Deterministic and stochastic regimes of asexual evolution on rugged fitness landscapes

We study the adaptation dynamics of an initially maladapted asexual population with genotypes represented by binary sequences of length $L$. The population evolves in a maximally rugged fitness landscape with a large number of local optima. We find that whether the evolutionary trajectory is deterministic or stochastic depends on the effective mutational distance $d_{\mathrm{eff}}$ upto which the population can spread in genotype space. For $d_{\mathrm{eff}}=L$, the deterministic quasispecies theory operates while for $d_{\mathrm{eff}} < 1$, the evolution is completely stochastic. Between these two limiting cases, the dynamics are described by a local quasispecies theory below a crossover time $T_{\times}$ while above $T_{\times}$, the population gets trapped at a local fitness peak and manages to find a better peak either via stochastic tunneling or double mutations. In the stochastic regime $d_\mathrm{eff} < 1$, we identify two subregimes associated with clonal interference and uphill adaptive walks, respectively. We argue that our findings are relevant to the interepretation of evolution experiments with microbial populations.
Revised version, to appear in Genetics. Note on the role of selection in defining d_eff added; new figure 4 included