The Distribution of Gene Frequencies in Populations

"The Theory was Beautiful Indeed": Rise, Fall and Circulation of Maximizing Methods in Population Genetics (1930-1980)

Describing the theoretical population geneticists of the 1960s, Joseph Felsenstein reminisced: "our central obsession was finding out what function evolution would try to maximize. Population geneticists used to think, following Sewall Wright, that mean relative fitness, W, would be maximized by natural selection" (Felsenstein 2000). The present paper describes the genesis, diffusion and fall of this "obsession", by giving a biography of the mean fitness function in population genetics. This modeling method devised by Sewall Wright in the 1930s found its heyday in the late 1950s and early 1960s, in the wake of Motoo Kimura's and Richard Lewontin's works. It seemed a reliable guide in the mathematical study of deterministic effects (the study of natural selection in populations of infinite size, with no drift), leading to powerful generalizations presenting law-like properties. Progress in population genetics theory, it then seemed, would come from the application of this method to the study of systems with several genes. This ambition came to a halt in the context of the influential objections made by the Australian mathematician Patrick Moran in 1963. These objections triggered a controversy between mathematically- and biologically-inclined geneticists, with affected both the formal standards and the aims of population genetics as a science. Over the course of the 1960s, the mean fitness method withered with the ambition of developing the deterministic theory. The mathematical theory became increasingly complex. Kimura re-focused his modeling work on the theory of random processes; as a result of his computer simulations, Lewontin became the staunchest critic of maximizing principles in evolutionary biology. The mean fitness method then migrated to other research areas, being refashioned and used in evolutionary quantitative genetics and behavioral ecology.

EVIDENCE FOR LIMITED INTERCONTINENTAL GENE FLOW IN THE COSMOPOLITAN MUSHROOM, SCHIZOPHYLLUM COMMUNE

The genetic structure of populations of Schizophyllum commune was inferred from electrophoretic variation among 136 individuals at 11 polymorphic allozyme loci to determine the extent of geographic differentiation in this widespread mushroom species. The majority of the genetic variation was contained within populations; however, considerable genetic differentiation was observed among populations (global G_ST = 0.214). Clustering analysis demonstrated that genetic distance was correlated with geographic distance and that a large component of the genetic variation was due to allele frequency differences among populations from the eastern and western hemispheres. Our results also suggest that populations are large and geographically widespread. The lack of fixed genetic differences among intercontinental populations at any of the allozyme loci suggests that long-distance spore dispersal may counter the effects of genetic drift in this cosmopolitan species. These results are contrasted with a previous description of the same collection, in which the mating allele distribution of the species displayed no population substructure at any geographic scale (Raper et al. 1958). Broader implications of this study are that both species and mating allele distributions may not be correlated with long-distance gene flow in basidiomycete fungi.

THE MAINTENANCE OF POLYGENIC VARIATION IN FINITE POPULATIONS

Models of the maintenance of genetic variance in a polygenic trait have usually assumed that population size is infinite and that selection is weak. Consequently, they will overestimate the amount of variation maintained in finite populations. I derive approximations for the equilibrium genetic variance, V^G in finite populations under weak stabilizing selection for triallelic loci and for an infinite "rare alleles" model. These are compared to results for neutral characters, to the "Gaussian allelic" model, and to Wright's approximation for a biallelic locus under arbitrary selection pressures. For a variety of parameter values, the three-allele, Gaussian, and Wrightian approximations all converge on the neutral model when population size is small. As expected, far less equilibrium genetic variance can be maintained if effective population size, N, is on the order of a few hundred than if N is infinite. All of the models predict that comparisons among populations with N less than about 10⁴ should show substantial differences in V^G. While it is easier to maintain absolute V^G when alleles interact to yield dominance or overdominance for fitness, less additivity also makes V^G more susceptible to differences in N. I argue that experimental data do not seem to reflect the predicted degree of relationship between N and V^G. This calls into question the ability of mutation-selection balance or simple balancing selection to explain observed V^G. The dependence of V^G on N could be used to test the adequacy of mutation-selection balance models.

Theories that narrate the world: Ronald A. Fisher's mass selection and Sewall Wright's shifting balance

Theories are composed of multiple interacting components. I argue that some theories have narratives as essential components, and that narratives function as integrative devices of the mathematical components of theories. Narratives represent complex processes unfolding in time as a sequence of stages, and hold the mathematical elements together as pieces in the investigation of a given process. I present two case studies from population genetics: R. A. Fisher's "mas selection" theory, and Sewall Wright's shifting balance theory. I apply my analysis to an early episode of the "R. A. Fisher - Sewall Wright controversy."

Differential influences of local subpopulations on regional diversity and differentiation for greater sage-grouse (Centrocercus urophasianus)

The distribution of spatial genetic variation across a region can shape evolutionary dynamics and impact population persistence. Local population dynamics and among-population dispersal rates are strong drivers of this spatial genetic variation, yet for many species we lack a clear understanding of how these population processes interact in space to shape within-species genetic variation. Here, we used extensive genetic and demographic data from 10 subpopulations of greater sage-grouse to parameterize a simulated approximate Bayesian computation (ABC) model and (i) test for regional differences in population density and dispersal rates for greater sage-grouse subpopulations in Wyoming, and (ii) quantify how these differences impact subpopulation regional influence on genetic variation. We found a close match between observed and simulated data under our parameterized model and strong variation in density and dispersal rates across Wyoming. Sensitivity analyses suggested that changes in dispersal (via landscape resistance) had a greater influence on regional differentiation, whereas changes in density had a greater influence on mean diversity across all subpopulations. Local subpopulations, however, varied in their regional influence on genetic variation. Decreases in the size and dispersal rates of central populations with low overall and net immigration (i.e. population sources) had the greatest negative impact on genetic variation. Overall, our results provide insight into the interactions among demography, dispersal and genetic variation and highlight the potential of ABC to disentangle the complexity of regional population dynamics and project the genetic impact of changing conditions.

Testing neutrality at copy-number-variable loci under the finite-allele and finite-site models

Copy-number variation (CNV) is an important form of DNA structural variation because a certain proportion of genomes in many eukaryotic species can contribute to such variations. Owing to the differences between CNVs and single nucleotide polymorphisms (SNPs) in size, mutation rate and maintaining mechanism, it is more realistic to characterize CNV evolution under the finite-allele and finite-site models. Here, we propose a method to test multiple CNVs neutrality under the finite-allele and finite-site models and the assumption of mutation-drift process. The statistical property of the method is evaluated through Monte Carlo simulations under the effects of the sample size, the scaled mutation rates, the number of CNVs, the population demographic change, and selection. Different from Tajima's D test, a bootstrap or a permutation approach is suggested to conduct a neutrality test. Application of this method is illustrated using the diploid CNV genotypes measured in discrete copy numbers in 11 HapMap phase III populations. The results show that the mutation-drift process can explain the variation of genome-wide CNVs among 1184 individuals (856 CNVs, ∼0.02Mb on average in size), irrespective of the historical demographic changes. Patterns from allele-frequency-spectrum analysis also support the hypothesis of neutral CNVs. Our results suggest that most human chromosomal changes in healthy individuals via unbalanced rearrangements of the segments with certain sizes are neutral.

Time reversal and age distributions, I. Discrete-time Markov chains

The connection between the age distribution of a discrete-time Markov chain and a certain time-reversed Markov chain is exhibited. A method for finding properties of age distributions follows simply from this approach. The results, which have application in several areas in applied probability, are illustrated by examples from population genetics.

Time reversal and age distributions, I. Discrete-time Markov chains

The connection between the age distribution of a discrete-time Markov chain and a certain time-reversed Markov chain is exhibited. A method for finding properties of age distributions follows simply from this approach. The results, which have application in several areas in applied probability, are illustrated by examples from population genetics.

Diffusion models in population genetics

Population genetics is that branch of genetics, whose object is the study of the genetical make-up of natural populations. By investigating the laws which govern the genetic structure of natural populations, we intend to clarify the mechanism of evolution.

Diffusion models in population genetics

Population genetics is that branch of genetics, whose object is the study of the genetical make-up of natural populations. By investigating the laws which govern the genetic structure of natural populations, we intend to clarify the mechanism of evolution.

A General Approximation for the Dynamics of Quantitative Traits

Selection, mutation, and random drift affect the dynamics of allele frequencies and consequently of quantitative traits. While the macroscopic dynamics of quantitative traits can be measured, the underlying allele frequencies are typically unobserved. Can we understand how the macroscopic observables evolve without following these microscopic processes? This problem has been studied previously by analogy with statistical mechanics: the allele frequency distribution at each time point is approximated by the stationary form, which maximizes entropy. We explore the limitations of this method when mutation is small (4Nμ < 1) so that populations are typically close to fixation, and we extend the theory in this regime to account for changes in mutation strength. We consider a single diallelic locus either under directional selection or with overdominance and then generalize to multiple unlinked biallelic loci with unequal effects. We find that the maximum-entropy approximation is remarkably accurate, even when mutation and selection change rapidly.

Speciation in peripheral populations: effects of drift load and mating systems

Speciation in peripheral populations has long been considered one of the most plausible scenarios for speciation with gene flow. In this study, however we identify two additional problems of peripatric speciation, as compared to the parapatric case, that may impede the completion of the speciation process for most parameter regions. First, with (predominantly) unidirectional gene flow, there is no selection pressure to evolve assortative mating on the continent. We discuss the implications of this for different mating schemes. Second, genetic load can build up in small populations. This can lead to extinction of the peripheral species, or generate selection pressure for lower assortative mating to avoid inbreeding. In this case, either a stable equilibrium with intermediate assortment evolves or there is cycling between phases of hybridization and phases of complete isolation.

Approximated prediction of genomic selection accuracy when reference and candidate populations are related

Background

Genomic selection is still to be evaluated and optimized in many species. Mathematical modeling of selection schemes prior to their implementation is a classical and useful tool for that purpose. These models include formalization of a number of entities including the precision of the estimated breeding value. To model genomic selection schemes, equations that predict this reliability as a function of factors such as the size of the reference population, its diversity, its genetic distance from the group of selection candidates genotyped, number of markers and strength of linkage disequilibrium are needed. The present paper aims at exploring new approximations of this reliability.

Results

Two alternative approximations are proposed for the estimation of the reliability of genomic estimated breeding values (GEBV) in the case of non-independence between candidate and reference populations. Both were derived from the Taylor series heuristic approach suggested by Goddard in 2009. A numerical exploration of their properties showed that the series were not equivalent in terms of convergence to the exact reliability, that the approximations may overestimate the precision of GEBV and that they converged towards their theoretical expectations. Formulae derived for these approximations were simple to handle in the case of independent markers. A few parameters that describe the markers’ genotypic variability (allele frequencies, linkage disequilibrium) can be estimated from genomic data corresponding to the population of interest or after making assumptions about their distribution. When markers are not in linkage equilibrium, replacing the real number of markers and QTL by the “effective number of independent loci”, as proposed earlier is a practical solution. In this paper, we considered an alternative, i.e. an “equivalent number of independent loci” which would give a GEBV reliability for unrelated individuals by considering a sub-set of independent markers that is identical to the reliability obtained by considering the full set of markers.

Conclusions

This paper is a further step towards the development of deterministic models that describe breeding plans based on the use of genomic information. Such deterministic models carry low computational burden, which allows design optimization through intensive numerical exploration.

Electronic supplementary material

The online version of this article (doi:10.1186/s12711-016-0183-3) contains supplementary material, which is available to authorized users.

αIIbβ3 variants defined by next-generation sequencing: predicting variants likely to cause Glanzmann thrombasthenia

Next-generation sequencing is transforming our understanding of human genetic variation but assessing the functional impact of novel variants presents challenges. We analyzed missense variants in the integrin αIIbβ3 receptor subunit genes ITGA2B and ITGB3 identified by whole-exome or -genome sequencing in the ThromboGenomics project, comprising ∼32,000 alleles from 16,108 individuals. We analyzed the results in comparison with 111 missense variants in these genes previously reported as being associated with Glanzmann thrombasthenia (GT), 20 associated with alloimmune thrombocytopenia, and 5 associated with aniso/macrothrombocytopenia. We identified 114 novel missense variants in ITGA2B (affecting ∼11% of the amino acids) and 68 novel missense variants in ITGB3 (affecting ∼9% of the amino acids). Of the variants, 96% had minor allele frequencies (MAF) < 0.1%, indicating their rarity. Based on sequence conservation, MAF, and location on a complete model of αIIbβ3, we selected three novel variants that affect amino acids previously associated with GT for expression in HEK293 cells. αIIb P176H and β3 C547G severely reduced αIIbβ3 expression, whereas αIIb P943A partially reduced αIIbβ3 expression and had no effect on fibrinogen binding. We used receiver operating characteristic curves of combined annotation-dependent depletion, Polyphen 2-HDIV, and sorting intolerant from tolerant to estimate the percentage of novel variants likely to be deleterious. At optimal cut-off values, which had 69-98% sensitivity in detecting GT mutations, between 27% and 71% of the novel αIIb or β3 missense variants were predicted to be deleterious. Our data have implications for understanding the evolutionary pressure on αIIbβ3 and highlight the challenges in predicting the clinical significance of novel missense variants.

Purifying selection, drift, and reversible mutation with arbitrarily high mutation rates

Some species exhibit very high levels of DNA sequence variability; there is also evidence for the existence of heritable epigenetic variants that experience state changes at a much higher rate than sequence variants. In both cases, the resulting high diversity levels within a population (hyperdiversity) mean that standard population genetics methods are not trustworthy. We analyze a population genetics model that incorporates purifying selection, reversible mutations, and genetic drift, assuming a stationary population size. We derive analytical results for both population parameters and sample statistics and discuss their implications for studies of natural genetic and epigenetic variation. In particular, we find that (1) many more intermediate-frequency variants are expected than under standard models, even with moderately strong purifying selection, and (2) rates of evolution under purifying selection may be close to, or even exceed, neutral rates. These findings are related to empirical studies of sequence and epigenetic variation.

EvoSNP-DB: A database of genetic diversity in East Asian populations

Genome-wide association studies (GWAS) have become popular as an approach for the identification of large numbers of phenotype-associated variants. However, differences in genetic architecture and environmental factors mean that the effect of variants can vary across populations. Understanding population genetic diversity is valuable for the investigation of possible population specific and independent effects of variants. EvoSNP-DB aims to provide information regarding genetic diversity among East Asian populations, including Chinese, Japanese, and Korean. Non-redundant SNPs (1.6 million) were genotyped in 54 Korean trios (162 samples) and were compared with 4 million SNPs from HapMap phase II populations. EvoSNP-DB provides two user interfaces for data query and visualization, and integrates scores of genetic diversity (Fst and VarLD) at the level of SNPs, genes, and chromosome regions. EvoSNP-DB is a web-based application that allows users to navigate and visualize measurements of population genetic differences in an interactive manner, and is available online at [http://biomi.cdc.go.kr/EvoSNP/]. [BMB Reports 2013; 46(8): 416-421]

Allelic diversity and its implications for the rate of adaptation

Genetic variation is usually estimated empirically from statistics based on population gene frequencies, but alternative statistics based on allelic diversity (number of allelic types) can provide complementary information. There is a lack of knowledge, however, on the evolutionary implications attached to allelic-diversity measures, particularly in structured populations. In this article we simulated multiple scenarios of single and structured populations in which a quantitative trait subject to stabilizing selection is adapted to different fitness optima. By forcing a global change in the optima we evaluated which diversity variables are more strongly correlated with both short- and long-term adaptation to the new optima. We found that quantitative genetic variance components for the trait and gene-frequency-diversity measures are generally more strongly correlated with short-term response to selection, whereas allelic-diversity measures are more correlated with long-term and total response to selection. Thus, allelic-diversity variables are better predictors of long-term adaptation than gene-frequency variables. This observation is also extended to unlinked neutral markers as a result of the information they convey on the demographic population history. Diffusion approximations for the allelic-diversity measures in a finite island model under the infinite-allele neutral mutation model are also provided.

Allelic diversity and its implications for the rate of adaptation

Genetic variation is usually estimated empirically from statistics based on population gene frequencies, but alternative statistics based on allelic diversity (number of allelic types) can provide complementary information. There is a lack of knowledge, however, on the evolutionary implications attached to allelic-diversity measures, particularly in structured populations. In this article we simulated multiple scenarios of single and structured populations in which a quantitative trait subject to stabilizing selection is adapted to different fitness optima. By forcing a global change in the optima we evaluated which diversity variables are more strongly correlated with both short- and long-term adaptation to the new optima. We found that quantitative genetic variance components for the trait and gene-frequency-diversity measures are generally more strongly correlated with short-term response to selection, whereas allelic-diversity measures are more correlated with long-term and total response to selection. Thus, allelic-diversity variables are better predictors of long-term adaptation than gene-frequency variables. This observation is also extended to unlinked neutral markers as a result of the information they convey on the demographic population history. Diffusion approximations for the allelic-diversity measures in a finite island model under the infinite-allele neutral mutation model are also provided.

Hamiltonian inclusive fitness: a fitter fitness concept

In 1963-1964 W. D. Hamilton introduced the concept of inclusive fitness, the only significant elaboration of Darwinian fitness since the nineteenth century. I discuss the origin of the modern fitness concept, providing context for Hamilton's discovery of inclusive fitness in relation to the puzzle of altruism. While fitness conceptually originates with Darwin, the term itself stems from Spencer and crystallized quantitatively in the early twentieth century. Hamiltonian inclusive fitness, with Price's reformulation, provided the solution to Darwin's 'special difficulty'-the evolution of caste polymorphism and sterility in social insects. Hamilton further explored the roles of inclusive fitness and reciprocation to tackle Darwin's other difficulty, the evolution of human altruism. The heuristically powerful inclusive fitness concept ramified over the past 50 years: the number and diversity of 'offspring ideas' that it has engendered render it a fitter fitness concept, one that Darwin would have appreciated.

Fitness landscape for nucleosome positioning

Histone-DNA complexes, so-called nucleosomes, are the building blocks of DNA packaging in eukaryotic cells. The histone-binding affinity of a local DNA segment depends on its elastic properties and determines its accessibility within the nucleus, which plays an important role in the regulation of gene expression. Here, we derive a fitness landscape for intergenic DNA segments in yeast as a function of two molecular phenotypes: their elasticity-dependent histone affinity and their coverage with transcription factor binding sites. This landscape reveals substantial selection against nucleosome formation over a wide range of both phenotypes. We use it as the core component of a quantitative evolutionary model for intergenic DNA segments. This model consistently predicts the observed diversity of histone affinities within wild Saccharomyces paradoxus populations, as well as the affinity divergence between neighboring Saccharomyces species. Our analysis establishes histone binding and transcription factor binding as two separable modes of sequence evolution, each of which is a direct target of natural selection.

Population genetics of gene function

This paper shows that differentiating the lifetimes of two phenotypes independently from their fertility can lead to a qualitative change in the equilibrium of a population: since survival and reproduction are distinct functional aspects of an organism, this observation contributes to extend the population-genetical characterisation of biological function. To support this statement a mathematical relation is derived to link the lifetime ratio T₁/T₂, which parameterizes the different survival ability of two phenotypes, with population variables that quantify the amount of neutral variation underlying a population's phenotypic distribution.

The frequency-dependent Wright-Fisher model: diffusive and non-diffusive approximations

We study a class of processes that are akin to the Wright-Fisher model, with transition probabilities weighted in terms of the frequency-dependent fitness of the population types. By considering an approximate weak formulation of the discrete problem, we are able to derive a corresponding continuous weak formulation for the probability density. Therefore, we obtain a family of partial differential equations for the evolution of the probability density, and which will be an approximation of the discrete process in the joint large population, small time-steps and weak selection limit. If the fitness functions are sufficiently regular, we can recast the weak formulation in a more standard formulation, without any boundary conditions, but supplemented by a number of conservation laws. The equations in this family can be purely diffusive, purely hyperbolic or of convection-diffusion type, with frequency dependent convection. The particular outcome will depend on the assumed scalings. The diffusive equations are of the degenerate type; using a duality approach, we also obtain a frequency dependent version of the Kimura equation without any further assumptions. We also show that the convective approximation is related to the replicator dynamics and provide some estimate of how accurate is the convective approximation, with respect to the convective-diffusion approximation. In particular, we show that the mode, but not the expected value, of the probability distribution is modelled by the replicator dynamics. Some numerical simulations that illustrate the results are also presented.

Effects of genetic drift and gene flow on the selective maintenance of genetic variation

Explanations for the genetic variation ubiquitous in natural populations are often classified by the population-genetic processes they emphasize: natural selection or mutation and genetic drift. Here we investigate models that incorporate all three processes in a spatially structured population, using what we call a construction approach, simulating finite populations under selection that are bombarded with a steady stream of novel mutations. As expected, the amount of genetic variation compared to previous models that ignored the stochastic effects of drift was reduced, especially for smaller populations and when spatial structure was most profound. By contrast, however, for higher levels of gene flow and larger population sizes, the amount of genetic variation found after many generations was greater than that in simulations without drift. This increased amount of genetic variation is due to the introduction of slightly deleterious alleles by genetic drift and this process is more efficient when migration load is higher. The incorporation of genetic drift also selects for fitness sets that exhibit allele-frequency equilibria with larger domains of attraction: they are "more stable." Moreover, the finiteness of populations strongly influences levels of local adaptation, selection strength, and the proportion of allele-frequency vectors that can be distinguished from the neutral expectation.

On measures of association among genetic variables

Systems involving many variables are important in population and quantitative genetics, for example, in multi-trait prediction of breeding values and in exploration of multi-locus associations. We studied departures of the joint distribution of sets of genetic variables from independence. New measures of association based on notions of statistical distance between distributions are presented. These are more general than correlations, which are pairwise measures, and lack a clear interpretation beyond the bivariate normal distribution. Our measures are based on logarithmic (Kullback-Leibler) and on relative ‘distances’ between distributions. Indexes of association are developed and illustrated for quantitative genetics settings in which the joint distribution of the variables is either multivariate normal or multivariate-t, and we show how the indexes can be used to study linkage disequilibrium in a two-locus system with multiple alleles and present applications to systems of correlated beta distributions. Two multivariate beta and multivariate beta-binomial processes are examined, and new distributions are introduced: the GMS-Sarmanov multivariate beta and its beta-binomial counterpart.

Microsatellites as targets of natural selection

The ability to survey polymorphism on a genomic scale has enabled genome-wide scans for the targets of natural selection. Theory that connects patterns of genetic variation to evidence of natural selection most often assumes a diallelic locus and no recurrent mutation. Although these assumptions are suitable to selection that targets single nucleotide variants, fundamentally different types of mutation generate abundant polymorphism in genomes. Moreover, recent empirical results suggest that mutationally complex, multiallelic loci including microsatellites and copy number variants are sometimes targeted by natural selection. Given their abundance, the lack of inference methods tailored to the mutational peculiarities of these types of loci represents a notable gap in our ability to interrogate genomes for signatures of natural selection. Previous theoretical investigations of mutation-selection balance at multiallelic loci include assumptions that limit their application to inference from empirical data. Focusing on microsatellites, we assess the dynamics and population-level consequences of selection targeting mutationally complex variants. We develop general models of a multiallelic fitness surface, a realistic model of microsatellite mutation, and an efficient simulation algorithm. Using these tools, we explore mutation-selection-drift equilibrium at microsatellites and investigate the mutational history and selective regime of the microsatellite that causes Friedreich's ataxia. We characterize microsatellite selective events by their duration and cost, note similarities to sweeps from standing point variation, and conclude that it is premature to label microsatellites as ubiquitous agents of efficient adaptive change. Together, our models and simulation algorithm provide a powerful framework for statistical inference, which can be used to test the neutrality of microsatellites and other multiallelic variants.

Purely competitive evolutionary dynamics for games

We introduce and analyze a purely competitive dynamics for the evolution of an infinite population subject to a three-strategy game. We argue that this dynamics represents a characterization of how certain systems, both natural and artificial, are governed. In each period, the population is randomly sorted into pairs, which engage in a once-off play of the game; the probability that a member propagates its type to its offspring is proportional only to its payoff within the pair. We show that if a type is dominant (obtains higher payoffs in games with both other types), its "pure" population state, comprising only members of that type, is globally attracting. If there is no dominant type, there is an unstable "mixed" fixed point; the population state eventually oscillates between the three near-pure states. We then allow for mutations, where offspring have a nonzero probability of randomly changing their type. In this case, the existence of a dominant type renders a point near its pure state globally attracting. If no dominant type exists, a supercritical Hopf bifurcation occurs at the unique mixed fixed point, and above a critical (typically low) mutation rate, this fixed point becomes globally attracting: the implication is that even very low mutation rates can stabilize a system that would, in the absence of mutations, be unstable.

A general population genetic framework for antagonistic selection that accounts for demography and recurrent mutation

Antagonistic selection--where alleles at a locus have opposing effects on male and female fitness ("sexual antagonism") or between components of fitness ("antagonistic pleiotropy")--might play an important role in maintaining population genetic variation and in driving phylogenetic and genomic patterns of sexual dimorphism and life-history evolution. While prior theory has thoroughly characterized the conditions necessary for antagonistic balancing selection to operate, we currently know little about the evolutionary interactions between antagonistic selection, recurrent mutation, and genetic drift, which should collectively shape empirical patterns of genetic variation. To fill this void, we developed and analyzed a series of population genetic models that simultaneously incorporate these processes. Our models identify two general properties of antagonistically selected loci. First, antagonistic selection inflates heterozygosity and fitness variance across a broad parameter range--a result that applies to alleles maintained by balancing selection and by recurrent mutation. Second, effective population size and genetic drift profoundly affect the statistical frequency distributions of antagonistically selected alleles. The "efficacy" of antagonistic selection (i.e., its tendency to dominate over genetic drift) is extremely weak relative to classical models, such as directional selection and overdominance. Alleles meeting traditional criteria for strong selection (N(e)s >> 1, where N(e) is the effective population size, and s is a selection coefficient for a given sex or fitness component) may nevertheless evolve as if neutral. The effects of mutation and demography may generate population differences in overall levels of antagonistic fitness variation, as well as molecular population genetic signatures of balancing selection.

A general definition of the heritable variation that determines the potential of a population to respond to selection

Genetic selection is a major force shaping life on earth. In classical genetic theory, response to selection is the product of the strength of selection and the additive genetic variance in a trait. The additive genetic variance reflects a population's intrinsic potential to respond to selection. The ordinary additive genetic variance, however, ignores the social organization of life. With social interactions among individuals, individual trait values may depend on genes in others, a phenomenon known as indirect genetic effects. Models accounting for indirect genetic effects, however, lack a general definition of heritable variation. Here I propose a general definition of the heritable variation that determines the potential of a population to respond to selection. This generalizes the concept of heritable variance to any inheritance model and level of organization. The result shows that heritable variance determining potential response to selection is the variance among individuals in the heritable quantity that determines the population mean trait value, rather than the usual additive genetic component of phenotypic variance. It follows, therefore, that heritable variance may exceed phenotypic variance among individuals, which is impossible in classical theory. This work also provides a measure of the utilization of heritable variation for response to selection and integrates two well-known models of maternal genetic effects. The result shows that relatedness between the focal individual and the individuals affecting its fitness is a key determinant of the utilization of heritable variance for response to selection.

Likelihood-based inferences under isolation by distance: two-dimensional habitats and confidence intervals

Likelihood-based methods of inference of population parameters from genetic data in structured populations have been implemented but still little tested in large networks of populations. In this work, a previous software implementation of inference in linear habitats is extended to two-dimensional habitats, and the coverage properties of confidence intervals are analyzed in both cases. Both standard likelihood and an efficient approximation are considered. The effects of misspecification of mutation model and dispersal distribution, and of spatial binning of samples, are considered. In the absence of model misspecification, the estimators have low bias, low mean square error, and the coverage properties of confidence intervals are consistent with theoretical expectations. Inferences of dispersal parameters and of the mutation rate are sensitive to misspecification or to approximations inherent to the coalescent algorithms used. In particular, coalescent approximations are not appropriate to infer the shape of the dispersal distribution. However, inferences of the neighborhood parameter (or of the product of population density and mean square dispersal rate) are generally robust with respect to complicating factors, such as misspecification of the mutation process and of the shape of the dispersal distribution, and with respect to spatial binning of samples. Likelihood inferences appear feasible in moderately sized networks of populations (up to 400 populations in this work), and they are more efficient than previous moment-based spatial regression method in realistic conditions.

The statistical mechanics of a polygenic character under stabilizing selection, mutation and drift

By exploiting an analogy between population genetics and statistical mechanics, we study the evolution of a polygenic trait under stabilizing selection, mutation and genetic drift. This requires us to track only four macroscopic variables, instead of the distribution of all the allele frequencies that influence the trait. These macroscopic variables are the expectations of: the trait mean and its square, the genetic variance, and of a measure of heterozygosity, and are derived from a generating function that is in turn derived by maximizing an entropy measure. These four macroscopics are enough to accurately describe the dynamics of the trait mean and of its genetic variance (and in principle of any other quantity). Unlike previous approaches that were based on an infinite series of moments or cumulants, which had to be truncated arbitrarily, our calculations provide a well-defined approximation procedure. We apply the framework to abrupt and gradual changes in the optimum, as well as to changes in the strength of stabilizing selection. Our approximations are surprisingly accurate, even for systems with as few as five loci. We find that when the effects of drift are included, the expected genetic variance is hardly altered by directional selection, even though it fluctuates in any particular instance. We also find hysteresis, showing that even after averaging over the microscopic variables, the macroscopic trajectories retain a memory of the underlying genetic states.

Heterozygosity-fitness correlations among wild populations of European tree frog (Hyla arborea) detect fixation load

Quantifying the impacts of inbreeding and genetic drift on fitness traits in fragmented populations is becoming a major goal in conservation biology. Such impacts occur at different levels and involve different sets of loci. Genetic drift randomly fixes slightly deleterious alleles leading to different fixation load among populations. By contrast, inbreeding depression arises from highly deleterious alleles in segregation within a population and creates variation among individuals. A popular approach is to measure correlations between molecular variation and phenotypic performances. This approach has been mainly used at the individual level to detect inbreeding depression within populations and sometimes at the population level but without consideration about the genetic processes measured. For the first time, we used in this study a molecular approach considering both the interpopulation and intrapopulation level to discriminate the relative importance of inbreeding depression vs. fixation load in isolated and non-fragmented populations of European tree frog (Hyla arborea), complemented with interpopulational crosses. We demonstrated that the positive correlations observed between genetic heterozygosity and larval performances on merged data were mainly caused by co-variations in genetic diversity and fixation load among populations rather than by inbreeding depression and segregating deleterious alleles within populations. Such a method is highly relevant in a conservation perspective because, depending on how populations lose fitness (inbreeding vs. fixation load), specific management actions may be designed to improve the persistence of populations.

An analytical contrast between fitness maximization and selection for mixability

It has been recently shown numerically that sex enables selection for alleles that perform well across different genetic contexts, i.e., selection for mixability. Here we capture this result analytically in a simple case. This serves a dual purpose. First, it provides a clear distinction between fitness maximization and selection for mixability. Second, it points out a limitation of the traditional analytical approach as applied to mixability.

Surprising fitness consequences of GC-biased gene conversion. II. Heterosis

Heterosis is a widespread phenomenon corresponding to the increase in fitness following crosses between individuals from different populations or lines relative to their parents. Its genetic basis has been a topic of controversy since the early 20th century. The masking of recessive deleterious mutations in hybrids likely explains a substantial part of heterosis. The dynamics and consequences of these mutations have thus been studied in depth. Recently, it was suggested that GC-biased gene conversion (gBGC) might strongly affect the fate of deleterious mutations and may have significant fitness consequences. gBGC is a recombination-associated process mimicking selection in favor of G and C alleles, which can interfere with selection, for instance by increasing the frequency of GC deleterious mutations. I investigated how gBGC could affect the amount and genetic structure of heterosis through an analysis of the interaction between gBGC and selection in subdivided populations. To do so, I analyzed the infinite island model both by numerical computations and by analytical approximations. I showed that gBGC might have little impact on the total amount of heterosis but could greatly affect its genetic basis.

Pacific salmon and the coalescent effective population size

Pacific salmon include several species that are both commercially important and endangered. Understanding the causes of loss in genetic variation is essential for designing better conservation strategies. Here we use a coalescent approach to analyze a model of the complex life history of salmon, and derive the coalescent effective population (CES). With the aid of Kronecker products and a convergence theorem for Markov chains with two time scales, we derive a simple formula for the CES and thereby establish its existence. Our results may be used to address important questions regarding salmon biology, in particular about the loss of genetic variation. To illustrate the utility of our approach, we consider the effects of fluctuations in population size over time. Our analysis enables the application of several tools of coalescent theory to the case of salmon.