Percolation on the fitness hypercube and the evolution of reproductive isolation.

We study the structure and properties of adaptive landscapes arising from the assumption that genotype fitness can only be 0 (inviable genotype) or 1 (viable genotype). An appropriate image of resulting ("holey") fitness landscapes is a (multidimensional) flat surface with many holes. We have demonstrated that in the genotype space there are clusters of viable genotypes whose members can evolve from any member by single substitutions and that there are "species" defined according to the biological species concept. Assuming that the number of genes, n, is very large while the proportion of viable genotypes among all possible genotypes, p, is very small, we have deduced many qualitative and quantitative properties of holey adaptive landscapes which may be related to the patterns of speciation. Relationship between p and n determines two qualitatively different regimes: subcritical and supercritical. The subcritical regime takes place if p is extremely small. In this case, the largest clusters of viable genotypes in the genotype space have size of order n and there are many of such size; typical members of a cluster are connected by a single ("evolutionary") path; the number of different (biological) species in the cluster has order n; the expected number of different species in the cluster within k viable substitutions from any its member is of order k. The supercritical regime takes place if p is small but not extremely small. In this case, there exists a cluster of viable genotypes (a "giant" component) that has size of order 2n/n; the giant component comes "near" every point of the genotype space; typical members of the giant component are connected by many evolutionary paths; the number of different (biological) species on the "giant" component has at least order n2; the expected number of different species on the "giant" component within k viable substitution from any its member is at least of order kn. At the boundary of two regimes all properties of adaptive landscapes undergo dramatic changes, a physical analogy of which is a phase transition. We have considered the most probable (within the present framework) scenario of biological evolution on holey landscapes assuming that it starts on a genotype from the largest connected component and proceeds along it by mutation and genetic drift. In this scenario, there is no need to cross any "adaptive valleys"; reproductive isolation between populations evolves as a side effect of accumulating different mutations. The rate of divergence is very fast: a few substitutions are sufficient to result in a new biological species. We argue that macroevolution and speciation on "rugged" fitness landscapes proceed according to the properties of the corresponding holey landscapes.