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We study the Langevin dynamics of the adaptive immune system, modelled by alymphocyte network in which the B cells are interacting with the T cells andantigen. We assume that B clones and T clones are evolving in different thermalnoise environments and on different timescales. We derive stationarydistributions and use statistical mechanics to study clonal expansion of Bclones in this model when the B and T clone sizes are assumed to be the slowand fast variables respectively and vice versa. We derive distributions of Bclone sizes and use general properties of ferromagnetic systems to predictcharacteristics of these distributions, such as the average B cellconcentration, in some regimes where T cells can be modelled as binaryvariables. This analysis is independent of network topologies and its resultsare qualitatively consistent with experimental observations. In order to obtainfull distributions we assume that the network topologies are random and locallyequivalent to trees. The latter allows us to employ the Bethe-Peierls approachand to develop a theoretical framework which can be used to predict thedistributions of B clone sizes. As an example we use this theory to computedistributions for the models of immune system defined on random regularnetworks.

Statistical mechanics of clonal expansion in lymphocyte networks modelled with slow and fast variables