ANZIAM J.
46 (2005), 495505

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



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 diseaseproducing
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
timeevolution of the antibodyligand
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 onedimensional diffusion equation.
Thus wellknown diffusion theory can be applied
to antibodyligand 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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