The population genetics of adaptation on correlated fitness landscapes: the block model

Several recent theoretical studies of the genetics of adaptation have focused on the mutational landscape model, which considers evolution on rugged fitness landscapes (i.e., ones having many local optima). Adaptation in this model is characterized by several simple results. Here I ask whether these results also hold on correlated fitness landscapes, which are smoother than those considered in the mutational landscape model. In particular, I study the genetics of adaptation in the block model, a tunably rugged model of fitness landscapes. Considering the scenario in which adaptation begins from a high fitness wild-type DNA sequence, I use extreme value theory and computer simulations to study both single adaptive steps and entire adaptive walks. I show that all previous results characterizing single steps in adaptation in the mutational landscape model hold at least approximately on correlated landscapes in the block model; many entire-walk results, however, do not.     

The distribution of fitness effects among beneficial mutations

We know little about the distribution of fitness effects among new beneficial mutations, a problem that partly reflects the rarity of these changes. Surprisingly, though, population genetic theory allows us to predict what this distribution should look like under fairly general assumptions. Using extreme value theory, I derive this distribution and show that it has two unexpected properties. First, the distribution of beneficial fitness effects at a gene is exponential. Second, the distribution of beneficial effects at a gene has the same mean regardless of the fitness of the present wild-type allele. Adaptation from new mutations is thus characterized by a kind of invariance: natural selection chooses from the same spectrum of beneficial effects at a locus independent of the fitness rank of the present wild type. I show that these findings are reasonably robust to deviations from several assumptions. I further show that one can back calculate the mean size of new beneficial mutations from the observed mean size of fixed beneficial mutations.     

Properties of selected mutations and genotypic landscapes under Fisher's geometric model

The fitness landscape—the mapping between genotypes and fitness—determines properties of the process of adaptation. Several small genotypic fitness landscapes have recently been built by selecting a handful of beneficial mutations and measuring fitness of all combinations of these mutations. Here, we generate several testable predictions for the properties of these small genotypic landscapes under Fisher's geometric model of adaptation. When the ancestral strain is far from the fitness optimum, we analytically compute the fitness effect of selected mutations and their epistatic interactions. Epistasis may be negative or positive on average depending on the distance of the ancestral genotype to the optimum and whether mutations were independently selected, or coselected in an adaptive walk. Simulations show that genotypic landscapes built from Fisher's model are very close to an additive landscape when the ancestral strain is far from the optimum. However, when it is close to the optimum, a large diversity of landscape with substantial roughness and sign epistasis emerged. Strikingly, small genotypic landscapes built from several replicate adaptive walks on the same underlying landscape were highly variable, suggesting that several realizations of small genotypic landscapes are needed to gain information about the underlying architecture of the fitness landscape.     

Fisher's geometric model predicts the effects of random mutations when tested in the wild

Fisher's geometric model of adaptation (FGM) has been the conceptual foundation for studies investigating the genetic basis of adaptation since the onset of the neo Darwinian synthesis. FGM describes adaptation as the movement of a genotype toward a fitness optimum due to beneficial mutations. To date, one prediction of FGM, the probability of improvement is related to the distance from the optimum, has only been tested in microorganisms under laboratory conditions. There is reason to believe that results might differ under natural conditions where more mutations likely affect fitness, and where environmental variance may obscure the expected pattern. We chemically induced mutations into a set of 19 Arabidopsis thaliana accessions from across the native range of A. thaliana and planted them alongside the premutated founder lines in two habitats in the mid‐Atlantic region of the United States under field conditions. We show that FGM is able to predict the outcome of a set of random induced mutations on fitness in a set of A. thaliana accessions grown in the wild: mutations are more likely to be beneficial in relatively less fit genotypes. This finding suggests that FGM is an accurate approximation of the process of adaptation under more realistic ecological conditions.     

Beneficial fitness effects are not exponential for two viruses

The distribution of fitness effects for beneficial mutations is of paramount importance in determining the outcome of adaptation. It is generally assumed that fitness effects of beneficial mutations follow an exponential distribution, for example, in theoretical treatments of quantitative genetics, clonal interference, experimental evolution, and the adaptation of DNA sequences. This assumption has been justified by the statistical theory of extreme values, because the fitnesses conferred by beneficial mutations should represent samples from the extreme right tail of the fitness distribution. Yet in extreme value theory, there are three different limiting forms for right tails of distributions, and the exponential describes only those of distributions in the Gumbel domain of attraction. Using beneficial mutations from two viruses, we show for the first time that the Gumbel domain can be rejected in favor of a distribution with a right-truncated tail, thus providing evidence for an upper bound on fitness effects. Our data also violate the common assumption that small-effect beneficial mutations greatly outnumber those of large effect, as they are consistent with a uniform distribution of beneficial effects.     

The fitness effect of mutations across environments: Fisher's geometrical model with multiple optima

When are mutations beneficial in one environment and deleterious in another? More generally, what is the relationship between mutation effects across environments? These questions are crucial to predict adaptation in heterogeneous conditions in a broad sense. Empirical evidence documents various patterns of fitness effects across environments but we still lack a framework to analyze these multivariate data. In this article, we extend Fisher's geometrical model to multiple environments determining distinct peaks. We derive the fitness distribution, in one environment, among mutants with a given fitness in another and the bivariate distribution of random mutants’ fitnesses across two or more environments. The geometry of the phenotype‐fitness landscape is naturally interpreted in terms of fitness trade‐offs between environments. These results may be used to fit/predict empirical distributions or to predict the pattern of adaptation across heterogeneous conditions. As an example, we derive the genomic rate of substitution and of adaptation in a metapopulation divided into two distinct habitats in a high migration regime and show that they depend critically on the geometry of the phenotype‐fitness landscape.     

The distribution of fitness effects of new beneficial mutations in Pseudomonas fluorescens

Theoretical studies of adaptation emphasize the importance of understanding the distribution of fitness effects (DFE) of new mutations. We report the isolation of 100 adaptive mutants—without the biasing influence of natural selection—from an ancestral genotype whose fitness in the niche occupied by the derived type is extremely low. The fitness of each derived genotype was determined relative to a single reference type and the fitness effects found to conform to a normal distribution. When fitness was measured in a different environment, the rank order changed, but not the shape of the distribution. We argue that, even with detailed knowledge of the genetic architecture underpinning the adaptive types (as is the case here), the DFEs remain unpredictable, and we discuss the possibility that general explanations for the shape of the DFE might not be possible in the absence of organism-specific biological details.     

Properties of adaptive walks on uncorrelated landscapes under strong selection and weak mutation

We examine properties of adaptive walks on uncorrelated (i.e. random) fitness landscapes starting from moderately fit genotypes under strong selection weak mutation. As an extension of Orr's model for a single step in an adaptive walk under these conditions, we show that the fitness rank of the dominant genotype in a population after the fixation of a beneficial mutation is, on average, (i+6)/4, where i is the fitness rank of the starting genotype. This accounts for the change in rank due to acquiring a new set of single-mutation neighbors after fixing a new allele through natural selection. Under this scenario, adaptive walks can be modeled as a simple Markov chain on the space of possible fitness ranks with an absorbing state at i = 1, from which no beneficial mutations are accessible. We find that these walks are typically short and are often completed in a single step when starting from a moderately fit genotype. As in Orr's original model, these results are insensitive to both the distribution of fitness effects and most biological details of the system under consideration.     

A general multivariate extension of Fisher's geometrical model and the distribution of mutation fitness effects across species

The evolution of complex organisms is a puzzle for evolutionary theory because beneficial mutations should be less frequent in complex organisms, an effect termed "cost of complexity." However, little is known about how the distribution of mutation fitness effects (f(s)) varies across genomes. The main theoretical framework to address this issue is Fisher's geometric model and related phenotypic landscape models. However, it suffers from several restrictive assumptions. In this paper, we intend to show how several of these limitations may be overcome. We then propose a model of f(s) that extends Fisher's model to account for arbitrary mutational and selective interactions among n traits. We show that these interactions result in f(s) that would be predicted by a much smaller number of independent traits. We test our predictions by comparing empirical f(s) across species of various gene numbers as a surrogate to complexity. This survey reveals, as predicted, that mutations tend to be more deleterious, less variable, and less skewed in higher organisms. However, only limited difference in the shape of f(s) is observed from Escherichia coli to nematodes or fruit flies, a pattern consistent with a model of random phenotypic interactions across many traits. Overall, these results suggest that there may be a cost to phenotypic complexity although much weaker than previously suggested by earlier theoretical works. More generally, the model seems to qualitatively capture and possibly explain the variation of f(s) from lower to higher organisms, which opens a large array of potential applications in evolutionary genetics.     

Pleiotropic effects of beneficial mutations in Escherichia coli

Micromutational models of adaptation have placed considerable weight on antagonistic pleiotropy as a mechanism that prevents mutations of large effect from achieving fixation. However, there are few empirical studies of the distribution of pleiotropic effects, and no studies that have examined this distribution for a large number of adaptive mutations. Here we examine the form and extent of pleiotropy associated with beneficial mutations in Escherichia coli. To do so, we used a collection of independently evolved genotypes, each of which contains a beneficial mutation that confers increased fitness in a glucose-limited environment. To determine the pleiotropic effects of these mutations, we examined the fitnesses of the mutants in five novel resource environments. Our results show that the majority of mutations had significant fitness effects in alternative resources, such that pleiotropy was common. The predominant form of this pleiotropy was positive--that is, most mutations that conferred increased fitness in glucose also conferred increased fitness in novel resources. We did detect some deleterious pleiotropic effects, but they were primarily limited to one of the five resources, and within this resource, to only a subset of mutants. Although pleiotropic effects were generally positive, fitness levels were lower and more variable on resources that differed most in their mechanisms of uptake and catabolism from that of glucose. Positive pleiotropic effects were strongly correlated in magnitude with their direct effects, but no such correlation was found among mutants with deleterious pleiotropic effects. Whereas previous studies of populations evolved on glucose for longer periods of time showed consistent declines on some of the resources used here, our results suggest that deleterious pleiotropic effects were limited to only a subset of the beneficial mutations available.     

The distribution of fitness effects in an uncertain world

The distribution of fitness effects (DFE) among new mutations plays a critical role in adaptive evolution and the maintenance of genetic variation. Although fitness landscape models predict several key features of the DFE, most theory to date focuses on predictable environmental conditions, while ignoring stochastic environmental fluctuations that feature prominently in the ecology of many organisms. Here, we derive an extension of Fisher's geometric model that incorporates two common effects of environmental variation: (1) nonadaptive genotype‐by‐environment interactions (G × E), in which the phenotype of a given genotype varies across environmental contexts; and (2) random fluctuation of the fitness optimum, which generates fluctuating selection. We show that both factors cause a mismatch between the DFE within single generations and the distribution of geometric mean fitness effects (averaged over multiple generations) that governs long‐term evolutionary change. Such mismatches permit strong evolutionary constraints—despite an abundance of beneficial fitness variation within single environmental contexts—and to conflicting DFE estimates from direct versus indirect inference methods. Finally, our results suggest an intriguing parallel between the genetics and ecology of evolutionary constraints, with environmental fluctuations and pleiotropy placing qualitatively similar limits on the availability of adaptive genetic variation.     

The distribution of fitness effects of new mutations

The distribution of fitness effects (DFE) of new mutations is a fundamental entity in genetics that has implications ranging from the genetic basis of complex disease to the stability of the molecular clock. It has been studied by two different approaches: mutation accumulation and mutagenesis experiments, and the analysis of DNA sequence data. The proportion of mutations that are advantageous, effectively neutral and deleterious varies between species, and the DFE differs between coding and non-coding DNA. Despite these differences between species and genomic regions, some general principles have emerged: advantageous mutations are rare, and those that are strongly selected are exponentially distributed; and the DFE of deleterious mutations is complex and multi-modal.     

Environmental variation alters the fitness effects of rifampicin resistance mutations in Pseudomonas aeruginosa

The fitness effects of antibiotic resistance mutations in antibiotic‐free conditions play a key role in determining the long‐term maintenance of resistance. Although resistance is usually associated with a cost, the impact of environmental variation on the cost of resistance is poorly understood. Here, we test the impact of heterogeneity in temperature and resource availability on the fitness effects of antibiotic resistance using strains of the pathogenic bacterium Pseudomonas aeruginosa carrying clinically important rifampicin resistance mutations. Although the rank order of fitness was generally maintained across environments, fitness effects relative to the wild type differed significantly. Changes in temperature had a profound impact on the fitness effects of resistance, whereas changes in carbon substrate had only a weak impact. This suggests that environmental heterogeneity may influence whether the costs of resistance are likely to be ameliorated by second‐site compensatory mutations or by reversion to wild‐type rpoB. Our results highlight the need to consider environmental heterogeneity and genotype‐by‐environment interactions for fitness in models of resistance evolution.     

The genetics of speciation: Insights from Fisher's geometric model

Research in speciation genetics has uncovered many robust patterns in intrinsic reproductive isolation, and fitness landscape models have been useful in interpreting these patterns. Here, we examine fitness landscapes based on Fisher's geometric model. Such landscapes are analogous to models of optimizing selection acting on quantitative traits, and have been widely used to study adaptation and the distribution of mutational effects. We show that, with a few modifications, Fisher's model can generate all of the major findings of introgression studies (including “speciation genes” with strong deleterious effects, complex epistasis and asymmetry), and the major patterns in overall hybrid fitnesses (including Haldane's Rule, the speciation clock, heterosis, hybrid breakdown, and male–female asymmetry in the F1). We compare our approach to alternative modeling frameworks that assign fitnesses to genotypes by identifying combinations of incompatible alleles. In some cases, the predictions are importantly different. For example, Fisher's model can explain conflicting empirical results about the rate at which incompatibilities accumulate with genetic divergence. In other cases, the predictions are identical. For example, the quality of reproductive isolation is little affected by the manner in which populations diverge.     

The distribution of beneficial and fixed mutation fitness effects close to an optimum

The distribution of the selection coefficients of beneficial mutations is pivotal to the study of the adaptive process, both at the organismal level (theories of adaptation) and at the gene level (molecular evolution). A now famous result of extreme value theory states that this distribution is an exponential, at least when considering a well-adapted wild type. However, this prediction could be inaccurate under selection for an optimum (because fitness effect distributions have a finite right tail in this case). In this article, we derive the distribution of beneficial mutation effects under a general model of stabilizing selection, with arbitrary selective and mutational covariance between a finite set of traits. We assume a well-adapted wild type, thus taking advantage of the robustness of tail behaviors, as in extreme value theory. We show that, under these general conditions, both beneficial mutation effects and fixed effects (mutations escaping drift loss) are beta distributed. In both cases, the parameters have explicit biological meaning and are empirically measurable; their variation through time can also be predicted. We retrieve the classic exponential distribution as a subcase of the beta when there are a moderate to large number of weakly correlated traits under selection. In this case too, we provide an explicit biological interpretation of the parameters of the distribution. We show by simulations that these conclusions are fairly robust to a lower adaptation of the wild type and discuss the relevance of our findings in the context of adaptation theories and experimental evolution.     

The distribution of fitness effects of new deleterious amino acid mutations in humans

The distribution of fitness effects of new mutations is a fundamental parameter in genetics. Here we present a new method by which the distribution can be estimated. The method is fairly robust to changes in population size and admixture, and it can be corrected for any residual effects if a model of the demography is available. We apply the method to extensively sampled single-nucleotide polymorphism data from humans and estimate the distribution of fitness effects for amino acid changing mutations. We show that a gamma distribution with a shape parameter of 0.23 provides a good fit to the data and we estimate that >50% of mutations are likely to have mild effects, such that they reduce fitness by between one one-thousandth and one-tenth. We also infer that <15% of new mutations are likely to have strongly deleterious effects. We estimate that on average a nonsynonymous mutation reduces fitness by a few percent and that the average strength of selection acting against a nonsynonymous polymorphism is approximately 9 x 10(-5). We argue that the relaxation of natural selection due to modern medicine and reduced variance in family size is not likely to lead to a rapid decline in genetic quality, but that it will be very difficult to locate most of the genes involved in complex genetic diseases.     

Linking temperature dependence of fitness effects of mutations to thermal niche adaptation

Fitness effects of mutations may generally depend on temperature that influences all rate-limiting biophysical and biochemical processes. Earlier studies suggested that high temperatures may increase the availability of beneficial mutations (‘more beneficial mutations’), or allow beneficial mutations to show stronger fitness effects (‘stronger beneficial mutation effects’). The ‘more beneficial mutations’ scenario would inevitably be associated with increased proportion of conditionally beneficial mutations at higher temperatures. This in turn predicts that populations in warm environments show faster evolutionary adaptation but suffer fitness loss when faced with cold conditions, and those evolving in cold environments become thermal-niche generalists (‘hotter is narrower’). Under the ‘stronger beneficial mutation effects’ scenario, populations evolving in warm environments would show faster adaptation without fitness costs in cold environments, leading to a ‘hotter is (universally) better’ pattern in thermal niche adaptation. We tested predictions of the two competing hypotheses using an experimental evolution study in which populations of two model bacterial species, Escherichia coli and Pseudomonas fluorescens, evolved for 2400 generations at three experimental temperatures. Results of reciprocal transplant experiments with our P. fluorescens populations were largely consistent with the ‘hotter is narrower’ prediction. Results from the E. coli populations clearly suggested stronger beneficial mutation effects at higher assay temperatures, but failed to detect faster adaptation in populations evolving in warmer experimental environments (presumably because of limitation in the supply of genetic variation). Our results suggest that the influence of temperature on mutational effects may provide insight into the patterns of thermal niche adaptation and population diversification across thermal conditions.     

An exact steady state solution of Fisher's geometric model and other models

Because nearly neutral substitutions are thought to contribute substantially to molecular evolution, and much of our insight about the workings of nearly neutral evolution relies on theory, solvable models of this process are of particular interest. Here, I present an analytical method for solving models of nearly neutral evolution at steady state. The steady state solution applies to any constant fitness landscape under a dynamic of successive fixations, each of which occurs on the background of the population's most recent common ancestor. Because this dynamic neglects the effects of polymorphism in the population beyond the mutant allele under consideration, the steady state solution provides a decent approximation of evolutionary dynamics when the population mutation rate is low (Nu<1). To demonstrate the method, I apply it to two examples: Fisher's geometric model (FGM), and a simple model of molecular evolution. Since recent papers have studied the steady state behavior of FGM under this dynamic, I analyze its behavior in detail and compare the results with previous work.     

On adaptation: A reduction of the Kauffman-Levin model to a problem in graph theory and its consequences

It is shown that complex adaptations are best modelled as discrete processes represented on directed weighted graphs. Such a representation captures the idea that problems of adaptation in evolutionary biology are problems in a discrete space, something that the conventional representations using continuous adaptive landscapes does not. Further, this representation allows the utilization of well-known algorithms for the computation of several biologically interesting results such as the accessibility of one allele from another by a specified number of point mutations, the accessibility of alleles at a local maximum of fitness, the accessibility of the allele with the globally maximum fitness, etc. A reduction of a model due to Kauffman and Levin to such a representation is explicitly carried out and it is shown how this reduction clarifies the biological questions that are of interest.Thanks are due to William Wimsatt, James F. Crow, and the referees for Biology and Philosophy for comments on an earlier version of this paper. Remarks by members of the audience, especially Abner Shimony, of a seminar at Boston University, February 19, 1988, were also very helpful. The diagrams were prepared with the assistance of Tracy Lubas.