There are many processes in cell biology that can be modeled in terms of particles dif fusing in a two-dimensional (2D) or three-dimensional (3D) bounded domain Ω ⊂ Rd
containing a set of small subdomains or interior compartments Uj , j = 1,..., N
(singularly-perturbed diffusion problems). The domain 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 ∂Uj . This
allows us to incorporate a variety of previous studies of singularly perturbed diffu sion 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.