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The cofilin activity pathway in metastasizing mammary tumour cells Prosk, Erin 2009

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The Cofilin Activity Pathway in Metastasizing Mammary Tumour Cells by Erin Prosk  B.Sc., McGill University, 2007  A THESIS SUBMITTED IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF MASTER OF SCIENCE in The Faculty of Graduate Studies (Mathematics)  THE UNIVERSITY OF BRITISH COLUMBIA (Vancouver) August 2009 c Erin Prosk 2009  Abstract The activity of cofilin has been identified as a critical determinant of the metastatic potential of carcinoma cells in vivo [15, 23]. The burst cofilin-mediated barbed end production following stimulation of a cancer cell with EGF is not yet completely understood [7, 24]. This motivates the use of mathematical models to test experimental hypotheses and propose areas for future experimental consideration.  In this thesis, I outline the initial temporal models of the cofilin activity pathway in metastasizing mammary tumour cells developed by myself and my supervisor Leah Edelstein-Keshet. This work results from a collaboration with experimentalist Dr. John Condeelis (Albert Einstein College of Medicine of Yeshiva University). The project is hierarchical, building from a reduced model of cofilin-barbed end interaction (Chapter 2), to include distinct cofilin forms (Chapter 3) and compartmental considerations (Chapter 4). In each model, we investigate essential mechanisms of the cofilin pathway required to reproduce the barbed end peak observed in experiment. The models presented in Chapters 2-4 represent the initial step in the modeling analysis of the cofilin activity pathway. The work serves to validate current hypotheses about the cofilin activity pathway and identify important interactions and considerations for future experimental and theoretical development.  ii  Table of Contents Abstract  . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .  ii  Table of Contents . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .  iii  List of Tables  . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .  v  List of Figures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .  vi  Acknowledgements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .  vii  1 Introduction . . . . . . . . . . . 1.1 Cell Motility . . . . . . . . . 1.2 Review of Actin Dynamics . 1.2.1 Role of Cofilin . . . . 1.3 Review of Experimental Data  . . . .  . . . . .  . . . . .  . . . . .  . . . . .  . . . . .  . . . . .  . . . . .  . . . . .  . . . . .  . . . . .  . . . . .  . . . . .  . . . . .  . . . . .  . . . . .  . . . . .  . . . . .  . . . . .  . . . . .  . . . . .  . . . . .  . . . . .  . . . . .  . . . . .  . . . . .  . . . . .  . . . . .  1 1 3 4 7  2 Reduced Model . . . . . . . . . . . . . . . . . . . . . . 2.1 Motivation . . . . . . . . . . . . . . . . . . . . . . . 2.2 Barbed End Model . . . . . . . . . . . . . . . . . . 2.3 Cofilin Barbed End Model Details . . . . . . . . . . 2.3.1 Cofilin Model Assumptions . . . . . . . . . . 2.4 Linear Filament Severing . . . . . . . . . . . . . . . 2.4.1 Analysis of Reduced Linear Model . . . . . . 2.4.2 Model Scaling and Parameter Analysis . . . 2.4.3 Simulation of Reduced Linear Model . . . . 2.4.4 Time-Dependent Stimulation . . . . . . . . . 2.5 Nonlinear Filament Severing . . . . . . . . . . . . . 2.5.1 Simulation of Reduced Nonlinear Model . . . 2.5.2 Nonlinear Filament Severing with Saturation  . . . . . . . . . . . . .  . . . . . . . . . . . . .  . . . . . . . . . . . . .  . . . . . . . . . . . . .  . . . . . . . . . . . . .  . . . . . . . . . . . . .  . . . . . . . . . . . . .  . . . . . . . . . . . . .  . . . . . . . . . . . . .  . . . . . . . . . . . . .  . . . . . . . . . . . . .  . . . . . . . . . . . . .  . . . . . . . . . . . . .  . . . . . . . . . . . . .  . . . . . . . . . . . . .  . . . . . . . . . . . . .  11 11 11 12 13 15 15 17 17 17 21 23 26  3 Well-mixed Cofilin Model . . . . . . . . . . . . 3.1 Introduction . . . . . . . . . . . . . . . . . . . 3.1.1 Assumptions . . . . . . . . . . . . . . . 3.2 Model Development . . . . . . . . . . . . . . . 3.3 Full Normalized Model . . . . . . . . . . . . . 3.4 Model Parameter Analysis . . . . . . . . . . . 3.4.1 PLC and PIP2 Parameters . . . . . . . 3.4.2 Cofilin Data Review . . . . . . . . . . . 3.4.3 Cofilin Model Parameter Analysis . . . 3.5 Summary of Model Equations and Parameters 3.6 Simulations of Well-Mixed Model . . . . . . .  . . . . . . . . . . .  . . . . . . . . . . .  . . . . . . . . . . .  . . . . . . . . . . .  . . . . . . . . . . .  . . . . . . . . . . .  . . . . . . . . . . .  . . . . . . . . . . .  . . . . . . . . . . .  . . . . . . . . . . .  . . . . . . . . . . .  . . . . . . . . . . .  . . . . . . . . . . .  . . . . . . . . . . .  . . . . . . . . . . .  . . . . . . . . . . .  28 28 30 31 39 40 40 43 44 49 50  . . . . . . . . . . .  . . . . .  . . . . . . . . . . .  . . . . . . . . . . .  iii  Table of Contents 4 Cofilin Compartmental Model . . . . . . . . . . . . 4.1 Introduction . . . . . . . . . . . . . . . . . . . . . 4.1.1 General Assumptions . . . . . . . . . . . . 4.2 Model Details . . . . . . . . . . . . . . . . . . . . 4.3 Model Equations . . . . . . . . . . . . . . . . . . . 4.4 Compartmental Model Summary . . . . . . . . . . 4.5 Model Parameter Analysis . . . . . . . . . . . . . 4.5.1 Steady State Analysis . . . . . . . . . . . . 4.5.2 Model Parameter Estimation . . . . . . . . 4.5.3 Distinct Transition Rates in Compartments 4.6 Further Proposed Simplifying Assumptions . . . . 4.7 Reduced Model Summary . . . . . . . . . . . . . . 4.8 Simulations of Revised Compartmental Model . . 4.8.1 Discussion of Compartmental Model . . . .  . . . . . . . . . . . . . .  . . . . . . . . . . . . . .  . . . . . . . . . . . . . .  . . . . . . . . . . . . . .  . . . . . . . . . . . . . .  . . . . . . . . . . . . . .  . . . . . . . . . . . . . .  . . . . . . . . . . . . . .  . . . . . . . . . . . . . .  . . . . . . . . . . . . . .  . . . . . . . . . . . . . .  . . . . . . . . . . . . . .  . . . . . . . . . . . . . .  . . . . . . . . . . . . . .  . . . . . . . . . . . . . .  . . . . . . . . . . . . . .  . . . . . . . . . . . . . .  55 55 58 59 62 66 66 67 69 70 71 73 75 75  5 Discussion . . . . . . . . . . . . . . . . . . . . 5.1 Discussion of Temporal Model Results . . . 5.2 Proposed Future Considerations . . . . . . 5.2.1 Issues for Experimental Investigation 5.2.2 Future Modeling Work . . . . . . . 5.2.3 Other Perspectives . . . . . . . . .  . . . . . .  . . . . . .  . . . . . .  . . . . . .  . . . . . .  . . . . . .  . . . . . .  . . . . . .  . . . . . .  . . . . . .  . . . . . .  . . . . . .  . . . . . .  . . . . . .  . . . . . .  . . . . . .  . . . . . .  80 80 81 81 82 83  Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .  84  Appendix . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .  88  . . . . . . . . . . . . . . . . . . . . . . .  iv  List of Tables 2.1 2.2  Linear reduced model parameters . . . . . . . . . . . . . . . . . . . . . . . . . . . Nonlinear reduced model parameters . . . . . . . . . . . . . . . . . . . . . . . . .  18 24  3.1 3.2 3.3  Definitions of cofilin pools for the well-mixed model . . . . . . . . . . . . . . . . . Table 3.1 reprinted . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Parameter estimates for the well-mixed model . . . . . . . . . . . . . . . . . . . .  29 49 49  4.1 4.2 4.3  Definitions of cofilin forms for the compartmental model . . . . . . . . . . . . . . Table 4.1 repeated with volume calculations . . . . . . . . . . . . . . . . . . . . . Parameter estimates for the revised compartmental cofilin model . . . . . . . . .  56 73 74  v  List of Figures 1.1 1.2 1.3 1.4 1.5  The common cofilin activity cycle in invasive cells and inflammatory cells Experimental data of PLC and PIP2 dynamics . . . . . . . . . . . . . . . Increase in filament-bound cofilin . . . . . . . . . . . . . . . . . . . . . . . Relative increase in barbed end density in the leading edge . . . . . . . . Relative increase in barbed ends: control vs PLC-inhibited . . . . . . . .  . . . . .  . . . . .  . . . . .  . . . . .  6 7 8 9 10  2.1 2.2 2.3 2.4 2.5 2.6 2.7  Results of reduced linear model with step function stimulation rate . . Phase space linear model results with step stimulation . . . . . . . . . Results of reduced linear model with linear stimulation rate . . . . . . Phase space linear results with linear stimulation . . . . . . . . . . . . Simulation results for reduced nonlinear model with linear stimulation Phase space results for nonlinear model with linear stimulation . . . . Results of nonlinear saturated model in parameter space . . . . . . . .  . . . . . . .  . . . . . . .  . . . . . . .  . . . . . . .  . . . . . . .  . . . . . . .  18 19 19 20 24 25 27  3.1 3.2 3.3 3.4 3.5 3.6 3.7 3.8 3.9 3.10 3.11  Schematic of well-mixed temporal model of cofilin pathway . . . PLC fit of model parameters . . . . . . . . . . . . . . . . . . . . PIP2 fit of model parameters . . . . . . . . . . . . . . . . . . . . Simulation results for two PIP2 hypotheses . . . . . . . . . . . . PLC and PIP2 simulation results for two cases of PIP2 reduction Cofilin pool simulation results for 60% reduction of PIP2 . . . . . Close up of low-level cofilin results for 60% reduction of PIP2 . . Barbed end simulation results for 60% reduction of PIP2 . . . . . Cofilin pool simulation results for 95% reduction of PIP2 . . . . . Close up of low-level cofilin results for 95% reduction of PIP2 . . Barbed end simulation results for 95% reduction of PIP2 . . . . .  . . . . . . . . . . .  . . . . . . . . . . .  . . . . . . . . . . .  . . . . . . . . . . .  . . . . . . . . . . .  . . . . . . . . . . .  . . . . . . . . . . .  . . . . . . . . . . .  29 41 42 42 51 51 52 52 53 53 54  4.1 4.2 4.3 4.4 4.5  Schematic of temporal compartmental model . . . . . . . . . . . . . Schematic of revised cofilin compartmental model . . . . . . . . . . . Cofilin results of the compartmental model in relative concentrations Cofilin results of compartmental model in molecular fractions . . . . Barbed end results for the compartmental model . . . . . . . . . . .  . . . . .  . . . . .  . . . . .  . . . . .  . . . . .  . . . . .  . . . . .  57 72 77 78 79  . . . . . . . . . . .  vi  Acknowledgements This work is part of a very wonderful experience in graduate studies at UBC. I would like to thank the Department of Mathematics and the Institute of Applied Math for the incredible support, amazing learning opportunities, and stimulating working environment. I appreciate the many interesting and helpful discussions with members of the Math Biology group, visitors and other colleagues, including Sasha Jilkine, Stan Maree, and Jacco van Rheenen. Thank you to our collaborator John Condeelis for his invaluable contributions to the project. I would like to thank my research supervisor Leah Edelstein-Keshet for her unwavering support, enthusiasm and patience. Thank you to the IGTC Math Biology Program for the financial support to make my graduate experience possible.  vii  Chapter 1  Introduction 1.1  Cell Motility  Directed cell motility is a fundamental process in many biological and pathological contexts. In human physiology, cell movement encompasses such critical processes as cellular differentiation during embryogenesis, chemotaxis of immune cells, fibroblast migration during wound healing, and invasion and metastatic behaviour of tumour cells. Each of these processes demands the ability of a cell to detect and respond to external signals quickly and with relative precision.  Playing such a critical role in the maintainence of biological and physiological well-being, cell motility has been stationed at the forefront of both experimental and theoretical study for many years. However, much of this work has been focused on the motility of biological systems, chemotaxis of amoeboid Dictyostelium cells and fish keratocytes. These cell types, and their mechanism of movement are inherently different than the human physiological counterparts of which a surprisingly small number of systems have been studied extensively.  Most surprisingly, theoretical modeling of the migration and invasion of cancer, our population’s leading cause of death, has seen little attention until recently. Consequently, the cellular mechanisms causing and driving the progression of this dangerous disease are very poorly understood. Modern advancements in fluorescence microscopy imaging and parallel work in the analysis of genetic networks which determine metastatic potential of tumour cells have shone light onto the cellular pathways important for metastatic behaviour. These recent experimental discoveries demand theoretical analysis to expand on previous cell motility theories in light of the experimentally predicted similarities and distinct differences of cancer cell migration. Such understanding will be critical to determine the important cellular targets to prevent invasive  1  1.1. Cell Motility migration of cancerous cells.  General cell movement occurs by fine coordination of several distinct and important steps. Cells initiate protrusion at the front in response to an external stimulus. They attach to available substrate at the site of protrusion before contracting at the back and releasing their trailing edge. This process is repeated as the cell moves in the direction of protrusion toward an external stimulus. It is a fundamental sequence of events consistent across the wide range of motile cells. In each case, the initial step in the motility cycle requires the sensing of chemotactic signals by receptors on the surface of the cell. The activation is relayed to the interior of the cell, kicking off a complex signaling pathway. Activity culminates in polymerization of new actin at the front of the cell, generating a protrusive force to extend the membrane in the direction of motion. In this manner, the activity of the signaling pathway determines the directionality of cell movement by designating the location of the initial protrusion.  The goal of this thesis is to examine one such signaling pathway, the cofilin activity cycle, which has been recently identified as a key component of the metastatic phenotype of mammary tumour cells [15, 29, 32]. This work is the result of a collaboration with Dr. John Condeelis, an experimental cell biologist, professor and co-chair of the Department of Anatomy and Structural Biology at the Albert Einstein College of Medicine of Yeshiva University. His research lab has produced much of the recent experimental literature of breast cancer study in vivo, and supports the important role of the cofilin activity cycle.  The inherent complexity of cancer cell study in vivo limits the extent of experimental understanding. Discussions with John Condeelis identified the importance of deciphering an early peak of barbed end density produced by the stimulated activity cycle and critical to the directed migration of cancer cells. The current experimental shortcomings motivate our initial modeling efforts. We focus initially on reproducing the barbed end peak by examining the underlying cofilin interactions in a simplified, temporal framework. We hope to gain insight into the important interactions within the cofilin activity cycle, however, determining model parameters is a major challenge toward accurate representation of the system. This thesis outlines the important initial step in the analysis of the primary regulators of cancer cell metastasis. The work  2  1.2. Review of Actin Dynamics described here helps to simplify the complex spatiotemporal system and provide a foundation for later models. In the next section, I will motivate our recent modeling efforts by describing some experimental work from the Condeelis Lab.  1.2  Review of Actin Dynamics  Actin exists in the cell in two very closely linked forms. As long, thin filaments, actin comprises the primary component of the cell’s cytoskeleton. The filaments are crosslinked into a network of structural scaffolding determining cell shape and maintaining mechanical properties, internal force and resistance within the cell.  The cytoskeleton and each individual filament are dynamic structures constantly polymerizing and depolymerizing via addition or loss of the monomer form of the protein, G-actin. G-actin diffuses freely in the cell and associates to the nucleotide ATP in the cytosol which hydrolyzes over time to ADP. Actin filaments are polar structures with distinctly higher monomer addition rates and an affinity to ATP-bound G-actin at their barbed ends, and increased rate of monomer dissociation of ADP-bound G-actin at the pointed ends. In this manner, polymerizing actin filaments will exhibit a newly formed cap of ATP-actin at the free ”barbed end” with a tail of older ADP-bound monomers trailing to the disassembling ”pointed end”. An extensive analysis of the dynamics of the actin forms and associated actin binding proteins has been studied in Mogilner and Edelstein-Keshet (2002) [21].  The dynamics of the actin cytoskeleton are controlled by a complex network of signalling pathways in the cell. The activity of these networks can be spatially separated, allowing a cell to polarize in response to an external signal. Signalling molecules such as phosphoinositides and Rho proteins can influence the activity of local actin binding proteins (ABPs) which up or downregulate actin polymerization in a local region of the cell. Some ABPs, which upregulation actin, do so by creating additional polymerizing barbed ends. The enhanced polymerization generates force against the membrane and causes the cell to protrude at the site of activity.  The cell has three general mechanisms to create new free barbed ends: (1) uncapping of inhibited barbed ends by release of capping protein or release of bound gelsolin [18, 30], (2) nu3  1.2. Review of Actin Dynamics cleation of new filaments by the Arp2/3 complex [26], or (3) severing of existing older filaments by cofilin [7].  The activity of these proteins is tightly regulated in a cell to maintain dynamic stability of the actin cytoskeleton. The regulation of each protein can occur via pathways which are strongly interconnected, or by independent mechanisms, depending on the cell type studied and on the system of chemoattractant sensing within the cell. Here we study the regulation of just one such protein, the control of cofilin activity in the rat MTLn3 mammary adenocarcinoma cell line, an experimental model for the study of breast cancer cells.  1.2.1  Role of Cofilin  As cancer develops in the body, carcinoma cells undergo many stages of mutations. Early cell mutations increase proliferation at the cancer site, creating tumours. As the disease progresses, later mutations promote the abnormal migration of these normally non-motile cells, which can metastasize, or invade blood vessels causing the disease to spread to other tissues. The activity status of cofilin has been shown to be distinctly different between strictly proliferating nonmotile tumour cells and those which exhibit metastatic potential [24, 32]. Furthermore, recent work in vivo has identified that local cofilin activity is necessary for directional sensing and determines direction of cell motility [15, 23].  Cofilin binding to actin filaments occurs via its Ser-3 binding site. Under certain cellular conditions, the filament-bound cofilin can instill sufficient force on the filament to create a break. Cofilin associates to older segments of actin filaments where monomers are ADP-associated. This means that unlike binding and nucleation activity of the Arp2/3 complex, filament severing by cofilin is not restricted to the newly assembled filament cap, and does not require filaments to be actively polymerizing at all. Cofilin can bind and sever a capped mother filament, creating a new barbed end site for polymerizing of the daughter filament. This introduces the possibility of synergistic interactions between cofilin and Arp2/3 activity whereby the recently severed and polymerizing filament tips produced by cofilin are available for Arp2/3 nucleation [13].  Cofilin has been shown in vitro to bind filaments cooperatively [1, 3, 11]. This behaviour  4  1.2. Review of Actin Dynamics is yet to be proven in living cells. However, cooperative effects are expected to occur due to the increased torsional flexibility of actin filaments upon cofilin binding [27]. It is strongly hypothesized by experimentalists that between 5 and 7 bound cofilin molecules are required in vivo to generate sufficient force to create a break (informal discussions with J. Condeelis and J. van Rheenen). This is supported by studies that examine structural force dynamics of cofilin binding [1, 27].  Inactivation of cofilin occurs by inhibition of the Ser-3 site, preventing binding to actin filaments and any consequent severing activity [22]. Cellular cofilin binds both G-actin monomers, and phosphate molecules at the same Ser-3 site with high affinity [28, 31]. Furthermore, cofilin is known to bind PIP2 at the membrane of a cell. Structural studies have shown the PIP2 -binding site to overlap with the filament binding Ser-3 site [16]. In this manner, cellular cofilin can be inactivated, that is, its filament-severing activity is inhibited, by monomer-binding, phosphorylation, and PIP2 -binding at the membrane. These inactive forms of cofilin are relatively stable in a resting cell. In general, reactivation of cofilin requires a recycling process involving cellular complexes to strip monomers (SSH or CIN), facilitate dephosphorylation (LIM kinase) or cleave PIP2 (PLC-γ).  Gradient sensing and chemotaxis of tumour cells is hypothesized to occur via a local excitation, global inhibition (LEGI) model [23, 25, 34]. Local activation of cofilin activity has been shown to occur by a PLC-mediated release of PIP2 -bound cofilin at the membrane of the cell [33], and is not spatially or temporally correlated to dephosphorylation [31]. Upon stimulation with chemoattractant, epidermal growth factor, or EGF, LIM kinase activity increases globally in the cell, increasing the phosphorylation and deactivation of cofilin [31]. Such a LEGI model facilitates a tight control of protrusion sites and allows the cell to accomplish precise directional sensing.  The cofilin activity pathway of metastasizing tumour cells is well motivated and explained in the recent review by van Rheenen et. al (2009) [32]. We use the qualitative structure described in Figure 1.1 together with data from recent experimental work (outlined in the Section 1.3) to quantify cellular interactions and underlying mechanisms of the observed burst of barbed end  5  1.2. Review of Actin Dynamics density. These initial simple models provide a framework on which to extend further work in the field.  Figure 1.1: Cofilin molecules cycle through three compartments in the cell; the cytosol, the actin, and the PM compartments. Cofilin at the membrane (PM compartment) initially translocates to the F-actin compartment upon EGF mediated PIP2 reduction. Cofilin binds and severs actin filaments resulting in filaments with free barbed ends, and cofilinG-actin complex. The cofilinGactin complex cannot bind actin or PM, and therefore diffuses to the cytosol compartment. In the cytosol compartment, cofilin is phosphorylated by LIM-kinase (LIMK), resulting in the release of cofilin from the cofilinG-actin complex. Upon cofilin dephosphorylation by SSH, cofilin can reenter the membrane (PM) or F-actin compartment, starting a new cycle. The cycling of cofilin through the three compartments increases free barbed ends, resulting in newly polymerized actin filaments. The Arp2/3 complex prefers to bind to these newly formed actin filaments, which amplifies the cofilin-induced actin polymerization, resulting in protrusion formation [32, 33]. Reprinted with permission from J van Rheenen: EGF-induced PIP2 hydrolysis releases and activates cofilin locally in carcinoma cells. Journal of Cell Biology 179(6):1247-59 (2007).  6  1.3. Review of Experimental Data  1.3  Review of Experimental Data  Recent work has identified and quantified the impact of cofilin activity in the rat MTLn3 carcinoma cell line. This cell type has been chosen for study due to the PLC-dependency of cofilin-induced barbed end formation, protrusion and chemotaxis [23], and the correlation of cofilin activity with the metastatic potential of the cell [34].  Experimental work utilizing recent developments in multiphoton fluorescence imaging and MTLn3 cell properties has demonstrated the critical role of the cofilin pathway in invasive tumour cells in vivo. The cycle initiates via an increase in activity of PLC-γ in the membrane of the cell upon EGF stimulation [24]. This induces a rapid and significant reduction of PIP2 into diacylglycerol (DAG) and inositoltrisphosphate (IP3 ) in the membrane.  Cofilin has been shown to be in rapid equilibrium with PIP2 in a strong and stable configuration at the membrane of a cell. This relatively high fraction of cellular cofilin is reduced significantly upon EGF stimulation. The experimental time profiles of these phenomena are shown in Figure 1.2. We use these to fit PLC and PIP2 dynamics of our models described in Chapters 3 and 4 of this thesis.  Figure 1.2: Left: Dynamics of PLC from Figure 2 A of Mouneimne (2004) [24]. Right: Dynamics of PIP2 from Figure 1 A of van Rheennen (2007) [33]. Reprinted with permission from G Mouneimne: Phospholipase C and cofilin are required for carcinoma cell directionality in response to EGF stimulation. Journal of Cell Biology 166(5):697-708 (2004), and J van Rheenen: EGF-induced PIP2 hydrolysis releases and activates cofilin locally in carcinoma cells. Journal of Cell Biology 179(6):1247-59 (2007).  7  1.3. Review of Experimental Data Until recent developments in microscopy, cofilin dynamics were studied in vitro where cellular interactions and pathway dynamics may be significantly different. Consequently, though the dynamics we study here are strongly supported by experimental studies [32], the collection of quantified temporal data of cofilin in its various cellular forms is less complete. We are cautious of using the extensive in vitro library of work to validate our model. Much of the study of filament binding dynamics, especially the extensive analysis of cofilin binding and severing actin filaments has been under in vitro conditions [1, 4, 11, 17].  Instead, we focus on the filament binding data described in van Rheenen (2007) where cofilin is released from the membrane upon EGF stimulation, and binds rapidly to membrane-associated filaments (< 200nm from the membrane) [33]. We work under the assumption motivated by data in van Rheenen (2007), whereby filament-bound cofilin increases less than two-fold following stimulation with EGF. This is shown in Figure 1.3.  Figure 1.3: Relative increase in filament-bound cofilin at 0s and 60s following stimulation with EGF, from Figure 5 A of van Rheennen (2007) [33]. Reprinted with permission from J van Rheenen EGF-induced PIP2 hydrolysis releases and activates cofilin locally in carcinoma cells. Journal of Cell Biology 179(6):1247-59 (2007).  As described in previous sections, this relatively small fold increase in cofilin activity (in the form of bound cofilin) induces an amplification of barbed end production in the cell. Within 60s following stimulation with EGF, the relative number of free polymerizing barbed ends in the cell increases to a magnitude up to a 12-fold greater than the barbed end density at rest [24]. This increase is highly dependent on experimental conditions such as temperature, and pH changes induced by the activity of other signaling networks in the cell, The increase in barbed 8  1.3. Review of Experimental Data end density is stated as an average of 2.7 and 3.2 fold increases in other studies [15, 19]. Within a simple modeling framework, we attempt to reproduce these quantified barbed end amplifications. An example of maximal barbed end amplification, for optimal cellular conditions is shown in Figure 1.4, from Mouneimne (2004) [24].  The early peak of barbed end density at 60s following stimulation with EGF has been shown to be a product of PLC-mediated cofilin activity in the cell. The later peak, at 180s after stimulation, is dependent on the Arp2/3 pathway regulated by PI3 kinase (PI3K) in both amoeboid D. discoideum and carcinoma cells [8]. Figure 1.5 exhibits this phenomenon by quantifying barbed ends, under two conditions: control vs PLC-inhibited. The early peak of barbed ends is cofilin dependent and is sufficient to determine direction of cell motility in carcinoma cells. This motivates the focus our attention on the analysis of the cofilin pathway and resulting peak of barbed end density 60s following stimulation with EGF [15].  Figure 1.4: Relative increase in barbed end density by quantifying ATP-actin fluorescence increase at the leading edge of the cell, from Figure 1 C of Mouneimne (2004) [24]. Reprinted with permission from G Mouneimne: Phospholipase C and cofilin are required for carcinoma cell directionality in response to EGF stimulation. Journal of Cell Biology 166(5):697-708 (2004).  The experimental data outlined here, together with the hypothesized cofilin interactions in the pathway outlined in the recent review from van Rheenen et. al (2009) forms the foundation of our model development [32]. The following chapters describe the initial analysis of the pathway dynamics. In each case, we use the relevant data shown in the figures here to fit parameters, and both motivate and validate model assumptions.  9  1.3. Review of Experimental Data  Figure 1.5: Relative increase in barbed ends at the cell edge under control and PLC-inhibited conditions from Figure 3 B of Mouneimne (2004) [24]. Reprinted with permission from G Mouneimne: Spatial and temporal control of cofilin activity is required for directional sensing during chemotaxis. Current Biology 16(22):2193-205 (2006).  In the following chapters, I will describe our efforts to quantify the effect of the cofilin activity cycle on the actin cytoskeleton at the leading edge. The models are hierarchical, expanding from a simple reduced cofilin-barbed end model, to a complex model framework describing interactions between distinct cofilin forms and transitions between cellular compartments. Chapter 2 first describes a reduced model which examines the interaction between the density of a single cofilin form, termed active cofilin for its filament severing ability, and production of barbed ends. Chapter 3 introduces a closed temporal model of cofilin transitioning between various forms in a well-mixed cell, and the consequent impact on cytoskeletal dynamics. Finally, Chapter 4 expands the cofilin cycle model to include some spatial considerations by dividing the cell into edge and interior compartments, while still maintaining temporal dynamics in the equations. I will also outline the different levels of complexity in the number of cofilin interactions we have examined in each model. Finally, in Chapter 5, I will discuss the conclusions made from this work, and the proposed areas for both experimental and theoretical future study.  10  Chapter 2  Reduced Cofilin Barbed End Model 2.1  Motivation  Cofilin is known to sever filaments under specific cell conditions. The severing events create an increase in free barbed ends which can polymerize and create protrusive forces. It has been shown that cofilin activity is directly linked to spatial localization of directed movement of the cell [15]. In this chapter, I outline our examination of the relationship between cofilin activity and free barbed ends with a simplified model. Using the information outlined in Section 1.3, we attempt to reproduce the signal amplification using a variety of modeling techniques to capture the biological properties of the cofilin activity cycle.  2.2  Barbed End Model  Under the most general and simplest considerations, free, polymerizing barbed ends are produced by a variety of actin-binding proteins, including cofilin, in the cell. The rate of barbed end production is tightly regulated in a resting cell, such that the base production rate PB,rest can be considered constant. Growing barbed ends are lost due to binding of capping proteins at a rate proportional to barbed end density. We consider a constant capping rate, kcap based on a near constant lifetime of barbed ends in the cell, kcap ≈ 1s−1 [26]. We define barbed end density in the cell as B(t), expressed in units of number per µm3 . Barbed end dynamics in a resting cell are described by the proposed equation dB = PB,rest − kcap B. dt  (2.1)  11  2.3. Cofilin Barbed End Model Details This allows a simple identification of the resting steady state level of barbed ends, since if B˙ = 0, then Brest =  PB,rest . kcap  Without an applied stimulus, the system approaches this resting state asymptotically, from any initial level of barbed ends B(0).  Suppose that upon stimulation, production of barbed ends increases from PB,rest to PB,stim . Through the same analysis, the steady state barbed end level under stimulated conditions will be Bstim =  PB,stim , kcap  where PB,stim > PB,rest . Here we ignore any transition period between the two states of production, considering a stepwise increase in production rate from rest to stimulated conditions. This models an instantaneous increase in activity of actin-binding proteins in the cell. The stimulated steady state is also asymptotically stable, and barbed end levels will approach Bstim as long as the production rate remains at PB,stim . If the stimulus is removed and barbed end production returns to the level PB,rest , the barbed end density decreases back to Brest .  The magnitude of increase in barbed end production and barbed end level is dependent on the ratio between resting and stimulated states since the condition PB,stim Bstim ≤ Brest PB,rest  (2.2)  must be satisfied for all time. Thus, the ratio of production rates, PB,stim /PB,rest , determines the maximum amplification of barbed ends. Barbed end density will reach maximum amplification only if the stimulus is applied for sufficient time.  2.3  Cofilin Barbed End Model Details  Shown in Figure 1.2 in Section 1.3, EGF stimulation of MTLn3 cells can induce up to a 10-15 fold increase in barbed end density within 60s following stimulation. This has been shown to be dependent on the cofilin pathway and independent of the activity status of other ABPs in the 12  2.3. Cofilin Barbed End Model Details cell [13]. We define the cofilin concentration that is active in severing filaments as C(t) usually expressed in units of µM . We propose the following general formulation for a mini model of cofilin and barbed ends dC dt dB dt  = f (C, B), = g(C, B),  where B(t) is the density of barbed ends (usually in number per µm2 ).  2.3.1  Cofilin Model Assumptions  • The early and late peaks of barbed end levels, at 60s and 180s following stimulus with EGF are dependent on cofilin and Arp2/3 respectively [13, 24]. The respective pathways are independent of one another, as shown in the data reprinted here in Figure 1.5 [13]. • Barbed ends are produced by the activity of actin-binding proteins in the cell at rest. There is no apparent feedback between barbed ends and their own rate of production (as discussed previously). The increase in barbed end production under stimulated conditions is assumed to be strictly due to increased activity of cofilin. Our simple barbed end equation (Equation 2.1) applies here with a cofilin-dependent barbed end production rate, PB,stim = PB,stim (C). • The observed 10-15 fold increase at 60s demands amplification upstream of the barbed end dynamics. We assume here that this stems from the activity of cofilin. • Severing events occur at low frequency in resting cells as several (between 5-7) bound cofilin molecules are required to sever an actin filament. Cofilin has been shown in vitro to exhibit cooperative binding and severing properties [1, 11]. Following a severing event, cofilin molecules are left in an inactive, monomer-bound form which must be recycled on a longer time scale before being reactivated [31, 33]. In this simplified model, the cofilin concentration C(t) (in µM ) is in a form bound to the actin filaments, available to cut provided required conditions are met. We address distinct cofilin forms, active and inactive, in Chapters 3 and 4.  13  2.3. Cofilin Barbed End Model Details We identify that cofilin can be activated at some time dependent rate in the cell, AC (t), which we assume to be dependent on some external factor. Cofilin is deactivated at a rate kof f C proportional to its concentration. We assume that cofilin also loses its activated status after a severing event at a rate dependent on its concentration Fsev (C), as in Figure 1.1 from van Rheenen (2009), cofilin is released from the daughter filament in an inactive, monomerbound form. We use a barbed end equation of a form similar to Equation 2.1. We introduce the production of barbed ends by cofilin severing at a rate proportional to the rate of cofilin loss αFsev (C).  We propose the following system to represent interactions between cofilin activity and barbed ends in the cell dC dt dB dt  =  AC − kof f C − Fsev (C),  =  PB + αFsev (C) − kcap B.  (2.3)  Here AC = AC (t) is the stimulus dependent activation rate of severing cofilin, kof f is a deactivation rate constant, Fsev (C) is the cofilin-dependent severing rate which removes active cofilin from the system and increases barbed end density, α is a conversion factor between units of cofilin and barbed end density, and converts loss of cofilin activity into barbed end density produced (calculations shown in the Appendix).  We recognize that cofilin can be reactivated via a recycling process of inactivated cofilin, but we neglect this effect in this model as we consider this contribution to cofilin activity to be low within the relatively short time frame of the early barbed end peak [31, 33].  14  2.4. Linear Filament Severing  2.4  Linear Filament Severing  We propose a constant, basal level of cofilin activity in a resting cell, AC,rest . We first examine a simple linear severing rate such that new barbed ends are produced at a rate proportional to the concentration of active cofilin dC dt dB dt  = AC − kof f C − ksev C, = PB + αksev C − kcap B.  (2.4)  where ksev is the severing constant. Based on this severing rate and Equations 2.4, the basal level of cofilin activity in a resting cell is  Crest =  AC,rest . kof f + ksev  (2.5)  Cofilin activity increases upon stimulation with EGF [33]. We model this phenomenon by introducing temporal EGF dependence to the activation rate  AC (t) =     AC,stim · EGF  when stimulated by EGF,    AC,rest  otherwise,  (2.6)  where AC,stim · EGF > AC,rest . While the EGF stimulus is switched on, the concentration of active cofilin approaches the stimulated steady state value  Cstim =  2.4.1  AC,stim . kof f + ksev  Analysis of Reduced Linear Model  Similar to the barbed end relationship described previously, under these assumptions the maximum increase in cofilin activity is dependent on the relative increase in activation rates C(t) Cstim AC,stim ≤ = . Crest Crest AC,rest  15  2.4. Linear Filament Severing Equations 2.4 imply a resting barbed end production rate  PB,rest = PB + α · ksev Crest ,  where PB > 0 is the production of barbed ends by actin-binding proteins in the cell. The overall barbed end production increases upon stimulation to  PB,stim = PB + α · ksev Cstim .  From Equations 2.4, the resting barbed end density is defined as  Brest =  PB + αksev Crest , kcap  (2.7)  and stimulated steady state Bstim =  PB + α · ksev Cstim . kcap  Thus, the relative increase in barbed ends is bounded by the relationship Bstim PB + α · ksev Cstim B(t) ≤ = . Brest Brest PB + α · ksev Crest In order to capture the 10-15 fold increase in barbed ends shown in Figure 1.4, B(t) ≈ 10-15. Brest However, from Figure 2 of van Rheenen (2007), shown in Figure 1.3 in Chapter 1, the cofilin activity, recognized as the concentration of cofilin that is filament-bound and ready to sever, increases only by 2-fold during the first 60s after EGF stimulation [33]. This restricts the magnification of barbed ends to be less than a 2-fold increase, since B(t) PB + 2 · α · ksev Crest ≤ ≤ 2. Brest PB + α · ksev Crest This analytic analysis suggests a limitation of the linear model. We predict that the model will fail to generate barbed end increases of of greater than two-fold for the applied two-fold increase in cofilin concentration. These predictions can be tested by model simulations. 16  2.4. Linear Filament Severing  2.4.2  Model Scaling and Parameter Analysis  We scale the reduced linear model, Equations 2.4, by their rest steady states, shown in Equations 2.5 and 2.7. After some algebra and simplifications, we obtain simplified equations in nondimensional variables, C(t) = Crest · c(t) and B(t) = Brest · b(t), to obtain simplified equations in nondimensional variables dc dt db dt  =  (kof f + ksev )[A˜C − c]  = Aksev [c − 1] − kcap [b − 1]  (2.8)  where kof f , ksev are as described previously (in s−1 ), A˜C = AC,stim /AC,rest is the scaled cofilin stimulation rate, and the scaling parameter A is  A=  αCrest . Brest  These simplifications reduce the number of parameters required for simulations. We test a range of scaled stimulation rates to obtain cofilin fold increase of 1.2-2.0 upon stimulation, as suggested in van Rheenen (2007) (shown in Figure 1.3 of this thesis). We approximate the cofilin off-rate, kof f , to generate these cofilin profiles. We use a low severing rate parameter (ksev ) as motivated by discussions with J. Condeelis and similar to the predicted value in vitro [1, 11]. The parameter A is calculated from work shown in the Appendix and approximations from the literature [19, 26]. These values are summarized in Table 2.1  2.4.3  Simulation of Reduced Linear Model  Using the parameters outlined in Table 2.1, results of linear model simulations are shown in Figures 2.1 and 2.2. The barbed end dynamics follow the cofilin concentrations, but exhibit an increase of less than two-fold. This is governed by the low increase in the linear severing rate.  2.4.4  Time-Dependent Stimulation  A more accurate representation of the EGF-induced increase in cofilin activity can be obtained by applying a stimulation rate which is linearly time dependent. This introduces a more gradual  17  2.4. Linear Filament Severing Parameter kof f ksev A˜stim kcap A  Description cofilin-filament unbinding cofilin severing rate scaled cofilin stimulation barbed end capping rate scaling parameter  Value 1.5s−1 0.011s−1 0.1-5.0 1s−1 1000  Source estimated in model Andrianantoandro 2006 [1] van Rheenen 2007 [33] Pollard 2000 [26] Pollard 2000, Lorenz 2004 [19, 26]  Table 2.1: Parameter estimates for reduced Cofilin compartmental model with a linear severing rate function. 2  C  1.5 1 0.5  0  20  40  60  80  100  120  140  160  180  200  0  20  40  60  80  100  120  140  160  180  200  0  20  40  60  80  100  120  140  160  180  200  ! Fsev  150  100  50 2  B  1.5 1 0.5  time  Figure 2.1: Results of simulations of Equations 2.8 with parameters as outlined in Table 2.1: time profiles of cofilin concentration relative to rest concentration (top curve), rate of actin filament severing (middle curve), and barbed end dynamics normalised by resting levels (bottom curve), with linear severing rate function. The stimulus is a step function AC (t) given by Equation 2.6, shown for increasing levels of stimulation Astim = 0.1, 1.0, 2.0, 3.0, 4.0, 5.0s−1 .  increase in cofilin activity as observed in in vitro studies of cofilin binding to actin [1, 11]. The activation rate AC (in s−1 ) has the form  AC (t) =     AC,stim (t − tEGF ) · EGF  while EGF is on    AC,rest  otherwise  (2.9)  where AC,stim > AC,rest and tEGF represents the time of EGF stimulation.  The results of simulations, shown in Figures 2.3 and 2.4 are similar to previous step-function  18  2.4. Linear Filament Severing  2  1.8  Barbed ends  1.6  1.4  1.2  1  0.8 0.8  1  1.2  1.4  1.6  1.8  2  C  Figure 2.2: Phase space dynamics of cofilin activity and barbed end peak density (Equations 2.8 with parameters from Table 2.1), and linear severing rate function. Increases to the magnitude of stimulation, Astim = 0.1, 1.0, 2.0, 3.0, 4.0, 5.0s−1 , induce increases in magnitudes of cofilin concentration and barbed ends (loops in phase space).  stimulus, exhibiting very little amplification of barbed end density. The reduced model with linear severing rate is unable to generate the 10-12 fold barbed end amplification observed in experiment.  2  C  1.5 1 0.5  0  20  40  60  80  100  120  140  160  180  200  0  20  40  60  80  100  120  140  160  180  200  0  20  40  60  80  100  120  140  160  180  200  ! Fsev  150  100  50 2  B  1.5 1 0.5  time  Figure 2.3: Results of simulations of Equations 2.8 with parameters as outlined in Table 2.1: Time profiles of cofilin (top curve), filament severing rate (middle curve) and barbed end dynamics (bottom curve), with linear severing rate function. The stimulus is applied as a linear function of time, see Equation 2.9, and scaled stimulation rate A˜stim = 0.001-0.008.  19  2.4. Linear Filament Severing  1.9 1.8 1.7  Barbed ends  1.6 1.5 1.4 1.3 1.2 1.1 1 0.9 0.8  1  1.2  1.4  1.6  1.8  2  C  Figure 2.4: Phase space dynamics of cofilin activity and barbed end peak density (from Equations 2.8), with linear severing rate function and linear stimulus function with scaled stimulation A˜stim = 0.001-0.008s−1 .  20  2.5. Nonlinear Filament Severing  2.5  Nonlinear Filament Severing  Based on the limitations of the cofilin model with linear severing rate, we propose a nonlinear severing rate as predicted by experimental work [1]. It has been identified that more than one bound cofilin molecule may be required to apply the torsional force necessary to create a break in an actin filament [27]. The kinetics of a such a severing event can be described by the simple schematic nC + F → B + nCi , where n is the number of molecules required to create a break, usually 5-7. The notation Ci represents an inactive form of cofilin, a product of a severing event, the dynamics of which will not be considered here.  These kinetics suggest the existence of a cooperative effect within the severing function, and motivate a nonlinear severing function. For example, consider a resting cofilin activity concentration where the equilibrium ready to sever state is 4 molecules per filament. Severing events would occur rarely at rest. An increase in the density of bound molecules upon stimulation to a state of 6 molecules per filament causes a substantial increase in severing frequency.  Typical biochemical assumptions motivate saturable kinetics in the form of a Hill function. We believe cofilin dynamics to be far from the region of saturation in vivo, and here use a severing rate proportional to C n . Our examination of saturable kinetics is described briefly in Section 2.5.2, and provides supporting evidence for our assumptions.  We propose a nonlinear severing rate function where severing is low at rest conditions and increases when cofilin is elevated above rest concentrations. These conditions suggest an appropriate severing function for the cofilin-barbed end model in Equation 2.3 of the form  Fsev (C) = ksev Crest  C Crest  n  ,  (2.10)  where n ≈5-7 and ksev is a small constant. We assume the severing rate at rest, Fsev (Crest ) = ksev Crest to be low.  21  2.5. Nonlinear Filament Severing Under these considerations, the cofilin-barbed end model takes the form dC dt dB dt  = AC − kof f C − ksev Crest = PB + αksev Crest  C  C Crest  n  ,  n  Crest  − kcap B..  (2.11)  At rest, barbed end production rate proposed earlier is of similar form  PB,rest = PB + α · ksev Crest , where if ksev << 1s−1 , cofilin severing events are insignificant compared to the barbed end production by other actin binding proteins.  Notice that under such a model, even slight increases in cofilin activity above resting levels will be amplified so that here the stimulated severing rate is higher than the linear rate function for given resting severing rate ksev . Solving for the resting steady state of active cofilin, we find the same relationship for rest state cofilin activity  Crest =  AC,rest , kof f + ksev  and at rest the barbed end density is  Brest =  PB + α · ksev Crest . kcap  Similar to the procedure in the previous section, scaling Equations 2.11 by the resting concentrations of cofilin and barbed ends, so that C = Crest · c and B = Brest · b, simplifying and rearranging terms, obtains the system dc dt db dt  =  kof f [A˜C − c] + ksev [A˜C − cn ],  =  Aksev [cn − 1] − kcap [b − 1].  (2.12)  where A˜C = AC /AC,rest and steady states of these equations are now c = 1 and b = 1.  22  2.5. Nonlinear Filament Severing Expressed in this form, we notice the impact of the nonlinear severing rate. If the severing rate constant ksev is low, as expected from experimental work [33], cofilin severing activity is low at rest conditions in the cell. Upon stimulation, A˜C > 1, and cofilin activity increases above rest levels. This increases severing of filaments significantly and barbed end density is amplified. While the cofilin stimulus is turned on, the system is unstable and activity will increase exponentially, thereby causing exponential increase in barbed end density. This continues until the stimulus is removed and severing activity reduces back to basal resting level.  2.5.1  Simulation of Reduced Nonlinear Model  Simulations of Equations 2.4 are shown in Figures 2.5 and 2.6, with an applied nonlinear severing rate, linear stimulus as in Section 2.4.4, and with parameter values described in Table 2.2, can capture the 10-15 fold barbed end amplification. The amplification can be achieved with biologically reasonable parameter values and for increases in cofilin activity in the range of 1.5-2.0 fold increase. The results are robust to model parameters kof f and ksev . The impact of the nonlinear assumption is exemplified in the profile of the function αFsev (C). The magnitude of the severing rate, approximately one at rest conditions, increases several orders of magnitude with elevated cofilin activity. This generates the observed amplification of barbed end density.  This model represents a system far from saturation where filament-binding of cofilin is unlimited. The results can be viewed in the cofilin-barbed end phase plane Figure 2.6, where barbed end amplification is demonstrated with respect to cofilin activity. Notice increases in barbed end production are slight for cofilin activity just above resting levels. However, nonlinear effects take over as cofilin activity is increased up to the two-fold amplification .  23  2.5. Nonlinear Filament Severing  Parameter kof f ksev A˜stim n kcap α A  Description cofilin unbinding cofilin severing parameter scaled cofilin stimulation cooperativity parameter barbed end capping rate unit conversion factor scaling factor  Value 1.5s−1 0.011s−1 1-8·10−6 7 1s−1 600µM −1 µm−3 1000  Source Pollard 2000 [26] Andrianantoandro 2006 [1] van Rheenen 2007 [33] assumed in model Pollard 2000 [26] Mogilner, LEK 2002 [21] Pollard 2000, Lorenz 2004 [19, 26]  Table 2.2: Parameter estimates for reduced cofilin model with nonlinear, unsaturated severing rate function  C  1.5  1  0.5  0  20  40  60  80  100  120  140  160  180  200  0  20  40  60  80  100  120  140  160  180  200  0  20  40  60  80  100  120  140  160  180  200  ! Fsev  150 100 50 0 15  B  10 5 0  time  Figure 2.5: Results of simulations of Equations 2.12: Time profiles of cofilin concentration (top), scaled severing rate (middle), and barbed end density relative to basal levels(bottom) for nonlinear, unsaturated severing rate described in Section 2.5 with ksev = 0.011s−1 , A˜C = 18 · 10−6 and other parameter values as outlined in Table 2.2  24  2.5. Nonlinear Filament Severing  12  10  Barbed ends  8  6  4  2  0 0.9  1  1.1  1.2  1.3  1.4  1.5  C  Figure 2.6: Phase plane dynamics of cofilin activity and barbed end density (from Equations 2.12) normalized by resting concentrations in the cell, with nonlinear, unsaturated severing rate with ksev = 0.011s−1 , A˜C = 1-8 · 10−6 and other parameter values as outlined in Table 2.2  25  2.5. Nonlinear Filament Severing  2.5.2  Nonlinear Filament Severing with Saturation  We considered the possibility of saturation in the dependence of actin filament severing on cofilin concentration. Saturable kinetics can be incorporated into the filament severing rate by introducing a Hill function of the form  Fsev (C) = γmax ksev Crest  Cn , Cn + K  (2.13)  where n = 7 is the nonlinear factor or Hill coefficient, K is a saturation coefficient in units µM , and γmax is a nondimensional parameter to scale the resting severing rate ksev Crest of Section 2.5 to the saturated or maximum severing rate γmax ksev Crest .  The reduced model can be scaled and simplified in terms of the saturable severing rate, following a similar procedure to that described in the previous section. Forgoing the algebraic details, we emphasize that under this severing rate assumption, barbed end amplification loses its robustness to parameters. Barbed end amplification in the 10-15 fold range is obtained strictly within a small window of the γmax ksev parameter space. This range is shown in the left panel of Figure 2.7 where relatively high parameter settings are required (compared to experimental predictions [1, 11]).  The source of this model limitation is due to an increase in the severing rate at rest, and reduction of stimulated severing in certain parameter regions. This prevents increases of several orders of magnitude required to generate substantial amplification of barbed end density. The filament severing rate for rest concentration of cofilin is shown in the right panel of Figure 2.7 for the same range of parameter space.  The results of this applied severing rate, together with the experimental studies of filament binding concentrations and severing rates in vitro suggest that the increases in cofilin activity observed in living cells may be far from the potential regions of saturation. These conclusions suggest that the nonlinear severing rate described in Section 2.5 is a plausible severing function for future model expansions.  26  2.5. Nonlinear Filament Severing  Resting Severing Rate ! Fsev  Max Barbed ends 2  2 140  11  1.8  1.8  10 1.6  120  1.6 9  1.4  1.4  100  8 1.2  7  K  K  1.2  80  1  6  1  0.8  5  0.8  0.6  4  0.6  3  0.4  60  40  0.4  20  2 0.2  0.2 0.005  0.01  0.015  ksev  0.02  0.025  0.03  0.005  0.01  0.015  ksev  0.02  0.025  0.03  Figure 2.7: Results of simulation of barbed end model using saturated filament severing rate described in Section 2.5.2. Left: Maximum barbed end peak in parameter space of saturation coefficients K and combined severing rate constant γmax ksev . NOTE: x-axis labeled ksev should be labeled γmax ksev Right: Magnitude of scaled severing rate at rest conditions in the cell.  27  Chapter 3  Well-mixed Temporal Cofilin Model 3.1  Introduction  It has been shown that stimulation of an MTLn3 cell with EGF produces two large transient peaks of actin filament barbed end density. According to John Condeelis, and as outlined in several publications from his group [13, 24, 31, 32], the early transient peak at 60s after EGF stimulation is due to an increase in cofilin activity caused by a PLC-mediated release of PIP2 bound inactive cofilin at the membrane of the cell (shown in Figure 1.5 in Section 1.3). The cofilin released upon stimulation is in an active form, ready to sever actin. It immediately translocates to bind to uninhibited membrane-associated actin filaments [33].  In this chapter, we expand the reduced cofilin activity model of Chapter 2 to analyse the dynamics of cofilin in its various cellular forms. We introduce cellular interactions and transitions between cofilin forms, some active in filament severing, others inactive, to quantify the mechanisms responsible for the observed cofilin-dependent protrusion. We follow the conclusions of the work in van Rheenen et. al (2007), and use recent supporting data to reproduce the observed peak of barbed ends. We apply the results of the reduced model of Chapter 2 to identify the filament-severing rate necessary to generate sufficient signal amplification.  We work with a closed model of the cofilin cycle, whereby cofilin exists in one of several proposed forms. We include slow recycling steps of the inactive cofilin forms via actin monomerstripping and phosphorylation-dephosphorylation processes. These processes are facilitated by complexes including phosphatases such as SSH or CIN, and on LIM kinase (LIMK), but we will  28  3.1. Introduction consider constant complex activity here. We use temporal data of cofilin interactions to validate model assumptions where available. Following Figure 1.1 from van Rheenen (2007), we propose a temporal model of the cofilin cycle as outlined in Figure 3.1. Cofilin Pool C2 CA CF CM CP Ctot  Description PIP2 -bound cofilin active cofilin F-actin-bound cofilin G-actin-monomer-bound phosphorylated cofilin total cellular cofilin  Resting Level 0.10 · Ctot 0.01 · Ctot 0.02 · Ctot 0.67 · Ctot 0.20 · Ctot 10µM  Source van Rheenen 2007 [33] approximated in model van Rheenen 2007 [33] conservation assumption Song 2006 [31] Pollard 2000 [26]  Table 3.1: Definitions of the forms of cofilin modeled in this chapter with rest steady state concentrations  EGF  PLC  IP3  PIP2  + DAG membrane  C2 CF  CA  CP  CM  Figure 3.1: Schematic of proposed well-mixed temporal model of cofilin pathway: EGF stimulation activates PLC, causing the membrane lipid PIP2 to be hydrolyzed. This releases the cofilin from the membranous PIP2 -bound pool (C2 ) to an active form (pool CA ), from which it can attach to actin filaments (pool CF ). Once the filament is severed, cofilin comes off, carrying an actin monomer (pool CM ) with it. It is phosphorylated by LIM kinase into the form CP , and then stripped of its monomer, so that it could reattach to PIP2 . Here we have also assumed that the active cofilin can exchange with the filament bound form or be phosphorylated.  29  3.1. Introduction  3.1.1  Assumptions  Expanding on the assumptions of the reduced cofilin barbed end model of Chapter 2, we work with the following conditions. • We consider here only temporal dynamics, and are thereby approximating the cell as a well-mixed system where cofilin concentrations are averaged across the cell, and molecules move freely between the cytosolic and peripheral compartments of the cell. In this model, we neglect spatial considerations. • Cofilin is conserved in the system, so that cellular cofilin exists in one of the forms proposed here and their sum is constant, equal to the total concentration of cofilin in the cell. That is, for all time t, our system satisfies the equation  C2 (t) + CA (t) + CF (t) + CM (t) + CP (t) = Ctot .  (3.1)  It is convenient to relate all cofilin pools to the fraction of total cofilin in the cell. Experimental data is expressed relative to average cofilin concentration Ctot , rather than in absolute concentrations. Scaling respective cofilin forms with respect to Ctot , such that ci (t) = Ctot · Ci (t) obtains the dimensionless conservation equation  c2 (t) + ca (t) + cf (t) + cm (t) + cp (t) = 1,  (3.2)  for all time t.  At this point we neglect compartmental volume considerations, all cofilin forms are assumed to be well-mixed within the domain. This conservation equation will be examined and modified in further models. However, here we use this simple equation to gain rough insight into model behaviour. • We assume that actin filaments are abundant and at constant density within the domain. We consider an actin binding rate dependent only on available cofilin, and combine the rate constant as (kon F ). We assume filament severing to be independent of filament density. • Stimulated production of barbed ends is assumed to be cofilin dependent. We assume, as 30  3.2. Model Development before, that barbed ends do not feed back on their own production. We use the equations proposed in Section 2.5 to govern barbed end dynamics.  3.2  Model Development  The model considered here is closely based on Figure 3.1. We identify the cofilin concentration active in filament-severing, C, of the reduced model, as the concentration of filament bound cofilin CF , in this model framework. In this section, I motivate and describe the equations of component of the system. Readers can skip to Section 3.3 for a summary of model equations.  Under the simplest assumptions, for a resting cell, we consider basal rates of activation, IG , and basal deactivation rates proportional to concentrations, −dG G. A typical substance with concentration G(t) would satisfy the equation dG = IG − dG G. dt To such equations, we add various linear and nonlinear terms to account for interconversion and stimulation effects. Unless otherwise stated, component concentrations are measured in micro-molar values, (µM ), where 1µM ≈ 6 · 1017 molecL−1 . We also consider a closed system of cofilin, so that the sum of cofilin in all forms is a constant, as in Equation 3.1. 1. Dynamics of PLC Activity: We start analysis by quantifying phospholipase-C (PLC) activity and PtdIns(4,5)P2 (PIP2 ) dynamics, as the experimental time profiles of these regulators are well-documented, shown in Figure 1.2. Furthermore, PLC activity represents the kicking-off point for our simulations, initiating the cascade of downstream transient states. We assume a basal level of production and decay of PLC activity in a resting cell. We introduce an enhanced production rate of active PLC, which is dependent on presence of an EGF stimulus. Once the EGF signal is turned off, the active PLC concentration settles down to its resting state, and consequently each cofilin pool returns to resting state on their respective time scales.  31  3.2. Model Development The proposed equation for PLC dynamics is dP LC = dt  + IP LC − dP LC P LC ,  IstimP LC (t) stimulated activation  (3.3)  basal rates  where IP LC is a constant base level of PLC activation in the cell and dP LC is a deactivation rate. The term IstimP LC (t) represents the enhanced PLC activation rate due to stimulation by external factors, here, EGF. We propose a stimulation term of the form  IstimP LC (t) =     Istim · EGF  when stimulated by EGF,    0  otherwise.  ,  where stimulation by EGF is represented by a Heaviside (step) function EGF (t) = H(t − ton ) − H(t − tof f ). In the absence of external stimulation, IstimP LC = 0, and PLC activity is at its resting level  P LCrest =  IP LC . dP LC  We normalize equation 3.3 in terms of this resting value, P LC(t) = P LCrest · plc(t), where plc(t) is a dimensionless variable. Equation 3.3 becomes dplc = dP LC (I˜stimP LC (t) + 1 − plc), dt  (3.4)  where IstimP LC (t) I˜stimP LC (t) = , IP LC is a ratio of elevated activation rate to basal activation. These parameters can be fit well using data from Mouneimne (2004) [24], outlined in Section 3.4.1. 2. Dynamics of PIP2 : Similarly, we assume that a basal membrane concentration of the phosphoinositide PIP2 exists in a resting cell. PIP2 is known to bind and inactivate cofilin molecules at the membrane. Furthermore, the release of this pool of PIP2 -bound cofilin upon stimulation of the cell is proposed as the source of enhanced cofilin activity [33].  32  3.2. Model Development We assume PIP2 to be in equilibrium under rest conditions of the cell, and that resting rates of production and decay are low [33]. Upon EGF stimulation, increased PLC activity catalyzes increased hydrolysis of PIP2 at the membrane [24, 33]. We represent this phenomenon by introducing a PLC-mediated hydrolysis rate dhyd , in effect when PLC activity is elevated above rest levels. At this point, we consider only the effect of PLC-mediated enhanced PIP2 hydrolysis. We currently neglect other dynamics of the phosphoinositide pathway for simplicity.  We propose the following equation of PIP2 dynamics dP2 = IP2 − dP 2 P2 − dt  dhyd P LCrest  (P LC − P LCrest )P2 .  (3.5)  basal rates P LC−mediated hydrolysis  At rest, Equation 3.5 leads to  P2,rest =  IP 2 . dP 2  Normalizing P2 in Equation 3.5 by its resting steady state level, P2 (t) = P2,rest · p2 (t), where p2 (t) is dimensionless, obtains the scaled equation dp2 = dP 2 (1 − p2 − d˜hyd (plc − 1)p2 ), dt  (3.6)  where dhyd d˜hyd = . dP 2 Similar to I˜stimP LC , the parameter d˜hyd is a dimensionless ratio of stimulated and resting rates, in this case (PLC-mediated hydrolysis of PIP2 )/(basal decay rates). The estimation of PIP2 parameters (Section 3.4.1) relies on the rough PIP2 temporal data shown in Figure 1.2 (from van Rheenen (2007) [33]). 3. Dynamics of cofilin bound to PIP2 at the membrane: Experimental work has verified that a population of cofilin molecules is bound to PIP2 at the membrane of a resting cell [14, 33]. We assume that cofilin binds to PIP2 in a 1:1 binding ratio, and is thus released from the membrane upon EGF stimulation at the same rate as PIP2 hydrolysis. 33  3.2. Model Development We assume that membrane-bound cofilin, C2 , is formed by dephosphorylation of inactive phospho-cofilin, CP . For now, we assume C2 recovers at a constant dephosphorylation rate and binds to PIP2 at the membrane with mass-action kinetics.  The equation for membrane-bound cofilin is dC2 = dt  kP 2 P2,rest  P2 CP  −  dC2 C2  −  dhyd P LCrest  (P LC − P LCrest )C2 .  basal unbinding phospho−cof ilin binds P IP 2  Loss via P IP 2 hydrolysis  (3.7) The first term represents a simplified PIP2 binding rate of phosphorylated cofilin that is stripped of its phosphate group.  Here we approximate the re-binding of cofilin to the membrane, neglecting the distinct processes of cofilin dephosphorylation and PIP2 binding. The recycling process rebuilds membrane-associated cofilin slowly as shown in the time profiles of phosphorylated cofilin in Fig 2 of Song (2006) [31], and the PIP2 recovery in Figure 1.2 of this theisis (from van Rheenen (2007) [33]).  Normalizing the cofilin variables by the average concentration of cofilin in the cell Ci (t) = Ctot · ci (t), and substituting the normalized PLC and PIP2 variables, plc and p2 , the fraction of cellular cofilin that is bound to PIP2 is described by dc2 = kP 2 p2 cp − dC2 c2 − dhyd (plc − 1)c2 . dt  (3.8)  4. Dynamics of active cofilin: We assume that the cofilin released from PIP2 is in a free, unligated state. We consider this uninhibited state to be active, CA , as it is ready to bind actin filaments. We assume that active cofilin is released at the membrane from the PIP2 -bound (C2 ) pool. Active cofilin can be phosphorylated (to CP ) by LIM-Kinase, and consequently deactivated. We also assume active cofilin to be in equilibrium with actin filaments in a resting cell, with on and off rates proportional to concentrations in respective pools.  34  3.2. Model Development  The resulting equation for active cofilin is dCA = dC2 C2 + kof f CF − (kon F )CA − dt of f −on actin f ilaments  kp CA  +  dhyd P LCrest  (P LC −P LCrest )C2 ,  phosphorylation  (3.9) where (kon F ), and kof f are the filament binding and unbinding rate constants, kp is the rate constant of phosphorylation by LIMK, dC2 is the decay of PIP2 -bound cofilin, and dhyd is the rate constant of PLC-mediated PIP2 hydrolysis described previously, all in units s−1 .  Scaling by the average concentration of cellular cofilin, so CA (t) = Ctot · ca (t), results in the equation for the active cofilin fraction, in dimensionless form dca = dC2 c2 + kof f cf − (kon F )ca − kp ca + dhyd (plc − 1)c2 . dt  (3.10)  5. Dynamics of filament-bound cofilin: Cofilin that is bound to F-actin makes up the pool CF . We assume that severing of filaments occurs in a resting cell at a basal rate proportional to the rest concentration of filament-bound cofilin, with rate constant ksev . We assume loss of filament-bound cofilin is proportional to the concentration of bound cofilin.  We use a nonlinear unsaturated severing rate function Fsev (CF ), as described in Section 2.5 Fsev (CF ) = ksev CF,rest  CF CF,rest  n  ,  where n = 7, and ksev << 1s−1 so that barbed end production by cofilin severing is low at rest conditions. We assume that following a severing event, the bound cofilin molecules are released from the filament in an actin-monomer-bound inactive form which must be recycled before rebinding to filaments.  35  3.2. Model Development The dynamics of the filament-bound cofilin compartment are described, at this stage by dCF = (kon F )CA − kof f CF − ksev CF,rest dt  n  CF  .  CF,rest  (3.11)  nonlinear f ilament severing  Under these assumptions, the term ksev CF,rest is the only source of inactive monomerbound cofilin. The model expansion outlined in Chapter 4 investigates the possibility of active cofilin binding to G-actin monomers in the cytosol as another potential source of monomer-bound cofilin.  Normalizing CF in Equation 3.11 by the average cofilin concentration in the cell, so CF (t) = Ctot · cf (t), leads to dcf = (kon F )ca − kof f cf − ksev φF dt  cf φF  n  ,  (3.12)  where φF =  CF,rest = cf,rest . Ctot  6. Dynamics of G-actin monomer-bound cofilin: After an actin filament severing event, cofilin is left bound to an actin monomer, termed CM , as described in van Rheenen (2007) [33]. Cofilin in this form is inactive, unable to bind and sever actin again until it is recycled to the membrane. This requires an actin monomer-stripping and phosphorylationdephosphorylation process. We use the approximation that monomer-bound cofilin transitions into the phosphorylated pool at a rate proportional to CM concentration, with a rate constant equal to the phosphorylation rate of active cofilin, kp .  The proposed equation is dCM = ksev CF,rest dt  CF  n  −  CF,rest  kp CM  ,  (3.13)  phosphorylation source  with normalized form dcm = ksev φF dt  cf φF  n  − kp cm .  (3.14) 36  3.2. Model Development Here we neglect the monomer-stripping step of the recovery process for simplicity. Thus, we work under the assumption that the monomer is stripped and cofilin is phosphorylated in one step, whereby phosphorylation is the rate-determining step of the process. We examine the implications of this assumption in Chapter 4. 7. Dynamics of phosphorylated cofilin: Cofilin is inactivated in the cytosol by phosphorylation via LIM-Kinase. Cofilin in this CP pool is inhibited from binding both P IP2 , and actin filaments. Phosphorylation of two different cofilin pools, active cofilin (CA ) and monomer-bound cofilin (CM ) are considered, assuming with the same rate kp . Phosphorylated cofilin rebinds PIP2 at the membrane via a dephosphorylation process which requires a stripping proteinase SSH or CIN [31, 32]. We assume here a (slow) dephosphorylation rate proportional to concentration of phospho-cofilin and mass action binding kinetics proportional to concentrations of phosphorylated cofilin and PIP2 .  The proposed equation governing dynamics of the phosphorylated cofilin pool is dCP = kp CA + kp CM − dt  kP 2 P2,rest  P2 C P ,  (3.15)  with scaled form dcp = kp (ca + cm ) − kP 2 p2 cp . dt  (3.16)  8. Dynamics of Barbed Ends: Following the reduced model outlined in Chapter 2 and Section 2.5, we propose a barbed end equation where barbed end production in a stimulated cell is dependent on filament-bound cofilin CF . Barbed end dynamics are governed by the equation dB = dt  PB − kcap B  + αksev CF,rest  CF CF,rest  n  ,  (3.17)  basal source−sink cof ilin−induced rate  where the rates PB and ksev CF,rest represent barbed end production in a resting cell, kcap is the capping rate constant, n is the nonlinear severing factor and α is the estimated conversion factor between cofilin used and barbed ends created, and converts between units of cofilin concentration (µM ) and barbed end density (number per µm3 ). 37  3.2. Model Development  Here we apply an assumption that the basal severing rate by filament-bound cofilin is negligible with respect to the production rate PB , so that αksev CF,rest << PB . Consequently, the resting barbed end concentration  Brest =  PB + αksev CF,rest , kcap  is approximated by  Brest ≈  PB . kcap  We normalize the barbed end density by the approximate rest steady state and scale cofilin concentration by the average cofilin concentration in the cell to obtain db = Aksev φF dt  cf φF  n  − kcap (b − 1),  (3.18)  where φF = cf,rest , and the conversion parameter α is combined with scaling factors in the parameter  A=α  Ctot , Brest  as in Chapter 2. The approximation step described here must be verified with cofilin parameters and information from the literature, that is, whether the filament-severing rate is much lower than other basal rates of barbed end production, αksev CF,rest < PB . The model simulation results using a non-approximated barbed end equation are outlined in Chapter 4.  38  3.3. Full Normalized Model  3.3  Full Normalized Model  The assumptions and simplifications described in Section 3.2 allow us to propose a simple model framework for a closed, temporal model of the cofilin activity cycle. The collected equations for the full model are displayed in the Appendix.  The equations of the scaled model are: dplc dt dp2 dt dc2 dt dca dt dcf dt dcm dt dcp dt db dt  = dP LC (I˜stimP LC (t) + 1 − plc),  (3.19)  = dP 2 (1 − p2 − d˜hyd (plc − 1)p2 ),  (3.20)  = kP 2 p2 cp − dC2 c2 − dhyd (plc − 1)c2 ,  (3.21)  = dC2 c2 + kof f cf − (kon F )ca − kp ca + dhyd (plc − 1)c2 ,  (3.22)  =  (kon F )ca − kof f cf − ksev φF cf φF  cf φF  n  ,  (3.23)  n  − kp cm ,  (3.24)  = kp ca + kp cm − kP 2 p2 cp ,  (3.25)  = ksev φF  = kcap (1 − b) + Aksev φF  cf φF  n  .  (3.26)  The dynamics of PLC and PIP2 , Equations 3.19 and 3.20 are decoupled from the rest of the system. The cofilin equations, 3.21-3.25 represent the closed cofilin activity cycle and are strongly coupled by interconversion terms. The solution of the barbed end equation, 3.26 is a system readout which is dependent on the profile of filament bound cofilin, cf (t), the solution of Equation 3.23 (recall that φF = cf,rest ).  In the absence of a stimulus, the system has a basal steady state where PLC and PIP2 are at their respective basal levels plc = 1 and p2 = 1. Similarly, barbed end density is at its resting steady state, b = 1. The equations of cofilin concentrations, 3.21-3.25, are expressed as fractions of the average concentration of the cell, the respective steady state are given in Table 3.1.  If a stimulus is introduced, such that I˜stimP LC (t) = I˜stim > 0, a stimulated steady state exists where PLC activity is enhanced above resting levels, plc > 1. The elevated activity of  39  3.4. Model Parameter Analysis PLC induces a transient response in the cofilin cycle, which is known experimentally to increase filament-severing, and create a burst of barbed end density. To quantify this transient response with the above model, we require parameter estimates obtained from the experimental literature.  3.4  Model Parameter Analysis  The equations of the model represent a downstream cascade of effects initialized by PLC and PIP2 dynamics at the cell membrane. The cofilin cycle returns to resting steady state on a slow time scale following EGF stimulation. This simplifies model analysis, as dynamics of PLC activity and PIP2 can be fit to data independently of cofilin considerations. Cofilin transition rates can be examined based on PLC and PIP2 results, model assumptions and available data about respective cofilin fractions.  3.4.1  PLC and PIP2 Parameters  We first study the most upstream entity, PLC activity. In the right panel Figure 3.2, we reproduce the PLC activity profile subject to EGF stimulation shown in the left panel of the same figure (from Mouneimne et al (2004) [24]).  The model equation 3.4 can produce a good representation of experimental results, as shown in Figure 3.2. To obtain accurate time scale and relative proportions of stimulated and basal levels, we assign parameter values dP LC = 0.018s−1 and use scaled stimulation rates in the range I˜stim = 1.3-1.6, such that during stimulation, the scaled stimulated production of PLC activity is IstimP LC (t) = 1.3-1.6. In view of the reasonable agreement, we use the values  dP LC = 0.018s−1 ,  I˜stim = 1.5.  The simple combined model consisting of Equations 3.19 and 3.20 facilitates the analysis of the decay rate of PIP2 , dP 2 , and the nondimensional PLC-mediated PIP2 hydrolysis rate, d˜hyd . In combination, these determine the PIP2 reduction due to hydrolysis as well as the recovery time back to basal levels.  From van Rheenen et al (2007), PIP2 decreases to a minimum of 40-60% of its basal level 40  3.4. Model Parameter Analysis upon EGF stimulation [33]. Figure 1 A in that paper, shown here in the left panel of Figure 3.3, shows PIP2 levels recover half of the lost concentration on a time scale of approximately 360s. This suggests a decay rate dP 2 ≈ ln(2)/(360) ≈ 0.0018s−1 . Comparison between the simulations in the right panel of Figure 3.3 with the PIP2 data in Figure 1.2, suggests a reasonable value of d˜hyd in the range of 10-20. This means that the PLC-dependent hydrolysis of PIP2 is 10-20 times greater than resting background decay rate of PIP2 due to other processes. For model simulations, we adopt the parameter settings  dP 2 = 0.0018s−1 ,  d˜hyd = 20.  2.2  2  1.8  PLC  1.6  1.4  1.2  1  0.8  0  50  100  150  200  250  300  time  Figure 3.2: Left: Dynamics of PLC from Figure 2 A of Mouneimne (2004) [24]. Reprinted with permission from G Mouneimne: Spatial and temporal control of cofilin activity is required for directional sensing during chemotaxis. Current Biology 16(22):2193-205 (2006). Right: Dynamics of PLC resulting from Eqn. 3.19, with dP LC = 0.018s−1 and I˜stim = 1.3 (blue), I˜stim = 1.4 (green), I˜stim = 1.5, (red), and I˜stim = 1.6 (cyan). We notice that the model is robust in these parameter settings. I˜stim determines maximum peak of PLC activity, dP LC = 0.018s−1 gives a good fit to experimental results.  41  3.4. Model Parameter Analysis  1  PIP2  0.8  0.6  0.4  0.2  0  0  50  100  150  200  250  300  time  1  1  0.8  0.8  0.6  0.6  PIP2  PIP2  Figure 3.3: Dynamics of PIP2 resulting from equation 3.20 for various values of the parameters. Right: Dynamics of PIP2 from Figure 1 A of van Rheennen (2007) [33]. Reprinted with permission from J van Rheenen EGF-induced PIP2 hydrolysis releases and activates cofilin locally in carcinoma cells. Journal of Cell Biology 179(6):1247-59 (2007). Right: PIP2 profiles for fixed dP 2 = 0.018s−1 and varying PLC-dependent hydrolysis strengths, d˜hyd = 2 (blue), d˜hyd = 4 (green), d˜hyd = 10 (red), and d˜hyd = 20 (cyan). The parameter dP 2 = 0.018s−1 was chosen to give our estimated best fit of PIP2 recovery. We can compare these curves to the experimental data by van Rheenen’s Figure 1A to select parameters for our (rough) approximation of PIP2 dynamics  0.4  0.4  0.2  0.2  0  0  50  100  150  time  200  250  300  0  0  50  100  150  200  250  300  time  Figure 3.4: Model simulations of PIP2 dynamics with dP 2 = 0.018s−1 , and Left: d˜hyd = 20 so that PIP2 is almost completely reduced in the periphery, as per discussions with J. Condeelis, and Right: d˜hyd = 5, a fit to the van Rheenen data.  42  3.4. Model Parameter Analysis  3.4.2  Cofilin Data Review  We can use the resting state concentrations, together with data from the literature to satisfy the at rest steady state equations and solve for unknown model parameter values. In this section, I first outline available time series information, then compare with the parameter constraints imposed by the model.  We interpret Figure 6 A of van Rheenen (2007) as an exponential fit of cofilin residence time, or decay, bound to either PIP2 (recovery tc2 = 3.7s) or actin filaments (recovery tcf = 26s) [33]. We recognize that under a linear model, these recovery times represent the half-life of the respective cofilin binding, such that the decay rate of C2 is given by  dc2 =  ln(2) ≈ 0.19s−1 . tc2  (3.27)  We can estimate the loss of filament-bound cofilin, which under our model occurs by the sum of rates kof f and ksev , both linear at rest  dcf = kof f + ksev =  ln(2) ≈ 0.027s−1 . tcf  (3.28)  We expect severing events to be rare in a resting cell, as barbed end production is tightly regulated and severing occurs primarily as stochastic fluctuations of the system. This implies that the off-rate, kof f , of cofilin being released from actin filaments without causing a severing event should be significantly higher that the severing parameter ksev . This information is consistent with our assumption kof f >ksev together with in vitro filament severing rates given in the literature [1, 2] where ksev ≈ 0.010-0.012s−1 .  If we interpret Figure 6 B of van Rheenen (2007) as the fit of a linear combination of the C2 and CF exponential decay equations, then  y = A1 exp(−dc2 t) + A2 exp(−dcf t).  43  3.4. Model Parameter Analysis Using this method, the relative ratios of PIP2 -bound cofilin and filament-bound cofilin can be obtained for both a resting and stimulated cell. We obtain the at rest ratio from this figure, implemented as initial conditions in our model, C2,rest 0.85 = . CF,rest 0.15 This ratio decreases upon stimulation with EGF to 0.38 C2,stim = , CF,stim 0.62 which is important for time profile fitting of our model simulations. Furthermore, Figure 2 in van Rheenen (2007) shows an observed 2-fold increase of filament-bound cofilin CF upon stimulation with EGF.  Figure 2 of Song et al. (2006) [31], exhibits the time profile of phosphorylated cofilin before and after stimulation of the cell with EGF. The resting level is identified as CP,rest = 0.18(±4) · Ctot , and stimulated level, which we take to be the maximum of the phosphorylated cofilin pool is CP,max = 0.38(±6) · Ctot . Further time profiles of phospho-cofilin can be used to fit model simulations [31], but this is beyond the scope of this thesis. We use compare the experimental data of a stimulated cell to model simulations, shown in Section 4.8, and discuss conclusions in Chapter 5.  3.4.3  Cofilin Model Parameter Analysis  Here we outline the analysis of model assumptions and equations to combine with available cofilin data from Section 3.4.2 to gain information about unknown model parameters.  The information given in Song et. al (2006), [31], implies the rest fraction of phosphorylated cofilin in the cell to be  cp,rest = 0.20. Through analysis of Figure 6 of van Rheenen (2007) [33], the ratio of cofilin bound to F-actin  44  3.4. Model Parameter Analysis and PIP2 at the periphery of the cell is identified as cf,rest 0.15 ≈ , c2,rest 0.85 which gives c2,rest ≈ 5.8 · cf,rest . From personal communication with J. Condeelis, we estimate the fraction of PIP2 -bound cofilin at rest as c2,rest ≈ 0.10. This supports resting level fractions of filament-bound, and PIP2 -bound cofilin to be approximately  cf,rest  =  0.02,  c2,rest  =  0.10.  The conservation assumption Equation 3.2 also implies that, at rest, the normalized cofilin pools sum up to 1 c2,rest + ca,rest + cf,rest + cm,rest + cp,rest = 1.  (3.29)  Thus the sum of at rest steady state fractions of active and monomer-bound cofilin is approximately  ca,rest + cm,rest  =  1 − c2,rest − cf,rest − cp,rest  =  1 − 0.10 − 0.02 − 0.20  =  0.68.  (3.30)  We first identify steady state relationships between resting cofilin pool levels. We set derivatives of Equations 3.21-3.25 to zero, and use the assumption ksev << 1s−1 . From equation 3.21, parameters must satisfy the relationship  c2,rest =  kP 2 · p2,rest · cp,rest , dC2  (3.31)  where ci,rest is the scaled cofilin level and p2,rest = 1 is scaled P IP2 level in the resting cell.  Equation 3.31, together with the rest levels data of PIP2 -bound and phosphorylated cofilin, 45  3.4. Model Parameter Analysis provides a relationship between the dephosphorylation rate of the model kP 2 and the decay rate of PIP2 cofilin dC2 . We have  kP 2 =  c2,rest 0.10 · dC2 = · dC2 , p2,rest · cp,rest 1 · 0.20  so that kP 2 = 0.5 · dC2 .  Since PIP2 -bound cofilin decays as PIP2 decays at the membrane, we assume dC2 = dP 2 . The PIP2 decay rate constant, dP 2 approximated in Section 3.4.1. We then set parameter values dC2 = 0.002s−1 ,  kP 2 = 0.001s−1 .  Furthermore, from equation 3.25  cp,rest =  kp · (ca,rest + cm,rest ). kP 2  (3.32)  From above, the rest fraction of phospho-cofilin is cp,rest = 0.20, and from Equation 3.30 ca,rest + cm,rest = 0.68. This gives the relation 0.68 kP 2 = , 0.20 kp and kP 2 = 3.40 · kp . The phosphorylation rate can be approximated as  kp = 2.94 · 10−4 s−1 . The steady state equation of monomer-bound cofilin, Equation 3.24 can be simplified, since  46  3.4. Model Parameter Analysis φF = CF,rest /Ctot = cf,rest . This identifies the relationship  cm,rest =  ksev · cf,rest . kp  (3.33)  Notice that the ratio between monomer-bound and filament-bound cofilin at resting conditions is determined here by the ratio between basal severing and phosphorylation rates. This implies ksev cm,rest = . cf,rest kp  (3.34)  We identify this parameter ksev , (for which there are estimates only from in vitro work and other modeling efforts [1, 6]), as a free parameter to be tested by model simulations. It is identified by the constraints of Equation 3.34 once the rest steady state level of monomer-bound cofilin, cm,rest is assigned.  From discussion with J. Condeelis, we work under the assumption that the fraction of active cofilin is very small at any time t. Condeelis predicts this cofilin fraction to be near zero in a resting cell, therefore, we estimate the rest level of active cofilin pool of cofilin as ca,rest ≈ 0.01. By conservation, Equation 3.30, this gives cm,rest = 0.67. This identifies the severing model parameter, since to satisfy Equation 3.34, we have  ksev =  0.67 · kp , 0.02  so ksev = 0.0099s−1 . In line with the conservation equation 3.30,  cm,rest  =  0.67,  ca,rest  =  0.01.  The cofilin equations 3.22 and 3.23 introduce two actin-binding parameters, kof f and simplified binding rate constant (kon F ). From Equation 3.28, and the linear decay of filament-bound 47  3.4. Model Parameter Analysis cofilin from Equation 3.23, we obtain the relationship between rate constants of loss of filamentbound cofilin  kof f = dcf − ksev  =  0.027s−1 − 0.0099s−1 ,  (3.35)  =  0.0171s−1  (3.36)  based on previous assumptions. Thus the model assumptions satisfy experimental predictions that kof f > ksev .  We can gain further information about parameters by using the relationship between steady states of Equations 3.10 and 3.12  ca,rest =  1 · [dC2 c2,rest + kof f cf,rest ], kp + (kon F )  (3.37)  kon F · ca,rest . kof f + ksev  (3.38)  and cf,rest ≈  Setting the relations for ca,rest equal to one another and simplifying obtains the relation c2,rest kp ksev + kp kof f + (kon F )ksev = , (kon F )dC2 cf,rest which can be further simplified to be in terms previously identified parameter ratios cm,rest ksev + kof f dC2 c2,rest · − = . kp cf,rest cf,rest (kon F )  (3.39)  From Equation 3.38, together with previous assumptions about resting concentrations and resulting parameter estimates, we find that the filament association and dissociation parameters must satisfy 6.80 · 5.0 − 33.50 =  ksev + kof f ca,rest = . (kon F ) cf,rest  Here we have defined ca,rest = 0.01, so  (kon F ) = 2 · (ksev + kof f ).  48  3.5. Summary of Model Equations and Parameters Using experimental data from van Rheenen (2007) (Section 3.4.2), the filament binding rate is estimated as  (kon F ) = 2 ·  3.5  ln(2) −1 s = 0.054s−1 . 26  Summary of Model Equations and Parameters  We simulate the system of ODEs, Equations 3.19-3.26 using parameter values described in Section 3.4.3 and summarized in Table 3.2 and 3.3. The system is initialized at the respective rest steady states. The stimulus is introduced as an EGF step function turned on at t = 60s and off at t = 120s, such that  I˜stimP LC (t) = I˜stim (Heaviside(t − 60) − Heaviside(t − 120)). Cofilin Pool C2 CA CF CM CP Ctot  Description PIP2 -bound cofilin active cofilin F-actin-bound cofilin G-actin-monomer-bound phosphorylated cofilin total cellular cofilin  Resting Level 0.10 · Ctot 0.01 · Ctot 0.02 · Ctot 0.67 · Ctot 0.20 · Ctot 10µM  Source van Rheenen 2007 [33] approximated in model van Rheenen 2007 [33] conservation assumption Song 2006 [31] Pollard 2000 [26]  Table 3.2: Definitions of the forms of cofilin modeled in this chapter with rest steady state concentrations Parameter I˜stim dP LC dP 2 dhyd dC2 kof f kon F ksev kp kP 2 kcap α A  Description scaled PLC activation rate PLC decay rate PIP2 resting hydrolysis PLC-PIP2 hydrolysis rate PIP2 -cofilin decay rate Cofilin-F-actin off-rate Filament binding rate Cofilin severing rate Cofilin phosphorylation Dephosphorylation Barbed end capping rate unit conversion factor scaling factor  Value 1.5 0.018s−1 0.002s−1 0.01s−1 0.002s−1 0.0171s−1 0.054s−1 0.0099s−1 2.94 · 10−4 s−1 0.001s−1 1s−1 600µM −1 µm−3 1000  Source Mouneimne 2004 [24] Mouneimne 2004 [24] van Rheenen 2007 [33] van Rheenen 2007 [33] assumption van Rheenen 2007 [33] estimated by model estimated by model estimated by model estimated by model Dawes 2006 [9] Mogilner, LEK 2002 [21] Pollard 2000 [26]  Table 3.3: Parameter estimates for the well-mixed temporal model, Equations 3.19-3.26  49  3.6. Simulations of Well-Mixed Model  3.6  Simulations of Well-Mixed Model  The following figures exhibit an example of simulation results of the well-mixed temporal cofilin system for proposed model assumptions and parameter estimates. We simulate two scenarios proposed by experiments. First, we use a PLC-hydrolysis parameter of dhyd = 0.01s−1 to produce a 60% reduction of PIP2 levels, shown in the left panel of Figure 3.5. This represents the overall reduction of PIP2 as given in van Rheenen (2007). The model simulations results are shown in Figures 3.6-3.8. Secondly, model simulation with dhyd = 0.04s−1 produces almost complete reduction of PIP2 , shown in the right panel of Figure 3.5. Motivated by discussions with J. Condeelis, this represents cellular activity in local regions near an EGF receptor. The results of model simulations are given in Figures 3.9-3.11.  The simulations of a 60% reduction in PIP2 levels result in a similar reduction in the level of PIP2 -bound cofilin. The resulting barbed end density reaches a maximum increase of less than 5-fold over resting barbed end levels (Figure 3.8). This peak is enhanced by increasing the PIP2 reduction, since this induces a much larger release of PIP2 -bound cofilin (Figure 3.10). This generates a barbed end peak of approximately 8-9 fold increase over resting levels (Figure 3.11). While these results reproduce the experimental increase in barbed ends, up to 10-12 fold, at a respectable level, both scenarios fail to generate the 2-fold increase in filament-bound cofilin shown in van Rheenen (2007).  In the next chapter, I outline our work to reformulate the well-mixed cell approximation. By working within a model framework which better represents the physical properties of a cell, we aim to analyze the assumptions made in this chapter and verify model results.  50  3.6. Simulations of Well-Mixed Model  2.5  2.5 PLC PIP2  2  2  1.5  1.5  1  1  0.5  0.5  0  0  50  100  150  200  250  0  300  0  50  100  time  150  200  250  300  time  Figure 3.5: Simulation profiles of PLC (green) and PIP2 (magenta) with parameter settings from Table 3.3 Left: Results for dhyd = 0.01s−1 . Right: Results for dhyd = 0.04s−1 .  0.8 EGF added  C2 CA CF CM CP  EGF removed  0.7  0.6  C*/Ctotal  0.5  0.4  0.3  0.2  0.1  0  0  50  100  150  200  250  300  time  Figure 3.6: Cofilin time profiles as fractions of average cofilin concentration, for parameter settings outlined in Table 3.3.  51  3.6. Simulations of Well-Mixed Model  0.12 EGF added  C2 CA CF  EGF removed  0.1  C*/Ctotal  0.08  0.06  0.04  0.02  0  0  50  100  150  200  250  300  time  Figure 3.7: As in Figure 3.6, magnified to show a close up of low-level cofilin fraction profiles for parameter settings outlined in Table 3.3.  0.04  A Fsev  0.03 0.02 0.01 0  0  50  100  150  200  250  300  200  250  300  time  Barbed ends  5 4 3 2 1 0  0  50  100  150  time  Figure 3.8: Rate of severing (top) and barbed end time profiles (bottom) produced by cofilin activity dynamics and using parameter settings outlined in Table 3.3. Notice small barbed end amplification of approximately 3-fold increase.  52  3.6. Simulations of Well-Mixed Model  0.8 C2 CA CF CM CP  EGF added 0.7 EGF removed 0.6  C*/Ctotal  0.5  0.4  0.3  0.2  0.1  0  0  50  100  150  200  250  300  time  Figure 3.9: Cofilin time profiles as fractions of average cofilin concentration, for parameter settings outlined in Table 3.3 and dhyd = 0.04s−1 .  0.12 EGF added  C2 CA CF  EGF removed  0.1  C*/Ctotal  0.08  0.06  0.04  0.02  0  0  50  100  150  200  250  300  time  Figure 3.10: As in Figure 3.9, magnified to show a close up of low-level cofilin fraction profiles for parameter settings outlined in Table 3.3 and dhyd = 0.04s−1 .  53  3.6. Simulations of Well-Mixed Model  0.4  A Fsev  0.3 0.2 0.1 0  0  50  100  150  200  250  300  200  250  300  time  Barbed ends  10 8 6 4 2 0  0  50  100  150  time  Figure 3.11: Rate of severing (top) and barbed end time profiles (bottom), shown as relative density over resting barbed end concentration, for parameter settings outlined in Table 3.3 and dhyd = 0.04s−1 . Notice a barbed end amplification of over 8-fold with 60s of addition of EGF stimulus.  54  Chapter 4  Temporal Cofilin Compartmental Model 4.1  Introduction  In this chapter, I outline our examination of a compartmental model, challenging the well-mixed assumption of Chapter 3. We introduce two volume compartments, one to represent the small region at the cell edge and another to represent the large cellular interior. Compartmental considerations are supported by the desire to differentiate the cofilin forms identified to be bound at the cell edge by fluorescence-tagged membrane experiments versus those that diffuse freely in the cytosol [24, 33]. Cofilin bound to PIP2 (C2 ) has been shown to be stationary and membrane-associated [33]. Cofilin binding to F-actin is known experimentally to be confined to the periphery of the cell by inhibition of tropomyosin binding to filaments in the interior [12, 33]. These model revisions quantify relative local concentrations of cofilin in the periphery and cytosolic compartments by identifying the different volumes of the edge and interior compartments, which may be up to 1 or 2 orders of magnitude larger. This requires revision of the model equations to represent transition between compartments.  We also consider whether interactions previously neglected could be significant, so we include the reactivation of cofilin via direct dephosphorylation of CP , re-binding of active cofilin CA back onto PIP2 at the membrane (into C2 pool), and a new equilibrium relationship between active and monomer-bound cofilin. The revised definitions of cofilin pools in respective compartments are outlined in Table 4.1 and interactions between cofilin forms shown in the schematic, Figure 4.1.  55  4.1. Introduction This model version serves as the final step in the analysis of the temporal dynamics of the cofilin pathway before embarking on a spatio-temporal model. Here, we address all possible interactions between the identified cofilin forms with the goal of identifying the critical interactions to include in the spatial framework. A future one-dimensional (PDE) model should then simulate the distribution of cofilin as distance from the cell edge. Thus, the edge and interior compartments examined in this chapter will determine the boundary conditions of a radial model. This final temporal model also facilitates the verification of appropriate and necessary model assumptions to simplify future work. Cofilin Pool C2E E CA CFE E CM CPE I CA I CM CPI Ctot  Description PIP2 -bound cofilin active cofilin in edge F-actin-bound cofilin G-actin-monomer-bound in edge phosphorylated cofilin in edge active cofilin in interior G-actin-monomer-bound in interior phosphorylated cofilin in interior average cofilin concentration  Resting Level 2.00 · Ctot 10−4 · Ctot 0.40 · Ctot 0.67 · Ctot 0.20 · Ctot 10−4 · Ctot 0.67 · Ctot 0.20 · Ctot 10µM  Source discussion approximated in model van Rheenen 2007 conservation assumption Song 2006 approximated in model conservation assumption Song 2006 Pollard 2000  Table 4.1: Cofilin forms in the edge and interior compartments, with rest concentrations defined relative to average cellular cofilin concentration.  56  4.1. Introduction  EGF  PLC  IP3  PIP2  membrane  C2  CF  CA CP  CM edge CA  CP  CM interior  Figure 4.1: Schematic of proposed temporal compartmental model of cofilin pathway: Dynamics are separated into those at the edge of the cell, and interactions in the cytosol, or interior of the cell. At the membrane, EGF stimulation activates PLC, causing the membrane lipid PIP2 to be hydrolyzed. This releases the cofilin from the membranous PIP2 -bound pool (C2E ) to an E active form (pool CA ) at the edge of the cell, from which it can attach to actin filaments (pool E E CF ), or bind to G-actin (CM ), phosporylated (to CPE ), or rebind to PIP2 . Once the filament E is severed, cofilin comes off, carrying an actin monomer (pool CM ) with it. Filament dynamics are assumed to occur near the membrane, in the edge compartment. Following a severing event, the G-actin-cofilin is stripped of its monomer, and phosphorylated by LIM kinase into the form CP , so that it could reattach to PIP2 . The freely diffusing forms of cofilin diffuse between the I I edge and interior compartment, to CA , CM , CPI .  57  4.1. Introduction  4.1.1  General Assumptions  • We will consider here only temporal dynamics, and thereby approximate the cell as a wellmixed system where cofilin molecules move freely between the cytosolic and peripheral compartments of the cell. • Cellular cofilin is divided into several forms in the edge compartment (within 200 nm of the cell membrane) and in the interior compartment. Freely diffusing forms can transition between the two compartments, whereas cofilin molecules bound to PIP2 at the membrane, or bound to filaments at the edge of the cell are confined to the edge compartment. We assume that the edge and interior compartments have fixed volumes, V E and V I respectively. • The total amount of cofilin is conserved in the cell. We use the notation CiE and CiI to denote concentrations in micromolar units (µM ) of cofilin in various forms. The total number of molecules of cofilin is given by the conservation statement  E E I I V E ·(C2E +CA +CFE +CM +CPE )+V I ·(CA +CM +CPI ) = Vtot ·Ctot = (V E +V I )·Ctot , (4.1)  where V E , V I are volumes of compartments, usually in µm3 , Ctot is the total average cofilin concentration in the cell ( 10µM ), and the cofilin forms, Ci , are as defined in Table 4.1. • Actin filament density is constant and abundant within the domain. We consider an actin binding rate dependent only on available cofilin, and assume filament severing to be independent of filament density for now. • Barbed end dynamics are governed by equations proposed in earlier works and do not feed back on their own production. We assume that active barbed ends observed at 60s following stimulation with EGF are due to amplification of cofilin severing.  58  4.2. Model Details  4.2  Model Details  Motivated by experimental conclusions, we define the edge compartment as the region within 200nm of the cell membrane, and the interior compartment as the remaining fraction of the cell [33]. This implies a large difference in compartmental volumes, V I >> V E , confirmed by calculations outlined in the Appendix. Since some cofilin forms, C2 and CF exist only at the edge of the cell, whereas the other cofilin forms, CA , CM , and CP are present at both the edge, and in the interior of the cell, we must consider the effective local densities of each cofilin fraction in the given compartment. We will denote relative local cofilin concentrations of respective forms as CiE in the edge compartment, and CiI in the interior. For example, monomer-bound cofilin concentration in the edge compartment is described by  CiE =  molar quantity of monomer-bound cofilin in edge . volume of edge compartment  We ensure that balance equations are written in terms of conservation of mass or of numbers of molecules. When compartment volumes differ, as in the case we describe here, this introduces some correction factors from dilution effects. To avoid confusion about relative local concentrations, we outline several different notations here. These all convey the same message, conservation of total cofilin, Equation 4.1. The quantity Vtot · Ctot has units µm3 · µM since these are common units of measure.. Converting this quantity into number of molecules of cofilin requires a conversion factor η, described in the Appendix. The total number of cofilin molecules in the cell is given by #molecules = Vtot · Ctot · η, where η = 600 molecules µM −1 µm−3 , since 1µM ≈ 600 molecules per µm3 , details of this calculation are outlined in the Appendix.  The available data that describes the resting cofilin concentrations in a cell is given as the proportion of total cofilin in the cell. We take these to be proportions of the total molecular content (as in fluorescence data). We use the average total cellular cofilin concentration of 10µM [26]. This allows us to calculate the total molecular content in our model cell approximation, 59  4.2. Model Details after choosing a volume estimate.  We can nondimensionalize Equation 4.1 by dividing by Vtot · Ctot , obtaining the relationship VE Vtot  C2E CE CE CE CE + A + F + M + P Ctot Ctot Ctot Ctot Ctot  +  VI Vtot  I CA CI CI + M + P Ctot Ctot Ctot  = 1.  We define compartment volumes by total volume of the cell, and relative cofilin concentrations scaled by average cofilin concentration Ctot , defining vE =  VE , Vtot  vI =  VI , Vtot  ci =  Ci , Ctot  This leads to the the nondimensional conservation equation  E E E E I I I I v E · (cE 2 + ca + cf + cm + cp ) + v · (ca + cm + cp ) = 1.  (4.2)  Finally, if we combine relative local concentrations with their respective volume fraction, such that E E c˜E i = v · ci ,  c˜Ii = v I · cIi ,  the conservation equation can be expressed as  (˜ cE ˜E ˜E ˜E ˜E cIa + c˜Im + c˜Ip ) = 1. 2 +c a +c f +c m+c p ) + (˜  Notice that the c˜i expression represents the fraction of total molecular cofilin in each respective form.  These modifications in methodology and model development introduce several new aspects to the model. First, available data gives estimates of resting cofilin fractions of respective forms [24, 31, 33]. We take these to be proportional to the average cofilin concentration in the cell. The fractions which exist in strictly the edge compartment must be scaled to obtain the relative local concentrations of each cofilin form. For example, if Ci , in units of µM , is the average cellular concentration of cofilin in form i, we can convert this to the relative local concentration in the edge compartment using the relationship  60  4.2. Model Details  CiE =  Vtot · Ci . VE  Based on the exchange between compartments, we must now incorporate 3 additional ODEs I I in the system to describe the dynamics of CA , CM and CPI in the interior. The rate of change  of each respective cofilin form in each compartment is described qualitatively by d [molecules of form i in compartment] = rate of exchange + source terms − loss terms. dt Currently, we assume that exchange between compartments follows simple diffusion. Transitions between compartments of each respective cofilin form occur at a rate proportional to the concentration difference. We utilize known cofilin diffusion coefficient (D ≈ 10µm2 s−1 ) [26] to give the rate of molecular exchange and use a width parameter of the cell (ω = 0.2µm) [21]. These parameters scale the compartmental transition terms and match units in Equations 4.3 and 4.4 The rate of change of cofilin in given forms satisfies d(V E CiE ) dt d(V I CiI ) dt  =  ωD[CiI − CiE ] + source terms − loss terms,  (4.3)  =  −ωD[CiI − CiE ] + source terms − loss terms.  (4.4)  The considerations of compartmental volumes and transitions affect only the diffusing forms that exchange between the edge and interior, not C2 or CF . We assume a constant diffusion coefficient for all forms of cofilin, though the coefficient describing diffusion of CM may be slightly lower due to the larger structural size of the complex.  We also take into account several interactions neglected in previous modelling efforts. These include equilibrium binding of G-actin on/off active cofilin (CA ↔ CM ), rebinding of active cofilin to PIP2 at the membrane, and dephosphorylation of phospho-cofilin at locations other than the membrane. These interactions are included in the full model schematic shown in Figure 4.1. The addition of these interactions in the absence of further experimental data will require extensive simplifying model assumptions. However, we hope that the process will facilitate some insight into the importance of each respective interaction effect.  61  4.3. Model Equations  4.3  Model Equations  For the most part, equations will closely follow their forms in Equations 3.19-3.26. However, in this version, we expand the model to incorporate a more complete collection of potential cellular interactions.  We keep the same assumptions and equations for PLC and PIP2 , since these system components are not involved in the conservation equation. We proceed with the scaled equations as experimental data is commonly given in a normalized form as fold increases over resting levels.  I outline here the revised model equations for PIP2 -bound cofilin, C2E , and both edge and I E , where new terms are expressed in red font. The and CA interior forms of active cofilin, CA  Equations 3.23-3.25 are revised in a similar manner to include new terms from Figure 4.1 and exchange between compartments. 1. Dynamics of cofilin bound to PIP2 at the membrane: Since PIP2 is a membraneassociated lipid, cofilin bound to PIP2 is assumed to exist only in the membrane compartment. We quantify the cofilin bound to the membrane at any time t, as #molesC2E = V E · C2E . The equation for membrane-bound cofilin dynamics is given by d(V E C2E ) = dt  kP p2 P2,rest  P2 V E CPE  kAp2 P2,rest  +  E − dC2 V E C2E P2 V E C A basal unbinding  phospho−cof ilin binds P IP 2  active cof ilin binds P IP 2  −  dhyd P LCrest  (P LC − P LCrest )V E C2E ,  (4.5)  Loss when P IP 2 hydrolyzed  where terms are as previously with the addition of PIP2 -binding rates from phosphorylated and active cofilin forms, kP p2 and kAp2 , respectively. We then normalize the molecular level of PIP2 -bound cofilin by the total amount of cofilin in the cell to obtain fractions of total cellular cofilin, v E · cE i =  V E CiE · . Vtot Ctot  (4.6)  Since fractioned edge compartment volume, v E = V E /Vtot , is assumed to be approximately constant in time, and appears in all terms here, we can divide it from both sides of  62  4.3. Model Equations Equation 4.5. Using normalized PLC and PIP2 variables, (P LC(t) = P LCrest · plc(t)), the nondimensional concentration of PIP2 -bound cofilin (C2 (t) = Ctot · c2 ) is dcE 2 E E E = kP p2 p2 cE p + kAp2 p2 ca − dC2 c2 − dhyd (plc − 1)c2 . dt  (4.7)  2. Dynamics of active cofilin at the periphery: Cofilin liberated from the membrane compartment is released into the edge compartment of the cell. We consider this cofilin E form to be active (pool CA ), as it can bind to available and uninhibited actin filaments.  Active cofilin diffuses between the edge and interior compartments of the cell, at a rate proportional to its concentration difference. Here we assume G-actin monomers with respective on and off rates proportional to concentrations in respective pools. Expanding on assumptions of previous models, here we consider the reactivation of cofilin by dephosphorylation (from CP ) or loss of the actin monomer (from CM ). These considerations were motivated by conversations with J. Condeelis, and work shown in Mogilner, Keshet (2002) [21] respectively. The revised equation of active cofilin is E dV E CA I E E E = ωD CA − CA +dC2 V E C2E + kof f V E CFE − (kon F )V E CA − kp V E CA dt dif f usion  +  kdp V E CPE  E E − + kma V E CM − kam V E CA  dephosphorylation  of f −on−G−actin−monomer  kAp2 P2,rest  E P2 V E C A  active cof ilin binds P IP 2  +  dhyd P LCrest  (P LC − P LCrest )V E C2E , (4.8)  where dC2 , kon F , and kof f , kp , and dhyd are defined as before (in s−1 ). The constants ω (µm) and D (µm2 s−1 ) are the cell width and diffusion parameters described previously, kam and kma are G-actin-monomer binding and un-binding rates, and kdp is the rate of dephosphorylation via various cellular phosphatases (s−1 ).  We scale the equation for active cofilin dynamics as in Equation 4.6 to obtain the equation  63  4.3. Model Equations for nondimensional active cofilin in the edge compartment dcE ωD a E E E E E E = E [cIa − cE a ] + dC2 c2 + kof f cf − (kon F )ca − kp ca + kdp cp + kma cm dt V E E −kam cE a − kAp2 p2 ca + dhyd (plc − 1)c2 .  (4.9)  Note that the compartment volume consideration affects the scaled equation only by a concentration, or ”dilution” effect in the exchange term. 3. Dynamics of active cofilin in the interior: The cofilin released into the edge compartment can be contained there by to membrane or filament associated dynamics, but can also transition into the interior compartment of the cell. We consider the large interior compartment of the cell to be well-mixed, as characterized by fast diffusion rates in the cytosol of the cell.  In the interior compartment, we assume cofilin cycles between different forms based on equilibrium interactions between active, monomer-bound, and phospho-cofilin. We assume transition rate constants to be equivalent to transition rates proposed at the edge of the cell. Since the relative volumes of the interior to the edge compartments can differ by up to two orders of magnitude, molecules that transition into the interior compartment are diffused due to the increase in relative volume. Thus, concentrations of interior pools will be relatively constant in the absence of a large release of cofilin molecules from on of the edge pools (ie. such as the release of C2 following EGF stimulation).  The equation for active cofilin in the interior compartment is I d(V I CA ) = dt  I E −ωD CA − CA source/loss due to dif f usion  I I I − kp V I C A + kdp V I CPI + kma V I CM − kam V I CA , phos−dephosphorylation  of f −on−monomer  (4.10) where transition rates are as described previously. We scale Equation 4.10, as in Equation 4.6, to obtain the equation for nondimensional local concentration of active interior cofilin dcIa ωD I I I I = − I [cIa − cE a ] − kp ca + kdp cp + kma cm − kam ca . dt V  (4.11)  64  4.3. Model Equations 4. Dynamics of Barbed Ends: We use the full (non-approximated) barbed end equation from Section 2.5 of Chapter 2 to a generate a more complete picture of barbed end dynamics. We work with the non-approximated barbed end steady state equation  Brest =  PB + αksev CF,rest , kcap  and normalize Equation 3.17 , similar to the method of Section 2.5, such that B(t) = Brest · b(t) db cf = Aksev [ dt φF  n  − 1] − kcap [b − 1]  (4.12)  where parameters A, ksev , and kcap are as described previously, and φF is the scaled resting steady state of filament-bound cofilin, φF = cf,rest .  The analysis of model parameters in described briefly in Section 4.5, alternatively readers can skip to the outline of parameters used in Tables 4.2 and 4.3.  65  4.4. Compartmental Model Summary  4.4  Compartmental Model Summary  With these simplifications, we can rewrite the model equations in their reduced and normalized form dplc dt dp2 dt dcE 2 dt dcE a dt  = dP LC (I˜stimP LC (t) + 1 − plc),  (4.13)  = dP 2 (1 − p2 − d˜hyd (plc − 1)p2 ),  (4.14)  E = kP p2 p2 cE p + kAp2 p2 ca − dC2 c2 − dhyd (plc − 1)c2 ,  (4.15)  =  ωD I E E E E [c − cE a ] + dC2 c2 + kof f cf − (kon F )ca − kp ca + kdp cp + kma cm VE a E −kam cE a − kAp2 p2 ca + dhyd (plc − 1)c2 ,  dcE f dt dcE m dt dcE p dt dcIa dt dcIm dt dcIp dt db dt  =  (kon F )ca − kof f cf − ksev φF  =  ωD I [c − cE m ] + ksev φF VE m  = = = = =  cf φF  (4.16)  n  cf φF  ,  (4.17)  n E E E − kmp cE m + kpm cp − kma cm , +kam ca (4.18)  ωD I E E E E E [c − cE p ] + kp ca − kdp cp + kmp cm − kpm cp − kP p2 p2 cp , VE p ωD I I I I I − I [cIp − cE p ] − kp ca + kdp cp + kma cm − kam ca , V ωD I I I I − I [cIm − cE m ] − kmp cm + kpm cp − kma cm + kam ca , V ωD I I I I − I [cIp − cE p ] + kp ca − kdp cp + kmp cm − kpm cp , V n cf kcap (1 − b) + Aksev φF . φF  (4.19) (4.20) (4.21) (4.22) (4.23)  Notice that the reduced cofilin equations form a closed system when multiplied by the respective volume fractions of each compartment, such that  vE ·  4.5  dcE dc2 dcE dcf dcE p + a + + m+ dt dt dt dt dt  + vI ·  dcIp dcIa dcI + m+ dt dt dt  = 0.  Model Parameter Analysis  As outlined in Section 3.4.2, we have information referring to the concentrations of various cofilin pools at rest.  We first make some assumptions necessary for model identification.  66  4.5. Model Parameter Analysis • We first consider the assumption that the freely diffusing cofilin forms are at equal concentrations in the edge and interior compartments at rest. This allows us to reduce the compartment exchange term from the steady state equations.1 • We assume that transition rates between cofilin forms are equivalent in the respective compartments. Effectively, we assume that the cellular complexes required for each transition are equally abundant and active in the edge and the interior of the cell. This assumption may be accurate for some cofilin processes (such as LIM kinase-mediated phosphorylation) and very inaccurate for others (ie. monomer-binding). The transition rates will be considered independently in a later model reduction.  4.5.1  Steady State Analysis  The steady state equations provide constraints on parameters. We set derivatives of Equations 4.15-4.22 to zero and examine rest state conditions. From Equation 4.15  cE 2,rest =  1 dC2  E · [kP p2 cE p,rest + kAp2 ca,rest ].  (4.24)  Under the assumption cE a,rest << 1 as described in Chapter 3, this gives the approximate relationship cE kp2 2,rest , ≈ E dC2 cp,rest indicating that the binding rate of phospho-cofilin to PIP2 at the membrane must be significantly higher than the C2 decay rate in order to maintain the large local concentration of PIP2 -bound cofilin. We assume here that free diffusing forms of cofilin have the same concentration in the edge and interior compartments. The steady state equations of these forms account only for the source and loss terms of the pool, so from Equation 4.16, the resting level of active cofilin in the edge compartment is  cE a,rest =  E dC2 c2,rest + kof f cf,rest + kma cE m,rest + kdp cp,rest . kAp2 + kon F + kam + kp  (4.25)  1 The no net flux simplification will have to be reviewed in later models. It is possible that rest conditions could lead to opposed standing gradients at the cell edge such that the net flux between compartments is zero.  67  4.5. Model Parameter Analysis The steady state equation for filament-bound cofilin, from Equation 4.17, is much simpler, since our model includes just one known source  cE f,rest =  kon F · cE , kof f + ksev a,rest  (4.26)  which can be rearranged to obtain the relationship cE f,rest cE a,rest  =  kon F . kof f + ksev  This implies that binding parameters must exceed the loss rates by some magnitude to maintain E the relatively high local concentration of filament-bound cofilin, (cE f,rest >> ca,rest ) in the edge  compartment at rest.  Based on Equation 4.18, we find  cE m,rest =  E E ksev cE f,rest + kam ca,rest + kpm cm,rest , kmp + kma  (4.27)  where the severing term is expected to be low at rest.  Equation 4.19 implies the concentration of phosphorylated cofilin in the edge compartment at rest satisfies  cE p,rest =  E kp cE a,rest + kmp cm,rest . kdp + kpm + kp2  (4.28)  The equations of cofilin forms in the interior compartment each have two source terms and two loss terms. Based on Equation 4.20, the relative concentration of active cofilin in the interior is given by  cIa,rest =  kdp cIp,rest + kma cIm,rest . kp + kam  (4.29)  Similarly, from Equation 4.21, the interior monomer-bound cofilin at rest satisfies  cIm,rest =  kpm cIp,rest + kam cIa,rest . kmp + kma  (4.30)  68  4.5. Model Parameter Analysis And finally, the concentration of phospho-cofilin in the interior, from Equation 4.22, must satisfy  cIp,rest =  4.5.2  kp cIa,rest + kmp cIm,rest . kdp + kpm  (4.31)  Model Parameter Estimation  Under the assumption of equal concentrations in the edge and interior compartments, we can combine equations for respective forms of cofilin and reduce parameters. Equating the edge and interior concentrations of monomer-bound cofilin at rest, Equations 4.27 and 4.30, and reducing the equivalent loss terms obtains the relationship  E E I I ksev cE f,rest + kam ca,rest + kpm cp,rest = kpm cp,rest + kam ca,rest ,  (4.32)  which reduces to ksev cE f,rest = 0. Since cE f,rest > 0, this implies ksev = 0. This zero severing rate contradiction could be a result of any one of our model assumptions. Equation 4.32 suggests that one or more of the simplifications should be reconsidered in future models. Some suggested modifications are: 1. Variation in parameters between the edge and interior compartments. This revision to our initial assumptions is supported by the dependence of cellular complexes on the rate of cofilin transition between forms. The concentrations of such complexes may vary in the edge and interior of the cell. 2. The assumption of equal concentrations of cofilin forms between the edge and interior should be reviewed. This may be most easily approached in a one-dimensional spatial framework as here we work with local average cofilin concentrations in each compartment. Here I address the contradiction from Equation 4.32 by considering the first modification, and revise the parameter consistency assumption in the following section.  69  4.5. Model Parameter Analysis  4.5.3  Distinct Transition Rates in Compartments  In Equation 4.32, both the monomer-binding rate kam , and the combined dephosphorylationmonomer-binding rate kpm are assumed to be equivalent in the edge and interior of the cell. However, both of these processes are dependent on the concentration of G-actin available for binding. It has been shown in both experimental and analytical work that concentrations of G-actin are higher in the cell interior due to higher filaments breakdown, and are depleted at the cell edge where there is an increased polymerization rate [26? ].  We assume that the cofilin-G-actin association rate is negligible in the edge compartment, such that  E kam ≈ 0,  E kpm ≈ 0.  (4.33)  I I , to differentiate between these and kpm We rewrite Equation 4.32 with the notation kam  rates. Therefore,  E E E E I I I I ksev cE f,rest + kam ca,rest + kpm cm,rest = kpm cp,rest + kam ca,rest .  This approximation obtains the relationship  I I I I ksev cE f,rest ≈ kpm cp,rest + kam ca,rest .  (4.34)  The positivity of rate constants and the condition imposed by Equation 4.34 set restrictions on parameter magnitudes. As outlined in previous chapters, we expect filament severing by cofilin to be low in the cell at rest. Here, the magnitude of ksev must be significant enough to maintain I I kpm > 0 and kam > 0.  Outlined in previous chapters, we assume a very low concentration of active cofilin in the resting cell. In Chapter 3, we used the value ca,rest = 0.01 and determined the resulting severing parameter, ksev , by applying model constraints (Equation 3.34). Here we resolve the constraint of Equation 4.34 by applying a lower rest state concentration of active cofilin ca,rest ≈ 10−4 . This is motivated and supported by personal communication with J. Condeelis where he predicts  70  4.6. Further Proposed Simplifying Assumptions a near-zero rest concentration of active cofilin.  In a similar manner, the remaining rest state relationships, Equations 4.28 and 4.31 to obtain the phosphorylation and dephosphorylation parameter relationships. The remaining parameters are constrained by Equations 4.25 and 4.29. The parameter estimates outlined in Table 4.3 were calculated using the temporal data described in Section 3.4.2, the volume assumptions outlined in the Appendix, and the scaled resting concentrations, and parameter relationships described here.  4.6  Further Proposed Simplifying Assumptions  The number of parameters introduced by this expansion creates a model identification problem in the absence of additional experimental data. To proceed with parameter estimation and model simulations, we must make some critical assumptions to simplify equations and reduce the number of unknown parameters to be analyzed. Here we start by identifying a large number (upper bound) of simplifications. These should be reviewed in future works.  We summarize the parameter simplifications, motivated by assumptions described in Section 4.5.3, and by conversations with J. Condeelis. 1. Following Section 4.5.3, we assume that monomer binding of both active and phosphocofilin occurs at a very low rate in the edge compartment. We differentiate between edge and interior parameters, as in Equation 4.33, such that  E kam ≈ 0,  E kpm ≈ 0.  Interior rates are denoted as I kam ,  I kpm .  2. From conversations with J. Condeelis, we identify that phosphorylation rates of active and monomer-bound cofilin are indistinguishable in the cell. Effectively, this is an assumption that the monomer-stripping step is negligible within the phosphorylation process, possible  71  4.6. Further Proposed Simplifying Assumptions due to a combined LIM kinase-SSH complex. The phosphorylation rates are expressed as  kmp ≈ kp .  3. We also identify that rebinding of active cofilin to PIP2 at the membrane occurs at a very low rate, due to the instability of active cofilin, and its tendency to diffuse away from membrane contact after its release. We assume  kAp2 ≈ 0,  eliminating this interaction from the model. We apply these simplifying assumptions to the compartmental model. The revised schematic of cofilin interconversion between forms is shown in Figure 4.2. Parameter values obtained from calculations in this section, including these simplifying assumptions are outlined in Table 4.3. EGF  PLC  IP3  PIP2  membrane  C2  CF  CA CP  CM edge CA  CP  CM interior  Figure 4.2: Schematic of proposed reduced temporal model of cofilin pathway with reduced assumptions outlined in Section 4.6.  72  4.7. Reduced Model Summary  4.7  Reduced Model Summary  Here we summarize model variables and their resting state levels in Table 4.2, and parameter values used to simulate the model equations in Table 4.3. Cofilin Pool C2E E CA CFE E CM CPE I CA I CM CPI Ctot VE VI  Description PIP2 -bound cofilin active cofilin in edge F-actin-bound cofilin G-actin-bound cofilin in edge phosphorylated cofilin in edge active cofilin in interior G-actin-bound cofilin in interior phosphorylated cofilin in interior average cofilin concentration volume of edge compartment volume of interior compartment  Resting Level 2.00 · Ctot 10−4 · Ctot 0.40 · Ctot 0.68 · Ctot 0.20 · Ctot 10−4 · Ctot 0.68 · Ctot 0.20 · Ctot 10µM 50µm3 950µm3  Source discussion approximated in model van Rheenen 2007 conservation assumption Song 2006 approximated in model conservation assumption Song 2006 Pollard 2000 Mogilner Keshet 2002 Mogilner Keshet 2002  Table 4.2: Cofilin forms in the edge and interior compartments, as in Table 4.1, repeated here for the purpose of summarizing conclusions. We designate volume of the edge and interior compartments (shown in the Appendix) and summarize rest concentrations defined relative to average cellular cofilin concentration. Note that the local concentration, C2E is higher than the average cellular cofilin concentration due to the large number of cofilin molecules crowded in the small edge compartment.  73  4.7. Reduced Model Summary  Parameter I˜stim dP LC dP 2 dhyd Vtot VE kP p2 kAp2 dC2 kof f  Description PLC activation rate PLC decay rate PIP2 resting hydrolysis PLC-PIP2 hydrolysis rate total cell volume edge compartment volume PIP2 -binding of phospho-cofilin PIP2 -binding of active cofilin PIP2 -cofilin decay rate cofilin-F-actin off-rate  Value 1.5 0.018s−1 0.002s−1 0.01s−1 1000µm3 50µm3 0.020s−1 0 0.002s−1 0.016s−1  kon F ksev kma I kam E kam kp kmp kdp kcap α A  Filament binding rate Cofilin severing rate monomer-cofilin off-rate monomer-cofilin binding in interior monomer-cofilin binding in edge phosphorylation of active cofilin phosphorylation of monomer-cofilin cofilin dephosphorylation barbed end capping rate unit conversion factor scaling factor  106.6s−1 0.0112s−1 0.0083s−1 4.79s−1 0 0.0118s−1 0.0118s−1 0.020s−1 1s−1 600µM −1 µm−3 1000  Source Mouneimne 2004 Mouneimne 2004 van Rheenen 2007 van Rheenen 2007 estimated in Appendix estimated in Appendix estimated by model assumption assumption van Rheenen 2007 required by model estimated by model required by model estimated in model estimated in model assumption estimated by model assumption estimated by model Dawes 2006 Mogilner, LEK 2002 Lorenz 2004  Table 4.3: Parameter estimates for revised compartmental cofilin pathway model  74  4.8. Simulations of Revised Compartmental Model  4.8  Simulations of Revised Compartmental Model  We show some preliminary results here based on our current parameter estimates. We initialize the system at the resting steady states outlined in Table 4.2, and stimulate using parameter values in Table 4.3 via addition of an EGF signal at 60s, removed at 120s.  We obtain the same PLC and PIP2 results, shown in Figure 3.5 and simulate the effect of these dynamics on the cofilin pools in the edge and interior compartments, shown in the top and bottom panels of Figure 4.8.1 respectively. Figure 4.4 shows these same cofilin results, but disE plays profiles as fractions of total molecules of cofilin, for example v E · cE · CiE /Vtot · Ctot . i =V  Two cases of barbed end simulation results are shown in Figure 4.5 for parameters in Table 4.3. The top panel uses low hydrolization rate dhyd = 0.01s−1 , as estimated by experimental results. The bottom panel displays results for the increased hydrolysis, dhyd = 0.04s−1 , as per discussions with J. Condeelis.  4.8.1  Discussion of Compartmental Model  The compartmental model expands the well-mixed model in several respects. The compartmental framework provides a more accurate representation of the cell, accounting for the high local concentrations of PIP2 -bound and filament-bound cofilin. The added complexity of interactions and cellular geometry considerations introduce challenges in parameter constraints. Furthermore, the lack of available experimental data requires assumptions to simplify the model and identify parameters.  However, in view of these increased complexities, and the distinct differences in the dynamics of cofilin pools, the barbed end results of the compartmental and well-mixed models, Figures 3.8, and 4.5 are remarkably similar. This may indicate that the interactions introduced in the compartmental are not essentia to generate the barbed end amplification. The simplified cofilin interactions of the well-mixed system should be investigated under the compartmental framework to confirm this hypothesis. The compartmental model also exhibits similar limitations as those described in Chapter 3. The increase of filament-bound cofilin increases only to a maximum of 1.4 relative to rest levels, failing to reach the 2-fold increase shown in Figure 1.3 from 75  4.8. Simulations of Revised Compartmental Model van Rheenen (2007). We propose several experimental considerations to investigate these issues in the discussion of Section 5.2.1 of Chapter 5.  76  4.8. Simulations of Revised Compartmental Model  2.5 caE  EGF added  cmE cpE c2E  2  cfE  Ci/Ctot  1.5  1  0.5  0  0  50  100  150  200  250  300  time  0.8 caI  EGF added  cmI  0.7  cpI  0.6  Ci/Ctot  0.5  0.4  0.3  0.2  0.1  0  0  50  100  150  200  250  300  time  Figure 4.3: Simulations of cofilin concentrations, Equations 4.15-4.22, in the edge and interior compartment relative to average cofilin concentration Ctot for parameter settings found in Table 4.3 and discussed in the text. Cofilin pools are initiated at resting concentrations, a stimulus (EGF) is added at 60s and removed at 120s. Top: Dynamics of cofilin pools in the edge compartment. Bottom: Dynamics of cofilin pools in interior. Notice small and shallow increase in filament bound cofilin at the edge. This is due to a high rest severing rate, and relatively high phosphorylation of active cofilin. The resulting barbed end peak (Figure 4.5) is significantly less than the 12 fold increase (even with n = 7) and does not exhibit the sharp increase and decrease observed in experiments (Mouneimne 2004) [24].  77  4.8. Simulations of Revised Compartmental Model  0.12 vEcaE  EGF added  vEcmE vEcpE  0.1  vEc2E vEcfE  0.06  i i  VC / V C  tot tot  0.08  0.04  0.02  0  0  50  100  150  200  250  300  time  0.8 vIcaI  EGF added  vIcmI  0.7  vIcpI  ViCi / VtotCtot  0.6  0.5  0.4  0.3  0.2  0.1  0  0  50  100  150  200  250  300  time  Figure 4.4: Time profiles of amount of cofilin as fractions of total cellular cofilin (# of molecules) for parameters from Table 4.3. Top: Edge compartment dyamics, Bottom: Interior compartment dynamics. Simulations are the same as in Figure 4.8.1, but concentrations are scaled by relative volumes of compartments.  78  4.8. Simulations of Revised Compartmental Model  0.02  A Fsev  0.015 0.01 0.005 0  0  50  100  150  200  250  300  200  250  300  200  250  300  200  250  300  time  Barbed ends  4 3 2 1 0  0  50  100  150  time  0.04  A Fsev  0.03 0.02 0.01 0  0  50  100  150  time  Barbed ends  8 6 4 2 0  0  50  100  150  time  Figure 4.5: Results of the compartmental model for parameters in Table 4.3, demonstrating two cases of PIP2 hydrolysis, 60% decrease (Top), and 95% decrease (Bottom). Top: Rate of filament-severing by cofilin (top curve), and resulting barbed end profiles (bottom curve) for dhyd = 0.01s−1 , shown as relative density over resting barbed end concentration. Notice the shallow barbed end amplification of approximately 3-fold increase within 60s of addition of EGF stimulus with slow decay back to resting barbed end density. Bottom: Rate of filament-severing by cofilin (top curve), and resulting fold increase in barbed ends (bottom curve) for increased PIP2 hydrolysation rate dhyd = 0.04s−1 . Amplification increase to approximately 8-fold, similar to the result of the well-mixed model of Chapter 3.  79  Chapter 5  Discussion of Cofilin Model Results Recent experimental work indicates that cofilin activity determines the sites of actin polymerization and resulting cell movement in metastasizing mammary tumour cells [7, 15]. Previously hypothesized to be important only for filament disassembly, as strictly an actin depolymerizing factor [4, 5], cofilin is now unexpectedly at the cell motility forefront as a key determinant of a cell’s ability to detect chemoattractant gradients and direct cell protrusion. Furthermore, cofilin may interact synergistically with the Arp2/3 complex to promote actin nucleation, and thereby influence overall cell polarity.  Though substantial ground has been made experimentally to identify the underlying mechanisms of cofilin activation in the cell [31, 33], there are many open questions and difficult obstacles to overcome in order to gain a more complete understanding of cancer cell metastasis. The modeling work described in this thesis represents one such initial step in the critical analysis of this important activity cycle.  5.1  Discussion of Temporal Model Results  The models discussed in Chapters 2-4 determine important interactions with the cofilin activity cycle and propose the significant model considerations for simulation of the initial cofilindependent peak of barbed ends in a stimulated cell. Both the well-mixed and compartmental models of Chapters 3 and 4 respectively can reproduce temporally accurate barbed end peaks observed in recent experiments [15, 19, 23] (reproduced in Figure 1.4). In both cases, the magnitude of the barbed end readout of model simulations is robust to most parameters, including  80  5.2. Proposed Future Considerations severing rate parameter assumptions. A barbed end peak of the 10-fold magnitude observed under certain experimental conditions [23], requires almost complete reduction of PIP2 levels (Figures 3.11, and 4.5).  The similar barbed end results obtained from the well-mixed and compartmental models indicate that some of the complexities introduced in Chapter 4 do not play a significant role in barbed end generation. These should be investigated in future work such that a spatial model can be appropriately simplified to include only the essential level of complexity.  5.2  Proposed Future Considerations  The models described in this thesis facilitate insight about future considerations for both modeling and experimental work.  5.2.1  Issues for Experimental Investigation  The models described in this thesis facilitate insight about future considerations for both modeling and experimental work. A residual challenge stemmed from the discrepancy between model parameter constraints and experimentally proposed relative magnitudes of the rate that cofilin unbinds from (kof f ) and severs (ksev ) an actin filament. An example of such a constraint relationship is outlined in Section 4.5.2. In lieu of conclusive experimental data, modelers must enforce system assumptions to resolve constraints. Experimental work to investigate the relationship between filament-binding, severing and G-actin binding of cofilin in the cell periphery would be invaluable to either support or negate model assumptions.  The shortfall of both models to reproduce the observed 2-fold amplification of filament-bound cofilin at 60s following stimulation requires further attention. We suppose that this issue stems from the filament severing rate used in both models, motivated in Section 2.5. Through personal communication with J. Condeelis, and supported by recent work outlined in Frantz (2008), we identify that filament severing in vivo is highly dependent on pH levels in the cell [14]. Stimulation with EGF induces a pH change through a pathway independent of the cofilin activity  81  5.2. Proposed Future Considerations cycle studied in our model. This effect is proposed to delay the effect of the amplified severing, and could result in a build up of filament-bound cofilin before the rapid filament-severing occurs. Such a hypothesis requires further experimental motivation. For example, a time profile of filament-bound cofilin density would be beneficial to test model validity. Currently the only in vivo CF data describe the density at 0s and 60s following EGF stimulation (Figure 1.3 from van Rheenen (2007)).  5.2.2  Future Modeling Work  Further research into the cofilin activity cycle is imperative. Due to the complexity and cost of the experimental study of mammary tumour cells in vivo, the medical and societal importance of understanding the metastatic phenotype notwithstanding, research in this field demands the collaboration of experimentalists and modelers. As stated, we propose the models outlined in this thesis as a foundational study into the temporal dynamics of the cofilin activity pathway. We focused here on the mechanisms necessary to produce the EGF-induced burst of barbed ends produced by increased cofilin activity. There are many potential directions for future modeling work.  A logical first step would be to extend the model to include spatial properties, by analysing a one dimensional transect of the cell. A model of this form would facilitate a more accurate representation of the dynamic filament density of the cell. Spatial profile data of the barbed end density produced by the cofilin cycle has been developed recently [23, 24]. An important distinction from previous spatial models of cell motility in keratocytes [9, 10], the barbed end peak produced by cofilin is transient, and would not generate steady state traveling wave solutions. This can be traced to the inherently different methods of migration of keratocytes and cancer cells.  Further investigation into the role of the cofilin activity cycle in the overall metastatic phenotype of cancer cells is also critically important. The advancement of the big picture of how cancer cells migrate and invade is critical to the identification of the important targets to inhibit the spread, and consequent fatality of cancer in the body.  82  5.2. Proposed Future Considerations  5.2.3  Other Perspectives  The cofilin activity cycle has emerged as a critical regulator of membrane protrusion; however, tumour cell migration and invasion has been shown to depend on the activity status of several signalling pathways, including the N-Wasp and MENA pathway, critical for both protrusion and the necessary breakdown of the extracellular matrix [20]. It is through fine coordination of the respective signalling networks that cancer cells are able to produce sufficient protrusive force and enzymatic activity to migrate into blood vessels and invade body tissues. As the understanding of N-Wasp and MENA signalling pathways develop experimentally, theoretical investigation of the relationships within the regulation network will be important to test hypotheses and propose interactions to be analysed experimentally.  83  Bibliography [1] Ernesto Andrianantoandro and Thomas D Pollard. Mechanism of actin filament turnover by severing and nucleation at different concentrations of adf/cofilin. Mol Cell, 24(1):13–23, Oct 2006. [2] L Blanchoin and T D Pollard. Mechanism of interaction of acanthamoeba actophorin (adf/cofilin) with actin filaments. J Biol Chem, 274(22):15538–46, May 1999. [3] Wenxiang Cao, Jim P Goodarzi, and Enrique M De La Cruz. Energetics and kinetics of cooperative cofilin-actin filament interactions. J Mol Biol, 361(2):257–67, Aug 2006. [4] M F Carlier, V Laurent, J Santolini, R Melki, D Didry, G X Xia, Y Hong, N H Chua, and D Pantaloni. Actin depolymerizing factor (adf/cofilin) enhances the rate of filament turnover: implication in actin-based motility. J Cell Biol, 136(6):1307–22, Mar 1997. [5] M F Carlier and D Pantaloni. Control of actin dynamics in cell motility. J Mol Biol, 269(4):459–67, Jun 1997. [6] A E Carlsson.  Stimulation of actin polymerization by filament severing.  Biophys J,  90(2):413–22, Jan 2006. [7] A Y Chan, M Bailly, N Zebda, J E Segall, and J S Condeelis. Role of cofilin in epidermal growth factor-stimulated actin polymerization and lamellipod protrusion. J Cell Biol, 148(3):531–42, Feb 2000. [8] Lingfeng Chen, Chris Janetopoulos, Yi Elaine Huang, Miho Iijima, Jane Borleis, and Peter N Devreotes. Two phases of actin polymerization display different dependencies on pi(3,4,5)p3 accumulation and have unique roles during chemotaxis. Mol Biol Cell, 14(12):5028–37, Dec 2003.  84  Bibliography [9] Adriana T Dawes, G Bard Ermentrout, Eric N Cytrynbaum, and Leah Edelstein-Keshet. Actin filament branching and protrusion velocity in a simple 1d model of a motile cell. J Theor Biol, 242(2):265–79, Sep 2006. [10] Adriana T Dawes and Leah Edelstein-Keshet. Phosphoinositides and rho proteins spatially regulate actin polymerization to initiate and maintain directed movement in a onedimensional model of a motile cell. Biophys J, 92(3):744–68, Feb 2007. [11] Enrique M De La Cruz. Cofilin binding to muscle and non-muscle actin filaments: isoformdependent cooperative interactions. J Mol Biol, 346(2):557–64, Feb 2005. [12] Vera DesMarais, Ilia Ichetovkin, John Condeelis, and Sarah E Hitchcock-DeGregori. Spatial regulation of actin dynamics: a tropomyosin-free, actin-rich compartment at the leading edge. J Cell Sci, 115(Pt 23):4649–60, Dec 2002. [13] Vera DesMarais, Frank Macaluso, John Condeelis, and Maryse Bailly. Synergistic interaction between the arp2/3 complex and cofilin drives stimulated lamellipod extension. J Cell Sci, 117(Pt 16):3499–510, Jul 2004. [14] Christian Frantz, Gabriela Barreiro, Laura Dominguez, Xiaoming Chen, Robert Eddy, John Condeelis, Mark J S Kelly, Matthew P Jacobson, and Diane L Barber. Cofilin is a ph sensor for actin free barbed end formation: role of phosphoinositide binding. J Cell Biol, 183(5):865–79, Dec 2008. [15] Mousumi Ghosh, Xiaoyan Song, Ghassan Mouneimne, Mazen Sidani, David S Lawrence, and John S Condeelis. Cofilin promotes actin polymerization and defines the direction of cell motility. Science, 304(5671):743–6, Apr 2004. [16] Vitaliy Y Gorbatyuk, Neil J Nosworthy, Scott A Robson, Naresh P S Bains, Mark W Maciejewski, Cris G Dos Remedios, and Glenn F King. Mapping the phosphoinositidebinding site on chick cofilin explains how pip2 regulates the cofilin-actin interaction. Mol Cell, 24(4):511–22, Nov 2006. [17] Ravine A Gungabissoon and James R Bamburg. Regulation of growth cone actin dynamics by adf/cofilin. J Histochem Cytochem, 51(4):411–20, Apr 2003.  85  Bibliography [18] Philip A Kuhlman. Dynamic changes in the length distribution of actin filaments during polymerization can be modulated by barbed end capping proteins. Cell Motil Cytoskeleton, 61(1):1–8, May 2005. [19] Mike Lorenz, Vera DesMarais, Frank Macaluso, Robert H Singer, and John Condeelis. Measurement of barbed ends, actin polymerization, and motility in live carcinoma cells after growth factor stimulation. Cell Motil Cytoskeleton, 57(4):207–17, Apr 2004. [20] Mike Lorenz, Hideki Yamaguchi, Yarong Wang, Robert H Singer, and John Condeelis. Imaging sites of n-wasp activity in lamellipodia and invadopodia of carcinoma cells. Curr Biol, 14(8):697–703, Apr 2004. [21] Alex Mogilner and Leah Edelstein-Keshet. Regulation of actin dynamics in rapidly moving cells: a quantitative analysis. Biophys J, 83(3):1237–58, Sep 2002. [22] K Moriyama, K Iida, and I Yahara. Phosphorylation of ser-3 of cofilin regulates its essential function on actin. Genes Cells, 1(1):73–86, Jan 1996. [23] Ghassan Mouneimne, Vera DesMarais, Mazen Sidani, Eliana Scemes, Weigang Wang, Xiaoyan Song, Robert Eddy, and John Condeelis. Spatial and temporal control of cofilin activity is required for directional sensing during chemotaxis. Curr Biol, 16(22):2193–205, Nov 2006. [24] Ghassan Mouneimne, Lilian Soon, Vera DesMarais, Mazen Sidani, Xiaoyan Song, Shu-Chin Yip, Mousumi Ghosh, Robert Eddy, Jonathan M Backer, and John Condeelis. Phospholipase c and cofilin are required for carcinoma cell directionality in response to egf stimulation. J Cell Biol, 166(5):697–708, Aug 2004. [25] Michiru Nishita, Chinatsu Tomizawa, Masahiro Yamamoto, Yuji Horita, Kazumasa Ohashi, and Kensaku Mizuno. Spatial and temporal regulation of cofilin activity by lim kinase and slingshot is critical for directional cell migration. J Cell Biol, 171(2):349–59, Oct 2005. [26] T D Pollard, L Blanchoin, and R D Mullins. Molecular mechanisms controlling actin filament dynamics in nonmuscle cells. Annu Rev Biophys Biomol Struct, 29:545–76, 2000. [27] Ewa Prochniewicz, Neal Janson, David D Thomas, and Enrique M De la Cruz. Cofilin increases the torsional flexibility and dynamics of actin filaments. J Mol Biol, 353(5):990– 1000, Nov 2005. 86  Bibliography [28] F Ressad, D Didry, G X Xia, Y Hong, N H Chua, D Pantaloni, and M F Carlier. Kinetic analysis of the interaction of actin-depolymerizing factor (adf)/cofilin with g- and f-actins. comparison of plant and human adfs and effect of phosphorylation. J Biol Chem, 273(33):20894–902, Aug 1998. [29] Mazen Sidani, Deborah Wessels, Ghassan Mouneimne, Mousumi Ghosh, Sumanta Goswami, Corina Sarmiento, Weigang Wang, Spencer Kuhl, Mirvat El-Sibai, Jonathan M Backer, Robert Eddy, David Soll, and John Condeelis. Cofilin determines the migration behavior and turning frequency of metastatic cancer cells. J Cell Biol, 179(4):777–91, Nov 2007. [30] P Silacci, L Mazzolai, C Gauci, N Stergiopulos, H L Yin, and D Hayoz. Gelsolin superfamily proteins: key regulators of cellular functions. Cell Mol Life Sci, 61(19-20):2614–23, Oct 2004. [31] Xiaoyan Song, Xiaoming Chen, Hideki Yamaguchi, Ghassan Mouneimne, John S Condeelis, and Robert J Eddy. Initiation of cofilin activity in response to egf is uncoupled from cofilin phosphorylation and dephosphorylation in carcinoma cells. J Cell Sci, 119(Pt 14):2871–81, Jul 2006. [32] Jacco van Rheenen, John Condeelis, and Michael Glogauer. A common cofilin activity cycle in invasive tumor cells and inflammatory cells. J Cell Sci, 122(Pt 3):305–11, Feb 2009. [33] Jacco van Rheenen, Xiaoyan Song, Wies van Roosmalen, Michael Cammer, Xiaoming Chen, Vera Desmarais, Shu-Chin Yip, Jonathan M Backer, Robert J Eddy, and John S Condeelis. Egf-induced pip2 hydrolysis releases and activates cofilin locally in carcinoma cells. J Cell Biol, 179(6):1247–59, Dec 2007. [34] Weigang Wang, Ghassan Mouneimne, Mazen Sidani, Jeffrey Wyckoff, Xiaoming Chen, Anastasia Makris, Sumanta Goswami, Anne R Bresnick, and John S Condeelis. The activity status of cofilin is directly related to invasion, intravasation, and metastasis of mammary tumors. J Cell Biol, 173(3):395–404, May 2006.  87  Appendix Full Model Equations from Chapter 3 The full model equations in unscaled form are dP LC dt dP2 dt dC2 dt dCA dt dCF dt dCM dt dCP dt dB dt  =  IP LC + IstimP LC (t) − dP LC P LC,  =  IP2 − dP 2 P2 −  =  kdp P2,rest  dhyd P LCrest  (5.1)  (P LC − P LCrest )P2 ,  P2 CP − dC2 C2 −  dhyd P LCrest  (P LC − P LCrest )C2 ,  =  dC2 C2 + kof f CF − (kon F )CA − kp CA +  =  (kon F )CA − kof f CF − ksev CF,rest  =  ksev CF,rest  =  kp CA + kp CM −  =  PB − kcap B + αksev CF,rest  CF,rest  IstimP LC (t) =  dhyd P LCrest  CF CF,rest  (5.3)  (P LC − P LCrest )C2(5.4) ,  n  ,  (5.5)  n  CF  with  (5.2)  ,  − kp CM ,  kdp P2,rest  (5.6)  P2 CP , CF CF,rest  (5.7) n  .  (5.8)     Istim · EGF  when stimulated by EGF    0  otherwise  and Fsev (CF ) = ksev CF,rest  CF CF,rest  n  .  By adding together the equations for the cofilin forms, we find that dC2 dCA dCF dCM dCP + + + + =0 dt dt dt dt dt  88  Appendix which is a check for conservation of the total cofilin, i.e. for the fact that  C2 + CA + CF + CM + CP = Ctot = constant.  Calculation of Conversion Factor α Here we describe calculations to obtain the model conversion parameter α. This parameter serves two purposes: to convert between common units of cofilin concentration measurement (µM ) and barbed end density (number per µm2 ), and to scale the number of cofilin molecules used in a severing event into the number of barbed ends produced.  We first proceed as described in Mogilner and Edelstein-Keshet 2002, [21], to estimate conversion between units of cellular concentration, µM , and common units of barbed end quantification, #µm−2 . We have  1µM ≈ 10−6 ·  6x1023 molec 6x1017 molec 10−15 dm3 = · , L dm3 1µm3  and therefore, 1µM ≈ 600  moleculesperµm3 .  This relationship gives us a working estimate for conversion between unit volumes,  ν = 600µM −1 µm−3 .  Experimental approximations of barbed end density in a resting cell, such as those given in Lorenz (2004) [19] are generally given in units of number per µm2 . This means that we must multiply the parameter ν by the approximate width of the lamellipod. If we approximate the lamellipod width as ω ≈ 0.200µm, we obtain a unit conversion parameter  ν˜ = ω · ν ≈ 120µM −1 µm−2 . It is proposed experimentally that a finite number n, where n ≈ 5-7, of bound cofilin molecules are required to force a filament break. We use this assumption throughout Chapters  89  Appendix 2-4, and use n = 7 in most simulations. Therefore, we approximate the conversion parameter as  α=  ν˜ ≈ 17.1µM −1 µm−2 . n  Cell Geometry Calculations from Chapter 4 We now give rough calculations for the volumes of the edge and interior compartments use in the compartmental model of Chapter 4. First we calculate the total volume of the cell, approximating a resting cell as a half-sphere of radius 5 − 10µm. The total cellular volume is  Vtot =  1 4π 3 2π · r ≈ (5-10µm)3 , 2 3 3  and therefore, Vtot ≈ 250-2000µm3 . From van Rheenen (2007, 2008), we divide the cell into a periphery or edge compartment, reaching a distance 200nm into the cell. We denote this volume as V E . We obtain a range of magnitudes for V E depending on assumptions about the geometric representation of this periphery compartment, that is whether data describes the periphery as a thin ring around the edge of the cell (as if a cross-section), or a thin film that covers the entire surface of the cell. (This is still to be discussed) Working with assumption 2,  VE ≈  1 4πr2 · ω = 2π(5-10µm)2 · 0.200µm, 2  which gives a range of V E ≈ 40-156µm3 . Assumption 1 would result in a much smaller value of V E , we will work with this value for now. We will denote the volume of the interior compartment as V I , it comprises the remaining volume fraction V I = Vtot − V E ≈ 200-1850µm3  90  Appendix  MATLAB Code for Simulations of Temporal Model in Chapter 3 %% Cofilin ODE - FULL with constant filament density clear; close all % ---Variables--% % PLC = PLC normalized to PLC(rest) % P2 = PIP2 normalized to P2(rest) % % C2 = Cofilin inactivated by PIP2 at membrane % CA = Cofilin active free diffusing % CF = Cofilin ready-to-sever filament bound % CM = Cofilin inactivated by G-actin monomer post-severing % CP = Cofilin inactivated by phosphate added by LIMK - freely diffusing % %---F = F-actin density normalized to F-actin(rest) % B = barbed end density - active polymerizing barbed ends  % ---Parameters---  % PLC d_plc=0.018;  % decay rate  I_stim=d_plc*1.5;  % scaled stimulation  % PIP2 d_p2=0.002;  % decay  d_hyd=0.01; %d_hyd=0.04;  % scaled stim hydrolysis ~20*(resting decay)  psi=1;  % binding ratio of C2:PIP2  91  Appendix % C2 d_c2=0.002;  % base decay rate (assume = PIP2 decay)  k_dp=0.5*d_c2;  % dephos rate (onto membrane) Assume slow  % CF k_sev=0.67/0.02*kp;  % base proportional severing rate  % CA kp=k_dp/3.4;  % satisfy rest states  k_off=0.08; k_onF=(k_sev + k_off)/(0.68-k_sev/kp);  % CM %kp_Act=0.01;  % enhanced phosphorylation  %b_SSH=0.01;  % reduction in phosphorylation rate due to G-actin removal  % CP  % B k_cap=1;  % capping rate  A=1000;  % ratio a*C_tot/B_rest  % Other n=7;  % non-linear factor  sat=0; %1;  % saturation on/off  K=1;  % saturation coefficient  t_stim=60;  % length of stimulus  t_off=120;  CP_re=0.20;  % Song (2006)  92  Appendix C2_re=0.10; % discussion CF_re=0.02;  % van Rheenen (2007)  CM_re=0.67; CA_re=0.01; % estimate  C_rest = [C2_re CA_re CF_re CM_re CP_re];  phi_F = CF_re;  % ---Initials---  C2_check = k_dp/(d_c2)*CP_re;  CA_check = 1/(kp + k_onF)*(d_c2*C2_re + k_off*CF_re);  CM_check = k_sev/kp*CF_re;  CF_check = k_onF/(k_off+k_sev)*CA_re;  CP_check = kp/k_dp*(CA_re + CM_re); % verify match defined rest states  % B_re = alpha*k_sev/kappa*CF_re;  C_check = [C2_check CA_check CF_check CM_check CP_check];  % print resting concentrations  disp(C_check) % ---Equations---  Fsev = @(x) k_sev*phi_F*(x./phi_F).^n;  % severing function  %Fsev = @(x) k_sev*phi_A*(x./phi_A).^n./(K^n+sat*(x./CF_re).^n);  % severing function  %Fsev = @(x) k_sev*x;  93  Appendix IPLC = @(x) I_stim*x;  KP_ACT = @(x) kp_Act*x;  EGF = @(x) (x>60)*(x<120);  % EGF stimulus  % ---ODEs---  % S = [PLC PIP2 C2 CA CF CM CP B T] (9)!  dSdt = @(t,S)[  IPLC(EGF(S(9))) - d_plc*(S(1)-1);  d_p2*(1-S(2)) - d_hyd*(S(1)-1)*S(2);  k_dp*S(2)*S(7) - d_c2*S(3) - d_hyd*(S(1)-1)*S(3);  d_c2*S(3) - kp*S(4) - k_onF*S(4) + k_off*S(5) + d_hyd*(S(1)-1)*S(3);  k_onF*S(4) - k_off*S(5) - k_sevb*S(5) - Fsev(S(5));  k_sevb*S(5) - kp*S(6) + Fsev(S(5));  kp*S(6) + kp*S(4) - k_dp*S(2)*S(7);  %k_cap*(1-S(8)) + A*Fsev(S(5)); %k_cap*(Fsev(S(5))/(k_sev*phi_F)-S(8)); A*Fsev(S(5)) - k_cap*S(8) 1];  % system ODEs  % ---Integration---  94  Appendix  options = odeset(’abstol’, 1e-28);  Sinit = [1 1 C2_re CA_re CF_re CM_re CP_re 1 0];  tfinal = 300;  % initial conditions  % final time in s  [T,S] = ode45(dSdt, [0 tfinal], Sinit, options);  95  

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