I will present a model of adaptation under mutation selection and genetic drift (wright fisher diffusion approximation). The goal is to derive analytic predictions that can be fitted to observables from microbial experimental evolution. Typically, a purely asexual clone ('ancestor') is introduced into a new environment where it undergoes regular cycles of growth and dilution for a long period of time, in replicates. The mean fitness of replicate populations and (sometimes) the genetic distance to the ancestor (at both neutral markers and selected sites) are measured. In all such experiments (e.g. Lenski's LTEE), the trajectories of mean fitness (and genetic distance) are non-linear with time, suggesting the prevalence of diminishing returns epistasis (the rate and/or effects of beneficial mutations decrease as the population adapts).
I propose to study this problem assuming a simple one peak phenotype-fitness landscape (Fisher's geometrical model) that entails epistasis of a form consistent with mutant fitness data. I follow the dynamics of the expected cumulant generating function (CGF) of the traits (fitness and genetic distances), over time and over stochastic replicates. As the action of the diffusion generator on this function is not linear, the dynamics are closed by assuming that stochastic deviations between replicates are small (leading order perturbation). This yields a (nonlocal first order) pde for the CGF of an infinite population, coupled with a pde for the expected deviation from this CGF within finite populations. Explicit approximate solutions are obtained by linear perturbations of the pdes in two extreme regimes, where either mutation or selection rates are small relative to one another. The results are checked against numerical solutions and stochastic individual based simulations.
If time allows, I will discuss extensions to other mutation models and simple results on the rate of change in genetic distance once mutation-selection-drift balance is reached (hopefully relevant to phylogenetic studies).
