title: A new bell-shaped function for idiotypic interactions based on cross-linking.
abstract: Most recent models of the immune network are based upon a phenomenological log bell-shaped interaction function. This function depends on a single parameter, the "field," which is the sum of all ligand concentrations weighted by their respective affinities. The typical behavior of these models is dominated by percolation, a phenomenon in which a local stimulus spreads globally throughout the network. The usual reason for employing a log bell-shaped interaction function is that B cells are activated by cross-linking of their surface immunoglobulin receptors. Here we formally derive a new phenomenological log bell-shaped function from the chemistry of receptor cross-linking by bivalent ligand. Specifying how this new function depends on the ligand concentrations requires two fields: a binding field and a cross-linking field. When we compare the activation functions for ligand-receptor pairs with different affinities, the one-field and the two-field functions differ markedly. In the case of the one-field activation function, its graph is shifted to increasingly higher concentration as the affinity decreases but keeps its width and height. In the case of the two-field activation function, the graph of a low-affinity interaction is nested within the graphs of all higher-affinity interactions. We show that this difference in the relations among activation functions for different affinities radically changes the network behavior. In models that described B cell proliferation using the one-field activation function, network behavior was dominated by low-affinity interactions. Conversely, in our new model, the high-affinity interactions are the most significant. As a consequence, percolation is no longer the only typical network behavior.
authors.affiliation: Theoretical Biology, Utrecht University, Netherlands. rdb@alive.biol.ruu.nl
authors.forename: R J
authors.initials: RJ
authors.lastname: De Boer
authors.forename: M C
authors.initials: MC
authors.lastname: Boerlijst
authors.forename: B
authors.initials: B
authors.lastname: Sulzer
authors.forename: A S
authors.initials: AS
authors.lastname: Perelson
mesh_terms: D000818:Animals; D001402:B-Lymphocytes / Q000276:immunology*; D006801:Humans; D007130:Immunoglobulin Idiotypes; D008024:Ligands; D008433:Mathematics; D018448:Models, Immunological; D016011:Normal Distribution; D011971:Receptors, Immunologic
chemical_list: D007130:Immunoglobulin Idiotypes; D008024:Ligands; D011971:Receptors, Immunologic