A simple idiotypic network model with complex dynamics

Abstract

According to the network theory of the immune response, the various B and T lymphocyte clones that comprise the immune system respond to antigen in a coordinated way due to idiotypic interactions. Many models of idiotypic interactions have ignored the fact that cells communicate by secreting soluble molecules, and that the chemical reactions between these soluble molecules and cell surface receptors control key features of the immune response. Here we develop and analyze a set of simple idiotypic network models in which cells and molecules are distinguished, and which incorporate realistic chemical interactions among the various cellular and molecular species. Our first, most complete model involves a set of eleven ordinary differential equations. Using two time scale analysis we find that on a long time scale the chemical reactions reactions come to equilibrium, and we need only consider a set of six differential equations with algebraic constraints. We then show that a reduced four differential equation model captures most of the dynamic features of the full model. Using numerical bifurcation techniques we study the steady state and dynamic behavior of the reduced model for a range of realistic parameter values. Our models exhibit multiple steady states, limit cycle oscillations and chaotic behavior. Comparison with experimental data collected in vivo show that these dynamical features are characteristic of certain immune phenomena.