The stochastic edge in adaptive evolution

For small asexual populations and low mutation rates, the speed of adaptation is primarily limited by the availability of beneficial mutations: a mutation has the time to reach fixation before the next mutation occurs. Therefore, in this case the speed of adaptation increases linearly with population size and mutation rate. By contrast, for large asexual populations or high mutation rates, beneficial mutations are abundant. In this case, the main limit to adaptation is that many beneficial mutations are wasted: when arising on different genetic backgrounds, they cannot recombine and thus are in competition with each other. The theoretical prediction of the speed of adaptation in the latter case is a formidable challenge even for the simplest models. The earliest attempts to predict this speed go back to Maynard Smith (1971), and in recent years several groups have improved upon and extended this work (Barton 1995;Tsimring et al. 1996;Prügel-Bennett 1997;Kessler et al. 1997;Gerrish and Lenski 1998;Orr 2000;Rouzine et al. 2003Rouzine et al. , 2008;;Wilke 2004;Desai and Fisher 2007). The recent works can be broadly subdivided into two classes: (i) so-called "clonal-interference models" (Gerrish and Lenski 1998;Orr 2000;Wilke 2004;Park and Krug 2007), which emphasize that different beneficial mutations have different-sized effects, and that mutations with large beneficial effects tend to outcompete mutations with small beneficial effects, and (ii) models in which all mutations have the same effect s (Tsimring et al. 1996;Kessler et al. 1997;Rouzine et al. 2003Rouzine et al. , 2008;;Desai and Fisher 2007). The latter type of models emphasize that in large populations, multiple beneficial mutations frequently occur in quick succession on the same genetic background. These models, however, neglect clonal-interference effects.

For the second class of models, where all mutations have the same fitness effect, each individual can be conveniently described by the number k of beneficial mutations it holds. The whole adapting population can then be seen as a traveling wave (Tsimring et al. 1996;Rouzine et al. 2003Rouzine et al. , 2008) ) moving with time through fitness space towards increasing values of k. In the traveling-wave approach, the bulk of the population, for which each k value is occupied by many individuals, can be accurately described using a deterministic partial differential equation. However, the partial differential equation breaks down for the rare mutants that have the highest fitness in the population, because these rare mutants are subject to substantial genetic drift and stochasticity. Therefore, the description of this stochastic edge must be approached differently, and must be coupled with the description of the bulk of the population. Specifically, the deterministic equation admits a traveling-wave solution for any velocity. The high-fitness tail of that solution ends at a finite point, which is identified with the stochastic edge. To select one solution (and thus determine the wave speed), Rouzine et al. (2003Rouzine et al. ( , 2008) ) estimated the average size of the stochastic edge using a stochastic argument, and matched this size to the solution of the deterministic equation.

Recently, Desai and Fisher (2007) have proposed a new method to calculate the speed of adaptation for the same model. They mainly carry out an elaborate treatment of the stochastic edge, with little attention paid to the bulk of the population. The full-population model is effectively replaced with a two-class model consisting of the best-fit and the secondbest-fit classes only; the best-fit class is treated stochastically, whereas the next-best class is assumed to increase exponentially in time due to selection. Beneficial mutations are neglected compared to the effect of selection, except for mutations into the best-fit class. At the very end of the derivation, the sizes of other fitness classes are estimated to provide a normalization condition.

Both Rouzine et al. (2003Rouzine et al. ( , 2008) ) and Desai and Fisher (2007) calculate the speed of adaptation in steady state, when mutation-selection balance maintains the shape of the traveling wave. The transient dynamics generally happen on a short timescale but are hard to quantify analytically (Tsimring et al. 1996;Desai and Fisher 2007). Rouzine et al. (2003Rouzine et al. ( , 2008) ) define the speed of adaptation as the change of the population’s mean number of mutations over time, V = d k /dt. Desai and Fisher (2007) consider instead the change in the population’s mean fitness, v = sV . Both approaches consider as an intermediate quantity the lead q, defined as the difference between the number of mutations of the best fit individuals and the average number of mutations in the population, and write a relation between q and the mean establishment time τ = 1/V of a new fitness class at the stochastic edge of the population. (Note that Rouzine et al. (2003Rouzi

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