Geometric Methods in the Elastic... | Buy

This book contains a comprehensive description of the mechanical equilibrium and deformation of membranes as a surface problem in differential geometry. Following the pioneering work by W Helfrich, the fluid membrane is seen as a nematic or smectic -- A liquid crystal film and its elastic energy form is deduced exactly from the curvature elastic theory of the liquid crystals. With surface variation the minimization of the energy at fixed osmotical pressure and surface tension gives a completely new surface equation in geometry that involves potential interest in mathematics. The investigations of the rigorous solution of the equation that have been carried out in recent years by the authors and their co-workers are presented here, among which the torus and the discocyte (the normal shape of the human red blood cell) may attract attention in cell biology. Within the framework of our mathematical model by analogy with cholesteric liquid crystals, an extensive investigation is made of the formation of the helical structures in a tilted chiral lipid bilayer, which

Introduction to Liquid Crystal Biomembranes

1

(28)

Liquid Crystals

1

(7)

Mysterious Matter

1

(2)

Orientational Order

3

(1)

Classification of Thermotropics

4

(3)

Classification of Lyotropics

7

(1)

Amphiphiles and Lyotropic Liquid Crystals

8

(5)

Amphiphile

8

(2)

Monolayer

10

(1)

Micelle

11

(1)

Phase Diagram

12

(1)

Phase Transitions in Biomembranes

13

(6)

Fluid Mosaic Model

13

(1)

Lipid Bilayer

14

(1)

Phase Transitions in Bilayer

15

(2)

Classification of Lipid Bilayer Phase

17

(2)

Simple Biochemistry of LC Biomembranes

19

(5)

Effect of Chain Length

19

(1)

Double Bond Effect

20

(1)

Effect of Ionic Condition

21

(1)

Cholesterol Effect

22

(1)

Complexity in Biomembrane

23

(1)

Artificial Bilayers and Vesicles

24

(5)

Lipids for Artificial Vesicles

24

(1)

Multilamellar Vesicles

24

(1)

Single-Bilayered Vesicles

25

(1)

References

26

(3)

Curvature Elasticity Theory of Fluid Membranes

29

(42)

Shape Problem in Red Blood Cell

29

(4)

Membranes in Cell

29

(2)

High Deformability of Cell Membranes

31

(1)

Difficulty in the Explanation of Discocyte Shape

32

(1)

Differential Geometry of Surface

33

(14)

Lipid-Bilayer Vesicle Viewed as a Closed Surface

33

(1)

Space Curve

33

(2)

Surface and Parametric Curves

35

(1)

First Fundamental Form

36

(2)

Area

38

(1)

The Normal and the Tangent Plane

39

(1)

Second Fundamental Form

39

(1)

Christoffel Symbols

40

(1)

Curves and Directions on a Surface

41

(1)

Normal Curvature of a Curve on a Surface

42

(2)

Principal Directions, Line of Curvature and Principal Curvatures

44

(1)

The Mean Curvature and the Gaussian Curvature

44

(3)

Differential Invariants on a Surface

47

(5)

Gradient of a Scalar Field

47

(2)

Divergence of a Vector Field

49

(1)

Laplace-Beltrami Operator on a Scalar Function

50

(1)

Two-Dimensional Curl of a Vector

50

(1)

Other Differential Invariants

51

(1)

Curvature Elasticity of Fluid Membranes in Liquid Crystal Phase

52

(19)

Fluid Membranes Viewed as Liquid Crystals

52

(1)

Helfrich's Approach

53

(4)

A Derivation by Way of 2D Differential Invariants

57

(4)

The Spontaneous Curvature Viewed from Landau-de Gennes Theory

61

(3)

Discussion of Helfrich Bending Energy on Liquid Crystal Point of View

64

(2)

Spontaneous Curvature and Flexoelectric Effect

66

(2)

References

68

(3)

General Vesicle Shape Equation of Helfrich, Spontaneous Curvature Theory

71

(26)

Mathematical Preliminary

71

(11)

General Shape Equation

82

(4)

Spherical Vesicles

86

(4)

Circular Cylindrical Vesicles

90

(2)

Third Order Energy Variation

92

(5)

References

96

(1)

Solutions of General Shape Equation

97

(78)

Clifford Torus

97

(7)

Dupin Cyclide

104

(6)

Circular Biconcave Discoid

110

(4)

Surfaces of Revolution with Constant Mean Curvature and Extended Surfaces

114

(6)

Noncircular Cylindrical Surfaces

120

(22)

Euler-Lagrange Shape Equations for Axisymmetric Vesicles

142

(10)

Instability and Periodic Deformation in Bilayer Membranes

152

(7)

Numerical Observation of Nonaxisymmetric Vesicles

159

(16)

Model, Software, and the Procedure

160

(3)

Main Results

163

(7)

Discussion

170

(1)

References

170

(5)

Theory of Tilted Chiral Lipid Bilayers

175

(40)

Free Energy Expression for TCLB with Strong Chirality

175

(4)

Tilt-Equilibrium and Surface-Equilibrium Equations in Case of Strong Chiral Effect

179

(3)

The Wound Ribbon Helix

182

(6)

The Twisted Strip Helix

188

(5)

Spherical Vesicle

193

(3)

General Formula of Free Energy of TCLB

196

(5)

The Effect of Other Elastic Constants

201

(3)

High- and Low-Pitch Helical Structures of TCLBs

204

(11)

References

211

(4)

APPENDIX

215

(8)

I. Tensor Calculus

215

(5)

II. The Gradient Operator

220

(3)

Subject Index

223

(8)

Author Index

231

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