  The vertebrate adaptive immune system provides a flexible and diverse set of
molecules to neutralize pathogens. Yet, viruses such as HIV can cause chronic
infections by evolving as quickly as the adaptive immune system, forming an
evolutionary arms race. Here we introduce a mathematical framework to study the
coevolutionary dynamics of antibodies with antigens within a host. We focus on
changes in the binding interactions between the antibody and antigen
populations, which result from the underlying stochastic evolution of genotype
frequencies driven by mutation, selection, and drift. We identify the critical
viral and immune parameters that determine the distribution of antibody-antigen
binding affinities. We also identify definitive signatures of coevolution that
measure the reciprocal response between antibodies and viruses, and we
introduce experimentally measurable quantities that quantify the extent of
adaptation during continual coevolution of the two opposing populations. Using
this analytical framework, we infer rates of viral and immune adaptation based
on time-shifted neutralization assays in two HIV-infected patients. Finally, we
analyze competition between clonal lineages of antibodies and characterize the
fate of a given lineage in terms of the state of the antibody and viral
populations. In particular, we derive the conditions that favor the emergence
of broadly neutralizing antibodies, which may be useful in designing a vaccine
against HIV.

Populations and Evolution