1 Introduction to Liquid Crystal Biomembranes
1.1 Liquid Crystals
1.1.1 Mysterious Matter
1.1.2 Orientational Order
1.1.3 Classification of Thermotropics
1.1.4 Classification of Lyotropics
1.2 Amphiphiles and Lyotropic Liquid Crystals
1.2.1 Amphiphile
1.2.2 Monolayer
1.2.3 Micelle
1.2.4 Phase Diagram
1.3 Phase Transitions in Biomembranes
1.3.1 Fluid Mosaic Model
1.3.2 Lipid Bilayer
1.3.3 Phase Transitions in Bilayer
1.3.4 Classification of Lipid Bilayer Phase
1.4 Simple Biochemistry of LC Biomembranes
1.4.1 Effect of Chain Length
1.4.2 Double Bond Effect
1.4.3 Effect of Ionic Condition
1.4.4 Cholesterol Effect
1.4.5 Complexity in Biomembrane
1.5 Artificial Bilayers and Vesicles
1.5.1 Lipids for Artificial Vesicles
1.5.2 Multilamellar Vesicles
1.5.3 Single-Bilayered Vesicles
References
2 Curvature Elasticity of Fluid Membranes
2.1 Shape Problem in Red Blood Cell
2.1.1 Membranes in Cell
2.1.2 High Deformability of Cell Membranes
2.1.3 Difficulty in the Explanation of Discocyte Shape
2.2 Classic Differential Geometry of Surface
2.2.1 Lipid-Bilayer Vesicle Viewed as a Closed Surface
2.2.2 Space Curve
2.2.3 Surface and Parametric Curves
2.2.4 First Fundamental Form
2.2.5 Area
2.2.6 The Normal and the Tangent Plane
2.2.7 Second Fundamental Form
2.2.8 Christoffel Symbols
2.2.9 Curves and Directions on a Surface
2.2.10 Normal Curvature of a Curve on a Surface
2.2.11 Principal Directions, Line of Curvature and Principal Curvatures
2.2.12 The Mean Curvature and the Gaussian Curvature
2.3 Differential Invariants on a Surface
2.3.1 Gradient of a Scalar Field
2.3.2 Divergence of a Vector Field
2.3.3 Laplace-Beltrami Operator on a Scalar Function
2.3.4 Two-Dimensional Curl of a Vector
2.3.5 Other Differential Invariants
2.4 Curvature Elasticity of Fluid Membranes in Liquid Crystal Phase
2.4.1 Fluid Membranes Viewed as Liquid Crystals
2.4.2 Helfrich’’s Approach
2.4.3 A Derivation by Way of 2D Differential Invariants
2.4.4 The Spontaneous Curvature Viewed from Landau-de Gennes Theory
2.4.5 Discussion of Helfrich Bending Energy on Liquid Crystal Point of View
2.4.6 Spontaneous Curvature and Flexoelectric Effect
References
3 Shape Equation of Lipid Vesicles and Its Solutions
3.1 Mathematical Preliminary
3.2 General Shape Equation
3.3 Spherical Vesicles
3.4 Nearly Spherical Vesicles and Third Order Energy Variation
3.5 Circular Cylindrical Vesicles
3.6 Noncircular Cylindrical Vesicles
3.7 Clifford Torus
3.8 Dupin Cyclide
3.9 Shape Equation for Axisymmetric Vesicles
3.10 Circular Biconcave Discoid
3.11 Surfaces of Revolution with Constant Mean Curvature and
Extended Surfaces
3.12 Challenge
References
4 Governing Equations for Open Lipid Membranes and
Their Solutions
4.1 Mathematical Preliminary
4.1.1 Surface Theory Based on Moving Frame Method
4.1.2 Hodge Star, Stokes Theorem and some Important Geometric
Relations
4.1.3 Variational Theory Based on the Moving Frame Method
4.2 Governing Equations for Open Lipid Membranes
4.2.1 Governing Equations
4.2.2 Compatibility Conditions
4.3 Analytic Solutions
4.4 Quasi-Exact Solutions
4.5 Challenge
4.5.1 Neck Condition of Two-Phase Vesicles in the Budding State.
4.5.2 Minimal Surface with Boundary Curve
References
5 Theory of Tilted Chiral Lipid Bilayers
5.1 Theory of Tilted Chiral Lipid Bilayers with Strong Chirality
5.1.1 Geometric Description and Free Energy
5.1.2 Tilt-Equilibrium and Surface-Equilibrium Equations in Case of Strong Chiral Effect
5.1.3 Wound Ribbon Helix
5.1.4 Twisted Ribbon
5.1.5 Spherical Vesicle
5.2 General Theory for TCLB
5.2.1 General Formula of Chiral Pree Energy
5.2.2 The Effect of Other Elastic Constants
5.2.3 High- and Low-Pitch Helical Structures of TCLBs
5.3 Concise theory for Chiral Lipid Membranes
5.3.1 Brief Introduction to Moving Frame Method and Exterior Differential Forms
5.3.2 Constructing the Free Energy
5.3.3 Governing Equations to Describe Equilibrium Configurations of CLMs without Free Edges
5.3.4 Governing Equations to Describe Equilibrium Configurations of CLMs with Free Edges
5.3.5 Solutions to Governing Equations of CLMs without Free Edges
5.4 Challenge
References
6 Some Untouched Topics
6.1 Nonlocal Theory of Membrane Elasticity
6.1.1 Area-difference Elasticity
6.1.2 Membrane with Nonlocal Interactions
6.2 Numerical Simulations
6.2.1 Surface Evolver
6.2.2 Phase Field
6.3 Effects of Membrane Skeleton
6.3.1 Composite Shell Model of Cell Membranes
6.3.2 Stability of Cell Membranes and the Function of Membrane Skeleton
6.4 Application in Elasticity of Low-Dimensional Carbon Materials
6.4.1 Revised Lenosky Model
