Numerical investigations of the bulk-surface wave pinning model

TL;DR

This paper uses numerical methods to explore how the bulk diffusion rate influences cell polarization patterns in a bulk-surface reaction-diffusion model, revealing the importance of spatial heterogeneity in the bulk component.

Contribution

It demonstrates through simulations that bulk diffusion rates and heterogeneity significantly affect cell polarization, highlighting scenarios where reduced models may be insufficient.

Findings

Bulk diffusion rate impacts polarization response.

Spatial heterogeneity of the bulk can trigger polarization.

Reduced models may overlook critical heterogeneity effects.

Abstract

The bulk-surface wave pinning model is a reaction-diffusion system for studying cell polarisation. It is constituted by a surface reaction-diffusion equation, coupled to a bulk diffusion equation with a non-linear boundary condition. Cell polarisation arises as the surface component develops specific patterns. Since proteins diffuse much faster in the cell interior than on the membrane, in the literature, the bulk component is often assumed to be spatially homogeneous. {\re Therefore, the model can be reduced to a single surface equation}. However, {\re in real applications} a spatially non-uniform bulk component might be an important player to take into account. In this paper, we study, through numerical computations, the role of the bulk component and, more specifically, how different bulk diffusion rates might affect the polarisation response. We find that the bulk component is indeed a key factor in {\re determining the surface polarisation response}. Moreover, for certain geometries, it is the spatial heterogeneity of the bulk component that triggers the polarisation response, which might not be possible in a reduced model. Understanding how polarisation depends on bulk diffusivity might be crucial when studying models of migrating cells, which are naturally subject to domain deformation.