Compatibility between shape equation and boundary conditions of lipid membranes with free edges

TL;DR

This paper investigates the compatibility of shape equations and boundary conditions in lipid membranes with free edges, deriving conditions for existence, proving non-existence of certain membrane shapes, and validating solutions with experimental data.

Contribution

It establishes the conditions under which shape equations and boundary conditions are compatible, including the vanishing of the first integral and non-existence theorems for specific membrane shapes.

Findings

First integral of shape equation vanishes for axisymmetric membranes

Non-existence of certain open membrane shapes (torus, biconcave discodal)

Numerical solutions align with experimental data

Abstract

Only some special open surfaces satisfying the shape equation of lipid membranes can be compatible with the boundary conditions. As a result of this compatibility, the first integral of the shape equation should vanish for axisymmetric lipid membranes, from which two theorems of non-existence are verified: (i) There is no axisymmetric open membrane being a part of torus satisfying the shape equation; (ii) There is no axisymmetric open membrane being a part of a biconcave discodal surface satisfying the shape equation. Additionally, the shape equation is reduced to a second-order differential equation while the boundary conditions are reduced to two equations due to this compatibility. Numerical solutions to the reduced shape equation and boundary conditions agree well with the experimental data [A. Saitoh \emph{et al.}, Proc. Natl. Acad. Sci. USA \textbf{95}, 1026 (1998)].