The outcomes of evolution are determined by a stochastic dynamical process that governs how mutations arise and spread through a population. Here, we analyze the dynamics of molecular evolution in twelve experimental populations of Escherichia coli, using whole-genome metagenomic sequencing at 500-generation intervals through 60,000 generations. Despite a declining rate of fitness gain, molecular evolution continues to be characterized by signatures of rapid adaptation, with multiple beneficial variants simultaneously competing for dominance in each population. Interactions between ecological and evolutionary processes play an important role, as long-term quasi-stable coexistence arises spontaneously in most populations, and evolution continues within each clade. We also present new evidence that the targets of natural selection change over time, as epistasis and historical contingency alter the strength of selection on different genes. Together, these results show that long-term adaptation to a constant environment can be a more complex and dynamic process than is often assumed.

Although widely used in practice, the behavior and accuracy of the popular module identification technique called modularity maximization is not well understood in practical contexts. Here, we present a broad characterization of its performance in such situations. First, we revisit and clarify the resolution limit phenomenon for modularity maximization. Second, we show that the modularity function Q exhibits extreme degeneracies: it typically admits an exponential number of distinct high-scoring solutions and typically lacks a clear global maximum. Third, we derive the limiting behavior of the maximum modularity Qmax for one model of infinitely modular networks, showing that it depends strongly both on the size of the network and on the number of modules it contains. Finally, using three real-world metabolic networks as examples, we show that the degenerate solutions can fundamentally disagree on many, but not all, partition properties such as the composition of the largest modules and the distribution of module sizes. These results imply that the output of any modularity maximization procedure should be interpreted cautiously in scientific contexts. They also explain why many heuristics are often successful at finding high-scoring partitions in practice and why different heuristics can disagree on the modular structure of the same network. We conclude by discussing avenues for mitigating some of these behaviors, such as combining information from many degenerate solutions or using generative models.

Gut microbiota are shaped by a combination of ecological and evolutionary forces. While the ecological dynamics have been extensively studied, much less is known about how species of gut bacteria evolve over time. Here, we introduce a model-based framework for quantifying evolutionary dynamics within and across hosts using a panel of metagenomic samples. We use this approach to study evolution in approximately 40 prevalent species in the human gut. Although the patterns of between-host diversity are consistent with quasi-sexual evolution and purifying selection on long timescales, we identify new genealogical signatures that challenge standard population genetic models of these processes. Within hosts, we find that genetic differences that accumulate over 6-month timescales are only rarely attributable to replacement by distantly related strains. Instead, the resident strains more commonly acquire a smaller number of putative evolutionary changes, in which nucleotide variants or gene gains or losses rapidly sweep to high frequency. By comparing these mutations with the typical between-host differences, we find evidence that some sweeps may be seeded by recombination, in addition to new mutations. However, comparisons of adult twins suggest that replacement eventually overwhelms evolution over multi-decade timescales, hinting at fundamental limits to the extent of local adaptation. Together, our results suggest that gut bacteria can evolve on human-relevant timescales, and they highlight the connections between these short-term evolutionary dynamics and longer-term evolution across hosts.

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...

Pervasive natural selection can strongly influence observed patterns of genetic variation, but these effects remain poorly understood when multiple selected variants segregate in nearby regions of the genome. Classical population genetics fails to account for interference between linked mutations, which grows increasingly severe as the density of selected polymorphisms increases. Here, we describe a simple limit that emerges when interference is common, in which the fitness effects of individual mutations play a relatively minor role. Instead, similar to models of quantitative genetics, molecular evolution is determined by the variance in fitness within the population, defined over an effectively asexual segment of the genome (a “linkage block”). We exploit this insensitivity in a new “coarse-grained” coalescent framework, which approximates the effects of many weakly selected mutations with a smaller number of strongly selected mutations that create the same variance in fitness. This approximation generates accurate and efficient predictions for silent site variability when interference is common. However, these results suggest that there is reduced power to resolve individual selection pressures when interference is sufficiently widespread, since a broad range of parameters possess nearly identical patterns of silent site variability.

Purifying selection reduces genetic diversity, both at sites under direct selection and at linked neutral sites. This process, known as background selection, is thought to play an important role in shaping genomic diversity in natural populations. Yet despite its importance, the effects of background selection are not fully understood. Previous theoretical analyses of this process have taken a backward-time approach based on the structured coalescent. While they provide some insight, these methods are either limited to very small samples or are computationally prohibitive. Here, we present a new forward-time analysis of the trajectories of both neutral and deleterious mutations at a nonrecombining locus. We find that strong purifying selection leads to remarkably rich dynamics: neutral mutations can exhibit sweep-like behavior, and deleterious mutations can reach substantial frequencies even when they are guaranteed to eventually go extinct. Our analysis of these dynamics allows us to calculate analytical expressions for the full site frequency spectrum. We find that whenever background selection is strong enough to lead to a reduction in genetic diversity, it also results in substantial distortions to the site frequency spectrum, which can mimic the effects of population expansions or positive selection. Because these distortions are most pronounced in the low and high frequency ends of the spectrum, they become particularly important in larger samples, but may have small effects in smaller samples. We also apply our forward-time framework to calculate other quantities, such as the ultimate fates of polymorphisms or the fitnesses of their ancestral backgrounds.

Natural environments are never truly constant, but the evolutionary implications of temporally varying selection pressures remain poorly understood. Here we investigate how the fate of a new mutation in a fluctuating environment depends on the dynamics of environmental variation and on the selective pressures in each condition. We find that even when a mutation experiences many environmental epochs before fixing or going extinct, its fate is not necessarily determined by its time-averaged selective effect. Instead, environmental variability reduces the efficiency of selection across a broad parameter regime, rendering selection unable to distinguish between mutations that are substantially beneficial and substantially deleterious on average. Temporal fluctuations can also dramatically increase fixation probabilities, often making the details of these fluctuations more important than the average selection pressures acting on each new mutation. For example, mutations that result in a trade-off between conditions but are strongly deleterious on average can nevertheless be more likely to fix than mutations that are always neutral or beneficial. These effects can have important implications for patterns of molecular evolution in variable environments, and they suggest that it may often be difficult for populations to maintain specialist traits, even when their loss leads to a decline in time-averaged fitness.population genetics | fixation probability | fluctuating environment | effective diffusion E volutionary trade-offs are widespread: Adaptation to one environment often leads to costs in other conditions. For example, drug resistance mutations often carry a cost when the dosage of the drug decays (1), and seasonal variations in climate can differentially select for certain alleles in the summer or winter (2). Similarly, laboratory adaptation to specific temperatures (3, 4) or particular nutrient sources (5, 6) often leads to declines in fitness in other conditions. Related trade-offs apply to any specialist phenotype or regulatory system that incurs a general cost to confer benefits in specific environmental conditions (7). Despite the ubiquity of these trade-offs, it is not always easy to predict when a specialist phenotype can evolve and persist. How useful must a trait be on average to be maintained? How regularly does it need to be useful? How much easier is it to maintain in a larger population compared with a smaller one?The answers to these questions depend on two major factors. First, how often do new mutations create or destroy a specialist phenotype, and what are their typical costs and benefits across environmental conditions? This is fundamentally an empirical question, which depends on the costs and benefits of the trait in question, as well as its genetic architecture (e.g., the target size for loss-of-function mutations that disable a regulatory system). In this paper, we focus instead on the second major factor: given that a particular mutation occurs, how does its long-term fate depend on its fitn...

Genetic interactions can strongly influence the fitness effects of individual mutations, yet the impact of these epistatic interactions on evolutionary dynamics remains poorly understood. Here we investigate the evolutionary role of epistasis over 50,000 generations in a well-studied laboratory evolution experiment in Escherichia coli. The extensive duration of this experiment provides a unique window into the effects of epistasis during long-term adaptation to a constant environment. Guided by analytical results in the weak-mutation limit, we develop a computational framework to assess the compatibility of a given epistatic model with the observed patterns of fitness gain and mutation accumulation through time. We find that a decelerating fitness trajectory alone provides little power to distinguish between competing models, including those that lack any direct epistatic interactions between mutations. However, when combined with the mutation trajectory, these observables place strong constraints on the set of possible models of epistasis, ruling out many existing explanations of the data. Instead, we find that the data are consistent with a "two-epoch" model of adaptation, in which an initial burst of diminishing-returns epistasis is followed by a steady accumulation of mutations under a constant distribution of fitness effects. Our results highlight the need for additional DNA sequencing of these populations, as well as for more sophisticated models of epistasis that are compatible with all of the experimental data.A central feature of evolutionary adaptation is that the space of potential innovations can vary with the evolutionary history of a population. Examples are common in the microbial world: the ability to import a nutrient may be beneficial only if a mechanism has evolved to utilize it (Quandt et al. 2014), while a previously advantageous drug resistance mutation can be rendered obsolete by the acquisition of a second resistance allele (Weinreich et al. 2006). This capacity for evolutionary feedback is quantified in terms of epistasis, which measures how the effect of a mutation depends on the genetic background in which it arises. In principle, epistasis can lead to widespread historical contingency and can fundamentally alter the dynamics of adaptation (Wright 1932;Gould 1989). But in practice, the long-term evolutionary impact of epistasis remains largely uncharacterized.Empirical patterns of epistasis are most commonly measured using a direct approach (see de Visser and Krug 2014 for a recent review). Candidate mutations are introduced into a set of genetic backgrounds via crossing or other genetic reconstruction techniques, and the fitnesses of the reconstructed genotypes are measured using competitive fitness assays or related proxies. These data yield a functional relationship between the fitness effect of a mutation and its genetic background, with the traditional pairwise epistasis emerging as a special case when the backgrounds are single mutants. We use the term microscopic epista...