Cellular diffusion processes in singularly perturbed domains
Paul C. Bressloff · Journal of Mathematical Biology · 2024
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 $$\varOmega \subset {\mathbb {R}}^d$$ Ω ⊂ R d containing a set of small subdomains or interior compartments $${\mathcal {U}}_j$$ U j , $$j=1,\ldots ,N$$ j = 1 , … , N (singularly-perturbed diffusion problems). The domain $$\varOmega $$ Ω 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 $$\partial {\mathcal {U}}_j$$ ∂ U 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.