The competition between simple and complex evolutionary trajectories in asexual populations

Abstract

BackgroundOn rugged fitness landscapes where sign epistasis is common, adaptation can often involve either individually beneficial “uphill” mutations or more complex mutational trajectories involving fitness valleys or plateaus. The dynamics of the evolutionary process determine the probability that evolution will take any specific path among a variety of competing possible trajectories. Understanding this evolutionary choice is essential if we are to understand the outcomes and predictability of adaptation on rugged landscapes.ResultsWe present a simple model to analyze the probability that evolution will eschew immediately uphill paths in favor of crossing fitness valleys or plateaus that lead to higher fitness but less accessible genotypes. We calculate how this probability depends on the population size, mutation rates, and relevant selection pressures, and compare our analytical results to Wright-Fisher simulations.ConclusionWe find that the probability of valley crossing depends nonmonotonically on population size: intermediate size populations are most likely to follow a “greedy” strategy of acquiring immediately beneficial mutations even if they lead to evolutionary dead ends, while larger and smaller populations are more likely to cross fitness valleys to reach distant advantageous genotypes. We explicitly identify the boundaries between these different regimes in terms of the relevant evolutionary parameters. Above a certain threshold population size, we show that the probability that the population finds the more distant peak depends only on a single simple combination of the relevant parameters.

Mutation supply and the repeatability of selection for antibiotic resistance

“…Highly beneficial alleles with multiple mutations clearly exist, but are probably rare and only occasionally sampled even in the largest libraries. There are therefore clear parallels between our work and model predictions on the decrease repeatability at large population sizes [28,29]. However, as we performed a single round of selection on standing genetic variation, the putative scarcity of these alleles can in itself account for the observed patterns and the accessibility of evolutionary trajectories will not play a role.…”

“…A non-monotonic relationship between population size and the repeatability of evolution has been predicted in theoretical work [28,29], including decreases in repeatability in large populations due to the sampling of rare but highly beneficial combinations of multiple mutations. Here we could not show the existence of such a non-monotonic relationship due to the large variation between biological replicates when the full dataset is considered (Figure 3).…”

“…These large populations will sometimes sample rare, highly beneficial combinations of multiple mutations that would otherwise be evolutionarily inaccessible due to epistasis [28]. Genetic drift will also allow valley crossing in some small populations, whereas intermediate-size populations will instead greedily converge on a local fitness peak most of the time [27,29]. Examples of non-monotonic relations between population size and evolutionary predictability have also been observed experimentally, although they implicate a different underlying mechanism.…”

“…On the other hand, recent studies have shown that the relationship between population size and the repeatability of evolution can be more complex. Theoretical work has shown that the effects of population size on the repeatability of evolution can be non-monotonic: for very large populations adapting on a rugged fitness landscape the repeatability of evolution can decrease [28,29]. These large populations will sometimes sample rare, highly beneficial combinations of multiple mutations that would otherwise be evolutionarily inaccessible due to epistasis [28].…”

Genetic draft and valley crossing

“…On its own, valley crossing becomes more likely as the population size increases (Weissman et al 2009). However, recent work suggests that valley crossing relative to the fixation of a simple beneficial mutation is least likely at intermediate population sizes where tunneling occurs Ochs and Desai (2015). This suggests that tunneling may be the most difficult mode of valley crossing.…”