A bulk-surface reaction-diffusion system for cell polarization

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.

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

“…We focus on a minimal model for the amplification step that has been proposed in [21]. The significance of the suggested model stems from the fact that in a suitable parameter regime an asymptotic reduction leads to a generalized obstacle-type problem that allows for a clear and mathematically tractable characterization of polarized states.…”

“…The significance of the suggested model stems from the fact that in a suitable parameter regime an asymptotic reduction leads to a generalized obstacle-type problem that allows for a clear and mathematically tractable characterization of polarized states. In [21] we have analyzed stationary states and the onset of polarization. The present paper continues this analysis by considering the time-dependent problem.…”

“…The model proposed in [21] consists of a system of PDEs, motivated by the GTPase cycle model presented in [25,29]. We consider a protein that can be in an active or an inactive state, where the inactive protein moreover can be bound to the cell membrane or be in a cytosolic state, i.e.…”

“…This concentration in general may vary with space and time. Most of these processes are modeled by linear kinetic laws, except for parts of the activation and deactivation processes that need the catalyzation by enzymes and are described by simple Michaelis-Menten type rate laws, see [21] for more details on the model derivation.…”

“…A particularly convenient form is presented in Proposition 2.7. Stationary solutions of model ( 1)-( 4) and the corresponding scaling limits have already been studied in [21]. In particular, in addition to well-posedness, the onset of polarization is studied in [21] for sufficiently small (rescaled) mass of protein.…”