Cellular diffusion processes in singularly perturbed domains

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 \(\varOmega \subset {\mathbb {R}}^d\) $\Omega \subset {\mathbb{R}}^d$ containing a set of small subdomains or interior compartments \({\mathcal {U}}_j\) ${\mathcal{U}}_j$ , \(j=1,\ldots ,N\) $j=1,\dots ,N$ (singularly-perturbed diffusion problems). The domain \(\varOmega \) $\Omega$ 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 \(\partial {\mathcal {U}}_j\) $\partial {\mathcal{U}}_j$ . 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.