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Long-Term Dynamics of Adaptation in Asexual Populations

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Science  13 Dec 2013:
Vol. 342, Issue 6164, pp. 1364-1367
DOI: 10.1126/science.1243357

The Fit Get Fitter

Advances in modern biology have allowed us to measure evolutionary fitness and estimate the rate of fixation of beneficial mutations. Drawing on the Long-Term Evolution Experiment, studying the evolution of Escherichia coli in a constant environment, Wiser et al. (p. 1364, published online 14 November) demonstrate that even after 50,000 generations over 20 years, gains in fitness show no evidence of leveling off. Instead, fitness is following a power-law relationship that is dependent on epistasis and clonal interference.

Abstract

Experimental studies of evolution have increased greatly in number in recent years, stimulated by the growing power of genomic tools. However, organismal fitness remains the ultimate metric for interpreting these experiments, and the dynamics of fitness remain poorly understood over long time scales. Here, we examine fitness trajectories for 12 Escherichia coli populations during 50,000 generations. Mean fitness appears to increase without bound, consistent with a power law. We also derive this power-law relation theoretically by incorporating clonal interference and diminishing-returns epistasis into a dynamical model of changes in mean fitness over time.

The dynamics of evolving populations are often discussed in terms of movement on an adaptive landscape, where peaks and valleys are states of high and low fitness, respectively. There is considerable interest in the structure of these landscapes (17). Recent decades have seen tremendous growth in experiments using microbes to address fundamental questions about evolution (8), but most have been short in duration. The Long-Term Evolution Experiment (LTEE) with Escherichia coli provides the opportunity to characterize the dynamics of adaptive evolution over long periods under constant conditions (1, 9, 10). Twelve populations were founded from a common ancestor in 1988 and have been evolving for >50,000 generations, with samples frozen every 500 generations. The frozen bacteria remain viable, and we use this “fossil record” to assess whether fitness continues to increase and to characterize mean fitness trajectories (11).

We first performed 108 competitions, in the same conditions as the LTEE, between samples from nine populations at 40,000 and 50,000 generations against marked 40,000-generation clones (11). Three populations were excluded for technical reasons (11). Fitness was quantified as the dimensionless ratio of the competitors’ realized growth rates. Most populations experienced significant improvement (Fig. 1A), and the grand mean fitness increased by 3.0% (Fig. 1B).

Fig. 1 Fitness changes in nine E. coli populations between 40,000 and 50,000 generations.

(A) Filled symbols: six populations whose improvement was significant (P < 0.05); open symbols: three populations without significant improvement. (B) Grand-mean fitness at 40,000 and 50,000 generations relative to 40,000-generation competitor and the ratio of means showing overall gain. Error bars are 95% confidence limits based on replicate assays (A) or populations (B).

To examine the shape of the fitness trajectory, we competed samples from all 12 populations and up to 41 time points against the ancestor (11). We compared the fit of two alternative models with the fitness trajectories. The hyperbolic model describes a decelerating trajectory with an asymptote. The power law also decelerates (provided the exponent is <1), but fitness has no upper limit.

Hyperbolic model Embedded ImagePower law

Embedded Image

Mean fitness is Embedded Image, time in generations is t, and each model has two parameters, a and b. Both models are constrained such that the ancestral fitness is 1, hence the offset of +1 in the power law. The hyperbolic model was fit to the first 10,000 generations of the LTEE (9), but others suggested an alternative nonasymptotic trajectory (12). The grand mean fitness values and the trajectory for each model are shown in Fig. 2A and the individual populations in fig. S1. Both models fit the data very well; the correlation coefficients for the grand means and model trajectories are 0.969 and 0.986 for the hyperbolic and power-law models, respectively. When Bayesian information criterion scores (11) are used, the power law outperforms the hyperbolic model with a posterior odds ratio of ~30 million (table S1). The superior performance of the power law also holds when populations are excluded because of incomplete time series or evolved hypermutability (table S1). The power law provides a better fit to the grand-mean fitness than the hyperbolic model in early, middle, and late generations (fig. S2). The power law is supported (odds ratios >10) in six individual populations, whereas none supports the hyperbolic model to that degree (table S2). The power law also predicts fitness gains more accurately than the hyperbolic model. When fit to data for the first 20,000 generations only, the hyperbolic model badly underestimates later measurements, whereas the power-law trajectory predicts them accurately (Fig. 2B and fig. S3).

Fig. 2 Comparison of hyperbolic and power-law models.

(A) Hyperbolic (red) and power-law (blue) models fit to the set of mean fitness values (black symbols) from all 12 populations. (B) Fit of hyperbolic (solid red) and power-law (solid blue) models to data from first 20,000 generations only (filled symbols), with model predictions (dashed lines) and later data (open symbols). Error bars are 95% confidence limits based on the replicate populations.

The power law describes the fitness trajectories well, but it is not explanatory. We have derived a dynamical model of asexual populations with clonal interference and diminishing-returns epistasis, which generates mean-fitness trajectories that agree well with the experimental data. Clonal interference refers to competition among organisms with different beneficial mutations, which impedes their spread in asexual populations (1316). Diminishing-returns epistasis occurs when the marginal improvement from a beneficial mutation declines with increasing fitness (5, 6). We outline key points of the model below (11).

We used a coarse-grained approach that describes the magnitudes and time scales of fixation events (13). Beneficial mutations of advantage s are exponentially distributed with probability density αeαs, where 1/α is the mean advantage. This distribution is for mathematical convenience; the theory of clonal interference is robust to the form of the distribution (13). We assume that deleterious mutations do not appreciably affect the dynamics; deleterious mutations occur at a higher rate than beneficial mutations, but the resulting load is very small relative to the fitness increase measured over the course of the LTEE (17).

We assume the distribution of available benefits declines after a mutation with advantage Embedded Image fixes, such that α increases by a factor linearly related to Embedded Image:Embedded Imagewhere g > 0 is the diminishing-returns parameter, Embedded Image is beneficial effect of the nth fixed mutation, and αn is α after n fixations. Then, the mean fitness of an asexual population adapting to a constant environment is approximated by (11):Embedded Imagewhere Embedded Image and Embedded Image are the beneficial effect and fixation time, respectively, for the first fixed mutation.

Comparing this formula with the power law, g = 1/2a. The value of g estimated for the six populations that retained the low ancestral mutation rate throughout 50,000 generations is 6.0 (95% confidence interval 5.3 to 6.9). In the LTEE, the beneficial effect of the first fixation, Embedded Image, is typically ~0.1 (1, 9, 10). It follows that the distribution of beneficial effects immediately after the first fixation is shifted such that the mean advantage is Embedded Image of its initial value (11). This estimate of g also accords well with epistasis observed for early mutations in one of the populations (fig. S4). In principle, g might vary among populations if some fixed mutations lead to regions of the fitness landscape with different epistatic tendencies (18). However, an analysis of variance shows no significant heterogeneity in g among the six populations that maintained the ancestral mutation rate (P = 0.3478) (table S3). The g values tend to be lower for several populations that evolved hypermutability (table S4). However, these fits are confounded by the change in mutation rate; we show below that it is not necessary to invoke a difference in diminishing-returns epistasis between the hypermutable populations and those that retained the low ancestral mutation rate.

Diminishing-returns epistasis generates the power-law dynamics through the relation between a and g. Clonal interference affects the dynamics through the parameter b, which depends on Embedded Image and Embedded Image, which in turn are functions of the population size N, beneficial mutation rate μ, and initial mean beneficial effect 1/α0 (11). For the LTEE, N = 3.3 × 107, which takes into account the daily dilutions and regrowth (1). However, μ and α0 are unknown. Pairs of values that all match the best fit to the populations that retained the low mutation rate are shown in Fig. 3A. The expected values for beneficial effects and fixation times across a range of pairs are shown in Fig. 3B. The dynamics are similar among pairs with high beneficial mutation rates (μ > 10−8), giving Embedded Image and Embedded Image generations for the first fixation, which agree well with observations from the LTEE (1, 9, 10). At lower values of μ, adaptation becomes limited by the supply of beneficial mutations, and fixation times are inconsistent with the LTEE. This model also predicts that the rate of adaptation decelerates more sharply than the rate of genomic evolution (fig. S5), which is qualitatively consistent with observations (10, 11). The model assumes that individual beneficial mutations sweep sequentially, although “cohorts” of beneficial mutations may co-occur, especially at high μ (11, 15, 16, 19). However, the inferred role of diminishing returns in generating population mean-fitness dynamics is unaffected by this complication, because the power-law exponent is independent of μ. Moreover, we have verified by numerical simulations that co-occurring beneficial mutations have no appreciable affect on long-term fitness trajectories over the range of parameters considered here (fig. S6).

Fig. 3 Theoretical model generating power-law dynamics.

(A) Parameter pairs for μ and α0 that match best fit of power law to fitness trajectories for populations that retained ancestral mutation rate for 50,000 generations. (B) Expected times and beneficial effects of successive fixations for different pairs that match the best fit. The α0 values corresponding to each μ are shown in (A). In both panels g = 6.0, and N = 3.3 × 107.

Six populations evolved hypermutator phenotypes that increased their point-mutation rates by ~100-fold (11). Three of them became hypermutable early in the LTEE (between ~2500 and ~8500 generations) and had measurable fitness trajectories through at least 30,000 generations (table S2). Our model predicts these populations should adapt faster than those that retained the ancestral mutation rate. We pooled the data from these early hypermutators and confirmed that their composite fitness trajectory was substantially higher than that of the populations with the low mutation rate (Fig. 4). If the hypermutators’ beneficial mutation rate also increased by ~100-fold, the difference in trajectories is best fit by an ancestral rate μ = 1.7 × 10−6 (95% confidence interval 2.5 × 10−7 to 6.1 × 10−5), although higher values cannot be ruled out (11). Note that this fit was obtained by using the same initial distribution of fitness effects, α0, and epistasis parameter, g, for the hypermutators and the populations that retained the ancestral mutation rate.

Fig. 4 Effect of hypermutability on observed and predicted fitness trajectories.

Black circles: mean fitness of six populations that retained low ancestral mutation rate. Green triangles: mean fitness of three populations that evolved hypermutability early in the LTEE, including one with measurable values through 30,000 generations only. The hypermutators have higher mean fitness at 28 of 31 time points from 5000 to 50,000 generations. Black curve: Predicted trajectory of dynamic model with μ = 1.7 × 10−6, α0 = 85, g = 6.0, and N = 3.3 × 107. Green curve: Predicted trajectory with μ increased 100-fold starting at 4667 generations and all other parameters unchanged.

Both our empirical and theoretical analyses imply that adaptation can continue for a long time for asexual organisms, even in a constant environment. The 50,000 generations studied here occurred in one scientist’s laboratory in ~21 years. Now imagine that the experiment continues for 50,000 generations of scientists, each overseeing 50,000 bacterial generations, for 2.5 billion generations total. At that time, the predicted fitness relative to the ancestor is ~4.7 based on the power-law parameters estimated from all 12 populations (table S4). The ancestor’s doubling time in the glucose-limited minimal medium of the LTEE was ~55 min, and its growth commenced after a lag phase of ~90 min (20). If the bacteria eliminate the lag, a fitness of 4.7 implies a doubling time of ~23 min (fig. S7). Although that is fast for a minimal medium where cells must synthesize most constituents, it is slower than the 10 min that some species can achieve in nutrient-rich media (21). At some distant time, biophysical constraints may come into play, but the power-law fit to the LTEE does not predict implausible growth rates even far into the future. Also, some equilibrium might eventually be reached between the fitness-increasing effects of beneficial mutations and fitness-reducing effects of deleterious mutations (22), although it is impossible to predict when for realistic scenarios with heterogeneous selection coefficients, compensatory mutations, reversions, and changing mutation rates.

Fitness may continue to increase because even very small advantages become important over very long time scales in large populations. Consider a mutation with an advantage s = 10−6. The probability that this mutation escapes drift loss is ~4s for asexual binary fission (13), so it would typically have to occur 2.5 × 105 times before finally taking hold. Given a mutation rate of 10−10 per base pair per generation (23) and effective population size of ~3.3 × 10−7, it would require ~108 generations for that mutation to escape drift and millions more to fix. Also, pleiotropy and epistasis might allow a sustained supply of advantageous mutations, because many net-beneficial mutations have maladaptive side effects that create opportunities for compensatory mutations to ameliorate those effects.

The LTEE uses a simple, constant environment to minimize complications and thus illuminates the fundamental dynamics of adaptation by natural selection in asexual populations. The medium has one limiting resource and supports low population densities (for bacteria) to minimize the potential for cross-feeding on, or inhibition by, secreted by-products. Frequency-dependent interactions are weak in most populations, although stronger in some others (24). Also, such interactions should favor organisms that are more fit than their immediate predecessor, but they are not expected to amplify gains relative to a distant ancestor, as fitness was measured here. In fact, such interactions may cause fitness to fall relative to a distant ancestor (25). In any case, small-effect beneficial mutations should allow fitness to increase far into the future.

At present, the evidence that fitness can increase for tens of thousands of generations in a constant environment is limited to the LTEE, but these findings have broader implications for understanding evolutionary dynamics and the structure of fitness landscapes. It might be worthwhile to examine fitness trajectories from other evolution experiments in light of our results, although data from short-term experiments may not suffice to discriminate between asymptotic and nonasymptotic trajectories. We hope other teams will perform long experiments similar to the LTEE and that theoreticians will refine our models as appropriate.

Supplementary Materials

www.sciencemag.org/content/342/6164/1364/suppl/DC1

Materials and Methods

Supplementary Text

Figs. S1 to S7

Tables S1 to S4

References (2640)

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  38. Acknowledgments: This work was supported by grants from the National Science Foundation (DEB-1019989) including the BEACON Center for the Study of Evolution in Action (DBI-0939454), and by funds from the Hannah Chair Endowment at Michigan State University. We thank three reviewers for comments; I. Dworkin, J. Krug, A. McAdam, C. Wilke, and L. Zaman for discussions; and N. Hajela for technical assistance. R.E.L. will make strains available to qualified recipients, subject to completion of a material transfer agreement that can be found at www.technologies.msu.edu/inventors/mta-cda/mta/mta-forms. Datasets and analysis scripts are available at the Dryad Digital Repository (doi:10.5061/dryad.0hc2m).
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