  Affinity Maturation (AM) is the process through which the immune system is
able to develop potent antibodies against new pathogens it encounters, and is
at the base of the efficacy of vaccines. At its core AM is analogous to a
Darwinian evolutionary process, where B-cells mutate and are selected on the
base of their affinity for an Antigen (Ag), and Ag availability tunes the
selective pressure. In cases when this selective pressure is high the number of
B-cells might quickly decrease and the population might risk extinction in what
is known as a population bottleneck. Here we study the probability for a B-cell
lineage to survive this bottleneck scenario as a function of the progenitor
affinity for the Ag. Using recursive relations and probability generating
functions we derive expressions for the average extinction time and progeny
size for lineages that go extinct. We then extend our results to the full
population, both in the absence and presence of competition for T-cell help,
and quantify the population survival probability as a function of Ag
concentration and initial population size. Our study suggests the population
bottleneck phenomenology might represent a limit case in the space of
biologically plausible maturation scenarios, whose characterization could help
guide the process of vaccine development.
