Equilibrium Theory and Geometrical Constraint Equation for Two-Component Lipid Bilayer Vesicles

This paper aims at the general mathematical framework for the equilibrium theory of two-component lipid bilayer vesicles. To take into account the influences of the local compositions togethers with the mean mean curvature and Gaussian curvature of the membrane surface, a general potential functional is constructed. We introduce two kinds of virtual displacement modes: the normal one and the tangential one. By minimizing the potential functional, the equilibrium differential equation and the boundary conditions of two-component lipid vesicles are derived. Additionally, the geometrical Constraint equation and geometrically permissible condition for the two-component lipid vesicles are presented. The physical, mathematical, and biological meanings of the equilibrium differential equations and the geometrical constraint equations are discussed. The influences of physical parameters oil the geometrically permissible phase diagrams are predicted. Numerical results call be used to explain recent experiments.