  Chronic infections of the human immunodeficiency virus (HIV) create a very
complex co-evolutionary process, where the virus tries to escape the
continuously adapting host immune system. Quantitative details of this process
are largely unknown and could help in disease treatment and vaccine
development. Here we study a longitudinal dataset of ten HIV-infected people,
where both the B-cell receptors and the virus are deeply sequenced. We focus on
simple measures of turnover, which quantify how much the composition of the
viral strains and the immune repertoire change between time points. At the
single-patient level, the viral-host turnover rates do not show any
statistically significant correlation, however they correlate if the
information is aggregated across patients. In particular, we identify an
anti-correlation: large changes in the viral pool composition come with small
changes in the B-cell receptor repertoire. This result seems to contradict the
naive expectation that when the virus mutates quickly, the immune repertoire
needs to change to keep up. However, we show that the observed anti-correlation
naturally emerges and can be understood in terms of simple population-genetics
models.
