Modeling and computation of two phase geometric biomembranes using surface finite elements

Abstract

Biomembranes consisting of multiple lipids may involve phase separation phenomena leading to coexisting domains of different lipid compositions. The modeling of such biomembranes involves an elastic or bending energy together with a line energy associated with the phase interfaces. This leads to a free boundary problem for the phase interface on the unknown equilibrium surface which minimizes an energy functional subject to volume and area constraints. In this paper we propose a new computational tool for computing equilibria based on an L 2 relaxation flow for the total energy in which the line energy is approximated by a surface Ginzburg-Landau phase field functional. The relaxation dynamics couple a nonlinear fourth order geometric evolution equation of Willmore flow type for the membrane with a surface Allen-Cahn equation describing the lateral decomposition. A novel system is derived involving second order elliptic operators where the field variables are the positions of material points of the surface, the mean curvature vector and the surface phase field function. The resulting variational formulation uses H 1 spaces, and we employ triangulated surfaces and H 1 conforming quadratic surface finite elements for approximating solutions. Together with a semi-implicit time discretization of the evolution equations an iterative scheme is obtained essentially requiring linear solvers only. Numerical experiments are presented which exhibit convergence and the power of this new method for two component geometric biomembranes by computing equilibria such as dumbbells, discocytes and starfishes with lateral phase separation.

Convergence of the point integral method for Laplace–Beltrami equation on point cloud

“…is the ith component of η(x, y) defined in (21). To simplify the notation, we drop the variable (x, y) in the function η(x, y).…”

“…In many problems, including material science [10,20], fluid flow [22,25], biology and biophysics [2,3,21,37], people need to study the physical process, for instance diffusion and convection, in curved surfaces which introduce different kinds of PDEs in surfaces. It has been several decades to develop numerical methods for solving PDEs in surfaces.…”

A Fully Discrete Evolving Surface Finite Element Method

“…Partial differential equations on moving surfaces appear in a wide range of applications such as material science [3,7], fluid flow [10,12], and biology and biophysics [2,8,14,16]. Numerical approaches include level set methods, surface finite elements, finite volume methods, and diffuse interface methods; see [1,4,5,9,13,15,17] and the references cited therein.…”

Passing from bulk to bulk-surface evolution in the Allen–Cahn equation

“…[14]). In particular, the behavior of two solutions of (AC bulk ) near the boundary Γ is depicted for subsequent times in case of the dynamical boundary condition (14) and the Neumann boundary The dynamic boundary condition models a strong interaction between the wall and the mixture components described by the order parameter u (resp., U ). In particular, due to the potential W s and the surface diffusion we have an accumulation of the phase U = 1 at the boundary Γ, which can be clearly seen in the pictures.…”

“…As a longterm goal we shall apply this approach to the related problem of deriving interface conditions in reaction-diffusion systems. Nonstandard interface and transmission conditions in semiconductor heterostructures and biological systems are of great importance (see [14,18,36]). Especially in organic photovoltaics interfaces are the fundamental building block, see [28,Sect.…”