The genetic basis of phenotypic adaptation II : The 1 distribution of adaptive substitutions in the moving 2 optimum model

1 We consider a population that adapts to a gradually changing environment. Our 2 aim is to describe how ecological and genetic factors combine to determine the genetic 3 basis of adaptation. Specifically, we consider the evolution of a polygenic trait that is 4 under stabilizing selection with a moving optimum. The ecological dynamics is defined 5 by the strength of selection, σ̃, and the speed of the optimum, ṽ; the key genetic param6 eters are the mutation rate Θ and the variance of the effects of new mutations, ω. We 7 develop analytical approximations within an “adaptive walk” framework and describe 8 how selection acts as a sieve that transforms a given distribution of new mutations into 9 the distribution of adaptive substitutions. Our analytical results are complemented by 10 individual based simulations. We find that: (i) The ecological dynamics has a strong 11 effect on the distribution of adaptive substitutions. Its impact depends largely on a 12 single composite measure γ = ṽ/(σ̃Θω3), which combines the ecological and genetic 13 parameters. (ii) Depending on γ, we can distinguish two distinct adaptive regimes: For 14 large γ, the adaptive process is mutation-limited and dominated by genetic constraints, 15 whereas for small γ, it is environmentally limited and dominated by the external eco16 logical dynamics. (iii) Deviations from the adaptive walk approximation occur for large 17 mutation rates, when different mutant alleles interact via linkage or epistasis. (iv) In 18 contrast to predictions from previous models assuming constant selection, the distribu19 tion of adaptive substitutions is generally not exponential. 20