The effect of genetic drift on the variance/covariance components generated by multilocus additive x additive epistatic systems

The effect of population bottlenecks on the components of the genetic variance/covariance generated by n neutral independent additive x additive loci has been studied theoretically. In its simplest version, this situation can be modelled by specifying the allele frequencies and homozygous effects at each locus, and an additional factor measuring the strength of the n-th order epistatic interaction. The variance/covariance components in an infinitely large panmictic population (ancestral components) were compared with their expected values at equilibrium over replicates randomly derived from the base population, after t bottlenecks of size N (derived components). Formulae were obtained giving the derived components (and the between-line variance) as functions of the ancestral ones (alternatively, in terms of allele frequencies and effects) and the corresponding inbreeding coefficient F(t). The n-th order derived component of the genetic variance/covariance is continuously eroded by inbreeding, but the remaining components may increase initially until a critical F(t) value is attained, which is inversely related to the order of the pertinent component, and subsequently decline to zero. These changes can be assigned to the between-line variances/covariances of gene substitution and epistatic effects induced by drift. Numerical examples indicate that: (1) the derived additive variance/covariance component will generally exceed its ancestral value unless epistasis is weak; (2) the derived epistatic variance/covariance components will generally exceed their ancestral values unless allele frequencies are extreme; (3) for systems showing equal ancestral additive and total non-additive variance/covariance components, those including a smaller number of epistatic loci may generate a larger excess in additive variance/covariance after bottlenecks than others involving a larger number of loci, provided that F(t) is low. Our results indicate that it is unlikely that the rate of evolution may be significantly accelerated after population bottlenecks, in spite of occasional increments of the derived additive variance over its ancestral value.     

Persistence of changes in the genetic covariance matrix after a bottleneck

Abstract.— Genetic variance, phenotypic variance, and the genetic covariance matrix ( G ) can change as a result of genetic drift. These changes will persist over time to some extent and will continue if population size remains relatively small. Nine populations founded by a single pair of Drosophila melanogaster were measured for a series of six morphological characteristics for a large number of parent-offspring families at both the third generation after the bottlenecks and after 20 generations. From these data, the phenotypic variance, additive genetic variance, and G were estimated for each line at each generation. Phenotypic and genetic variances were highly correlated over time, so that the measurements made at the third generation were predictive of the state of the population 17 generations later. Genetic covariances were also somewhat stable over time; however, the G matrices of some lines changed significantly over the intervening generations. This change did not return the populations toward their original state before the population bottlenecks. We conclude that the genetic covariance matrix can change as a result of mild genetic drift over a short span of time.     

Effects of genetic drift on variance components under a general model of epistasis

We analyze the changes in the mean and variance components of a quantitative trait caused by changes in allele frequencies, concentrating on the effects of genetic drift. We use a general representation of epistasis and dominance that allows an arbitrary relation between genotype and phenotype for any number of diallelic loci. We assume initial and final Hardy-Weinberg and linkage equilibrium in our analyses of drift-induced changes. Random drift generates transient linkage disequilibria that cause correlations between allele frequency fluctuations at different loci. However, we show that these have negligible effects, at least for interactions among small numbers of loci. Our analyses are based on diffusion approximations that summarize the effects of drift in terms of F, the inbreeding coefficient, interpreted as the expected proportional decrease in heterozygosity at each locus. For haploids, the variance of the trait mean after a population bottleneck is var(delta(z)) = sigma(n)k=1 FkV(A(k)), where n is the number of loci contributing to the trait variance, V(A(1)) = V(A) is the additive genetic variance, and V(A(k)) is the kth-order additive epistatic variance. The expected additive genetic variance after the bottleneck, denoted (V*(A)), is closely related to var(delta(z)); (V*(A)) = (1 - F) sigma(n)k=1 kFk-1V(A(k)). Thus, epistasis inflates the expected additive variance above V(A)(1 - F), the expectation under additivity. For haploids (and diploids without dominance), the expected value of every variance component is inflated by the existence of higher order interactions (e.g., third-order epistasis inflates (V*(AA. This is not true in general with diploidy, because dominance alone can reduce (V*(A)) below V(A)(1 - F) (e.g., when dominant alleles are rare). Without dominance, diploidy produces simple expressions: var(delta(z)) = sigma(n)k=1 (2F)kV(A(k)) and (V(A)) = (1 - F) sigma(n)k=1 k(2F)k-1V(A(k)). With dominance (and even without epistasis), var(delta(z)) and (V*(A)) no longer depend solely on the variance components in the base population. For small F, the expected additive variance simplifies to (V*(A)) approximately equal to (1 - F)V(A) + 4FV(AA) + 2FV(D) + 2FC(AD), where C(AD) is a sum of two terms describing covariances between additive effects and dominance and additive X dominance interactions. Whether population bottlenecks lead to expected increases in additive variance depends primarily on the ratio of nonadditive to additive genetic variance in the base population, but dominance precludes simple predictions based solely on variance components. We illustrate these results using a model in which genotypic values are drawn at random, allowing extreme and erratic epistatic interactions. Although our analyses clarify the conditions under which drift is expected to increase V(A), we question the evolutionary importance of such increases.     

Effects of partial selfing on the equilibrium genetic variance,mutation load,and inbreeding depression under stabilizing selection

The mating system of a species is expected to have important effects on its genetic diversity. In this article, we explore the effects of partial selfing on the equilibrium genetic variance V_g, mutation load L, and inbreeding depression δ under stabilizing selection acting on a arbitrary number n of quantitative traits coded by biallelic loci with additive effects. When the ratio is low (where U is the total haploid mutation rate on selected traits) and effective recombination rates are sufficiently high, genetic associations between loci are negligible and the genetic variance, mutation load, and inbreeding depression are well predicted by approximations based on single‐locus models. For higher values of and/or lower effective recombination, moderate genetic associations generated by epistasis tend to increase V_g, L, and δ, this regime being well predicted by approximations including the effects of pairwise associations between loci. For yet higher values of and/or lower effective recombination, a different regime is reached under which the maintenance of coadapted gene complexes reduces V_g, L, and δ. Simulations indicate that the values of V_g, L, and δ are little affected by assumptions regarding the number of possible alleles per locus.     

The additive genetic variance after bottlenecks is affected by the number of loci involved in epistatic interactions

Abstract We investigated the role of the number of loci coding for a neutral trait on the release of additive variance for this trait after population bottlenecks. Different bottleneck sizes and durations were tested for various matrices of genotypic values, with initial conditions covering the allele frequency space. We used three different types of matrices. First, we extended Cheverud and Routman's model by defining matrices of "pure" epistasis for three and four independent loci; second, we used genotypic values drawn randomly from uniform, normal, and exponential distributions; and third we used two models of simple metabolic pathways leading to physiological epistasis. For all these matrices of genotypic values except the dominant metabolic pathway, we find that, as the number of loci increases from two to three and four, an increase in the release of additive variance is occurring. The amount of additive variance released for a given set of genotypic values is a function of the inbreeding coefficient, independently of the size and duration of the bottleneck. The level of inbreeding necessary to achieve maximum release in additive variance increases with the number of loci. We find that additive-by-additive epistasis is the type of epistasis most easily converted into additive variance. For a wide range of models, our results show that epistasis, rather than dominance, plays a significant role in the increase of additive variance following bottlenecks.     

Stability of genetic variance and covariance for reproductive characters in the face of climate change in a wild bird population

Global warming has had numerous effects on populations of animals and plants, with many species in temperate regions experiencing environmental change at unprecedented rates. Populations with low potential for adaptive evolutionary change and plasticity will have little chance of persistence in the face of environmental change. Assessment of the potential for adaptive evolution requires the estimation of quantitative genetic parameters, but it is as yet unclear what impact, if any, global warming will have on the expression of genetic variances and covariances. Here we assess the impact of a changing climate on the genetic architecture underlying three reproductive traits in a wild bird population. We use a large, long-term, data set collected on great tits (Parus major) in Wytham Woods, Oxford, and an 'animal model' approach to quantify the heritability of, and genetic correlations among, laying date, clutch size and egg mass during two periods with contrasting temperature conditions over a 40-year period (1965-1988 [cooler] vs. 1989-2004 [warmer]). We found significant additive genetic variance and heritability for all traits under both temperature regimes. We also found significant negative genetic covariances and correlations between clutch size and egg weight during both periods, and among laying date and clutch size in the colder years only. The overall G matrix comparison among periods, however, showed only a minor difference among periods, thus suggesting that genotype by environment interactions are negligible in this context. Our results therefore suggest that despite substantial changes in temperature and in mean laying date phenotype over the last decades, and despite the large sample sizes available, we are unable to detect any significant change in the genetic architecture of the reproductive traits studied.     

The evolution of genetic architecture under frequency-dependent disruptive selection

We propose a model to analyze a quantitative trait under frequency-dependent disruptive selection. Selection on the trait is a combination of stabilizing selection and intraspecific competition, where competition is maximal between individuals with equal phenotypes. In addition, there is a density-dependent component induced by population regulation. The trait is determined additively by a number of biallelic loci, which can have different effects on the trait value. In contrast to most previous models, we assume that the allelic effects at the loci can evolve due to epistatic interactions with the genetic background. Using a modifier approach, we derive analytical results under the assumption of weak selection and constant population size, and we investigate the full model by numerical simulations. We find that frequency-dependent disruptive selection favors the evolution of a highly asymmetric genetic architecture, where most of the genetic variation is concentrated on a small number of loci. We show that the evolution of genetic architecture can be understood in terms of the ecological niches created by competition. The phenotypic distribution of a population with an adapted genetic architecture closely matches this niche structure. Thus, evolution of the genetic architecture seems to be a plausible way for populations to adapt to regimes of frequency-dependent disruptive selection. As such, it should be seen as a potential evolutionary pathway to discrete polymorphisms and as a potential alternative to other evolutionary responses, such as the evolution of sexual dimorphism or assortative mating.     

Stabilizing selection,mutational bias,and the evolution of sex*

Stabilizing selection around a fixed phenotypic optimum is expected to disfavor sexual reproduction, since asexually reproducing organisms can maintain a higher fitness at equilibrium, while sex disrupts combinations of compensatory mutations. This conclusion rests on the assumption that mutational effects on phenotypic traits are unbiased, that is, mutation does not tend to push phenotypes in any particular direction. In this article, we consider a model of stabilizing selection acting on an arbitrary number of polygenic traits coded by bialellic loci, and show that mutational bias may greatly reduce the mean fitness of asexual populations compared with sexual ones in regimes where mutations have weak to moderate fitness effects. Indeed, mutation and drift tend to push the population mean phenotype away from the optimum, this effect being enhanced by the low effective population size of asexual populations. In a second part, we present results from individual‐based simulations showing that positive rates of sex are favored when mutational bias is present, while the population evolves toward complete asexuality in the absence of bias. We also present analytical (QLE) approximations for the selective forces acting on sex in terms of the effect of sex on the mean and variance in fitness among offspring.     

The Kuroshio Current influences genetic diversity and population genetic structure of a tropical seagrass,Enhalus acoroides

Information on genetic diversity and differentiation of seagrass populations is essential for the conservation of coastal ecosystems. However, little is known about the seagrasses in the Indo‐West Pacific Ocean, where the world's highest diversity of seagrasses occurs. The influence of sea currents on these populations is also unknown. We estimated the genetic diversity and population genetic structure and identified reproductive features in Enhalus acoroides populations from the Yaeyama Islands, Hainan Island and the Philippines. The Philippines are situated at the centre of the E. acoroides range, Yaeyama and Hainan are peripheral populations, and the Yaeyama population is at the northern limit of the species range. The powerful Kuroshio Current flows from the Philippines to Yaeyama. Genetic analyses using nine microsatellite markers indicated that reproduction of E. acoroides is mostly sexual. Clonal diversity does not decrease in northern populations, although genetic diversity does. However, the genetic diversity of the Yaeyama populations is greater than that of the Hainan populations. Significant genetic differentiation among most populations was evident; however, the Yaeyama and north‐east Philippines populations were genetically similar, despite being separated by ~1100 km. An assignment test suggested that recruitment occurs from the north‐east Philippines to Yaeyama. The strong current in this region is probably responsible for the extant genetic diversity and recruitment patterns.     

The evolution of genetic architecture. I. Diversification of genetic backgrounds by genetic drift

The genetic architecture of a phenotype plays a critical role in determining phenotypic evolution through its effects on patterns of genetic variation. Genetic architecture is often considered to be constant in evolutionary quantitative genetic models. However, genetic architecture may be variable and itself evolve when there are dominance and epistatic interactions among alleles at the same and different loci, respectively. The evolution of genetic architecture by genetic drift is examined here by testing the breeding value of four standard inbred mouse strains mated across a set of 26 related recombinant quasi-inbred (RqI) lines generated from the intercross of the Large (LG/J) and Small (SM/J) inbred mouse strains. Phenotypes of interest include age-specific body weights, growth, and adult body composition. If the genetic architecture of these traits has differentiated by genetic drift during the production of the RqI strains, we should observe interactions between tester strain and RqI strain. The breeding values of the tester strains will change relative to one another depending on which RqI strain they are crossed to. The study included an average of 15.1 offspring per cross, over a total of 100 different crosses. Multivariate and univariate analyses of variance indicate that there is strongly significant interaction for all traits. Interaction is more pronounced in males than in females and accounted for an average of about 40% of the explained variation in males and 30% in females. These results indicate that the genetic architecture of these traits has differentiated by genetic drift in the RqI strains since their isolation from a common founder population. Further analysis indicates that this differentiation results in changes in the order of tester strain effects so that common patterns of selection in these differentiated populations could result in the fixation of different alleles.     

The effect of population bottlenecks on mutation rate evolution in asexual populations

In the absence of recombination, a mutator allele can spread through a population by hitchhiking with beneficial mutations that appear in its genetic background. Theoretical studies over the past decade have shown that the survival and fixation probability of beneficial mutations can be severely reduced by population size bottlenecks. Here, we use computational modelling and evolution experiments with the yeast S. cerevisiae to examine whether population bottlenecks can affect mutator dynamics in adapting asexual populations. In simulation, we show that population bottlenecks can inhibit mutator hitchhiking with beneficial mutations and are most effective at lower beneficial mutation supply rates. We then subjected experimental populations of yeast propagated at the same effective population size to three different bottleneck regimes and observed that the speed of mutator hitchhiking was significantly slower at smaller bottlenecks, consistent with our theoretical expectations. Our results, thus, suggest that bottlenecks can be an important factor in mutation rate evolution and can in certain circumstances act to stabilize or, at least, delay the progressive elevation of mutation rates in asexual populations. Additionally, our findings provide the first experimental support for the theoretically postulated effect of population bottlenecks on beneficial mutations and demonstrate the usefulness of studying mutator frequency dynamics for understanding the underlying dynamics of fitness‐affecting mutations.     

Characterizing the evolution of genetic variance using genetic covariance tensors

Determining how genetic variance changes under selection in natural populations has proved to be a very resilient problem in evolutionary genetics. In the same way that understanding the availability of genetic variance within populations requires the simultaneous consideration of genetic variance in sets of functionally related traits, determining how genetic variance changes under selection in natural populations will require ascertaining how genetic variance–covariance (G) matrices evolve. Here, we develop a geometric framework using higher-order tensors, which enables the empirical characterization of how G matrices have diverged among populations. We then show how divergence among populations in genetic covariance structure can then be associated with divergence in selection acting on those traits using key equations from evolutionary theory. Using estimates of G matrices of eight male sexually selected traits from nine geographical populations of Drosophila serrata, we show that much of the divergence in genetic variance occurred in a single trait combination, a conclusion that could not have been reached by examining variation among the individual elements of the nine G matrices. Divergence in G was primarily in the direction of the major axes of genetic variance within populations, suggesting that genetic drift may be a major cause of divergence in genetic variance among these populations.     

WILL POPULATION BOTTLENECKS AND MULTILOCUS EPISTASIS INCREASE ADDITIVE GENETIC VARIANCE?

We apply new analytical methods to understand the consequences of population bottlenecks for expected additive genetic variance. We analyze essentially all models for multilocus epistasis that have been numerically simulated to demonstrate increased additive variance. We conclude that for biologically plausible models, large increases in expected additive variance--attributable to epistasis rather than dominance--are unlikely. Naciri-Graven and Goudet (2003) found that as the number of epistatically interacting loci increases, additive variance tends to be inflated more after a bottleneck. We argue that this result reflects biologically unrealistic aspects of their models. Specifically, as the number of loci increases, higher-order epistatic interactions become increasingly important in these models, with an increasing fraction of the genetic variance becoming nonadditive, contrary to empirical observations. As shown by Barton and Turelli (2004), without dominance, conversion of nonadditive to additive variance depends only on the variance components and not on the number of loci per se. Numerical results indicating that more inbreeding is needed to produce maximal release of additive variance with more loci follow directly from our analytical results, which show that high levels of inbreeding (F > 0.5) are needed for significant conversion of higher-order components. We discuss alternative approaches to modeling multilocus epistasis and understanding its consequences.     

The change in quantitative genetic variation with inbreeding

Inbreeding is known to reduce heterozygosity of neutral genetic markers, but its impact on quantitative genetic variation is debated. Theory predicts a linear decline in additive genetic variance (V(A)) with increasing inbreeding coefficient (F) when loci underlying the trait act additively, but a nonlinear hump-shaped relationship when dominance and epistasis are important. Predictions for heritability (h2) are similar, although the exact shape depends on the value of h2 in the absence of inbreeding. We located 22 published studies in which the level of genetic variation in experimentally inbred populations (measured by V(A) or h2) was compared with that in outbred control populations. For life-history traits, the data strongly supported a nonlinear change in genetic variation with increasing F. V(A) and h2 were, respectively, 244% and 50% higher at F = 0.4 than in outbred populations, and dominance plus epistatic variance together exceeded additive variance by a factor of four. For nonfitness traits the decline was linear and estimates of nonadditive variance were small. These results confirm that population bottlenecks frequently increase V(A) in some traits, and imply that life-history traits are underlain by substantial dominance or epistasis. However, the importance of drift-induced genetic variation in conservation or evolutionary biology is questionable, in part because inbreeding depression usually accompanies inbreeding.     

The role of genetic structure in the adaptive divergence of populations experiencing saltwater intrusion due to relative sea‐level rise

Saltwater intrusion into estuaries creates stressful conditions for nektonic species. Previous studies have shown that Gambusia affinis populations with exposure to saline environments develop genetic adaptations for increased survival during salinity stress. Here, we evaluate the genetic structure of G. affinis populations, previously shown to have adaptations for increased salinity tolerance, and determine the impact of selection and gene flow on structure of these populations. We found that gene flow was higher between populations experiencing different salinity regimes within an estuary than between similar marsh types in different estuaries, suggesting the development of saline‐tolerant phenotypes due to local adaptation. There was limited evidence of genetic structure along a salinity gradient, and only some of the genetic variation among sites was correlated with salinity. Our results suggest limited structure, combined with selection to saltwater intrusion, results in phenotypic divergence in spite of a lack of physical barriers to gene flow.     

Sunshine versus gold: The effect of population age on genetic structure of an invasive mosquito

The genetic diversity and structure of invasive species are affected by the time since invasion, but it is not well understood how. We compare likely the oldest populations of Aedes aegypti in continental North America with some of the newest to illuminate the range of genetic diversity and structure that can be found within the invasive range of this important disease vector. Aedes aegypti populations in Florida have probably persisted since the 1600‐1700s, while populations in southern California derive from new invasions that occurred in the last 10 years. For this comparison, we genotyped 1,193 individuals from 28 sites at 12 highly variable microsatellites and a subset of these individuals at 23,961 single nucleotide polymorphisms (SNPs). This is the largest sample analyzed for genetic structure for either region, and it doubles the number of southern California populations previously analyzed. As predicted, the older populations (Florida) showed fewer indicators of recent founder effect and bottlenecks; in particular, these populations have dramatically higher genetic diversity and lower genetic structure. Geographic distance and driving distance were not good predictors of genetic distance in either region, especially southern California. Additionally, southern California had higher levels of genetic differentiation than any comparably sized documented region throughout the worldwide distribution of the species. Although population age and demographic history are likely driving these differences, differences in climate and transportation practices could also play a role.     

The effect of population structure on the rate of evolution

Ecological factors exert a range of effects on the dynamics of the evolutionary process. A particularly marked effect comes from population structure, which can affect the probability that new mutations reach fixation. Our interest is in population structures, such as those depicted by ‘star graphs’, that amplify the effects of selection by further increasing the fixation probability of advantageous mutants and decreasing the fixation probability of disadvantageous mutants. The fact that star graphs increase the fixation probability of beneficial mutations has lead to the conclusion that evolution proceeds more rapidly in star-structured populations, compared with mixed (unstructured) populations. Here, we show that the effects of population structure on the rate of evolution are more complex and subtle than previously recognized and draw attention to the importance of fixation time. By comparing population structures that amplify selection with other population structures, both analytically and numerically, we show that evolution can slow down substantially even in populations where selection is amplified.     

Genetic variance in fitness and its cross-sex covariance predict adaptation during experimental evolution

The additive genetic variation (V_A) of fitness in a population is of particular importance to quantify its adaptive potential and predict its response to rapid environmental change. Recent statistical advances in quantitative genetics and the use of new molecular tools have fostered great interest in estimating fitness V_A in wild populations. However, the value of V_A for fitness in predicting evolutionary changes over several generations remains mostly unknown. In our study, we addressed this question by combining classical quantitative genetics with experimental evolution in the model organism Tribolium castaneum (red flour beetle) in three new environmental conditions (Dry, Hot, Hot-Dry). We tested for potential constraints that might limit adaptation, including environmental and sex genetic antagonisms captured by negative genetic covariance between environments and female and male fitness, respectively. Observed fitness changes after 20 generations mainly matched our predictions. Given that body size is commonly used as a proxy for fitness, we also tested how this trait and its genetic variance (including nonadditive genetic variance) were impacted by environmental stress. In both traits, genetic variances were sex and condition dependent, but they differed in their variance composition, cross-sex and cross-environment genetic covariances, as well as in the environmental impact on V_A.     

Prediction of effects of genetic drift on variance components under a general model of epistasis

For a model of diallelic loci with arbitrary epistasis, Barton and Turelli [2004. Effects of genetic drift on variance components under a general model of epistasis. Evolution 58, 2111-2132] gave results for variances among and within replicate lines obtained by inbreeding without selection. Here, we discuss the relation between their population genetic methods and classical quantitative genetic arguments. In particular, we consider the case of no dominance using classical identity by descent arguments, which generalizes their results from two alleles to multiple alleles. To clarify the connections between the alternative methods, we obtain the same results using an intermediate method, which explicitly identifies the statistical effects of sets of loci. We also discuss the effects of population bottlenecks on covariances among relatives.     

Argentine population genetic structure: large variance in Amerindian contribution

Argentine population genetic structure was examined using a set of 78 ancestry informative markers (AIMs) to assess the contributions of European, Amerindian, and African ancestry in 94 individuals members of this population. Using the Bayesian clustering algorithm STRUCTURE, the mean European contribution was 78%, the Amerindian contribution was 19.4%, and the African contribution was 2.5%. Similar results were found using weighted least mean square method: European, 80.2%; Amerindian, 18.1%; and African, 1.7%. Consistent with previous studies the current results showed very few individuals (four of 94) with greater than 10% African admixture. Notably, when individual admixture was examined, the Amerindian and European admixture showed a very large variance and individual Amerindian contribution ranged from 1.5 to 84.5% in the 94 individual Argentine subjects. These results indicate that admixture must be considered when clinical epidemiology or case control genetic analyses are studied in this population. Moreover, the current study provides a set of informative SNPs that can be used to ascertain or control for this potentially hidden stratification. In addition, the large variance in admixture proportions in individual Argentine subjects shown by this study suggests that this population is appropriate for future admixture mapping studies.