Heritable genetic variation via mutation-selection balance: Lerch's zeta meets the abdominal bristle

Natural selection acting on complex traits hampers the predictive accuracy of polygenic scores in ancient samples

The prediction of phenotypes from ancient humans has gained interest due to its potential to investigate the evolution of complex traits. These predictions are commonly performed using polygenic scores computed with DNA information from ancient humans along with genome-wide association studies (GWAS) data from present-day humans. However, numerous evolutionary processes could impact the prediction of phenotypes from ancient humans based on polygenic scores. In this work we investigate how natural selection impacts phenotypic predictions on ancient individuals using polygenic scores. We use simulations of an additive trait to analyze how natural selection impacts phenotypic predictions with polygenic scores. We simulate a trait evolving under neutrality, stabilizing selection and directional selection. We find that stabilizing and directional selection have contrasting effects on ancient phenotypic predictions. Stabilizing selection accelerates the loss of large-effect alleles contributing to trait variation. Conversely, directional selection accelerates the loss of small and large-effect alleles that drive individuals farther away from the optimal phenotypic value. These effects result in specific shared genetic variation patterns between ancient and modern populations which hamper the accuracy of polygenic scores to predict phenotypes. Furthermore, we conducted simulations that include realistic strengths of stabilizing selection and heritability estimates to show how natural selection could impact the predictive accuracy of ancient polygenic scores for two widely studied traits: height and body mass index. We emphasize the importance of considering how natural selection can decrease the reliability of ancient polygenic scores to perform phenotypic predictions on an ancient population.

Polygenic selection to a changing optimum under self-fertilisation

Many traits are polygenic, affected by multiple genetic variants throughout the genome. Selection acting on these traits involves co-ordinated allele-frequency changes at these underlying variants, and this process has been extensively studied in random-mating populations. Yet many species self-fertilise to some degree, which incurs changes to genetic diversity, recombination and genome segregation. These factors cumulatively influence how polygenic selection is realised in nature. Here, we use analytical modelling and stochastic simulations to investigate to what extent self-fertilisation affects polygenic adaptation to a new environment. Our analytical solutions show that while selfing can increase adaptation to an optimum, it incurs linkage disequilibrium that can slow down the initial spread of favoured mutations due to selection interference, and favours the fixation of alleles with opposing trait effects. Simulations show that while selection interference is present, high levels of selfing (at least 90%) aids adaptation to a new optimum, showing a higher long-term fitness. If mutations are pleiotropic then only a few major-effect variants fix along with many neutral hitchhikers, with a transient increase in linkage disequilibrium. These results show potential advantages to self-fertilisation when adapting to a new environment, and how the mating system affects the genetic composition of polygenic selection.

Conditional frequency spectra as a tool for studying selection on complex traits in biobanks

Natural selection on complex traits is difficult to study in part due to the ascertainment inherent to genome-wide association studies (GWAS). The power to detect a trait-associated variant in GWAS is a function of frequency and effect size - but for traits under selection, the effect size of a variant determines the strength of selection against it, constraining its frequency. To account for GWAS ascertainment, we propose studying the joint distribution of allele frequencies across populations, conditional on the frequencies in the GWAS cohort. Before considering these conditional frequency spectra, we first characterized the impact of selection and non-equilibrium demography on allele frequency dynamics forwards and backwards in time. We then used these results to understand conditional frequency spectra under realistic human demography. Finally, we investigated empirical conditional frequency spectra for GWAS variants associated with 106 complex traits, finding compelling evidence for either stabilizing or purifying selection. Our results provide insight into polygenic score portability and other properties of variants ascertained with GWAS, highlighting the utility of conditional frequency spectra.

Back to the fundamentals: a reply to Basener and Sanford 2018

Fisher's fundamental theorem of natural selection has haunted theoretical population genetic literature since it was proposed in 1930, leading to numerous interpretations. Most of the confusion stemmed from Fisher's own obscure presentation. By the 1970s, a clearer view of Fisher's theorem had been achieved and it was found that, regardless of its utility or significance, it represents a general theorem of evolutionary biology. Basener and Sanford (J Math Biol 76:1589-1622, 2018) writing in JOMB, however, paint a different picture of the fundamental theorem as one hindered by its assumptions and incomplete due to its failure to explicitly incorporate mutational effects. They argue that Fisher saw his theorem as a "mathematical proof of Darwinian evolution". In this reply, we show that, contrary to Basener and Sanford, Fisher's theorem is a general theorem that applies to any evolving population, and that, far from their assertion that it needed to be expanded, the theorem already implicitly incorporates ancestor-descendant variation. We also show that their numerical simulations produce unrealistic results. Lastly, we argue that Basener and Sanford's motivations were in undermining not merely Fisher's theorem, but the concept of universal common descent itself.

Evolution of Phenotypic Variance Provides Insights into the Genetic Basis of Adaptation

Most traits are polygenic, and the contributing loci can be identified by genome-wide association studies. The genetic basis of adaptation (adaptive architecture) is, however, difficult to characterize. Here, we propose to study the adaptive architecture of traits by monitoring the evolution of their phenotypic variance during adaptation to a new environment in well-defined laboratory conditions. Extensive computer simulations show that the evolution of phenotypic variance in a replicated experimental evolution setting can distinguish between oligogenic and polygenic adaptive architectures. We compared gene expression variance in male Drosophila simulans before and after 100 generations of adaptation to a novel hot environment. The variance change in gene expression was indistinguishable for genes with and without a significant change in mean expression after 100 generations of evolution. We suggest that the majority of adaptive gene expression evolution can be explained by a polygenic architecture. We propose that tracking the evolution of phenotypic variance across generations can provide an approach to characterize the adaptive architecture.

Abundant pleiotropy across neuroimaging modalities identified through a multivariate genome-wide association study

Genetic pleiotropy is abundant across spatially distributed brain characteristics derived from one neuroimaging modality (e.g. structural, functional or diffusion magnetic resonance imaging [MRI]). A better understanding of pleiotropy across modalities could inform us on the integration of brain function, micro- and macrostructure. Here we show extensive genetic overlap across neuroimaging modalities at a locus and gene level in the UK Biobank (N = 34,029) and ABCD Study (N = 8607). When jointly analysing phenotypes derived from structural, functional and diffusion MRI in a genome-wide association study (GWAS) with the Multivariate Omnibus Statistical Test (MOSTest), we boost the discovery of loci and genes beyond previously identified effects for each modality individually. Cross-modality genes are involved in fundamental biological processes and predominantly expressed during prenatal brain development. We additionally boost prediction of psychiatric disorders by conditioning independent GWAS on our multimodal multivariate GWAS. These findings shed light on the shared genetic mechanisms underlying variation in brain morphology, functional connectivity, and tissue composition.

Pervasive G × E interactions shape adaptive trajectories and the exploration of the phenotypic space in artificial selection experiments

Quantitative genetics models have shown that long-term selection responses depend on initial variance and mutational influx. Understanding limits of selection requires quantifying the role of mutational variance. However, correlative responses to selection on nonfocal traits can perturb the selection response on the focal trait; and generations are often confounded with selection environments so that genotype by environment (G×E) interactions are ignored. The Saclay divergent selection experiments (DSEs) on maize flowering time were used to track the fate of individual mutations combining genotyping data and phenotyping data from yearly measurements (DSEYM) and common garden experiments (DSECG) with four objectives: (1) to quantify the relative contribution of standing and mutational variance to the selection response, (2) to estimate genotypic mutation effects, (3) to study the impact of G×E interactions in the selection response, and (4) to analyze how trait correlations modulate the exploration of the phenotypic space. We validated experimentally the expected enrichment of fixed beneficial mutations with an average effect of +0.278 and +0.299 days to flowering, depending on the genetic background. Fixation of unfavorable mutations reached up to 25% of incoming mutations, a genetic load possibly due to antagonistic pleiotropy, whereby mutations fixed in the selection environment (DSEYM) turned to be unfavorable in the evaluation environment (DSECG). Global patterns of trait correlations were conserved across genetic backgrounds but exhibited temporal patterns. Traits weakly or uncorrelated with flowering time triggered stochastic exploration of the phenotypic space, owing to microenvironment-specific fixation of standing variants and pleiotropic mutational input.

Adaptation of a quantitative trait to a changing environment: New analytical insights on the asexual and infinitesimal sexual models

Predicting the adaptation of populations to a changing environment is crucial to assess the impact of human activities on biodiversity. Many theoretical studies have tackled this issue by modeling the evolution of quantitative traits subject to stabilizing selection around an optimal phenotype, whose value is shifted continuously through time. In this context, the population fate results from the equilibrium distribution of the trait, relative to the moving optimum. Such a distribution may vary with the shape of selection, the system of reproduction, the number of loci, the mutation kernel or their interactions. Here, we develop a methodology that provides quantitative measures of population maladaptation and potential of survival directly from the entire profile of the phenotypic distribution, without any a priori on its shape. We investigate two different systems of reproduction (asexual and infinitesimal sexual models of inheritance), with various forms of selection. In particular, we recover that fitness functions such that selection weakens away from the optimum lead to evolutionary tipping points, with an abrupt collapse of the population when the speed of environmental change is too high. Our unified framework allows deciphering the mechanisms that lead to this phenomenon. More generally, it allows discussing similarities and discrepancies between the two systems of reproduction, which are ultimately explained by different constraints on the evolution of the phenotypic variance. We demonstrate that the mean fitness in the population crucially depends on the shape of the selection function in the infinitesimal sexual model, in contrast with the asexual model. In the asexual model, we also investigate the effect of the mutation kernel and we show that kernels with higher kurtosis tend to reduce maladaptation and improve fitness, especially in fast changing environments.

Sexually antagonistic selection maintains genetic variance when sexual dimorphism evolves

Genetic variance (VG) in fitness related traits is often unexpectedly high, evoking the question how VG can be maintained in the face of selection. Sexually antagonistic (SA) selection favouring alternative alleles in the sexes is common and predicted to maintain VG, while directional selection should erode it. Both SA and sex-limited directional selection can lead to sex-specific adaptations but how each affect VG when sexual dimorphism evolves remain experimentally untested. Using replicated artificial selection on the seed beetle Callosobruchus maculatus body size we recently demonstrated an increase in size dimorphism under SA and male-limited (ML) selection by 50% and 32%, respectively. Here we test their consequences on genetic variation. We show that SA selection maintained significantly more ancestral, autosomal additive genetic variance than ML selection, while both eroded sex-linked additive variation equally. Ancestral female-specific dominance variance was completely lost under ML, while SA selection consistently sustained it. Further, both forms of selection preserved a high genetic correlation between the sexes (rm,f). These results demonstrate the potential for sexual antagonism to maintain more genetic variance while fuelling sex-specific adaptation in a short evolutionary time scale, and are in line with predicted importance of sex-specific dominance reducing sexual conflict over alternative alleles.

Dissecting the loci underlying maturation timing in Atlantic salmon using haplotype and multi-SNP based association methods

Characterizing the role of different mutational effect sizes in the evolution of fitness-related traits has been a major goal in evolutionary biology for a century. Such characterization in a diversity of systems, both model and non-model, will help to understand the genetic processes underlying fitness variation. However, well-characterized genetic architectures of such traits in wild populations remain uncommon. In this study, we used haplotype-based and multi-SNP Bayesian association methods with sequencing data for 313 individuals from wild populations to test the mutational composition of known candidate regions for sea age at maturation in Atlantic salmon (Salmo salar). We detected an association at five loci out of 116 candidates previously identified in an aquaculture strain with maturation timing in wild Atlantic salmon. We found that at four of these five loci, variation explained by the locus was predominantly driven by a single SNP suggesting the genetic architecture of this trait includes multiple loci with simple, non-clustered alleles and a locus with potentially more complex alleles. This highlights the diversity of genetic architectures that can exist for fitness-related traits. Furthermore, this study provides a useful multi-SNP framework for future work using sequencing data to characterize genetic variation underlying phenotypes in wild populations.

Maintenance of quantitative genetic variance in complex, multi-trait phenotypes: The contribution of rare, large effect variants in two Drosophila species

The interaction of evolutionary processes to determine quantitative genetic variation has implications for contemporary and future phenotypic evolution, as well as for our ability to detect causal genetic variants. While theoretical studies have provided robust predictions to discriminate among competing models, empirical assessment of these has been limited. In particular, theory highlights the importance of pleiotropy in resolving observations of selection and mutation, but empirical investigations have typically been limited to few traits. Here, we applied high-dimensional Bayesian Sparse Factor Genetic modeling to gene expression datasets in 2 species, Drosophila melanogaster and Drosophila serrata, to explore the distributions of genetic variance across high-dimensional phenotypic space. Surprisingly, most of the heritable trait covariation was due to few lines (genotypes) with extreme [>3 interquartile ranges (IQR) from the median] values. Intriguingly, while genotypes extreme for a multivariate factor also tended to have a higher proportion of individual traits that were extreme, we also observed genotypes that were extreme for multivariate factors but not for any individual trait. We observed other consistent differences between heritable multivariate factors with outlier lines vs those factors without extreme values, including differences in gene functions. We use these observations to identify further data required to advance our understanding of the evolutionary dynamics and nature of standing genetic variation for quantitative traits.

Population differentiation of polygenic score predictions under stabilizing selection

Given the many small-effect loci uncovered by genome-wide association studies (GWAS), polygenic scores have become central to genomic medicine, and have found application in diverse settings including evolutionary studies of adaptation. Despite their promise, polygenic scores have been found to suffer from limited portability across human populations. This at first seems in conflict with the observation that most common genetic variation is shared among populations. We investigate one potential cause of this discrepancy: stabilizing selection on complex traits. Counterintuitively, while stabilizing selection constrains phenotypic evolution, it accelerates the loss and fixation of alleles underlying trait variation within populations (GWAS loci). Thus even when populations share an optimum phenotype, stabilizing selection erodes the variance contributed by their shared GWAS loci, such that predictions from GWAS in one population explain less of the phenotypic variation in another. We develop theory to quantify how stabilizing selection is expected to reduce the prediction accuracy of polygenic scores in populations not represented in GWAS samples. In addition, we find that polygenic scores can substantially overstate average genetic differences of phenotypes among populations. We emphasize stabilizing selection around a common optimum as a useful null model to connect patterns of allele frequency and polygenic score differentiation. This article is part of the theme issue 'Celebrating 50 years since Lewontin's apportionment of human diversity'.

Species-specific partial gene duplication in Arabidopsis thaliana evolved novel phenotypic effects on morphological traits under strong positive selection

Gene duplication is increasingly recognized as an important mechanism for the origination of new genes, as revealed by comparative genomic analysis. However, how new duplicate genes contribute to phenotypic evolution remains largely unknown, especially in plants. Here, we identified the new gene EXOV, derived from a partial gene duplication of its parental gene EXOVL in Arabidopsis thaliana. EXOV is a species-specific gene that originated within the last 3.5 million years and shows strong signals of positive selection. Unexpectedly, RNA-sequencing analyses revealed that, despite its young age, EXOV has acquired many novel direct and indirect interactions in which the parental gene does not engage. This observation is consistent with the high, selection-driven substitution rate of its encoded protein, in contrast to the slowly evolving EXOVL, suggesting an important role for EXOV in phenotypic evolution. We observed significant differentiation of morphological changes for all phenotypes assessed in genome-edited and T-DNA insertional single mutants and in double T-DNA insertion mutants in EXOV and EXOVL. We discovered a substantial divergence of phenotypic effects by principal component analyses, suggesting neofunctionalization of the new gene. These results reveal a young gene that plays critical roles in biological processes that underlie morphological evolution in A. thaliana.

Evolution of polygenic traits under global vs local adaptation

Observations about the number, frequency, effect size, and genomic distribution of alleles associated with complex traits must be interpreted in light of evolutionary process. These characteristics, which constitute a trait's genetic architecture, can dramatically affect evolutionary outcomes in applications from agriculture to medicine, and can provide a window into how evolution works. Here, I review theoretical predictions about the evolution of genetic architecture under spatially homogeneous, global adaptation as compared with spatially heterogeneous, local adaptation. Due to the tension between divergent selection and migration, local adaptation can favor "concentrated" genetic architectures that are enriched for alleles of larger effect, clustered in a smaller number of genomic regions, relative to expectations under global adaptation. However, the evolution of such architectures may be limited by many factors, including the genotypic redundancy of the trait, mutation rate, and temporal variability of environment. I review the circumstances in which predictions differ for global vs local adaptation and discuss where progress can be made in testing hypotheses using data from natural populations and lab experiments. As the field of comparative population genomics expands in scope, differences in architecture among traits and species will provide insights into how evolution works, and such differences must be interpreted in light of which kind of selection has been operating.

Polygenic adaptation after a sudden change in environment

Polygenic adaptation is thought to be ubiquitous, yet remains poorly understood. Here, we model this process analytically, in the plausible setting of a highly polygenic, quantitative trait that experiences a sudden shift in the fitness optimum. We show how the mean phenotype changes over time, depending on the effect sizes of loci that contribute to variance in the trait, and characterize the allele dynamics at these loci. Notably, we describe the two phases of the allele dynamics: The first is a rapid phase, in which directional selection introduces small frequency differences between alleles whose effects are aligned with or opposed to the shift, ultimately leading to small differences in their probability of fixation during a second, longer phase, governed by stabilizing selection. As we discuss, key results should hold in more general settings and have important implications for efforts to identify the genetic basis of adaptation in humans and other species.

Pleiotropy or linkage? Their relative contributions to the genetic correlation of quantitative traits and detection by multitrait GWA studies

Genetic correlations between traits may cause correlated responses to selection. Previous models described the conditions under which genetic correlations are expected to be maintained. Selection, mutation, and migration are all proposed to affect genetic correlations, regardless of whether the underlying genetic architecture consists of pleiotropic or tightly linked loci affecting the traits. Here, we investigate the conditions under which pleiotropy and linkage have different effects on the genetic correlations between traits by explicitly modeling multiple genetic architectures to look at the effects of selection strength, degree of correlational selection, mutation rate, mutational variance, recombination rate, and migration rate. We show that at mutation-selection(-migration) balance, mutation rates differentially affect the equilibrium levels of genetic correlation when architectures are composed of pairs of physically linked loci compared to architectures of pleiotropic loci. Even when there is perfect linkage (no recombination within pairs of linked loci), a lower genetic correlation is maintained than with pleiotropy, with a lower mutation rate leading to a larger decrease. These results imply that the detection of causal loci in multitrait association studies will be affected by the type of underlying architectures, whereby pleiotropic variants are more likely to be underlying multiple detected associations. We also confirm that tighter linkage between nonpleiotropic causal loci maintains higher genetic correlations at the traits and leads to a greater proportion of false positives in association analyses.

Evolution of phenotypic variance in response to a novel hot environment

Shifts in trait means are widely considered as evidence for adaptive responses, but the impact on phenotypic variance remains largely unexplored. Classic quantitative genetics provides a theoretical framework to predict how selection on phenotypic mean affects the variance. In addition to this indirect effect, it is also possible that the variance of the trait is the direct target of selection, but experimentally characterized cases are rare. Here, we studied gene expression variance of Drosophila simulans males before and after 100 generations of adaptation to a novel hot laboratory environment. In each of the two independently evolved populations, the variance of 125 and 97 genes was significantly reduced. We propose that the drastic loss in environmental complexity from nature to the laboratory may have triggered selection for reduced variance. Our observation that selection could drive changes in the variance of gene expression could have important implications for studies of adaptation processes in natural and experimental populations.

Histone H3K27 methylation perturbs transcriptional robustness and underpins dispensability of highly conserved genes in fungi

Epigenetic modifications are key regulators of gene expression and underpin genome integrity. Yet, how epigenetic changes affect the evolution and transcriptional robustness of genes remains largely unknown. Here, we show how the repressive histone mark H3K27me3 underpins the trajectory of highly conserved genes in fungi. We first performed transcriptomic profiling on closely related species of the plant pathogen Fusarium graminearum species complex. We determined transcriptional responsiveness of genes across environmental conditions to determine expression robustness. To infer evolutionary conservation, we used a framework of 23 species across the Fusarium genus including three species covered with histone methylation data. Gene expression variation is negatively correlated with gene conservation confirming that highly conserved genes show higher expression robustness. In contrast, genes marked by H3K27me3 do not show such associations. Furthermore, highly conserved genes marked by H3K27me3 encode smaller proteins, exhibit weaker codon usage bias, higher levels of hydrophobicity, show lower intrinsically disordered regions, and are enriched for functions related to regulation and membrane transport. The evolutionary age of conserved genes with H3K27me3 histone marks falls typically within the origins of the Fusarium genus. We show that highly conserved genes marked by H3K27me3 are more likely to be dispensable for survival during host infection. Lastly, we show that conserved genes exposed to repressive H3K27me3 marks across distantly related Fusarium fungi are associated with transcriptional perturbation at the microevolutionary scale. In conclusion, we show how repressive histone marks are entangled in the evolutionary fate of highly conserved genes across evolutionary timescales.

Evolvability: A Quantitative-Genetics Perspective

The concept of evolvability emerged in the early 1990s and soon became fashionable as a label for different streams of research in evolutionary biology. In evolutionary quantitative genetics, evolvability is defined as the ability of a population to respond to directional selection. This differs from other fields by treating evolvability as a property of populations rather than organisms or lineages and in being focused on quantification and short-term prediction rather than on macroevolution. While the term evolvability is new to quantitative genetics, many of the associated ideas and research questions have been with the field from its inception as biometry. Recent research on evolvability is more than a relabeling of old questions, however. New operational measures of evolvability have opened possibilities for understanding adaptation to rapid environmental change, assessing genetic constraints, and linking micro- and macroevolution.

The Biology of Chernobyl

Environmental disasters offer the unique opportunity for landscape-scale ecological and evolutionary studies that are not possible in the laboratory or small experimental plots. The nuclear accident at Chernobyl (1986) allows for rigorous analyses of radiation effects on individuals and populations at an ecosystem scale. Here, the current state of knowledge related to populations within the Chernobyl region of Ukraine and Belarus following the largest civil nuclear accident in history is reviewed. There is now a significant literature that provides contrasting and occasionally conflicting views of the state of animals and how they are affected by this mutagenic stressor. Studies of genetic and physiological effects have largely suggested significant injuries to individuals inhabiting the more radioactive areas of the Chernobyl region. Most population censuses for most species suggest that abundances are reduced in the more radioactive areas.

Interplay between extreme drift and selection intensities favors the fixation of beneficial mutations in selfing maize populations

Population and quantitative genetic models provide useful approximations to predict long-term selection responses sustaining phenotypic shifts, and underlying multilocus adaptive dynamics. Valid across a broad range of parameters, their use for understanding the adaptive dynamics of small selfing populations undergoing strong selection intensity (thereafter High Drift-High selection regime, HDHS) remains to be explored. Saclay Divergent Selection Experiments (DSEs) on maize flowering time provide an interesting example of populations evolving under HDHS, with significant selection responses over 20 generations in two directions. We combined experimental data from Saclay DSEs, forward individual-based simulations, and theoretical predictions to dissect the evolutionary mechanisms at play in the observed selection responses. We asked two main questions: How do mutations arise, spread, and reach fixation in populations evolving under HDHS? How does the interplay between drift and selection influence observed phenotypic shifts? We showed that the long-lasting response to selection in small populations is due to the rapid fixation of mutations occurring during the generations of selection. Among fixed mutations, we also found a clear signal of enrichment for beneficial mutations revealing a limited cost of selection. Both environmental stochasticity and variation in selection coefficients likely contributed to exacerbate mutational effects, thereby facilitating selection grasp and fixation of small-effect mutations. Together our results highlight that despite a small number of polymorphic loci expected under HDHS, adaptive variation is continuously fueled by a vast mutational target. We discuss our results in the context of breeding and long-term survival of small selfing populations.

Structural properties of generalised Planck distributions

A family of generalised Planck (GP) laws is defined and its structural properties explored. Sometimes subject to parameter restrictions, a GP law is a randomly scaled gamma law; it arises as the equilibrium law of a perturbed version of the Feller mean reverting diffusion; the density functions can be decreasing, unimodal or bimodal; it is infinitely divisible. It is argued that the GP law is not a generalised gamma convolution. Characterisations are obtained in terms of invariance under random contraction of a weighted version of a related law. The GP law is a particular instance of equilibrium laws obtained from a recursion suggested by a genetic mutation-selection balance model. Some related infinitely divisible laws are exhibited.

Is Evolution in Response to Extreme Events Good for Population Persistence?

AbstractClimate change is predicted to increase the severity of environmental perturbations, including storms and droughts, which act as strong selective agents. These extreme events are often of finite duration (pulse disturbances). Hence, while evolution during an extreme event may be adaptive, the resulting phenotypic changes may become maladaptive when the event ends. Using individual-based models and analytic approximations that fuse quantitative genetics and demography, we explore how heritability and phenotypic variance affect population size and extinction risk in finite populations under an extreme event of fixed duration. Since more evolution leads to greater maladaptation and slower population recovery following an extreme event, greater heritability can increase extinction risk when the extreme event is short. Alternatively, when an extreme event is sufficiently long, heritability often helps a population persist. We also find that when events are severe, the buffering effect of phenotypic variance can outweigh the increased load it causes.

Assortative mating can help adaptation of flowering time to a changing climate: Insights from a polygenic model

Several empirical studies report fast evolutionary changes in flowering time in response to contemporary climate change. Flowering time is a polygenic trait under assortative mating, since flowering time of mates must overlap. Here, we test whether assortative mating, compared with random mating, can help better track a changing climate. For each mating pattern, our individual‐based model simulates a population evolving in a climate characterized by stabilizing selection around an optimal flowering time, which can change directionally and/or fluctuate. We also derive new analytical predictions from a quantitative genetics model for the expected genetic variance at equilibrium, and its components, the lag of the population to the optimum and the population mean fitness. We compare these predictions between assortative and random mating, and to our simulation results. Assortative mating, compared with random mating, has antagonistic effects on genetic variance: it generates positive associations among similar allelic effects, which inflates the genetic variance, but it decreases genetic polymorphism, which depresses the genetic variance. In a stationary environment with substantial stabilizing selection, assortative mating affects little the genetic variance compared with random mating. In a changing climate, assortative mating however increases genetic variance compared to random mating, which diminishes the lag of the population to the optimum, and in most scenarios translates into a fitness advantage relative to random mating. The magnitude of this fitness advantage depends on the extent to which genetic variance limits adaptation, being larger for faster environmental changes and weaker stabilizing selection.

A white noise approach to evolutionary ecology

Although the evolutionary response to random genetic drift is classically modelled as a sampling process for populations with fixed abundance, the abundances of populations in the wild fluctuate over time. Furthermore, since wild populations exhibit demographic stochasticity and since random genetic drift is in part due to demographic stochasticity, theoretical approaches are needed to understand the role of demographic stochasticity in eco-evolutionary dynamics. Here we close this gap for quantitative characters evolving in continuously reproducing populations by providing a framework to track the stochastic dynamics of abundance density across phenotypic space using stochastic partial differential equations. In the process we develop a set of heuristics to operationalize the powerful, but abstract theory of white noise and diffusion-limits of individual-based models. Applying these heuristics, we obtain stochastic ordinary differential equations that generalize classical expressions of ecological quantitative genetics. In particular, by supplying growth rate and reproductive variance as functions of abundance densities and trait values, these equations track population size, mean trait and additive genetic variance responding to mutation, demographic stochasticity, random genetic drift, deterministic selection and noise-induced selection. We demonstrate the utility of our approach by formulating a model of diffuse coevolution mediated by exploitative competition for a continuum of resources. In addition to trait and abundance distributions, this model predicts interaction networks defined by niche-overlap, competition coefficients, or selection gradients. Using a high-richness approximation, we find linear selection gradients and competition coefficients are uncorrelated, but magnitudes of linear selection gradients and quadratic selection gradients are both positively correlated with competition coefficients. Hence, competing species that strongly affect each other's abundance tend to also impose selection on one another, but the directionality is not predicted. This approach contributes to the development of a synthetic theory of evolutionary ecology by formalizing first principle derivations of stochastic models tracking feedbacks of biological processes and the patterns of diversity they produce.

Elevated temperature increases genome-wide selection on de novo mutations

Adaptation in new environments depends on the amount of genetic variation available for evolution, and the efficacy by which natural selection discriminates among this variation. However, whether some ecological factors reveal more genetic variation, or impose stronger selection pressures than others, is typically not known. Here, we apply the enzyme kinetic theory to show that rising global temperatures are predicted to intensify natural selection throughout the genome by increasing the effects of DNA sequence variation on protein stability. We test this prediction by (i) estimating temperature-dependent fitness effects of induced mutations in seed beetles adapted to ancestral or elevated temperature, and (ii) calculate 100 paired selection estimates on mutations in benign versus stressful environments from unicellular and multicellular organisms. Environmental stress per se did not increase mean selection on de novo mutation, suggesting that the cost of adaptation does not generally increase in new ecological settings to which the organism is maladapted. However, elevated temperature increased the mean strength of selection on genome-wide polymorphism, signified by increases in both mutation load and mutational variance in fitness. These results have important implications for genetic diversity gradients and the rate and repeatability of evolution under climate change.

The evolution of group differences in changing environments

The selection pressures that have shaped the evolution of complex traits in humans remain largely unknown, and in some contexts highly contentious, perhaps above all where they concern mean trait differences among groups. To date, the discussion has focused on whether such group differences have any genetic basis, and if so, whether they are without fitness consequences and arose via random genetic drift, or whether they were driven by selection for different trait optima in different environments. Here, we highlight a plausible alternative: that many complex traits evolve under stabilizing selection in the face of shifting environmental effects. Under this scenario, there will be rapid evolution at the loci that contribute to trait variation, even when the trait optimum remains the same. These considerations underscore the strong assumptions about environmental effects that are required in ascribing trait differences among groups to genetic differences.

How does the strength of selection influence genetic correlations?

Genetic correlations between traits can strongly impact evolutionary responses to selection, and may thus impose constraints on adaptation. Theoretical and empirical work has made it clear that without strong linkage and with random mating, genetic correlations at evolutionary equilibrium result from an interplay of correlated pleiotropic effects of mutations, and correlational selection favoring combinations of trait values. However, it is not entirely clear how change in the overall strength of stabilizing selection across traits (breadth of the fitness peak, given its shape) influences this compromise between mutation and selection effects on genetic correlation. Here, we show that the answer to this question crucially depends on the intensity of genetic drift. In large, effectively infinite populations, genetic correlations are unaffected by the strength of selection, regardless of whether the genetic architecture involves common small-effect mutations (Gaussian regime), or rare large-effect mutations (House-of-Cards regime). In contrast in finite populations, the strength of selection does affect genetic correlations, by shifting the balance from drift-dominated to selection-dominated evolutionary dynamics. The transition between these domains depends on mutation parameters to some extent, but with a similar dependence of genetic correlation on the strength of selection. Our results are particularly relevant for understanding how senescence shapes patterns of genetic correlations across ages, and genetic constraints on adaptation during colonization of novel habitats.

Low Additive Genetic Variation in a Trait Under Selection in Domesticated Rice

Quantitative traits are important targets of both natural and artificial selection. The genetic architecture of these traits and its change during the adaptive process is thus of fundamental interest. The fate of the additive effects of variants underlying a trait receives particular attention because they constitute the genetic variation component that is transferred from parents to offspring and thus governs the response to selection. While estimation of this component of phenotypic variation is challenging, the increasing availability of dense molecular markers puts it within reach. Inbred plant species offer an additional advantage because phenotypes of genetically identical individuals can be measured in replicate. This makes it possible to estimate marker effects separately from the contribution of the genetic background not captured by genotyped loci. We focused on root growth in domesticated rice, Oryza sativa, under normal and aluminum (Al) stress conditions, a trait under recent selection because it correlates with survival under drought. A dense single nucleotide polymorphism (SNP) map is available for all accessions studied. Taking advantage of this map and a set of Bayesian models, we assessed additive marker effects. While total genetic variation accounted for a large proportion of phenotypic variance, marker effects contributed little information, particularly in the Al-tolerant tropical japonica population of rice. We were unable to identify any loci associated with root growth in this population. Models estimating the aggregate effects of all measured genotypes likewise produced low estimates of marker heritability and were unable to predict total genetic values accurately. Our results support the long-standing conjecture that additive genetic variation is depleted in traits under selection. We further provide evidence that this depletion is due to the prevalence of low-frequency alleles that underlie the trait.

Individual variation in dispersal and fecundity increases rates of spatial spread

Dispersal and fecundity are two fundamental traits underlying the spread of populations. Using integral difference equation models, we examine how individual variation in these fundamental traits and the heritability of these traits influence rates of spatial spread of populations along a one-dimensional transect. Using a mixture of analytic and numerical methods, we show that individual variation in dispersal rates increases spread rates and the more heritable this variation, the greater the increase. In contrast, individual variation in lifetime fecundity only increases spread rates when some of this variation is heritable. The highest increases in spread rates occur when variation in dispersal positively co-varies with fecundity. Our results highlight the importance of estimating individual variation in dispersal rates, dispersal syndromes in which fecundity and dispersal co-vary positively and heritability of these traits to predict population rates of spatial spread.

Genetic Paths to Evolutionary Rescue and the Distribution of Fitness Effects Along Them

The past century has seen substantial theoretical and empirical progress on the genetic basis of adaptation. Over this same period, a pressing need to prevent the evolution of drug resistance has uncovered much about the potential genetic basis of persistence in declining populations. However, we have little theory to predict and generalize how persistence-by sufficiently rapid adaptation-might be realized in this explicitly demographic scenario. Here, we use Fisher's geometric model with absolute fitness to begin a line of theoretical inquiry into the genetic basis of evolutionary rescue, focusing here on asexual populations that adapt through de novo mutations. We show how the dominant genetic path to rescue switches from a single mutation to multiple as mutation rates and the severity of the environmental change increase. In multi-step rescue, intermediate genotypes that themselves go extinct provide a "springboard" to rescue genotypes. Comparing to a scenario where persistence is assured, our approach allows us to quantify how a race between evolution and extinction leads to a genetic basis of adaptation that is composed of fewer loci of larger effect. We hope this work brings awareness to the impact of demography on the genetic basis of adaptation.

Evolutionary perspectives on polygenic selection, missing heritability, and GWAS

Genome-wide association studies (GWAS) have successfully identified many trait-associated variants, but there is still much we do not know about the genetic basis of complex traits. Here, we review recent theoretical and empirical literature regarding selection on complex traits to argue that "missing heritability" is as much an evolutionary problem as it is a statistical problem. We discuss empirical findings that suggest a role for selection in shaping the effect sizes and allele frequencies of causal variation underlying complex traits, and the limitations of these studies. We then use simulations of selection, realistic genome structure, and complex human demography to illustrate the results of recent theoretical work on polygenic selection, and show that statistical inference of causal loci is sharply affected by evolutionary processes. In particular, when selection acts on causal alleles, it hampers the ability to detect causal loci and constrains the transferability of GWAS results across populations. Last, we discuss the implications of these findings for future association studies, and suggest that future statistical methods to infer causal loci for genetic traits will benefit from explicit modeling of the joint distribution of effect sizes and allele frequencies under plausible evolutionary models.

Polygenic Adaptation to an Environmental Shift: Temporal Dynamics of Variation Under Gaussian Stabilizing Selection and Additive Effects on a Single Trait

Predictions about the effect of natural selection on patterns of linked neutral variation are largely based on models involving the rapid fixation of unconditionally beneficial mutations. However, when phenotypes adapt to a new optimum trait value, the strength of selection on individual mutations decreases as the population adapts. Here, I use explicit forward simulations of a single trait with additive-effect mutations adapting to an "optimum shift." Detectable "hitchhiking" patterns are only apparent if (i) the optimum shifts are large with respect to equilibrium variation for the trait, (ii) mutation rates to large-effect mutations are low, and (iii) large-effect mutations rapidly increase in frequency and eventually reach fixation, which typically occurs after the population reaches the new optimum. For the parameters simulated here, partial sweeps do not appreciably affect patterns of linked variation, even when the mutations are strongly selected. The contribution of new mutations vs. standing variation to fixation depends on the mutation rate affecting trait values. Given the fixation of a strongly selected variant, patterns of hitchhiking are similar on average for the two classes of sweeps because sweeps from standing variation involving large-effect mutations are rare when the optimum shifts. The distribution of effect sizes of new mutations has little effect on the time to reach the new optimum, but reducing the mutational variance increases the magnitude of hitchhiking patterns. In general, populations reach the new optimum prior to the completion of any sweeps, and the times to fixation are longer for this model than for standard models of directional selection. The long fixation times are due to a combination of declining selection pressures during adaptation and the possibility of interference among weakly selected sites for traits with high mutation rates.

Structural variants exhibit widespread allelic heterogeneity and shape variation in complex traits

It has been hypothesized that individually-rare hidden structural variants (SVs) could account for a significant fraction of variation in complex traits. Here we identified more than 20,000 euchromatic SVs from 14 Drosophila melanogaster genome assemblies, of which ~40% are invisible to high specificity short-read genotyping approaches. SVs are common, with 31.5% of diploid individuals harboring a SV in genes larger than 5kb, and 24% harboring multiple SVs in genes larger than 10kb. SV minor allele frequencies are rarer than amino acid polymorphisms, suggesting that SVs are more deleterious. We show that a number of functionally important genes harbor previously hidden structural variants likely to affect complex phenotypes. Furthermore, SVs are overrepresented in candidate genes associated with quantitative trait loci mapped using the Drosophila Synthetic Population Resource. We conclude that SVs are ubiquitous, frequently constitute a heterogeneous allelic series, and can act as rare alleles of large effect.

Thinking About the Evolution of Complex Traits in the Era of Genome-Wide Association Studies

Many traits of interest are highly heritable and genetically complex, meaning that much of the variation they exhibit arises from differences at numerous loci in the genome. Complex traits and their evolution have been studied for more than a century, but only in the last decade have genome-wide association studies (GWASs) in humans begun to reveal their genetic basis. Here, we bring these threads of research together to ask how findings from GWASs can further our understanding of the processes that give rise to heritable variation in complex traits and of the genetic basis of complex trait evolution in response to changing selection pressures (i.e., of polygenic adaptation). Conversely, we ask how evolutionary thinking helps us to interpret findings from GWASs and informs related efforts of practical importance.

Compensatory trans-regulatory alleles minimizing variation in TDH3 expression are common within Saccharomyces cerevisiae

Heritable variation in gene expression is common within species. Much of this variation is due to genetic differences outside of the gene with altered expression and is trans-acting. This trans-regulatory variation is often polygenic, with individual variants typically having small effects, making the genetic architecture and evolution of trans-regulatory variation challenging to study. Consequently, key questions about trans-regulatory variation remain, including the variability of trans-regulatory variation within a species, how selection affects trans-regulatory variation, and how trans-regulatory variants are distributed throughout the genome and within a species. To address these questions, we isolated and measured trans-regulatory differences affecting TDH3 promoter activity among 56 strains of Saccharomyces cerevisiae, finding that trans-regulatory backgrounds varied approximately twofold in their effects on TDH3 promoter activity. Comparing this variation to neutral models of trans-regulatory evolution based on empirical measures of mutational effects revealed that despite this variability in the effects of trans-regulatory backgrounds, stabilizing selection has constrained trans-regulatory differences within this species. Using a powerful quantitative trait locus mapping method, we identified ∼100 trans-acting expression quantitative trait locus in each of three crosses to a common reference strain, indicating that regulatory variation is more polygenic than previous studies have suggested. Loci altering expression were located throughout the genome, and many loci were strain specific. This distribution and prevalence of alleles is consistent with recent theories about the genetic architecture of complex traits. In all mapping experiments, the nonreference strain alleles increased and decreased TDH3 promoter activity with similar frequencies, suggesting that stabilizing selection maintained many trans-acting variants with opposing effects. This variation may provide the raw material for compensatory evolution and larger scale regulatory rewiring observed in developmental systems drift among species.

Parallel genetic evolution and speciation from standing variation

Adaptation often proceeds from standing variation, and natural selection acting on pairs of populations is a quantitative continuum ranging from parallel to divergent. Yet, it is unclear how the extent of parallel genetic evolution during adaptation from standing variation is affected by the difference in the direction of selection between populations. Nor is it clear whether the availability of standing variation for adaptation affects progress toward speciation in a manner that depends on the difference in the direction of selection. We conducted a theoretical study investigating these questions and have two primary findings. First, the extent of parallel genetic evolution between two populations rapidly declines as selection changes from fully parallel toward divergent, and this decline is steeper in organisms with more traits (i.e., greater dimensionality). This rapid decline happens because small differences in the direction of selection greatly reduce the fraction of alleles that are beneficial in both populations. For example, populations adapting to optima separated by an angle of 33° might have only 50% of potentially beneficial alleles in common. Second, relative to when adaptation is from only new mutation, adaptation from standing variation improves hybrid fitness under parallel selection and reduces hybrid fitness under divergent selection. Under parallel selection, genetic parallelism from standing variation reduces the phenotypic segregation variance in hybrids, thereby increasing mean fitness in the parental environment. Under divergent selection, larger pleiotropic effects of alleles fixed from standing variation cause maladaptive transgressive phenotypes when combined in hybrids. Adaptation from standing genetic variation therefore slows progress toward speciation under parallel selection and facilitates progress toward speciation under divergent selection.

Metapop: An individual-based model for simulating the evolution of tree populations in spatially and temporally heterogeneous landscapes

Metapop is a stochastic individual-based simulation program. It uses quantitative genetics theory to produce an explicit description of the typical life cycle of monoecious and hermaphroditic plant species. Genome structure, the relationship between genotype and phenotype, and the effects of landscape heterogeneity on each individual can be finely parameterized by the user. Unlike most existing simulation packages, Metapop can simulate phenotypic plasticity, which may have a genetic component, and assortative mating, two important features of tree species. Each simulation is parameterized through text files, and raw data are generated recurrently, describing the allelic state of each quantitative trait locus involved in phenotypic variability. The data can be generated in Genepop or Fstat format, and may thus be analysed with other existing packages. Metapop also automatically computes a range of populations statistics, enabling the user to monitor evolutionary dynamics directly, from gene to metapopulation level.

Genetic architecture and selective sweeps after polygenic adaptation to distant trait optima

Understanding the genetic basis of phenotypic adaptation to changing environments is an essential goal of population and quantitative genetics. While technological advances now allow interrogation of genome-wide genotyping data in large panels, our understanding of the process of polygenic adaptation is still limited. To address this limitation, we use extensive forward-time simulation to explore the impacts of variation in demography, trait genetics, and selection on the rate and mode of adaptation and the resulting genetic architecture. We simulate a population adapting to an optimum shift, modeling sequence variation for 20 QTL for each of 12 different demographies for 100 different traits varying in the effect size distribution of new mutations, the strength of stabilizing selection, and the contribution of the genomic background. We then use random forest regression approaches to learn the relative importance of input parameters in determining a number of aspects of the process of adaptation, including the speed of adaptation, the relative frequency of hard sweeps and sweeps from standing variation, or the final genetic architecture of the trait. We find that selective sweeps occur even for traits under relatively weak selection and where the genetic background explains most of the variation. Though most sweeps occur from variation segregating in the ancestral population, new mutations can be important for traits under strong stabilizing selection that undergo a large optimum shift. We also show that population bottlenecks and expansion impact overall genetic variation as well as the relative importance of sweeps from standing variation and the speed with which adaptation can occur. We then compare our results to two traits under selection during maize domestication, showing that our simulations qualitatively recapitulate differences between them. Overall, our results underscore the complex population genetics of individual loci in even relatively simple quantitative trait models, but provide a glimpse into the factors that drive this complexity and the potential of these approaches for understanding polygenic adaptation.

A Large Panel of Drosophila simulans Reveals an Abundance of Common Variants

The rapidly expanding availability of large NGS data sets provides an opportunity to investigate population genetics at an unprecedented scale. Drosophila simulans is the sister species of the model organism Drosophila melanogaster, and is often presumed to share similar demographic history. However, previous population genetic and ecological work suggests very different signatures of selection and demography. Here, we sequence a new panel of 170 inbred genotypes of a North American population of D. simulans, a valuable complement to the DGRP and other D. melanogaster panels. We find some unexpected signatures of demography, in the form of excess intermediate frequency polymorphisms. Simulations suggest that this is possibly due to a recent population contraction and selection. We examine the outliers in the D. simulans genome determined by a haplotype test to attempt to parse the contribution of demography and selection to the patterns observed in this population. Untangling the relative contribution of demography and selection to genomic patterns of variation is challenging, however, it is clear that although D. melanogaster was thought to share demographic history with D. simulans different forces are at work in shaping genomic variation in this population of D. simulans.

Quantifying how constraints limit the diversity of viable routes to adaptation

Convergent adaptation occurs at the genome scale when independently evolving lineages use the same genes to respond to similar selection pressures. These patterns of genetic repeatability provide insights into the factors that facilitate or constrain the diversity of genetic responses that contribute to adaptive evolution. A first step in studying such factors is to quantify the observed amount of repeatability relative to expectations under a null hypothesis. Here, we formulate a novel index to quantify the constraints driving the observed amount of repeated adaptation in pairwise contrasts based on the hypergeometric distribution, and then generalize this for simultaneous analysis of multiple lineages. This index is explicitly based on the probability of observing a given amount of repeatability by chance under a given null hypothesis and is readily compared among different species and types of trait. We also formulate an index to quantify the effective proportion of genes in the genome that have the potential to contribute to adaptation. As an example of how these indices can be used to draw inferences, we assess the amount of repeatability observed in existing datasets on adaptation to stress in yeast and climate in conifers. This approach provides a method to test a wide range of hypotheses about how different kinds of factors can facilitate or constrain the diversity of genetic responses observed during adaptive evolution.

How many ways can evolution solve the same adaptive problem? While convergent adaptation is evident in many organisms at the phenotypic level, we are only beginning to understand how commonly this convergence extends to the genome scale. Quantifying the repeatability of adaptation at the genome scale is therefore critical for assessing how constraints affect the diversity of viable genetic responses. Here, we develop probability-based indices to quantify the deviation between observed repeatability and expectations under a range of null hypotheses, and an estimator of the proportion of loci in the genome that can contribute to adaptation. We demonstrate the usage of these indices with individual-based simulations and example datasets from yeast and conifers and discuss how they differ from previously developed approaches to studying repeatability. Because these indices are unitless, they provide a general approach to quantifying and comparing how constraints drive convergence at the genome scale across a wide range of traits and taxa.

Effect of migration and environmental heterogeneity on the maintenance of quantitative genetic variation: a simulation study

The paradox of high genetic variation observed in traits under stabilizing selection is a long-standing problem in evolutionary theory, as mutation rates appear too low to explain observed levels of standing genetic variation under classic models of mutation-selection balance. Spatially or temporally heterogeneous environments can maintain more standing genetic variation within populations than homogeneous environments, but it is unclear whether such conditions can resolve the above discrepancy between theory and observation. Here, we use individual-based simulations to explore the effect of various types of environmental heterogeneity on the maintenance of genetic variation (V_A ) for a quantitative trait under stabilizing selection. We find that V_A is maximized at intermediate migration rates in spatially heterogeneous environments and that the observed patterns are robust to changes in population size. Spatial environmental heterogeneity increased variation by as much as 10-fold over mutation-selection balance alone, whereas pure temporal environmental heterogeneity increased variance by only 45% at max. Our results show that some combinations of spatial heterogeneity and migration can maintain considerably more variation than mutation-selection balance, potentially reconciling the discrepancy between theoretical predictions and empirical observations. However, given the narrow regions of parameter space required for this effect, this is unlikely to provide a general explanation for the maintenance of variation. Nonetheless, our results suggest that habitat fragmentation may affect the maintenance of V_A and thereby reduce the adaptive capacity of populations.

Phylogenetic divergence of cell biological features

Most cellular features have a range of states, but understanding the mechanisms responsible for interspecific divergence is a challenge for evolutionary cell biology. Models are developed for the distribution of mean phenotypes likely to evolve under the joint forces of mutation and genetic drift in the face of constant selection pressures. Mean phenotypes will deviate from optimal states to a degree depending on the effective population size, potentially leading to substantial divergence in the absence of diversifying selection. The steady-state distribution for the mean can even be bimodal, with one domain being largely driven by selection and the other by mutation pressure, leading to the illusion of phenotypic shifts being induced by movement among alternative adaptive domains. These results raise questions as to whether lineage-specific selective pressures are necessary to account for interspecific divergence, providing a possible platform for the establishment of null models for the evolution of cell-biological traits.

A population genetic interpretation of GWAS findings for human quantitative traits

Human genome-wide association studies (GWASs) are revealing the genetic architecture of anthropomorphic and biomedical traits, i.e., the frequencies and effect sizes of variants that contribute to heritable variation in a trait. To interpret these findings, we need to understand how genetic architecture is shaped by basic population genetics processes-notably, by mutation, natural selection, and genetic drift. Because many quantitative traits are subject to stabilizing selection and because genetic variation that affects one trait often affects many others, we model the genetic architecture of a focal trait that arises under stabilizing selection in a multidimensional trait space. We solve the model for the phenotypic distribution and allelic dynamics at steady state and derive robust, closed-form solutions for summary statistics of the genetic architecture. Our results provide a simple interpretation for missing heritability and why it varies among traits. They predict that the distribution of variances contributed by loci identified in GWASs is well approximated by a simple functional form that depends on a single parameter: the expected contribution to genetic variance of a strongly selected site affecting the trait. We test this prediction against the results of GWASs for height and body mass index (BMI) and find that it fits the data well, allowing us to make inferences about the degree of pleiotropy and mutational target size for these traits. Our findings help to explain why the GWAS for height explains more of the heritable variance than the similarly sized GWAS for BMI and to predict the increase in explained heritability with study sample size. Considering the demographic history of European populations, in which these GWASs were performed, we further find that most of the associations they identified likely involve mutations that arose shortly before or during the Out-of-Africa bottleneck at sites with selection coefficients around s = 10-3.

Evolutionary Rescue over a Fitness Landscape

Evolutionary rescue describes a situation where adaptive evolution prevents the extinction of a population facing a stressing environment. Models of evolutionary rescue could in principle be used to predict the level of stress beyond which extinction becomes likely for species of conservation concern, or, conversely, the treatment levels most likely to limit the emergence of resistant pests or pathogens. Stress levels are known to affect both the rate of population decline (demographic effect) and the speed of adaptation (evolutionary effect), but the latter aspect has received less attention. Here, we address this issue using Fisher's geometric model of adaptation. In this model, the fitness effects of mutations depend both on the genotype and the environment in which they arise. In particular, the model introduces a dependence between the level of stress, the proportion of rescue mutants, and their costs before the onset of stress. We obtain analytic results under a strong-selection-weak-mutation regime, which we compare to simulations. We show that the effect of the environment on evolutionary rescue can be summarized into a single composite parameter quantifying the effective stress level, which is amenable to empirical measurement. We describe a narrow characteristic stress window over which the rescue probability drops from very likely to very unlikely as the level of stress increases. This drop is sharper than in previous models, as a result of the decreasing proportion of stress-resistant mutations as stress increases. We discuss how to test these predictions with rescue experiments across gradients of stress.