On The Equivalence of Local and Global Area-constraint Formulations for Lipid Bilayer Vesicles

TL;DR

This paper proves that global and local area constraints in lipid bilayer vesicle models are mathematically equivalent for smooth, genus-zero surfaces, unifying two approaches in membrane modeling.

Contribution

It demonstrates the equivalence of global and local area constraint formulations in lipid vesicle models for genus-zero surfaces, including phase-field models.

Findings

Global and local constraints are equivalent for smooth, genus-zero vesicles.

The equivalence holds at the level of equilibrium equations and second variations.

The result applies to a broad class of membrane models.

Abstract

Lipid bilayer membranes are commonly modeled as area-preserving fluid surfaces that resist bending. There appear to be two schools of thought in the literature concerning the actual area constraint. In some works the total or global area (GA) of the vesicle is a prescribed constant, while in others the local area ratio is assigned to unity. In this work we demonstrate the equivalence of these ostensibly distinct approaches in the specific case when the equilibrium configuration is a smooth, closed surface of genus zero. We accomplish this in the context of the Euler-Lagrange equilibrium equations, constraint equations and the second-variation with admissibility conditions, for a broad class of models - including the phase-field type.