Background selection and population differentiation

A general analytical formula is derived, which predicts the effects of background selection on population differentiation at a neutral locus as a result of its linkage with selected loci of deleterious mutations. The theory is based on the assumptions of random mating, multiplicative fitness, and weak selection in hermaphrodite plants in the island model of population structure. The analytical results show that Fst at the neutral locus increases as a result of the effects of background selection, regardless of the dependence or independence among linked background selective loci. The increment in Fst is closely related to the magnitude of linkage disequilibria between the neutral locus and selected loci, and can be estimated by the ratio of Fst with background selection to Fst without background selection minus one. The steady-state linkage disequilibrium between a neutral locus and a selected locus in subpopulations, primarily attained by gene flow, decreases with the recombination rate, and can be enhanced when there are dependence among linked selected loci. Monte Carlo computer simulations with two- and three-locus models show that the analytical formulae perform well under general conditions. Application of the present theory may aid in analyzing the genome-wide mapping of the effect of background selection in terms of Fst.     

Selection and segregation distortion in a sex-differentiated population

We extend the classical model for selection at an autosomal locus in a sex-differentiated population to include segregation distortion. The equations remain the same, but the fitness parameters are interpreted differently and refer to alleles instead of genotypes. We derive conditions for internal and external stability of the equilibria, i.e., stability with respect to perturbations of alleles that are already present at equilibrium and stability with respect to invasion attempts by newly arising alleles. We show that, in a sex-differentiated population, external stability of an equilibrium can be judged on the basis of Shaw--Mohler criteria. Throughout, we compare the situation in populations with and without sex differentiation. Interestingly, internal stability is more difficult to achieve in a population without sex differentiation than in a population in which selection and segregation distortion are restricted to one sex. In a companion paper we show how the general results of the present paper can lead to new insights into specific systems such as the t complex of the house mouse.     

Natural selection and density-dependent population growth

Natural selection was studied in the context of density-dependent population growth using a single locus, continuous time model for the rates of change of population size and allele frequency. The maximization principle of density-dependent selection was applied to a class of fitness expressions with explicit recruitment and mortality terms. Three general results were obtained: First, at low population densities, the genetic basis of selection is the difference between the mean recruitment rate and the mean mortality rate. Second, at densities much higher than the equilibrium population size, selection is expected to act to minimize the mean mortality rate. Third, as the population approaches its equilibrium density, selection is predicted to maximize the ratio of the mean recruitment rate to the mean mortality rate.     

Antagonistic selection factors induce a continuous population divergence in a polymorphism

Understanding the relative importance of selection and stochastic factors in population divergence of adaptive traits is a classical topic in evolutionary biology. However, it is difficult to separate these factors and detect the effects of selection when two or more contrasting selective factors are simultaneously acting on a single locus. In the damselfly Ischnura senegalensis, females exhibit color dimorphism and morph frequencies change geographically. We here evaluated the role of selection and stochastic factors in population divergence of morph frequencies by comparing the divergences in color locus and neutral loci. Comparisons between population pairwise F_ST for neutral loci and for the color locus did not detect any stochastic factors affecting color locus. Although comparison between population divergence in color and neutral loci using all populations detected only divergent selection, we detected two antagonistic selective factors acting on the color locus, that is, balancing and divergent selection, when considering geographical distance between populations. Our results suggest that a combination of two antagonistic selective factors, rather than stochastic factors, establishes the geographic cline in morph frequency in this system.     

Perfect simulation from population genetic models with selection

We consider using the ancestral selection graph (ASG) to simulate samples from population genetic models with selection. Currently the use of the ASG to simulate samples is limited. This is because the computational requirement for simulating samples increases exponentially with the selection rate and also due to needing to simulate a sample of size one from the population at equilibrium. For the only case where the distribution of a sample of size one is known, that of parent-independent mutations, more efficient simulation algorithms exist. We will show that by applying the idea of coupling from the past to the ASG, samples can be simulated from a general K-allele model without knowledge of the distribution of a sample of size one. Furthermore, the computation involved in generating such samples appears to be less than that of simulating the ASG until its ultimate ancestor. In particular, in the case of genic selection with parent-independent mutations, the computational requirement increases only quadratically with the selection rate. The algorithm is demonstrated by simulating samples at a microsatellite locus.     

Hitchhiking effect of a beneficial mutation spreading in a subdivided population

A central problem in population genetics is to detect and analyze positive natural selection by which beneficial mutations are driven to fixation. The hitchhiking effect of a rapidly spreading beneficial mutation, which results in local removal of standing genetic variation, allows such an analysis using DNA sequence polymorphism. However, the current mathematical theory that predicts the pattern of genetic hitchhiking relies on the assumption that a beneficial mutation increases to a high frequency in a single random-mating population, which is certainly violated in reality. Individuals in natural populations are distributed over a geographic space. The spread of a beneficial allele can be delayed by limited migration of individuals over the space and its hitchhiking effect can also be affected. To study this effect of geographic structure on genetic hitchhiking, we analyze a simple model of directional selection in a subdivided population. In contrast to previous studies on hitchhiking in subdivided populations, we mainly investigate the range of sufficiently high migration rates that would homogenize genetic variation at neutral loci. We provide a heuristic mathematical analysis that describes how the genealogical structure at a neutral locus linked to the locus under selection is expected to change in a population divided into two demes. Our results indicate that the overall strength of genetic hitchhiking--the degree to which expected heterozygosity decreases--is diminished by population subdivision, mainly because opportunity for the breakdown of hitchhiking by recombination increases as the spread of the beneficial mutation across demes is delayed when migration rate is much smaller than the strength of selection. Furthermore, the amount of genetic variation after a selective sweep is expected to be unequal over demes: a greater reduction in expected heterozygosity occurs in the subpopulation from which the beneficial mutation originates than in its neighboring subpopulations. This raises a possibility of detecting a "hidden" geographic structure of population by carefully analyzing the pattern of a selective sweep.     

The effect of hitch-hiking on genes linked to a balanced polymorphism in a subdivided population

The effect of multi-allelic balancing selection on nucleotide diversity at linked neutral sites was investigated by simulations of subdivided populations. The motivation is to understand the behaviour of self-recognition systems such as the MHC and plant self-incompatibility. For neutral sites, two types of subdivision are present: (1) into demes (connected by migration), and (2) into classes defined by different functional alleles at the selected locus (connected by recombination). Previous theoretical studies of each type of subdivision separately have shown that each increases diversity, and decreases the relative frequencies of low-frequency variants, at neutral sites or loci. We show here that the two types of subdivision act non-additively when sampling is at the whole population level, and that subdivision produces some non-intuitive results. For instance, in highly subdivided populations, genetic diversity at neutral sites may decrease with tighter linkage to a selected locus or site. Another conclusion is that, if there is population subdivision, balancing selection leads to decreased expected FST values for neutral sites linked to the selected locus. Finally, we show that the ability to detect balancing selection by its effects on linked variation, using tests such as Tajima's D, is reduced when genes in a subdivided population are sampled from the total population, rather than within demes.     

The role of population size in molecular evolution.

The results of a computer simulation study of the role of population size in population genetical models of molecular evolution are presented. If the mutation rate and strength of selection are held fixed and the population size increased, the eight models examined fall into three domains based on their rates of substitution. In the Ohta domain, the rate of substitution decreases with increasing population size; in the Kimura domain, the rate of substitution remains close to the mutation rate; in the Darwin domain, the rate of substitution increases without bound. In the Kimura and Darwin domains, the rate of substitution is much less sensitive to the population size than suggested by two-allele theories. Remarkably, the overdominance model converges to the neutral model with increasing N. The variation at a neutral locus linked to a selected locus is found to be insensitive to the population size for certain models of selection. A selected locus can actually cause the rate of substitution of deleterious alleles at a linked locus to increase with increasing population size. These unexpected results illustrate that intuition based on two-allele theory is often misleading.     

Mutation and selection in a large population

In this paper we study a large, but finite population, in which mutation and selection occur at a single genetic locus in a diploid organism. We provide theoretical results for the equilibrium allele frequencies, their variances and covariances and their equilibrium distribution, when the population size is larger than the reciprocal of the mean allelic mutation rate. We are also able to infer that the equilibrium distribution of allele frequencies takes the form of a constrained multivariate Gaussian distribution. Our results provide a rapid way of obtaining useful information in the case of complex mutation and selection schemes when the population size is large. We present numerical simulations to test the applicability of our theoretical formulations. The results of these simulations are in very reasonable agreement with the theoretical predictions.     

The effects of the mating system on the evolution of migration in a spatially heterogeneous population

Summary Verbal explanations for the evolution of migration and dispersal often invoke inbreeding depression as an important force. Experimental work on plant populations indicates that while inbreeding depression may favor increased migration rates, adaptation to local environments may reduce the advantage to migrants. We formalize and test this hypothesis using a two-locus genetic model that incorporates lowered fitness in offspring produced by self-fertilization, and habitat differentiation. We also use the model to address questions about the general theory of genetic modifiers and the modifier reduction principle. We find that even under conditions when migration would increase the mean fitness of a population, migration may not be favored. This result is due to the associations that develop between genotypes at a locus subject to overdominant selection and at a neutral locus controlling the migration rate. Thus, it appears that, in this model, the forces of local adaptation, which favor a reduction in the migration rate, overwhelm those of inbreeding depression, which may favor dispersal.     

Negative effects of individual heterozygosity on reproductive success in a wild bird population

The evolutionary consequences of individual genetic diversity are frequently studied by assessing heterozygosity–fitness correlations (HFCs). The prevalence of positive and negative HFCs and the predominance of general versus local effects in wild populations are far from understood, partly because comprehensive studies testing for both inbreeding and outbreeding depression are lacking. We studied a genetically diverse population of blue tits in southern Germany using a genome‐wide set of 87 microsatellites to investigate the relationship between proxies of reproductive success and measures of multilocus and single‐locus individual heterozygosity (MLH and SLH). We used complimentary measures of MLH and partitioned markers into functional categories according to their position in the blue tit genome. HFCs based on MLH were consistently negative for functional loci, whereas correlations were rather inconsistent for loci found in nonfunctional areas of the genome. Clutch size was the only reproductive variable showing a general effect. We found evidence for local effects for three measures of reproductive success: arrival date at the breeding site, the probability of breeding at the study site and male reproductive success. For these, we observed consistent, and relatively strong, negative effects at one functional locus. Remarkably, this marker had a similar effect in another blue tit population from Austria (~400 km to the east). We suggest that a genetic local effect on timing of arrival might be responsible for most negative HFCs detected, with carry‐over effects on other reproductive traits. This effect could reflect individual differences in the distance between overwintering areas and breeding sites.     

Duplication and population dynamics shape historic patterns of selection and genetic variation at the major histocompatibility complex in rodents

Genetic variation at the major histocompatibility complex (MHC) is vitally important for wildlife populations to respond to pathogen threats. As natural populations can fluctuate greatly in size, a key issue concerns how population cycles and bottlenecks that could reduce genetic diversity will influence MHC genes. Using 454 sequencing, we characterized genetic diversity at the DRB Class II locus in montane voles (Microtus montanus), a North American rodent that regularly undergoes high‐amplitude fluctuations in population size. We tested for evidence of historic balancing selection, recombination, and gene duplication to identify mechanisms maintaining allelic diversity. Counter to our expectations, we found strong evidence of purifying selection acting on the DRB locus in montane voles. We speculate that the interplay between population fluctuations and gene duplication might be responsible for the weak evidence of historic balancing selection and strong evidence of purifying selection detected. To further explore this idea, we conducted a phylogenetically controlled comparative analysis across 16 rodent species with varying demographic histories and MHC duplication events (based on the maximum number of alleles detected per individual). On the basis of phylogenetic generalized linear model‐averaging, we found evidence that the estimated number of duplicated loci was positively related to allelic diversity and, surprisingly, to the strength of purifying selection at the DRB locus. Our analyses also revealed that species that had undergone population bottlenecks had lower allelic richness than stable species. This study highlights the need to consider demographic history and genetic structure alongside patterns of natural selection to understand resulting patterns of genetic variation at the MHC.     

Comparative analysis of population structure across environments and geographical scales at major histocompatibility complex and microsatellite loci in Atlantic salmon (Salmo salar)

Evidence of selection acting on major histocompatibility complex (MHC) genes has been illustrated with the analysis of their nucleotide sequences and allele frequency distribution. Comparing the patterns of population differentiation at neutral markers and MHC genes in the wild may provide further insights about the relative role of selection and neutrality in shaping their diversity. In this study, we combine both methods to assess the role of selection on a MHC gene in Atlantic salmon. We compare variation at a MHC class II B locus and microsatellites among 14 samples from seven different rivers and seven subpopulations within a single river system covering a variety of habitats and different geographical scales. We show that diversifying selection is acting on the sites involved in antigen presentation and that balancing selection maintains a high level of polymorphism within populations. Despite important differences in habitat type, the comparison of the population structure at MHC and microsatellites on large geographical scales reveals a correlation between patterns of differentiation, indicating that drift and migration have been more important than selection in shaping population differentiation at the MHC locus. In contrast, strong discrepancies between patterns of population differentiation at the two types of markers provides support for the role of selection in shaping population structure within rivers. Together, these results confirm that natural selection is influencing MHC gene diversity in wild Atlantic salmon although neutral forces may also be important in their evolution.     

Recent selection for self‐compatibility in a population of Leavenworthia alabamica

The evolution of self‐compatibility (SC) is the first step in the evolutionary transition in plants from outcrossing enforced by self‐incompatibility (SI) to self‐fertilization. In the Brassicaceae, SI is controlled by alleles of two tightly linked genes at the S‐locus. Despite permitting inbreeding, mutations at the S‐locus leading to SC may be selected if they provide reproductive assurance and/or gain a transmission advantage in a population when SC plants self‐ and outcross. Positive selection can leave a genomic signature in the regions physically linked to the focus of selection when selection has occurred recently. From an SC population of Leavenworthia alabamica with a known nonfunctional mutation at the S‐locus, we collected sequence data from a ～690 Kb region surrounding the S‐locus, as well as from regions not linked to the S‐locus. To test for recent positive selection acting at the S‐locus, we examined polymorphism and the site‐frequency spectra. Using forward simulations, we demonstrate that recent selection of the strength expected for SC at a locus formerly under balancing selection can generate patterns similar to those seen in our empirical data.     

Homozygosity in a population of variable size and mutation rate

A formula is obtained for the probability that two genes at a single locus, sampled at random from a population at time t, are of particular types. The model assumed is a diffusion approximation to a neutral Wright-Fisher model in which mutation is not necessarily symmetric and the population size is a function of time. It is shown that for symmetric mutation in a population undergoing a step-function type bottleneck, homozygosity increases with decreasing population size. A formula is given for the distribution of the number of segregating sites occurring in two randomly sampled sequences of completely linked sites, with general mutation at a site and identical mutation structure between sites.We give similar results for a population of fixed size but for which the mutation rate is a function of time, and not necessarily symmetric. We confirm the intuitively clear effect that increasing the mutation rate decreases homozygosity.     

Intense selection in an age-structured population

In a population with overlapping generations, intense selection can perturb the age distribution and thus affect the rate of increase of an advantageous allele. We found that the age-specific nature of intense selection, such as that generated by many diseases, can affect the outcome of selection on loci, such as those conferring disease resistance. We also found that the temporal dynamics of selection alter the speed of evolution, particularly when selection is intense, and even more so when it is age-specific. We relate our model and results to selection for disease resistance, although the results have broader implications for inferences about past selection pressures in general.     

Detecting population structure in a high gene-flow species,Atlantic herring (Clupea harengus): direct,simultaneous evaluation of neutral vs putatively selected loci

In many marine fish species, genetic population structure is typically weak because populations are large, evolutionarily young and have a high potential for gene flow. We tested whether genetic markers influenced by natural selection are more efficient than the presumed neutral genetic markers to detect population structure in Atlantic herring (Clupea harengus), a migratory pelagic species with large effective population sizes. We compared the spatial and temporal patterns of divergence and statistical power of three traditional genetic marker types, microsatellites, allozymes and mitochondrial DNA, with one microsatellite locus, Cpa112, previously shown to be influenced by divergent selection associated with salinity, and one locus located in the major histocompatibility complex class IIA (MHC-IIA) gene, using the same individuals across analyses. Samples were collected in 2002 and 2003 at two locations in the North Sea, one location in the Skagerrak and one location in the low-saline Baltic Sea. Levels of divergence for putatively neutral markers were generally low, with the exception of single outlier locus/sample combinations; microsatellites were the most statistically powerful markers under neutral expectations. We found no evidence of selection acting on the MHC locus. Cpa112, however, was highly divergent in the Baltic samples. Simulations addressing the statistical power for detecting population divergence showed that when using Cpa112 alone, compared with using eight presumed neutral microsatellite loci, sample sizes could be reduced by up to a tenth while still retaining high statistical power. Our results show that the loci influenced by selection can serve as powerful markers for detecting population structure in high gene-flow marine fish species.     

Genetic variation in the acorn barnacle from allozymes to population genomics

Understanding the patterns of genetic variation within and among populations is a central problem in population and evolutionary genetics. We examine this question in the acorn barnacle, Semibalanus balanoides, in which the allozyme loci Mpi and Gpi have been implicated in balancing selection due to varying selective pressures at different spatial scales. We review the patterns of genetic variation at the Mpi locus, compare this to levels of population differentiation at mtDNA and microsatellites, and place these data in the context of genome-wide variation from high-throughput sequencing of population samples spanning the North Atlantic. Despite considerable geographic variation in the patterns of selection at the Mpi allozyme, this locus shows rather low levels of population differentiation at ecological and trans-oceanic scales (F(ST)?～?5%). Pooled population sequencing was performed on samples from Rhode Island (RI), Maine (ME), and Southwold, England (UK). Analysis of more than 650 million reads identified approximately 335,000 high-quality SNPs in 19 million base pairs of the S. balanoides genome. Much variation is shared across the Atlantic, but there are significant examples of strong population differentiation among samples from RI, ME, and UK. An F(ST) outlier screen of more than 22,000 contigs provided a genome-wide context for interpretation of earlier studies on allozymes, mtDNA, and microsatellites. F(ST) values for allozymes, mtDNA and microsatellites are close to the genome-wide average for random SNPs, with the exception of the trans-Atlantic F(ST) for mtDNA. The majority of F(ST) outliers were unique between individual pairs of populations, but some genes show shared patterns of excess differentiation. These data indicate that gene flow is high, that selection is strong on a subset of genes, and that a variety of genes are experiencing diversifying selection at large spatial scales. This survey of polymorphism in S. balanoides provides a number of genomic tools that promise to make this a powerful model for ecological genomics of the rocky intertidal.     

A multilocus-multiallele analysis of frequency-dependent selection induced by intraspecific competition

The study of the mechanisms that maintain genetic variation has a long history in population genetics. We analyze a multilocus-multiallele model of frequency- and density-dependent selection in a large randomly mating population. The number of loci and the number of alleles per locus are arbitrary. The n loci are assumed to contribute additively to a quantitative character under stabilizing or directional selection as well as under frequency-dependent selection caused by intraspecific competition. We assume the strength of stabilizing selection to be weak, whereas the strength of frequency dependence may be arbitrary. Density-dependence is induced by population regulation. Our main result is a characterization of the equilibrium structure and its stability properties in terms of all parameters. It turns out that no equilibrium exists with more than two alleles segregating per locus. We give necessary and sufficient conditions on the strength of frequency dependence to ensure the maintenance of multilocus polymorphism. We also give explicit formulas on the number of polymorphic loci maintained at equilibrium. These results are based on the assumption that selection is sufficiently weak compared with recombination, so that linkage equilibrium can be assumed. If additionally the population size is assumed to be constant, we prove that the dynamics of the model form a generalized gradient system. For the model in its general form we are able to derive necessary and sufficient conditions for the stability of the monomorphic equilibria. Furthermore, we briefly analyze a special symmetric two-locus two-allele model for a constant population size but allowing for linkage disequilibrium. Finally, we analyze a single diallelic locus with dominance to illustrate the complications that can occur if the assumption of additivity is relaxed.     

The diffusion model for migration and selection in a dioecious population

The diffusion approximation is derived for migration and selection at a multiallelic locus in a dioecious population subdivided into a lattice of panmictic colonies. Generations are discrete and nonoverlapping; autosomal and X-linked loci are analyzed. The relation between juvenile and adult subpopulation numbers is very general and includes both soft and hard selection; the zygotic sex ratio is the same in every colony. All the results hold for both adult and juvenile migration. If ploidy-weighted average selection, drift, and diffusion coefficients are used, then the ploidy-weighted average allelic frequencies satisfy the corresponding partial differential equation for a monoecious population. The boundary conditions and the unidimensional transition conditions for coincident discontinuities in the carrying capacity and migration rate extend identically. The previous unidimensional formulation and analysis of symmetric, nearest-neighbor migration of a monoecious population across a geographical barrier is generalized to symmetric migration of arbitrary finite range, and the transition conditions are shown to hold for a dioecious population. Thus, the entire theory of clines and of the wave of advance of favorable alleles is applicable to dioecious populations.This work was supported by National Science Foundation grant BSR-9006285