Domain formation via phase separation for spherical biomembranes with small deformations

TL;DR

This paper models lipid raft formation on biological membranes by coupling curvature and composition, deriving a surface Cahn-Hilliard functional, and demonstrating the emergence of stable raft-like structures through analysis and simulations.

Contribution

It introduces a novel energy functional coupling membrane curvature and composition, and analyzes the formation and stability of raft-like structures on deformed biomembranes.

Findings

Raft-like structures can emerge and stabilize over time.

Existence and uniqueness of minimizers and gradient flow solutions are proven.

Parameter dependence of raft formation is numerically explored.

Abstract

We derive and analyse an energy to model lipid raft formation on biological membranes involving a coupling between the local mean curvature and the local composition. We apply a perturbation method recently introduced by Fritz, Hobbs and the first author to describe the geometry of the surface as a graph over an undeformed Helfrich energy minimising surface. The result is a surface Cahn-Hilliard functional coupled with a small deformation energy We show that suitable minimisers of this energy exist and consider a gradient flow with conserved Allen-Cahn dynamics, for which existence and uniqueness results are proven. Finally, numerical simulations show that for the long time behaviour raft-like structures can emerge and stablise, and their parameter dependence is further explored.