EPISTASIS AND THE EFFECT OF FOUNDER EVENTS ON THE ADDITIVE GENETIC VARIANCE

Models of founder events have focused on the reduction in the genetic variation following a founder event. However, recent work (Bryant et al., 1986; Goodnight, 1987) suggests that when there is epistatic genetic variance in a population, the total genetic variance within demes may actually increase following a founder event. Since the additive genetic variance is a statistical property of a population and can change with the level of inbreeding, some of the epistatic genetic variance may be converted to additive genetic variance during a founder event. The model presented here demonstrates that some of the additive-by-additive epistatic genetic variance is converted to additive genetic variance following a founder event. Furthermore, the amount of epistasis converted to additive genetic variance is a function of the recombination rate and the propagule size. For a single founder event of two individuals, as much as 75% of the epistatic variance in the ancestral population may become additive genetic variance following the founder event. For founder events involving two individuals with free recombination, the relative contribution of epistasis to the additive genetic variance following a founder event is equal to its proportion of the total genetic variance prior to the founder event. Traits closely related to fitness are expected to have relatively little additive genetic variance but may have substantial nonadditive genetic variance. Founder events may be important in the evolution of fitness traits, not because they lead to a reduction in the genetic variance, but rather because they lead to an increase in the additive genetic variance.     

The reality and importance of founder speciation in evolution

A founder event occurs when a new population is established from a small number of individuals drawn from a large ancestral population. Mayr proposed that genetic drift in an isolated founder population could alter the selective forces in an epistatic system, an observation supported by recent studies. Carson argued that a period of relaxed selection could occur when a founder population is in an open ecological niche, allowing rapid population growth after the founder event. Selectable genetic variation can actually increase during this founder-flush phase due to recombination, enhanced survival of advantageous mutations, and the conversion of non-additive genetic variance into additive variance in an epistatic system, another empirically confirmed prediction. Templeton combined the theories of Mayr and Carson with population genetic models to predict the conditions under which founder events can contribute to speciation, and these predictions are strongly confirmed by the empirical literature. Much of the criticism of founder speciation is based upon equating founder speciation to an adaptive peak shift opposed by selection. However, Mayr, Carson and Templeton all modeled a positive interaction of selection and drift, and Templeton showed that founder speciation is incompatible with peak-shift conditions. Although rare, founder speciation can have a disproportionate importance in adaptive innovation and radiation, and examples are given to show that "rare" does not mean "unimportant" in evolution. Founder speciation also interacts with other speciation mechanisms such that a speciation event is not a one-dimensional process due to either selection alone or drift alone.     

PERSPECTIVE: THE THEORIES OF FISHER AND WRIGHT IN THE CONTEXT OF METAPOPULATIONS: WHEN NATURE DOES MANY SMALL EXPERIMENTS

We critically review the two major theories of adaptive evolution developed early in this century, Wright's shifting balance theory and Fisher's large population size theory, in light of novel findings from field observations, laboratory experiments, and theoretical research conducted over the past 15 years. Ecological studies of metapopulations have established that the processes of local extinction and colonization of demes are relatively common in natural populations of many species and theoretical population genetic models have shown that these ecological processes have genetic consequences within and among local demes. Within demes, random genetic drift converts nonadditive genetic variance into additive genetic variance, increasing, rather than limiting, the potential for adaptation to local environments. For this reason, the genetic differences that arise by drift among demes, can be augmented by local selection. The resulting adaptive differences in gene combinations potentially contribute to the genetic origin of new species. These and other recent findings were not discussed by either Wright or Fisher. For example, although Wright emphasized epistatic genetic variance, he did not discuss the conversion process. Similarly, Fisher did not discuss how the average additive effect of a gene varies among demes across a metapopulation whenever there is epistasis. We discuss the implications of such recent findings for the Wright-Fisher controversy and identify some critical open questions that require additional empirical and theoretical study.     

Contribution of genetic effects to genetic variance components with epistasis and linkage disequilibrium

Background

Cockerham genetic models are commonly used in quantitative trait loci (QTL) analysis with a special feature of partitioning genotypic variances into various genetic variance components, while the F_∞ genetic models are widely used in genetic association studies. Over years, there have been some confusion about the relationship between these two type of models. A link between the additive, dominance and epistatic effects in an F_∞ model and the additive, dominance and epistatic variance components in a Cockerham model has not been well established, especially when there are multiple QTL in presence of epistasis and linkage disequilibrium (LD).

Results

In this paper, we further explore the differences and links between the F_∞ and Cockerham models. First, we show that the Cockerham type models are allelic based models with a special modification to correct a confounding problem. Several important moment functions, which are useful for partition of variance components in Cockerham models, are also derived. Next, we discuss properties of the F_∞ models in partition of genotypic variances. Its difference from that of the Cockerham models is addressed. Finally, for a two-locus biallelic QTL model with epistasis and LD between the loci, we present detailed formulas for calculation of the genetic variance components in terms of the additive, dominant and epistatic effects in an F_∞ model. A new way of linking the Cockerham and F_∞ model parameters through their coding variables of genotypes is also proposed, which is especially useful when reduced F_∞ models are applied.

Conclusion

The Cockerham type models are allele-based models with a focus on partition of genotypic variances into various genetic variance components, which are contributed by allelic effects and their interactions. By contrast, the F_∞ regression models are genotype-based models focusing on modeling and testing of within-locus genotypic effects and locus-by-locus genotypic interactions. When there is no need to distinguish the paternal and maternal allelic effects, these two types of models are transferable. Transformation between an F_∞ model's parameters and its corresponding Cockerham model's parameters can be established through a relationship between their coding variables of genotypes. Genetic variance components in terms of the additive, dominance and epistatic genetic effects in an F_∞ model can then be calculated by translating formulas derived for the Cockerham models.

     

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.     

The mapping of epistatic effects onto a genealogical tree in haploid populations

In this paper we present a model that maps epistatic effects onto a genealogical tree for a haploid population. Prior work has demonstrated that genealogical structure causes the genotypic values of individuals to covary. Our results indicate that epistasis can reduce genotypic covariance that is caused by genealogical structure. Genotypic effects (both additive and epistatic) occur along the branches of a genealogical tree, from the base of the tree to its tips. Epistasis reduces genotypic covariance because there is a reweighting of the contribution of branches to the states of genotypes compared to the additive case. Branches near the tips of a genealogical tree contribute proportionally more genetic effects with epistasis than without epistasis. Epistatic effects are most numerous at basal positions in a genealogical tree when a population is constant in size and experiencing no selection, optimizing selection, diversifying selection or directional selection, indicating that epistatic effects are typically old. For a population that is growing in size, epistatic effects are most numerous at midpoints in a genealogical tree, indicating epistatic effects are of moderate age. Our results are important in that they suggest epistatic effects may typically explain deep (old) divergences and broad patterns of divergence that exist in populations, except in growing populations. In a growing population, epistatic effects may cause more within group divergence higher up in a tree and less between group divergence that is deep in a tree. The distribution of the number of epistatic effects and the expected variance and covariance in the number of epistatic effects is also provided assuming neutrality.     

Increased genetic variance after a population bottleneck

A conventional view holds that population bottlenecks cause massive losses of genetic variability, thus endangering the viability of the derived population. Although some alleles that were infrequent in the parent population may be lost new empirical evidence from Drosophila and housefly populations has demonstrated that genetic variance available to selection may actually increase following a single severe bottleneck. Several theoretical models support this view, and suggest that the increase may result from conversion of balanced epistatic variance to additive variance that becomes immediately available to selection. These effects appear to be greatest on the inheritance of quantitative characters, releasing new variance through the disruption of covariance matrices that underlie and interrelate quantitative traits. Thus, character change in adaptation and speciation may, in some instances, be promoted by founder events.     

Identifying quantitative trait locus by genetic background interactions in association studies

Association studies are designed to identify main effects of alleles across a potentially wide range of genetic backgrounds. To control for spurious associations, effects of the genetic background itself are often incorporated into the linear model, either in the form of subpopulation effects in the case of structure or in the form of genetic relationship matrices in the case of complex pedigrees. In this context epistatic interactions between loci can be captured as an interaction effect between the associated locus and the genetic background. In this study I developed genetic and statistical models to tie the locus by genetic background interaction idea back to more standard concepts of epistasis when genetic background is modeled using an additive relationship matrix. I also simulated epistatic interactions in four-generation randomly mating pedigrees and evaluated the ability of the statistical models to identify when a biallelic associated locus was epistatic to other loci. Under additive-by-additive epistasis, when interaction effects of the associated locus were quite large (explaining 20% of the phenotypic variance), epistasis was detected in 79% of pedigrees containing 320 individuals. The epistatic model also predicted the genotypic value of progeny better than a standard additive model in 78% of simulations. When interaction effects were smaller (although still fairly large, explaining 5% of the phenotypic variance), epistasis was detected in only 9% of pedigrees containing 320 individuals and the epistatic and additive models were equally effective at predicting the genotypic values of progeny. Epistasis was detected with the same power whether the overall epistatic effect was the result of a single pairwise interaction or the sum of nine pairwise interactions, each generating one ninth of the epistatic variance. The power to detect epistasis was highest (94%) at low QTL minor allele frequency, fell to a minimum (60%) at minor allele frequency of about 0.2, and then plateaued at about 80% as alleles reached intermediate frequencies. The power to detect epistasis declined when the linkage disequilibrium between the DNA marker and the functional polymorphism was not complete.     

A Genome-Wide Association Analysis Reveals Epistatic Cancellation of Additive Genetic Variance for Root Length in Arabidopsis thaliana

Efforts to identify loci underlying complex traits generally assume that most genetic variance is additive. Here, we examined the genetics of Arabidopsis thaliana root length and found that the genomic narrow-sense heritability for this trait in the examined population was statistically zero. The low amount of additive genetic variance that could be captured by the genome-wide genotypes likely explains why no associations to root length could be found using standard additive-model-based genome-wide association (GWA) approaches. However, as the broad-sense heritability for root length was significantly larger, and primarily due to epistasis, we also performed an epistatic GWA analysis to map loci contributing to the epistatic genetic variance. Four interacting pairs of loci were revealed, involving seven chromosomal loci that passed a standard multiple-testing corrected significance threshold. The genotype-phenotype maps for these pairs revealed epistasis that cancelled out the additive genetic variance, explaining why these loci were not detected in the additive GWA analysis. Small population sizes, such as in our experiment, increase the risk of identifying false epistatic interactions due to testing for associations with very large numbers of multi-marker genotypes in few phenotyped individuals. Therefore, we estimated the false-positive risk using a new statistical approach that suggested half of the associated pairs to be true positive associations. Our experimental evaluation of candidate genes within the seven associated loci suggests that this estimate is conservative; we identified functional candidate genes that affected root development in four loci that were part of three of the pairs. The statistical epistatic analyses were thus indispensable for confirming known, and identifying new, candidate genes for root length in this population of wild-collected A. thaliana accessions. We also illustrate how epistatic cancellation of the additive genetic variance explains the insignificant narrow-sense and significant broad-sense heritability by using a combination of careful statistical epistatic analyses and functional genetic experiments.     

Epistasis and Its Contribution to Genetic Variance Components

We present a new parameterization of physiological epistasis that allows the measurement of epistasis separate from its effects on the interaction (epistatic) genetic variance component. Epistasis is the deviation of two-locus genotypic values from the sum of the contributing single-locus genotypic values. This parameterization leads to statistical tests for epistasis given estimates of two-locus genotypic values such as can be obtained from quantitative trait locus studies. The contributions of epistasis to the additive, dominance and interaction genetic variances are specified. Epistasis can make substantial contributions to each of these variance components. This parameterization of epistasis allows general consideration of the role of epistasis in evolution by defining its contribution to the additive genetic variance.     

The effect of epistasis on the excess of the additive and nonadditive variances after population bottlenecks

The effect of population bottlenecks on the components of the genetic variance generated by two neutral independent epistatic loci has been studied theoretically (VA, additive; VD, dominant; VAA, additive x additive; VAD, additive x dominant; VDD; dominant x dominant components of variance). Nonoverdominance and overdominance models were considered, covering all possible types of marginal gene action at the single locus level. The variance components in an infinitely large panmictic population (ancestral components) were compared with their expected values at equilibrium, after t consecutive bottlenecks of equal size N (derived components). Formulae were obtained in terms of allele frequencies and effects at each locus and the corresponding epistatic value. An excess of VA after bottlenecks can be assigned to two sources: (1) the spatiotemporal changes in the marginal average effects of gene substitution alpha(i), which are equal to zero only for additive gene action within and between loci; and (2) the covariance between alpha2(i) and the heterozygosity at the loci involved, which is generated by dominance, with or without epistasis. Numerical examples were analyzed, indicating that an increase in VA after bottlenecks will only occur if its ancestral value is minimal or very small. For the nonoverdominance model with weak reinforcing epistasis, that increase has been detected only for extreme frequencies of the negative allele at one or both loci. With strong epistasis, however, this result can be extended to a broad range of intermediate frequencies. With no epistasis, the same qualitative results were found, indicating that dominance can be considered as the primary cause of an increase in VA following bottlenecks. In parallel, the derived total nonadditive variance exceeded its ancestral value (V(NA) = V(D) + V(AA) + V(AD) + V(DD)) for a range of combinations of allele frequencies covering those for an excess of VA and for very large frequencies of the negative allele at both loci. For the overdominance model, an increase in V(A) and V(NA) was respectively observed for equilibrium (intermediate) frequencies at one or both loci or for extreme frequencies at both loci. For all models, the magnitude of the change of V(A) and V(NA) was inversely related to N and t. At low levels of inbreeding, the between-line variance was not affected by the type of gene action. For the models considered, the results indicate that it is unlikely that the rate of evolution may be accelerated after population bottlenecks, in spite of occasional increments of the derived V(A) over its ancestral value.     

Partitioning of population genetic variance under multiplicative-epistatic gene action

Determining the way in which different QTLs interact (epistasis) in their effects on the phenotype is crucial to many areas in population genetics and evolutionary biology. For example, in the founder event, a separated population readapts to a new environment through the release of cryptic gene-gene interactions. In hybrid zones, hybrid speciation must be subjected to natural selection for epistasis resulting from genomic recombinations between different species. However, there is a severe shortage of relevant methodologies to estimate epistatic genetic effects and variances. A statistical model has recently been proposed to estimate the number of QTLs, their genetic effects and allelic frequencies in segregating populations. This model is based on multiplicative gene action and derived from a two-level intra- and interspecific mating design. In this paper, we formulate a statistical procedure for partitioning the genetic variance into additive, dominant and various kinds of epistatic components in an intra- or mixed intra- and interspecific hybrid population. The procedure can be used to study the genetic architecture of fragmented populations and hybrid zones, thus allowing for a better recognition of the role of epistasis in evolution and hybrid speciation. A real example for two Populus species, P. tremuloides and P. tremula, is provided to illustrate the procedure. In this example, we found that considerable new genetic variation is formed through genomic recombination between two aspen species. Received: 1 May 1999 / Accepted: 27 July 1999     

Epistasis and the temporal change in the additive variance-covariance matrix induced by drift

The effect of population bottlenecks on the components of the genetic covariance generated by two neutral independent epistatic loci has been studied theoretically (additive, covA; dominance, covD; additive-by-additive, covAA; additive-by-dominance, covAD; and dominance-by-dominance, covDD). The additive-by-additive model and a more general model covering all possible types of marginal gene action at the single-locus level (additive/dominance epistatic model) were considered. The 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 consecutive bottlenecks of equal size N (derived components). Formulae were obtained in terms of the allele frequencies and effects at each locus, the corresponding epistatic effects and the inbreeding coefficient Ft. These expressions show that the contribution of nonadditive loci to the derived additive covariance (covAt) does not linearly decrease with inbreeding, as in the pure additive case, and may initially increase or even change sign in specific situations. Numerical examples were also analyzed, restricted for simplicity to the case of all covariance components being positive. For additive-by-additive epistasis, the condition covAt > covA only holds for high frequencies of the allele decreasing the metric traits at each locus (negative allele) if epistasis is weak, or for intermediate allele frequencies if it is strong. For the additive/dominance epistatic model, however, covAt > covA applies for low frequencies of the negative alleles at one or both loci and mild epistasis, but this result can be progressively extended to intermediate frequencies as epistasis becomes stronger. Without epistasis the same qualitative results were found, indicating that marginal dominance induced by epistasis can be considered as the primary cause of an increase of the additive covariance after bottlenecks. For all models, the magnitude of the ratio covAt/covA was inversely related to N and t.     

Detecting epistatic genetic variance with a clonally replicated design: models for lowvs high-order nonallelic interaction

A quantitative genetic model, that uses known family structure with clonal replicates to separate genetic variance into its additive, dominance and epistatic components, is available in the current literature. Making use of offspring testing, this model is based on the theory that components of variance from the linear model of an experimental design may be expressed in terms of expected covariances among relatives. However, if interactions between a pair of quantitative trait loci (QTLs) explain a large proportion of the total epistasis, it will seriously overestimate the additive and dominance variances but underestimate the epistatic variance. In the present paper, a new model is developed to manipulate this problem by combining parental and offspring material into the same test. Under the condition described above, the new model can provide an accurate estimate for additive x additive variances. Also, its accuracy in estimating dominance and total epistatic variances is much greater than the accuracy of the previous model. However, if there is obvious evidence showing the major contribution of high-order interactions, especially among 4QTLs, to the total epistasis, the previous model is more appropriate to partition the genetic variance for a quantitative trait. The re-analysis of an example from a factorial mating design in poplar shows large differences in estimating variance components between the new and previous models when two different assumptions (lowvs high-order epistatic interactions) are used. The new model will be an alternative to estimating the mode of quantitative inheritance for species, especially for longlived, predominantly outcrossing forest trees, that can be clonally replicated.     

Modeling epistasis of quantitative trait loci using Cockerham's model

We use the orthogonal contrast scales proposed by Cockerham to construct a genetic model, called Cockerham's model, for studying epistasis between genes. The properties of Cockerham's model in modeling and mapping epistatic genes under linkage equilibrium and disequilibrium are investigated and discussed. Because of its orthogonal property, Cockerham's model has several advantages in partitioning genetic variance into components, interpreting and estimating gene effects, and application to quantitative trait loci (QTL) mapping when compared to other models, and thus it can facilitate the study of epistasis between genes and be readily used in QTL mapping. The issues of QTL mapping with epistasis are also addressed. Real and simulated examples are used to illustrate Cockerham's model, compare different models, and map for epistatic QTL. Finally, we extend Cockerham's model to multiple loci and discuss its applications to QTL mapping.     

Evolutionary Divergence of the Genetic Architecture Underlying Photoperiodism in the Pitcher-Plant Mosquito, Wyeomyia Smithii

We determine the contribution of composite additive, dominance, and epistatic effects to the genetic divergence of photoperiodic response along latitudinal, altitudinal, and longitudinal gradients in the pitcher-plant mosquito, Wyeomyia smithii. Joint scaling tests of crosses between populations showed wide-spread epistasis as well as additive and dominance differences among populations. There were differences due to epistasis between an alpine population in North Carolina and populations in Florida, lowland North Carolina, and Maine. Longitudinal displacement resulted in differences due to epistasis between Florida and Alabama populations separated by 300 km but not between Maine and Wisconsin populations separated by 2000 km. Genetic differences between New Jersey and Ontario did not involve either dominance or epistasis and we estimated the minimum number of effective factors contributing to a difference in mean critical photoperiod of 5 SD between them as n(E) = 5. We propose that the genetic similarity of populations within a broad northern region is due to their more recent origin since recession of the Laurentide Ice Sheet and that the unique genetic architecture of each population is the result of both mutation and repeated migration-founder-flush episodes during the dispersal of W. smithii in North America. Our results suggest that differences in composite additive and dominance effects arise early in the genetic divergence of populations while differences due to epistasis accumulate after more prolonged isolation.     

Quantitative Genetics of Doubled Haploid Populations and Application to the Theory of Line Development

The line value of a genotype is defined as the expected value of all lines that can be derived from this genotype. Specific genetic effects are defined for this value: only additive and additive by additive epistatic effects are necessary. There is no dominance effect for such a value. A general expression for the covariances between related lines is given. From a design with several lines per haplodiploidized plant taken at random from a population it is possible to estimate the additive variance for line value and the variance of additive by additive epistasis for line value. Variances of higher order epistasis can be estimated with a two-factor mating design in which a cross is replaced by the population of lines that can be derived from it. With a diallel or a factorial design a direct test for the presence of homozygous by homozygous epistasis is possible. The application of the concept of line value to the theory of line development leads to simple expressions of genetic advance in one cycle of recurrent selection according to the testing system. A brief consideration of these expressions leads to the conclusion that single doubled haploid descent recurrent selection will be one of the most efficient methods for low heritabilities and with a rapid development of doubled haploid lines.     

Epistasis can increase multivariate trait diversity in haploid non-recombining populations

We evaluate the effect of epistasis on genetically-based multivariate trait variation in haploid non-recombining populations. In a univariate setting, past work has shown that epistasis reduces genetic variance (additive plus epistatic) in a population experiencing stabilizing selection. Here we show that in a multivariate setting, epistasis also reduces total genetic variation across the entire multivariate trait in a population experiencing stabilizing selection. But, we also show that the pattern of variation across the multivariate trait can be more even when epistasis occurs compared to when epistasis is absent, such that some character combinations will have more genetic variance when epistasis occurs compared to when epistasis is absent. In fact, a measure of generalized multivariate trait variation can be increased by epistasis under weak to moderate stabilizing selection conditions, as well as neutral conditions. Likewise, a measure of conditional evolvability can be increased by epistasis under weak to moderate stabilizing selection and neutral conditions. We investigate the nature of epistasis assuming a multivariate-normal model genetic effects and investigate the nature of epistasis underlying the biophysical properties of RNA. Increased multivariate diversity occurs for populations that are infinite in size, as well as populations that are finite in size. Our model of finite populations is explicitly genealogical and we link our findings about the evenness of eigenvalues with epistasis to prior work on the genealogical mapping of epistatic effects.