# A bulk-surface reaction-diffusion system for cell polarization

## TL;DR

This paper models cell polarization through a PDE system involving proteins on the cell surface and interior, analyzing steady states and their relation to obstacle problems, with results on conditions for polarization based on protein mass.

## Contribution

It introduces a novel PDE model for cell polarization and analyzes the limiting behavior, establishing conditions for polarization and characterizing critical mass thresholds.

## Key findings

- Steady states converge to obstacle problem solutions in certain regimes.
- Polarization occurs if the total protein mass is below a critical threshold.
- Critical mass for polarization can be explicitly characterized in some variants.

## Abstract

We propose a model for cell polarization as a response to an external signal which results in a system of PDEs for different variants of a protein on the cell surface and interior respectively. We study stationary states of this model in certain parameter regimes in which several reaction rates on the membrane as well as the diffusion coefficient within the cell are large. It turns out that in suitable scaling limits steady states converge to solutions of some obstacle type problems. For these limiting problems we prove the onset of polarization if the total mass of protein is sufficiently small. For some variants we can even characterize precisely the critical mass for which polarization occurs.