在生物細胞中，細胞膜通常會耦合到actin cortex (主要由一種細胞骨架—微絲
(actin filaments)—所組成)︒這將形成一種層狀結構並決定大部分細胞的運動︒
自從科學家在十九世紀發現了紅血球細胞膜上的震動現象 (flickering phenomena) 後，有些物理學家 (現在我們稱這些人為生物物理學家) 開始對細胞
的動力學產生興趣並且開始構築一些模型去解釋觀察到的現象︒在此論文中，我將提出一些理論模型去探討細胞膜耦合到actin cortex的動力學︒在我們最新
的模型中，我們考慮了肌凝蛋白 (myosin protein/motor) 會在actin cortex 產生收縮性的應力 (contractile stress)，這些肌凝蛋白作用就是提供actin cortex一個力偶極子 (force dipole) 的分布 (distribution)︒我們發現肌凝蛋白的收縮性在長波長的時候對系統很重要，並且會使的細胞膜進入不平衡的狀態 (non-equilibrium states)︒此外，臨界波長 (crossover wavelength) 與系統中肌凝蛋白的根號濃度 (concentration) 倒數成正比︒從系統的特徵時間(characteristic time scale) 可以觀察到從純黏滯到肌凝蛋白的收縮性主導的區間 (from pure viscous to contractility-dominant
regimes)︒最後，我們也會討論這些臨界波長和各區間的物理意義︒

Keywords: actin cortex， biomembrane (生物膜)， soft matter (軟物質)， active matter

In a biological cell, the cell membrane is coupled to the actin cortex. This forms an outer layer that determines much of the mechanics of the cell. Since scientists discovered the flickering phenomena in red blood cells in the nineteenth century, biophysicists constructed models that were expected to explain the dynamics of cell membranes. In this thesis, I present theoretical models to investigate the dynamics of a membrane coupled to an actin cortex. The main feature of our lastest model is that the contractility provided by the myosin motors in the actin cortex is included. This provides a distribution of active force dipoles in the actin cortex. We found that the contractility from the myosin becomes significant in the long wavelength limit, bringing the membrane into non-equilibrium states. The crossover wavelength is proportional to the inverse square root of myosin concentration.
Also, the characteristic time scale for the cell membrane dynamics shows crossover from purely viscous to contractility-dominant regimes. Last but not the least, the physical meaning of the crossovers are discussed in details.

Keywords: actin cortex, biomembrane, soft matter, active matter.
1

1 Introduction 5
2 Background knowledges 9
2.1 Lipid bilayer . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9
2.1.1 Monge parametrization . . . . . . . . . . . . . . . . . . . . . 9
2.1.2 Free energy of a lipid bilayer . . . . . . . . . . . . . . . . . . 10
2.2 Actin cortex . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11
2.3 Incompressibility condition for actin gel and fluid outside the membrane
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13
3 A passive membrane embedded in a fluid above a static wall 14
3.1 Free energy, hydrodynamic equation, and boundary conditions . . . 14
3.2 Hydrodynamic decay rate . . . . . . . . . . . . . . . . . . . . . . . 17
3.3 Mean square displacement (MSD) . . . . . . . . . . . . . . . . . . . 20
4 Model I: a membrane coupled to a passive actin cortex 23
4.1 Free energy, hydrodynamic equations, and boundary conditions . . 23
4.2 Hydrodynamic decay rate . . . . . . . . . . . . . . . . . . . . . . . 26
4.3 Mean square displacement (MSD) . . . . . . . . . . . . . . . . . . . 31
5 Model II: a membrane coupled to a active actin cortex 37
5.1 Free energy, hydrodynamic equations, and boundary conditions . . 37
5.2 Hydrodynamic decay rate . . . . . . . . . . . . . . . . . . . . . . . 42
5.3 Mean Square Displacement (MSD) . . . . . . . . . . . . . . . . . . 49
3
6 Summary and future works 52
APPENDICES 54
A Fourier transform of membrane free energy 55
B Solving the velocity and pressure around a membrane 57
C The velocity and the pressure field in Model II 62
Bibliography 64

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