Generation and motion of interfaces in a mass-conserving reaction-diffusion system

TL;DR

This paper analyzes a mass-conserving reaction-diffusion model for cell polarization on curved membranes, revealing interface formation, growth/shrinkage, and long-term steady states through asymptotic analysis.

Contribution

It introduces a minimal, mass-conserving model and characterizes its multi-timescale dynamics, including interface generation and evolution on curved geometries.

Findings

Interface generation occurs under certain conditions.

Domains grow or shrink based on global parameters.

Long-term behavior follows area-preserving geodesic curvature flow.

Abstract

Reaction-diffusion models with nonlocal constraints naturally arise as limiting cases of coupled bulk-surface models of intracellular signalling. In this paper, a minimal, mass-conserving model of cell-polarization on a curved membrane is analyzed in the limit of slow surface diffusion. Using the tools of formal asymptotics and calculus of variations, we study the characteristic wave-pinning behavior of this system on three dynamical timescales. On the short timescale, generation of an interface separating high- and low-concentration domains is established under suitable conditions. Intermediate timescale dynamics is shown to lead to a uniform growth or shrinking of these domains to sizes which are fixed by global parameters. Finally, the long time dynamics reduces to area-preserving geodesic curvature flow that may lead to multi-interface steady state solutions. These results provide a foundation for studying cell polarization and related phenomena in biologically relevant geometries.