Adaptive walks by the fittest among finite random mutants on a Mt. Fuji-type fitness landscape. II. Effect of small non-additivity.

We examined properties of adaptive walks by the fittest on "rough Mt. Fuji-type" fitness landscapes, which are modeled by superposing small uncorrelated random component on an additive fitness landscape. A single adaptive walk is carried out by repetition of the evolution cycle composed of (1) mutagenesis process that produces random d-fold point mutants of population size N and (2) selection process that picks out the fittest mutant among them. To comprehend trajectories of the walkers, the fitness landscape is mapped into a (x, y, z)-space, where x, y and z represent, respectively, normalized Hamming distance from the peak on the additive fitness landscape, scaled additive fitness and scaled nonadditive fitness. Thus a single adaptive walk is expressed as the dynamics of a particle in this space. We drew the "hill-climbing" vector field, where each vector represents the most probable step for a walker in a single step. Almost all of the walkers are expected to move along streams of vectors existing on a particular surface that overlies the (x, y)-plane, toward the neighborhood of a characteristic point at which a mutation-selection-random drift balance is reached. We could theoretically predict this reachable point in the case of random sampling search strategy.