Article: Zhao, SM; Healey, T; Li, QD; “Direct Computation of Two-Phase Icosahedral Equilibria of Lipid Bilayer Vesicles”, Computer Methods in Applied Mechanics and Engineering, 314: 164-179; SI
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 vesicle 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. (C) 2016 Elsevier B.V. All rights reserved.
Funding Acknowledgement: National Science Foundation [DMS-1312377]
Funding Text: This work was supported in part of the National Science Foundation through grant DMS-1312377, which is gratefully acknowledged. We also thank Jim Jenkins, Chris Earls and Sanjay Dharmaravam for valuable discussions concerning this work.