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.     

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.

     

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     

Estimates of genetic variance in an F2 maize population

Maize (Zea mays L.) breeders have used several genetic-statistical models to study the inheritance of quantitative traits. These models provide information on the importance of additive, dominance, and epistatic genetic variance for a quantitative trait. Estimates of genetic variances are useful in understanding heterosis and determining the response to selection. The objectives of this study were to estimate additive and dominance genetic variances and the average level of dominance for an F2 population derived from the B73 x Mo17 hybrid and use weighted least squares to determine the importance of digenic epistatic variances relative to additive and dominance variances. Genetic variances were estimated using Design III and weighted least squares analyses. Both analyses determined that dominance variance was more important than additive variance for grain yield. For other traits, additive genetic variance was more important than dominance variance. The average level of dominance suggests either overdominant gene effects were present for grain yield or pseudo-overdominance because of linkage disequilibrium in the F2 population. Epistatic variances generally were not significantly different from zero and therefore were relatively less important than additive and dominance variances. For several traits estimates of additive by additive epistatic variance decreased estimates of additive genetic variance, but generally the decrease in additive genetic variance was not significant.     

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.     

Epistasis plays an important role as genetic basis of heterosis in rice

Ｔｈｅｇｅｎｅｔｉｃｂａｓｉｓｏｆｈｅｔｅｒｏｓｉｓｉｓｓｔｉｌｌａｄｅｂａｔｉｎｇｉｓｓｕｅ.Ｔｗｏｈｙｐｏｔｈｅｓｅｓ,ｔｈｅｄｏｍｉｎａｎｃｅｈｙｐｏｔｈｅｓｉｓａｎｄｔｈｅｏｖｅｒｄｏｍｉｎａｎｃｅｈｙｐｏｔｈｅｓｉｓ,ｂｏｔｈｐｒｏｐｏｓｅｄｉｎ1908[1—3],ｈａｖｅｃｏｍｐｅｔｅｄｆｏｒｍｏｓｔｐａｒｔｏｆｔｈｉｓｃｅｎｔｕｒｙ.Ａｌｔｈｏｕｇｈｍａｎｙｒｅｓｅａｒｃｈｅｒｓｐｒｅｆｅｒｏｎｅｈｙｐｏｔｈｅｓｉｓｔｏｔｈｅｏｔｈｅｒ,ｅｘｐｅｒｉｍｅｎｔａｌｄａｔａａｌｌｏｗｉｎｇｆｏｒｃｒ…     

Analysis of cytoplasmic and maternal effects. II. Genetic models for triploid endosperms

Genetic models for quantitative traits of triploid endosperms are proposed for the analysis of direct gene effects, cytoplasmic effects, and maternal gene effects. The maternal effect is partitioned into maternal additive and dominance components. In the full genetic model, the direct effect is partitioned into direct additive and dominance components and high-order dominance component, which are the cumulative effects of three-allele interactions. If the high-order dominance effects are of no importance, a reduced genetic model can be used. Monte Carlo simulations were conducted in this study for demonstrating unbiasedness of estimated variance and covariance components from the MINQUE (0/1) procedure, which is a minimum norm quadratic unbiased estimation (MINQUE) method setting 0 for all the prior covariances and 1 for all the prior variances. Robustness of estimating variance and covariance components for the genetic models was tested by simulations. Both full and reduced genetic models are shown to be robust for estimating variance and covariance components under several situations of no specific effects. Efficiency of predicting random genetic effects for the genetic models by the MINQUE (0/1) procedure was compared with the best linear unbiased prediction (BLUP). A worked example is given to illustrate the use of the reduced genetic model for kernel growth characteristics in corn (Zea mays L.).     

Genetic expectations of quantitative trait loci main and interaction effects obtained with the triple testcross design and their relevance for the analysis of heterosis

Interpretation of experimental results from quantitative trait loci (QTL) mapping studies on the predominant type of gene action can be severely affected by the choice of statistical model, experimental design, and provision of epistasis. In this study, we derive quantitative genetic expectations of (i) QTL effects obtained from one-dimensional genome scans with the triple testcross (TTC) design and (ii) pairwise interactions between marker loci using two-way analyses of variance (ANOVA) under the F(2)- and the F(infinity)-metric model. The theoretical results show that genetic expectations of QTL effects estimated with the TTC design are complex, comprising both main and epistatic effects, and that genetic expectations of two-way marker interactions are not straightforward extensions of effects estimated in one-dimensional scans. We also demonstrate that the TTC design can partially overcome the limitations of the design III in separating QTL main effects and their epistatic interactions in the analysis of heterosis and that dominance x additive epistatic interactions of individual QTL with the genetic background can be estimated with a one-dimensional genome scan. Furthermore, we present genetic expectations of variance components for the analysis of TTC progeny tested in a split-plot design, assuming digenic epistasis and arbitrary linkage.     

An epistatic genetic basis for fluctuating asymmetry of tooth size and shape in mice

Although there typically is little additive genetic variation for fluctuating asymmetry (FA), or variation in nondirectional differences between left and right sides of bilateral characters, several investigators have hypothesized that FA may have an epistatic genetic basis. We tested this hypothesis by conducting a whole genome scan of FA of size and shape of the mandibular molars in house mice from an F2 intercross population generated from crossing the Large (LG/J) and Small (SM/J) inbred strains. Although no individual genes (QTLs=quantitative trait loci) on any of the 19 autosomes significantly affected FA for centroid size, and only two affected shape FA, a number of pairwise combinations of QTLs exhibited significant epistasis for FA in both molar size and shape. The QTLs involved in these interactions differed for FA in molar size versus FA in molar shape, but their epistatic contributions to the total variance was nearly the same (about 20%) for FA in both molar characters. It was noted that the genetic architecture of FA in the molar characters, consisting of little or no additive genetic variance but an abundance of epistatic genetic variance, is consistent with that of other typical fitness components such as litter size.     

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.     

A quantitative genetic model for mixed diploid and triploid hybrid progenies in tree breeding and evolution

Interspecific hybridization has played a critical role in tree evolution and breeding. The findings of triploidy in forest trees stimulate the development of a quantitative genetic model to estimate the nature of gene action. The model is based on clonally replicated triploid progenies derived from a two-level population and individual-within-population mating design in which offspring have a double dose of alleles from the parent and a single dose of alleles from the other parent. With the same genetic assumptions of a diploid model, except non-Mendelian behavior at meiosis, and the experimental variances estimated from a linear statistical model, total genetic variances in the triploid progenies are separated into additive, dominance, and epistatic components. In addition, by combining the new model with the already existing model based on disomic expression, the partitioning of additive, dominant, and epistatic variances can be obtained for a mixed diploid/triploid F₁ progeny population. This paper provides an alternative technique to study the modes of quantitative inheritance for outcrossing, long-lived forest trees in which inbred lines cannot be easily generated. The accuracy for estimating gene action using this technique is discussed.     

Genomic prediction of hybrid crops allows disentangling dominance and epistasis

We revisited, in a genomic context, the theory of hybrid genetic evaluation models of hybrid crosses of pure lines, as the current practice is largely based on infinitesimal model assumptions. Expressions for covariances between hybrids due to additive substitution effects and dominance and epistatic deviations were analytically derived. Using dense markers in a GBLUP analysis, it is possible to split specific combining ability into dominance and across-groups epistatic deviations, and to split general combining ability (GCA) into within-line additive effects and within-line additive by additive (and higher order) epistatic deviations. We analyzed a publicly available maize data set of Dent × Flint hybrids using our new model (called GCA-model) up to additive by additive epistasis. To model higher order interactions within GCAs, we also fitted “residual genetic” line effects. Our new GCA-model was compared with another genomic model which assumes a uniquely defined effect of genes across origins. Most variation in hybrids is accounted by GCA. Variances due to dominance and epistasis have similar magnitudes. Models based on defining effects either differently or identically across heterotic groups resulted in similar predictive abilities for hybrids. The currently used model inflates the estimated additive genetic variance. This is not important for hybrid predictions but has consequences for the breeding scheme—e.g. overestimation of the genetic gain within heterotic group. Therefore, we recommend using GCA-model, which is appropriate for genomic prediction and variance component estimation in hybrid crops using genomic data, and whose results can be practically interpreted and used for breeding purposes.     

A General Model for Multilocus Epistatic Interactions in Case-Control Studies

Background

Epistasis, i.e., the interaction of alleles at different loci, is thought to play a central role in the formation and progression of complex diseases. The complexity of disease expression should arise from a complex network of epistatic interactions involving multiple genes.

Methodology

We develop a general model for testing high-order epistatic interactions for a complex disease in a case-control study. We incorporate the quantitative genetic theory of high-order epistasis into the setting of cases and controls sampled from a natural population. The new model allows the identification and testing of epistasis and its various genetic components.

Conclusions

Simulation studies were used to examine the power and false positive rates of the model under different sampling strategies. The model was used to detect epistasis in a case-control study of inflammatory bowel disease, in which five SNPs at a candidate gene were typed, leading to the identification of a significant three-locus epistasis.     

The Contribution of Quantitative Trait Loci and Neutral Marker Loci to the Genetic Variances and Covariances among Quantitative Traits in Random Mating Populations

Using Cockerham's approach of orthogonal scales, we develop genetic models for the effect of an arbitrary number of multiallelic quantitative trait loci (QTLs) or neutral marker loci (NMLs) upon any number of quantitative traits. These models allow the unbiased estimation of the contributions of a set of marker loci to the additive and dominance variances and covariances among traits in a random mating population. The method has been applied to an analysis of allozyme and quantitative data from the European oyster. The contribution of a set of marker loci may either be real, when the markers are actually QTLs, or apparent, when they are NMLs that are in linkage disequilibrium with hidden QTLs. Our results show that the additive and dominance variances contributed by a set of NMLs are always minimum estimates of the corresponding variances contributed by the associated QTLs. In contrast, the apparent contribution of the NMLs to the additive and dominance covariances between two traits may be larger than, equal to or lower than the actual contributions of the QTLs. We also derive an expression for the expected variance explained by the correlation between a quantitative trait and multilocus heterozygosity. This correlation explains only a part of the genetic variance contributed by the markers, i.e., in general, a combination of additive and dominance variances and, thus, provides only very limited information relative to the method supplied here.     

The effect of epistasis between linked genes on quantitative trait locus analysis

The effect of epistasis between linked genes on quantitative trait locus (QTL) analysis was studied as a function of their contribution to the phenotypic variance and their genetic distance by simulation of F2 (at least 200 individuals) and recombinant inbred line (RIL) populations. Data sets were replicated 100 times. For F2 populations, the presence of epistasis improves the detection of QTLs having effects in opposite directions. Epistasis between linked QTLs (26.5 cM) was poorly detected even when its contribution was relatively high compared to the main effects, and was null for heritabilities lower than 0.10. The detection of false-positive main effects is strongly affected by the distance between epistatic QTLs. The closer they are (≤11.5 cM), the higher the probability of detecting false-positive main-effect QTLs and the lower the probability of detecting epistatic effects. In this case, the presence of main-effect QTLs is due to the deviation of the heterozygote from the homozygotes at each linked interacting QTL and is algebraically explained by the joint effect of the linkage and the additive-by-additive interaction, resulting in a heterosis at a single genomic region in the absence of simulated dominant genetic effects. The number of false-positive main effects only reached nominal levels at about 100 cM. For RIL populations, the number of false positives or the detection of existing epistasis does not depend on the distance, and the power to detect epistatic QTLs is much higher even with small sample sizes and low contributions to the trait. RIL populations are highly recommended to detect epistatic QTLs and to better infer the genetic architecture of a quantitative trait.     

Use of randomization testing to detect multiple epistatic QTLs

Here, we describe a randomization testing strategy for mapping interacting quantitative trait loci (QTLs). In a forward selection strategy, non-interacting QTLs and simultaneously mapped interacting QTL pairs are added to a total genetic model. Simultaneous mapping of epistatic QTLs increases the power of the mapping strategy by allowing detection of interacting QTL pairs where none of the QTL can be detected by their marginal additive and dominance effects. Randomization testing is used to derive empirical significance thresholds for every model selection step in the procedure. A simulation study was used to evaluate the statistical properties of the proposed randomization tests and for which types of epistasis simultaneous mapping of epistatic QTLs adds power. Least squares regression was used for QTL parameter estimation but any other QTL mapping method can be used. A genetic algorithm was used to search for interacting QTL pairs, which makes the proposed strategy feasible for single processor computers. We believe that this method will facilitate the evaluation of the importance at epistatic interaction among QTLs controlling multifactorial traits and disorders.     

A method for marker-assisted selection based on QTLs with epistatic effects

A method for marker-assisted selection (MAS) based on quantitative trait loci (QTLs) with epistatic effects is proposed. The efficiency of such method is investigated by simulations under a wide range of situations. In the presence of epistasis, MAS generally yields longer persistence response than that based exclusively on additive or additive and dominance. Neglecting epistasis could result in considerable loss in response, and more pronounced at later generations. In addition to population size and trait heritability, genetic variance configurations play an important role in determining both the short- and long-term efficiencies of MAS. MAS using breeding values not only achieves higher response, but also tends to have smaller standard error than other methods in most cases. Errors in QTL detection cause distinct reductions in responses to MAS in most cases. It is thus concluded that verifications of putative QTL and its magnitude of effect and accurate map chromosome location are imperative to realize the potentials of MAS.     

Signals of demographic expansion in Drosophila virilis

Background

The antagonistic co-evolution of hosts and their parasites is considered to be a potential driving force in maintaining host genetic variation including sexual reproduction and recombination. The examination of this hypothesis calls for information about the genetic basis of host-parasite interactions – such as how many genes are involved, how big an effect these genes have and whether there is epistasis between loci. We here examine the genetic architecture of quantitative resistance in animal and plant hosts by concatenating published studies that have identified quantitative trait loci (QTL) for host resistance in animals and plants.

Results

Collectively, these studies show that host resistance is affected by few loci. We particularly show that additional epistatic interactions, especially between loci on different chromosomes, explain a majority of the effects. Furthermore, we find that when experiments are repeated using different host or parasite genotypes under otherwise identical conditions, the underlying genetic architecture of host resistance can vary dramatically – that is, involves different QTLs and epistatic interactions. QTLs and epistatic loci vary much less when host and parasite types remain the same but experiments are repeated in different environments.

Conclusion

This pattern of variability of the genetic architecture is predicted by strong interactions between genotypes and corroborates the prevalence of varying host-parasite combinations over varying environmental conditions. Moreover, epistasis is a major determinant of phenotypic variance for host resistance. Because epistasis seems to occur predominantly between, rather than within, chromosomes, segregation and chromosome number rather than recombination via cross-over should be the major elements affecting adaptive change in host resistance.