Search

The fitness landscapes of genetic sequences are high-dimensional and “rugged” due to sign epistasis. Empirical limitations and the abstractness of many landscape models limit our understanding of how ruggedness shapes the mode and tempo... The “fitness” landscapes of genetic sequences are characterized by high dimensionality and “ruggedness” due to sign epistasis. Ascending from low to high fitness on such landscapes can be difficult because adaptive trajectories get stuck at low-fitness local peaks. Compounding matters, recent theoretical arguments have proposed that extremely long, winding adaptive paths may be required to reach even local peaks: a “maze-like” landscape topography. The extent to which peaks and mazes shape the mode and tempo of evolution is poorly understood, due to empirical limitations and the abstractness of many landscape models. We explore the prevalence, scale, and evolutionary consequences of landscape mazes in a biophysically grounded computational model of protein evolution that captures the “frustration” between “stability” and aggregation propensity. Our stability-aggregation landscape exhibits extensive sign epistasis and local peaks galore. Although this frequently obstructs adaptive ascent to high fitness and virtually eliminates reproducibility of evolutionary outcomes, many adaptive paths do successfully complete the ascent from low to high fitness, with hydrophobicity a critical mediator of success. These successful paths exhibit maze-like properties on a global landscape scale, in which taking an indirect path helps to avoid low-fitness local peaks. This delicate balance of “hard but possible” adaptation could occur more broadly in other biological settings where competing interactions and frustration are important.

Adaptation dynamics on fitness landscapes is often studied theoretically in the strong-selection, weak-mutation regime. However, in a large population, multiple beneficial mutants can emerge before any of them fixes in the population. Competition between mutants is known as clonal interference, and while it is known to slow down the rate of adaptation (when compared to the strong-selection, weak-mutation model with the same parameters), how it affects the shape of long-term fitness trajectories in the presence of epistasis is an open question. Here, by considering how changes in fixation probabilities arising from weak clonal interference affect the dynamics of adaptation on fitness-parameterized landscapes, we find that the change in the shape of fitness trajectory arises only through changes in the supply of beneficial mutations (or equivalently, the beneficial mutation rate). Furthermore, a depletion of beneficial mutations as a population climbs up the fitness landscape can speed up the rescaled fitness trajectory (where adaptation speed is measured relative to its value at the start of the experiment), while an enhancement of the beneficial mutation rate does the opposite of slowing it down. Our findings suggest that by carrying out evolution experiments in both regimes (with and without clonal interference), one could potentially distinguish the different sources of macroscopic epistasis (fitness effect of mutations vs change in fraction of beneficial mutations).

Novel genotypes evolve under selection through mutations in pre-existing genes. However, mutations have pleiotropic phenotypic effects that influence the fitness of emerging genotypes in complex ways. The evolution of antimicrobial resistance is mediated by selection of mutations in genes coding for antibiotic-target proteins. Drug-resistance is commonly associated with a fitness cost due to the impact of resistance-conferring mutations on protein function and/or stability. These costs are expected to prohibit the selection of drug-resistant mutations at low drug pressures. Using laboratory evolution of rifampicin resistance in Escherichia coli, we show that when exposed intermittently to low concentration (0.1 × minimal inhibitory concentration) of rifampicin, the evolution of canonical drug resistance was indeed unfavorable. Instead, these bacterial populations adapted by evolving into small-colony variants that displayed enhanced pellicle-forming ability. This shift in lifestyle from planktonic to pellicle-like was necessary for enhanced fitness at low drug pressures, and was mediated by the genetic activation of the fim operon promoter, which allowed expression of type I fimbriae. Upon continued low drug exposure, these bacteria evolved exclusively into high-level drug-resistant strains through mutations at a limited set of loci within the rifampicin-resistance determining region of the rpoB gene. We show that our results are explained by mutation-specific epistasis, resulting in differential impact of lifestyle switching on the competitive fitness of different rpoB mutations. Thus, lifestyle-alterations that are selected at low selection pressures have the potential to modify the fitness effects of mutations, change the genetic structure, and affect the ultimate fate of evolving populations.

Gene interactions, or epistasis, play a large role in determining evolutionary outcomes. The ruggedness of fitness landscapes, and thus the predictability of evolution and the accessibility of high-fitness genotypes, is determined largely by the pervasiveness of epistasis and the degree of correlation between similar genotypes. We created all possible pairings of three sets of five beneficial first-step mutations fixed during adaptive walks under three different regimes: selection on growth rate alone, on growth rate and thermal stability, and on growth rate and pH stability. All 30 double-mutants displayed negative, antagonistic epistasis with regard to growth rate and fitness, but positive epistasis and additivity were common for the stability phenotypes. This suggested that biophysically simple phenotypes, such as capsid stability, may on average behave more additively than complex phenotypes like viral growth rate. Growth rate epistasis was also smaller in magnitude when the individual effects of single mutations were smaller. Significant sign epistasis, such that the effect of a mutation that is beneficial in the wild-type background is deleterious in combination with a second mutation, emerged more frequently in intragenic mutational pairings than in intergenic pairs, and was evident in nearly half of the double-mutants, indicating that the fitness landscape is moderately uncorrelated and of intermediate ruggedness. Together, our results indicated that mutations may interact additively with regard to phenotype when considered at a basic, biophysical level, but that epistasis arises as a result of pleiotropic interactions between the individual components of complex phenotypes and diminishing returns arising from intermediate phenotypic optima.

The fitness landscape is a concept that is widely used for understanding and predicting evolutionary adaptation. The topography of the fitness landscape depends critically on the environment, with potentially far-reaching consequences for evolution under changing conditions. However, few studies have assessed directly how empirical fitness landscapes change across conditions, or validated the predicted consequences of such change. We previously evolved replicate yeast populations in the presence of either gradually increasing, or constant high, concentrations of the heavy metals cadmium (Cd), nickel (Ni), and zinc (Zn), and analyzed their phenotypic and genomic changes. Here, we reconstructed the local fitness landscapes underlying adaptation to each metal by deleting all repeatedly mutated genes both by themselves and in combination. Fitness assays revealed that the height, and/or shape, of each local fitness landscape changed considerably across metal concentrations, with distinct qualitative differences between unconditionally (Cd) and conditionally toxic metals (Ni and Zn). This change in topography had particularly crucial consequences in the case of Ni, where a substantial part of the individual mutational fitness effects changed in sign across concentrations. Based on the Ni landscape analyses, we made several predictions about which mutations had been selected when during the evolution experiment. Deep sequencing of population samples from different time points generally confirmed these predictions, demonstrating the power of landscape reconstruction analyses for understanding and ultimately predicting evolutionary dynamics, even under complex scenarios of environmental change.

An important question in evolutionary theory is how adaptation might be hindered on a rugged fitness landscape in the presence of strong selection that.. Gene conversion is a ubiquitous phenomenon that leads to the exchange of genetic information between homologous DNA regions and maintains coevolving multi-gene families in most prokaryotic and eukaryotic organisms. In this paper, we study its implications for the evolution of a single functional gene with a silenced duplicate, using two different models of evolution on rugged fitness landscapes. Our analytical and numerical results show that, by helping to circumvent valleys of low fitness, gene conversion with a passive duplicate gene can cause a significant speedup of adaptation, which depends nontrivially on the frequency of gene conversion and the structure of the landscape. We find that stochastic effects due to finite population sizes further increase the likelihood of exploiting this evolutionary pathway. A universal feature appearing in both deterministic and stochastic analysis of our models is the existence of an optimal gene conversion rate, which maximizes the speed of adaptation. Our results reveal the potential for duplicate genes to act as a “scratch paper” that frees evolution from being limited to strictly beneficial mutations in strongly selective environments.

The snowball hypothesis suggests that the number of genetic incompatibilities between two populations will accumulate faster than linearly as they diverge... Genetic incompatibilities can emerge as a byproduct of genetic divergence. According to Dobzhansky and Muller, an allele that fixes in one population may be incompatible with an allele at a different locus in another population when the two alleles are brought together in hybrids. Orr showed that the number of Dobzhansky-Muller incompatibilities (DMIs) should accumulate faster than linearly—i.e., snowball—as two lineages diverge. Several studies have attempted to test the snowball effect using data from natural populations. One limitation of these studies is that they have focused on predictions of the Orr model, but not on its underlying assumptions. Here, we use a computational model of RNA folding to test both predictions and assumptions of the Orr model. Two populations are allowed to evolve in allopatry on a holey fitness landscape. We find that the number of inviable introgressions (an indicator for the number of DMIs) snowballs, but does so more slowly than expected. We show that this pattern is explained, in part, by the fact that DMIs can disappear after they have arisen, contrary to the assumptions of the Orr model. This occurs because DMIs become progressively more complex (i.e., involve alleles at more loci) as a result of later substitutions. We also find that most DMIs involve >2 loci, i.e., they are complex. Reproductive isolation does not snowball because DMIs do not act independently of each other. We conclude that the RNA model supports the central prediction of the Orr model that the number of DMIs snowballs, but challenges other predictions, as well as some of its underlying assumptions.

Recent studies of protein evolution contend that the longer an amino acid substitution is present at a site, the less likely it is to revert to the amino acid previously occupying that site. Here we study this phenomenon of decreasing reversion rates rigorously and in a much more general context. We show that, under weak mutation and for arbitrary fitness landscapes, reversion rates decrease with time for any site that is involved in at least one epistatic interaction. Specifically, we prove that, at stationarity, the hazard function of the distribution of waiting times until reversion is strictly decreasing for any such site. Thus, in the presence of epistasis, the longer a particular character has been absent from a site, the less likely the site will revert to its prior state. We also explore several examples of this general result, which share a common pattern whereby the probability of having reverted increases rapidly at short times to some substantial value before becoming almost flat after a few substitutions at other sites. This pattern indicates a characteristic tendency for reversion to occur either almost immediately after the initial substitution or only after a very long time.

The fitness landscape defines the relationship between genotypes and fitness in a given environment, and underlies fundamental quantities such as the distribution of selection coefficient, or the magnitude and type of epistasis. A better understanding of variation of landscape structure across species and environments is thus necessary to understand and predict how populations will adapt. An increasing number of experiments investigates the properties of fitness landscapes by identifying mutations, constructing genotypes with combinations of these mutations, and measuring the fitness of these genotypes. Yet these empirical landscapes represent a very small sample of the vast space of all possible genotypes, and this sample is often biased by the protocol used to identify mutations. Here we develop a rigorous statistical framework based on Approximate Bayesian Computation to address these concerns, and use this flexible framework to fit a broad class of phenotypic fitness models (including Fisher's model) to 26 empirical landscapes representing 9 diverse biological systems. In spite of uncertainty due to the small size of most published empirical landscapes, the inferred landscapes have similar structure in similar biological systems. Surprisingly, goodness of fit tests reveal that this class of phenotypic models, which has been successful so far in interpreting experimental data, is a plausible model in only 3 out of 9 biological systems. More precisely, although Fisher's model was able to explain several statistical properties of the landscapes - including mean and standard deviation of selection and epistasis coefficients -, it was often unable to explain the full structure of fitness landscapes., Comment: Added an alternative ABC algorithm based on a set of summary statistics

Fitness is a central quantity in evolutionary models of viruses. However, it remains difficult to determine viral fitness experimentally, and existing in vitro assays can be poor predictors of in vivo fitness of viral populations within their hosts. Next-generation sequencing can nowadays provide snapshots of evolving virus populations, and these data offer new opportunities for inferring viral fitness. Using the equilibrium distribution of the quasispecies model, an established model of intrahost viral evolution, we linked fitness parameters to the composition of the virus population, which can be estimated by next-generation sequencing. For inference, we developed a Bayesian Markov chain Monte Carlo method to sample from the posterior distribution of fitness values. The sampler can overcome situations where no maximum-likelihood estimator exists, and it can adaptively learn the posterior distribution of highly correlated fitness landscapes without prior knowledge of their shape. We tested our approach on simulated data and applied it to clinical human immunodeficiency virus 1 samples to estimate their fitness landscapes in vivo. The posterior fitness distributions allowed for differentiating viral haplotypes from each other, for determining neutral haplotype networks, in which no haplotype is more or less credibly fit than any other, and for detecting epistasis in fitness landscapes. Our implemented approach, called QuasiFit, is available at http://www.cbg.ethz.ch/software/quasifit.

Empirical evidence for diminishing fitness returns of beneficial mutations supports Fisher’s geometric model. We show that a similar pattern emerges through the phenomenon of regression to the mean and that few studies correct for it. Although biases are often small, regression to the mean has overemphasized diminishing returns and will hamper cross-study comparisons unless corrected for.

Evolutionary adaptations in metabolic networks are fundamental to evolution of microbial growth. Studies on unneeded-protein synthesis indicate reductions in fitness upon nonfunctional protein synthesis, showing that cell growth is limited by constraints acting on cellular protein content. Here, we present a theory for optimal metabolic enzyme activity when cells are selected for maximal growth rate given such growth-limiting biochemical constraints. We show how optimal enzyme levels can be understood to result from an enzyme benefit minus cost optimization. The constraints we consider originate from different biochemical aspects of microbial growth, such as competition for limiting amounts of ribosomes or RNA polymerases, or limitations in available energy. Enzyme benefit is related to its kinetics and its importance for fitness, while enzyme cost expresses to what extent resource consumption reduces fitness through constraint-induced reductions of other enzyme levels. A metabolic fitness landscape is introduced to define the fitness potential of an enzyme. This concept is related to the selection coefficient of the enzyme and can be expressed in terms of its fitness benefit and cost.

In relating genotypes to fitness, models of adaptation need to both be computationally tractable and qualitatively match observed data. One reason that tractability is not a trivial problem comes from a combinatoric problem whereby no matter in what order a set of mutations occurs, it must yield the same fitness. We refer to this as the bookkeeping problem. Because of their commutative property, the simple additive and multiplicative models naturally solve the bookkeeping problem. However, the fitness trajectories and epistatic patterns they predict are inconsistent with the patterns commonly observed in experimental evolution. This motivates us to propose a new and equally simple model that we call stickbreaking. Under the stickbreaking model, the intrinsic fitness effects of mutations scale by the distance of the current background to a hypothesized boundary. We use simulations and theoretical analyses to explore the basic properties of the stickbreaking model such as fitness trajectories, the distribution of fitness achieved, and epistasis. Stickbreaking is compared to the additive and multiplicative models. We conclude that the stickbreaking model is qualitatively consistent with several commonly observed patterns of adaptive evolution.

Because adaptation depends upon the fixation of novel beneficial mutations, the fitness effects of beneficial mutations that are substituted by selection are key to our understanding of the process of adaptation. In this study, we experimentally investigated the fitness effects of beneficial mutations that are substituted when populations of the pathogenic bacterium Pseudomonas aeruginosa adapt to the antibiotic rifampicin. Specifically, we isolated the first beneficial mutation to be fixed by selection when 96 populations of three different genotypes of P. aeruginosa that vary considerably in fitness in the presence of rifampicin were challenged with adapting to a high dose of this antibiotic. The simple genetics of rifampicin resistance allowed us to determine the genetic basis of adaptation in the majority of our populations. We show that the average fitness effects of fixed beneficial mutations show a simple and clear pattern of diminishing returns, such that selection tends to fix mutations with progressively smaller effects as populations approach a peak on the adaptive landscape. The fitness effects of individual mutations, on the other hand, are highly idiosyncratic across genetic backgrounds, revealing pervasive epistasis. In spite of this complexity of genetic interactions in this system, there is an overall tendency toward diminishing-returns epistasis. We argue that a simple overall pattern of diminishing-returns adaptation emerges, despite pervasive epistasis between beneficial mutations, because many beneficial mutations are available, and while the fitness landscape is rugged at the fine scale, it is smooth and regular when we consider the average over possible routes to adaptation. In the context of antibiotic resistance, these results show that acquiring mutations that confer low levels of antibiotic resistance does not impose any constraint on the ability to evolve high levels of resistance.

Understanding how interactions among deleterious mutations affect fitness may shed light on a variety of fundamental biological phenomena, including the evolution of sex, the buffering of genetic variations, and the topography of fitness landscapes. It remains an open question under what conditions and to what extent such interactions may be synergistic or antagonistic. To address this question, we employed a computer model for the intracellular growth of bacteriophage T7. We created in silico 90,000 mutants of phage T7, each carrying from 1 to 30 mutations, and evaluated the fitness of each by simulating its growth cycle. The simulations sought to account for the severity of single deleterious mutations on T7 growth, as well as the effect of the resource environment on our fitness measures. We found that mildly deleterious mutations interacted synergistically in poor-resource environments but antagonistically in rich-resource environments. However, severely deleterious mutations always interacted antagonistically, irrespective of environment. These results suggest that synergistic epistasis may be difficult to experimentally distinguish from nonepistasis because its effects appear to be most pronounced when the effects of mutations on fitness are most challenging to measure. Our approach demonstrates how computer simulations of developmental processes can be used to quantitatively study genetic interactions at the population level.

Much of the current theory of adaptation is based on Gillespie's mutational landscape model (MLM), which assumes that the fitness values of genotypes linked by single mutational steps are independent random variables. On the other hand, a growing body of empirical evidence shows that real fitness landscapes, while possessing a considerable amount of ruggedness, are smoother than predicted by the MLM. In the present article we propose and analyse a simple fitness landscape model with tunable ruggedness based on the Rough Mount Fuji (RMF) model originally introduced by Aita et al. [Biopolymers 54:64-79 (2000)] in the context of protein evolution. We provide a comprehensive collection of results pertaining to the topographical structure of RMF landscapes, including explicit formulae for the expected number of local fitness maxima, the location of the global peak, and the fitness correlation function. The statistics of single and multiple adaptive steps on the RMF landscape are explored mainly through simulations, and the results are compared to the known behavior in the MLM model. Finally, we show that the RMF model can explain the large number of second-step mutations observed on a highly-fit first step backgound in a recent evolution experiment with a microvirid bacteriophage [Miller et al., Genetics 187:185-202 (2011)]., Comment: 43 pages, 12 figures; revised version with new results on the number of fitness maxima

When polygenic traits are under stabilizing selection, many different combinations of alleles allow close adaptation to the optimum. If alleles have equal effects, all combinations that result in the same deviation from the optimum are equivalent. Furthermore, the genetic variance that is maintained by mutation-selection balance is $2 \mu/S$ per locus, where $\mu$ is the mutation rate and $S$ the strength of stabilizing selection. In reality, alleles vary in their effects, making the fitness landscape asymmetric, and complicating analysis of the equilibria. We show that that the resulting genetic variance depends on the fraction of alleles near fixation, which contribute by $2 \mu/S$, and on the total mutational effects of alleles that are at intermediate frequency. The interplay between stabilizing selection and mutation leads to a sharp transition: alleles with effects smaller than a threshold value of $2\sqrt{\mu / S}$ remain polymorphic, whereas those with larger effects are fixed. The genetic load in equilibrium is less than for traits of equal effects, and the fitness equilibria are more similar. We find that if the optimum is displaced, alleles with effects close to the threshold value sweep first, and their rate of increase is bounded by $\sqrt{\mu S}$. Long term response leads in general to well-adapted traits, unlike the case of equal effects that often end up at a sub-optimal fitness peak. However, the particular peaks to which the populations converge are extremely sensitive to the initial states, and to the speed of the shift of the optimum trait value., Comment: Accepted in Genetics

Fisher’s geometric model of adaptive evolution argues that adaptive evolution should generally result from the substitution of many mutations of small effect because advantageous mutations of small effect should be more common than those of large effect. However, evidence for both evolution by small steps and for Fisher’s model has been mixed. Here we report supporting results from a new experimental test of the model. We subjected the bacteriophage ϕ6 to intensified genetic drift in small populations and caused viral fitness to decline through the accumulation of a deleterious mutation. We then propagated the mutated virus at a range of larger population sizes and allowed fitness to recover by natural selection. Although fitness declined in one large step, it was usually recovered in smaller steps. More importantly, step size during recovery was smaller with decreasing size of the recovery population. These results confirm Fisher’s main prediction that advantageous mutations of small effect should be more common. We also show that the advantageous mutations of small effect are compensatory mutations whose advantage is conditional (epistatic) on the presence of the deleterious mutation, in which case the adaptive landscape of ϕ6 is likely to be very rugged.

We consider an asexual population under strong selection–weak mutation conditions evolving on rugged fitness landscapes with many local fitness peaks. Unlike the previous studies in which the initial fitness of the population is assumed to be high, here we start the adaptation process with a low fitness corresponding to a population in a stressful novel environment. For generic fitness distributions, using an analytic argument we find that the average number of steps to a local optimum varies logarithmically with the genotype sequence length and increases as the correlations among genotypic fitnesses increase. When the fitnesses are exponentially or uniformly distributed, using an evolution equation for the distribution of population fitness, we analytically calculate the fitness distribution of fixed beneficial mutations and the walk length distribution.

Adaptation in haploid organisms has been extensively modeled but little tested. Using a microvirid bacteriophage (ID11), we conducted serial passage adaptations at two bottleneck sizes (104 and 106), followed by fitness assays and whole-genome sequencing of 631 individual isolates. Extensive genetic variation was observed including 22 beneficial, several nearly neutral, and several deleterious mutations. In the three large bottleneck lines, up to eight different haplotypes were observed in samples of 23 genomes from the final time point. The small bottleneck lines were less diverse. The small bottleneck lines appeared to operate near the transition between isolated selective sweeps and conditions of complex dynamics (e.g., clonal interference). The large bottleneck lines exhibited extensive interference and less stochasticity, with multiple beneficial mutations establishing on a variety of backgrounds. Several leapfrog events occurred. The distribution of first-step adaptive mutations differed significantly from the distribution of second-steps, and a surprisingly large number of second-step beneficial mutations were observed on a highly fit first-step background. Furthermore, few first-step mutations appeared as second-steps and second-steps had substantially smaller selection coefficients. Collectively, the results indicate that the fitness landscape falls between the extremes of smooth and fully uncorrelated, violating the assumptions of many current mutational landscape models.

Codon usage bias (CUB) has been documented across a wide range of taxa and is the subject of numerous studies. While most explanations of CUB invoke some type of natural selection, most measures of CUB adaptation are heuristically defined. In contrast, we present a novel and mechanistic method for defining and contextualizing CUB adaptation to reduce the cost of nonsense errors during protein translation. Using a model of protein translation, we develop a general approach for measuring the protein production cost in the face of nonsense errors of a given allele as well as the mean and variance of these costs across its coding synonyms. We then use these results to define the nonsense error adaptation index (NAI) of the allele or a contiguous subset thereof. Conceptually, the NAI value of an allele is a relative measure of its elevation on a specific and well-defined adaptive landscape. To illustrate its utility, we calculate NAI values for the entire coding sequence and across a set of nonoverlapping windows for each gene in the Saccharomyces cerevisiae S288c genome. Our results provide clear evidence of adaptation to reduce the cost of nonsense errors and increasing adaptation with codon position and expression. The magnitude and nature of this adaptation are also largely consistent with simulation results in which nonsense errors are the only selective force driving CUB evolution. Because NAI is derived from mechanistic models, it is both easier to interpret and more amenable to future refinement than other commonly used measures of codon bias. Further, our approach can also be used as a starting point for developing other mechanistically derived measures of adaptation such as for translational accuracy.

Uncertain environments pose a tremendous challenge to populations: The selective pressures imposed by the environment can change so rapidly that adaptation by mutation alone would be too slow. One solution to this problem is given by the phenomenon of stochastic phenotype switching, which causes genetically uniform populations to be phenotypically heterogenous. Stochastic phenotype switching has been observed in numerous microbial species and is generally assumed to be an adaptive bet-hedging strategy to anticipate future environmental change. We use an explicit population genetic model to investigate the evolutionary dynamics of phenotypic switching rates. We find that whether or not stochastic switching is an adaptive strategy is highly contingent upon the fitness landscape given by the changing environment. Unless selection is very strong, asymmetric fitness landscapes—where the cost of being maladapted is not identical in all environments—strongly select against stochastic switching. We further observe a threshold phenomenon that causes switching rates to be either relatively high or completely absent, but rarely intermediate. Our finding that marginal changes in selection pressures can cause fundamentally different evolutionary outcomes is important in a wide range of fields concerned with microbial bet hedging.

We study the adaptation dynamics of an initially maladapted asexual population with genotypes represented by binary sequences of length $L$. The population evolves in a maximally rugged fitness landscape with a large number of local optima. We find that whether the evolutionary trajectory is deterministic or stochastic depends on the effective mutational distance $d_{\mathrm{eff}}$ upto which the population can spread in genotype space. For $d_{\mathrm{eff}}=L$, the deterministic quasispecies theory operates while for $d_{\mathrm{eff}} < 1$, the evolution is completely stochastic. Between these two limiting cases, the dynamics are described by a local quasispecies theory below a crossover time $T_{\times}$ while above $T_{\times}$, the population gets trapped at a local fitness peak and manages to find a better peak either via stochastic tunneling or double mutations. In the stochastic regime $d_\mathrm{eff} < 1$, we identify two subregimes associated with clonal interference and uphill adaptive walks, respectively. We argue that our findings are relevant to the interepretation of evolution experiments with microbial populations., Revised version, to appear in Genetics. Note on the role of selection in defining d_eff added; new figure 4 included

The so-called "Fundamental Theorem of Natural Selection", that the mean fitness of a population increases with time under natural selection, is known not to be true, as a mathematical theorem, when fitnesses depend on more than one locus. Although this observation may not have particular biological relevance, (so that mean fitness may well increase in the great majority of interesting situations), it does suggest that it is of interest to find an evolutionary result which is correct as a mathematical theorem, no matter how many loci are involved. The aim of the present note is to prove an evolutionary theorem relating to the variance in fitness, rather than the mean: this theorem is true for an arbitrary number of loci, as well as for arbitrary (fixed) fitness parameters and arbitrary linkage between loci. Connections are briefly discussed between this theorem and the principle of quasi-linkage equilibrium.

Speciation is fundamental to the process of generating the huge diversity of life on Earth. However, we are yet to have a clear understanding of its molecular-genetic basis. Here, we examine a computational model of reproductive isolation that explicitly incorporates a map from genotype to phenotype based on the biophysics of protein–DNA binding. In particular, we model the binding of a protein transcription factor to a DNA binding site and how their independent coevolution, in a stabilizing fitness landscape, of two allopatric lineages leads to incompatibilities. Complementing our previous coarse-grained theoretical results, our simulations give a new prediction for the monomorphic regime of evolution that smaller populations should develop incompatibilities more quickly. This arises as (1) smaller populations have a greater initial drift load, as there are more sequences that bind poorly than well, so fewer substitutions are needed to reach incompatible regions of phenotype space, and (2) slower divergence when the population size is larger than the inverse of discrete differences in fitness. Further, we find longer sequences develop incompatibilities more quickly at small population sizes, but more slowly at large population sizes. The biophysical model thus represents a robust mechanism of rapid reproductive isolation for small populations and large sequences that does not require peak shifts or positive selection. Finally, we show that the growth of DMIs with time is quadratic for small populations, agreeing with Orr’s model, but nonpower law for large populations, with a form consistent with our previous theoretical results.