Shell Transformation Model for Simulating Cell Surface Structure KOH TIONG SOON [B.Appl.Sci(Hons.)] A THESIS SUBMITTED FOR THE DEGREE OF DOCTOR OF PHILOSOPHY DEPARTMENT OF MATERIALS SCIENCE AND ENGINEERING NATIONAL UNIVERSITY OF SINGAPORE 2012 Acknowledgement I would like to express my great gratitude to my supervisor Dr Chiu Cheng-hsin for his invaluable guidance and encouragement during my Ph.D study I will also like to thank all the group members, Huang Zhijun, Gerard Paul Marcelo Leyson, and Lai Weng Soon for the insightful discussions and all the assistance Special thanks will be given to my parents for their remarkable patience and constant support It will not be possible for me to complete my study without them Finally, I want to acknowledge National University of Singapore for the research scholarship i Contents Acknowledgement i Contents ii Abstract vi List of Figures viii List of Symbols x Introduction 1.1 Overview of Cell Surface Structures 1.2 Literature Review of Cell Mechanics 1.3 Objective and Approach 1.4 Outline 10 Kinematics of Thin Shell 12 2.1 Thin Shell Model 12 2.2 Kirchhoff-Love Postulate 14 2.3 Fundamental Quantities of a Surface 15 ii Contents iii 2.4 Deformation Gradient 16 2.5 Lagrangian Strain 18 2.6 Lagrangian Strains for Infinitesimal Deflection and Deformation 20 2.7 Axial Symmetric Shell 22 2.8 Comparison of Bending Strains 26 2.9 Summary 29 Thin Elastic Shell Under Finite Elasticity 30 3.1 Finite Elasticity 30 3.2 Linear Elasticity 32 3.3 Stress Resultants and Stress-Couple Resultants 32 3.4 Equilibrium Equations 35 3.4.1 Free Energy of Shells 36 3.4.2 Variation δU0 37 3.4.3 Variation δUQ 41 3.4.4 Variation δW 43 3.4.5 Balance of Force and Moment 43 3.5 Equilibrium Equations for Axial Symmetric Shell 45 3.6 Summary 47 Contents iv Shell Transformation Model 48 4.1 Introduction 48 4.2 Deformation 49 4.3 Lagrangian Strains in Shell under Biaxial Transformation Strains 51 4.4 Linear Transformation Strain 52 4.5 Numerical Implementation 54 4.5.1 Expressions for u1 and u3 54 4.5.2 Residual Loading 55 4.5.3 Numerical Iterations 55 Summary 57 4.6 Pit Formation of Clathrin Mediated Endocytosis 58 5.1 Introduction 58 5.2 Model 61 5.3 Simulation for Pit Formation 63 5.4 Parametric Study on Pit Formation Mechanism 66 5.5 Pit Formation in Cells with Different Shapes 72 5.6 Simulation for Coat Protein Budding 77 5.7 Discussion 79 5.8 Summary 81 Contents v Invagination of Clathrin Mediated Endocytosis 83 6.1 Introduction 83 6.2 Model 85 6.3 Simulation for Plasma Membrane Remodeling 89 6.4 Simulation for Rocketing Actin Filaments 92 6.5 Simulation for Intrinsic Shear Dipole 97 6.6 Summary 102 Phagocytosis and Viral Budding 103 7.1 7.2 Simulation for Phagocytosis 103 7.3 Simulation for Viral Budding 107 7.4 Introduction 103 Summary 112 Conclusion 114 Abstract The morphology of biological cells changes significantly when the cells carries out biological processes These morphological changes are controlled by the plasma membrane, the biochemical signaling, and the actin network The roles of plasma membrane and biochemical signaling have been studied extensively in the literature, while the roles of the actin network during these biological processes are less understood For example, actin filaments are known to be active in clathrinmediated endocytosis, phagocytosis, and viral budding However, how a thin actin network is capable of producing the drastic morphological changes in these processes is still open question from the mechanics point of view In this thesis research, a model is developed for investigating the deformation mechanisms of the cell surface structures during the biological process that involves significant morphological changes The model consists of two parts: The first one is the mechanics of the cell surface structure, and this is taken into account by a thin shell theory that allows large deformation and finite elasticity in the system The second part, on other hand, describes the changes in the cell surface structures when the cell carries out the biological processes The changes are represented by transformation strains, forces, and dipoles in the shell The model is termed the shell transformation model in this thesis vi Abstract vii The shell transformation model is applied to examine the pit formation and invagination process during clathrin-mediated endocytosis, the viral budding, and the formation of pseudopodium during the phagocytosis Of particular interest are the mechanisms that lead to the unique morphology observed in the experiments of the biological processes List of Figures 1.1 Schematic diagrams of key components in the cell 2.1 Schematic diagram of the thin shell 13 2.2 Schematic diagram axial-symmetric thin shell in its reference state 23 2.3 Schematic diagram of thin shell subjected to in-plane strain 29 4.1 Schematic diagrams of the shell transformation states 50 5.1 Schematic diagrams of clathrin mediated endocytosis 59 5.2 Schematic diagram of a thin shell in its reference state 62 5.3 Effects of area mismatch and curvature mismatch on pit morphologies 64 5.4 Comparing the effect of area and curvature mismatch on pit formation 66 5.5 Effects of coating size on pit formation 5.6 Characteristic parameter of pit morphology due to size of strain layer 68 5.7 Effects of strained layer thickness on pit formation 71 5.8 Effects of strained layer position on pit formation 72 5.9 Pit morphology induced by area mismatch strain in prolate spheroid 73 67 5.10 Characteristic parameters of pit formation in prolate spheriod 75 5.11 Pit morphology induced by area mismatch strain in oblate spheriod 76 5.12 Characteristic parameters of pit formation in oblate spheriod 77 5.13 Budding morphology due to area and curvature mismatch 79 5.14 Variation of area mismatch with epsin concentration 80 6.1 Schematic diagram of a thin shell subject to in-plane force 86 6.2 Simulation for plasma membrane relaxation 90 viii List of Figures ix 6.3 Effects of area mismatch strain in plasma membrane 92 6.4 Effects of q1 with different φ0 on pocket morphology 93 6.5 Characteristic parameters of pocket morphology due to q1 with different φ0 95 6.6 Effects of q1 with different φw on pocket morphology 96 6.7 Characteristic parameters of pocket morphology due to q1 with different φw 97 6.8 Pocket morphology due to intrinsic shear dipole with different φ0 98 6.9 Effects of intrinsic shear dipole with different φ0 on the characteristic parameters of pocket morphology 99 6.10 Pocket morphology due to intrinsic shear dipole with different φw 101 6.11 Effects of intrinsic shear dipole with different φw on the characteristic parameters of pocket morphology 101 7.1 m Variation of E13 with φ 105 7.2 Simulation for phagocytosis by intrinsic shear dipole 106 7.3 Effects of φw and φ0 on cell surface 107 7.4 Schematic diagrams of viral budding 108 7.5 Simulation for viral budding by in-plane force and intrinsic shear dipole 110 Chapter 8: Conclusion 115 and is found to be consistent with the result in the literature The second part of Chapter 4, on the other hand, is the numerical implementation of our thin shell theory In particular, the numerical scheme solves the non-linear equations in the theory by using a pseudo dynamic formula to search for the shell morphology that minimizes the free energy of the system Chapter investigates the formation of clathrin coated pit by considering the effects of area mismatch and curvature mismatch After understanding the significance of the area mismatch, the focus of Chapter then turns to the dependence of the pit formation on three geometric factors of the clathrin coating, namely, the coating size, the effective thickness, and the position The results show that the depth of the pit can be increased by increasing the magnitude of any of the three factors Though affected by these factors, the pit remains shallow, suggesting that the clathrin coating can produce a pit but is unable to generate a deep pocket for subsequent invagination process Similar findings are also found in the simulation for the budding of vesicles by coat protein Chapter studies the development of deep pocket on cell surface during the invagination phase in clathrin mediated endocytosis The study is carried by modeling the two machanisms of the actin network by the in-plane force and the intrinsic shear dipole, respectively The results show that the in-plane force can generate deep pockets when the force is sufficiently large The intrinsic shear dipole on the other hand, is also able to form pockets but with a larger pocket opening Chapter applies the shell transformation model to the formation of pseudopodium during the phagocytosis and the viral budding The pseudopodium induced by motor proteins is modeled by the intrinsic shear dipole, and it was found that the shear dipole is an effective mechanism to generate the large surface deformation The simulation results also show that by varying the distribution of the loading region, the surface profile resembling that in experiments can be generated The study of viral budding focuses on the formation of shallow bumps by the viral proteins and the subsequent development of the budding by the actin Chapter 8: Conclusion 116 filament network The formation of shallow bump is similar to the generation of pits by the clathrin coating The difference between the two similar scenarios is that that the viral protein needs to exhibit a negative area mismatch strain in order to induce the bump, while the clathrin coating is characterized by a positive area mismatch to produce a pit The subsequent budding by actin filament can be achieved by rocketing of actin filaments and the shear forces induced by motor motion in the actin network The former is modeled by the in-plane force and the latter by the intrinsic shear dipole The simulation results show that both mechanisms can cause budding, while the size of the budding 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termed the shell. .. springs Shell with Liquid Core model Modeling cells as a shell with a liquid core includes the effects of the thin surface structure and the cytoplasm on the deformation of the cells In this model,