  Diffusion to capture is an ubiquitous phenomenon in many fields in biology
and physical chemistry, with implications as diverse as ligand-receptor binding
on eukaryotic and bacterial cells, nutrient uptake by colonies of unicellular
organisms and the functioning of complex core-shell nanoreactors. Whenever many
boundaries compete for the same diffusing molecules, they inevitably shield a
variable part of the molecular flux from each other. This gives rise to the
so-called diffusive interactions (DI), which can reduce substantially the
influx to a collection of reactive boundaries depending chiefly on their
geometrical configuration. In this review we provide a pedagogical discussion
of the main mathematical aspects underlying a rigorous account of DIs. Starting
from a striking and deep result on the mean-field description of ligand binding
to a receptor-covered cell, we develop little by little a rigorous mathematical
description of DIs in the stationary case through the use of translational
addition theorems for spherical harmonics. We provide several enlightening
illustrations of this powerful mathematical theory, including diffusion to
capture to ensembles of reactive boundaries within a spherical cavity.
