A probabilistic algorithm for optimising the steady-state diffusional flux into a partially absorbing body

Cells in an aqueous environment absorb diffusing nutrient molecules through nanoscale protein channels in their outer membranes. Assuming that there are constraints on the number of such channels a cell can produce, we ask the question: given a nondepleting source of nutrients, what is the optimal distribution of these channels over the cell surface? We coarse-grain this problem, phrasing it as a diffusion problem with position-dependent Robin boundary conditions on the surface. The aim is to maximize the steady-state total flux through the partially absorbing surface under an integral constraint on the local reactivities. We develop an algorithm to tackle this problem that uses the stored and processed results of a particle-based simulation with reflective boundary conditions to a posteriori estimate absorption flux at essentially negligible additional computational cost. We validate the algorithm against a few cases for which analytical or semi-analytical results are available. We apply it to two examples: a spherical cell in the presence of a point source and a spheroidal cell with an isotropic source at infinity. In the former case, there is a significant gain relative to the homogeneous case, while in the latter case the gain is only 1%.