A parabolic free boundary problem arising in a model of cell polarization

TL;DR

This paper analyzes a bulk-surface reaction-diffusion model for cell polarization, proving convergence to a parabolic obstacle problem and establishing stability and contraction properties for the simplified surface equation.

Contribution

It introduces a new asymptotic analysis linking the reaction-diffusion system to a parabolic obstacle problem and demonstrates stability results for stationary states.

Findings

Convergence to a parabolic obstacle problem in the asymptotic limit

L1-contraction property for the reduced surface equation

Stability of stationary states for time-constant signals

Abstract

The amplification of an external signal is a key step in direction sensing of biological cells. We consider a simple model for the response to a time-depending signal, which was previously proposed by the last three authors. The model consists of a bulk-surface reaction-diffusion model. We prove that in a suitable asymptotic limit the system converges to a bulk-surface parabolic obstacle type problem. For this model and a reduction to a nonlocal surface equation we show an L1-contraction property and, in the case of time-constant signals, the stability of stationary states.