Mathematical Modeling and Analysis of in vitro Actin Filament Dynamics and Cell Blebbing A DISSERTATION SUBMITTED TO THE FACULTY OF THE GRADUATE SCHOOL OF THE UNIVERSITY OF MINNESOTA BY Jifeng Hu IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF DOCTOR OF PHILOSOPHY Dr. Hans G. Othmer, Advisers December, 2009 c Jifeng Hu 2009 ° Acknowledgements I would like to express sincere thanks and gratitude to my thesis adviser Hans G. Othmer for the encouragement and guidance through these years of my PhD study. I would like to thank Tasos Matzavinos for his valuable discussion and help on the numerical simulations. I would like to thank all the members of Hans Othmer’s group for their comments and support. i Dedication Dedicated to my wife and my parents. ii Abstract The dynamics of the actin cytoskeleton are vital for cell motility observed in many biological processes, such as morphogenetic movements during embryotic development, fibroblast migra- tion during wound healing, and chemotactic movements of immune cells. To fulfill specific tasks, motile cells manipulate various actin structures within regions such as the lamellipodium, filopodia and stress fibers. A large pool of regulatory proteins and motor molecules coordinate the dynamic change of these structures to generate mechanical forces. In this thesis, we first investigate the temporal evolution of filament length distribution in a deterministic approach. The change of filament lengths is described by ordinary differential equations, and effects of diverse regulatory mechanisms are explored. We predict that the end- wise polymerization alone produces a long-lived Gaussian-like distribution of filament lengths, which eventually evolves to an exponential distribution. The introduction of fragmentation drastically leads to a Bessel-type equilibrium distribution. Our model confirms that profilin proteins slow filament growth, decreases the extent of polymerization, and promote filament treadmilling. Actin monomers are associated with nucleotide ATP. In a filament, ATP can hy- drolyze randomly into ADP-Pi, and subsequently release the phoshpate becoming ADP. Ran- dom ATP hydrolysis complicates the filament dynamics. The effect is analyzed in a stochastic model where each subunit within a filament is distinguished by associated nucleotide types. We theoretically predict a large length fluctuation occurring around the critical concentration of ATP-actin where the filament tip is bound intermittently by nucleotides ADP and ATP. By implementing an efficient stochastic simulation algorithm, we are able to track the evolution of iii length and nucleotide profile of single filament. We also investigate the phenomenon of cell blebbing, a typical membrane protrusion driven by actin dynamics and acto-myosin contraction. The major components of blebbing cells are recognized, and models for each component and their interaction are individually considered. Our analysis shows that a simple constraint on the membrane expansion rate relates the dynamic bleb size with the ring constricting the bleb. The properties of equilibrium state of blebbing are probed in a mechanical model whereby a uniform hydrostatic pressure is established by the balance of membrane tension in the contracting and expanding cell domains. We recognize that a potential membrane flow is important in establishing the blebbing equilibrium, and the influence of various flow types is compared. iv Contents Acknowledgements i Dedication ii Abstract iii List of Tables ix List of Figures x 1 Introduction 1 1.1 Outline of the thesis . 1 1.2 Biologicalbackground .............................. 2 1.2.1 Cell motility . 2 1.2.2 Biochemistry of actin and actin-binding proteins . 7 1.3 Review of the previous models . 12 1.3.1 Filament nucleation and elongation . 12 1.3.2 Fragmentation and annealing . 13 1.3.3 Large filament length fluctuations . 16 1.3.4 The nucleotide profile of filaments . 18 1.4 Motivation of the current work . 20 v 2 A deterministic model of pure actin polymerization 21 2.1 Introduction.................................... 21 2.2 Formulation of the nucleation-elongation model . 22 2.3 Analysis of the dynamics and steady state . 23 2.3.1 Existence, uniqueness and monotonicity of the steady state . 23 2.3.2 Asymptotic convergence to the steady state . 24 2.3.3 The time evolution of the first and second moments . 25 2.4 The time evolution of the filament length distribution . 27 2.4.1 The phases of filament polymerization . 27 2.4.2 The filament nucleation phase . 28 2.4.3 Theconvectivephase........................... 30 2.4.4 Thediffusivephase............................ 32 3 The dynamics of pure filament polymerization with fragmentation 38 3.1 Introduction.................................... 38 3.2 Formulation of the fragmentation model . 39 3.3 Analysis of the steady state of filament polymerization . 40 3.3.1 The monomer pool at the steady state . 40 3.3.2 The steady-state distribution of filament lengths . 41 3.4 Analysis of the dynamics of filament length distribution . 44 3.4.1 Overview of the time evolution of filament lengths . 44 3.4.2 Temporal evolution of filament mean length . 45 3.5 A continuum approximation of fragmentation . 50 4 Actin dynamics in the presence of profilin proteins 54 4.1 Introduction.................................... 54 4.2 Modeldescription................................. 55 4.3 Theoretical analysis . 56 4.3.1 Profilin slows the initial filament elongation . 56 vi 4.3.2 The steady-state filament length distribution . 58 4.3.3 The steady state of various protein populations . 63 4.4 Numericalresults ................................. 65 4.4.1 Time evolution of profilin-regulated filament lengths . 65 4.4.2 Protein concentrations and endwise polymerization . 67 4.4.3 The dependence of protein concentrations on Ptot ............ 70 4.4.4 The dependence of protein concentrations on Ftot ............ 73 5 A stochastic approach to actin filament polymerization with profilin 74 5.1 Introduction.................................... 74 5.2 Results of the stochastic model of pure actin dynamics . 77 5.2.1 Dynamics of the filament length distribution . 77 5.2.2 Dynamics of the nucleotide composition . 79 5.2.3 Effect from the filament population . 82 5.3 Results of the stochastic model of actin dynamics with profilin . 83 5.3.1 Dynamics of G-actin and F-actin . 84 5.3.2 Filament treadmilling and length fluctuations . 85 5.3.3 Dynamics of filament nucleotide profile . 87 6 Large length fluctuations: a single-filament approach 90 6.1 Introduction.................................... 90 6.2 The single-state filament model . 92 6.3 The two-state filament model . 94 6.3.1 The multi-step side walk model . 94 6.3.2 One-step side walk model . 106 6.4 The three-state filament model . 111 6.4.1 The master equation for filament states . 111 6.4.2 Result of numerical simulations . 114 vii 7 Cell blebbing 118 7.1 Background.................................... 118 7.1.1 Cellshape................................. 118 7.1.2 Cellblebbing ............................... 121 7.1.3 Blebbing of M2 cells . 122 7.1.4 Blebbing of NZ-treated detached fibroblast cells . 124 7.2 Mechanical elements of cell blebbing . 125 7.2.1 Plasma membranes . 126 7.2.2 Actin-myosin cortex . 127 7.2.3 Cytoplasm ................................ 128 7.2.4 Interactions between components . 128 7.3 Constraints on the cell configuration during blebbing . 129 7.4 A mechanical model for the equilibrium state of blebbing . 134 7.4.1 Modeldescription ............................ 134 7.4.2 The steady state and its stability . 137 7.4.3 The case with no cross-domain membrane flow . 139 7.4.4 The case with positive cross-domain membrane flow . 141 7.4.5 The case with uniform membrane tension . 143 8 Conclusions and future work 147 Bibliography 151 viii List of Tables 2.1 Terms in the perturbation expansion of the eigenvalues . 36 4.1 The kinetic rate constants used in the profilin-mediate filament model . 57 5.1 The kinetic rate constants for the stochastic profilin-actin model . 76 5.2 The kinetic rate constants for the stochastic profilin-actin model (continued) . 77 6.1 The kinetic rate constants used in the single filament model . 116 6.2 The kinetic rate constants used in the single filament model (continued) . 117 ix List of Figures 1.1 Actin-driven movement of different types of cells . 3 1.2 A schematic of cell movement cycle . 4 1.3 The microscopy of dense actin structures . 5 1.4 A dendritic nucleation treadmilling model of actin filament network . 7 1.5 The kinetics of actin monomers with different nucleotides . 10 1.6 The filament polymerization at the barbed end and pointed end . 11 1.7 Diverse actin-binding proteins . 12 2.1 Dynamics of filament lengths in the nucleation-elongation model . 27 2.2 A schematic of the network for nucleation and filament growth. 28 2.3 The time evolution of trimers and filament population . 29 2.4 The time evolution of G-actin concentration . 31 2.5 The convective phase of actin filament growth . 32 2.6 The time evolution of the mean and variance of the filament length distribution. 33 2.7 Dynamics of the components of filament length distribution . 36 3.1 Filament length distributions in in vitro assays.................. 39 3.2 The steady-state filament lengths at various fragmentation rates . 43 6 1 3.3 The fragmentation-regulated filament lengths. kf = 10− s− ......... 44 3.4 Dynamics of actin monomers and filament mean length with fragmentation . 46 3.5 The differentiated factors on average filament lengths . 49 3.6 The dynamics of average length of actin filaments under fragmentation . 50 x 3.7 The curve of function u(x) ............................ 52 3.8 The dependence of filament mean length on the fragmentation activities. 53 4.1 A deterministic model of profilin-regulated actin filament polymerization . 55 4.2 Actin dynamics in the absence and presence of profilin . 65 4.3 Statistics of filament lengths in the presence of profilin .