Existence and convergence of stochastic processes underlying a thin layer approximation of a coupled bulk-surface PDE

Abstract

We study a system of coupled bulk-surface partial differential equations (BS-PDEs), describing changes in concentration of certain proteins (Rho GTPases) in a living cell. These proteins, when activated, are bound to the plasma membrane where they diffuse and react with the inactive species; inactivated species diffuse inside the cell cortex; these react with the activated species when they are close to the plasma membrane. For our case study, we model the cell cortex as an annulus, and the plasma membrane as its outer circle. Mathematically, the aim of the paper is twofold: Firstly, we show the master equation for the changes in concentration of Rho GTPases is the Kolmogorov forward differential equation for an underlying Feller stochastic process, and, in particular, the related Cauchy problem is well-posed. Secondly, since the cell cortex is typically a rather thin domain, we study the situation where the thickness of the annulus modeling the cortex converges to 0. To this end, we note that letting the thickness of the annulus to 0 is equivalent to keeping it constant while increasing the rate of radial diffusion. As a result, in the limit, solutions to the master equation lose dependence on the radial coordinate and can be thought of as functions on the circle. Furthermore, the limit master equation can be seen as describing diffusion on two copies of the circle with jumps from one copy to the other.