Modeling quantitative trait Loci and interpretation of models

A quantitative genetic model relates the genotypic value of an individual to the alleles at the loci that contribute to the variation in a population in terms of additive, dominance, and epistatic effects. This partition of genetic effects is related to the partition of genetic variance. A number of models have been proposed to describe this relationship: some are based on the orthogonal partition of genetic variance in an equilibrium population. We compare a few representative models and discuss their utility and potential problems for analyzing quantitative trait loci (QTL) in a segregating population. An orthogonal model implies that estimates of the genetic effects are consistent in a full or reduced model in an equilibrium population and are directly related to the partition of the genetic variance in the population. Linkage disequilibrium does not affect the estimation of genetic effects in a full model, but would in a reduced model. Certainly linkage disequilibrium would complicate the detection of QTL and epistasis. Using different models does not influence the detection of QTL and epistasis. However, it does influence the estimation and interpretation of genetic effects.     

Epistatic pleiotropy and the genetic architecture of covariation within early and late-developing skull trait complexes in mice

The role of epistasis as a source of trait variation is well established, but its role as a source of covariation among traits (i.e., as a source of "epistatic pleiotropy") is rarely considered. In this study we examine the relative importance of epistatic pleiotropy in producing covariation within early and late-developing skull trait complexes in a population of mice derived from an intercross of the Large and Small inbred strains. Significant epistasis was found for several pairwise combinations of the 21 quantitative trait loci (QTL) affecting early developing traits and among the 20 QTL affecting late-developing traits. The majority of the epistatic effects were restricted to single traits but epistatic pleiotropy still contributed significantly to covariances. Because of their proportionally larger effects on variances than on covariances, epistatic effects tended to reduce within-group correlations of traits and reduce their overall degree of integration. The expected contributions of single-locus and two-locus epistatic pleiotropic QTL effects to the genetic covariance between traits were analyzed using a two-locus population genetic model. The model demonstrates that, for single-locus or epistatic pleiotropy to contribute to trait covariances in the study population, both traits must show the same pattern of single-locus or epistatic effects. As a result, a large number of the cases where loci show pleiotropic effects do not contribute to the covariance between traits in this population because the loci show a different pattern of effect on the different traits. In general, covariance patterns produced by single-locus and epistatic pleiotropy predicted by the model agreed well with actual values calculated from the QTL analysis. Nearly all single-locus and epistatic pleiotropic effects contributed positive components to covariances between traits, suggesting that genetic integration in the skull is achieved by a complex combination of pleiotropic effects.     

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.     

Multilocus Population Genetics with Weak Epistasis. II. Equilibrium Properties of Multilocus Models: What Is the Unit of Selection?

Using perturbation techniques, I study the equilibrium of deterministic discrete time multilocus models with weak epistasis. The most important results are on the relationship between epistasis and disequilibrium. Disequilibrium involving a particular set of loci reflects only epistasis simultaneously involving those loci. Moreover, all the disequilibria of all orders vary approximately as the inverse of the probability of at least one recombination event among the loci involved. Finally, higher order disequilibria among loci will be lower than lower order ones, even if the level of epistasis is the same at all orders. In this sense, the unit of selection is small. However, given the larger number of higher order disequilibria, these higher order disequilibria may play an important role in the computation of gametic frequencies from allelic frequencies in models with a large number of loci. Finally, I show that epistasis between blocks of loci will be averages of epistatic effects, not additions of epistatic effects. Thus, failure to find significant epistasis on a chromosomal basis does not rule out the importance of epistatic effects.     

Both additivity and epistasis control the genetic variation for fruit quality traits in tomato

The effect of a gene involved in the variation of a quantitative trait may change due to epistatic interactions with the overall genetic background or with other genes through digenic interactions. The classical populations used to map quantitative trait loci (QTL) are poorly efficient to detect epistasis. To assess the importance of epistasis in the genetic control of fruit quality traits, we compared 13 tomato lines having the same genetic background except for one to five chromosome fragments introgressed from a distant line. Six traits were assessed: fruit soluble solid content, sugar content and titratable acidity, fruit weight, locule number and fruit firmness. Except for firmness, a large part of the variation of the six traits was under additive control, but interactions between QTL leading to epistasis effects were common. In the lines cumulating several QTL regions, all the significant epistatic interactions had a sign opposite to the additive effects, suggesting less than additive epistasis. Finally the re-examination of the segregating population initially used to map the QTL confirmed the extent of epistasis, which frequently involved a region where main effect QTL have been detected in this progeny or in other studies.     

Epistatic interactions of genes influence within-individual variation of physical activity traits in mice

A number of quantitative trait loci (QTLs) recently have been discovered that affect various activity traits in mice, but their collective impact does not appear to explain the consistently moderate to high heritabilities for these traits. We previously suggested interactions of genes, or epistasis, might account for additional genetic variability of activity, and tested this for the average distance, duration and speed run by mice during a 3 week period. We found abundant evidence for epistasis affecting these traits, although, recognized that epistatic effects may well vary within individuals over time. We therefore conducted a full genome scan for epistatic interactions affecting these traits in each of seven three-day intervals. Our intent was to assess the extent and trends in epistasis affecting these traits in each of the intervals. We discovered a number of epistatic interactions of QTLs that influenced the activity traits in the mice, the majority of which were not previously found and appeared to affect the activity traits (especially distance and speed) primarily in the early or in the late age intervals. The overall impact of epistasis was considerable, its contribution to the total phenotypic variance varying from an average of 22–35% in the three traits across all age intervals. It was concluded that epistasis is more important than single-locus effects of genes on activity traits at specific ages and it is therefore an essential component of the genetic architecture of physical activity.     

水稻株高上位性效应和ＱＥ互作效应的ＱＴＬ遗传研究

利用基因混合模型的ＱＴＬ定位方法研究了由籼稻品种ＩＲ６４和粳稻品种Ａzucena杂交衍生的ＤＨ群体在４个环境中的ＱＴＬ上位性效应和环境互作效应,结果表明,上位性是数量性状的重要遗传基础,并揭示了上位性的几个重要特点,所有的ＱＴＬ都参与了上位性效应的形成,６４％的ＱＴＬ还具有本身的加性效应,因此传统方法对ＱＴＬ加性效应的估算会由于上位性的影响而有偏,其他３６％的ＱＴＬ没有本身的加性效应,却参与了４８％的上位性互作用,这些位点可能通过诱发和修饰其他位点而起作用,上位性的特点还包括,经常发现了一个ＱＴＬ与多个ＱＴＬ发生互作;大效应的ＱＴＬ也参与上位性互作;上位性互作受环境影响,ＱＴＬ与环境的互效应比ＱＴＬ的主效应更多地被检测到,表明数量性状基因的表达易受环境影响。     

COMPARISON OF NON‐GAUSSIAN QUANTITATIVE GENETIC MODELS FOR MIGRATION AND STABILIZING SELECTION

The balance between stabilizing selection and migration of maladapted individuals has formerly been modeled using a variety of quantitative genetic models of increasing complexity, including models based on a constant expressed genetic variance and models based on normality. The infinitesimal model can accommodate nonnormality and a nonconstant genetic variance as a result of linkage disequilibrium. It can be seen as a parsimonious one‐parameter model that approximates the underlying genetic details well when a large number of loci are involved. Here, the performance of this model is compared to several more realistic explicit multilocus models, with either two, several or a large number of alleles per locus with unequal effect sizes. Predictions for the deviation of the population mean from the optimum are highly similar across the different models, so that the non‐Gaussian infinitesimal model forms a good approximation. It does, however, generally estimate a higher genetic variance than the multilocus models, with the difference decreasing with an increasing number of loci. The difference between multilocus models depends more strongly on the effective number of loci, accounting for relative contributions of loci to the variance, than on the number of alleles per locus.     

Little epistasis for anxiety-related measures in the DeFries strains of laboratory mice

Recent advances in methodologies for testing epistatic interactions, combined with several successes in demonstrating genetic interaction effects in animal and human genetics, have rekindled interest in the role of epistatic influences on complex traits. It has even been suggested that the unacknowledged presence of epistasis vitiates the genetic dissection of human and animal behavior. Here we report a genome-wide interaction analysis of 1636 F₂ mice to show that epistasis is of minimal importance in an animal model of anxiety. By using a sufficiently large sample of F₂ animals, we provide evidence that interaction effects between any two loci contribute less than 5% to the total phenotypic variance in multiple tests of anxiety. We conclude that interactions between loci do not necessarily vitiate the genetic analysis of behavior in at least one animal model of anxiety.     

Effect on linkage disequilibrium of selection for a quantitative character with epistasis

Summary Selection for a character controlled by additive genes induces linkage disequilibrium which reduces the additive genetic variance usable for further selective gains. Additive x additive epistasis contributes to selection response through development of linkage disequilibrium between interacting loci. To investigate the relative importance of the two effects of linkage disequilibrium, formulae are presented and results are reported of simulations using models involving additive, additive x additive and dominance components. The results suggest that so long as epistatic effects are not large relative to additive effects, and the proportion of pairs of loci which show epistasis is not very high, the predominant effect of linkage disequilibrium will be to reduce the rate of selection response.     

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.     

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 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.     

Efficiency of triple test cross for detecting epistasis with marker information

The triple test cross (TTC) is an experimental design for detecting epistasis and estimating the components of genetic variance for quantitative traits. In this paper, we extend the analysis to include molecular information. The statistical power of the mating design was assessed under a model assuming that a finite number of loci affect the trait in question. Formulae are developed for the analysis with or without marker information relating to the recombination fraction between loci, the genetical properties of quantitative trait controlled by the quantitative trait loci (QTL), the linkage phases of the parents and population size. Application of these formulae showed that the recombination fraction between genes and the magnitude and the types of epistasis have important interactions in their effects on power. The results demonstrate that the TTC may have increased power to detect epistasis when marker information is present. However, the simulation experiments show that the standard deviation of the estimated expected mean square was higher with one marker than that with two, whereas the corresponding value without marker information was the lowest. In addition, we demonstrate that the relative position of QTL and markers and the number of markers can both affect the power of epistatic detection.     

Statistical epistasis is a generic feature of gene regulatory networks

Functional dependencies between genes are a defining characteristic of gene networks underlying quantitative traits. However, recent studies show that the proportion of the genetic variation that can be attributed to statistical epistasis varies from almost zero to very high. It is thus of fundamental as well as instrumental importance to better understand whether different functional dependency patterns among polymorphic genes give rise to distinct statistical interaction patterns or not. Here we address this issue by combining a quantitative genetic model approach with genotype-phenotype models capable of translating allelic variation and regulatory principles into phenotypic variation at the level of gene expression. We show that gene regulatory networks with and without feedback motifs can exhibit a wide range of possible statistical genetic architectures with regard to both type of effect explaining phenotypic variance and number of apparent loci underlying the observed phenotypic effect. Although all motifs are capable of harboring significant interactions, positive feedback gives rise to higher amounts and more types of statistical epistasis. The results also suggest that the inclusion of statistical interaction terms in genetic models will increase the chance to detect additional QTL as well as functional dependencies between genetic loci over a broad range of regulatory regimes. This article illustrates how statistical genetic methods can fruitfully be combined with nonlinear systems dynamics to elucidate biological issues beyond reach of each methodology in isolation.     

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.