ANZIAM  J.  46 (2005), 495-505

Diffusion theory can be applied to antibodies attaching to ligand sites

D. P. Wilson   School of Mathematical Sciences   Queensland University of Technology   GPO Box 2434   Brisbane Qld 4001   Australia   and D. L. S. McElwain   School of Mathematical Sciences   Queensland University of Technology   GPO Box 2434   Brisbane Qld 4001   Australia     s.mcelwain@qut.edu.au

Abstract

Humoral immunity is that aspect of specific immunity that is mediated by B lymphocytes and involves the neutralising of disease-producing microorganisms, called pathogens, by means of antibodies attaching to the pathogen's binding sites. This inhibits the pathogen's entry into target cells. We present a master equation in both discrete and in continuous form for a ligand bound at n sites becoming a ligand bound at m sites in a given interaction time. To track the time-evolution of the antibody-ligand 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-ligand interactions. We consider three distinct cases 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.

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