The Distribution of Gene Frequencies in Populations

Study design and the sampling of deleterious rare variants in biobank-scale datasets

One key component of study design in population genetics is the "geographic breadth" of a sample (i.e., how broad a region across which individuals are sampled). How the geographic breadth of a sample impacts observations of rare, deleterious variants is unclear, even though such variants are of particular interest for biomedical and evolutionary applications. Here, in order to gain insight into the effects of sample design on ascertained genetic variants, we formulate a stochastic model of dispersal, genetic drift, selection, mutation, and geographically concentrated sampling. We use this model to understand the effects of the geographic breadth of sampling effort on the discovery of negatively selected variants. We find that samples which are more geographically broad will discover a greater number of variants as compared to geographically narrow samples (an effect we label "discovery"); though the variants will be detected at lower average frequency than in narrow samples (e.g., as singletons, an effect we label "dilution"). Importantly, these effects are amplified for larger sample sizes and fitness effects. We validate these results using both population genetic simulations and empirical analyses in the UK Biobank. Our results are particularly important in two contexts: the association of large-effect rare variants with particular phenotypes and the inference of negative selection from allele frequency data. Overall, our findings emphasize the importance of considering geographic breadth when designing and carrying out genetic studies, especially at biobank scale.

Effect of assortative mating and sexual selection on polygenic barriers to gene flow

Assortative mating and sexual selection are widespread in nature and can play an important role in speciation by facilitating the buildup and maintenance of reproductive isolation (RI). However, their contribution to genome-wide suppression of gene flow during RI is rarely quantified. Here, we consider a polygenic "magic" trait that is divergently selected across two populations connected by migration, while also serving as the basis of assortative mating, thus generating sexual selection on one or both sexes. We obtain theoretical predictions for divergence at individual trait loci by assuming that the effect of all other loci on any locus can be encapsulated via an effective migration rate, which bears a simple relationship to measurable fitness components of migrants and various early-generation hybrids. Our analysis clarifies how "tipping points" (characterized by an abrupt collapse of adaptive divergence) arise, and when assortative mating can shift the critical level of migration beyond which divergence collapses. We quantify the relative contributions of viability and sexual selection to genome-wide barriers to gene flow and discuss how these depend on existing divergence levels. Our results suggest that effective migration rates provide a useful way of understanding genomic divergence, even in scenarios involving multiple, interacting mechanisms of RI.

Isolating selective from non-selective forces using site frequency ratios

A new method is introduced for estimating the distribution of mutation fitness effects using site frequency spectra. Unlike previous methods, which make assumptions about non-selective factors, or that try to incorporate such factors into the underlying model, this new method mostly avoids non-selective effects by working with the ratios of counts of selected sites to neutral sites. An expression for the likelihood of a set of selected/neutral ratios is found by treating the ratio of two Poisson random variables as the ratio of two gaussian random variables. This approach also avoids the need to estimate the relative mutation rates of selected and neutral sites. Simulations over a wide range of demographic models, with linked selection effects show that the new SFRatios method performs well for statistical tests of selection, and it performs well for estimating the distribution of selection effects. Performance was better with weak selection models and for expansion and structured demographic models than for bottleneck models. Applications to two populations of Drosophila melanogaster reveal clear but very weak selection on synonymous sites. For nonsynonymous sites, selection was found to be consistent with previous estimates and stronger for an African population than for one from North Carolina.

Study design and the sampling of deleterious rare variants in biobank-scale datasets

One key component of study design in population genetics is the "geographic breadth" of a sample (i.e., how broad a region across which individuals are sampled). How the geographic breadth of a sample impacts observations of rare, deleterious variants is unclear, even though such variants are of particular interest for biomedical and evolutionary applications. Here, in order to gain insight into the effects of sample design on ascertained genetic variants, we formulate a stochastic model of dispersal, genetic drift, selection, mutation, and geographically concentrated sampling. We use this model to understand the effects of the geographic breadth of sampling effort on the discovery of negatively selected variants. We find that samples which are more geographically broad will discover a greater number variants as compared geographically narrow samples (an effect we label "discovery"); though the variants will be detected at lower average frequency than in narrow samples (e.g. as singletons, an effect we label "dilution"). Importantly, these effects are amplified for larger sample sizes and moderated by the magnitude of fitness effects. We validate these results using both population genetic simulations and empirical analyses in the UK Biobank. Our results are particularly important in two contexts: the association of large-effect rare variants with particular phenotypes and the inference of negative selection from allele frequency data. Overall, our findings emphasize the importance of considering geographic breadth when designing and carrying out genetic studies, especially at biobank scale.

Selfing Shapes Fixation of a Mutant Allele Under Flux Equilibrium

Sexual reproduction with alternative generations in a life cycle is an important feature in eukaryotic evolution. Partial selfing can regulate the efficacy of purging deleterious alleles in the gametophyte phase and the masking effect in heterozygotes in the sporophyte phase. Here, we develop a new theory to analyze how selfing shapes fixation of a mutant allele that is expressed in the gametophyte or the sporophyte phase only or in two phases. In an infinitely large population, we analyze a critical selfing rate beyond which the mutant allele tends to be fixed under equilibrium between irreversible mutation and selection effects. The critical selfing rate varies with genes expressed in alternative phases. In a finite population with partial self-fertilization, we apply Wright's method to calculate the fixation probability of the mutant allele under flux equilibrium among irreversible mutation, selection, and drift effects and compare it with the fixation probability derived from diffusion model under equilibrium between selection and drift effects. Selfing facilitates fixation of the deleterious allele expressed in the gametophyte phase only but impedes fixation of the deleterious allele expressed in the sporophyte phase only. Selfing facilitates or impedes fixation of the deleterious allele expressed in two phases, depending upon how phase variation in selection occurs in a life cycle. The overall results help to understand the adaptive strategy that sexual reproductive plant species evolve through the joint effects of partial selfing and alternative generations in a life cycle.

The genetic architecture of polygenic local adaptation and its role in shaping barriers to gene flow

We consider how the genetic architecture underlying locally adaptive traits determines the strength of a barrier to gene flow in a mainland-island model. Assuming a general life cycle, we derive an expression for the effective migration rate when local adaptation is due to genetic variation at many loci under directional selection on the island, allowing for arbitrary fitness and dominance effects across loci. We show how the effective migration rate can be combined with classical single-locus diffusion theory to accurately predict multilocus differentiation between the mainland and island at migration-selection-drift equilibrium and determine the migration rate beyond which local adaptation collapses, while accounting for genetic drift and weak linkage. Using our efficient numerical tools, we then present a detailed study of the effects of dominance on barriers to gene flow, showing that when total selection is sufficiently strong, more recessive local adaptation generates stronger barriers to gene flow. We then study how heterogeneous genetic architectures of local adaptation affect barriers to gene flow, characterizing adaptive differentiation at migration-selection balance for different distributions of fitness effects. We find that a more heterogeneous genetic architecture generally yields a stronger genome-wide barrier to gene flow and that the detailed genetic architecture underlying locally adaptive traits can have an important effect on observable differentiation when divergence is not too large. Lastly, we study the limits of our approach as loci become more tightly linked, showing that our predictions remain accurate over a large biologically relevant domain.

Disentangling the Factors Selecting for Unicellular Programmed Cell Death

AbstractThe widespread occurrence of genetically programmed cell death (PCD) in unicellular species poses an evolutionary puzzle. While kin selection theory predicts that the fitness benefits of cell suicide must be preferentially directed toward genetic relatives, it does not predict the nature of these benefits. Furthermore, cell suicide must be conditionally expressed, leaving open the question of what conditions optimally regulate expression. Here we formalize several verbal hypotheses for the ecological function of unicellular PCD. We show that self-sacrifice by healthy cells cannot evolve. Instead, PCD evolution requires that damaged cells sense impending death and then (1) expedite this death to spare resources for groupmates, (2) prepare cellular contents so that necrotic toxins are not released upon death, or initiate autolysis in order to (3) release beneficial compounds or (4) release anticompetitior toxins. The prerequisite ability to predict death is a severe cell biological constraint as well as an ecological constraint that restricts PCD evolution to species with specific sources of mortality. We show that the specific type of PCD that will evolve, though, differs on the basis of a species' ecology, life history, and genetic structure.

Did Dissociative Amnesia Evolve?

Dissociative amnesia is a diagnosis category that implies a proposed mechanism (often called dissociation) by which amnesia is caused by psychogenic means, such as trauma, and that amnesia is reversible later. Dissociative amnesia is listed in some of the most influential diagnostic manuals. Authors have noted the similarities in definition to repressed memories. Dissociative amnesia is a disputed category and phenomenon, and here I discuss the plausibility that this cognitive mechanism evolved. I discuss some general conditions by which cognitive functions will evolve, that is, the relatively continuous adaptive pressure by which a cognitive ability would clearly be adaptive if variation produced it. I discuss how adaptive gene mutations typically spread from one individual to the whole species. The article also discusses a few hypothetical scenarios and several types of trauma, to examine the likely adaptive benefits of blocking out memories of trauma, or not. I conclude that it is unlikely that dissociative amnesia evolved, and invite further development of these ideas and scenarios by others.

Tracking clonal evolution of drug resistance in ovarian cancer patients by exploiting structural variants in cfDNA

Drug resistance is the major cause of therapeutic failure in high-grade serous ovarian cancer (HGSOC). Yet, the mechanisms by which tumors evolve to drug resistant states remains largely unknown. To address this, we aimed to exploit clone-specific genomic structural variations by combining scaled single-cell whole genome sequencing with longitudinally collected cell-free DNA (cfDNA), enabling clonal tracking before, during and after treatment. We developed a cfDNA hybrid capture, deep sequencing approach based on leveraging clone-specific structural variants as endogenous barcodes, with orders of magnitude lower error rates than single nucleotide variants in ctDNA (circulating tumor DNA) detection, demonstrated on 19 patients at baseline. We then applied this to monitor and model clonal evolution over several years in ten HGSOC patients treated with systemic therapy from diagnosis through recurrence. We found drug resistance to be polyclonal in most cases, but frequently dominated by a single high-fitness and expanding clone, reducing clonal diversity in the relapsed disease state in most patients. Drug-resistant clones frequently displayed notable genomic features, including high-level amplifications of oncogenes such as CCNE1, RAB25, NOTCH3, and ERBB2. Using a population genetics Wright-Fisher model, we found evolutionary trajectories of these features were consistent with drug-induced positive selection. In select cases, these alterations impacted selection of secondary lines of therapy with positive patient outcomes. For cases with matched single-cell RNA sequencing data, pre-existing and genomically encoded phenotypic states such as upregulation of EMT and VEGF were linked to drug resistance. Together, our findings indicate that drug resistant states in HGSOC pre-exist at diagnosis and lead to dramatic clonal expansions that alter clonal composition at the time of relapse. We suggest that combining tumor single cell sequencing with cfDNA enables clonal tracking in patients and harbors potential for evolution-informed adaptive treatment decisions.

Latent mutations in the ancestries of alleles under selection

We consider a single genetic locus with two alleles A1 and A2 in a large haploid population. The locus is subject to selection and two-way, or recurrent, mutation. Assuming the allele frequencies follow a Wright-Fisher diffusion and have reached stationarity, we describe the asymptotic behaviors of the conditional gene genealogy and the latent mutations of a sample with known allele counts, when the count n1 of allele A1 is fixed, and when either or both the sample size n and the selection strength |α| tend to infinity. Our study extends previous work under neutrality to the case of non-neutral rare alleles, asserting that when selection is not too strong relative to the sample size, even if it is strongly positive or strongly negative in the usual sense (α→-∞ or α→+∞), the number of latent mutations of the n1 copies of allele A1 follows the same distribution as the number of alleles in the Ewens sampling formula. On the other hand, very strong positive selection relative to the sample size leads to neutral gene genealogies with a single ancient latent mutation. We also demonstrate robustness of our asymptotic results against changing population sizes, when one of |α| or n is large.

Limits to species' range: the tension between local and global adaptation

We know that heritable variation is abundant, and that selection causes all but the smallest populations to rapidly shift beyond their original trait distribution. So then, what limits the range of a species? There are physical constraints and also population genetic limits to the effectiveness of selection, ultimately set by population size. Global adaptation, where the same genotype is favoured over the whole range, is most efficient when based on a multitude of weakly selected alleles and is effective even when local demes are small, provided that there is some gene flow. In contrast, local adaptation is sensitive to gene flow and may require alleles with substantial effect. How can populations combine the advantages of large effective size with the ability to specialise into local niches? To what extent does reproductive isolation help resolve this tension? I address these questions using eco-evolutionary models of polygenic adaptation, contrasting discrete demes with continuousspace.

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.

The evolution of chemodiversity in plants-From verbal to quantitative models

Plants harbour a great chemodiversity, that is diversity of specialised metabolites (SMs), at different scales. For instance, individuals can produce a large number of SMs, and populations can differ in their metabolite composition. Given the ecological and economic importance of plant chemodiversity, it is important to understand how it arises and is maintained over evolutionary time. For other dimensions of biodiversity, that is species diversity and genetic diversity, quantitative models play an important role in addressing such questions. Here, we provide a synthesis of existing hypotheses and quantitative models, that is mathematical models and computer simulations, for the evolution of plant chemodiversity. We describe each model's ingredients, that is the biological processes that shape chemodiversity, the scales it considers and whether it has been formalized as a quantitative model. Although we identify several quantitative models, not all are dynamic and many influential models have remained verbal. To fill these gaps, we outline our vision for the future of chemodiversity modelling. We identify quantitative models used for genetic variation that may be adapted for chemodiversity, and we present a flexible framework for the creation of individual-based models that address different scales of chemodiversity and combine different ingredients that bring this chemodiversity about.

Analyses of allele age and fitness impact reveal human beneficial alleles to be older than neutral controls

A classic population genetic prediction is that alleles experiencing directional selection should swiftly traverse allele frequency space, leaving detectable reductions in genetic variation in linked regions. However, despite this expectation, identifying clear footprints of beneficial allele passage has proven to be surprisingly challenging. We addressed the basic premise underlying this expectation by estimating the ages of large numbers of beneficial and deleterious alleles in a human population genomic data set. Deleterious alleles were found to be young, on average, given their allele frequency. However, beneficial alleles were older on average than non-coding, non-regulatory alleles of the same frequency. This finding is not consistent with directional selection and instead indicates some type of balancing selection. Among derived beneficial alleles, those fixed in the population show higher local recombination rates than those still segregating, consistent with a model in which new beneficial alleles experience an initial period of balancing selection due to linkage disequilibrium with deleterious recessive alleles. Alleles that ultimately fix following a period of balancing selection will leave a modest 'soft' sweep impact on the local variation, consistent with the overall paucity of species-wide 'hard' sweeps in human genomes.

How development affects evolution

Natural selection acts on developmentally constructed phenotypes, but how does development affect evolution? This question prompts a simultaneous consideration of development and evolution. However, there has been a lack of general mathematical frameworks mechanistically integrating the two, which may have inhibited progress on the question. Here, we use a new mathematical framework that mechanistically integrates development into evolution to analyse how development affects evolution. We show that, while selection pushes genotypic and phenotypic evolution up the fitness landscape, development determines the admissible evolutionary pathway, such that evolutionary outcomes occur at path peaks rather than landscape peaks. Changes in development can generate path peaks, triggering genotypic or phenotypic diversification, even on constant, single-peak landscapes. Phenotypic plasticity, niche construction, extra-genetic inheritance, and developmental bias alter the evolutionary path and hence the outcome. Thus, extra-genetic inheritance can have permanent evolutionary effects by changing the developmental constraints, even if extra-genetically acquired elements are not transmitted to future generations. Selective development, whereby phenotype construction points in the adaptive direction, may induce adaptive or maladaptive evolution depending on the developmental constraints. Moreover, developmental propagation of phenotypic effects over age enables the evolution of negative senescence. Overall, we find that development plays a major evolutionary role.

Polygenic adaptation dynamics in large, finite populations

Although many phenotypic traits are determined by a large number of genetic variants, how a polygenic trait adapts in response to a change in the environment is not completely understood. In the framework of diffusion theory, we study the steady state and the adaptation dynamics of a large but finite population evolving under stabilizing selection and symmetric mutations when selection and mutation are moderately large. We find that in the stationary state, the allele frequency distribution at a locus is unimodal if its effect size is below a threshold effect and bimodal otherwise; these results are the stochastic analog of the deterministic ones where the stable allele frequency becomes bistable when the effect size exceeds a threshold. It is known that following a sudden shift in the phenotypic optimum, in an infinitely large population, selective sweeps at a large-effect locus are prevented and adaptation proceeds exclusively via subtle changes in the allele frequency; in contrast, we find that the chance of sweep is substantially enhanced in large, finite populations and the allele frequency at a large-effect locus can reach a high frequency at short times even for small shifts in the phenotypic optimum.

Reference genome of the color polymorphic desert annual plant sandblossoms, Linanthus parryae

Sandblossoms, Linanthus parryae is a widespread annual plant species found in washes and sandy open habitats across the Mojave Desert and Eastern Sierra Nevada of California. Studies in this species have played a central role in evolutionary biology, serving as the first test cases of the shifting balance theory of evolution, models of isolation by distance, and metrics to describe the genetic structure of natural populations. Despite the importance of L. parryae in the development of landscape genetics and phylogeography, there are no genomic resources available for the species. Through the California Conservation Genomics Project, we assembled the first genome in the genus Linanthus. Using PacBio HiFi long reads and Hi-C chromatin conformation capture, we assembled 123 scaffolds spanning 1.51 Gb of the 1.96 Gb estimated genome, with a contig N50 of 18.7 Mb and a scaffold N50 of 124.8 Mb. This assembly, with a BUSCO completeness score of 88.7%, will allow us to revisit foundational ideas central to our understanding of how evolutionary forces operate in a geographic landscape. In addition, it will be a new resource to uncover adaptations to arid environments in the fragile desert habitat threatened by urban and solar farm development, climate change, and off-road vehicles.

Identifying rare variants inconsistent with identity-by-descent in population-scale whole-genome sequencing data

Analyses of genetic variation typically assume that rare variants within a population are inherited from a single common ancestral event identity-by-descent (IBD). However, there are genetic and technical processes through which rare variants in population genetic data may deviate from this simple evolutionary model, including recurrent mutations, gene conversions and genotyping error. All these processes can decrease the expected length of shared background haplotype surrounding a rare variant if that variant was inherited from a single event descending from a common ancestor. No method exists to computationally infer rare variants inconsistent with this simple model-denoted here as 'IBD-inconsistent'-using unphased population sequencing data.We hypothesized that the difference in shared haplotype background length can distinguish variants consistent and inconsistent with this simple IBD transmission population sequencing data without pedigree information. We implemented a Bayesian hierarchical model and used Gibbs sampling to estimate the posterior probability of IBD state for rare variants, using simulated recurrent mutations to demonstrate that our approach accurately distinguishes rare variants consistent and inconsistent with a simple IBD inheritance model.Applying our method to whole-genome sequencing data from 3,621 human individuals in the UK10K consortium, we found that IBD-inconsistent variants correlated with higher local mutation rates and genomic features like replication timing. Using a heuristic to categorize IBD-inconsistent variants as gene conversions, we found that potential gene conversions had expected properties such as enriched local GC content.By identifying IBD-inconsistent variants, we can better understand the spectrum of recent mutations in human populations, a source of genetic variation driving evolution and a key factor in understanding recent demographic history.

Revisiting the number of self-incompatibility alleles in finite populations: From old models to new results

Under gametophytic self-incompatibility (GSI), plants are heterozygous at the self-incompatibility locus (S-locus) and can only be fertilized by pollen with a different allele at that locus. The last century has seen a heated debate about the correct way of modelling the allele diversity in a GSI population that was never formally resolved. Starting from an individual-based model, we derive the deterministic dynamics as proposed by Fisher (The genetical theory of natural selection - A complete, Variorum edition, Oxford University Press, 1958) and compute the stationary S-allele frequency distribution. We find that the stationary distribution proposed by Wright (Evolution, 18, 609, 1964) is close to our theoretical prediction, in line with earlier numerical confirmation. Additionally, we approximate the invasion probability of a new S-allele, which scales inversely with the number of resident S-alleles. Lastly, we use the stationary allele frequency distribution to estimate the population size of a plant population from an empirically obtained allele frequency spectrum, which complements the existing estimator of the number of S-alleles. Our expression of the stationary distribution resolves the long-standing debate about the correct approximation of the number of S-alleles and paves the way for new statistical developments for the estimation of the plant population size based on S-allele frequencies.

Reproductive isolation via polygenic local adaptation in sub-divided populations: Effect of linkage disequilibria and drift

This paper considers how polygenic local adaptation and reproductive isolation between hybridizing populations is influenced by linkage disequilibria (LD) between loci, in scenarios where both gene flow and genetic drift counteract selection. It shows that the combined effects of multi-locus LD and genetic drift on allele frequencies at selected loci and on heterozygosity at neutral loci are predicted accurately by incorporating (deterministic) effective migration rates into the diffusion approximation (for selected loci) and into the structured coalescent (for neutral loci). Theoretical approximations are tested against individual-based simulations and used to investigate conditions for the maintenance of local adaptation on an island subject to one-way migration from a differently adapted mainland, and in an infinite-island population with two habitats under divergent selection. The analysis clarifies the conditions under which LD between sets of locally deleterious alleles allows these to be collectively eliminated despite drift, causing sharper and (under certain conditions) shifted migration thresholds for loss of adaptation. Local adaptation also has counter-intuitive effects on neutral (relative) divergence: F_ST is highest for a pair of subpopulations belonging to the same (rare) habitat, despite the lack of reproductive isolation between them.

Environmental adaptation often involves spatially heterogeneous selection at many genetic loci. Thus, the evolutionary consequences of hybridisation between populations adapted to different environments depend on the coupled dynamics of multiple loci under selection, migration and genetic drift, making them challenging to predict. Here, I introduce theoretical approximations that accurately capture the effect of such coupling on allele frequencies at individual loci, while also accounting for the stochastic effects of genetic drift. I then use these approximations to study hybridisation in a metapopulation consisting of many interconnected subpopulations, where each subpopulation belongs to one of two habitats under divergent selection. The analysis clarifies how subpopulations belonging to a rare habitat can maintain local adaptation despite high levels of migration if net selection against multi-locus genotypes is stronger than a threshold which depends on the relative abundances of the two habitats. Further, local adaptation in a metapopulation can significantly elevate F_ST between subpopulations belonging to the same habitat, even though these are not reproductively isolated. These findings highlight the importance of carefully considering the genetic architecture and spatial context of divergence when interpreting patterns of genomic differentiation between speciating populations.

Surviving environmental change: when increasing population size can increase extinction risk

Populations threatened by an abrupt environmental change-due to rapid climate change, pathogens or invasive competitors-may survive if they possess or generate genetic combinations adapted to the novel, challenging condition. If these genotypes are initially rare or non-existent, the emergence of lineages that allow a declining population to survive is known as 'evolutionary rescue'. By contrast, the genotypes required for survival could, by chance, be common before the environmental change. Here, considering both of these possibilities, we find that the risk of extinction can be lower in very small or very large populations, but peaks at intermediate population sizes. This pattern occurs when the survival genotype has a small deleterious effect before the environmental change. Since mildly deleterious mutations constitute a large fraction of empirically measured fitness effects, we suggest that this unexpected result-an intermediate size that puts a population at a greater risk of extinction-may not be unusual in the face of environmental change.

Fisher's historic 1922 paper On the dominance ratio

R.A. Fisher's 1922 paper On the dominance ratio has a strong claim to be the foundation paper for modern population genetics. It greatly influenced subsequent work by Haldane and Wright, and contributed 3 major innovations to the study of evolution at the genetic level. First, the introduction of a general model of selection at a single locus, which showed how variability could be maintained by heterozygote advantage. Second, the use of the branching process approach to show that a beneficial mutation has a substantial chance of loss from the population, even when the population size is extremely large. Third, the invention of the concept of a probability distribution of allele frequency, caused by random sampling of allele frequencies due to finite population size, and the first use of a diffusion equation to investigate the properties of such a distribution. Although Fisher was motivated by an inference that later turned out to lack strong empirical support (a substantial contribution of dominance to quantitative trait variability), and his use of a diffusion equation was marred by a technical mistake, the paper introduced concepts and methods that pervade much subsequent work in population genetics.

Clonal fitness inferred from time-series modelling of single-cell cancer genomes

Progress in defining genomic fitness landscapes in cancer, especially those defined by copy number alterations (CNAs), has been impeded by lack of time-series single-cell sampling of polyclonal populations and temporal statistical models1-7. Here we generated 42,000 genomes from multi-year time-series single-cell whole-genome sequencing of breast epithelium and primary triple-negative breast cancer (TNBC) patient-derived xenografts (PDXs), revealing the nature of CNA-defined clonal fitness dynamics induced by TP53 mutation and cisplatin chemotherapy. Using a new Wright-Fisher population genetics model8,9 to infer clonal fitness, we found that TP53 mutation alters the fitness landscape, reproducibly distributing fitness over a larger number of clones associated with distinct CNAs. Furthermore, in TNBC PDX models with mutated TP53, inferred fitness coefficients from CNA-based genotypes accurately forecast experimentally enforced clonal competition dynamics. Drug treatment in three long-term serially passaged TNBC PDXs resulted in cisplatin-resistant clones emerging from low-fitness phylogenetic lineages in the untreated setting. Conversely, high-fitness clones from treatment-naive controls were eradicated, signalling an inversion of the fitness landscape. Finally, upon release of drug, selection pressure dynamics were reversed, indicating a fitness cost of treatment resistance. Together, our findings define clonal fitness linked to both CNA and therapeutic resistance in polyclonal tumours.

Inferring Parameters of the Distribution of Fitness Effects of New Mutations When Beneficial Mutations Are Strongly Advantageous and Rare

Characterizing the distribution of fitness effects (DFE) for new mutations is central in evolutionary genetics. Analysis of molecular data under the McDonald-Kreitman test has suggested that adaptive substitutions make a substantial contribution to between-species divergence. Methods have been proposed to estimate the parameters of the distribution of fitness effects for positively selected mutations from the unfolded site frequency spectrum (uSFS). Such methods perform well when beneficial mutations are mildly selected and frequent. However, when beneficial mutations are strongly selected and rare, they may make little contribution to standing variation and will thus be difficult to detect from the uSFS. In this study, I analyze uSFS data from simulated populations subject to advantageous mutations with effects on fitness ranging from mildly to strongly beneficial. As expected, frequent, mildly beneficial mutations contribute substantially to standing genetic variation and parameters are accurately recovered from the uSFS. However, when advantageous mutations are strongly selected and rare, there are very few segregating in populations at any one time. Fitting the uSFS in such cases leads to underestimates of the strength of positive selection and may lead researchers to false conclusions regarding the relative contribution adaptive mutations make to molecular evolution. Fortunately, the parameters for the distribution of fitness effects for harmful mutations are estimated with high accuracy and precision. The results from this study suggest that the parameters of positively selected mutations obtained by analysis of the uSFS should be treated with caution and that variability at linked sites should be used in conjunction with standing variability to estimate parameters of the distribution of fitness effects in the future.

Applicability of the Mutation-Selection Balance Model to Population Genetics of Heterozygous Protein-Truncating Variants in Humans

The fate of alleles in the human population is believed to be highly affected by the stochastic force of genetic drift. Estimation of the strength of natural selection in humans generally necessitates a careful modeling of drift including complex effects of the population history and structure. Protein-truncating variants (PTVs) are expected to evolve under strong purifying selection and to have a relatively high per-gene mutation rate. Thus, it is appealing to model the population genetics of PTVs under a simple deterministic mutation-selection balance, as has been proposed earlier (Cassa et al. 2017). Here, we investigated the limits of this approximation using both computer simulations and data-driven approaches. Our simulations rely on a model of demographic history estimated from 33,370 individual exomes of the Non-Finnish European subset of the ExAC data set (Lek et al. 2016). Additionally, we compared the African and European subset of the ExAC study and analyzed de novo PTVs. We show that the mutation-selection balance model is applicable to the majority of human genes, but not to genes under the weakest selection.

Extreme Polygenicity of Complex Traits Is Explained by Negative Selection

Complex traits and common diseases are extremely polygenic, their heritability spread across thousands of loci. One possible explanation is that thousands of genes and loci have similarly important biological effects when mutated. However, we hypothesize that for most complex traits, relatively few genes and loci are critical, and negative selection-purging large-effect mutations in these regions-leaves behind common-variant associations in thousands of less critical regions instead. We refer to this phenomenon as flattening. To quantify its effects, we introduce a mathematical definition of polygenicity, the effective number of independently associated SNPs (Me), which describes how evenly the heritability of a trait is spread across the genome. We developed a method, stratified LD fourth moments regression (S-LD4M), to estimate Me, validating that it produces robust estimates in simulations. Analyzing 33 complex traits (average N = 361k), we determined that heritability is spread ∼4× more evenly among common SNPs than among low-frequency SNPs. This difference, together with evolutionary modeling of new mutations, suggests that complex traits would be orders of magnitude less polygenic if not for the influence of negative selection. We also determined that heritability is spread more evenly within functionally important regions in proportion to their heritability enrichment; functionally important regions do not harbor common SNPs with greatly increased causal effect sizes, due to selective constraint. Our results suggest that for most complex traits, the genes and loci with the most critical biological effects often differ from those with the strongest common-variant associations.

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.

Measuring intolerance to mutation in human genetics

In numerous applications, from working with animal models to mapping the genetic basis of human disease susceptibility, knowing whether a single disrupting mutation in a gene is likely to be deleterious is useful. With this goal in mind, a number of measures have been developed to identify genes in which protein-truncating variants (PTVs), or other types of mutations, are absent or kept at very low frequency in large population samples-genes that appear 'intolerant' to mutation. One measure in particular, the probability of being loss-of-function intolerant (pLI), has been widely adopted. This measure was designed to classify genes into three categories, null, recessive and haploinsufficient, on the basis of the contrast between observed and expected numbers of PTVs. Such population-genetic approaches can be useful in many applications. As we clarify, however, they reflect the strength of selection acting on heterozygotes and not dominance or haploinsufficiency.

From Fixation Probabilities to d-player Games: An Inverse Problem in Evolutionary Dynamics

The probability that the frequency of a particular trait will eventually become unity, the so-called fixation probability, is a central issue in the study of population evolution. Its computation, once we are given a stochastic finite population model without mutations and a (possibly frequency dependent) fitness function, is straightforward and it can be done in several ways. Nevertheless, despite the fact that the fixation probability is an important macroscopic property of the population, its precise knowledge does not give any clear information about the interaction patterns among individuals in the population. Here we address the inverse problem: from a given fixation pattern and population size, we want to infer what is the game being played by the population. This is done by first exploiting the framework developed in Chalub and Souza (J Math Biol 75:1735-1774, 2017), which yields a fitness function that realises this fixation pattern in the Wright-Fisher model. This fitness function always exists, but it is not necessarily unique. Subsequently, we show that any such fitness function can be approximated, with arbitrary precision, using d-player game theory, provided d is large enough. The pay-off matrix that emerges naturally from the approximating game will provide useful information about the individual interaction structure that is not itself apparent in the fixation pattern. We present extensive numerical support for our conclusions.

Allelic frequency estimation in presence of uncertain priors

In this paper, we assume that allele frequencies are random variables and follow certain statistical distributions. However, specifying an appropriate informative prior distribution with specific hyperparameters seems to be a major issue. Assuming that prior information varies over some classes of priors, we develop the concept of robust Bayes estimation into the context of allele frequency estimation. We first assume that the region of interest is a single locus and the prior information is represented in terms of a class of Beta distributions, and present explicit forms of the resulting Bayes and robust Bayes estimators. We then extend our results to biallelic k-loci and multi-allelic k-loci cases within the region of interest. We perform a simulation study to measure performance of the proposed robust Bayes estimators against some Bayes estimators associated with specific hyperparameters. The simulations reflect satisfactory performance of the proposed robust Bayes estimators when there is no evidence implying the actual prior distribution.

Haploids, polymorphisms and fluctuating selection

I analyze the joint impact of directional and fluctuating selection with reversible mutation in finite bi-allelic haploid populations using diffusion approximations of the Moran and chemostat models. Results differ dramatically from those of the classic Wright-Fisher diffusion. There, a strong dispersive effect attributable to fluctuating selection dissipates nascent polymorphisms promoted by a relatively weak emergent frequency dependent selective effect. The dispersive effect in the Moran diffusion with fluctuations every birth-death event is trivial. The same frequency dependent selective effect now dominates and polymorphism is promoted. The dispersive effect in the chemostat diffusion with fluctuations every generation is identical to that in the Wright-Fisher diffusion. Nevertheless, polymorphism is again promoted because the emergent frequency dependent effect is doubled, an effect attributable to geometric reproduction within generations. Fluctuating selection in the Moran and chemostat diffusions can also promote bi-allelic polymorphisms when one allele confers a net benefit. Rapid fluctuations within generations are highly effective at promoting polymorphism in large populations. The bi-allelic distribution is approximately Gaussian but becomes uniform and then U-shaped as the frequency of environmental fluctuations decreases to once a generation and then once every multiple generations. Trade-offs (negative correlations in fitness) help promote polymorphisms but are not essential. In all three models the frequency dependent effect raises the probability of ultimate fixation of new alleles, but less effectively in the Wright-Fisher diffusion. Individual-based forward simulations confirm the calculations.

A Simple Test Identifies Selection on Complex Traits

Important traits in agricultural, natural, and human populations are increasingly being shown to be under the control of many genes that individually contribute only a small proportion of genetic variation. However, the majority of modern tools in quantitative and population genetics, including genome-wide association studies and selection-mapping protocols, are designed to identify individual genes with large effects. We have developed an approach to identify traits that have been under selection and are controlled by large numbers of loci. In contrast to existing methods, our technique uses additive-effects estimates from all available markers, and relates these estimates to allele-frequency change over time. Using this information, we generate a composite statistic, denoted [Formula: see text] which can be used to test for significant evidence of selection on a trait. Our test requires pre- and postselection genotypic data but only a single time point with phenotypic information. Simulations demonstrate that [Formula: see text] is powerful for identifying selection, particularly in situations where the trait being tested is controlled by many genes, which is precisely the scenario where classical approaches for selection mapping are least powerful. We apply this test to breeding populations of maize and chickens, where we demonstrate the successful identification of selection on traits that are documented to have been under selection.

Statistical Inference in the Wright-Fisher Model Using Allele Frequency Data

The Wright–Fisher model provides an elegant mathematical framework for understanding allele frequency data. In particular, the model can be used to infer the demographic history of species and identify loci under selection. A crucial quantity for inference under the Wright–Fisher model is the distribution of allele frequencies (DAF). Despite the apparent simplicity of the model, the calculation of the DAF is challenging. We review and discuss strategies for approximating the DAF, and how these are used in methods that perform inference from allele frequency data. Various evolutionary forces can be incorporated in the Wright–Fisher model, and we consider these in turn. We begin our review with the basic bi-allelic Wright–Fisher model where random genetic drift is the only evolutionary force. We then consider mutation, migration, and selection. In particular, we compare diffusion-based and moment-based methods in terms of accuracy, computational efficiency, and analytical tractability. We conclude with a brief overview of the multi-allelic process with a general mutation model.

Generalization of the Ewens sampling formula to arbitrary fitness landscapes

In considering evolution of transcribed regions, regulatory sequences, and other genomic loci, we are often faced with a situation in which the number of allelic states greatly exceeds the size of the population. In this limit, the population eventually adopts a steady state characterized by mutation-selection-drift balance. Although new alleles continue to be explored through mutation, the statistics of the population, and in particular the probabilities of seeing specific allelic configurations in samples taken from the population, do not change with time. In the absence of selection, the probabilities of allelic configurations are given by the Ewens sampling formula, widely used in population genetics to detect deviations from neutrality. Here we develop an extension of this formula to arbitrary fitness distributions. Although our approach is general, we focus on the class of fitness landscapes, inspired by recent high-throughput genotype-phenotype maps, in which alleles can be in several distinct phenotypic states. This class of landscapes yields sampling probabilities that are computationally more tractable and can form a basis for inference of selection signatures from genomic data. Using an efficient numerical implementation of the sampling probabilities, we demonstrate that, for a sizable range of mutation rates and selection coefficients, the steady-state allelic diversity is not neutral. Therefore, it may be used to infer selection coefficients, as well as other evolutionary parameters from population data. We also carry out numerical simulations to challenge various approximations involved in deriving our sampling formulas, such as the infinite-allele limit and the “full connectivity” assumption inherent in the Ewens theory, in which each allele can mutate into any other allele. We find that, at least for the specific numerical examples studied, our theory remains sufficiently accurate even if these assumptions are relaxed. Thus our framework establishes both theoretical and practical foundations for inferring selection signatures from population-level genomic sequence samples.

On the proportional abundance of species: Integrating population genetics and community ecology

The frequency of genes in interconnected populations and of species in interconnected communities are affected by similar processes, such as birth, death and immigration. The equilibrium distribution of gene frequencies in structured populations is known since the 1930s, under Wright’s metapopulation model known as the island model. The equivalent distribution for the species frequency (i.e. the species proportional abundance distribution (SPAD)), at the metacommunity level, however, is unknown. In this contribution, we develop a stochastic model to analytically account for this distribution (SPAD). We show that the same as for genes SPAD follows a beta distribution, which provides a good description of empirical data and applies across a continuum of scales. This stochastic model, based upon a diffusion approximation, provides an alternative to neutral models for the species abundance distribution (SAD), which focus on number of individuals instead of proportions, and demonstrate that the relative frequency of genes in local populations and of species within communities follow the same probability law. We hope our contribution will help stimulate the mathematical and conceptual integration of theories in genetics and ecology.

Union of phylogeography and landscape genetics

Phylogeography and landscape genetics have arisen within the past 30 y. Phylogeography is said to be the bridge between population genetics and systematics, and landscape genetics the bridge between landscape ecology and population genetics. Both fields can be considered as simply the amalgamation of classic biogeography with genetics and genomics; however, they differ in the temporal, spatial, and organismal scales addressed and the methodology used. I begin by briefly summarizing the history and purview of each field and suggest that, even though landscape genetics is a younger field (coined in 2003) than phylogeography (coined in 1987), early studies by Dobzhansky on the "microgeographic races" of Linanthus parryae in the Mojave Desert of California and Drosophila pseudoobscura across the western United States presaged the fields by over 40 y. Recent advances in theory, models, and methods have allowed researchers to better synthesize ecological and evolutionary processes in their quest to answer some of the most basic questions in biology. I highlight a few of these novel studies and emphasize three major areas ripe for investigation using spatially explicit genomic-scale data: the biogeography of speciation, lineage divergence and species delimitation, and understanding adaptation through time and space. Examples of areas in need of study are highlighted, and I end by advocating a union of phylogeography and landscape genetics under the more general field: biogeography.

Multi-population Genomic Relationships for Estimating Current Genetic Variances Within and Genetic Correlations Between Populations

Different methods are available to calculate multi-population genomic relationship matrices. Since those matrices differ in base population, it is anticipated that the method used to calculate genomic relationships affects the estimate of genetic variances, covariances, and correlations. The aim of this article is to define the multi-population genomic relationship matrix to estimate current genetic variances within and genetic correlations between populations. The genomic relationship matrix containing two populations consists of four blocks, one block for population 1, one block for population 2, and two blocks for relationships between the populations. It is known, based on literature, that by using current allele frequencies to calculate genomic relationships within a population, current genetic variances are estimated. In this article, we theoretically derived the properties of the genomic relationship matrix to estimate genetic correlations between populations and validated it using simulations. When the scaling factor of across-population genomic relationships is equal to the product of the square roots of the scaling factors for within-population genomic relationships, the genetic correlation is estimated unbiasedly even though estimated genetic variances do not necessarily refer to the current population. When this property is not met, the correlation based on estimated variances should be multiplied by a correction factor based on the scaling factors. In this study, we present a genomic relationship matrix which directly estimates current genetic variances as well as genetic correlations between populations.

The population genetics of human disease: The case of recessive, lethal mutations

Do the frequencies of disease mutations in human populations reflect a simple balance between mutation and purifying selection? What other factors shape the prevalence of disease mutations? To begin to answer these questions, we focused on one of the simplest cases: recessive mutations that alone cause lethal diseases or complete sterility. To this end, we generated a hand-curated set of 417 Mendelian mutations in 32 genes reported to cause a recessive, lethal Mendelian disease. We then considered analytic models of mutation-selection balance in infinite and finite populations of constant sizes and simulations of purifying selection in a more realistic demographic setting, and tested how well these models fit allele frequencies estimated from 33,370 individuals of European ancestry. In doing so, we distinguished between CpG transitions, which occur at a substantially elevated rate, and three other mutation types. Intriguingly, the observed frequency for CpG transitions is slightly higher than expectation but close, whereas the frequencies observed for the three other mutation types are an order of magnitude higher than expected, with a bigger deviation from expectation seen for less mutable types. This discrepancy is even larger when subtle fitness effects in heterozygotes or lethal compound heterozygotes are taken into account. In principle, higher than expected frequencies of disease mutations could be due to widespread errors in reporting causal variants, compensation by other mutations, or balancing selection. It is unclear why these factors would have a greater impact on disease mutations that occur at lower rates, however. We argue instead that the unexpectedly high frequency of disease mutations and the relationship to the mutation rate likely reflect an ascertainment bias: of all the mutations that cause recessive lethal diseases, those that by chance have reached higher frequencies are more likely to have been identified and thus to have been included in this study. Beyond the specific application, this study highlights the parameters likely to be important in shaping the frequencies of Mendelian disease alleles.

What determines the frequencies of disease mutations in human populations? To begin to answer this question, we focus on one of the simplest cases: mutations that cause completely recessive, lethal Mendelian diseases. We first review theory about what to expect from mutation and selection in a population of finite size and generate predictions based on simulations using a plausible demographic scenario of recent human evolution. For a highly mutable type of mutation, transitions at CpG sites, we find that the predictions are close to the observed frequencies of recessive lethal disease mutations. For less mutable types, however, predictions substantially under-estimate the observed frequency. We discuss possible explanations for the discrepancy and point to a complication that, to our knowledge, is not widely appreciated: that there exists ascertainment bias in disease mutation discovery. Specifically, we suggest that alleles that have been identified to date are likely the ones that by chance have reached higher frequencies and are thus more likely to have been mapped. More generally, our study highlights the factors that influence the frequencies of Mendelian disease alleles.

When Does Frequency-Independent Selection Maintain Genetic Variation?

Frequency-independent selection is generally considered as a force that acts to reduce the genetic variation in evolving populations, yet rigorous arguments for this idea are scarce. When selection fluctuates in time, it is unclear whether frequency-independent selection may maintain genetic polymorphism without invoking additional mechanisms. We show that constant frequency-independent selection with arbitrary epistasis on a well-mixed haploid population eliminates genetic variation if we assume linkage equilibrium between alleles. To this end, we introduce the notion of frequency-independent selection at the level of alleles, which is sufficient to prove our claim and contains the notion of frequency-independent selection on haploids. When selection and recombination are weak but of the same order, there may be strong linkage disequilibrium; numerical calculations show that stable equilibria are highly unlikely. Using the example of a diallelic two-locus model, we then demonstrate that frequency-independent selection that fluctuates in time can maintain stable polymorphism if linkage disequilibrium changes its sign periodically. We put our findings in the context of results from the existing literature and point out those scenarios in which the possible role of frequency-independent selection in maintaining genetic variation remains unclear.