Diffusion to capture and the concept of diffusive interactions

Abstract: 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.