The genetic basis of phenotypic adaptation II: the distribution of adaptive substitutions in the moving optimum model.

We consider a population that adapts to a gradually changing environment. Our aim is to describe how ecological and genetic factors combine to determine the genetic basis of adaptation. Specifically, we consider the evolution of a polygenic trait that is under stabilizing selection with a moving optimum. The ecological dynamics are defined by the strength of selection, sigma, and the speed of the optimum, v; the key genetic parameters are the mutation rate Theta and the variance of the effects of new mutations, omega. We develop analytical approximations within an "adaptive-walk" framework and describe how selection acts as a sieve that transforms a given distribution of new mutations into the distribution of adaptive substitutions. Our analytical results are complemented by individual-based simulations. We find that (i) the ecological dynamics have a strong effect on the distribution of adaptive substitutions and their impact depends largely on a single composite measure gamma=v/(sigmaThetaomega(3)), which combines the ecological and genetic parameters; (ii) depending on gamma, we can distinguish two distinct adaptive regimes: for large gamma the adaptive process is mutation limited and dominated by genetic constraints, whereas for small gamma it is environmentally limited and dominated by the external ecological dynamics; (iii) deviations from the adaptive-walk approximation occur for large mutation rates, when different mutant alleles interact via linkage or epistasis; and (iv) in contrast to predictions from previous models assuming constant selection, the distribution of adaptive substitutions is generally not exponential.