Geometric theory on the elasticity of bio-membranes

TL;DR

This paper develops a geometric framework using exterior differential forms to analyze the shapes and stability of bio-membranes, offering a unified approach to various membrane configurations.

Contribution

It introduces a geometric scheme for deriving shape and stability equations of bio-membranes based on variational problems with exterior differential forms.

Findings

Derived shape equations for closed and open lipid bilayers

Analyzed stability conditions of bio-membranes

Unified treatment of different membrane types using geometric methods

Abstract

The purpose of this paper is to study the shapes and stabilities of bio-membranes within the framework of exterior differential forms. After a brief review of the current status in theoretical and experimental studies on the shapes of bio-membranes, a geometric scheme is proposed to discuss the shape equation of closed lipid bilayers, the shape equation and boundary conditions of open lipid bilayers and two-component membranes, the shape equation and in-plane strain equations of cell membranes with cross-linking structures, and the stabilities of closed lipid bilayers and cell membranes. The key point of this scheme is to deal with the variational problems on the surfaces embedded in three-dimensional Euclidean space by using exterior differential forms.