Direct computation of two-phase icosahedral equilibria of lipid bilayer vesicles

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.

A Lagrangian Thin-Shell Finite Element Method for Interacting Particles on Fluid Membranes

“…Some methods impose local area incompressibility (instead of global area constraint) [ 26 ] to suppress zero-energy modes, while others [ 25 , 39 ] dampen tangential motion by introducing ad hoc in-plane energies whose contributions are iteratively decreased to zero. Monge representation [ 29 ] or a radial graph ansatz [ 28 , 40 ]—where the surface is parametrized using a single unknown function—have also been used to circumvent reparametrization invariance. In this work, we employ the gauge-fixing procedure recently proposed in [ 27 ], which is computationally efficient for topologically spherical surfaces and does not require any iterative reduction in ad hoc energy terms that change the physics of the model.…”

“…We applied this framework to study interacting particles on a spherical fluid shell, employing a spectral Galerkin method and accordingly discretizing the surface deformation map using a spherical harmonic expansion. To circumvent computational issues stemming from the in-plane fluidity of the substrate [ 25 , 26 , 27 ], a radial graph ansatz [ 28 ] was implemented. According to this ansatz, the displacement is parametrized along the radial direction from the center of the reference sphere.…”

Numerical Methods for Biomembranes: conforming subdivision methods versus non-conforming PL methods

“…These numerical methods continue to be used extensively in the study of phenomena in biomembranes, see, e.g. [67,4,37,38,2] and references therein. Similar geometric variational problems show up in other scientific areas.…”

“…In principle, we can optimize in a symmetry preserving way by optimizing only over the degrees of freedom that determine the control mesh up to the desired symmetry, as is done in Brakke's Surface Evolver or [67]. This has the added advantage of reducing the dimensionality of the problem, but requires an extra effort in coding and algorithmic development.…”

Direct construction of symmetry-breaking directions in bifurcation problems with spherical symmetry

“…The latter are useful, e.g., in understanding/visualizing symmetry-breaking behavior, for numerical bifurcation purposes, cf. [1], [4], [15], etc.…”