Detecting epistasis from an ensemble of adapting populations

The role that epistasis plays during adaptation remains an outstanding problem, which has received considerable attention in recent years. Most of the recent empirical studies are based on ensembles of replicate populations that adapt in a fixed, laboratory controlled condition. Researchers often seek to infer the presence and form of epistasis in the fitness landscape from the time evolution of various statistics averaged across the ensemble of populations. Here, we provide a rigorous analysis of what quantities, drawn from time series of such ensembles, can be used to infer epistasis for populations evolving under weak mutation on finite-site fitness landscapes. First, we analyze the mean fitness trajectory-that is, the time course of the ensemble average fitness. We show that for any epistatic fitness landscape and starting genotype, there always exists a non-epistatic fitness landscape that produces the exact same mean fitness trajectory. Thus, the presence of epistasis is not identifiable from the mean fitness trajectory. By contrast, we show that two other ensemble statistics-the time evolution of the fitness variance across populations, and the time evolution of the mean number of substitutions-can detect certain forms of epistasis in the underlying fitness landscape.