The non-stationary dynamics of fitness distributions: asexual model with epistasis and standing variation

Various models describe asexual evolution by mutation, selection and drift. Some focus directly on fitness, typically modelling drift but ignoring or simplifying both epistasis and the distribution of mutation effects (travelling wave models). Others follow the dynamics of quantitative traits determining fitness (Fisher’s geometrical model), imposing a complex but fixed form of mutation effects and epistasis, and often ignoring drift. In all cases, predictions are typically obtained in high or low mutation rate limits and for long-term stationary regimes, thus loosing information on transient behaviors and the effect of initial conditions. Here, we connect fitness-based and trait-based models into a single framework, and seek explicit solutions even away from stationarity. The expected fitness distribution is followed over time via its cumulant generating function, using a deterministic approximation that neglects drift. In several cases, explicit trajectories for the full fitness distribution are obtained, for arbitrary mutation rates and standing variance. For non-epistatic mutation, especially with beneficial mutations, this approximation fails over the long term but captures the early dynamics, thus complementing stationary stochastic predictions. The approximation also handles several diminishing return epistasis models (e.g. with an optimal genotype): it can then apply at and away from equilibrium. General results arise at equilibrium, where fitness distributions display a ‘phase transition’ with mutation rate. Beyond this phase transition, in Fisher’s geometrical model, the full trajectory of fitness and trait distributions takes simple form, robust to details of the mutant phenotype distribution. Analytical arguments are explored for why and when the deterministic approximation applies.Significance statementHow fast do asexuals evolve in new environments? Asexual fitness dynamics are well documented empirically. Various corresponding theories exist, to which they may be compared, but most typically describe stationary regimes, thus losing information on the shorter timescale of experiments, and on the impact of the initial conditions set by the experimenter. Here, a general deterministic approximation is proposed that encompasses many previous models as subcases, and shows surprising accuracy when compared to stochastic simulations. It can yield predictions over both short and long timescales, hopefully fostering the quantitative test of alternative models, using data from experimental evolution in asexuals.