Thermodynamical interpretation of an adaptive walk on a Mt. Fuji-type fitness landscape: Einstein relation-like formula holds in a stochastic evolution

We have theoretically studied the statistical properties of adaptive walks (or hill-climbing) on a Mt. Fuji-type fitness landscape in the multi-dimensional sequence space through mathematical analysis and computer simulation. The adaptive walk is characterized by the “mutation distance” d as the step-width of the walker and the “population size” N as the number of randomly generated d-fold point mutants to be screened. In addition to the fitness W, we introduced the following quantities analogous to thermodynamical concepts: “free fitness” G(W)≡W+T×S(W), where T is the “evolutionary temperature” T∝d/ln N and S(W) is the entropy as a function of W, and the “evolutionary force” X≡d(G(W)/T)/dW, that is caused by the mutation and selection pressure. It is known that a single adaptive walker rapidly climbs on the fitness landscape up to the stationary state where a “mutation–selection–random drift balance” is kept. In our interpretation, the walker tends to the maximal free fitness state, driven by the evolutionary force X. Our major findings are as follows: First, near the stationary point W∗, the “climbing rate” J as the expected fitness change per generation is described by J≈L×X with L≈V/2, where V is the variance of fitness distribution on a local landscape. This simple relationship is analogous to the well-known Einstein relation in Brownian motion. Second, the “biological information gain” (ΔG/T) through adaptive walk can be described by combining the Shannonʹs information gain (ΔS) and the “fitness information gain” (ΔW/T).