A Minimum on the Mean Number of Steps Taken in Adaptive Walks

I consider the adaptation of a DNA sequence when mutant fitnesses are drawn randomly from a probability distribution. I focus on “gradient” adaptation in which the population jumps to the best mutant sequence available at each substitution. Given a random starting point, I derive the distribution of the number of substitutions that occur during adaptive walks to a locally optimal sequence. I show that the mean walk length is a constant:L̄ =e−1, where e≈2.72. I argue that this result represents a limit on what is possible under any form of adaptation. No adaptive algorithm on any fitness landscape can arrive at a local optimum in fewer than a mean of L̄=e−1 steps when starting from a random sequence. Put differently, evolution must try out at least e wild-type sequences during an average bout of adaptation.