Dynamics of one-pass germinal center models: Implications for affinity maturation

During an immune response, the affinity of antibodies that react with the antigen that triggered the response increases with time, a phenomenon known as affinity maturation. The molecular basis of affinity maturation has been partially elucidated. It involves the somatic mutation of immunoglobulin V-region genes within antigen-stimulated germinal center B cells and the subsequent selection of high affinity variants. This mutation and selection process is extremely efficient and produces large numbers of high affinity variants. Studies of the architecture of germinal centers suggested that B cells divide in the dark zone of the germinal center, then migrate to the light zone, where they undergo selection based on their interaction with antigen-loaded follicular dendritic cells, after which they exit the germinal center through the mantle zone. Kepler and Perelson questioned this architecturally driven view of the germinal center reaction. They, as well as others, argued that the large number of point mutations observed in germinal center B cell V-region genes, frequently 5 to 10 and sometimes higher, would most likely render cells incapable of binding the antigen, if no selection step was interposed between rounds of mutations. To clarify this issue, we address the question of whether a mechanism in which mutants are generated and then selected in one pass, with no post-selection amplification, can account for the observed efficiency of affinity maturation. We analyse a set of one-pass models of the germinal center reaction, with decaying antigen, and mutation occurring at transcription or at replication. We show that under all the scenarios, the proportion of high affinity cells in the output of a germinal center varies logarithmically with their selection probability. For biologically realistic parameters, the efficiency of this process is in clear disagreement with the experimental data. Furthermore, we discuss a set of, possibly counterintuitive, more general features of one-pass selection models that follow from our analysis. We believe that these results may also provide useful intuitions in other cases where a population is subjected to selection mediated by a selective force that decays over time. (C) 2000 Society for Mathematical Biology.