Diffusion Interactions for a Pair of Reactive Spheres

In this treatise competition and mutualism interactions are evaluated for two different spheres in an infinite medium with constant Fickian diffusion. Various source and sink reaction types, reaction rates, and size differences are considered. The competition problem is evaluated for the first order surface reactions, and the mutualism problem is modeled with a zeroth order surface source while the sink is either diffusion-limited, first order surface, or a volume distributed reactor. The reaction rate and concentration are evaluated using the bispherical expansion or the Bispherical coordinate system. The bispherical expansion involves one infinite sum, for the mutualism problem and two nested infinite sums for the competition problem. A matrix elimination technique is used to obtain an exact analytical solution from the bispherical expansion. Only one infinite sum is required in the calculation of the reaction rate and concentration for the bispherical coordinate system. In either case the solution is completely convergent, and often converges rapidly over the full range of parameters. In the competition study three effects were displayed: blocking, competition and a combination of the two. When the two effects were combined and both sinks are diffusion-limited the reaction rate for the larger sink goes through a minimum as the much smaller sink approaches it. The mutualism study, when both the sink and source are impenetrable, produced a surprising maximum in sink reaction as it approaches the larger source when the sink is very reactive and much smaller than the source. The reaction rate of a diffusion-limited sink when exposed to a source with a constant surface concentration is also included; it diverges when the two spheres touch but as the sink and source move further apart it approaches the diffusion-limited single sphere result. Finally, the mutualism problem for a permeable first order sink impermeable zeroth order source is compared with its effective surface reaction counterpart. When the sink is strong there is very good agreement, but when the sink is weak the two solutions diverge. Interestingly, this weak sink diverging region is the most likely physical chemical condition for cellular interaction.