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.     

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.     

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.     

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.     

Multiallelic models of genetic effects and variance decomposition in non-equilibrium populations

Quantitative genetics stems from the theoretical models of genetic effects, which are re-parameterizations of the genotypic values into parameters of biological (genetic) relevance. Different formulations of genetic effects are adequate to address different subjects. We thus need to generalize and unify them under a common framework for enabling researchers to easily transform genetic effects between different biological meanings. The Natural and Orthogonal Interactions (NOIA) model of genetic effects has been developed to achieve this aim. Here, we further implement the statistical formulation of NOIA with multiple alleles under Hardy–Weinberg departures (HWD). We show that our developments are straightforwardly connected to the decomposition of the genetic variance and we point out several emergent properties of multiallelic quantitative genetic models, as compared to the biallelic ones. Further, NOIA entails a natural extension of one-locus developments to multiple epistatic loci under linkage equilibrium. Therefore, we present an extension of the orthogonal decomposition of the genetic variance to multiple epistatic, multiallelic loci under HWD. We illustrate this theory with a graphical interpretation and an analysis of published data on the human acid phosphatase (ACP1) polymorphism.     

Comparison of three QTL detection models on biochemical,sensory, and yield characters in Coffea canephora

Coffea canephora is subject to enormous competitive challenges from other crops, especially for farmer sustainability and consumer requirements. Coffee breeding programs have to focus on specific traits linked to these two key targets, such as quality character, largely depending on the bean’s biochemical composition and field yield. Two segregating populations A and B, from crosses between a hybrid (Congolese?×?Guinean) FRT58 parental clone and a Congolese FRT51 genotype and between two Congolese parents FRT67 and FRT51, respectively, were used to characterize the quantitative trait loci (QTL) involved in agronomic and biochemical traits. A consensus genetic map was established using 249 SSRs covering 1,201 cM. Three QTL detection models per population with MapQTL (model I) and MCQTL (model II) followed by a connected population approach with MCQTL (model III) were compared based on their efficiency, precision for QTL detection, and their genetic effect assessment (additive, dominance, and parental-favorable allele). The analysis detected a total of 143 QTLs, 60 of which were shared between the three models; 28 found with two models; and two, 13, and 40 specific from models I, II, and III, respectively. The last model III based on connected populations is much more efficient in detecting QTLs with low variance explained and led to the genetic characterization of favorable allele. Thanks to this comparison of three QTL detection models on our quantitative genetic study, we will give a new insight for coffee breeding programs dedicated to managing complex agronomic or qualitative traits.     

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 genotype-environment interaction for quantitative trait loci affecting flowering time in Arabidopsis thaliana

A major goal of evolutionary biology is to understand the genetic architecture of the complex quantitative traits that may lead to adaptations in natural populations. Of particular relevance is the evaluation of the frequency and magnitude of epistasis (gene–gene and gene–environment interaction) as it plays a controversial role in models of adaptation within and among populations. Here, we explore the genetic basis of flowering time in Arabidopsis thaliana using a series of quantitative trait loci (QTL) mapping experiments with two recombinant inbred line (RIL) mapping populations [Columbia (Col) x Landsberg erecta (Ler), Ler x Cape Verde Islands (Cvi)]. We focus on the response of RILs to a series of environmental conditions including drought stress, leaf damage, and apical damage. These data were explicitly evaluated for the presence of epistasis using Bayesian based multiple-QTL genome scans. Overall, we mapped fourteen QTL affecting flowering time. We detected two significant QTL–QTL interactions and several QTL–environment interactions for flowering time in the Ler x Cvi population. QTL–environment interactions were due to environmentally induced changes in the magnitude of QTL effects and their interactions across environments – we did not detect antagonistic pleiotropy. We found no evidence for QTL interactions in the Ler x Col population. We evaluate these results in the context of several other studies of flowering time in Arabidopsis thaliana and adaptive evolution in natural populations.     

Inclusive composite interval mapping (ICIM) for digenic epistasis of quantitative traits in biparental populations

It has long been recognized that epistasis or interactions between non-allelic genes plays an important role in the genetic control and evolution of quantitative traits. However, the detection of epistasis and estimation of epistatic effects are difficult due to the complexity of epistatic patterns, insufficient sample size of mapping populations and lack of efficient statistical methods. Under the assumption of additivity of QTL effects on the phenotype of a trait in interest, the additive effect of a QTL can be completely absorbed by the flanking marker variables, and the epistatic effect between two QTL can be completely absorbed by the four marker-pair multiplication variables between the two pairs of flanking markers. Based on this property, we proposed an inclusive composite interval mapping (ICIM) by simultaneously considering marker variables and marker-pair multiplications in a linear model. Stepwise regression was applied to identify the most significant markers and marker-pair multiplications. Then a two-dimensional scanning (or interval mapping) was conducted to identify QTL with significant digenic epistasis using adjusted phenotypic values based on the best multiple regression model. The adjusted values retain the information of QTL on the two current mapping intervals but exclude the influence of QTL on other intervals and chromosomes. Epistatic QTL can be identified by ICIM, no matter whether the two interacting QTL have any additive effects. Simulated populations and one barley doubled haploids (DH) population were used to demonstrate the efficiency of ICIM in mapping both additive QTL and digenic interactions.     

A Unified Framework Integrating Parent-of-Origin Effects for Association Study

Genetic imprinting is the most well-known cause for parent-of-origin effect (POE) whereby a gene is differentially expressed depending on the parental origin of the same alleles. Genetic imprinting is related to several human disorders, including diabetes, breast cancer, alcoholism, and obesity. This phenomenon has been shown to be important for normal embryonic development in mammals. Traditional association approaches ignore this important genetic phenomenon. In this study, we generalize the natural and orthogonal interactions (NOIA) framework to allow for estimation of both main allelic effects and POEs. We develop a statistical (Stat-POE) model that has the orthogonal estimates of parameters including the POEs. We conducted simulation studies for both quantitative and qualitative traits to evaluate the performance of the statistical and functional models with different levels of POEs. Our results showed that the newly proposed Stat-POE model, which ensures orthogonality of variance components if Hardy-Weinberg Equilibrium (HWE) or equal minor and major allele frequencies is satisfied, had greater power for detecting the main allelic additive effect than a Func-POE model, which codes according to allelic substitutions, for both quantitative and qualitative traits. The power for detecting the POE was the same for the Stat-POE and Func-POE models under HWE for quantitative traits.     

High throughput analyses of epistasis for swine body dimensions and organ weights

High throughput analyses were performed to detect epistatic QTL in 17 body dimension and organ weight traits from a large F₂ pig population derived from a White Duroc and Erhualian intercross. The analyses used a nested test framework to handle multiple tests and a combined search algorithm to map epistatic QTL with empirical genome‐wide thresholds derived via prior permutation. Alternative statistical models (e.g. including vs. excluding carcass weight as a covariate) were tested to develop an in‐depth understanding of the role of epistasis in these kinds of traits. Epistasis signals were detected in only two or three traits under each statistical model studied. The interaction component of each pair of epistatic QTL explained a small proportion (0.7 to 2.1%) of the phenotypic variance in general. About half of the detected epistatic QTL pairs involved one of the two major QTL on porcine chromosomes 7 and 4. In those traits, the Erhualian allele consistently increased the phenotypes for the chromosome 7 QTL but decreased them for the chromosome 4 QTL. Models including carcass weight as covariate detected epistasis in body dimension traits whereas those excluding carcass weight found epistasis in organ weight traits. In addition, the epistasis results suggested that a QTL on chromosome 14 could be important for a number of organ weight traits. Using the high‐throughput analysis tool to examine different statistical models was essential for the generation of a complete picture of epistasis in a whole category of traits.     

Optimization of a crossing system using mate selection

A simple model based on one single identified quantitative trait locus (QTL) in a two-way crossing system was used to demonstrate the power of mate selection algorithms as a natural means of opportunistic line development for optimization of crossbreeding programs over multiple generations. Mate selection automatically invokes divergent selection in two parental lines for an over-dominant QTL and increased frequency of the favorable allele toward fixation in the sire-line for a fully-dominant QTL. It was concluded that an optimal strategy of line development could be found by mate selection algorithms for a given set of parameters such as genetic model of QTL, breeding objective and initial frequency of the favorable allele in the base populations, etc. The same framework could be used in other scenarios, such as programs involving crossing to exploit breed effects and heterosis. In contrast to classical index selection, this approach to mate selection can optimize long-term responses.     

Multiple-interval mapping for ordinal traits

Many statistical methods have been developed to map multiple quantitative trait loci (QTL) in experimental cross populations. Among these methods, multiple-interval mapping (MIM) can map QTL with epistasis simultaneously. However, the previous implementation of MIM is for continuously distributed traits. In this study we extend MIM to ordinal traits on the basis of a threshold model. The method inherits the properties and advantages of MIM and can fit a model of multiple QTL effects and epistasis on the underlying liability score. We study a number of statistical issues associated with the method, such as the efficiency and stability of maximization and model selection. We also use computer simulation to study the performance of the method and compare it to other alternative approaches. The method has been implemented in QTL Cartographer to facilitate its general usage for QTL mapping data analysis on binary and ordinal traits.     

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.     

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 unified statistical model for functional mapping of environment-dependent genetic expression and genotype x environment interactions for ontogenetic development

The effects of quantitative trait loci (QTL) on phenotypic development may depend on the environment (QTL x environment interaction), other QTL (genetic epistasis), or both. In this article, we present a new statistical model for characterizing specific QTL that display environment-dependent genetic expressions and genotype x environment interactions for developmental trajectories. Our model was derived within the maximum-likelihood-based mixture model framework, incorporated by biologically meaningful growth equations and environment-dependent genetic effects of QTL, and implemented with the EM algorithm. With this model, we can characterize the dynamic patterns of genetic effects of QTL governing growth curves and estimate the global effect of the underlying QTL during the course of growth and development. In a real example with rice, our model has successfully detected several QTL that produce differences in their genetic expression between two contrasting environments. These detected QTL cause significant genotype x environment interactions for some fundamental aspects of growth trajectories. The model provides the basis for deciphering the genetic architecture of trait expression adjusted to different biotic and abiotic environments and genetic relationships for growth rates and the timing of life-history events for any organism.     

Dissecting Genetic Architecture Underlying Seed Traits in Multiple Environments

The seeds of flowering plants develop from double fertilization and play a vital role in reproduction and supplying human and animal food. The genetic variation of seed traits is influenced by multiple genetic systems, e.g., maternal, embryo, and/or endosperm genomes. Understanding the genetic architecture of seed traits is a major challenge because of this complex mechanism of multiple genetic systems, especially the epistasis within or between different genomes and their interactions with the environment. In this study, a statistical model was proposed for mapping QTL with epistasis and QTL-by-environment (QE) interactions underlying endosperm and embryo traits. Our model integrates the maternal and the offspring genomes into one mapping framework and can accurately analyze maternal additive and dominant effects, endosperm/embryo additive and dominant effects, and epistatic effects of two loci in the same or two different genomes, as well as interaction effects of each genetic component of QTL with environment. Intensive simulations under different sampling strategies, heritabilities, and model parameters were performed to investigate the statistical properties of the model. A set of real cottonseed data was analyzed to demonstrate our methods. A software package, QTLNetwork-Seed-1.0.exe, was developed for QTL analysis of seed traits.     

Optimal use of regression models in genome-wide association studies

The performance of linear regression models in genome-wide association studies is influenced by how marker information is parameterized in the model. Considering the impact of parameterization is especially important when using information from multiple markers to test for association. Properties of the population, such as linkage disequilibrium (LD) and allele frequencies, will also affect the ability of a model to provide statistical support for an underlying quantitative trait locus (QTL). Thus, for a given location in the genome, the relationship between population properties and model parameterization is expected to influence the performance of the model in providing evidence for the position of a QTL. As LD and allele frequencies vary throughout the genome and between populations, understanding the relationship between these properties and model parameterization is of considerable importance in order to make optimal use of available genomic data. Here, we evaluate the performance of regression-based association models using genotype and haplotype information across the full spectrum of allele frequency and LD scenarios. Genetic marker data from 200 broiler chickens were used to simulate genomic conditions by selecting individual markers to act as surrogate QTL (sQTL) and then investigating the ability of surrounding markers to estimate sQTL genotypes and provide statistical support for their location. The LD and allele frequencies of markers and sQTL are shown to have a strong effect on the performance of models relative to one another. Our results provide an indication of the best choice of model parameterization given certain scenarios of marker and QTL LD and allele frequencies. We demonstrate a clear advantage of haplotype-based models, which account for phase uncertainty over other models tested, particularly for QTL with low minor allele frequencies. We show that the greatest advantage of haplotype models over single-marker models occurs when LD between markers and the causal locus is low. Under these situations, haplotype models have a greater accuracy of predicting the location of the QTL than other models tested.     

Multiple interval mapping for quantitative trait loci.

A new statistical method for mapping quantitative trait loci (QTL), called multiple interval mapping (MIM), is presented. It uses multiple marker intervals simultaneously to fit multiple putative QTL directly in the model for mapping QTL. The MIM model is based on Cockerham's model for interpreting genetic parameters and the method of maximum likelihood for estimating genetic parameters. With the MIM approach, the precision and power of QTL mapping could be improved. Also, epistasis between QTL, genotypic values of individuals, and heritabilities of quantitative traits can be readily estimated and analyzed. Using the MIM model, a stepwise selection procedure with likelihood ratio test statistic as a criterion is proposed to identify QTL. This MIM method was applied to a mapping data set of radiata pine on three traits: brown cone number, tree diameter, and branch quality scores. Based on the MIM result, seven, six, and five QTL were detected for the three traits, respectively. The detected QTL individually contributed from approximately 1 to 27% of the total genetic variation. Significant epistasis between four pairs of QTL in two traits was detected, and the four pairs of QTL contributed approximately 10.38 and 14.14% of the total genetic variation. The asymptotic variances of QTL positions and effects were also provided to construct the confidence intervals. The estimated heritabilities were 0.5606, 0.5226, and 0. 3630 for the three traits, respectively. With the estimated QTL effects and positions, the best strategy of marker-assisted selection for trait improvement for a specific purpose and requirement can be explored. The MIM FORTRAN program is available on the worldwide web (http://www.stat.sinica.edu.tw/chkao/).