On a Model for Phase Separation on Biological Membranes and its Relation to the Ohta-Kawasaki Equation

TL;DR

This paper analyzes a mathematical model for phase separation on biological membranes, extending the Cahn-Hilliard equation with active transport terms, and explores its asymptotic limits connecting it to the Ohta-Kawasaki equation.

Contribution

It provides rigorous mathematical results on existence, regularity, and long-term behavior of solutions, and links the model to the Ohta-Kawasaki equation through asymptotic analysis.

Findings

Existence and regularity of solutions established.

Long-time behavior characterized.

Reduction to surface equations with non-local terms in certain regimes.

Abstract

We provide a detailed mathematical analysis of a model for phase separation on biological membranes which was recently proposed by Garcke, R\"atz, R\"oger and the second author. The model is an extended Cahn-Hilliard equation which contains additional terms to account for the active transport processes. We prove results on the existence and regularity of solutions, their long-time behaviour, and on the existence of stationary solutions. Moreover, we investigate two different asymptotic regimes. We study the case of large cytosolic diffusion and investigate the effect of an infinitely large affinity between membrane components. The first case leads to the reduction of coupled bulk-surface equations in the model to a system of surface equations with non-local contributions. Subsequently, we recover a variant of the well-known Ohta-Kawasaki equation as the limit for infinitely large affinity between membrane components.