The population genetics of adaptation: multiple substitutions on a smooth fitness landscape

Much recent work in the theoretical study of adaptation has focused on the so-called strong selection-weak mutation (SSWM) limit, wherein adaptation is due to new mutations of definite selective advantage. This work, in turn, has focused on the first step (substitution) during adaptive evolution. Here we extend this theory to allow multiple steps during adaptation. We find analytic solutions to the probability that adaptation follows a certain path during evolution as well as the probability that adaptation arrives at a given genotype regardless of the path taken. We also consider the probability of parallel adaptation and the proportion of the total increase in fitness caused by the first substitution. Our key assumption is that there is no epistasis among beneficial mutations.