  The evolution of the adaptive immune system is characterized by changes in
the relative abundances of the B- and T-cell clones that make up its
repertoires. To fully capture this evolution, we need to describe the complex
dynamics of the response to pathogenic and self-antigenic stimulations, as well
as the statistics of novel lymphocyte receptors introduced throughout life.
Recent experiments, ranging from high-throughput immune repertoire sequencing
to quantification of the response to specific antigens, can help us
characterize the effective dynamics of the immune response. Here we describe
mathematical models informed by experiments that lead to a picture of clonal
competition in a highly stochastic context. We discuss how different types of
competition, noise and selection shape the observed clone-size distributions,
and contrast them with predictions of a neutral theory of clonal evolution.
These mathematical models show that memory and effector immune repertoire
evolution is far from neutral, and is driven by the history of the pathogenic
environment, while naive repertoire dynamics are consistent with neutral theory
and competition in a fixed antigenic environment. Lastly, we investigate the
effect of long-term clonal selection on repertoire aging.
