Geometric theory for adhering lipid vesicles

This paper aims at constructing a general mathematical model for the equilibrium theory of adhering lipid vesicles from a geometrical point of view. Based on the generalized potential functional, a few differential operators and their integral theorems on curved surfaces, the general normal and tangential equilibrium differential equations and boundary conditions are given at the first time for inhomogeneous lipid vesicles. A general boundary condition ψ ˙ = 2 ( w − γ / R ) / k c is first put forward including line tension. No assumptions are made either on the symmetry of the vesicle or on that of the substrate. The physical and biological meaning of the equilibrium differential equations and the boundary conditions are discussed. Numerical simulation results based on the Helfrich energy for adhering lipid vesicles under the axial symmetric condition show the effectiveness and convenience of the present theory.