Global existence for a bulk/surface model for active-transport-induced polarisation in biological cells

TL;DR

This paper proves the global existence of solutions for a coupled bulk/surface model describing chemical interactions and cytoskeletal dynamics in biological cells, addressing boundary coupling challenges and steady-state solutions.

Contribution

It establishes global existence results for the model with small and arbitrary data, and constructs steady-state solutions parametrized by total membrane-bound molecule mass.

Findings

Global existence of classical solutions for small data

Global existence for certain regularised boundary couplings with arbitrary data

Existence of steady-state solutions parametrized by total mass

Abstract

We consider a coupled bulk/surface model for advection and diffusion of interacting chemical species in biological cells. Specifically, we consider a signalling protein that can exist in both a cytosolic and a membrane-bound state, along with a variable that gives a coarse-grained description of the cytoskeleton. The main focus of our work is on the well-posedness of the model, whereby the coupling at the boundary is the main source of analytical difficulty. A priori $L^p$-estimates, together with classical Schauder theory, deliver global existence of classical solutions for small data on bounded, Lipschitz domains. For two physically reasonable regularised versions of the boundary coupling, we are able to prove global existence of solutions for arbitrary data. In addition, we prove the existence of a family of steady-state solutions of the main model which are parametrised by the total mass of the membrane-bound signal molecule.