A coupled surface-Cahn--Hilliard bulk-diffusion system modeling lipid raft formation in cell membranes

TL;DR

This paper introduces a coupled bulk-surface model for lipid raft formation in cell membranes, combining thermodynamic principles, asymptotic analysis, and numerical simulations to explore the dynamics of lipid phase separation and cholesterol interaction.

Contribution

It presents a novel coupled bulk-surface Cahn--Hilliard model with thermodynamic justification and analyzes its behavior, including asymptotic limits and long-term dynamics.

Findings

Raft-like structures emerge in non-equilibrium models.

Equilibrium models tend to form only macrodomains.

Numerical simulations confirm theoretical predictions.

Abstract

We propose and investigate a model for lipid raft formation and dynamics in biological membranes. The model describes the lipid composition of the membrane and an interaction with cholesterol. To account for cholesterol exchange between cytosol and cell membrane we couple a bulk-diffusion to an evolution equation on the membrane. The latter describes a relaxation dynamics for an energy taking lipid-phase separation and lipid-cholesterol interaction energy into account. It takes the form of an (extended) Cahn--Hilliard equation. Different laws for the exchange term represent equilibrium and non-equilibrium models. We present a thermodynamic justification, analyze the respective qualitative behavior and derive asymptotic reductions of the model. In particular we present a formal asymptotic expansion near the sharp interface limit, where the membrane is separated into two pure phases of saturated and unsaturated lipids, respectively. Finally we perform numerical simulations and investigate the long-time behavior of the model and its parameter dependence. Both the mathematical analysis and the numerical simulations show the emergence of raft-like structures in the non-equilibrium case whereas in the equilibrium case only macrodomains survive in the long-time evolution.