Kinks in two-phase lipid bilayer membranes

Abstract

Abstract. Common models for two-phase lipid bilayer membranes are based on an energy that consists of an elastic term for each lipid phase and a line energy at interfaces. Although such an energy controls only the length of interfaces, the membrane surface is usually assumed to be at least C 1 across phase boundaries. We consider the spontaneous curvature model for closed rotationally symmetric two-phase membranes without excluding tangent discontinuities at interfaces a priorily. We introduce a family of energies for smooth surfaces and phase fields for the lipid phases and derive a sharp interface limit that coincides with the Γ-limit on all reasonable membranes and extends the classical model by assigning a bending energy also to tangent discontinuities. The theoretical result is illustrated by numerical examples.

Thermodynamic equilibrium of binary mixtures on curved surfaces

“…in Refs. [37][38][39][40][41][42][43][44], and also by us in Ref. [18]) and appears to be the most popular choice within the mathematics-oriented literature.…”

Gradient flow dynamics of two-phase biomembranes: Sharp interface variational formulation and finite element approximation

“…There have been several studies on theoretical and numerical aspects of two-phase membranes taking curvature elasticity and line energy into account, see e.g. [28,29,39,11,40,15,30,22,23,24,25,26,13,32,14,9], which we discuss in the following.…”

“…In [29] it is assumed that the surface Γ = Γ 1 ∪ γ ∪ Γ 2 is a C 1 -surface, meaning in particular that the normal to Γ is continuous across the phase boundary γ. The works [25,26,27], on the other hand, also allow for discontinuities of the normal at γ. The first variation of the energy E in (1.1) has been derived in [22] for the C 1 -case and in [41] for the C 1 -and the C 0 -case.…”

“…Line energy in this context is replaced by a Ginzburg-Landau energy like in the classical Cahn-Hilliard theory. We refer to [40,30,22,23,24,25,26,32,31] for numerical results based on the phase field approach. The above papers use a gradient flow approach to obtain equilibrium shapes in the large time limit.…”

“…So far only results for the axisymmetric case are known. We refer to the work [13], where the existence of global minimizers for axisymmetric multi-phase membranes was shown, and the works [25,26,27], where the sharp interface limit of the phase field approach in an axisymmetric situation was studied. Existence results for the evolution problem are not available in the literature so far and should be addressed in the future.…”