Lipid membranes with free edges

TL;DR

This paper develops a mathematical framework using differential forms to analyze lipid membranes with free edges, deriving equilibrium equations and solutions that explain experimental observations.

Contribution

It introduces a novel approach using exterior differential forms to model lipid membranes with free edges, deriving equilibrium equations and solutions.

Findings

Derived equilibrium equations and boundary conditions for membranes with free edges.

Obtained analytical and numerical solutions under axisymmetric conditions.

Results explain experimental observations by Saitoh et al.

Abstract

Lipid membrane with freely exposed edge is regarded as smooth surface with curved boundary. Exterior differential forms are introduced to describe the surface and the boundary curve. The total free energy is defined as the sum of Helfrich's free energy and the surface and line tension energy. The equilibrium equation and boundary conditions of the membrane are derived by taking the variation of the total free energy. These equations can also be applied to the membrane with several freely exposed edges. Analytical and numerical solutions to these equations are obtained under the axisymmetric condition. The numerical results can be used to explain recent experimental results obtained by Saitoh \emph{et al}. [Proc. Natl. Acad. Sci. \textbf{95}, 1026 (1998)].