How to perform meaningful estimates of genetic effects

Although the genotype-phenotype map plays a central role both in Quantitative and Evolutionary Genetics, the formalization of a completely general and satisfactory model of genetic effects, particularly accounting for epistasis, remains a theoretical challenge. Here, we use a two-locus genetic system in simulated populations with epistasis to show the convenience of using a recently developed model, NOIA, to perform estimates of genetic effects and the decomposition of the genetic variance that are orthogonal even under deviations from the Hardy-Weinberg proportions. We develop the theory for how to use this model in interval mapping of quantitative trait loci using Halley-Knott regressions, and we analyze a real data set to illustrate the advantage of using this approach in practice. In this example, we show that departures from the Hardy-Weinberg proportions that are expected by sampling alone substantially alter the orthogonal estimates of genetic effects when other statistical models, like F₂ or G2A, are used instead of NOIA. Finally, for the first time from real data, we provide estimates of functional genetic effects as sets of effects of natural allele substitutions in a particular genotype, which enriches the debate on the interpretation of genetic effects as implemented both in functional and in statistical models. We also discuss further implementations leading to a completely general genotype-phenotype map.     

Applying QTL analysis to conservation genetics

Both analytical and molecular tools currently exist that can be used to prolifically apply quantitative trait loci (QTL) analysis to the study of natural populations. In this communication, we review and exemplify the use of QTL mapping tools and genetic modeling for conservation geneticists. We simulate populations inspired by relevant cases that can be encountered in the field and analyze them using the recently developed flexible intercross analysis (FIA) method. We then reanalyze these results with the also recently developed natural and orthogonal interactions (NOIA) model of genetic effects. Next, we further exemplify the potential of genetic modeling for the interpretation of the output of QTL analyses by reviewing studies on hybrids between wild individuals and their domesticated relatives. Based on the results here presented we emphasize several points that are pertinent in conservation genetics including (i) the advantages of FIA as a powerful tool to be applied to line crosses in which the parental lines are not inbred, (ii) the importance of obtaining estimates of genetic effects that are adequate to address the research issue under consideration, (iii) the versatility of genetic modeling, particularly NOIA, to dissect complex genetic architectures and (iv) the possibility of using currently available methods to address non-equilibrium multiallelic systems.     

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.     

Influence of Gene Interaction on Complex Trait Variation with Multilocus Models

Although research effort is being expended into determining the importance of epistasis and epistatic variance for complex traits, there is considerable controversy about their importance. Here we undertake an analysis for quantitative traits utilizing a range of multilocus quantitative genetic models and gene frequency distributions, focusing on the potential magnitude of the epistatic variance. All the epistatic terms involving a particular locus appear in its average effect, with the number of two-locus interaction terms increasing in proportion to the square of the number of loci and that of third order as the cube and so on. Hence multilocus epistasis makes substantial contributions to the additive variance and does not, per se, lead to large increases in the nonadditive part of the genotypic variance. Even though this proportion can be high where epistasis is antagonistic to direct effects, it reduces with multiple loci. As the magnitude of the epistatic variance depends critically on the heterozygosity, for models where frequencies are widely dispersed, such as for selectively neutral mutations, contributions of epistatic variance are always small. Epistasis may be important in understanding the genetic architecture, for example, of function or human disease, but that does not imply that loci exhibiting it will contribute much genetic variance. Overall we conclude that theoretical predictions and experimental observations of low amounts of epistatic variance in outbred populations are concordant. It is not a likely source of missing heritability, for example, or major influence on predictions of rates of evolution.     

A unified model for functional and statistical epistasis and its application in quantitative trait Loci analysis

Interaction between genes, or epistasis, is found to be common and it is a key concept for understanding adaptation and evolution of natural populations, response to selection in breeding programs, and determination of complex disease. Currently, two independent classes of models are used to study epistasis. Statistical models focus on maintaining desired statistical properties for detection and estimation of genetic effects and for the decomposition of genetic variance using average effects of allele substitutions in populations as parameters. Functional models focus on the evolutionary consequences of the attributes of the genotype-phenotype map using natural effects of allele substitutions as parameters. Here we provide a new, general and unified model framework: the natural and orthogonal interactions (NOIA) model. NOIA implements tools for transforming genetic effects measured in one population to the ones of other populations (e.g., between two experimental designs for QTL) and parameters of statistical and functional epistasis into each other (thus enabling us to obtain functional estimates of QTL), as demonstrated numerically. We develop graphical interpretations of functional and statistical models as regressions of the genotypic values on the gene content, which illustrates the difference between the models--the constraint on the slope of the functional regression--and when the models are equivalent. Furthermore, we use our theoretical foundations to conceptually clarify functional and statistical epistasis, discuss the advantages of NOIA over previous theory, and stress the importance of linking functional and statistical models.     

Natural and orthogonal interaction framework for modeling gene-environment interactions with application to lung cancer

Objectives: We aimed at extending the Natural and Orthogonal Interaction (NOIA) framework, developed for modeling gene-gene interactions in the analysis of quantitative traits, to allow for reduced genetic models, dichotomous traits, and gene-environment interactions. We evaluate the performance of the NOIA statistical models using simulated data and lung cancer data. Methods: The NOIA statistical models are developed for additive, dominant, and recessive genetic models as well as for a binary environmental exposure. Using the Kronecker product rule, a NOIA statistical model is built to model gene-environment interactions. By treating the genotypic values as the logarithm of odds, the NOIA statistical models are extended to the analysis of case-control data. Results: Our simulations showed that power for testing associations while allowing for interaction using the NOIA statistical model is much higher than using functional models for most of the scenarios we simulated. When applied to lung cancer data, much smaller p values were obtained using the NOIA statistical model for either the main effects or the SNP-smoking interactions for some of the SNPs tested. Conclusion: The NOIA statistical models are usually more powerful than the functional models in detecting main effects and interaction effects for both quantitative traits and binary traits.     

An empirical Bayes method for estimating epistatic effects of quantitative trait loci

Summary .   The genetic variance of a quantitative trait is often controlled by the segregation of multiple interacting loci. Linear model regression analysis is usually applied to estimating and testing effects of these quantitative trait loci (QTL). Including all the main effects and the effects of interaction (epistatic effects), the dimension of the linear model can be extremely high. Variable selection via stepwise regression or stochastic search variable selection (SSVS) is the common procedure for epistatic effect QTL analysis. These methods are computationally intensive, yet they may not be optimal. The LASSO (least absolute shrinkage and selection operator) method is computationally more efficient than the above methods. As a result, it has been widely used in regression analysis for large models. However, LASSO has never been applied to genetic mapping for epistatic QTL, where the number of model effects is typically many times larger than the sample size. In this study, we developed an empirical Bayes method (E-BAYES) to map epistatic QTL under the mixed model framework. We also tested the feasibility of using LASSO to estimate epistatic effects, examined the fully Bayesian SSVS, and reevaluated the penalized likelihood (PENAL) methods in mapping epistatic QTL. Simulation studies showed that all the above methods performed satisfactorily well. However, E-BAYES appears to outperform all other methods in terms of minimizing the mean-squared error (MSE) with relatively short computing time. Application of the new method to real data was demonstrated using a barley dataset.     

Natural and orthogonal model for estimating gene–gene interactions applied to cutaneous melanoma

Epistasis, or gene–gene interaction, results from joint effects of genes on a trait; thus, the same alleles of one gene may display different genetic effects in different genetic backgrounds. In this study, we generalized the coding technique of a natural and orthogonal interaction (NOIA) model for association studies along with gene–gene interactions for dichotomous traits and human complex diseases. The NOIA model which has non-correlated estimators for genetic effects is important for estimating influence from multiple loci. We conducted simulations and data analyses to evaluate the performance of the NOIA model. Both simulation and real data analyses revealed that the NOIA statistical model had higher power for detecting main genetic effects and usually had higher power for some interaction effects than the usual model. Although associated genes have been identified for predisposing people to melanoma risk: HERC2 at 15q13.1, MC1R at 16q24.3 and CDKN2A at 9p21.3, no gene–gene interaction study has been fully explored for melanoma. By applying the NOIA statistical model to a genome-wide melanoma dataset, we confirmed the previously identified significantly associated genes and found potential regions at chromosomes 5 and 4 that may interact with the HERC2 and MC1R genes, respectively. Our study not only generalized the orthogonal NOIA model but also provided useful insights for understanding the influence of interactions on melanoma risk.     

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.     

Genome‐wide association and epistasis studies unravel the genetic architecture of sudden death syndrome resistance in soybean

Soybean [Glycine max (L.) Merr.] is an economically important crop that is grown worldwide. Sudden death syndrome (SDS), caused by Fusarium virguliforme, is one of the top yield‐limiting diseases in soybean. However, the genetic basis of SDS resistance, especially with respect to epistatic interactions, is still unclear. To better understand the genetic architecture of soybean SDS resistance, genome‐wide association and epistasis studies were performed using a population of 214 germplasm accessions and 31 914 SNPs from the SoySNP50K Illumina Infinium BeadChip. Twelve loci and 12 SNP–SNP interactions associated with SDS resistance were identified at various time points after inoculation. These additive and epistatic loci together explained 24–52% of the phenotypic variance. Disease‐resistant, pathogenesis‐related and chitin‐ and wound‐responsive genes were identified in the proximity of peak SNPs, including stress‐induced receptor‐like kinase gene 1 (SIK1), which is pinpointed by a trait‐associated SNP and encodes a leucine‐rich repeat‐containing protein. We report that the proportion of phenotypic variance explained by identified loci may be considerably improved by taking epistatic effects into account. This study shows the necessity of considering epistatic effects in soybean SDS resistance breeding using marker‐assisted and genomic selection approaches. Based on our findings, we propose a model for soybean root defense against the SDS pathogen. Our results facilitate identification of the molecular mechanism underlying SDS resistance in soybean, and provide a genetic basis for improvement of soybean SDS resistance through breeding strategies based on additive and epistatic effects.     

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.     

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.

     

Epistasis of quantitative trait loci under different gene action models

Modeling and detecting nonallelic (epistatic) effects at multiple quantitative trait loci (QTL) often assume that the study population is in zygotic equilibrium (i.e., genotypic frequencies at different loci are products of corresponding single-locus genotypic frequencies). However, zygotic associations can arise from physical linkages between different loci or from many evolutionary and demographic processes even for unlinked loci. We describe a new model that partitions the two-locus genotypic values in a zygotic disequilibrium population into equilibrium and residual portions. The residual portion is of course due to the presence of zygotic associations. The equilibrium portion has eight components including epistatic effects that can be defined under three commonly used equilibrium models, Cockerham's model, F2-metric, and F(infinity)-metric models. We evaluate our model along with these equilibrium models theoretically and empirically. While all the equilibrium models require zygotic equilibrium, Cockerham's model is the most general, allowing for Hardy-Weinberg disequilibrium and arbitrary gene frequencies at individual loci whereas F2-metric and F(infinity)-metric models require gene frequencies of one-half in a Hardy-Weinberg equilibrium population. In an F2 population with two unlinked loci, Cockerham's model is reduced to the F2-metric model and thus both have a desirable property of orthogonality among the genic effects; the genic effects under the F(infinity)-metric model are not orthogonal but they can be easily translated into those under the F2-metric model through a simple relation. Our model is reduced to these equilibrium models in the absence of zygotic associations. The results from our empirical analysis suggest that the residual genetic variance arising from zygotic associations can be substantial and may be an important source of bias in QTL mapping studies.     

The effect of non-additive genetic interactions on selection in multi-locus genetic models

Additive genetic variance might usually be expected to decrease in a finite population because of genetic drift. However, both theoretical and empirical studies have shown that the additive genetic variance of a population could, in some cases, actually increase owing to the action of genetic drift in presence of non-additive effects. We used Monte-Carlo simulations to address a less-well-studied issue: the effects of directional truncation selection on a trait affected by non-additive genetic variation. We investigated the effects on genetic variance and the response to selection. We compared two different genetic models, representing various numbers of loci. We found that the additive genetic variance could also increase in the case of truncation selection, when dominance and epistasis was present. Additive-by-additive epistatic effects generally gave a higher increase in additive variance compared to dominance. However, the magnitude of the increase differed depending on the particular model and on the number of loci.     

Evidence for the genetic basis and epistatic interactions underlying ocean‐ and river‐maturing ecotypes of Pacific Lamprey (Entosphenus tridentatus) returning to the Klamath River,California

Surveys of genomic variation have improved our understanding of the relationship between fitness‐related phenotypes and their underlying genetic basis. In some cases, single large‐effect genes have been found to underlie important traits; however, complex traits are expected to be under polygenic control and elucidation of multiple gene interactions may be required to fully understand the genetic basis of the trait. In this study, we investigated the genetic basis of the ocean‐ and river‐maturing ecotypes in anadromous Pacific lamprey (Entosphenus tridentatus). In Pacific lamprey, the ocean‐maturing ecotype is distinguished by advanced maturity of females (e.g., large egg mass) at the onset of freshwater migration relative to immature females of the river‐maturing ecotype. We examined a total of 219 adult Pacific lamprey that were collected at‐entry to the Klamath River over a 12‐month period. Each individual was genotyped at 308 SNPs representing known neutral and adaptive loci and measured at morphological traits, including egg mass as an indicator of ocean‐ and river‐maturing ecotype for females. The two ecotypes did not exhibit genetic structure at 148 neutral loci, indicating that ecotypic diversity exists within a single population. In contrast, we identified the genetic basis of maturation ecotypes in Pacific lamprey as polygenic, involving two unlinked gene regions that have a complex epistatic relationship. Importantly, these gene regions appear to show stronger effects when considered in gene interaction models than if just considered additive, illustrating the importance of considering epistatic effects and gene networks when researching the genetic basis of complex traits in Pacific lamprey and other species.     

Bayesian analysis for genetic architecture of dynamic traits

The dissection of the genetic architecture of quantitative traits, including the number and locations of quantitative trait loci (QTL) and their main and epistatic effects, has been an important topic in current QTL mapping. We extend the Bayesian model selection framework for mapping multiple epistatic QTL affecting continuous traits to dynamic traits in experimental crosses. The extension inherits the efficiency of Bayesian model selection and the flexibility of the Legendre polynomial model fitting to the change in genetic and environmental effects with time. We illustrate the proposed method by simultaneously detecting the main and epistatic QTLs for the growth of leaf age in a doubled-haploid population of rice. The behavior and performance of the method are also shown by computer simulation experiments. The results show that our method can more quickly identify interacting QTLs for dynamic traits in the models with many numbers of genetic effects, enhancing our understanding of genetic architecture for dynamic traits. Our proposed method can be treated as a general form of mapping QTL for continuous quantitative traits, being easier to extend to multiple traits and to a single trait with repeat records.     

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.     

GENETIC ARCHITECTURE AND POSTZYGOTIC REPRODUCTIVE ISOLATION: EVOLUTION OF BATESON–DOBZHANSKY–MULLER INCOMPATIBILITIES IN A POLYGENIC MODEL

The Bateson–Dobzhansky–Muller model predicts that postzygotic isolation evolves due to the accumulation of incompatible epistatic interactions, but few studies have quantified the relationship between genetic architecture and patterns of reproductive divergence. We examined how the direction and magnitude of epistatic interactions in a polygenic trait under stabilizing selection influenced the evolution of hybrid incompatibilities. We found that populations evolving independently under stabilizing selection experienced suites of compensatory allelic changes that resulted in genetic divergence between populations despite the maintenance of a stable, high‐fitness phenotype. A small number of loci were then incompatible with multiple alleles in the genetic background of the hybrid and the identity of these incompatibility loci changed over the evolution of the populations. For F₁ hybrids, reduced fitness evolved in a window of intermediate strengths of epistatic interactions, but F₂ and backcross hybrids evolved reduced fitness across weak and moderate strengths of epistasis due to segregation variance. Strong epistatic interactions constrained the allelic divergence of parental populations and prevented the development of reproductive isolation. Because many traits with varying genetic architectures must be under stabilizing selection, our results indicate that polygenetic drift is a plausible hypothesis for the evolution of postzygotic reproductive isolation.     

Epistatic Module Detection for Case-Control Studies: A Bayesian Model with a Gibbs Sampling Strategy

The detection of epistatic interactive effects of multiple genetic variants on the susceptibility of human complex diseases is a great challenge in genome-wide association studies (GWAS). Although methods have been proposed to identify such interactions, the lack of an explicit definition of epistatic effects, together with computational difficulties, makes the development of new methods indispensable. In this paper, we introduce epistatic modules to describe epistatic interactive effects of multiple loci on diseases. On the basis of this notion, we put forward a Bayesian marker partition model to explain observed case-control data, and we develop a Gibbs sampling strategy to facilitate the detection of epistatic modules. Comparisons of the proposed approach with three existing methods on seven simulated disease models demonstrate the superior performance of our approach. When applied to a genome-wide case-control data set for Age-related Macular Degeneration (AMD), the proposed approach successfully identifies two known susceptible loci and suggests that a combination of two other loci—one in the gene SGCD and the other in SCAPER—is associated with the disease. Further functional analysis supports the speculation that the interaction of these two genetic variants may be responsible for the susceptibility of AMD. When applied to a genome-wide case-control data set for Parkinson's disease, the proposed method identifies seven suspicious loci that may contribute independently to the disease.