Direct Computation of Two-Phase Icosahedral Equilibria of Lipid Bilayer Vesicles

TL;DR

This paper develops a systematic computational approach to find icosahedral symmetric configurations of lipid-bilayer vesicles using a novel radial-graph formulation and symmetry reduction techniques, advancing understanding of membrane behavior.

Contribution

It introduces a radial-graph formulation and symmetry-based reduction method for directly computing icosahedral vesicle configurations in a complex lipid membrane model.

Findings

Successfully computed icosahedral vesicle configurations.

Demonstrated the effectiveness of symmetry reduction in complex membrane models.

Explored parameter effects on vesicle shapes.

Abstract

Correctly formulated continuum models for lipid-bilayer membranes present a significant challenge to computational mechanics. In particular, the mid-surface behavior is that of a 2-dimensional fluid, while the membrane resists bending much like an elastic shell. Here we consider a well-known Helfrich-Cahn-Hilliard model for two-phase lipid-bilayer vesicles, incorporating mid-surface fluidity, curvature elasticity and a phase field. We present a systematic approach to the direct computation of vesical configurations possessing icosahedral symmetry, which have been observed in experiment and whose mathematical existence has recently been established. We first introduce a radial-graph formulation to overcome the difficulties associated with fluidity within a conventional Lagrangian description. We use the so-called subdivision surface finite element method combined with an icosahedral-symmetric mesh. The resulting discrete equations are well-conditioned and inherit equivariance properties under a representation of the icosahedral group. We use group-theoretic methods to obtain a reduced problem that captures all icosahedral-symmetric solutions of the full problem. Finally we explore the behavior of our reduced model, varying numerous physical parameters present in the mathematical model.