Fisher's Geometric Model Research Articles

Competition and fixation of cohorts of adaptive mutations under Fisher geometrical model.

One of the simplest models of adaptation to a new environment is Fisher’s Geometric Model (FGM), in which populations move on a multidimensional landscape defined by the traits under selection. The predictions of this model have been found to be consistent with current observations of patterns of fitness increase in experimentally evolved populations. Recent studies investigated the dynamics of allele frequency change along adaptation of microbes to simple laboratory conditions and unveiled a dramatic pattern of competition between cohorts of mutations, i.e., multiple mutations simultaneously segregating and ultimately reaching fixation. Here, using simulations, we study the dynamics of phenotypic and genetic change as asexual populations under clonal interference climb a Fisherian landscape, and ask about the conditions under which FGM can display the simultaneous increase and fixation of multiple mutations—mutation cohorts—along the adaptive walk. We find that FGM under clonal interference, and with varying levels of pleiotropy, can reproduce the experimentally observed competition between different cohorts of mutations, some of which have a high probability of fixation along the adaptive walk. Overall, our results show that the surprising dynamics of mutation cohorts recently observed during experimental adaptation of microbial populations can be expected under one of the oldest and simplest theoretical models of adaptation—FGM.

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.

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.

Evolution of Fitness Trade-Offs in Locally Adapted Populations of Pseudomonas fluorescens

Local adaptation seems to be common in natural systems, but the genetic causes of its evolution remain poorly understood. Here we characterize the genetic causes of trade-offs generating local adaptation in populations of Pseudomonas fluorescens that had previously been evolved for specialization on three different carbon resources. We measured the fitness effects of mutations that arose during selection in that environment and in alternative environments to quantify the degree of specialization. We find that all mutations are beneficial in the environment of selection and that those arising later during adaptation are associated with increasingly antagonistic effects in alternative environments compared with those arising earlier, consistent with a multioptima version of Fisher's geometric model of adaptation. We also find that fitness of pairs of beneficial mutations are consistently less than additive in selection environments, producing a pattern of diminishing returns, but are more variable in alternative environments, being either positive or negative. Finally, we find that mutations in genes associated with loss of motility are beneficial across all environments, whereas mutations involving other functions, such as gene regulation, had more variable effects, being more environment specific. Taken together, these results provide a detailed account of the genetics of specialization and suggest that the evolution of trade-offs associated with local adaptation may often result from the antagonistic effects of beneficial mutations substituted later in adaptation.

Epigenetic mutations can both help and hinder adaptive evolution

Epigenetic variation is being integrated into our understanding of adaptation, yet we lack models on how epigenetic mutations affect evolution that includes de novo genetic change. We model the effects of epigenetic mutations on the dynamics and endpoints of adaptive walks-a process where a series of beneficial mutations move a population towards a fitness optimum. We use an individual-based model of an asexual population, where mutational effects are drawn from Fisher's geometric model. We find cases where epigenetic mutations speed adaptation or result in populations with higher fitness. However, we also find cases where they slow adaptation or result in populations with lower fitness. The effect of epigenetic mutations on adaptive walks depends crucially on their stability and fitness effects relative to genetic mutations, with small-effect epigenetic mutations generally speeding adaptation, and epigenetic mutations with the same fitness effects as genetic mutations slowing adaptation. Our work reveals a complex relationship between epigenetic mutations and natural selection and highlights the need for empirical data.

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 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 Utility of Fisher's Geometric Model in Evolutionary Genetics.

The accumulation of data on the genomic bases of adaptation has triggered renewed interest in theoretical models of adaptation. Among these models, Fisher Geometric Model (FGM) has received a lot of attention over the last two decades. FGM is based on a continuous multidimensional phenotypic landscape, but it is for the emerging properties of individual mutation effects that it is mostly used. Despite an apparent simplicity and a limited number of parameters, FGM integrates a full model of mutation and epistatic interactions that allows the study of both beneficial and deleterious mutations, and subsequently the fate of evolving populations. In this review, I present the different properties of FGM and the qualitative and quantitative support they have received from experimental evolution data. I later discuss how to estimate the different parameters of the model and outline some future directions to connect FGM and the molecular determinants of adaptation.

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.

Biased estimates of diminishing-returns epistasis? Empirical evidence revisited.

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.

Fisher's geometric model with a moving optimum.

Fisher's geometric model has been widely used to study the effects of pleiotropy and organismic complexity on phenotypic adaptation. Here, we study a version of Fisher's model in which a population adapts to a gradually moving optimum. Key parameters are the rate of environmental change, the dimensionality of phenotype space, and the patterns of mutational and selectional correlations. We focus on the distribution of adaptive substitutions, that is, the multivariate distribution of the phenotypic effects of fixed beneficial mutations. Our main results are based on an “adaptive-walk approximation,” which is checked against individual-based simulations. We find that (1) the distribution of adaptive substitutions is strongly affected by the ecological dynamics and largely depends on a single composite parameter γ, which scales the rate of environmental change by the “adaptive potential” of the population; (2) the distribution of adaptive substitution reflects the shape of the fitness landscape if the environment changes slowly, whereas it mirrors the distribution of new mutations if the environment changes fast; (3) in contrast to classical models of adaptation assuming a constant optimum, with a moving optimum, more complex organisms evolve via larger adaptive steps.

NICHE DIMENSIONALITY AND THE GENETICS OF ECOLOGICAL SPECIATION

Niche dimensionality is suggested to be a key determinant of ecological speciation ("multifarious selection" hypothesis), but genetic aspects of this process have not been investigated theoretically. We use Fisher's geometrical model to study how niche dimensionality influences the mean fitness of hybrids formed upon secondary contact between populations adapting in allopatry. Gaussian selection for an optimum generates two forms of reproductive isolation (RI): an extrinsic component due to maladaptation of the mean phenotype, and an intrinsic variance load resulting from what we term transgressive incompatibilities between mutations fixed in different populations. We show that after adaptation to a new environment, RI increases with (1) the mean initial maladaptation of diverging population, and (2) niche dimensionality, which increases the phenotypic variability of fixed mutations. Under mutation selection drift equilibrium in a constant environment, RI accumulates steadily with time, at a rate that also increases with niche dimensionality. A similar pattern can be produced by successive shifts in the optimum phenotype. Niche dimensionality thus has an effect per se on postzygotic isolation, beyond putative indirect effects (stronger selection, more genes). Our mechanism is consistent with empirical evidence about transgressive segregation in crosses between divergent populations, and with patterns of accumulation of RI with time in many taxa.

THE INEVITABILITY OF UNCONDITIONALLY DELETERIOUS SUBSTITUTIONS DURING ADAPTATION

Studies on the genetics of adaptation from new mutations typically neglect the possibility that a deleterious mutation might fix. Nonetheless, here we show that, in many regimes, the first mutation to fix is most often deleterious, even when fitness is expected to increase in the long term. In particular, we prove that this phenomenon occurs under weak mutation for any house-of-cards model with an equilibrium distribution. We find that the same qualitative results hold under Fisher's geometric model. We also provide a simple intuition for the surprising prevalence of unconditionally deleterious substitutions during early adaptation. Importantly, the phenomenon we describe occurs on fitness landscapes without any local maxima and is therefore distinct from "valley crossing." Our results imply that the common practice of ignoring deleterious substitutions leads to qualitatively incorrect predictions in many regimes. Our results also have implications for the substitution process at equilibrium and for the response to a sudden decrease in population size.

Influenza virus drug resistance: a time-sampled population genetics perspective.

The challenge of distinguishing genetic drift from selection remains a central focus of population genetics. Time-sampled data may provide a powerful tool for distinguishing these processes, and we here propose approximate Bayesian, maximum likelihood, and analytical methods for the inference of demography and selection from time course data. Utilizing these novel statistical and computational tools, we evaluate whole-genome datasets of an influenza A H1N1 strain in the presence and absence of oseltamivir (an inhibitor of neuraminidase) collected at thirteen time points. Results reveal a striking consistency amongst the three estimation procedures developed, showing strongly increased selection pressure in the presence of drug treatment. Importantly, these approaches re-identify the known oseltamivir resistance site, successfully validating the approaches used. Enticingly, a number of previously unknown variants have also been identified as being positively selected. Results are interpreted in the light of Fisher's Geometric Model, allowing for a quantification of the increased distance to optimum exerted by the presence of drug, and theoretical predictions regarding the distribution of beneficial fitness effects of contending mutations are empirically tested. Further, given the fit to expectations of the Geometric Model, results suggest the ability to predict certain aspects of viral evolution in response to changing host environments and novel selective pressures.

RATES OF FITNESS DECLINE AND REBOUND SUGGEST PERVASIVE EPISTASIS

Unraveling the factors that determine the rate of adaptation is a major question in evolutionary biology. One key parameter is the effect of a new mutation on fitness, which invariably depends on the environment and genetic background. The fate of a mutation also depends on population size, which determines the amount of drift it will experience. Here, we manipulate both population size and genotype composition and follow adaptation of 23 distinct Escherichia coli genotypes. These have previously accumulated mutations under intense genetic drift and encompass a substantial fitness variation. A simple rule is uncovered: the net fitness change is negatively correlated with the fitness of the genotype in which new mutations appear—a signature of epistasis. We find that Fisher's geometrical model can account for the observed patterns of fitness change and infer the parameters of this model that best fit the data, using Approximate Bayesian Computation. We estimate a genomic mutation rate of 0.01 per generation for fitness altering mutations, albeit with a large confidence interval, a mean fitness effect of mutations of −0.01, and an effective number of traits nine in mutS− E. coli. This framework can be extended to confront a broader range of models with data and test different classes of fitness landscape models.

Shifting fitness landscapes in response to altered environments.

The role of adaptation in molecular evolution has been contentious for decades. Here, we shed light on the adaptive potential in Saccharomyces cerevisiae by presenting systematic fitness measurements for all possible point mutations in a region of Hsp90 under four environmental conditions. Under elevated salinity, we observe numerous beneficial mutations with growth advantages up to 7% relative to the wild type. All of these beneficial mutations were observed to be associated with high costs of adaptation. We thus demonstrate that an essential protein can harbor adaptive potential upon an environmental challenge, and report a remarkable fit of the data to a version of Fisher's geometric model that focuses on the fitness trade-offs between mutations in different environments.

FISHER'S GEOMETRIC MODEL OF ADAPTATION MEETS THE FUNCTIONAL SYNTHESIS: DATA ON PAIRWISE EPISTASIS FOR FITNESS YIELDS INSIGHTS INTO THE SHAPE AND SIZE OF PHENOTYPE SPACE

The functional synthesis uses experimental methods from molecular biology, biochemistry and structural biology to decompose evolutionarily important mutations into their more proximal mechanistic determinants. However these methods are technically challenging and expensive. Noting strong formal parallels between R.A. Fisher's geometric model of adaptation and a recent model for the phenotypic basis of protein evolution, we sought to use the former to make inferences into the latter using data on pairwise fitness epistasis between mutations. We present an analytic framework for classifying pairs of mutations with respect to similarity of underlying mechanism on this basis, and also show that these data can yield an estimate of the number of mutationally labile phenotypes underlying fitness effects. We use computer simulations to explore the robustness of our approach to violations of analytic assumptions and analyze several recently published datasets. This work provides a theoretical complement to the functional synthesis as well as a novel test of Fisher's geometric model.

Evolution of clonal populations approaching a fitness peak

Populations facing novel environments are expected to evolve through the accumulation of adaptive substitutions. The dynamics of adaptation depend on the fitness landscape and possibly on the genetic background on which new mutations arise. Here, we model the dynamics of adaptive evolution at the phenotypic and genotypic levels, focusing on a Fisherian landscape characterized by a single peak. We find that Fisher's geometrical model of adaptation, extended to allow for small random environmental variations, is able to explain several features made recently in experimentally evolved populations. Consistent with data on populations evolving under controlled conditions, the model predicts that mean population fitness increases rapidly when populations face novel environments and then achieves a dynamic plateau, the rate of molecular evolution is remarkably constant over long periods of evolution, mutators are expected to invade and patterns of epistasis vary along the adaptive walk. Negative epistasis is expected in the initial steps of adaptation but not at later steps, a prediction that remains to be tested. Furthermore, populations are expected to exhibit high levels of phenotypic diversity at all times during their evolution. This implies that populations are possibly able to adapt rapidly to novel abiotic environments.

The strength of genetic interactions scales weakly with mutational effects

BackgroundGenetic interactions pervade every aspect of biology, from evolutionary theory, where they determine the accessibility of evolutionary paths, to medicine, where they can contribute to complex genetic diseases. Until very recently, studies on epistatic interactions have been based on a handful of mutations, providing at best anecdotal evidence about the frequency and the typical strength of genetic interactions. In this study, we analyze a publicly available dataset that contains the growth rates of over five million double knockout mutants of the yeast Saccharomyces cerevisiae.ResultsWe discuss a geometric definition of epistasis that reveals a simple and surprisingly weak scaling law for the characteristic strength of genetic interactions as a function of the effects of the mutations being combined. We then utilized this scaling to quantify the roughness of naturally occurring fitness landscapes. Finally, we show how the observed roughness differs from what is predicted by Fisher's geometric model of epistasis, and discuss the consequences for evolutionary dynamics.ConclusionsAlthough epistatic interactions between specific genes remain largely unpredictable, the statistical properties of an ensemble of interactions can display conspicuous regularities and be described by simple mathematical laws. By exploiting the amount of data produced by modern high-throughput techniques, it is now possible to thoroughly test the predictions of theoretical models of genetic interactions and to build informed computational models of evolution on realistic fitness landscapes.

ANTIBIOTIC RESISTANCE AND STRESS IN THE LIGHT OF FISHER'S MODEL

The role of mutations in evolution depends upon the distribution of their effects on fitness. This distribution is likely to depend on the environment. Indeed genotype-by-environment interactions are key for the process of local adaptation and ecological specialization. An important trait in bacterial evolution is antibiotic resistance, which presents a clear case of change in the direction of selection between environments with and without antibiotics. Here, we study the distribution of fitness effects of mutations, conferring antibiotic resistance to Escherichia coli, in benign and stressful environments without drugs. We interpret the distributions in the light of a fitness landscape model that assumes a single fitness peak. We find that mutation effects (s) are well described by a shifted gamma distribution, with a shift parameter that reflects the distance to the fitness peak and varies across environments. Consistent with the theoretical predictions of Fisher's geometrical model, with a Gaussian relationship between phenotype and fitness, we find that the main effect of stress is to increase the variance in s. Our findings are in agreement with the results of a recent meta-analysis, which suggest that a simple fitness landscape model may capture the variation of mutation effects across species and environments.