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/lnN 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 Shannon's information gain (
ΔS) and the “fitness information gain” (
ΔW/T).