Analysis and Approximation for Solving Highly Coupled Master Differential Equations of Receptor Interactions

Humoral immunity is one component of the human immune system and is the most important determinant of whether an invading pathogen (such as bacteria or viruses) establishes infection. This form of immunity is mediated by B lymphocytes and involves the neutralizing of pathogen receptor binding sites to inhibit the pathogen's entry into target cells. A master equation in both discrete and in continuous form is presented for a pathogen bound at n sites becoming a pathogen bound at m sites in a given interaction time. To track the time-evolution of the antibody-receptor interaction, it is shown that the process is most easily treated classically and that in this case the master equation can be reduced to an equivalent one-dimensional diffusion equation. Thus, well known diffusion theory can be applied to antibody-cell receptor interactions. Three distinct cases are considered depending on whether the probability of antibody binding compared to the probability of dissociation is relatively large, small or comparable and numerical solutions are given.