Diffusion from Disjoint Plane Domains 2. Arrays of Domains, with Application to the Response of a Muscle to a Drug
L. M. Hocking · IMA Journal of Applied Mathematics · 1971
The response of a muscle to the presence of a drug in its vicinity is governed by a chemical reaction between the drug and a large number of small receptors, occupying a patch on the surface of the muscle fibre. The time for the drug to act depends on the flux towards these receptors, produced by the concentration gradient set up by the reaction. The behaviour of the flux as a function of the time is determined for a plane array of receptors, using and extending the results obtained in Part 1, for a mathematically similar problem, solved there for one or two domains. The decrease in the flux through a receptor because of the influence of a large number of other receptors in the array is found, together with the time at which such decrease is operative. If the array is sufficiently large, there is a time-range during which it is effectively infinite, but eventually, the effect of the finite size of the array will alter the flux, and the patch will behave as if it were a continuous domain. The modifications produced when the supply of the reactant chemical in the receptors is finite are discussed in Section 4, and an estimate of the likely effect of an outer boundary to the region occupied by the drug is given in Section 5. By combining the results of the various sections, it is possible to describe the behaviour of the flux through a receptor or through the whole array for all likely ranges of values of the various lengths involved in the problem.