Numerical Analysis of a Model of Ligand-induced B-Cell Antigen-Receptor Clustering. Implications for Simple Models of B-Cell Activation in an Immune Network

B-cell activation driven by ligand-induced crosslinking of membrane immunoglobulin (mIg) is one of the most important processes in experimental and theoretical immunology. Although the activation of B cells through mIgs involves a complex series of intracellular processes, in immune network models it is usually assumed that there is a correlation between the degree of mIg crosslinking and the probability of B-cell activation. We explore the implications of this hypothesis by studying a model of ligand-induced B-cell receptor clustering proposed by Bell and further elaborated by Delisi and Perelson (BDP model). In this model, a critical time (tc) is defined at which the probability of infinite size complex formation (i.e., percolation) becomes non-zero. We use this variable, tc , as a means to characterize the degree of mIg crosslinking. To study the dependence of tc with respect to ligand valence, kinetic constants and ligand-receptor affinity × ligand concentration (K × C), we perform a systematic numerical study of the BDP model for parameter ranges including current empirical estimates for the kinetic constants. Concerning tc, we find that, for ranges of immunological interest (namely, those including current estimates of dissociation and receptor crosslinking rate constants), the curves obtained by plotting 1/tc vs. log(K × C) shift sensibly towards higher values of log(K × C), broadening and increasing its maximum amplitude, as the dissociation rate constant increases. As this finding suggests important consequences for immune network models, we further study the BDP model in an extended version for the case of two different ligands interacting simultaneously with a given B cell. The results indicate that for a B-cell activation function based on mIg-crosslinking, its behaviour can be far more rich and complex than previously assumed in immune network models. This activation function is reduced to an invariant bell-shaped curve only in the case of single ligands with a dissociation rate constant sufficiently large compared with the crosslinking rate constant.