Cellular diffusion processes in singularly perturbed domains

TL;DR

This paper reviews mathematical techniques for modeling particle diffusion in cell-like domains with small internal compartments, focusing on steady-state solutions and the challenges of time-dependent processes.

Contribution

It introduces a unified framework combining matched asymptotic analysis and Green's functions for singularly perturbed diffusion problems with Robin boundary conditions.

Findings

Provides analytical solutions for steady-state diffusion in complex cell geometries.

Highlights challenges in modeling time-dependent and stochastic diffusion processes.

Integrates various previous studies into a comprehensive mathematical approach.

Abstract

There are many processes in cell biology that can be modeled in terms of particles diffusing in a two-dimensional (2D) or three-dimensional (3D) bounded domain $\Omega \subset \R^d$ containing a set of small subdomains or interior compartments $\calU_j$, $j=1,\ldots,N$ (singularly-perturbed diffusion problems). The domain $\Omega$ could represent the cell membrane, the cell cytoplasm, the cell nucleus or the extracellular volume, while an individual compartment could represent a synapse, a membrane protein cluster, a biological condensate, or a quorum sensing bacterial cell. In this review we use a combination of matched asymptotic analysis and Green's function methods to solve a general type of singular boundary value problems (BVP) in 2D and 3D, in which an inhomogeneous Robin condition is imposed on each interior boundary $\partial \calU_j$. This allows us to incorporate a variety of previous studies of singularly perturbed diffusion problems into a single mathematical modeling framework. We mainly focus on steady-state solutions and the approach to steady-state, but also highlight some of the current challenges in dealing with time-dependent solutions and randomly switching processes