Distribution of fixed beneficial mutations and the rate of adaptation in asexual populations

Abstract

When large asexual populations adapt, competition between simultaneously segregating mutations slows the rate of adaptation and restricts the set of mutations that eventually fix. This phenomenon of interference arises from competition between mutations of different strengths as well as competition between mutations that arise on different fitness backgrounds. Previous work has explored each of these effects in isolation, but the way they combine to influence the dynamics of adaptation remains largely unknown. Here, we describe a theoretical model to treat both aspects of interference in large populations. We calculate the rate of adaptation and the distribution of fixed mutational effects accumulated by the population. We focus particular attention on the case when the effects of beneficial mutations are exponentially distributed, as well as on a more general class of exponential-like distributions. In both cases, we show that the rate of adaptation and the influence of genetic background on the fixation of new mutants is equivalent to an effective model with a single selection coefficient and rescaled mutation rate, and we explicitly calculate these effective parameters. We find that the effective selection coefficient exactly coincides with the most common fixed mutational effect. This equivalence leads to an intuitive picture of the relative importance of different types of interference effects, which can shift dramatically as a function of the population size, mutation rate, and the underlying distribution of fitness effects.E volutionary adaptation is driven by the accumulation of beneficial mutations, and yet many aspects of this process are still poorly understood. In asexual populations, this subject can be distilled into two main lines of inquiry: (i) what are the possible mutations available to the population? and (ii) which of these mutations are actually incorporated into the population, and what are the dynamics by which they fix?The first question is essentially an empirical matter. At any given instant in time, the set of accessible beneficial mutations is likely to depend on the history of the population as well as its environment and any epistatic interactions between mutations. Nonetheless, if history and epistatic effects do not significantly affect the statistics of the available mutations, we can define a constant distribution of fitness effects ρðsÞ that gives the relative probability of obtaining a mutation that increases the fitness of an individual by s.Gillespie (1) and Orr (2) have argued that there are general theoretical reasons to expect that ρðsÞ should follow an exponential distribution, although more recent theoretical work has challenged the ubiquity of this claim (3). Many experimental studies are roughly consistent with this exponential prediction (4-6), although here, too, we find significant exceptions (6-10). In the present work, we maintain a relatively agnostic view toward the precise form of ρðsÞ, although we devote special attention to the exponential case because of i...

Specificity of genome evolution in experimental populations of Escherichia coli evolved at different temperatures

“…Second, a phenomenon called clonal interference occurs in evolving asexual populations because beneficial mutations that arise in different lineages in the same population cannot be brought together by recombination. As a consequence, lineages with highly beneficial mutations outcompete other lineages with less beneficial mutations, such that only the most beneficial mutations available typically can achieve fixation on the timescale of this experiment (39)(40)(41). For example, even though mutations in nadR may be beneficial under all of the temperature regimes, they are able to reach high frequency in competition against other mutations only in the 32°C environment, whereas mutations in other genes will dominate the initial waves of adaptation at other temperatures.…”

The dynamics of molecular evolution over 60,000 generations

“…When beneficial mutations are common, mutations that would otherwise drive selective sweeps can be outcompeted by other lineages carrying superior beneficial mutations 23 . Further beneficial mutations can draw out this battle, resulting in allele-frequency trajectories with multiple inflection points 12,24,25 . Yet models of clonal interference predict that one lineage must eventually win, and so on long timescales the number of fixed mutations should grow at the same rate as the total allele frequency M p (t) .…”

“…These studies reveal complex dynamics, characterized by rapid adaptation, competition between beneficial mutations, diminishing-returns epistasis, and extensive genetic parallelism. These forces alter patterns of polymorphism 11 and influence which mutations ultimately fix 12–15 . However, it is unclear whether these dynamics are general or, instead, reflect the short timescales and novel environmental conditions of previous studies.…”

Collective Fluctuations in the Dynamics of Adaptation and Other Traveling Waves

“…Finally, there is also the possibility to invoke a particular dynamical constraint with the property that the dynamics exhibit a closed linear equation for the first moment. Importantly, this method, which has been called "model tuning" (Hallatschek 2011;Good et al 2012;Geyrhofer and Hallatschek 2013), reproduces the universal features in the large population size limit, which are independent of the chosen population control, ultimately because of the weakness of the population size dependence. Since this approach also reproduces the correct behavior in the small population size limit when mutational effect sizes are small, it may be viewed as a practical interpolation scheme for the entire range of population sizes (Hallatschek 2011).…”

“…Such an ad hoc approach, based on a growth rate cutoff, correctly reproduces the wave speed to the leading order but does not reveal other universal next-to leading-order corrections. One can also invoke a branching-process approximation for the tip of the wave, thereby neglecting effects of the nonlinear population size control, and then match this linearized description with a deterministic description of the bulk of the population (Rouzine et al 2003(Rouzine et al , 2008Schiffels et al 2011;Good et al 2012). Finally, there is also the possibility to invoke a particular dynamical constraint with the property that the dynamics exhibit a closed linear equation for the first moment.…”