Adaptive Walks by the Fittest among Finite Random Mutants on a Mt. Fuji-type Fitness Landscape

Abstract

Based on the theory of fitness distributions on a Mt. Fuji-type fitness landscape in a multivalued sequence space (Aita & Husimi, 1996J. theor. Biol.182, 469–485), we investigated the properties of adaptive walks on the ideal landscape in the case of a cloning-screening-type evolution experiment. We modeled that an adaptive walk is performed by repetition of the evolution cycle composed of the mutagenesis process generating randomd-fold point mutants of population sizeNand the selection process looking for the fittest mutant among them. While an adaptive walk is described in a sequence space, we simplified the description as follows. We mapped the landscape in anx–yplane, wherexandyrepresent a normalized Hamming distance from the global peak and a scaled fitness, respectively. An adaptive walk is described as a trajectory in the plane. The most certain step for a walker to move in a single evolution cycle is represented by a vector in the plane. Then, a walker moves along the streams in the vector field determined bydandN. The walker performs fast hill-climbing until a “trap-line”, which traverses the plane. Subsequently, the walker is likely to get trapped in an “apparent local optimum”. To continue the walk, apparent local optima must be eliminated by resettingdandNlarger. Therefore, for the fastest walk, the optimal schedule of thed-values (initially larged, then smalld) is effective, although the economical walk with high cost-performance is different. If a real landscape is just of the Mt. Fuji-type, the walk with the highest cost-performance will be performed by scanning site-directed optimization through all sites. However, in the case of the rough Mt. Fuji-type, which seems to be more realistic, the walking method we have examined will be effective for a walker to sidestep true local optima.