Convergence of an Approximation for Rotationally Symmetric Two-Phase Lipid Bilayer Membranes

Abstract

We consider a diffuse interface approximation for the lipid phases of rotationally symmetric two-phase bilayer membranes and rigorously derive its Γ-limit. In particular, we prove that limit vesicles are C 1 across interfaces, which justifies a regularity assumption that is widely made in formal asymptotic and numerical studies. Moreover, a limit membrane may consist of several topological spheres, which are connected at the axis of revolution and resemble complete buds of the vesicle. 6,12,13,15,22]. Here H and K are the mean curvature and the Gauss curvature of the membrane surface M , and µ is its area measure. The bending rigidities k ± > 0 and the Gauss rigidities k ± G are elastic material parameters, and H ± s -the so-called spontaneous curvatures -are supposed to reflect an asymmetry in the membrane. In the simplest case, the rigidities and spontaneous curvatures are constant within each lipid phase but different between the two phases. The length of the phase interfaces ∂M + = ∂M − is denoted by H 1 (∂M + ), and σ is a line tension parameter. among all closed surfacesIn [15] the Euler-Lagrange equations of (1.1) for axially symmetric two-phase membranes with exactly one interface are studied. The authors mention the possibility of different smoothness conditions at the interface, their analysis, however, is done for smooth membranes, which are C 1 across the interface, only. Phase field models for the lipid phases and also the membrane surface are introduced in [10,11,28,17] and studied numerically; convergence to the sharp interface limit is obtained by asymptotic expansion and under additional smoothness assumptions and topological restrictions.In this paper we are interested in the convergence of a diffuse interface approximation for the lipid phases in a rotationally symmetric setting without imposing smoothness or the topological structure of limit vesicles in advance. More precisely, for a closed surface arXiv:1603.05231v1 [math.AP]

Finite element approximation for the dynamics of fluidic two-phase biomembranes

“…Sharp interface limits of phase field approaches to two-phase membranes have been studied with the help of formal asymptotics by Elliott and Stinner (2010b) in the case of a C 1 -limiting surface, and rigorously by Helmers (2013) for axisymmetric two-phase membranes allowing for tangent discontinuities at interfaces. Later Helmers (2015) also showed a rigorous convergence result for the axisymmetric situation in the C 1 -case.…”

“…Through their first variation these energy contributions lead to driving forces for the evolution, which is given by a surface Navier-Stokes system, coupled to bulk dissipation of an ambient fluid, and a convective Cahn-Hilliard type equation, which is formulated on the evolving membrane. The fluid part of the model goes back to Arroyo and DeSimone (2009), whereas an evolution based on a Canham-Evans-Helfrich energy coupled to a Ginzburg-Landau energy on the surface has been studied in the context of gradient flows by Elliott andStinner (2010a,b, 2013); Helmers (2013); Mercker et al (2013); Helmers (2015); Mercker and Marciniak-Czochra (2015). However, a coupling, which will give the natural dynamics on the interface, is stated here for the first time, and we will show that physically reasonable energy dissipation inequalities hold.…”

Kinks in two-phase lipid bilayer membranes

“…Convergence to the sharp interface limit in these approaches is obtained by asymptotic expansion and under strong smoothness assumptions on the limit surface; in particular, the membrane is again assumed to be at least C 1 across interfaces. In [14] we prove that for rotationally symmetric membranes this regularity need not be assumed, but is included in the Γ-limit of an appropriate surface phase field approximation.…”

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

“…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.…”

“…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.…”