  Genetic interactions can strongly influence the fitness effects of individual
mutations, yet the impact of these epistatic interactions on evolutionary
dynamics remains poorly understood. Here we investigate the evolutionary role
of epistasis over 50,000 generations in a well-studied laboratory evolution
experiment in E. coli. The extensive duration of this experiment provides a
unique window into the effects of epistasis during long-term adaptation to a
constant environment. Guided by analytical results in the weak-mutation limit,
we develop a computational framework to assess the compatibility of a given
epistatic model with the observed patterns of fitness gain and mutation
accumulation through time. We find that a decelerating fitness trajectory alone
provides little power to distinguish between competing models, including those
that lack any direct epistatic interactions between mutations. However, when
combined with the mutation trajectory, these observables place strong
constraints on the set of possible models of epistasis, ruling out many
existing explanations of the data. Instead, we find that the data are
consistent with "two-epoch" model of adaptation, in which an initial burst of
diminishing returns epistasis is followed by a steady accumulation of mutations
under a constant distribution of fitness effects. Our results highlight the
need for additional DNA sequencing of these populations, as well as for more
sophisticated models of epistasis that are compatible with all of the
experimental data.
