k+ = liiM~ ls~l and T = !5uM in these graphs. It is worth noting that the unbinding rate, 7, does not depend on the filament length. The only way that the length could affect 7 is by restricting the maximum number of 83 Estimating Filament Rates of Diffusion cross-linkers attached. However, as seen in our example, the filaments would have to be very short (i.e. have only 5 available binding sites) for this to matter. Thus, only the concentration of cross-linkers and the kinetic rates of the cross-linker influence 7. 3 Es t imat ing Filament Rates of Diffusion The literature on rod-like polymers (Doi and Edwards, 1986) gives estimates for the filament rates of diffusion depending on the type of polymer solution. There are three different types of solutions of rod-like polymers that concern us: dilute, semi-dilute, and concentrated. When the filaments are spaced far enough apart so that they do not restrict each other, the solution is dilute. When the filaments are close enough to begin interfering with one another so as to only restrict one degree of freedom, the solution is semi-dilute. Tighter packing of filaments leads to concentrated solutions. The number of filaments per unit volume, v, (a quantity that can easily be measured, e.g. a concentration of 15 LIM of actin at length L has v = 24.4/L filaments per /i 3) is used to estimate the average distance between filaments, z^"1/3, and, thus, indicate various actin solution regimes. In the dilute solution regime, the average distance between filaments is greater than the length of the filaments so that v~ll3>L v<l/L3. (4.18) Thus, a critical point for dilute solutions is defined to be at v\ — (ir/6)L3. For v > v\, the solution is in the semi-dilute regime. The semi-dilute regime is valid up to the point where filaments interfere in more than one direction as shown in Figure 4.7. Thus, the semi-dilute regime is valid for the average volume, u"1, being larger than the volume enclosed in Figure 4.7, u~l>dL2 v<l/dL? (4.19) Thus, a critical point for semi-dilute solutions is defined to be v2 = 1/dL2. Therefore, if v < ui, the solution is dilute, if v\ < v < i / 2 , the solution is semi-dilute, and if v > v2, the 84 Estimating Filament Rates of Diffusion solution is concentrated. For a 15 LLM solution of actin, when the filament length is less than 0.2 LL the solution is dilute, when the filament length is between 0.2 LL and 5.5 LL the solution is semi-dilute, and when the filament length is greater than 5.5 \i the solution is concentrated. Thus, in most experimental situations, the actin filament solutions are in the semi-dilute regime. 2d Figure 4.7: The maximum volume between two tightly packed filaments is on the order of dl? where d is the diameter of the filament. In order to derive the rates of diffusion in the semi-dilute regime, the rates of diffusion in the dilute regime are needed. Filaments in a dilute solution behave independently of one another so that the diffusive properties can be ascertained by studying a single filament (or polymer). Because the polymers are assumed to be rigid (since we are dealing with lengths on the order of 1 LL), only two types of diffusion are possible, rotational and translational. The rotational diffusion coefficient is based on Brownian motion across a sphere. First, an estimate for the rotational friction constant, ( r, of a polymer is derived based on an external torque and the angular velocity it generates: 7T7? S L 3 Cr (4.20) 3 ( l n ( L / d - 6 ) ) ' This is then used to derive the Rotational diffusion coefficient in a dilute solution (Doi and Edwards, 1986), kbT 3kbTln(L/d-b) D no Cr 7r?] sL3 (4.21) 85 Estimating Filament Rates of Diffusion where L is the length of the polymer, d is its diameter, kb is Boltzmann's constant, T is the temperature (in degrees Kelvin), and i]s is the viscosity of the solvent, (b is a "generic", empirically determined constant.) The translational rate of diffusion can be broken into two components, movement parallel to the filament, D\\, and movement perpendicular to the filament, D± (see Figure 4.8). Both of these movements have a drag Figure 4.8: Filaments should have greater rates of diffusion parallel to their axis than perpendicular to their axis due to less drag. coefficient associated with them that are estimated to be 2irr)sL C" = l n ( W ( 4 " 2 2 ) C± = 2C||. (4.23) Thus, the Translational Diffusion Coefficient (Doi and Edwards, 1986) is W = WMt/<-H . C|| 37T7]SL In a semi-dilute solution, the movement of filaments is hampered. Thus, the effect of a single filament diffusing depends on the amount of space available to it in a semi-dilute solution. By considering an imaginary tube around a filament as in Figure 4.9 the restriction on the movements of a filament is modeled. For translational diffusion, diffusion in a direction perpendicular to the filament axis is nearly impossible because of the tight restriction of the tube. Thus, only diffusion parallel to the filament axis (along the tube) is possible. Because movement is restricted to one spatial dimension, the translational diffusion depends on the orientation of the filament as shown in Figure 86 Estimating Filament Rates of. Diffusion I Figure 4.9: In the semi-dilute regime, diffusion effects are restricted to an imag-inary tube surrounding the filament. 4.10. If the filament is parallel to the restricted corridor (i.e. (9 = 0, ±7r), then the corridor does not hamper parallel diffusion so that the diffusion coefficient is Ds. However, if the filament is not aligned along the corridor, then the translational diffusion is reduced. Therefore, an estimate for / i 2 is simply, Figure 4.10: For the one spatial dimension simulation, filaments can be viewed as sliding through a corridor. In a semi-dilute solution, filaments can only move in a parallel direction. The related movement in the ^-direction is obtained from a trigonometric relation. Rotational diffusion, on the other hand, is a multi-step process. Before the filament may begin to rotate it must first move out of the tube as shown in Figure 4.11. If it moves half its length, L/2,.then it is able to rotate up to an angle <p determined by the H2 ~ | cos 6\Ds • (4.25) 87 Estimating Filament Rates of Diffusion Figure 4.11: In the semi-dilute regime, filaments must first diffuse in the trans-lational direction, before they can rotate. The amount of rotation is restricted by the size of the tube. tube radius, r. Figure 4.11 shows that the angle <j> satisfies Because (p 1S small (the process occurs in a semi-dilute solution), sm(<f>) can be approxi-mated by cp so that (4-27) The amount of time, T , it takes for a filament to rotate by the angle 4> is then the same as the time it takes for the filament to move a distance L/2 through translational diffusion, r may be estimated as However, dividing Ds by L 2 produces a relation that depends on 1/L 3 just as in DQQ (Equation 4.21). Thus, r c< ^ - . (4.29) Because the time it takes a filament to rotate by the angle </> is r , the rotational diffusion coefficient in a semi-dilute solution can be approximated as 4> 2 ( T ) r2Dm Dn « — oc oc — - — . 4.30 The only unknown in equation 4.30 is the imagined tube radius, r. Estimating Filament Rates of Diffusion The tube radius, r, can be determined by considering the number of filaments that intersect the tube in a semi-dilute solution (call this function T(r)). r is defined to be that radius for which only one filament intersects the tube (i.e. when T(r) = 1). To determine T(r) , consider a small surface element on the tube, AS. Only filaments that lie within a distance L of the surface, can intersect it. Because the average number of filaments per unit volume is v, the average number within the volume L A S ' is uLAS. By summing over all surface elements, T(r) oc vLS = vL(2itrL) oc urL2. (4.31) Solving for T(r) = 1, gives an estimate for the tube radius, 1 r = vh2 (4.32) Having now found r, an estimate for the Rotational diffusion coefficient in a semi-dilute solution (Doi and Edwards, 1986), DQ, can be found by substituting r into Equation 4.30: ^ n ^ ' (4-33) where B is an empirically determined constant. If I consider a given, fixed, concentration of actin in i i M ' , say T, then the conversion from LIM (as outlined in the box at the beginning of the chapter) implies that v = l.fftT/L. Substituting this into the preceding equation yields y&HLH-h) i ( 4 3 4 ) TT?7s 1.63^ 2L 7 V ; Thus, rotational diffusion in a semi-dilute solution falls off as the seventh power of the filament length. This is directly related to the diffusion parameters in the model: Li\ = DQ in a semi-dilute solution and Li\ — Dno in a dilute solution. A summary of the parameters used in the diffusion terms as well as typical values for them are given in Table 4.1. The table shows that the rate of translational diffusion 89 Estimating Filament Rates of Diffusion in water of an actin filament of length 111 is 2.1 LI 2 j s and the rate of rotational diffusion is 41 rad 1 Is. Notice that these diffusion rates are in sharp contrast with the monomer diffusion rates: typical translational and rotational diffusion rates for an actin monomer in water are 90 LL 2/S and 1.1 x 107 rad 2/s, respectively (Dufort and Lumsden, 1993). Figure 4.12 shows effects of filament length on the translational and rotational diffusion rates in the semi-dilute regime. The sharp dependence of the rotational diffusion rate on the filament length is noticeable when compared to the translational diffusion rate. Parameter Meaning Value Source 0.1-1 fj, in vivo L filament length 4.9 n Wachsstock(93) in vitro d filament diameter 7.0 nm Lumsden (93) 8.0 nm Wachsstock (93) 0.01 Poise Wachsstock (93), (water) Vs solvent viscosity 0.55 Poise Dufort and Lumsden (1993) 100-1000 Poise Oster (cytoplasm) h Boltzmann's constant 1.38 x I O - 1 6 ergs/degree T absolute temperature 300 Kelvin room temperature B generic factor 1.3 x 103 Doi and Edwards (1986) b generic factor 0.8 Doi and Edwards (1986) Table 4.1: Parameter values for the diffusivities. Note that 1 erg = 1 gm cm2 s~2, 1 Poise = 1 gm cmT 1 s _ 1 . These results were based on filaments being rod-like which is the case for lengths on the order of one micron. At longer lengths, these results may be erroneous. Flexible polymers are more compact than rigid polymers leading to greater rates of diffusion in the dilute case. However, in the semi-dilute regime, the derivations for diffusion rates become much more difficult as entanglements and many different geometries complicate matters. Thus, the diffusion rates derived in this section can be used directly for filaments up to about 1.5 microns, but only as a lower bound for longer filament lengths. 90 Spatial and Angular Ranges of Interaction 1 0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0 x 10" diffusion constants for 100 Poise T : 1 r \ > cn "i r - mux - mua 1 1.1 1.2 1.3 1.4 1.5 1.6 1.7 1.8 length 1.9 Figure 4.12: Diffusion rates for translational and rotational diffusion of filaments in the semi-dilute regime,'as a function of filament length. Units are a2/s for the translational diffusion, and radians2/s for the rotational diffusion. The viscosity in this example is 100 poise and the actin concentration is 15fiM. 4 Spatial and Angula r Ranges of Interaction 4.1 General Remarks The kernel, K, is the key to modeling nonlocal effects. The kernels considered in the previous models of actin dynamics (Mogilner and Edelstein-Keshet, 1996) assumed that the relative position and orientation of filaments influence filament interactions indepen-dently so that K(9,x) = K1{9)K2(x). (4.35) 91 Spatial and Angular Ranges of Interaction Ki(9) is the probability that filaments that meet at an angle, 6, will interact. K2(x) is the probability that filaments whose centers of mass differ by x will interact. The basic shape of the kernels, Ki, such as symmetry properties and the number of maxima (typically one or two) influence the behavior most strongly. More precise details do not necessarily change the characteristics of the model significantly (Mogilner and Edelstein-Keshet, 1995). Also, the kernel can be adapted to a variety of cases, including interactions that tend to create parallel or orthogonal configurations (Civelekoglu and Edelstein-Keshet, 1994) as well as those that bunch filaments together (Section 1 in Chapter 3). Because K cannot be determined directly through experiments, we can at best sug-gest reasonable examples. The probability density function for the normal distribution, gives the basis for constructing the kernels, Ki. g has a maximum at u — 0 that smoothly decreases to zero as u goes to plus or minus infinity. The range is sufficiently described by the standard deviation, a: 68% of the distribution lies within one standard deviation and 95% of the distribution lies within two standard deviations. Thus, the effective size of the range of interactions is 2a. The Fourier transform of g has a closed form, allowing it to be substituted directly into the instability condition (Equation 3.36) for rough estimates of instability. K2(x) gives the probability that two filaments whose centers differ by x will interact. Figure 4.13a shows that when the centers are aligned, the number of cross-linkers that can bind filaments is at a maximum. Thus, the maximum probability for alignment should occur when x = 0. Figure 4.13b shows that as the distance between filaments (4.36) (4.37) 4.2 Spatial Part of the Kernel 92 Spatial and Angular Ranges of Interaction r (a) (b) L •H-(c) Figure 4.13: The distance between the centers of the filaments affects the prob-ability that filaments interact, (a) When the filaments overlap (so that there is centers are close), there are many sites for cross-linkers to bind, (b) As the over-lap decreases or the distance between the filaments increases, there is a smaller chance that a cross-linker may bind them, (c) The maximum distance between the centers of the filaments at which the filaments may still bind is the same as the length of the filament. increases, less cross-linkers are able to bind them together, and hence the probability of binding decreases. Figure 4.13c shows that the maximum range of interaction occurs when the centers of the filaments are separated by a distance equivalent to the length of the filaments. A l l of these considerations are accounted for with the normal distribution as in equation 4.36 with 2a = L : where m2 normalizes K2 over the appropriate domain. (In most C c L S G S 777.2 ~ \Z2vr, e.g. if the domain is the interval [—7r,7r] and L = 1, then m2 approximates J2~K to 9 decimal places.) 4.14 shows the spatial kernel, K2, for filaments of length 1 micron. (4.38) 93 Spatial and Angular Ranges of Interaction 0.9 h relative distance between filaments Figure 4.14: The spatial kernel, K2, is the normal probability density function with a = L/2. This graph shows the spatial kernel for filaments of length 1 \i. (The horizontal scale is in \x). 4.3 Angular Part of the Kernel Ki(8) gives the probability that two filaments whose orientation differ by 9 will interact. K~i depends even more heavily on the cross-linker configurations than K2. A recent paper by Janson and Taylor (1994) suggests that a-actinin can bend as far as 45° and favors an attachment angle of 90° as shown in Figure 4.15a. Because a-actinin favors a 90° attachment angle, the most preferred alignment of filaments occurs when the filaments are parallel, as in Figures 4.15b and 4.15f. Figure 4.15c shows that when one of the cross-linkers bend to 45°, the angle between the filaments increases to 45°. Figure 4.15d shows that when the other attachment bends to 45°, the angle between the filaments becomes 90° (in this most likely case). Therefore, a first approximation to the kernel for 94 Spatial and Angular Ranges of Interaction + Figure 4.15: The kernel, K\, depends on the orientation of the filaments, (a) Cross-Linkers like a-actinin can bend as far as 45°, but favor a 90° attachment. Thus, parallel alignment of filaments as in (b) and (f) is preferred, (c) When one of the attachments bends to its 45° maximum, the relative angle between the filaments is 45°. (d) When both of the attachments bend to 45° (the most unlikely case), the relative angle between the filaments is 90°. (e) Angles greater than 90° may separate the filaments, but this means that the cross-linker has changed locations. Depending on the cross-linker, parallel alignment (b) or anti-parallel alignment (f) may be preferred. 95 Spatial and Angular Ranges of Interaction angular alignment is the normal distribution with 2a = IT/2: *i.°W = ^ ( ^ r ) > (4-39) TT \ TT2 The angle between the filaments might seem to increase further than 90°. However, there is also an acute angle between these same filaments (see Figure 4.15e), so that we need not separately consider this case, but instead center K\JQ about ±ir: 4 / - e x p 7T \ '-8(0 + TT) 2 4 / - e x p TT \ ' - 8 ( 0 - vr) 2 , 7T2 if -7T < 0 < 0, (4.40) if 0 < 0 < TT. Because different types of a-actinin favor either parallel or anti-parallel alignment, Kii0, and KiiV may be adjusted. By assigning relative weights (w0 and wn) to the parallel and anti-parallel components, a general form for K\ is derived: KX(Q) = ^ - [w0Klt0(9) + w*KltV(9)} , (4.41) (where m i normalizes K\). 4.4 Angular Kernel for Specific a-actinin Types In the case of chicken smooth muscle a-actinin, anti-parallel alignment is favored four to one over parallel alignment (Meyer and Aebi, 1990), so that wv = 4 and i^n = 1 as shown in Figure 4.16. In the case of Acanthamoeba a-actinin, parallel alignment is favored four to one over anti-parallel alignment (Meyer and Aebi, 1990), so that w0 = 4 and wn = 1 as shown in Figure 4.17. 96 Spatial and Angular Ranges of Interaction -1 o 1 relative orientation between filaments Figure 4.16: The form of the kernel that we chose to reflect properties of the cross-linker, chicken oi-actinin, which favors anti-parallel alignment. The inten-sity of interactions between two filaments subtending angles between — TT and 7r radians are shown here. 97 Spatial and Angular Ranges of Interaction -1 0 1 relative orientation between filaments Figure 4.17: The form of the kernel that we chose to reflect properties of the cross-linker, Acanthamoeba a-actinin, which favors parallel alignment. The in-tensity of interactions between two filaments subtending angles between — TT and 7r radians are shown here. 98 Summary of Parameter Values 5 Summary of Parameter Values Parameter Meaning Value V Filament concentration (24.4/L) filaments per LI3 [a Cross-Linker concentration 0-8 fiM k+ for actin 0.61 x L per filament of length L LI3 s _ 1 for a-actinin 3.3 x 10~3 per cross-linker LI3 S - 1 k- Reverse rate constant 0.67-5.2 s"1 L filament length in vivo 0.1-1 LI filament length in vitro 4-6 \x d filament diameter 7 - 8 x IO" 3 LI hT/Vs in water 4.14 LI3 s~l in actin solution 0.075 LI? s~l in cytoplasm 4.14x x 10~4 ti3 s-1 C log term for diffusion H(L/d) - 0.8) rotational diffusion (dilute solution) in water 3.8(C/L 3) s-1 Dufort and Lumsden 0.07(C/L3) 5 - 1 cytoplasm 3.8 x 10- 4 (C/L 3 ) Du rotational diffusion (semi-dilute solution) in water 8.3(C/L 7) s-1 Dufort and Lumsden 0.15(C/L7) s~l cytoplasm 8.3 x 10~ 4(C/L 7) s-1 Ds translational diffusion in water 0.42(C/L) LI2 s-1 Dufort and Lumsden 7.8 x 10" 3 (C/L) ii 2 s~l cytoplasm 4.2 x 10" 5 (C/L) / i 2 s-1 Table 4.2: Parameter values in units consistent with the model. Table 4.2 lists the parameter values from the the literature and calculations described in this chapter, v is based on an actin solution of 15 jiM. The range of [a] is based on experiments (Wachsstock et a l , 1993; Wachsstock et al., 1994) where the total concen-tration of a-actinin varied between .1 LLM and 10 \xM. k+ and /c_ were converted from 99 Summary of Parameter Values Table 1.2. The rest of the values in the table give a quick reference for calculating the diffusion rates in different situations. Table 4.2 can then be used to determine values for the model parameters as deduced in the preceding sections. Table 4.3 gives the range of the model parameters along with a range of values based on typical values for the biological parameters. Once again, the amount of actin is fixed at 15 LLM. A typical value for the a-actinin concentration is 1 \iM. The kinetic rates of the a-actinin are k+ = 1 /iM" 1 s - 1 and = I s - 1 which roughly correspond to chicken a-actinin. Parameter Range Typical Value Units Comment L 0.1-5 1 M — Pi 0-2 0.1 per filament cone. s _ 1 section 1 7 0-5 0.9 s-1 section 2 Mi 10" 1 - 10" 5 Dn s-1 section 3 M2 l O " 1 - 10" 3 Ds U2 s~l section 3 o-i TT/8 - TT/4 TT/4 rad 2 s _ 1 section 4 0~2 0-3 L/2 M section 4 Table 4.3: The model parameters and ranges of values. Having determined how the biological parameters affect the model parameters, one can determine how a change in the biology can change the model parameters as shown in Table 4.4. Thus, the instability condition (Equation 3.36), Hiki + H2k22< M2K(I-K) , can be studied to determine how changes in the biology affect the network structure. In particular, a tendency for inhomogeneous networks is favored by increasing the cross-linker concentration, decreasing and increasing the filament length. In order to see what patterns develop, numerical simulations of the system are needed These are described in the next chapter. 100 Summary of Parameter Values Model Parameters Biological Parameters A L Pi / \ / 7 \ / M \ / \ Mi \ M2 \ instability / \ / Table 4.4: The arrows show how an increase in three biological parameters affect the model parameters. An increase in the total cross-linker concentration, A , causes an increase in Pi and a decrease in both 7 and M , increasing the likelihood of instability. An increase in k- (keeping k+ = 1) is the same as an increase in Kd = k-/k+. This decreases Pi but increases both 7 and M decreasing the likelihood of instability. Increasing the filament length, L, increases Pi and decreases M, p\ and i 4 2 ; thus, increasing the likelihood of instability. 101 Glossary of Parameters 6 Glossary of Parameters X Spatial position 0 Orientation L Average length of an actin filament d Diameter of an actin filament N(x 0,t) Number density of network (i.e. bound) filaments at x, 9 F(x, d,t) Number density of free filaments at x, 0 T Total concentration of actin (in all forms) in iiM Pi Effective binding rate for two free actin filaments via binding protein 02 Effective binding rate for free and network filaments via binding protein 7 Effective unbinding rate for network filaments A Total concentration of cross-linker such as a-actinin (in both bound and free forms) [a] Concentration of unbound cross-linker k+ Association rate constant for cross-linker and actin fc_ Dissociation rate, constant for cross-linker and actin Mi Rotational diffusion coefficient for actin filaments M2 Translational diffusion coefficient for actin filaments D m Rotational diffusion coefficient of rod-like polymer in dilute solution Dn Rotational diffusion coefficient of rod-like polymer in semi-dilute solution Ds Translational diffusion coefficient of rod-like polymer in semi-dilute solution V Total actin filament number concentration (number of filaments per unit volume.) Vl The critical concentration between dilute and semi-dilute solutions V2 The critical concentration between semi-dilute and concentrated solutions. Vs Solvent viscosity kbT Boltzmann's constant multiplied by temperature in degrees Kelvin 102 Glossary of Parameters T(r) The number of filaments that intersect a tube of radius r K2(x) Spatial dependence of the binding rate Ki(0) Angular dependence of the binding rate Parallel component of alignment kernel Anti-Parallel component of alignment kernel w0 Weight of parallel alignment w» Weight of anti-parallel alignment mi Normalization factor for K\ m2 Normalization factor for K2 103 Chapter 5 N u m e r i c a l S i m u l a t i o n s o f A c t i n F i l a m e n t s i n S p a c e a n d O r i e n t a t i o n Developing a numerical scheme to solve System 3.34 can be challenging. The convolu-tions (i.e. K * F) in the equations increase the complexity of some methods and are computationally intensive. The vast changes in the diffusion coefficients suggest the use of two different numerical methods. Since I am interested in the case where the ho-mogeneous steady state is unstable, the numerical scheme must be designed to handle rapid growth. Previous simulations have been carried out by (Civelekoglu and Edelstein-Keshet, 1994; Ladizhansky, 1994) for similar model(s) in the space-independent case, and by (Geigant and Stoll, 1996) for the angular-2D spatial case. However, this is the first case of simulations in which biologically realistic parameter values have been used. 1 Compu t ing the Convolutions Computing the convolutions is the most costly step in the simulation. Imagine the xO-space divided into a n n x n grid. Each grid point must have associated with it the value of the convolution at.that point. At the grid point (xi, 9j), the convolution for K * F is the same as the integral 104 Computing the Convolutions U s i n g a s imple in tegra l a p p r o x i m a t i o n such as the T r a p e z o i d m e t h o d for th i s i n t e g r a l at a s ingle g r i d p o i n t w o u l d result i n n2 c o m p u t a t i o n s because the value at a l l n2 g r i d po in t s mus t be used i n the c a l c u l a t i o n . S ince thi s must be done for each of the n2 g r i d po int s , the cost of c o m p u t i n g a l l the integrals for one t i m e step is 0(nA). I c an i m p r o v e o n thi s " p r i m i t i v e " m e t h o d b y t a k i n g advantage of the fact t h a t the in tegra l is a c o n v o l u t i o n . 1.1 The Fast Fourier Transform T h e c o n v o l u t i o n , K * F can be solved more easi ly i n F o u r i e r space. I denote : as the F o u r i e r t r a n s f o r m a n d /[•] as the inverse Four ie r t r ans form. T h e c o n v o l u t i o n K * F is equiva lent to by the C o n v o l u t i o n T h e o r e m . T h u s , c o m p u t i n g the c o n v o l u t i o n , K * F, c an be b r o k e n d o w n i n t o c o m p u t i n g the Four ie r t rans forms of K a n d F, t h e n m u l t i p l y i n g the t rans forms po in twi se i n F o u r i e r space, a n d finally t a k i n g the inverse Four ie r t r a n s f o r m . T h e cost of m u l t i p l y i n g the t rans forms pointwise i n Four ie r space is o n l y 0(n2) because there are n 2 po int s . T h u s , i f c o m p u t i n g the Four ie r t r ans fo rm a n d its inverse are less t h a n 0 ( n 4 ) , t h e n c o m p u t i n g the c o n v o l u t i o n i n Four ie r space w i l l be cheaper t h a n n u m e r i c a l l y i n t e g r a t i n g at each g r i d p o i n t on a large g r i d . C o m p u t i n g the discrete Four ie r t r a n s f o r m for a p e r i o d i c f u n c t i o n or a f u n c t i o n w i t h c o m p a c t s u p p o r t is done efficiently w i t h the Fast F o u r i e r T r a n s f o r m ( F F T ) a l g o r i t h m (Press et a l . , 1988; B u r d e n a n d Faires , 1993). T o see how this is done, let F(x) be a s ingle-var iable f u n c t i o n w i t h p e r i o d 27r. N e x t , d i v i d e the d o m a i n i n t o n pieces so t h a t Xj = j/(2ir) a n d Fj = F(XJ) for j = 0 , n — I. T h e n , the discrete F o u r i e r t r a n s f o r m is T h e n s ample d po int s give rise to n values i n Four ie r space, cok = 2irk/n for k = I[K*F]=I [KF dx 105 Computing the Convolutions 2TT —n/2,n/2 — 1, so that n ,to By letting computing the Fourier transform has been reduced to computing the summation, n-1 Fk = Yl  w k J F 3 , and then multiplying by 27r/n. Ordinarily, this would mean that the Fk values can be computed by multiplying the n-vector, F, by the n x n matrix, W', at a cost of 0(n2). However, the Danielson-Lanczos Lemma (Brigham, 1974) states that a discrete Fourier transform of length n is the same as the sum of two discrete Fourier transforms of length n/2 by breaking the summation into an "even" part and an "odd" part: n-1 Fk = ^2e2viik/nFj (5.1a) = n | ^ \ 2 « ( 2 i ) f e / n F 2 . + n | - 1 e 2 ^ ( 2 i + l ) f c / „ F 2 . + i ( 5 1 b ) j=0 j=0 n/2-1 n/2-1 = Y e2*ijkKn/2)F2j + Wk J2 e2*ijk'(-n'2)F2j+1 (5.1c) j=0 j=Q = P£ + WkF£. (5.1d) F ° is the "even" part and corresponds to the first summation in Equation 5.1c, while is the odd part and corresponds to the second summation in Equation 5.1c. If n is a power of 2, then the Danielson-Lanczos Lemma can be applied recursively to expand each successive part. The expansions of the even half and odd half in Equation 5.Id are p°k = F°° + WkFi1t Fl = Fkw + WkFk11, 106 Computing the Convolutions where Fk sums n/4 = n/22 terms as follows: ra/4-1 = ]T e 27r i j fc/ (n/4)^ ^ F f c 0 1 = n/J2 e 2 ^ k ^ F 4 j + 2 , 3=0 F f c n = e 2 ^ k / W 4 ) F , J + 3 . (Notice that if j = 0, the subscript of the F. term in the summation is the same as the superscript of Fk in binary representation if read backwards, (i.e. The "one's" digit is read before the "two's" digit.)) Similarly, these may be further expanded, _ ^pOOO + WkF%01 F°kl = p r + WkF%n FkW = p r + WkFk101 F}1 = p f c n o + WkFkul where 7^ " sums n/8 = n/2 3 terms. By making n a power of 2, n = 2P for some positive integer p. Thus, the reductions can be done p = log2 n times where the final summations consist of n/2p = 1 term. The Fourier transform of the single term is simply some Fj. Specifically, j is the reverse binary representation of b where b is the binary number in the final transform, F% (as pointed out above for the case where p = 2). The F F T algorithm works as follows for n a power of 2. First the array, Fj, is rearranged into bit-reversed order. The values in the array give the one-point transforms. By combining adjacent pairs, the two-point transforms are obtained. By combining these adjacent pairs, the four-point transforms are obtained. This continues, until the first and second halves are combined to give the final transform. Each level takes 0(n) 107 Computing the Convolutions computations to complete. Because there are 0(log 2 n) levels (one-point transforms up to n/2-point transforms), the complete F F T algorithm is done in 0 (n log 2 n) . (Figure 5.1 gives an example with n = 8). The Inverse Fast Fourier Transform (IFFT) works exactly like the F F T , so that it, too, is 0(nlog 2 n) . 0 1 2 3 4 5 6 7 000 001 01 0 011 100 101 110 111 ^ \ \ 000 001 010 011 100 101 110 111 3 levels of computations F | Figure 5.1: This figure shows the processes necessary for computing the FFT with n = 8. First the original data is reordered. (This is done once at the cost of 0(n).) Then the elements are combined pairwise at 3 or log2 n levels. At each level, the number of operations necessary are 0(n) due to the computations of Wk. Thus, the complete FFT may be calculated in <9(nlog2n) operations. In order to compute the F F T for a function of two-variables (as in System 3.34) a slight modification is necessary. Let F be a function that depends on the variables x and 8 so that Fjm = F(xj,8m). By applying the single-variable F F T to each row of F, the transform of the 0-space can be computed. Transposing the result and applying the single variable F F T to each row again, transforms the x-space as well. Because there are n rows of F, the F F T algorithm is applied n times, giving an overall cost of 0 ( n 2 log 2 n) operations in transforming the <9-space. Transforming the x-space is also 0(n2 log2ra). 108 Computing the Convolutions Since these are done in succession, the cost is additive so that the two-variable F F T is 0(n2 log 2 n), and similarly for the two-variable IFFT. Having determined the cost of computing the two-variable F F T and IFFT, the cost of computing the convolutions can be evaluated. The first step in computing the convo-lution, K * F, is to find the Fourier transform of both K and F. (Because the kernel, K, never changes within one simulation, computing K is only done once, unlike F that must be computed at every time step.) The cost of computing F is 0(n2 log 2 n). F and K are then multiplied together point-wise. Because there are n2 entries, the point-wise multiplication takes n2 operations. Finally, the convolution is obtained by taking the two-variable IFFT of the preceding result at a cost of 0 ( n 2 log2 n) operations. Therefore, the total cost of computing the convolution, K * F, is 0(n2 log2 n), because the steps are sequential, making the costs additive. 1.2 Comparison of Convolution Methods I computed the convolutions with both the integral and Fourier methods and compared the results. (Besides showing the savings of using the Fourier method, the comparison also gives one confidence that each method has been programmed correctly.) The comparison reveals the value of n for which the Fourier method is better than the integral method. Testing was done with K(x,9) = K1(x)K2(9) where and with F(x,9) = Fx(x)F2(9) where Fi(u) = sin(w) + 1. The convolution, K * F, was then computed 100 times with n being various powers of 2. Table 5.1 shows the run time and the value of the convolution for the position (—7T, —IT). By comparing the times between successive rows in the table, one verifies that 109 Computing the Convolutions the integral method is 0(n 4 ) and the Fourier method is 0(n2log2n). For example when n is increased from n = 16 to n = 32, the integral method increases as 73.18 4.68 while the Fourier Method increases as • 8.92 15.64 « 16 1.94 4.6 « 4 = 22(1) = 22log2(2) . One also sees that for n larger than 8, the Fourier method becomes increasingly better than the integral method. (This is very important for my simulations where n = 32.) computation time n Integral Fourier Convolution Method Method Value 4 0:00.04 0:00.08 0.9923 8 0:00.33 0:00.42 0.9949 16 0:04.68 0:01.94 0.9962 32 1:13.18 0:08.92 0.9965 64 19:53.67 0:42.98 0.9966 Table 5.1: The convolutions were computed using the integral method and the Fourier method for different grid size, n. At n = 8, the two methods are roughly the same. However, for n larger than 8, the Fourier method is much better. Also, note that comparing the ratio of times from n + 1 to n, confirms that the integral method is 0(n 4) and that the Fourier method is 0(n 2 log2 n). The factor normalizes a gaussian curve over the whole real line (—oo < x < oo) but is inappropriate for a finite domain. In the simulations, a normalization factor appropriate for the interval [—7r,7r] is used. 110 Selecting a Finite Difference Scheme 2 Selecting a Fin i te Difference Scheme Selecting a proper finite difference scheme is essential for solving the partial differential equation numerically. Big diffusion coefficients (when the filament lengths are small or the viscosity is close to that of water) suggest the use of an implicit method so that the stability of the scheme is independent of the time step. However, small diffusion coefficients (when the filament lengths are big or the viscosity is greater than that of water) make an implicit scheme unnecessary. My problem is such that both such regimes must be considered. Further, a method with a high order of accuracy in time is needed to compute the solution for an unstable homogeneous steady state. These criteria lead to the use of a fourth order Runge-Kutta method (Gerald and Wheatley, 1989) to solve the system of partial differential equations. Because it is explicit, this method takes some time to solve equations that have large diffusion coefficients but still solves them accurately. The numerical scheme was then tested for stability and accuracy. When working on a grid with a spatial step size of 8X and an angular step size of 8a, the parabolic part of the model equations requires the step size, h, to satisfy in order for the scheme to be stable (Strikwerda, 1989). I let h be half of the above minimum if 8t was less than 0.2, and 0.2 if 8t was greater than 0.2 in order for the method to be accurate in time. I then compared the results in two ways. I checked for stability by halving h and noticing that the solution was unchanged. I tested for accuracy by comparing the fourth order Runge-Kutta method with the first-order forward Euler method. The results agreed fairly closely on a short time scale, but as time increased the superiority of the fourth order Runge-Kutta method became apparent. Performing these tests also gave me confidence that my code was working. I l l Simulations and Results With a only a few other additions, I was able to develop my computer code to simulate the system of integro-PDE's. I used periodic boundary conditions in both the spatial and the angular variable. This is the natural boundary condition for the angular variable. For the spatial variable, it is a simplification that allows one to ignore the effects of boundaries. The diffusion terms were calculated with the standard second order differencing so that Because the Runge-Kutta method uses a weighted average of four parts, the diffusion terms as well as the convolutions with the Fourier method were calculated four times for every time step. This gave the basis for my computer code, and (with a few additional features like querying for parameter values from the user) I was ready to perform multi-ple simulations that tested how the biological parameters influenced the resulting actin networks. 3 Simulations and Results I used various parameters from one simulation to the next to study how the parameters affect the resulting actin network. In particular, I was able to test the effects of the filament length, the concentration of cross-linkers, the affinity of the cross-linker for actin, and the viscosity of the solution on the actin network. The only two parameters that stayed fixed in all simulations were the total concentration of actin (at 15 juM) and the cross-linker association rate constant, k+ (at 1 / i M _ 1 s _ 1 ) . The simulations were initialized by first finding the homogeneous steady state and then randomly perturbing the system from this state with a maximum perturbation of 10% of the steady state. In most simulations, the computer would work for 6 minutes in order to produce one minute of simulation time. (Some simulations took much longer than this because if the viscosity was low, or the filament length was short, the diffusion coefficient would be large, forcing 112 Simulations and Results the t i m e step to be s m a l l for n u m e r i c a l s tabi l i ty . ) T h e results of the s imula t ions were put i n t o the f o r m shown i n F i g u r e s 5.3 - 5.9 be low. E a c h rectangle represents a one-d imens iona l c o r r i d o r a p p r o x i m a t e l y 6 \i l ong , w i t h a c t i n filaments di f fus ing a n d i n t e r a c t i n g a long the l e n g t h of the cor r idor . B o t h the s p a t i a l a n d angu lar d i s t r i b u t i o n of filaments is shown i n these figures b y v i e w i n g t h e m as a set of t h i r t y two "angular h i s tograms" per frame, (arrayed a long the l e n g t h of the reg ion , at interva l s cor re spond ing to r o u g h l y 6 /32 = 0.1875^i). T h e l eng th of the spokes o n each of the wheels represents the l o c a l angu lar concentration of the a c t i n filaments a n d not the length of the filaments. F o r example , i f the ne twork is l o c a l l y i so t rop ic , s h o w i n g no d i r e c t i o n a l preference, the spokes are of equal l e n g t h as i n the t o p frame of F i g u r e s 5.3-5.5, regardless of the filament l eng th . If the a c t i n is " b u n d l e d " i n t o preferred d i rec t ions , some spokes are bigger t h a n others as i n the top frame of F igure s 5.7-5.9. T h e re la t ive dens i ty of filaments across the region is represented by the grey scale w i t h l i ght gray m e a n i n g l o w dens i ty a n d d a r k gray m e a n i n g h i g h density. F i g u r e 5.2 is a legend i n t e r p r e t i n g some of the "angular h i s tograms" f o u n d i n the results . Because the images o n l y show re la t ive differences, the m a x i m u m filament densi ty of the n u m b e r of filaments at one p o i n t a n d o r i e n t a t i o n is g iven for compar i sons between different d i agrams . T h e m i n i m u m filament dens i ty is also i n c l u d e d as a percentage of the m a x i m u m . 3.1 Effect of Filament Length T h e results show the effect of filament l e n g t h on the f o r m a t i o n of spa t io-angular pa t te rns w i t h two d i s t i n c t regimes. O n e t y p e of p a t t e r n , spa t i a l clusters , is f o r m e d w h e n filaments are short enough t h a t r o t a t i o n a l dif fusion dominates over t r a n s l a t i o n a l d i f fus ion. T h i s is s h o w n i n the t i m e sequence of F igures 5.3 - 5.6. F o r th i s sequence, filaments of l e n g t h 0.6, 0.7, 0.8, 0.9 a n d 1.0 \x ( top to b o t t o m of figure) were used. A t five minute s , there is not m u c h difference between the s imula t ions of different l e n g t h filaments ( F i g u r e 5.3). 113 Simulations and Results Isotropic, low density actin cluster Isotropic, intermediate density actin cluster Isotropic, high density actin cluster | Aligned, low density actin bundle | Aligned, intermediate density actin bundle j | Aligned, high density actin bundle Figure 5.2: Legend for figures 5.3 - 5.9. At 30 minutes into the dynamics, spatial clustering has begun to take place in the longer filament length simulations (Figure 5.4). By 90 minutes, the simulations of the longer filaments (0.8 — 1.0/x) show well defined clusters approximately 3 microns apart (Figure 5.5). At this same time, the simulations of the shorter filaments is still undetermined. Not until 150 minutes into the simulation can it be determined that filaments of length 0.6 and 0.7 LL form clusters that are approximately 1.5 microns apart. The number of clusters (i.e. the wavenumber corresponding to the spatial periodicity) is greater for the smallest filament lengths. (For filament lengths under 0.5 LL, the network remains homogeneous in space and angle (not shown).) 114 Simulations and Results Time = 5 minutes Length = 0.6 microns Minimum is 96.4% of the maximum: 0.69 Length - 0.7 microns Minimum is 96.2% of the maximum: 0.67 Length - 0.8 microns Minimum is 96.1% of the maximum: 0.65 Length - .9 microns Minimum is 95.7% of the maximum: 0.64 Length - 1.0 microns Minimum is 94.7% of the maximum: 0.63 Figure 5.3: Spatio-angular distribution of actin filament density for filament lengths in the range 0 . 6 - 1 / / at time = 5 minutes. The filaments remain uniform in their angular distribution, but are fairly uneven in their spatial distribution. 115 Simulations and Results Time = 30 minutes Length = 0.6 microns Minimum is 96.2% of the maximum: 0.69 Length = 0.7 microns Minimum is 94.8% of the maximum: 0.67 ;.- .:::!x. My, ii m « .. Length = 0.8 microns Minimum is 92.8% of the maximum: 0.67 Length = .9 microns Minimum is 89.3% of the maximum: 0.66 Length = 1.0 microns Minimum is 83.4% of the maximum: 0.67 * # i e # # > • * « » • # # # # # 1 1 i Fig ure 5.4: Same as Figure 5.3 at time = 30 minutes . The simulations using longer filament lengths show a stronger tendency to cluster at this point. 116 Simulations and Results Time = 90 minutes Length = 0.6 microns Minimum is 88.2% of the maximum: 0.72 Length = 0.7 microns Minimum is 69.0% of the maximum: 0.80 Length = 0.8 microns Minimum is 30.5% of the maximum: 1.23 • • « * * * • • « # # # # • 1 Length = .9 microns Minimum is 0.8% of the maximum: 6.28 Length = 1.0 microns Minimum is 0.0% of the maximum: 12.33 »# # Figure 5.5: Same as Figure 5.3 and 5.4 at time — 90 minutes. The simulations using longer filament lengths (0.9/z and 1.0fi) have formed clusters approximately 3 microns apart. The simulations using shorter filament lengths (0.6// and 0.7u) have still failed to show distinct clusters. The simulation using 0.8/t filaments can be seen as a transition between the two states at this time. 117 Simulations and Results Time = 150 minutes Lenj ^ th = 0.6 microns Minimum is 61.1% of the maximum: 0.87 - * * « • * • *; ft # » « • Len^ rth = 0.7 microns Minimum is 4.9% of the maximum: 3.05 * • •#• ^  •«• Lenj ^ th = 0.8 microns Minimum is 0.0% of the maximum: 12.00 Figure 5.6: Same as Figures 5.3 - 5.5 at time = 150 minutes. The simulations using shorter filament lengths (0.6/x and 0.7jti) are now seen to form clusters approximately 1.5 u apart. The simulation using 0.8/x filaments has now formed clusters approximately 3 p apart. 118 Simulations and Results Time = 15 minutes Length = 1.2 microns Minimum is 87.9% of the maximum: 0.62 Length = 1.4 microns Minimum is 80.2% of the maximum: 0.62 # # * /- ' * * - | 1 |f # w • - . . . • . , » Length = 1.6 microns Minimum is 72.7% of the maximum: 0.62 § t 1 1 1 1 1 * ' I t • • • * -'- * 9 9 : i '* * » » Length = 1.8 microns Minimum is 65.9% of the maximum: 0.62 § | I 1 j » • Length = 2.0 microns Minimum is 57.9% of the maximum: 0.62 i s H H % «. i ! | - * •? * * # • # 3fi ^ <j& ^ ';?* -:if • vjli :i*i* |j? $ (• Figure 5.7: Spatio-angular distribution of actin filament density for filament lengths in the range 1.2 — 2.0LI at time = 15 minutes. The simulations for longer filament lengths show that angular alignment is favored over spatial alignment at this time while the opposite is true for the simulations with shorter filament lengths. 119 Simulations and Results Time = 30 minutes Length = 1.2 microns Minimum is 55.6% of the maximum: 0.78 *####** • * « • • • Length = 1.4 microns Minimum is 19.1% of the maximum: 1.34 Length = 1.6 microns Minimum is 4.1% of the maximum: 2.55 * # # # 1 i * * * * * # # # $ • • » • * » • « » * * * Length = 1.8 microns Minimum is 0.9% of the maximum: 3.16 M i l l ; | 1 i v Length = 2.0 microns Minimum is 0.1% of the maximum: 7.83 i f f | 1 i * , * 1 * * | f * | | i * : , Figure 5.8: Same as Figure 5.7 at time = 30 minutes. At this time, the sim-ulations show that networks of shorter filaments prefer spatial ordering and networks of longer filaments prefer angular ordering. 120 Simulations and Results Time = 60 minutes Length = 1.2 microns Minimum is 0.0% of the maximum: 97.08 Length = 1.4 microns Minimum is 0.0% of the maximum: 177.41 Length = 1.6 microns Minimum is 0.0% of the maximum: 225.49 Length = 1.8 microns Minimum is 0.0% of the maximum: 292.28 Length = 2.0 microns Minimum is 0.0% of the maximum: 258.29 1 i Figure 5.9: Same as Figure 5.7 and 5.8 at time = 60 minutes. Al l the simulations for these lengths lead to the formation of bundles. Compare these results with Figure 4.12 to see how the different diffusion rates influence how bundles are formed. 121 Simulations and Results In the second time sequence, Figures 5.7- 5.9 filaments are longer than 111. At these longer filament lengths, network filaments not only form clusters, but they also align with one another (in angle) to form bundles (Figure 5.9). The corresponding magnitudes of the rotational and translational rates of diffusion in this regime are illustrated in Figure 4.12. An intersection of the two graphs occurs near 1.6/i. For smaller lengths, the rotational diffusion rate is faster, and thus tendency for (angular) alignment is suppressed in favor of spatial segregation. Above 1.6,u, translational diffusion is greater, so there is fast mixing spatially that favors angular alignment over clustering. 3.2 Effect of Viscosity The viscosity of actin solutions in vitro are generally assumed to be close to that of water, namely l c P = 0.01 Poise (Wachsstock et al., 1994), though this is an approximation that does not take into account the fact that the filaments themselves affect viscosity. In the cytoplasm, where there are a multitude of other particles, fibers, organelles, etc, viscosity is greater by orders of magnitude. (For example, (Oster, 1994) mentions a figure of 100-1000 Poise for the viscosity of the cytoplasm.) Viscosity, ns, influences both rotational and translational diffusion rates in the same way. (It appears in the denominator of the expressions). A high value of the viscosity leads to a low value of the diffusion rates, and hence of the LHS of the instability condition (Equation 3.36). Thus, a high viscosity makes it more likely that the system will become unstable at given wave numbers. A l l the simulations shown in Figures 5.3-5.9 were done with a value of viscosity much greater than that of water, e.g. 100 Poise. When the viscosity is close to that of water, e.g. 5cP (as in Lumsden (1993)), and all other parameters are held fixed as in Figures 5.3-5.9, no instability occurs for "small" filament lengths (< 2LL), and the solution remains homogeneous and isotropic. (Both diffusion rates are far too rapid). For longer lengths, (2 — 5//) we get bundling as shown in Figures 5.10 and 5.11. Thus, the viscosity influences 122 Simulations and Results which hlament lengths form patterns and what patterns form at these lengths. Time = 30 minutes Lenj jth = 2 microns Minimum is 100.0% of the maximum: 0.49 Len^ 5th = 3 microns Minimum is 94.1% of the maximum: 0.42 Lenj 5th = 4 microns Minimum is 0.0% of the maximum: i 5.08 , * i i i i t 1 * Lenj 5th = 5 microns Minimum is 0.0% of the maximum: ] 6.90 * I I I I J 1 ! , ' I t | | f 1 ' Figure 5.10: Simulations with viscosity ns = 5cP after 30 minutes. The tran-sition length between spatial and angular preference is now between 2 and 3 microns. 3.3 Effect of a-actinin Concentration Experiments by Wachsstock et al. (1993; 1994) showed that filaments were more likely to bundle at high concentrations of a-actinin as opposed to low concentrations. Figure 5.12 shows results after 60 minutes with filaments of length 2 microns at ILIM, 5LIM, and WfiM of a-actinin. At the low concentration, the network remains stable. At the intermediate concentration bundles have formed. At the high concentration more network filaments bundle as seen when comparing the maximum filament concentrations 123 Simulations and Results Time = 60 minutes Length == 2 microns Minimum is 100.0% of the maximum: 0.49 Length = 3 microns Minimum is 23.0% of the maximum: 0.86 ^ | | k .jij$k sip. *g& W Length = 4 microns Minimum is 0.0% of the maximum: 24.98 , t • * , * M * ' Length = 5 microns Minimum is 0.0% of the maximum: 25.89 < 1 1 I J i 1 i ! f i • Figure 5.11: Simulations with viscosity ns = 5cP after 60 minutes. Simulations for lengths between 3 and 5 p show the beginning of the formation of bundles. between the high and intermediate concentrations of a-actinin. These results correspond favorably with the experiments by Wachsstock et al. (1993; 1994). 3.4 Effect of Cross-Linker Affinity The type of cross-linker also affects the resulting actin network. Wachsstock et al. (1993; 1994) considered two different types of a-actinin, chicken and Acanthamoeba. Chicken a-actinin favors anti-parallel alignment and has a dissociation rate constant, /c_ = 0.7s~ :. Acanthamoeba a-actinin favors parallel alignment and has a dissociation rate constant, k- = 5. Using these parameters on different filament lengths and cross-linker concen-124 Simulations and Results A = lfiM M i n i m u m is 100.0% of the maximum: 0.62 M i n i m u m is A = bfiM 0.0% of the maximum: 179.83 # M i n i m u m is A = IOLLM 0.0% of the maximum: 478.08 / / Figure 5.12: When the viscosity was close to that of water (.05 Poise), we could see that increasing the total concentration of a-actinin lead to the formation of bundles over networks for filaments of 2u in length. (The concentrations of a-actinin are indicated above each corresponding picture.) trations, I tried duplicating the experimental results. (The viscosity was fixed at 5cP). Figures 5.13 -5.15 show the resulting networks after 60 minutes for various concentrations of chicken a-actinin. Figures 5.16 -5.18 show the resulting networks after 60 minutes for various concentrations of Acanthamoeba a-actinin. There are two main differences when comparing Figures 5.13 -5.15 with Figures 5.16 -5.18. For filament lengths of 2 and 3 microns, the transition to inhomogeneous networks occurs at greater cross-linker concentrations of Acanthamoeba a-actinin than of chicken a-actinin. For the 3 and 5 LL filament lengths, the chicken a-actinin forces bundles, but the Acanthamoeba a-actinin only could force filament alignment. Because Pi is inversely 125 Simulations and Results Lenj 5th = 2 microns Minimum is 100.0% of the maximum: 0.62 Lenj 5th = 3 microns Minimum is 0.0% of the maximum: 18.40 # # t , Leii;; 5th = 5 microns Minimum is 0.0% of the maximum: 30.33 . n i l ' H P Figure 5.13: Simulations with A = 1/xM of chicken a-actinin after 60 minutes for various filament lengths. Length = 2 microns Minimum is 0.0% of the maximum: 179.83 i Length = 3 microns Minimum is 0.0% of the maximum: 217.18 Length = 5 microns Minimum is 0.0% of the maximum: 90.46 1 1 1 1 / / / / Figure 5.14: Simulations with A = 5/xM of chicken a-actinin after 60 minutes for various filament lengths. 126 Simulations and Results Length = 2 microns M i n i m u m is 0.0% of the maximum: 478.08 / Length = 3 microns M i n i m u m is 0.0% of the maximum: 373.87 / / ' / Length = 5 microns M i n i m u m is 0.0% of the maximum: 105.68 > / / / / III' Figure 5.15: Simulations with A = WfiM of chicken a-actinin after 60 minutes for various filament lengths. Length = 2 microns M i n i m u m is 100.0% of the maximum: 0.15 Length - 3 microns M i n i m u m is 100.0% of the maximum: 0.14 gth - 5 microns M i n i m u m is 0.0% of the maximum: 2.89 t f 1 t f I ! t i ! } t t f ! i t t f f f f f t f | | * t I f I Figure 5.16: Simulations with A = 1/xM of Acanthamoeba a-actinin after 60 minutes for various filament lengths. 127 Simulations and Results Length = 2 microns Minimum is 100.0% of the maximum: 0.64 Length = 3 microns Minimum is 0.0% of the maximum: 7.55 1 1 1 1 1 \ \ » M M H I I 1 I 1 1 1 1 1 1 1 1 t I I 1 I t * Length = 5 microns Minimum is 0.0% of the maximum: 14.66 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 Figure 5.17: Simulations with A = 5fiM of Acanthamoeba a-actinin after 60 minutes for various filament lengths. Length = 2 microns Minimum is 0.0% of the maximum: 172.67 Length - 3 microns Minimum is 0.0% of the maximum: 17.14 / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / Length - 5 microns Minimum is 0.0% of the maximum: 19.97 I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I Figure 5.18: Simulations with A = 10\iM of Acanthamoeba a-actinin after 60 minutes for various filament lengths. 128 Simulations and Results proportional to /c_, an increase in (as done when changing from chicken a-actinin to Acanthamoeba a-actinin) causes a decrease in (see Equation 4.9) and an increase in 7 (see Figure 4.6). Decreasing Pi and increasing 7 decreases the chance of the instability condition (Equation 3.36) being satisfied. An increase in cross-linker concentration has the opposite effect on the instability condition. Thus, the transition to an inhomogeneous network may still occur for Acanthamoeba a-actinin, but only at greater cross-linker concentrations. Section 1.3 in Chapter 3 shows that increasing the filament length from small lengths causes clusters to form and as the length continues to increase, the clusters eventually disappear. The difference in for the two types of a-actinin affect the model parameters so that clusters are more likely to occur for chicken a-actinin than Acanthamoeba a-actinin. In the simulations, the Acanthamoeba a-actinin stops forming clusters for lengths less than 3 LI, while the turning point for the chicken a-actinin is greater than 5 LI. 3.5 Effect of K e r n e l The choice of the kernel also affected the final outcome as seen when comparing Figures 5.13 -5.15 with Figures 5.16 -5.18. Filaments were seen to align only in one direction when the parallel kernel was used as shown in Figures 5.16 -5.18. However, when the anti-parallel kernel was used (Figures 5.13 -5.15), filaments tended to align in two directions, 180° apart from one another. When the kernel's spread of influence was mildly changed, the overall results were not affected. Thus, the general shape of the kernel, such as its symmetry, greatly influences the final outcome, but small changes in the shape of the kernel have negligible effects (Mogilner and Edelstein-Keshet, 1995). 129 Conclusions 4 Conclusions The simulations show that filament length strongly affects the type of actin networks that form. These results can be understood partly in the context of the instability condition given by Expression 3.36. The diffusion rates in this expression influence what type of patterns can form. The rotational rate of diffusion, changes with L~7 causing vast changes in the angular component of the network (represented by the wavenumber ki) when the filament length is changed. When \ix is large, the patterns favored (if any) are those with k\ = 0 which causes clustering in space. The translational rate of diffusion, H2, only changes with L - 1 , so that the changes in the spatial pattern (represented by the wavenumber k2) are less sensitive to changes in filament length. When \x2 is large, patterns with k2 = 0 (bundling in angle) are favored. The cell is able to control the length of actin filaments through polymerization and depolymerization of actin by changing the concentration of available actin monomers (Zigmond, 1993). The cell may also control filament length by severing filaments with the aid of molecules such as gelsolin, fragmin and severin (Hartwig and Kwiatkowski, 1991). Recent modeling work by Edelstein-Keshet and Ermentrout (1998; 1998) shows how such processes influence both the distribution of filament lengths and the average length of the filaments. The results in this chapter along with the results of Section 1.3 from Chapter 3 suggest that the cell may be able to control transitions between random actin networks, actin clusters, and actin bundles by changing the length of its actin filaments. This would have profound effects on cell locomotion. Although the connection between filament length and filament order has been mentioned in previous papers (Coppin and Leavis, 1992; Furukawa et al., 1993; Suzuki et al., 1991) (mainly in regards to actin crystalline formation), no previous model has simulated the dynamics and made predictions based on biologically relevant parameter values. 130 Conclusions Because of the detailed derivations of the parameters and the ability of the model to describe non-local interactions, this model may be applied directly to other polymer interactions, where simple chemical-kinetic models fail to account for.the spatially dis-tributed nature of the interactions. However, even though agreement with experiments was obtained and the model suggests the intriguing possibility that filament lengths play a major role in network formation, the model has some shortcomings that limit its effectiveness. • In the model and parameter derivations, filaments were assumed to be stiff rods. However, longer actin filaments (several ii long) are flexible so that their orientation can no longer be simply defined and the derivation of the diffusion rates no longer holds. • The model assumes that the major component forming the cytoskeleton is the binding and unbinding of actin filaments (via cross-linkers). Other factors such as in situ polymerization, motor protein induced rearrangements, and boundary conditions are neglected. • Although effects of viscosity and other molecular clutter are included through the diffusion coefficients, conditions of crowding and mechanical forces exerted on the filaments are not included in the model. • The effects of fluid convection are not included in the model. For cells in which cytoplasmic streaming is a dominant effect, this would be a serious shortcoming. • The model assumes that the filaments are able to readily diffuse. This fits the in vitro experiments where the concentration of actin is near 15 \iM and thus the filaments are in the semi-dilute range. However, when the concentration of actin is closer to 100 \iM as seen in vivo, the solution becomes concentrated and seriously hampers the diffusion of filaments. Although, increasing the viscosity helps to 131 Conclusions handle these effects under the current model formulation, a different model may be more appropriate for greater actin concentrations. (Note: at 15 11M the lengths for the semi-dilute regime are 0.2/1 < L < 5.5/1 while at 100 \iM the lengths for the semi-dilute regime are 0.08/1 < L < 0.82/1.) • The model sets no bound on the filament concentration so that the number of filaments occupying some fixed volume could surpass the amount of room available. For example, by assuming that filaments are cylindrical and packed in a lattice, only about 17000 one micron filaments could occupy a volume of 1 /i3. Of course, the filaments could never pack so efficiently so that this number is much lower. If the concentrations begin to reach such levels, then crowding effects need to be added to the model which would also change the basic premises. • The model assumes that all filaments are of the same "average" length. The fact that filaments have an entire distribution of lengths may change the overall network. 132 Chapter 6 I n t e r a c t i n g F i l a m e n t s o f D i f f e r e n t L e n g t h s In Chapters 3 and 5 clustering of actin filaments and formation of networks were shown to depend on the length of the filament. However, in both chapters, the filaments were assumed to be of some average uniform length. In this chapter, I drop this assumption. Earlier results showed that in general, longer filaments tend to form bundles (or at least align) while shorter filaments do not (when all other factors are kept fixed). I now consider what happens in a solution that includes filaments of different lengths. I develop three models that address how different filament lengths in a solution influ-ence overall network formation. The first model shows the interaction of two populations of filaments (short and long). The second model shows the effects when one population of filaments is "frozen" in a fixed alignment and the other filaments are able to interact as before. Finally, a model is developed to study the interactions of an entire distribution of filament lengths. Before I develop the models, let us try to visualize what might happen in the following situations: 1. A solution with a high density of long filaments and a low density of short filaments. 2. A solution with a low density of long filaments and a high density of short filaments. 133 Model for Two Lengths 3. A s o l u t i o n t h a t has a n entire d i s t r i b u t i o n of f i laments of a l l lengths . In each of the above scenarios, one of three outcomes is poss ible : • A l l the f i laments become b u n d l e d . • A l l the f i laments f a i l to bundle . • F i l a m e n t s of a ce r t a in l eng th bundle whi l e others do not . In the first case, I w o u l d expect the l o n g filaments to d o m i n a t e so t h a t a l l the filaments become a l igned . I n the second case, I w o u l d expect t h a t e i ther a l l filaments f a i l to b u n d l e or o n l y the l o n g f i laments bundle . I n the t h i r d case, i t is h a r d t o pred ic t w h a t h a p p e n . M y guess w o u l d be t h a t some lengths w o u l d bund le whi l e others w o u l d not . B y s t u d y i n g these types of systems, one can explore whether the a s s u m p t i o n of a single l e n g t h i n the previous model s was over ly res tr ic t ive a n d whether any interes t ing b e h a v i o r was consequent ly missed. A l s o , one m a y discover i f a p o p u l a t i o n w i t h two types of behav ior (e.g. b u n d l i n g i n l o n g filaments b u t i so t rop ic short filaments) m i g h t occur , a result t h a t was not rea l i zed i n C h a p t e r 5. 1 M o d e l for Two Lengths I consider an a c t i n s o l u t i o n w i t h b o t h short a n d l o n g filaments i n var ious p r o p o r t i o n s . C o m p o s i t i o n of these so lut ions is governed b y one parameter , p, the r a t i o of shor t fila-ments to l o n g filaments. F o r case (1), p « 1, w h i l e for case (2), p » 1. T h e eventua l results are a l l based o n p. T h e m o d e l f r o m C h a p t e r 3 is extended to two f i lament lengths , short (s) a n d l o n g (I), as fol lows: Fs(9,t) = n u m b e r of free short f i laments per p? w i t h o r i e n t a t i o n 9 at t i m e t. Fi(6,t) — n u m b e r of free long filaments per p? w i t h o r i e n t a t i o n 9 at t i m e t. Ns(9,t) = n u m b e r of short network filaments per p? w i t h o r i e n t a t i o n 9 at t i m e t. Ni(9,t) = n u m b e r of long network filaments per p? w i t h o r i e n t a t i o n 9 at t i m e t. 134 Model for Two Lengths ( T h e n u m e r i c a l s i m u l a t i o n for a d i s t r i b u t i o n of f i laments i n space, angle, a n d l e n g t h were p r o h i b i t i v e , r e q u i r i n g very l o n g c o m p u t a t i o n a l t imes . F o r th i s reason, I d r o p the s p a t i a l heterogeneity.) A s before, the free f i laments m a y diffuse r o t a t i o n a l l y at rates LI(S) a n d LL(1) for short a n d l o n g f i laments , respectively. A l s o , K(9) is the p r o b a b i l i t y t h a t f i l aments whose o r i e n t a t i o n differ b y 9 b i n d . T h e m o d e l is a s imple genera l i z a t ion of those i n C h a p t e r 3. H a v i n g four classes of f i laments increases the n u m b e r of poss ible in terac t ions . T h e n o t a t i o n is now extended to inc lude short-to-short f i lament in terac t ions (ss) , short- to-l o n g f i l ament in terac t ions (si), a n d long-to- long f i lament in terac t ions (11) as denoted i n the super sc r ip t of the b i n d i n g rate parameter , (5. A s before, free-to-free f i l ament b i n d i n g occurs at the rate (3\ a n d free-to-network f i lament b i n d i n g occurs at the rate L32. A l l these in terac t ions are t h e n c o m b i n e d to f o r m the fo l lowing sys tem of i n t e g r o - P D E ' s : jtme,t)\ = lM(8)-^[Fa(01t)]-0'1'Fa(e,t)[K*F.(e1t)] -PilFs(9, t)[K * Fi(6, t)\ - ps2sFs(0, t)[K *NS(0, t)] -/3?Fs(6,t)[K*Nl(6,t)]+7Na(e,t), (6.1a) | [ * K M ) ] - ^ ^ m ^ - p f F t i e ^ K ^ F ^ t ) ] -$&($,t)[K * Ft(6,t)] - / # F , ( 0 , i ) [ t f * Ns(8,t)} -f32lFl(6,t)[K*Nl(8,t)]+7Nl(0,t), (6.1b) j t [Ns(9, t)} = P[aFs(9, t)[K * F,(9, t)} + P{lFt(e, t)[K * Fs(8, t)] - jNs(8, t) d +PlsNa(6, t)[K * Fs(8, t)] + fi%(d, t)[K * Fs(9, t], (6.1c) d t [Ni&t)] = PfFs(9,t)[K * Ft(9,t)} + P llFt(8,t)[K * Ft(9,t)} - -yN^t) +f3s2lNs(8, t)[K * Ft(8, t)} + pl2lNt(8, t)[K * Ft(8, t)}. (6.1d) S y s t e m 6.1 is the new m o d e l tha t describes the interac t ions of a c t i n filaments i n a s o l u t i o n of two lengths . 135 Model for Two Lengths 1.1 Preliminary Analysis Steady State Because the model is now a system of four integro-PDE's, the stability analysis becomes much more difficult as evidenced in the once simple task of finding the homogeneous steady state. Let the homogeneous steady state be denoted by (F s , Ft, Ns, Nt). Setting Equations 6.1c and 6.Id to zero leads to the following: P['F2a + Pl lFsFt + PS2 SFSNS + P s2 lFsNt = 7NS, (6.2a) P[ lF? + P[ lFsFi + 82FiNi + B s2 lFiNs = jNt. (6.2b) Because the filaments do not grow or shrink (following the same assumption as in Chapter 3), filament densities are conserved for short lengths and long lengths so that Fs + Ns = Ms (6.3a) Fi + Ni = Mt. (6.3b) In addition, the ratio of short to long filaments can be defined as P = ^ Ms=PMt. (6.4) The equations in System 6.3 may be solved for F and then substituted into System 6.2 to obtain a system of two nonlinear equations with two unknowns, jMs = [(P^-p^Fs+^f-P^F + ^M^j^Fs + p^M^, (6.5a) yMt = [{p ll-pl2 l)Fl + {pf-pf)Fs+[p llMl-1)}Fl + p s2 lMsFs. (6.5b) By solving this system, one may find the steady state solutions of the system. Unfortunately, the analytical solutions to System 6.5 (which are simplest when PI = P2, but still add little insight) are too cumbersome to be of any value. However, values of Fs and Fi can be found numerically (such as with a 2-D Newton's method) when numerical 136 Model for Two Lengths values for all the parameters are given. Also, graphical analysis of the equations similar to that done in Chapter 2 may be used once parameter values are given. Thus, System 6.5 is still useful once parameters are estimated. Instabilities The system of integro-PDE's may be linearized about the homogeneous steady state just as in Chapter 3 by first letting F s (0,i) = Fs + M9,t), (6.6a) Na(9,t) = Na + na(9,t), (6.6b) Ft(9,t) = + / , (# ,£) , (6.6c) Ni(9,t) = JV, + n,(0,*)- (6-6d) With these substitutions, System 6.1 (to first order) becomes ^ [/.(*>*)] = H(s)~[fs(e,t)] + 1ns(B,t) (6.7a) -P['Fafa(6, t) - P1°FS[K * fs(6, t)] - PfFtUd, t) - P?Fa[K * ft(9, t)] -6 S2 SNJS(8, t) - 8S2SFS[K * ns(9, t)] - P?Nrfs(9, t) - Bs2lFs[K * m{9, t)}, jtw,t)] = K O ^ W M ) ] + W M ) (6.7b) -0tFsfi(6, t) - p?Ft[K * fs(9, t)] - P[lm(e, t) - p llFL[K * ft(6, t)} -tfN.m t) - Ps2lFt[K * na(9, t)} - p^mie, t) - pl2lFt[K * nt(9, t)], f M M ) ] = -771.(61, t) (6.7c) +Pa1sFafa(9,t) + PisFs[K * fs(9,t)} + P[lFafi{9,t) + P[lFi[K * fa(9,t)\ +ps2sFsns(9, t) + P S2 SNS\K * fs(9, t)] + ps2lFsm(9, t) + P s2 lNt[K * fs(9, t)], jt[ni(9,t)} = -jni(9,t) (6.7d) +PllFlfa(9,t) + filFa[K*M6,t)} + fl^ +Ps2lFms(9, t) + p s2 lNa[K * ft(9, t)] + pl2lFini(9, t) + P l2 lNt[K * ft(9, t)]. 137 Model for Two Lengths Using System 6.7, I may consider several possible perturbations. To test for angular instabilities in long filaments alone (Case 1), I consider a pertur-bation of the form fs(6,t) na(6,t) fi(0,t) ni(9,t) 0 : 0, nl0eik9ext (6.8a) (6.8b) (6.8c) (6.8d) fl' > = B< > ni This reduces the system to two dependent variable much like the previous model. When substituted into System 6.7, the resulting linear system is (6.9) where _ I" -p(l)k2 - PfFs - p[lFt (l + K)- P?NS - pl2lNt [ P llFt + (pilF8 + P[ lFt + PllNs + Pl2lNt) K This system is very similar to Equation 3.23. In fact, if the P sl terms were not present, Equation 6.9 would be identical to Equation 3.23 as one might expect. The stability analysis is also the same. The trace of B is always negative and instabilities hinge on whether the determinant of B is negative or not. The instability condition now becomes 7 - P l2 lFik P l2 lF - 7 (6.10) fj,(l) ( i _ ^FA k2 < (p?FS + p°lNs) (k-i)+ pl2lNTK ( l - K) (6.11) Once again, if the P sl terms are removed, the instability condition reduces to the same form as before (Equation 3.28). Unfortunately, Equation 6.11 is not an easy condition to analyze, especially when trying to study the effects of the ratio of short to long filaments, p, that enters the analysis when substituting Equation 6.4 into System 6.5. p affects all steady state values, Fs, Ns, Fi and iV/. Thus, the analysis is best completed when 138 Model for Two Lengths parameter estimates are found that allow the homogeneous steady state to be easily solved. In Case 2, instabilities affecting both long and short filaments are tested by studying perturbations of the form (6.12a) (6.12b) (6.12c) (6.12d) The resulting linear system is where v is the vector [fs, fi,ns, n;] T and Q has the following form, fs(0,t) = f^elk6eXt nS(6,t) = nsQelk9eXt Mo,t) = fieikeext m(6,t) = nloeik0ext vt = Qv, Q Qu -PilW(k) 931 -Pl lFsK(k) P?FS 942 -PfFikik) PI SFS - 7 ffFi -Pl lFsK(k) -flftkik) + 7 P?FS Pl2 lFi - 7 (6.13) (6.14) with 9 n Q22 931 942 = -H(s)k2 - Pt'Fs ( l + K(kj) - P{1 Ft - fc sNs - p s2 lNh = -ii(l)k2 - p{'Ft ( l + K(k)) - P{ lF8 - p s2 lNs - p l2 lNt, Pi aFa (I + k{k)) + [pfp + p?N, + p*1^} k(k), PiF ( l + K{k)) + [p?Fs + p l2 lNt + p s2 lNs] K(k). (6.15a) (6.15b) (6.15c) (6.15d) Because of the size of the matrix Q (4x4), determining the eigenvalues is no simple task. The Hurwitz criteria can be used to determine whether the real part of the eigenvalues are all negative (the case where the system is stable), though this is somewhat cumbersome. 139 Model for Two Lengths Because I later find parameter estimates, the eigenvalues of Q may be found numerically for various values of the wavenumber, k. If the real part of at least one eigenvalue is nonnegative, the pattern associated with k is predicted to be unstable. 1.2 Parameter Evaluation Realistic parameter values are needed to complete the analysis in Section 1.1. These values may also be used in simulations to determine the complete network development. The estimates for the parameter values of the model in Chapter 3 derived in Chapter 4 aid in the development of estimates for the parameters of this model. Estimating Rates of Filament Binding In Chapter 4, it was shown that a 15/iM actin solution with cross-linkers (concentration, [a], and association rate constant, k+ = \\xM~1sec-1) leads to the following binding rate, Pi, for free filaments of length L, Pi ~ k [ • (6.16) This implies that for short filaments and long filaments (L = s and L = I, respectively) « • * ^ , (6.17) and / J ? « ^ . (6.18) Considering the possible binding sites along the filaments, we expect that the binding-rate of free filaments of length s to free filaments of length I (and vice-versa), Pf, would be in between the binding rate of short filaments and long filaments (i.e. P{s < Pf < /?"). Thus, some kind of interpolation or averaging is in order to determine Pf. I estimate Pf by simply taking the geometric mean, i.e. ^ = ^ = °^M^l. (6.I9) 140 Model for Two Lengths This estimate of 0f enables reductions in System 6.5 in certain situations. As for the binding rate between free filaments and network filaments, 02, the results from Chapter 4 suggest that 02 ~ j • (6.20) Using similar assumptions leads to ass osl oil P ? * ^ , P l2 1*^. . (6.21) Estimating Rotational Diffusion Rates When estimating the rotational diffusion rate of filaments in a semi-dilute solution in Section 3 of Chapter 4, we found that the rotational diffusion coefficient for a filament of length, L, was given by r2D< Dn oc , (6.22) where D™ = I T ' • (6-23) 2>kbT\n(L/d - b) TTVL 3 d is the polymer's diameter, kb is Boltzmann's constant, T is the temperature (in degrees Kelvin), r) is the viscosity of the solution, b is an empirically determined constant, and r is the radius of the imaginary tube that contains the filament of length L. A l l the steps that led up to this point still hold in the current setting. The only task left is to find the tube radius, r. As before, r is found by setting the number of filaments, T(r) , that intersect the tube to be 1. To determine T(r), once again consider the number of filaments that may intersect a surface element of the tube, AS. For a filament of length £ to intersect AS*, it must lie within its filament length, £, of the surface, or, equivalently, within the volume £AS\ If v is the average number of filaments per unit volume of length £, then the number of filaments that may intersect AS is on the order of u^AS. Thus, by denoting 141 Model for Two Lengths the average number of short filaments as vs and of long filaments as V[, the total number of filaments that intersect AS is on the order of vssAS + vtlAS. By integrating over all surface elements, T(r) oc (vss + uil) S = (uss + u{l) (2<KTL) . (6.24) Solving for Y(r) = 1, gives an estimate of the tube radius as roc -. — — - — . (6.25) This may then be substituted into Equation 6.22 to obtain the following new estimates: *BWH>li->l ( 6 , 6 ) irq(yss + vii) s' and = ™w*w-y, (6.27) TTV (VSS + Vil) I' where B is an empirically determined constant as before. These estimates adjust the term vL in earlier derivations to uss + i>il so that it is still a measure of total monomer concentration (or more precisely, total filament length). Thus, for a distribution of fila-ment lengths whose total length is S, the rotational diffusion of a filament of length, I, can be estimated as S B W M . A . - ^ ( 6 . 2 8 ) Computing Number of Filaments per /x3, u, in 15 \xM. Actin In order to compute the number of short filaments per /u3, i/s, and long filaments per LL3, VI, we must recall a few conversion factors from Chapter 4: 1 / i M (actin) — 602 (monomers of actin)//Li3 , (6.29) and 370 (monomers of actin) = 1/x filament. (6.30) 142 Model for Two Lengths U s i n g E q u a t i o n 6.30, one finds tha t the t o t a l n u m b e r of m o n o m e r s c o n t r i b u t e d b y short a n d l o n g f i laments is 370SZA, a n d 370/^;, respectively. T h i s c an be equated to the t o t a l n u m b e r of m o n o m e r s i n 15 pM of a c t i n by us ing E q u a t i o n 6.29, so t h a t 370s^ s + 370/zv, = 15(602) , or sus + Ivi = 24.4 . (6.31) F u r t h e r , p is the ra t io of the n u m b e r of short filaments to the n u m b e r of l o n g filaments, i.e. P = — => vs = m • (6.32) us m a y t h e n be s u b s t i t u t e d in to E q u a t i o n 6.31 to find ut, l + ps B y s u b s t i t u t i n g ui back i n t o E q u a t i o n 6.32, us c an be found , = (6.34) / + ps T h u s , us a n d v\ m a y be c o m p u t e d based on the short a n d l o n g lengths , the parameter p, a n d the a c t i n c o n c e n t r a t i o n (15 pM). Choosing Short and Long Lengths T h e c r i t e r i o n for choos ing the l eng th of short f i laments , s, a n d l o n g f i laments , I, is t h a t the short f i laments w i l l not a l i gn i n i so l a t ion whi l e the l o n g f i laments w i l l . T h e s i m u l a t i o n s i n C h a p t e r 5 i n d i c a t e t h a t under the assumed condi t ions , a filament l e n g t h of 0.8 p is shor t enough not to a l i gn (albeit near the c r i t i c a l l ength) , w h i l e a filament l e n g t h of 2.0 p is w e l l w i t h i n the range of lengths w h i c h cause a l i gnment . F igure s 6.1 a n d 6.2 show t h a t th i s s t i l l ho lds t rue w h e n the s p a t i a l cons iderat ions are d r o p p e d . T h u s , I cons ider s = 0.8 a n d I = 2.0. 143 Model for Two Lengths Short Filaments 2.1774 F | 2.1774 a 2.1774 time = 10 min time = 20 min Figure 6.1: Short filaments in isolation do not align, (length = 0.8 a) Long Filaments 0) E time = 10 min time = 20 min time = 30 min time = 40 min 0 1 2 3 4 e Figure 6.2: L o n g filaments in isolation align, (length = 2.0 fi) 144 Model for Two Lengths Estimates that Did Not Change Estimating the rate that network filaments dissociate, 7, should be the same as in Section 2 of Chapter 4, because the length of the filaments did not affect 7. Also, the angular component of the kernel derived in Section 4 of Chapter 4 can be applied directly so that K(6) = KX{9), (6.35) as defined by Equation 4.41. 1.3 Results Having now determined all the parameter values, I can study how the ratio of short to long filaments, p, affects the stability of the homogeneous steady state. Furthermore, I can numerically solve System 6.1 to see the specific results for various p. As a final step, I compare these simulations with those in which a single length occurs. The length is determined by averaging the short and long lengths at the specified ratio, p. The results obtained in this section are for an actin solution of 15 pM, a cross-linker solution of 1 pM, a cross-linker association rate constant, k+ = l / ^ M _ 1 s _ 1 , a cross-linker dissociation rate constant, = I s - 1 , and a kernel that favors parallel alignment. Analytical/Numerical Predictions The instability analysis that began in Section 1.1 can now be completed by finding the eigenvalues of B (Equation 6.10) for Case 1 and of Q (Equation 6.14) for Case 2. Case 1 tests for instabilities occurring only in long filaments. Case 2 tests for instabilities occurring in both short and long filaments. For each of these cases, I test whether filaments favor one direction (k = 1) or two directions (k = 2). Figure 6.3 shows the resulting patterns for all the different cases. (Because filament orientation is on a periodic domain, only integer values of the wave number, k, need be considered.) 145 Model for Two Lengths > Case l a ^ Case lb Case 2a ^ Case 2b Figure 6.3: Four different instabilities are tested based on their Case and the number of preferred directions. The graphic next to each case shows the insta-bility pattern associated with that case. A Matlab routine was set up to find the maximum of the real part of the eigenvalues for each case as p varied from 1CT2 to 1. The routine first computed the parameter values based on the specific biological quantities. Next, it found the corresponding homogeneous steady states. Finally, the routine found the maximum of the real part of the eigenvalues in each case. The results are given in Figure 6.4 and Table 6.1. (The table of values is included to show details that cannot be seen in the graph.) For p > 0.22, no eigenvalues will have a positive real part so that the network should be isotropic at p = 0.22, p = 0.5. At p = 0.1, only one type of instability is possible, so that both short and long filaments align in one direction. At p = 0.01, all types of instabilities are possible, but Case 2a should dominate. Also, when comparing results for p = 0.01 to those for p = 0.1, one 146 Model for Two Lengths would expect the patterns to form more quickly because of the larger real part of the eigenvalues in this case. x 10" Case 1 a x Case 1b * Case 2a + Case 2b Figure 6.4: A s p, the number of short to long filaments, increases, the m a x i m u m of the real part of the eigenvalues for each instabil i ty pattern eventually goes to zero. The maximum of the real part of the eigenvalues gives the rate at which a particular instability grows, and as such, indicates which pattern may dominate the others (based on the linear analysis). Figure 6.4 shows that the pattern with both long and short filaments favoring one direction (Case 2a) is most likely to dominate. It is surprising that for small values of p (near 0.01), patterns with two directions grow almost as quickly as those favoring one direction, even though the kernel favors parallel alignment. Figure 6.4 also indicates that Cases l a and lb will likely not be realized because they are dominated by Case 2a. Thus, based on the linearization, it is very unlikely that a solution will have 147 Model for Two Lengths p Case l a Case l b Case 2a Case 2b 0.01 1.99 x l O " 3 1.60 x 10- 3 2.21 x 10" 3 2.16 x 10" 3 0.035 1.15 x 10" 3 0 1.91 x l O " 3 1.14 x 10~3 0.07 0.01 x 10" 3 0 1.51 x l O " 3 0 0.1 0 0 1.20 x 10" 3 0 0.2 0 0 0.20 x l O - 3 0 0.22 0 0 0 0 Table 6.1: The maximum of the positive real part of the eigenvalues of the matrix B in Case 1 and the matrix Q in Case 2 at various values of p. (0 indicates that the maximum of the real part is not positive.) aligned long filaments with homogeneously aligned short filaments. The final insight gained from Figure 6.4 is that no patterns will form for p ; » 1 as they cease to exist near p RS 0.2. This means that as the number of short filaments increase with respect to the long filaments, instabilities are driven out and the network remains isotropic. Thus, a small concentration of long filaments with a large concentration of short filaments would be dominated by the short filaments, prohibiting the long filaments from aligning. Simulations To verify the instability findings, simulations were performed for four different values of p (0.01, 0.1, 0.22 and 0.5). The program was similar to the one in Chapter 5 where the initial state was chosen by randomly perturbing the system from the homogeneous steady state with a maximum perturbation 10% of the steady state value. Figures 6.5 -6.8 show the resulting distribution at various times for p = 0.01, p = 0.1, p = 0.22 and p = 0.5, respectively. The simulations show that the two lengths do indeed affect one another. When p = 0.01 (Figure 6.5), the very small number of short filaments quickly align themselves with the long filaments with an initial preference towards two directions before favoring 148 Model for Two Lengths Figure 6.5: Simulations for p = 0.01. After initially favoring two directions, one direction alignment quickly prevails in both (a) short and (b) long filaments with not much change after 40 minutes (not shown). 149 Model for Two Lengths Short Filaments Long Filaments 7h 6 -(b) I4" 3 -2b 1b 0 time = 20 min time = 40 min time = 60 min - 4 - 3 - 2 - 1 o 1 2 3 4 0 Figure 6.6: Simulations for p = 0.1. Single direction alignment is quickly evident in both (a) short and (b) long filaments without much change after 60 minutes (not shown). 150 Model for Two Lengths Figure 6.7: Simulations for p — 0.22. Single direction alignment is very slow to form in both (a) short and (b) long filaments. There is very little change after 360 minutes (not shown). At 360 minutes, the minimum filament density is approximately 15% of the maximum filament density. 151 Model for Two Lengths Short Filaments 0.2642 0.2642 0.2641 h (a) -g 0.2641 CD e 0.264 0.2639 time = 30 min time = 60 min time = 90 min -4 -3 0.7778 0.7776 (b) C 0.7774 CP "O • 0.7772 0.7768 Long Filaments time = 30 min time = 60 min time = 90 min o 9 Figure 6.8: Simulations for /9 = 0.5. Both (a) short and (b) long filaments quickly go to the homogeneous steady state. 152 Model for Two Lengths one. This agrees with Figure 6.4. (The maximum of the real part of the eigenvalues for the two direction alignment is fairly close to, but less than, that for one direction alignment in both filaments.) As the ratio increases to p = 0.1 (Figure6.6), the still relatively small number of short filaments align with the long filaments, but the pattern takes longer to develop. This corresponds to Figure 6.4 in that there is no indication of two directions of preference in Figure 6.6 (unlike with p — 0.01), and the maximum of the real part of the eigenvalues for Case 2a is smaller at p = .1 than at p = .01, indicating a longer time for the pattern to form. When p = 0.22 (Figure 6.7) a very interesting thing happens. According to the linear stability analysis (Figure 6.4), no patterns should form since there are no positive real parts of eigenvalues, albeit very close to the critical point. However, there is a very slow progression towards a favored angle in both long and short filaments. (The discrepancy between the linear analysis and nonlinear simulations is most likely due to slightly different kernels used in the different situations.) What makes this situation even more intriguing is that unlike the other simulations, the minimum density does not go to zero. Instead, the minimum density is approximately 15% of the maximum. This angular distribution remains well after 3 hours. Finally, when p = 0.5 (Figure 6.8), both short and long filaments return to the homogeneous steady state as Figure 6.4 indicates. In this case where the short filaments appropriate more actin, the mass of the long filaments may not be great enough to satisfy the instability condition (as indicated by M in Equation 3.28). In all four simulations, the short and long filaments behaved exactly the same, casting serious doubt on the possibility of patterns forming in long filaments that are different from those of short filaments (and vice versa). The simulations show that increasing the number of short filaments with respect to the number of long filaments decreases the likelihood of filament alignment. Increasing p is equivalent to decreasing the per-filament binding rate (since Bf s < Bf < Bf and more filaments will be binding at these lesser rates), increasing the overall rate of diffusion and 153 Model for Two Lengths putting less mass into the filaments that could align. A l l these effects would be expected to push the system away from alignment. Averaged Single Filaments The simulations may also be compared with simulations of a single length filament pop-ulation. This allows one to make some conclusions as to how effective taking the average filament length may be (as done in Chapter 4). The average filament length is derived from total filament length vss + number of filaments vs + Vi Using Equation 6.32 yields average filament length = vips + vii ps + / (6.36) (6.37) vtp + vl p + 1 Table 6.2 gives the average lengths for the ratios of short to long filaments used in the simulations. p average filament length 0.01 1.99 0.1 1.89 0.22 1.78 0.5 1.60 Table 6.2: The average filament length for short filaments (0.8 u) and long fila-ments (2.0 u) based on their ratio, p. The resulting single length filament simulations are shown in Figures 6.9-6.12. When p = 0.01 (Figure 6.9), alignment of the 1.99 p filaments occurs slightly more rapidly and with a marginally greater density than in the two filament simulation (Figure 6.5). For p = 0.1 (Figure 6.10), once again alignment is very pronounced and occurs rapidly. This result begins to show a difference between the two filament simulation (Figure 6.6) where the growth is slower and the peak is not as great nor as narrow. For p = 0.22 (Figure 154 Model for Two Lengths time = 10 min time = 20 min time = 30 min time = 40 min 9 Figure 6.9: Simulation with the average filament length of 1.99 u that corre-sponds to p = 0.01. Alignment occurs rapidly. 251 - i I i 1 1 1 , 1 time = 10 min time = 20 min time = 30 min time = 40 min - 4 - 3 - 2 - 1 0 1 2 3 4 9 Figure 6.10: Simulation with the average filament length of 1.89 p that corre-sponds to p = 0.1. Alignment occurs rapidly. 155 Model for Two Lengths 20 16 S12 CD " D O 10 £ CO time = 10 min time = 20 min time = 30 min time = 40 min Figure 6.11: Simulation with the average filament length of 1.78 u that corre-sponds to p = 0.22. Single direction alignment is favored. Figure 6.12: Simulation with the average filament length of 1.6 p that corre-sponds to p = 0.5. Single direction alignment is favored. 156 Model for Two Lengths 6.11), the alignment is still pronounced and occurs quickly. This is in sharp contrast with the two filament population (Figure 6.7) where alignment occurs very slowly and does not occur to the same extent as the one filament population. Finally, when p — 0.5 (Figure 6.12), single direction alignment is still favored. This conflicts greatly with the two filament population (Figure 6.8) where no alignment is seen. The biggest difference between the average length filament population and the two length filament population is seen in the diffusion term. The short filaments always diffuse at the rate 1.9 x 10~2rad/s. This can be compared with the diffusion rates for the average filament lengths as sum-marized in table 6.3. Because the rate of diffusion goes as 1/L 7 , these diffusion rates vary greatly. So whereas the short filaments are moving quickly enough to break up the alignment, the average filament lengths do not come close. Thus, the discrepancy in the results. p L p(L) 0.01 1.99 3.9 x IO" 5 0.1 1.89 5.5 x IO" 5 0.22 1.78 1.1 x 10" 4 0.5 1.60 1.7 x IO" 4 Table 6.3: The rates of diffusion for the average length solutions. 1.4 Conclusions The linear analysis of the model indicates that it is very unlikely for different patterns to form in short and long filaments. The simulations agree, showing that short and long filaments have the same time profiles and form the same patterns. Thus, a two length population model predicts that both long and short filaments will align in the same way so that the entire population of filaments behaves one way. This suggests, but does not prove, that an entire distribution of filament lengths would also strive for the same overall 157 Model for Two Lengths alignment in all lengths. When the results of the two length model are compared with one average length, one sees very different outcomes as the ratio of short to long filaments is increased. This indicates that taking an average filament length may miss some very important details. As pointed out earlier, one of the biggest differences is seen when comparing the diffusion coefficients of the short length with the average lengths. A possible correction may be to bias or weight the average length of a solution towards the shorter length to get similar results. Another difference between the different filament populations is the range of the dominant direction (as seen in the width of the peaks). In the two length models, this range is noticeably greater (wider peaks for greater p) than the single length model. If this holds true when spatial considerations are added, especially the result for p = 0.22, one may have a situation more closely resembling stuctures near the leading edge of a cell (Figure 3.2), a mix of bundles and isotropic networks that varies throughout space. Thus, the assumption of an average length may need to be relaxed for a wider range of patterns to form, although the patterns will probably be the same over all filament lengths. Increasing the ratio of short to long filaments (p) has a number of important effects. First, for a constant total mass of actin, this leads to a lower mass of long filaments. If left on their own, these long filaments would be less likely to aggregate and bundle (see M in Equation 3.36). With the "interference" by short filaments, it is even less likely that bundling occurs, since some significant number of bindings (between long and short filaments) are "unproductive". Further, increasing p increases the overal rate of diffusion and decreases the average binding rate, These effects would tend to drive the system towards a homogeneous steady state. In cases where p is small enough that bundling of filaments is possible, slight increases 158 Model for Two Lengths =—<e (a) Figure 6.13: (a) At low values of p, short filaments have little effect on the bundling of long filaments, (b) As p increases, the greater number of short filaments, may interfere with the long filaments, but are still attracted towards bundles, (c) As p increases further, the long filaments begin to bind with more short filaments instead of long filaments which spreads out the bundles. in p affect the shape of the density distribution by making the peaks broader as seen in Figures 6.5 - 6.7. This makes sense in view of the remark that increased p is associated with an increased overall rate of diffusion. We can also explain the result that both long and short filaments bundle together as shown in Figures 6.5 - 6.8. Since short filaments will not bundle alone (p = 0) under the conditions of the simulations, they are simply joining the bundles already created by the long filaments. This explains the results in the simulations when increasing p, and also may explain why long and short filaments always align the same way as Figure 6.13 tries to convey. A final point of concern is the low ratio at which patterns fail to form. Because in many instances filament length distributions are exponential (Edelstein-Keshet and Ermentrout, 1998), short filaments are most likely to outnumber long filaments, i.e. p > 1. Unfortunately, the simulations showed that no patterns form for ratios that large. However, the simulations are developed using just one set of parameters. By changing one of the biological parameters, such as increasing the cross-linker concentration, patterns 159 Fixed Filament Model may still form at these greater ratios. There may also be other criteria, such as fixed filaments (studied next), that enable patterns to form at greater values of p. 2 F i x e d Filament M o d e l The previous model may be adapted to see how two different filament populations behave when one population has a fixed orientation, for example due to other structural or geo-metric constraints. Because longer filaments present more binding sites and diffuse more slowly, they are more likely to be constrained and held fixed in the cell. To understand the effects of fixed long filaments I set Fi = 0 and Ni(0,t) = { Mi if90<9< dx (6.38) 0 otherwise where f90 to 9\ is the range of orientations for the fixed filaments. Substituting these values into System 6.1 produces the following model for short filaments: Ft(9,t) = pFee{8,t)-P°sF(8,t)[K*F(9,t)}-ps2sF{9,t)[K*N(9,t)} -ps2lF(8, t)[K * Ntf)] + jN(9, t), (6.39a) Nt(9,t) = PlsF(9,t)[K * F(9,t)] + Ps2sN(9,t)[K * F(9,t)] +P2slNl(9)[K * F{9,t)\ - jN{8,t). (6.39b) Because the definition of Ni(9) introduces a bias, there is no homogeneous steady state. However, the same code that produced the simulations in the previous section can easily be modified for use on System 6.39. The parameters and their derivations are the same as well. Thus, the steady state may be found numerically. Figures 6.14 to 6.17 show the resulting network when the ratio of short filaments (0.8 p) to fixed long filaments (2.0 p) is increased from p = 0.5 to p = 100. In all cases, the steady state is reached within 10 minutes. The figures show that the fixed orientation has 160 Fixed Filament Model Figure 6.14: Simulation with fixed long filaments where p = 0.5. The minimum is 27% of the maximum. some effect on the final outcome. However, there is no significant change in orientation for p > 100. 161 Fixed Filament Model -4 -3 Figure 6.15: Simulation with fixed long filaments where p = l. The minimum is 41% of the maximum. Figure 6.16: Simulation with fixed long filaments where p = 10. The minimum is 75% of the maximum. 162 Fixed Filament Model in c ' CD CD E CO 1h 0 I ' I 1 ' 1 1 1 1 - 4 - 3 - 2 - 1 0 1 2 3 4 9 Figure 6.17: Simulation with fixed long filaments where p = 100. The minimum is 95% of the maximum. 163 Length Distribution Model When comparing these results with the two filament population, one sees alignment of short filaments occurring at higher ratios, p. We also find that the pattern that develops consists of a mix of aligned and unaligned filaments. Thus, fixed filaments may be a key requirement in obtaining different patterns either in vivo or in vitro. 3 Length Dis t r ibu t ion M o d e l The final extension of the two filament model is to consider a continuum of lengths. Rather than looking at 2n equations for n different lengths, we could instead look at two equations where the length, I, of a filament becomes an independent variable. In this manner we can test the effects of an entire length distribution. I develop the system of equations here, but leave their analysis for future work. I first consider the actions and interactions of free filaments, of length / and orientation 9 at time t, F(l,8,t). Free filaments may be lost when they associate with other free filaments. Specifically, free filaments with length / and orientation 6 may bind with free filaments of length I' at 9'. By letting P(l, 9,1', 9') be the probability that such filaments may bind and assuming that binding of free filaments occurs at the rate f3\, then free filament binding is described by Ft(l, 9, t) = -/?! T / P(l, 9,1', 6')F{1,9)F(l\ 9') dl' dd1, (6.40) J-TT J A where A is the domain of filament lengths. The product Fi(9)F(l', 9') is the interaction term from the Law of Mass Action. The double integral is in place because, one must consider all free filaments of length /' and orientation 9'. In a similar manner free filaments may also be lost when they associate with network filaments of length /' at orientation 9'. Once again P(l, 9,1', 9') gives the probability of such a configuration occurring. Because network filaments are fixed there is a different rate of binding, /?2. Thus, free filaments 164 Length Distribution Model lost to bindings with network filaments is described by •Ft(l,9,t) = -p2 r I P(l,e,l',9')Fl(9)N(l',9')dl'd9'. (6.41) J —IT J A Free filaments of length I and orientation 9 may be gained when a like network filament dissociates from the network. Assuming this occurs at the rate 7, the free filaments may grow as Ft(l,9,t)=7N(l,9). (6.42) Finally, free filaments with length I may rotationally diffuse at a rate directly depen-dent on that length, LL(1). Combining this fact with Equations 6.40 - 6.42 produces the following model for free filaments: Ft{l,9,t) = ri(l)Fee(l,9)-(3^(1,9) T [ P(l, 9,1', 8')F(l', 9') dl'd9' - p2F{l, 9) f I P(l, 6,1', 9')N{1', 9') dl' <W' + ~/N(l, 9). (6.43) J — ix J A Network filaments of length I and orientation 9 are gained through free filament binding when filaments of length I and any orientation 9' bind with filaments whose orientation is 9, no matter what length, I'. The populations that interact in this case are F(l, 9') and F(l', 9) at the rate px leading to Nt(l, 8, t) =pi T I P(l', 8,1,8')F(l', 8)F{1,8') dl' d8'. (6.44) Network filaments of length I and orientation 8 are also gained when free filaments of length / and any orientation 8' bind to network filaments of any length /' at orientation 9, i.e. Nt(l,8,t) = p2 T f P(l',8,l,8')N(l',8)F(l,8l)dl'd8'. (6.45) J —TV JA Finally, network filaments of length / and orientation 8 steadily dissociate at the rate 7. Therefore, network filaments are modeled as Nt{l,9,t) = px ^ f P(l',9,l,8')F(l',8)F(l,8')dl'd8' J —TV JA + p2 T [ P(l\ 8, l, 8')N(l', 9)F{1,8') dl' d8' - <yN{l, 9). (6.46) J — TT J A 165 Length Distribution Model Equations 6.43 and 6.46 describe the new interactions for an entire distribution of filament lengths. Equation 6.28 already computes the rotational diffusion, fi(l), for this situation. The only requirement left is to chose a functional form for P. Unfortunately, this form will most likely not make the integral terms become convolutions. Thus, this system of integro-PDE's becomes much harder to analyze, and the numerical results take a longer time to compute. However, this approach shows how the current methodology may be used to develop new models. 166 Chapter 7 A c t i n N e t w o r k E x p e r i m e n t s 1 Rheology of Polymer Networks In order to understand the experimental results of actin networks with cross-linkers, one must first have some knowledge of the mechanical properties of polymer solutions. I give a general overview of these mechanical properties. A more detailed account can be found in Ferry (1980). Polymers in a solution behave like a viscoelastic substance, having properties of both liquids and solids. Ideal solids follow Hooke's Law: the strain or displacement, 7, of a material is proportional to the applied stress or force, a, (see Figure 7.1), i.e. The constant of proportionality, G, is called the shear modulus (or modulus of rigidity) a = G 7 . (7.1) Y Figure 7.1: A solid with shear stress, cr, will deform with some relative displace-ment, 7. 167 Rheology of Polymer Networks and measures the resistance to deformation (or elasticity) of a material. For example, stainless steel has a shear modulus G = 75GPa, while dry Douglas-fir timber has a shear modulus G = 0.7GPa (Beer and Johnston, 1992). A solid is said to be elastic, if the strain disappears when an applied stress is removed and if the object returns to its original form. A fluid, however, cannot return to its original form after an applied stress, since the displacement will be permanent. Equation 7.1 has little meaning in this setting. Instead, the rate of strain, 7, is measured against the applied stress, a. For a Newtonian fluid, the rate of strain is proportional to the stress, i.e. a = 7 7 7 • (7.2) In this case, the constant of proportionality, 77, is called the viscosity and measures the liquid's resistance to motion. For example, water has a viscosity of r] = l c P , while low grade crude oil has a viscosity of about 77 = lOcP (Rouse, 1946). Because viscoelastic materials, as the name implies, have both fluid (or viscous) and solid (or elastic) prop-erties, the relation between stress and strain over time becomes important, for neither Equation 7.1 nor Equation 7.2 hold exactly. One way of testing the physical behavior of a viscoelastic substance is to apply a fixed strain to the material and measure the resulting stress. If the material is believed to be more like a solid, a sudden strain may be applied. For example, consider holding a rubber band in some over-extended position. If rubber were a perfect solid, then the rubber band would continue to exert a constant force in the direction of its original state. If it were a viscoelastic solid, the stress would eventually begin to relax and go to zero as time goes to infinity. This possible relaxation is a key component of the viscoelastic behavior. The rate of strain, strain, and stress for a viscoelastic substance are shown over time in Figure 7.2 and described by the relation, a(t) = G(t)7. (7.3) 168 Rheology of Polymer Networks J y Time Time Time Figure 7.2: The rate of strain, 7, the strain, 7 and the stress, a over time for sudden strain. The stress relaxation over time indicates how the substance differs from a solid. G(t) is called the relaxation modulus and corresponds to the shear modulus in Equa-tion 7.1. (These are equal only if the substance is a solid.) If the material is believed to be more like a liquid, a constant rate of strain may be applied for some period of time and then released as in Figure 7.3. For an example, consider egg whites. While the rate of strain is kept hxed, the stress remains constant due to the fluid properties of the material. After the rate of strain is removed and the substance is fixed, the stress will not immediately go to zero, but take some time as the "solid" components relax. Other experiments are based on applying a fixed stress and measuring the resulting strain over time as in Figure 7.4. Imagine applying some fixed force to a rubber band. If the rubber band is a true solid, it should stretch only up to a certain point. If it were a 169 Rheology of Polymer Networks y Time Time Time Figure 7.3: A constant rate of strain, 7 may be applied to a substance believed to be like a liquid and then removed whiled keeping the strain, 7, fixed. How slowly the stress, a, goes to zero gives a measure of how solid the substance is. pure liquid, the displacement would grow linearly, since 7 = constant in this case. The combination of these two effects is described by, 7 (t) = J{t)a . (7.4) J(t) is the called the creep compliance. Only when the substance is a perfectly elastic solid will J = 1/G. In general, J(t) ^ l/G(t) as they change differently over time. A stress may also be applied over some time period and then released. The resulting strain over time may then be used to classify the substance as a viscoelastic solid, such as a cross-linked polymer, as in Figure 7.5 or a viscoelastic liquid, such as an uncross-linked polymer, as in Figure 7.6. Note the two main differences in the time profiles. The first difference is the behavior near the end of the (significant) stress period. The strain in a 170 Rheology of Polymer Networks r i i i i i i Y Figure 7.4: A sudden stress, a, may be applied to a substance while the strain, 7, is measured over time. The growth of the strain indicates how the substance differs from a solid. solid will approach a plateau while the strain in a liquid will have a noticeable tendency to increase. The second difference is seen in how the strain behaves once the stress is removed. The strain in a solid will go to zero while the strain in a liquid will approach a nonzero limit. Time Y Time Figure 7.5: The time profiles of stress, o, and strain, 7, showing the creep for a viscoelastic solid. 171 Rheology of Polymer Networks a Time Y Time Figure 7.6: The t ime profiles of stress, cr, and strain, 7, showing the creep for a viscoelastic l iquid . Because the mechanics of viscoelastic materials depend greatly on time, a third type of experiment is used. In this experiment, (see Figure 7.7), a strain is periodically applied at some frequency u, so that 7 = 70 sin (tot) . (7.5) The rate of strain is then 7 = 70W cos (cot) . (7.6) If the viscoelastic behavior is linear, then the stress will also be sinusoidal, but out of Figure 7.7: A strain, 7, may be applied to a substance sinusoidally. 172 Rheology of Polymer Networks phase with the strain as shown in Figure 7.8. In this case, the stress satisfies o = (T 0 sin (wt + 8) , (7.7) where 8 measures how in phase the stress is with the strain. If 8 = 0, then the substance is a pure solid, but if 8 = TC/2, then the substance is a pure liquid. By applying the trigonometric angle addition formula to Equation 7.7 one finds that a = <To cos 8 sin (ait) + tr0 sin 8 cos(ujt) — j0 (G' sin(a;t) + G" cos(tot)) , (7.8) where G' = — cos 8 and G" = — sin 8 (7.9) 7o To G'(co) is called the shear storage modulus and is associated with the solid part of the substance. G"(u>) is called the shear loss modulus and is associated with the liquid part of the substance. Because the mechanical effects of a viscoelastic substance depend on time (as in Equation 7.3), G' and G" vary with to. Small u> relate to long times where fluid properties are usually seen, and large to relate to short times where the response is usually dictated by the solid component of the substance. G' and G" have a second representation G* called the complex modulus, G* = G' + iG". (7.10) Knowing and 8 is the equivalent of knowing G' and G" as shown in Figure 7.9. \G*\ gives an overall measure of stiffness or resistance for the substance at a particular frequency. Over time, polymer solutions typically have three different responses corresponding to the scale where changes occur. The first response to a stress or strain happens on the microscopic scale as individual polymers change their shape (e.g. bent polymers may straighten). After the polymers change shape, the next response is governed by how the polymers are linked via cross-linkers or entanglements, typically, a response associated 173 Results of Experiments Stress Strain Figure 7.8: Top: Strain can be applied in a sinusoidal fashion while measuring the resulting stress. Bottom: The sinusoidal response of the stress can be broken into two components, one in phase and one out of phase with the strain. The component in phase measures the "solid" portion of the material and the one out of phase measures the "liquid" portion of the material. with solids. The third response corresponds to links breaking so that polymers slide past one another and the resulting mechanical properties are more typical of a fluid. Thus, one expects different characteristics for different frequencies, LO. 2 Results of Experiments Numerous experiments have been performed on concentrations of actin in the presence and absence of cross-linkers and other actin binding agents. Combined, they show a rich diversity in actin network development. Here, I show the results of various experiments 174 Results of Experiments Figure 7.9: The storage modulus, G', and loss modulus, G", can be represented in the complex modulus, G*. One can go back and forth between the two by using the magnitude \G*\ and the phase lag, 8. and attempt to explain how the results may be interpreted in view of our models wherever possible. The experiments change one of three biological parameters that affect the parameters in our models and hence, the model predictions. To understand the consequences of changing the model parameters, one should recall the value of the homogeneous steady state (Equation 3.10) when L3\ = jM F N=M-F= and the instability condition (Equation 3.36) P2M + 7 ' P2M2 f32M + j tokl + K%< ( ^ ) M 2 k ( i - k ) , (7.11) (7.12) (7.13) Increasing the binding rate, decreases the concentration of free filaments, increases the concentration of network filaments, and (by increasing the right-hand side of Equation 7.13) makes bundling more likely. Increasing the unbinding rate, 7, has the opposite effect - increasing the concentration of free filaments, decreasing the concentration of network filaments, and making bundling less likely. The estimates for the model parameters made in Chapter 4 depended on the biological parameters as follows: 175 Results of Experiments • Increasing the cross-linker dissociation rate constant, with respect to the asso-ciation rate constant, k+, (or equivalently, increasing the dissociation equilibrium constant, Kd = k_/k+) decreases fa (as in Equation 4.9) and increases 7 (as in Figure 4.6). • Increasing the cross-linker concentration increases $ (as in Equation 4.9) and de-creases 7 (as in Figure 4.6). Thus, changes in the biological parameters have the following Consequences according to our model: 1. Increasing (or Kd) decreases the concentration of bound filaments with respect to free filaments. 2. Increasing (or Kd) may remove instabilities so that the resulting network is a gel. 3. Increasing the cross-linker concentration increases the concentration of bound fila-ments with respect to free filaments. 4. Increasing the cross-linker concentration makes instabilities more likely so that the filament network bundles. 2.1 Janmey et al. (1994) To begin to understand the viscoelasticity of actin solutions, let us first consider experi-ments by Janmey et al. (1994). They conducted rheology experiments on actin-gelsolin solutions using oscillatory plate rheometers. (The same type of procedures are used in other studies described below, unless indicated otherwise.) By varying the amount of gelsolin, a filament severing molecule, one can change the average length of actin fila-ments in a solution. The results shown in their paper were carefully verified in several independent laboratories. Because no cross-linkers were present in their experiments, 176 Results of Experiments our model does not apply to this situation. However, one may understand the results based on the theory of polymer physics. Longer filaments are able to bend more so that as filament length grows, filaments are more likely to become entangled. The degree of entanglement is reflected in the stiffness of the solution. lOOOh I I I I I i a 2 4 6 8 10 L(4i) Figure 7.10: The storage modulus, G', at u = 1 for 2mg/ml « 8 5 / / M actin with various concentrations of gelsolin (as in Figure 2 of Janmey et a/.(1994)). By increasing the ratio of gelsolin to actin, one decreases the average length, L. Figure 7.10 (modified from Figure 2 in Janmey et al. (1994)) shows how the storage modulus, & , is affected by the filament length for an 85 LLM actin solution at 1 rad/s. As the filament length increases, so does G' so that the solid component of the actin solution becomes stiffer. Thus, as the filament length grows, entanglements are more likely. Figure 7.11 (modified from Figure 3 in Janmey et al. (1994)) shows the storage and loss moduli over a range of frequencies, co, for two different average filament lengths of a 34 LLM actin solution. G' remains fairly constant over all frequencies for the long filaments (Figure 7.11a) meaning that the solid-component does not change stiffness over time. The slight increase in G" at low frequencies shows that the solution is more viscous at longer times. It also implies that the ratio, G"/G' (a measure of the phase shift, 8), increases over time so that the solution behaves more like a fluid. However, when 177 Results of Experiments (a) <3 100h 10 (b) 0.01 0.1 1 10 100 CO (rad/s) a. 10 -1-0.1- 0.1 1 10 100 co (rad/s) Figure 7.11: The storage modulus, G' (solid line), and loss modulus, G" (dashed line), over a range of frequencies, LJ, for 34 fiM actin (a) without or (b) with gelsolin (as in Figure 3 of Janmey et aZ.(1994)). The average length in (a) is greater than 5 \i while the average length in (b) is less than 1 / i . the ratio, G"/G', of long filaments is compared to short filaments, one sees that the long filament solution is much more solid than the short filament solution. The fact that both G' and G" decrease over time means that the short filament solution becomes less resistant as time increases. As seen in electron micrographs (Gittes et al., 1993), the reason for this large contrast is that long filaments may bend greatly, affecting the degree of entanglement. This is an important result for our model where all filaments were assumed to be rigid. Thus, unless the average filament length is short (on the scale of 1 LL or less) or a stiffening agent, such as tropomyosin (Table 1.1), is present, some discrepancies may arise with comparisons of our models. 2.2 Sato et al. (1987) Sato et al. (1987) showed the difference between uncross-linked networks and a-actinin cross-linked networks. In Figure 1 of their paper (see Figure 7.12), they show the dynamic response of a solution with 24 fiM F-actin with and without 1.6 \iM a-actinin. The fact that both G' and G" are greater (which in turn means that |G* | is greater) over all frequencies for the solution with cross-linkers means that the cross-linkers make the actin 178 Results of Experiments frequency (Hz) Figure 7.12: The storage modulus, G' (solid lines), and loss modulus, G" (dashed lines), for 24 /iM F-actin alone (thin lines) or with 1.6 uM a-actinin (thick lines) over a range of frequencies (as in Figure 1 of Sato et a/.(1987)). network more resistant (viewed either as a fluid or a solid). An increased resistance is not surprising, as our model predicts more network filaments with an increase in cross-linker concentration (Consequence 3). Looking at the ratio G"/G', and hence, the phase shift, 8, indicates that at short times (high frequencies), solutions with and without the cross-linker behave more like solids, and at long times (short frequencies) both solutions behave more like fluids. In Figure 2 of the paper by Sato et al. (1987) (replicated in Figure 7.13), the effects of a-actinin on G' are shown at two different frequencies when varying the temperature. At a frequency of 0.6 Hz (Figure 7.13a), increasing the temperature noticeably decreases G' in the presence of cross-linkers. Increasing the temperature, as shown in Meyer and Aebi (1990), increases the dissociation equilibrium constant, Kd = k_/k+, of a-actinin. Thus, in terms of our model, this situation corresponds to Consequence 1, where the ratio of bound to unbound filaments decreases. This agrees with the results because the solutions become less stiff. When compared to the solution with no cross-linkers, one concludes that cross-links dominate the mechanical properties of the solution at short 179 Results of Experiments 400 (a) 0 200 0 (b) 0 20 40 0 20 40 Temperature (C) Temperature (C) Figure 7.13: The storage modulus, G', for 24 uM of F-actin alone (thin lines) or with 1.6 pM a-actinin (thick lines) at (a) 0.6 Hz and (b) 6 x 10 - 4 Hz as a function of temperature (as in Figure 2 of Sato et a/.(1987)). times. At longer times (i.e. shorter frequencies such as 6 x 10 - 4 Hz), the cross-linker has little effect on the solution (Figure 7.13b) implying that all cross-links have been broken. In Figure 2 of the paper by Wachsstock et al. (1993) (as sketched in Figure 7.14), and 8 are shown for 15 pM actin alone, with 10 pM amoeba a-actinin, and with 10 pM chicken a-actinin. At all frequencies, the chicken a-actinin is much more resistant than the amoeba a-actinin which is more resistant than the pure actin solution. The association rate constant, k+, for both chicken a-actinin and amoeba a-actinin is about 1 pM~ls~l. However, the dissociation rate constant, for chicken a-actinin is 0.7 s - 1 , while = 5.2s - 1 for amoeba a-actinin. Thus, the increased resistance is explained by Consequence 1 of our model. In Figure 3 of the paper by Wachsstock et al. (1993) (as sketched in Figure 7.15), \G*\ and 8 are shown at two different frequencies for various concentrations of amoeba and chicken a-actinin. We can interpret the way that the chicken a-actinin affects the 2.3 Wachsstock et al. (1993) and (1994) 180 Results of Experiments 1000Q .0001 .001 .01 .1 1 10 Frequency (Hz) Figure 7.14: The magnitude of the complex modulus, and the phase shift, 6, for 15 LIM actin alone (dotted line), with l O p M amoeba a-actinin (dashed line) and with 10>M chicken a-actinin (solid line) (as in Figure 2 of Wachsstock et o/.(1993)). mechanical properties of the solution in the context of the model in Chapter 3. Conse-quence 3 implies that the resistance should always grow as the cross-linker concentration increases, explaining the increase in |G* | . Consequence 4 implies that before some thresh-old concentration is reached, the network behaves like a gel, explaining the drop in 6. After the threshold concentration is reached, the filaments will bundle, explaining the more fluid-like behavior of an increased 6. As the cross-linker concentration continues to increase, the effects of the geometry of the network will submit to the effects of the in-creased concentration of network filaments, so that the solution becomes more solid once again. The results for the amoeba a-actinin, on the other hand, are quite perplexing. Because of Consequence 2, we would not expect bundles present for a concentration of amoeba a-actinin, unless they were also present for the same concentration of chicken a-actinin. By also taking Consequence 4 into consideration, one would expect the tran-181 Results of Experiments 0.0003 Hz 0.19 Hz 10000 r 1000-100-*~ 10 -1 g 1 -— to .5 -0" .01 .1 1 10 11 .01 .1 1 10 100 alpha-actinin (\xM) Figure 7.15: Rheological measurements are recorded at 0.0003 Hz (on left) and 0.19 Hz (on right) as amoeba a-actinin (dashed lines) and chicken a-actinin (solid lines) are added to 15 \xM of actin (as in Figure 3 of Wachsstock et aL(1993)). The left most point for each graph indicates the measurements for a pure actin solution. sition point for bundles with amoeba a-actinin to occur at a greater concentration than with chicken a-actinin. This does not follow from the results of the experiment. In fact, electron micrographs (Figures 4 and 5 in Wachsstock et al. (1993)) reveal that amoeba a-actinin bundles filaments at lower concentrations than chicken a-actinin. Thus, our model fails to explain these results. However, in other experiments such as those by Tempel et al. (1996) (to be discussed later), the model is in perfect agreement for the effects of increased Thus, I suspect that there may be some other differences besides the dissociation rate constants for chicken and amoeba a-actinin, and that these factors are not accounted for in the model. In Figure 2 of the paper by Wachsstock et al. (1994) (as rendered in Figure 7.16), fur-ther viscoelastic results were shown for different concentrations of chicken a-actinin and 182 Results of Experiments 10000 1000 100] 1 _ 1 "a s_ to .5 Chicken Alpha-Actinin Amoeba Alpha-Actinin -1 1 1 1 1 1 1 1 1 \:' * * — 1 1 - * * 1 ' .—. .0001 .001 .01 .1 1 .0001 .001 .01 Frequency (Hz) 10 Figure 7.16: The magnitude of the complex modulus, and the phase shift, <5, for 15 pM actin alone (faint dotted line) with chicken a-actinin (on the left) in concentrations of 0.03 pM (solid line), 0.1 pM (dashed line) and 0.3 pM (dotted line) or with amoeba a-actinin in concentrations of 0.1 pM (solid line), 0.3 pM (dashed line) and 1.0 pM (dotted line) (as in Figure 2 of Wachsstock et a/.(1994)). (Note that the phase shift 6 for 1.0 pM amoeba a-actinin was even more erratic than the other concentrations of amoeba a-actinin and has been omitted for clarity.) amoeba a-actinin. The results for these experiments that tested three different concen-trations of a-actinin over a wide range of frequencies can be compared with the results in Figure 7.15 that tested a wide range of concentrations at two different frequencies. The figures have 12 points in common (six for \G*\ and six for 8) for each cross-linker tested. In the case of chicken a-actinin, all 12 points are in agreement. Also, the near horizontal response of \G*\ for chicken a-actinin agrees with experiments by Sato et al. (1987) (Figure 7.12) and earlier results by Wachsstock et al. (1993) (Figure 7.14). Thus, there is a high degree of confidence in these results. However, for the case of amoeba a-actinin, several disagreements occur. Figure 7.15 shows that > 10 dyn/cm2 for 183 Results of Experiments 1 LIM amoeba a-actinin at 0.0003 Hz, but Figure 7.16 shows < 3 dyn/cm2 at the same point. Figure 7.15 also shows that 8 > 1.2 for 0.1 \iM amoeba a-actinin at 0.0003 Hz, but the other experiment shows that 8 < 0.7 at the same point. These discrepancies suggest that further experiments with amoeba a-actinin should be conducted to verify the findings. The results shown in Figure 7.16 for chicken a-actinin agree with some predictions of our model and disagree with others. At the low concentration of a-actinin one would expect a gel to exist because of Consequence 4. This agrees with the results for 0.03 \iM because 8 = 0 indicates a solid substance. Consequence 4 also indicates that bundling occurs at greater concentrations so that a more fluid substance is seen as with 0.1 LIM and 0.3 LLM concentrations. However, because bundling has occurred for the 0.1 LLM solution, Consequence 3 implies that the 0.3 iiM solution should be more solid. This prediction is not borne out by the experiments. The results from Chapter 6, may explain this discrepancy, though. If the 0.1 LLM concentration is near the bundling threshold, the distribution of filament lengths may introduce a deviation from perfect alignment. As the cross-linker concentration increases, networks favor alignment to a greater degree. In this case, 0.3 \iM a-actinin would be more fluid-like than the 0.1 LIM as the experiments show. Unfortunately, our models cannot explain why the solution is more fluid at high frequencies and solid at low frequencies. 2.4 Goldman et al. (1997) Goldman et al. (1997) show the effects of increasing cross-linker concentration for a different cross-linker, filamin (see Figure 7.17). As Table 1.1 indicates, the main differ-ence between filamin and a-actinin is that filamin is a gelating agent and a-actinin is a bundling agent. Their physical differences are seen in their size. Filamin is a flexible, 160 nm, 540 kDa dimer, while a-actinin is a rigid, 40 nm, 200 kDa dimer. To compare the 184 Results of Experiments 10 1 (a) 0.1 (b) 0.1 (c) 0.1 0.01 G' (Pa) G"(Pa) _J L_ _I l _ 10 (a) 0.1 0.1 0.1 -4 -3 -2 -1 0 10 10 10 10 10 i o3 i o 2 io1 10° 10 0.01 Frequency (Hz) Figure 7.17: The storage modulus, G' (shown on the left), and loss modulus, G" (shown on the right) for 10 fiM alone (dashed line) or with filamin (solid line) (as in Figure 3 of Goldman et o/.(1997)). The concentration of filamin is 0.03 uM in (a), 0.1 LLM in (b) and 0.2 uM in (c). The plots show relative changes of 15% in (a), 60% in (b) and a 4-fold increase in (c). The average filament length is about 5.5 u. results of filamin concentrations in Figure 7.17 with those of a-actinin in Figures 7.15 and 7.16, one must consider the ratio, G"/G', in order to obtain trends for the phase shift 8. At the low filamin concentration (molar ratio of actimfilamin = 333), the slight decrease in G" indicates a slightly more solid solution than pure actin. This is very dif-ferent than low concentrations of a-actinin which result in a very solid substance. At the intermediate filamin concentration (molar ratio of actimfilamin = 100), the noticeable increase in G', indicates that the solution is much more solid than before (although it is 185 Results of Experiments still more fluid at high frequencies because G" > G'). Using the pure actin solution as a benchmark, one sees that the filamin concentration shows a much more solid response at high frequencies than the a-actinin solution. At the high filamin concentration (molar ratio of actimfilamin = 50), the greater increase in G' over the increase in G" indicates an even more solid solution. This reveals an even larger discrepancy between filamin and a-actinin. The differences between filamin and a-actinin may be explained in terms of our model and the physical differences between the cross-linkers. Although, no specific values of the dissociation equilibrium constant, Kd, are given for filamin, Goldman notes that Kd = k-/k+ < 1 as is the case with chicken a-actinin. Thus, our model would predict similar instability transitions for both cross-linkers. At low concentrations of cross-linkers where the solution is a gel, the network is more solid for a-actinin than filamin. This might be attributed to filamin being more flexible so that there is little difference between a network with few filamin cross-links and no filamin at all. Because filamin is a gelating agent, at concentrations where the model predicts instabilities to occur, filaments would align in an orthogonal pattern as opposed the parallel alignment forced by a-actinin. This difference is seen physically when comparing the electron micrographs of the two solutions (Figures 4 and 5 in Wachsstock et al. (1993) for a-actinin and Figure 6 in Goldman et al. (1997) for filamin). Thus, the filamin solutions remain more solid than the pure actin solutions while the a-actinin solutions behave more fluid-like. In order for our model to duplicate such results, the kernel would have to be changed to more closely describe the effects of the filamin cross-linker (Civelekoglu and Edelstein-Keshet, 1994). 2.5 Tempel et al. (1996) Rather than using different types of a-actinin, Tempel et al. (1996) vary the temperature to change the dissociation equilibrium constant, Kd = k-/k+, of a-actinin. A n increase 186 Results of Experiments 100 a* 10 1 0.1 (1) (2) 0.1 1 10 100 1000 t(s) Figure 7.18: The creep, J(t), for a solution containing 9.5 LLM actin and 0.2 \xM a-actinin at 5°C7 (solid line) and 25°C (dashed line) (as in Figure 1 of Tempel et a/.(1996)). The average filament length, L, is 22 p. The numbers describe different regimes: (1) internal chain conformation dynamics, (2) rubber plateau, and (3) transition into fluid-like behavior. in temperature is equivalent to an increase in /c_ (with respect to a constant k+) as discussed earlier (Sato et al., 1987). Figure 7.18 shows how two temperatures affect the creep, J(t), for a solution with 9.5 fiM actin and 0.2 \iM a-actinin. Although, G(t) ^ 1/J{t) for viscoelastic substances, it is reasonable to believe that more creep, J , implies less stiffness, G. Because the creep for the solution is smaller at the lower temperature than at the higher temperature, we can infer that the solution is stiffer at lower temperatures. This follows from Consequence 1 of our model and agrees with the findings by Sato et al. (1987). This may also clarify the effects of /c_ on the network that were unclear for the two types of a-actinin studied by Wachsstock et al. (1994). Figure 7.19 shows the response of G'N, the elastic plateau modulus (corresponding to the time or frequency where cross-linker dynamics effect the network the most), and the phase shift, 8 (at small values of 8, tan(<5) £3 8), as functions of temperature. At high temperatures, the solution is like a gel, showing no angular or spatial preferences and behaving like a solid. As the temperature is decreased, it reaches a point, T s , where the gel becomes much more fluid-like. This corresponds to the onset of bundling as predicted 187 Results of Experiments 10-o . i to 0.25 0.2KT 26 21 17 13 9 5 T(C) Figure 7.19: The elastic plateau modulus, G'N (measured at 3.5 x 10~ 3 Hz), and tan(<5) as a function of temperature for 9.5 uM actin alone (dashed line) and with 0.95 uM a-actinin (as in Figure 2 of Tempel et a/.(1996)). The average filament length is 22 u. Where the phase shift, 8, has a maximum is the apparent gel-point temperature denoted Tg. by Consequence 2. As the temperature is decreased further, the solution becomes more solid as predicted by Consequence 1. Thus, our model clearly depicts how /c_ effects the gel-sol transition. Figure 7.20 shows the response of G'N, to temperature changes for different cross-linker concentrations. The gel point, Tg decreases for smaller cross-finker concentrations. Our model predicts this outcome as Consequences 2 and 4 are balanced under these circumstances. Consequence 4 implies that decreasing the cross-linker concentration makes the instability condition less likely. Thus, in order to keep the instability condition satisfied, one decreases (or temperature) so that Consequence 2 applies to make instabilities more likely. In this manner, the model parameters are similarly balanced at the bundling transition as are the gel-point temperature and cross-linker concentration in 188 Results of Experiments Figure 7.20: The elastic plateau modulus, G'N (measured at 3.5 x 10~3 Hz) as a function of temperature for 9.5 \iM actin with 0.95 LIM a-actinin (tightly-dotted line), 0.48 \s,M a-actinin (medium-dotted line) and 0.2 LIM a-actinin (sparsely-dotted line) (as in Figure 3 of Tempel et a/.(1996)). The average filament length is 22 ii in all cases. The gel-point temperature, Tg, is denoted by the arrowheads. the experiments. The agreement between the model and the experiments is very exciting. 2.6 Furukawa and Fechheimer (1996) Furukawa and Fechheimer (1996) take a different approach to study filament bundling. They use polarized light scattering to measure the ratio of the Hv/Vv scattered light intensities for different filament lengths and different actin concentrations. A n isotropic solution typically has a value of Hv/Vv = 0.025, while a liquid crystalline solution (see Section 3.1) has a value of Hv/Vv > 0.3. Using this technique they can determine at what concentration a particular solution enters the liquid crystalline regime. They found that a solution whose average length is 0.6 u never enters this regime for concentrations under 70 LIM, a solution whose average length is 1.5 LI enters this state at a concentration near 60 LIM, and a solution whose average length is 4.9 LI enters this regime at a concentration near 50 LIM. This follows the trend outlined in Section 3.1. In Figure 7.21, the polarized light scattering experiments are used to study how 189 Polymer Physics 0.00 1.4 2.8 4.2 L(ji) Figure 7.21: The ratio of two scattered light intensities (Hv/Vv) is found for a 24 uM solution of actin with 2 uM 30 kDa protein while the gelsolin concentration is changed so that the average filament length, L , changes (as in Figure 4b of Furukawa and Fechheimer (1996)). filament length affects bundling in the presence of the Dictyostelium 30 kDa protein. As Figure 7.21 indicates, there is an optimal intermediate length for filament bundling when the cross-linker and actin concentrations are fixed. This is exactly what our model predicts. At short filament lengths, the solution remains isotropic. As the filament length grows, instabilities in space may occur. As it grows further, instabilities in orientation are added to form bundles. As the filament length grows even further, the instabilities in space are lost, so that alignment in orientation is still possible which gives some ordering but not to the same extent as "bundling". Although, the optimal bundling length is quite small for the 30 kDa protein, this may be due to a lower dissociation equilibrium constant, Kd, or the size of the cross-linker which is much smaller than cv-actinin. 3 Po lymer Physics In the previous section, I summarized some of the results found in the literature and tried to explain how they related to our model. In the cases where the model could not explain the results, I included my own speculations. However, there is an entire field, 190 Polymer Physics polymer physics (Flory, 1971; deGennes, 1979; Doi and Edwards, 1986), that use different techniques to analyze the same problems. In this section, I will briefly explain some of the theory, how the theory relates to the experiments and compare it to our own model. 3.1 Regimes for Solutions of Rod-like Polymers Solutions of rod-like polymers of length L , may be classified as being in one of four different regimes based on the concentration of the number of filaments in the solution, v (Doi and Edwards, 1986). In Section 3 of Chapter 4, two regimes were described, dilute and semi-dilute. In dilute solutions, neighboring polymers do not interfere with one another so that v < fi ~ • (7.14) In semi-dilute solutions, neighboring polymers hamper rotational movement. This regime is described by 1 L>I<V<V2K,—, (7.15) where d is the diameter of the polymer. As the solution becomes more concentrated, two more regimes are encountered. The third regime, the isotropic concentrated solution, has polymers orienting in the same direction as their neighbors, but with no alignment preference overall. The fourth regime, called a liquid crystalline solution (or anisotropic concentrated solution), has polymers aligning globally. The critical value of v between these concentrated solutions, v*, is of the same order as v2. Because vx and v2 depend on L , a given concentration of actin may be classified based on its filament length. In this case, L < L \ constitutes a dilute solution, L \ < L < L 2 constitutes a semi-dilute solution, and L > L 2 constitutes a concentrated solution. For example, the critical lengths for 15 iiM F-actin, where v = 24.4/L, 0.2/i, and L2 fa 5.5LI. (The calculation of L\ follows from Equation 7.14 as follows nr*h ~  L 2 < m *  L<02-  (7- 16) 191 Polymer Physics A similar calculation can be done on Equation 7.15 to find L 2 . ) Calculating the critical lengths shows two trends: (1) increasing L increases the "crowdedness", and (2) increas-ing the actin concentration decreases the critical lengths. It is important to note that these calculations assumed that all of the actin was incorporated in filaments and that the filaments are assumed to be rigid. This theory agrees with results by Janmey et al. (1994) and Furukawa and Fech-heimer (1996). Both Figures 7.10 and 7.11 show increased lengths lead to more resistant solutions. This agrees with the above calculations that show increased lengths lead to more "crowded" solutions. The theory also suggests why the long length solution is more solid than the short length solution (Figure 7.11). It may be because long filaments are in a different regime than the short filaments, in which case the long filament solu-tion may be concentrated, while the short filament solution may be dilute or semi-dilute. The theory also agrees with the scattered light experiments by Furukawa and Fechheimer (1996) where liquid crystalline solutions were seen at smaller concentrations for increased lengths. Because this theory is used to develop estimates for the diffusion rates in our model, it can be viewed as a tool in our model development. Besides giving estimates for the diffusion rates, it also can be used to give ranges of filament lengths or concentrations where the model is valid. For example, if the solution is concentrated, then the free filaments would be unable to rotate. In this case, including a diffusion term, the driving force for bundling, would be erroneous. 3.2 Percolation Theory for Network Elasticity Nossal (1988) considers how a dimer such as a-actinin or actin-binding protein (ABP) affects the network elasticity by using results derived from percolation theory. He simpli-fies the problem by assuming that only cross-links made by the cross-linker will affect the 192 Polymer Physics elasticity. In this case, all cross-linkers connect two filaments as shown in Figure 7.22. He further assumes that there are n filaments in the solution and each filament is made of r monomers. The factor that determines whether a gel is formed is the fraction of all monomers, m, that are bound by cross-linkers. Percolation theory says that when m exceeds a critical value, m c , a "gel cluster" will form. As m grows, more filaments bind to the network. The critical cross-link fraction is mc = -^—. (7.17) r — 1 If C is the number of cross-linkers, then m is proportional to C/nr. Equation 7.17 can then be solved for a critical cross-linker concentration, Cc: C 1 TIT — oc => Cc oc =n. (7.18) nr r — 1 r — 1 Thus, when the number of cross-linkers exceeds the number of filaments, clusters form. This is similar to the trend seen in the experiments and in our model. -Free Filament ^--Dangling Ends Cross-linker Strands Figure 7.22: An ideal polymer network. All cross-linkers only bind two filaments (or four strands). Only strands that lie between two cross-linkers affect the elasticity. For m >^ m c , the shear modulus, G, may be calculated as V m 0 J V0 193 Polymer Physics where J\f is the number of strands in the network, V0 is the volume of the unstressed network, kb is the Boltzmann constant T is the temperature and m 0 = K Using Equation 7.17, one finds that m 0 = 2mc. (7.20) Thus, one would expect the shear modulus to grow linearly with the cross-linker con-centration in the asymptotic limit as shown in Figure 7.23. Comparing this result with Figure 7.15 at low frequencies shows that the resistance in the chicken cv-actinin may fol-low this relation (although it is difficult to determine the value of G', and then compute whether the log-log plot produces a linear relation for large cross-linker concentrations). Cc 2CC Number of Cross-linkers Figure 7.23: After the number of cross-linkers begin to form clusters, Cc, the shear modulus will eventually begin to grow linearly. 3.3 Percolation Theory for Phase Transitions Percolation theory was also used by Tempel et al. (1996) to determine phase transitions for their temperature experiments (Figure 7.20). They consider the relation between the mesh size, £ (the average distance between filament crossovers) and the average distance, d, between cross-links. When d becomes smaller than £, phase separation occurs. Furthermore, d can be related to the actual fraction of cross-links formed, p. Increasing p decreases d, or as the fraction of cross-linkers that bind to filaments increase, the average distance between the cross-links decrease. As previously discussed, decreasing 194 Polymer Physics temperature, decreases the dissociation equilibrium constant, Kd = k_/k+, which then causes the fraction of bound cross-linkers to increase. Thus, decreasing temperature increases p. A phase diagram may then be developed as shown in Figure 7.24. Our model predicts a similar phase diagram as discussed in Section 2.5. Figure 7.24: The phase of the actin network depends on the cross-linker concen-tration and temperature. If one were to begin at an intermediate cross-linker concentration and high temperature (or equivalently, high and then de-crease the temperature (or equivalently, decrease one would first encounter gels, then micro-gels or clusters and then bundles. 3.4 Mode l Comparisons Perhaps the main shortcoming in our model when compared to the polymer theory is that we must assume that filaments are rigid (in order to define orientations and derive diffusion rates). As discussed earlier, this is only true for shorter filament lengths (< 1 Li as seen in vivo). In this case, the strong cross-links that form in an actin solution are due more to the cross-linkers than to filament entanglements. In contrast, the polymer theory stresses entanglements and crossover points to a much greater extent. This is particularly useful when considering filaments that are long (> 5 LL as seen in vitro) and able to bend. The percolation theory and our model do agree to some extent on when clusters, gels, Cross-linker Concentration 195 Polymer Physics or bundles form (Tempel et al., 1996). However, when considering network development over time, such as I have done in my simulations, the polymer theory has trouble. The conditions for networks are based on equilibrium thermodynamics, that are unable to give the time course for the construction of the network. The theory is also unable to determine a precise network distribution throughout space, as our model does, but the distributions that our models predict are currently oversimplified. However, this may change if factors such as density limiting dependences are included. 196 B i b l i o g r a p h y Adams, R. A . 1991, Calculus of Several Variables, Addison-Wesley, second edition edition Alberts, B., Bray, D., Lewis, J., Raff, M . , Roberts, K., and Watson, J. D. 1989, Molecular Biology of the Cell, Second Edition, Garland, New York Beer, F. P. and Johnston, E. R. 1992, Mechanics of Materials, McGraw Hil l Inc., second edition edition Bray, D. 1992, Cell Movements, Garland Publishing, Inc. Brigham, E. O. 1974, The Fast Fourier Transform, Prentice-Hall, Englewood Cliffs, N.J. Burden, R. L. and Faires, J. D. 1993, Numerical Analysis, PWS Publishing Company, fifth edition edition Burridge, K. and Feramisco, J. R. 1981, Non-muscle a-actinins are calcium-sensitive actin-binding proteins, Nature 294:565-567 Civelekoglu, G. and Edelstein-Keshet, L. 1994, Models for the formation of actin struc-tures, Bull Math Biol 56:587-616 Colombo, R., DalleDonne, I., and Milzani, A . 1993, a-actinin increases actin filament end concentration by inhibiting annealing, Journal of Molecular Biology 230:1151-1158 Coppin, C. M . and Leavis, P. 1992, Quantitation of liquid-crystaline ordering in f-actin solutions, Biophysical Journal 63:794-807 deGennes, P. G. 1979, Scaling Concepts in Polymer Physics, Cornell University Press Doi, M . and Edwards, S. F. 1986, The Theory of Polymer Dynamics, Clarendon Press Dufort, P. A . and Lumsden, C. J. 1993, Cellular automaton model of the actin cytoskele-ton, Cell Motility and the Cytoskeleton 25:87-104 Edelstein-Keshet, L. and Ermentrout, G. B. 1990, Models for contact-mediated pattern formation: cells that form parallel arrays, Journal of Mathematical Biology 29:33-58 Edelstein-Keshet, L. and Ermentrout, G. B. 1998, Models for the length distribution of actin filaments i : Simple polymerization and fragmentation acting alone, Bulletin for Mathematical Biology 60:449-476 Ermentrout, G. B. and Edelstein-Keshet, L. 1998, Models for the length distribution of actin filaments i i : Polymerization and fragmentation by gelsolin acting together, Bulletin for Mathematical Biology 60:477-504 Ferry, J. D. 1980, Viscoelastic Properties of Polymers, John Wiley & Sons, Inc., third edition edition Fesce, R., Benfenati, F., Greengard, P., and Valtorta, F. 1992, Effects of the neuronal phosphoprotein synaptin I on actin polymerization: Ii analytic interpretation of ki-netic curves, The Journal of Biological Chemistry 267(16):11289-11299 Flory, P. 1971, Principles of Polymer Chemistry, Cornell University Press Frieden, C. 1983, Polymerization of actin: Mechanism of the M g 2 + - induced process at pH 8 and 20°C, Proc. Natl. Acad. Sci., USA 80:6513-6517 Furukawa, R. and Fechheimer, M . 1996, Role of the dictyostelium 30 kda protein in actin bundle formation, Biochemistry 35:7224-7232 Furukawa, R., Kundra, R., and Fechheimer, M . 1993, Formation of liquid crystals from 197 Bibliography actin filaments, Biochemistry 32:12346-12352 Geigant, E. and Stoll, M . 1996, A non-local model for alignment of oriented particles, Bonn University research summary Gerald, C. F. and Wheatley, P. O. 1989, Applied Numerical Analysis, Addison-Wesley Publishing Company, fourth edition edition Gittes, F., Mickey, B., Nettleton, J., and Howard, J. 1993, Flexural rigidity of micro-tubules and actin filaments measured from thermal fluctuations in shape, Journal of Cell Biology 120:923-934 Giuliano, K. A . and Taylor, D. L. 1995, Measurement and manipulation of cytoskeletal dynamics in living cells, Current Opinion in Cell Biology 7:4-12 Goldmann, W. H. , Tempel, M . , Sprenger, L, Isenberg, G., and Ezzell, R. M . 1997, Viscoelasticity of actin-gelsolin networks in the presence of filamin, European Journal of Biochemistry 246:373-379 Hartwig, J. H. and Kwiatkowski, D. J. 1991, Actin binding proteins, Current Opinion in Cell Biology 3:87-97 Jacquez, J. A . 1972, Compartmental Analysis in Biology and Medicine, Elsevier Pub-lishing Company Janmey, P. A . , Hvidt, S., Josef Kas, Lerche, D., Maggs, A . , Sackmann, E., Schliwa, M . , and Stossel, T. P. 1994, The mechanical properties of actin gels, Journal of Biological Chemistry 269:32503-32513 Janmey, P. A . , Peetermans, J., Zaner, K. S., Stossel, T. P., and Tanaka, T. 1986, Struc-ture and mobility of actin filaments as measured by quasielectric light scattering, viscometry and electron microscopy, J. Biol. Chem. 261(18):8357-8362 Janson, L. W . and Taylor, D. L. 1994, Actin-crosslinking protein regulation of filament movement in motility assays: a theoretical model, Biophysical Journal 67:973-982 Jockusch, B. M . and Isenberg, G. 1981, Interaction of a-actinin and vinculin with actin: Opposite effects on filament network formation, Proceedings of the National Academy of Sciences, USA 78:3005-3009 Korn, E. D., Carlier, M . , and Pantaloni, D. 1987, Actin polymerization and A T P hy-drolysis, Science 238:638-644 Kot, M . 1992, Discrete-time traveling waves: ecological examples, Journal of Mathemat-ical Biology 30:413-436 Ladizhansky, K. 1994, Distribution of generalized aspect with applications to actin fibers and social interactions, Technical report, Weizmann Institute of Science, Rehovot, Israel, MSc thesis, Lee A Segel, supervisor Levin, S. A . and Segel, L. A . 1985, Pattern generation in space and aspect., SI AM Review 27:45-67 Lewis, M . A . 1994, Spatial coupling of plant and herbivore dynamics, Journal of Math-ematical Biology 45:277-312 Luby-Phelps, K. 1994, Physical properties of cytoplasm, Current Opinion in Cell Biology 6:3-9 Maciver, S. K., Wachsstock, D. H. , Schwarz, W. H. , and Pollard, T. D. 1991, The actin filament severing protein acophorin promotes the formation of rigid bundles of actin 198 Bibliography filaments crosslinked with a-actinin, Journal of Cell Biology 115:1621-1628 Meyer, R. K. and Aebi, U. 1990, Bundling of actin filaments by a-actinin depends on its molecular length, Journal of Cell Biology 110:2013-2024 Milzani, A . , DalleDonne, I., and Colombo, R. 1995, N-ethylmaleimide-modified actin filaments do not bundle in the presence of a-actinin, Biochemical Cell Biology 73:116-122 Mogilner, A . and Edelstein-Keshet, L. 1995, Selecting a common direction i . how ori-entational order can arise from simple contact responses between interacting cells, J Math Biol 33:619-660 Mogilner, A . and Edelstein-Keshet, L. 1996, Spatio-angular order in populations of self-aligning objects: formation of oriented patches, Physica D 89:346-367 Murray, J. D. 1989, Mathematical Biology, Springer Verlag, Berlin Nossal, R. 1988, On the elasticity of cytoskeletal networks, Biophysical Journal 53:349-359 Oster, G. F. 1994, Biophysics of Cell Motility, University of California Berkeley, Lecture Notes Otto, J. J. 1994, Actin-bundling proteins, Current Opinion in Cell Biology 6:105-109 Parfenov, V . N . , Davis, D. S., Pochukalina, G. N. , Sample, C. E., Bugaeva, E. A. , and Murti , K. G. 1995, Nuclear actin filaments and their topological changes in frog oocytes, Experimental Cell Research 217:385-394 Press, W . PL, Flannery, B. P., Teukolsky, S. A. , and Vetterling, W. T. 1988, Numerical Recipes in C: The Art of Scientific Computing, Cambridge University Press Ross, S. M . 1989, Introduction to Probability Models, Academic Press, Inc., fourth edition edition Rouse, H. 1946, Elementary Mechanics of Fluids, John Wiley & Sons Inc. Sato, M . , Pollard, W. H. , and Pollard, T. D. 1987, Dependence of the mechanical properties of actin/a-actinin gels on deformation rate, Nature 325:828-830 Spiros, A . and Edelstein-Keshet, L. 1998, Testing a model for the dynamics of actin structures with biological parameter values, Bulletin of Mathematical Biology 60:275-306 Strikwerda, J. C. 1989, Finite Difference Schemes and Partial Differential Equations, Wadsworth & Brooks Suzuki, A. , Maeda, T., and Ito, T. 1991, Formation of liquid crystalline phase of actin filament solutions and its dependence on filament length as studied by optical bire-fringence, Biophysical Journal 59:25-30 Taylor, K. A . and Taylor, D. W. 1994, Formation of two-dimensional complexes of f-actin and crosslinking proteins on lipid monolayers: Demonstration of unipolar a-actinin-f-actin crosslinking, Biophysical Journal 67:1976-1983 Tempel, M . , Isenberg, G., and Sackmann, E. 1996, Temperature-induced sol-gel tran-sition and microgel formation in a-actinin cross-linked actin networks: a rheological study, Physical Review E 54:1802-1810 Theriot, J. A . 1994, Actin filament dynamics in cell motility, in J. E. Estes and P. J. Higgins (eds.), Actin:Biophysics, Biochemistry, and Cell Biology, pp 133-145, Plenum 199 Bibliography Press Tobacman, L. S. and Korn, E. D. 1983, The kinetics of actin nucleation and polymer-ization, The journal of Biological Chemistry 258:3207-3214 Wachsstock, D. H., Schwarz, W. PL, and Pollard, T. D. 1993, Affinity of a-actinin for actin determines the structure and mechanical properties of actin filament gels, Biophysical Journal 65:205-214 Wachsstock, D. H., Schwarz, W. PL, and Pollard, T. D. 1994, Cross-linker dynamics determine the mechanical properties of actin gels, Biophysical Journal 66:801-809 Zaner, K. S. 1995, Physics of actin networks, i . rheology of semi-dilute f-actin, Biophysical Journal 68:1019-1026 Zigmond, S. H. 1993, Recent quantitative studies of actin filament turnover during cell locomotion, Cell Motil. Cytoskel. 25:309-316 200 

Cite

Citation Scheme:

        

Citations by CSL (citeproc-js)

Usage Statistics

Share

Embed

Customize your widget with the following options, then copy and paste the code below into the HTML of your page to embed this item in your website.
                        
                            <div id="ubcOpenCollectionsWidgetDisplay">
                            <script id="ubcOpenCollectionsWidget"
                            src="{[{embed.src}]}"
                            data-item="{[{embed.item}]}"
                            data-collection="{[{embed.collection}]}"
                            data-metadata="{[{embed.showMetadata}]}"
                            data-width="{[{embed.width}]}"
                            async >
                            </script>
                            </div>
                        
                    
IIIF logo Our image viewer uses the IIIF 2.0 standard. To load this item in other compatible viewers, use this url:
http://iiif.library.ubc.ca/presentation/dsp.831.1-0080002/manifest

Comment

Related Items