PERCOLATION ON THE FITNESS HYPERCUBE AND THE EVOLUTION OF REPRODUCTIVE ISOLATION

We study the structure and properties of adaptive landscapes arising f rom the assumption that genotype fitness can only be 0 (inviable genot ype) or 1 (viable genotype). An appropriate image of resulting (''hole y'') fitness landscapes is a (multidimensional) flat surface with many holes. We have demonstrated that in the genotype space there are clus ters of viable genotypes whose members can evolve from any member by s ingle substitutions and that there are ''species'' defined according t o the biological species concept. Assuming that the number of genes, n , is very large while the proportion of viable genotypes among all pos sible genotypes, p, is very small, we have deduced many qualitative an d quantitative properties of holey adaptive landscapes which may be re lated to the patterns of speciation. Relationship between p and n dete rmines two qualitatively different regimes: subcritical and supercriti cal. The subcritical regime takes place if p is extremely small. In th is case, the largest clusters of viable genotypes in the genotype spac e have size of order n and there are many of such size; typical member s of a cluster are connected by a single (''evolutionary'') path; the number of different (biological) species in the cluster has order n; t he expected number of different species in the cluster within k viable substitutions from any its member is of order k. The supercritical re gime takes place if p is small but not extremely small. In this case, there exists a cluster of viable genotypes (a ''giant'' component) tha t has size of order 2(n)/n; the giant component comes ''near'' every p oint of the genotype space; typical members of the giant component are connected by many evolutionary paths; the number of different (biolog ical) species on the ''giant'' component has at least order n(2); 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 und ergo dramatic changes, a physical analogy of which is a phase transiti on. We have considered the most probable (within the present framework ) scenario of biological evolution on holey landscapes assuming that i t starts on a genotype from the largest connected component and procee ds along it by mutation and genetic drift. In this scenario, there is no need to cross any ''adaptive valleys''; reproductive isolation betw een populations evolves as a side effect of accumulating different mut ations. The rate of divergence is very fast: a few substitutions are s ufficient to result in a new biological species. We argue that macroev olution and speciation on ''rugged'' fitness landscapes proceed accord ing to the properties of the corresponding holey landscapes. (C) 1997 Academic Press Limited