Symmetry-Breaking Global Bifurcation in a Surface Continuum Phase-Field Model for Lipid Bilayer Vesicles

Abstract

We study a model for lipid-bilayer membrane vesicles exhibiting phase separation, incorporating a phase field together with membrane fluidity and bending elasticity. We prove the existence of a plethora of equilibria in the large, corresponding to symmetry-breaking solutions of the Euler-Lagrange equations, via global bifurcation from the spherical state. To the best of our knowledge, this constitutes the first rigorous existence results for this class of problems. We overcome several difficulties in carrying this out. Due to inherent surface fluidity combined with finite curvature elasticity, neither the Eulerian (spatial) nor the Lagrangian (material) description of the model lends itself well to analysis. This is resolved via a singularity-free radial-map description, which effectively eliminates the grossly underdetermined in-plane deformation. The resulting governing equations comprise a quasi-linear elliptic system with nonlinear constraints. We show the equivalence of the problem to that of finding the zeros of compact vector field. The latter is not routine. Using spectral and a-priori estimates together with Fredholm properties, we demonstrate that the principal part of the quasi-linear mapping defines an operator with compact resolvent. We then combine well known group-theoretic ideas for symmetry breaking with global bifurcation theory to obtain our results.

A finite element method to compute three-dimensional equilibrium configurations of fluid membranes: Optimal parameterization, variational formulation and applications

“…This fact, which motivates our choice of coordinate, has been widely recognized in the literature [31][32][33]. However, it does not appear to have been exploited in numerical methods for computing equilibrium membrane configurations.…”

“…The matrices K, L and M are evaluated using the choice of variations indicated in (33) and the expressions in (27). The argument w h in (33) again highlights the fact that the tangent matrices are configuration dependent, which is a consequence of the nonlinearity of the problem.…”

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

“…In the absence of the latter, the model reduces to the well-known Helfrich model [7]. The existence of a plethora of symmetry-breaking equilibria, bifurcating from the perfect spherical shape, 20 has been recently established for this class of phase-field models [8]. The results include configurations possessing icosahedral symmetry, which have been observed in experiments sometimes taking on rather surprising "soccer-ball" shapes [9].…”

“…Certainly a great deal of patient, trial-and-error "tweaking" is required. Here we present a systematic approach to computing any equilibria within the multitude of symmetry types uncovered in [8]. We focus here on icosahedral symmetry, while methodically exploring parameter space via numerical continuation.…”

“…As pointed out in [14], the resulting discrete equations are wildly ill-conditioned -a direct consequence of mid-plane fluidity. We get around this difficulty via the approach used in [4], [8], [15], introducing the deformation as a radial graph over the unit sphere.…”