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.     

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.

     

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.     

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.     

A complete classification of epistatic two-locus models

Background

The study of epistasis is of great importance in statistical genetics in fields such as linkage and association analysis and QTL mapping. In an effort to classify the types of epistasis in the case of two biallelic loci Li and Reich listed and described all models in the simplest case of 0/1 penetrance values. However, they left open the problem of finding a classification of two-locus models with continuous penetrance values.

Results

We provide a complete classification of biallelic two-locus models. In addition to solving the classification problem for dichotomous trait disease models, our results apply to any instance where real numbers are assigned to genotypes, and provide a complete framework for studying epistasis in QTL data. Our approach is geometric and we show that there are 387 distinct types of two-locus models, which can be reduced to 69 when symmetry between loci and alleles is accounted for. The model types are defined by 86 circuits, which are linear combinations of genotype values, each of which measures a fundamental unit of interaction.

Conclusion

The circuits provide information on epistasis beyond that contained in the additive × additive, additive × dominance, and dominance × dominance interaction terms. We discuss the connection between our classification and standard epistatic models and demonstrate its utility by analyzing a previously published dataset.     

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.     

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.     

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.     

Simultaneous fine mapping of closely linked epistatic quantitative trait loci using combined linkage disequilibrium and linkage with a general pedigree

Causal mutations and their intra- and inter-locus interactions play a critical role in complex trait variation. It is often not easy to detect epistatic quantitative trait loci (QTL) due to complicated population structure requirements for detecting epistatic effects in linkage analysis studies and due to main effects often being hidden by interaction effects. Mapping their positions is even harder when they are closely linked. The data structure requirement may be overcome when information on linkage disequilibrium is used. We present an approach using a mixed linear model nested in an empirical Bayesian approach, which simultaneously takes into account additive, dominance and epistatic effects due to multiple QTL. The covariance structure used in the mixed linear model is based on combined linkage disequilibrium and linkage information. In a simulation study where there are complex epistatic interactions between QTL, it is possible to simultaneously map interacting QTL into a small region using the proposed approach. The estimated variance components are accurate and less biased with the proposed approach compared with traditional models.     

Epistatic QTL pairs associated with meat quality and carcass composition traits in a porcine Duroc × Pietrain population

Background

Quantitative trait loci (QTL) analyses in pig have revealed numerous individual QTL affecting growth, carcass composition, reproduction and meat quality, indicating a complex genetic architecture. In general, statistical QTL models consider only additive and dominance effects and identification of epistatic effects in livestock is not yet widespread. The aim of this study was to identify and characterize epistatic effects between common and novel QTL regions for carcass composition and meat quality traits in pig.

Methods

Five hundred and eighty five F₂ pigs from a Duroc × Pietrain resource population were genotyped using 131 genetic markers (microsatellites and SNP) spread over the 18 pig autosomes. Phenotypic information for 26 carcass composition and meat quality traits was available for all F₂ animals. Linkage analysis was performed in a two-step procedure using a maximum likelihood approach implemented in the QxPak program.

Results

A number of interacting QTL was observed for different traits, leading to the identification of a variety of networks among chromosomal regions throughout the porcine genome. We distinguished 17 epistatic QTL pairs for carcass composition and 39 for meat quality traits. These interacting QTL pairs explained up to 8% of the phenotypic variance.

Conclusions

Our findings demonstrate the significance of epistasis in pigs. We have revealed evidence for epistatic relationships between different chromosomal regions, confirmed known QTL loci and connected regions reported in other studies. Considering interactions between loci allowed us to identify several novel QTL and trait-specific relationships of loci within and across chromosomes.     

QTLRel: an R Package for Genome-wide Association Studies in which Relatedness is a Concern

Background

Existing software for quantitative trait mapping is either not able to model polygenic variation or does not allow incorporation of more than one genetic variance component. Improperly modeling the genetic relatedness among subjects can result in excessive false positives. We have developed an R package, QTLRel, to enable more flexible modeling of genetic relatedness as well as covariates and non-genetic variance components.

Results

We have successfully used the package to analyze many datasets, including F₃₄ body weight data that contains 688 individuals genotyped at 3105 SNP markers and identified 11 QTL. It took 295 seconds to estimate variance components and 70 seconds to perform the genome scan on an Linux machine equipped with a 2.40GHz Intel(R) Core(TM)2 Quad CPU.

Conclusions

QTLRel provides a toolkit for genome-wide association studies that is capable of calculating genetic incidence matrices from pedigrees, estimating variance components, performing genome scans, incorporating interactive covariates and genetic and non-genetic variance components, as well as other functionalities such as multiple-QTL mapping and genome-wide epistasis.     

Genetic diversity and association mapping of seed vigor in rice (Oryza sativa L.)

Key message

Twenty-seven QTLs were identified for rice seed vigor, in which 16 were novel QTLs. Fifteen elite parental combinations were designed for improving seed vigor in rice.

Abstract

Seed vigor is closely related to direct seeding in rice (Oryza sativa L.). Previous quantitative trait locus (QTL) studies for seed vigor were mainly derived from bi-parental segregating populations and no report from natural populations. In this study, association mapping for seed vigor was performed on a selected sample of 540 rice cultivars (419 from China and 121 from Vietnam). Population structure was estimated on the basis of 262 simple sequence repeat (SSR) markers. Seed vigor was evaluated by root length (RL), shoot length (SL) and shoot dry weight in 2011 and 2012. Abundant phenotypic and genetic diversities were found in the studied population. The population was divided into seven subpopulations, and the levels of linkage disequilibrium (LD) ranged from 10 to 80 cM. We identified 27 marker–trait associations involving 18 SSR markers for three traits. According to phenotypic effects for alleles of the detected QTLs, elite alleles were mined. These elite alleles could be used to design parental combinations and the expected results would be obtained by pyramiding or substituting the elite alleles per QTL (apart from possible epistatic effects). Our results demonstrate that association mapping can complement and enhance previous QTL information for marker-assisted selection and breeding by design.     

Simultaneous mapping of epistatic QTL in DU6i × DBA/2 mice

We have mapped epistatic quantitative trait loci (QTL) in an F₂ cross between DU6i × DBA/2 mice. By including these epistatic QTL and their interaction parameters in the genetic model, we were able to increase the genetic variance explained substantially (8.8%–128.3%) for several growth and body composition traits. We used an analysis method based on a simultaneous search for epistatic QTL pairs without assuming that the QTL had any effect individually. We were able to detect several QTL that could not be detected in a search for marginal QTL effects because the epistasis cancelled out the individual effects of the QTL. In total, 23 genomic regions were found to contain QTL affecting one or several of the traits and eight of these QTL did not have significant individual effects. We identified 44 QTL pairs with significant effects on the traits, and, for 28 of the pairs, an epistatic QTL model fit the data significantly better than a model without interactions. The epistatic pairs were classified by the significance of the epistatic parameters in the genetic model, and visual inspection of the two-locus genotype means identified six types of related genotype–phenotype patterns among the pairs. Five of these patterns resembled previously published patterns of QTL interactions.     

Linear models for joint association and linkage QTL mapping

Background

Populational linkage disequilibrium and within-family linkage are commonly used for QTL mapping and marker assisted selection. The combination of both results in more robust and accurate locations of the QTL, but models proposed so far have been either single marker, complex in practice or well fit to a particular family structure.

Results

We herein present linear model theory to come up with additive effects of the QTL alleles in any member of a general pedigree, conditional to observed markers and pedigree, accounting for possible linkage disequilibrium among QTLs and markers. The model is based on association analysis in the founders; further, the additive effect of the QTLs transmitted to the descendants is a weighted (by the probabilities of transmission) average of the substitution effects of founders'' haplotypes. The model allows for non-complete linkage disequilibrium QTL-markers in the founders. Two submodels are presented: a simple and easy to implement Haley-Knott type regression for half-sib families, and a general mixed (variance component) model for general pedigrees. The model can use information from all markers. The performance of the regression method is compared by simulation with a more complex IBD method by Meuwissen and Goddard. Numerical examples are provided.

Conclusion

The linear model theory provides a useful framework for QTL mapping with dense marker maps. Results show similar accuracies but a bias of the IBD method towards the center of the region. Computations for the linear regression model are extremely simple, in contrast with IBD methods. Extensions of the model to genomic selection and multi-QTL mapping are straightforward.     

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.     

A method to optimize selection on multiple identified quantitative trait loci

A mathematical approach was developed to model and optimize selection on multiple known quantitative trait loci (QTL) and polygenic estimated breeding values in order to maximize a weighted sum of responses to selection over multiple generations. The model allows for linkage between QTL with multiple alleles and arbitrary genetic effects, including dominance, epistasis, and gametic imprinting. Gametic phase disequilibrium between the QTL and between the QTL and polygenes is modeled but polygenic variance is assumed constant. Breeding programs with discrete generations, differential selection of males and females and random mating of selected parents are modeled. Polygenic EBV obtained from best linear unbiased prediction models can be accommodated. The problem was formulated as a multiple-stage optimal control problem and an iterative approach was developed for its solution. The method can be used to develop and evaluate optimal strategies for selection on multiple QTL for a wide range of situations and genetic models.     

Different models of genetic variation and their effect on genomic evaluation

Background

The theory of genomic selection is based on the prediction of the effects of quantitative trait loci (QTL) in linkage disequilibrium (LD) with markers. However, there is increasing evidence that genomic selection also relies on "relationships" between individuals to accurately predict genetic values. Therefore, a better understanding of what genomic selection actually predicts is relevant so that appropriate methods of analysis are used in genomic evaluations.

Methods

Simulation was used to compare the performance of estimates of breeding values based on pedigree relationships (Best Linear Unbiased Prediction, BLUP), genomic relationships (gBLUP), and based on a Bayesian variable selection model (Bayes B) to estimate breeding values under a range of different underlying models of genetic variation. The effects of different marker densities and varying animal relationships were also examined.

Results

This study shows that genomic selection methods can predict a proportion of the additive genetic value when genetic variation is controlled by common quantitative trait loci (QTL model), rare loci (rare variant model), all loci (infinitesimal model) and a random association (a polygenic model). The Bayes B method was able to estimate breeding values more accurately than gBLUP under the QTL and rare variant models, for the alternative marker densities and reference populations. The Bayes B and gBLUP methods had similar accuracies under the infinitesimal model.

Conclusions

Our results suggest that Bayes B is superior to gBLUP to estimate breeding values from genomic data. The underlying model of genetic variation greatly affects the predictive ability of genomic selection methods, and the superiority of Bayes B over gBLUP is highly dependent on the presence of large QTL effects. The use of SNP sequence data will outperform the less dense marker panels. However, the size and distribution of QTL effects and the size of reference populations still greatly influence the effectiveness of using sequence data for genomic prediction.     

The role of epistatic gene interactions in the response to selection and the evolution of evolvability

It has been argued that the architecture of the genotype-phenotype map determines evolvability, but few studies have attempted to quantify these effects. In this article we use the multilinear epistatic model to study the effects of different forms of epistasis on the response to directional selection. We derive an analytical prediction for the change in the additive genetic variance, and use individual-based simulations to understand the dynamics of evolvability and the evolution of genetic architecture. This shows that the major determinant for the evolution of the additive variance, and thus the evolvability, is directional epistasis. Positive directional epistasis leads to an acceleration of evolvability, while negative directional epistasis leads to canalization. In contrast, pure non-directional epistasis has little effect on the response to selection. One consequence of this is that the classical epistatic variance components, which do not distinguish directional and non-directional effects, are useless as predictors of evolutionary dynamics. The build-up of linkage disequilibrium also has negligible effects. We argue that directional epistasis is likely to have major effects on evolutionary dynamics and should be the focus of empirical studies of epistasis.     

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.     

Genome-wide association mapping of yield and yield components of spring wheat under contrasting moisture regimes

Key message

A stable QTL that may be used in marker-assisted selection in wheat breeding programs was detected for yield, yield components and drought tolerance-related traits in spring wheat association mapping panel.

Abstract

Genome-wide association mapping has become a widespread method of quantitative trait locus (QTL) identification for many crop plants including wheat (Triticum aestivum L.). Its benefit over traditional bi-parental mapping approaches depends on the extent of linkage disequilibrium in the mapping population. The objectives of this study were to determine linkage disequilibrium decay rate and population structure in a spring wheat association mapping panel (n = 285–294) and to identify markers associated with yield and yield components, morphological, phenological, and drought tolerance-related traits. The study was conducted under fully irrigated and rain-fed conditions at Greeley, CO, USA and Melkassa, Ethiopia in 2010 and 2011 (five total environments). Genotypic data were generated using diversity array technology markers. Linkage disequilibrium decay rate extended over a longer genetic distance for the D genome (6.8 cM) than for the A and B genomes (1.7 and 2.0 cM, respectively). Seven subpopulations were identified with population structure analysis. A stable QTL was detected for grain yield on chromosome 2DS both under irrigated and rain-fed conditions. A multi-trait region significant for yield and yield components was found on chromosome 5B. Grain yield QTL on chromosome 1BS co-localized with harvest index QTL. Vegetation indices shared QTL with harvest index on chromosome 1AL and 5A. After validation in relevant genetic backgrounds and environments, QTL detected in this study for yield, yield components and drought tolerance-related traits may be used in marker-assisted selection in wheat breeding programs.