Prediction of genetic values of quantitative traits with epistatic effects in plant breeding populations

Though epistasis has long been postulated to have a critical role in genetic regulation of important pathways as well as provide a major source of variation in the process of speciation, the importance of epistasis for genomic selection in the context of plant breeding is still being debated. In this paper, we report the results on the prediction of genetic values with epistatic effects for 280 accessions in the Nebraska Wheat Breeding Program using adaptive mixed least absolute shrinkage and selection operator (LASSO). The development of adaptive mixed LASSO, originally designed for association mapping, for the context of genomic selection is reported. The results show that adaptive mixed LASSO can be successfully applied to the prediction of genetic values while incorporating both marker main effects and epistatic effects. Especially, the prediction accuracy is substantially improved by the inclusion of two-locus epistatic effects (more than onefold in some cases as measured by cross-validation correlation coefficient), which is observed for multiple traits and planting locations. This points to significant potential in using non-additive genetic effects for genomic selection in crop breeding practices.     

Genomic prediction of hybrid crops allows disentangling dominance and epistasis

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

The hybrid protein interactome contributes to rice heterosis as epistatic effects

Heterosis is the phenomenon in which hybrid progeny exhibits superior traits in comparison with those of their parents. Genomic variations between the two parental genomes may generate epistasis interactions, which is one of the genetic hypotheses explaining heterosis. We postulate that protein?protein interactions specific to F₁ hybrids (F₁‐specific PPIs) may occur when two parental genomes combine, as the proteome of each parent may supply novel interacting partners. To test our assumption, an inter‐subspecies hybrid interactome was simulated by in silico PPI prediction between rice japonica (cultivar Nipponbare) and indica (cultivar 9311). Four‐thousand, six‐hundred and twelve F₁‐specific PPIs accounting for 20.5% of total PPIs in the hybrid interactome were found. Genes participating in F₁‐specific PPIs tend to encode metabolic enzymes and are generally localized in genomic regions harboring metabolic gene clusters. To test the genetic effect of F₁‐specific PPIs in heterosis, genomic selection analysis was performed for trait prediction with additive, dominant and epistatic effects separately considered in the model. We found that the removal of single nucleotide polymorphisms associated with F₁‐specific PPIs reduced prediction accuracy when epistatic effects were considered in the model, but no significant changes were observed when additive or dominant effects were considered. In summary, genomic divergence widely dispersed between japonica and indica rice may generate F₁‐specific PPIs, part of which may accumulatively contribute to heterosis according to our computational analysis. These candidate F₁‐specific PPIs, especially for those involved in metabolic biosynthesis pathways, are worthy of experimental validation when large‐scale protein interactome datasets are generated in hybrid rice in the future.     

Multi-trait genomic prediction for nitrogen response indices in tropical maize hybrids

In maize breeding, genomic prediction may be an efficient tool for selecting single-crosses evaluated under abiotic stress conditions. In addition, a promising strategy is applying multiple-trait genomic prediction using selection indices (SIs), increasing genetics gains and reducing time per cycles. In this study, we aimed (i) to compare accuracy of single- and multi-trait genomic prediction (STGP; MTGP) in two maize datasets, (ii) to evaluate prediction of four selection indices that could contribute to the selection of tropical maize hybrids under contrasting nitrogen conditions, and (iii) to compare the use of linear (GBLUP) and nonlinear (RKHS/GK) kernels in STGP and MTGP analyses. For either single-trait GBLUP and RKHS analyses, the highest values obtained for accuracy were 0.40 and 0.41 using harmonic mean (HM), respectively. From multi-trait GBLUP and GK, using the combination of selection indices in MTGP seems to be suitable, increasing the accuracy. Adding grain yield and plant height in MTGP showed a slight improvement in accuracy compared to STGP. In general, there was a modest benefit of using single-trait RKHS and GK multi-trait, rather than GBLUP.     

Genome-enabled prediction of genetic values using radial basis function neural networks

The availability of high density panels of molecular markers has prompted the adoption of genomic selection (GS) methods in animal and plant breeding. In GS, parametric, semi-parametric and non-parametric regressions models are used for predicting quantitative traits. This article shows how to use neural networks with radial basis functions (RBFs) for prediction with dense molecular markers. We illustrate the use of the linear Bayesian LASSO regression model and of two non-linear regression models, reproducing kernel Hilbert spaces (RKHS) regression and radial basis function neural networks (RBFNN) on simulated data and real maize lines genotyped with 55,000 markers and evaluated for several trait-environment combinations. The empirical results of this study indicated that the three models showed similar overall prediction accuracy, with a slight and consistent superiority of RKHS and RBFNN over the additive Bayesian LASSO model. Results from the simulated data indicate that RKHS and RBFNN models captured epistatic effects; however, adding non-signal (redundant) predictors (interaction between markers) can adversely affect the predictive accuracy of the non-linear regression models.     

Partitioning of population genetic variance under multiplicative-epistatic gene action

Determining the way in which different QTLs interact (epistasis) in their effects on the phenotype is crucial to many areas in population genetics and evolutionary biology. For example, in the founder event, a separated population readapts to a new environment through the release of cryptic gene-gene interactions. In hybrid zones, hybrid speciation must be subjected to natural selection for epistasis resulting from genomic recombinations between different species. However, there is a severe shortage of relevant methodologies to estimate epistatic genetic effects and variances. A statistical model has recently been proposed to estimate the number of QTLs, their genetic effects and allelic frequencies in segregating populations. This model is based on multiplicative gene action and derived from a two-level intra- and interspecific mating design. In this paper, we formulate a statistical procedure for partitioning the genetic variance into additive, dominant and various kinds of epistatic components in an intra- or mixed intra- and interspecific hybrid population. The procedure can be used to study the genetic architecture of fragmented populations and hybrid zones, thus allowing for a better recognition of the role of epistasis in evolution and hybrid speciation. A real example for two Populus species, P. tremuloides and P. tremula, is provided to illustrate the procedure. In this example, we found that considerable new genetic variation is formed through genomic recombination between two aspen species. Received: 1 May 1999 / Accepted: 27 July 1999     

Epistasis and the mutation load: a measurement-theoretical approach

An approximate solution for the mean fitness in mutation-selection balance with arbitrary order of epistatic interaction is derived. The solution is based on the assumptions of coupling equilibrium and that the interaction effects are multilinear. We find that the effect of m-order epistatic interactions (i.e., interactions among groups of m loci) on the load is dependent on the total genomic mutation rate, U, to the mth power. Thus, higher-order gene interactions are potentially important if U is large and the interaction density among loci is not too low. The solution suggests that synergistic epistasis will decrease the mutation load and that variation in epistatic effects will elevate the load. Both of these results, however, are strictly true only if they refer to epistatic interaction strengths measured in the optimal genotype. If gene interactions are measured at mutation-selection equilibrium, only synergistic interactions among even numbers of genes will reduce the load. Odd-ordered synergistic interactions will then elevate the load. There is no systematic relationship between variation in epistasis and load at equilibrium. We argue that empirical estimates of gene interaction must pay attention to the genetic background in which the effects are measured and that it may be advantageous to refer to average interaction intensities as measured in mutation-selection equilibrium. We derive a simple criterion for the strength of epistasis that is necessary to overcome the twofold disadvantage of sex.     

Unraveling Additive from Nonadditive Effects Using Genomic Relationship Matrices

The application of quantitative genetics in plant and animal breeding has largely focused on additive models, which may also capture dominance and epistatic effects. Partitioning genetic variance into its additive and nonadditive components using pedigree-based models (P-genomic best linear unbiased predictor) (P-BLUP) is difficult with most commonly available family structures. However, the availability of dense panels of molecular markers makes possible the use of additive- and dominance-realized genomic relationships for the estimation of variance components and the prediction of genetic values (G-BLUP). We evaluated height data from a multifamily population of the tree species Pinus taeda with a systematic series of models accounting for additive, dominance, and first-order epistatic interactions (additive by additive, dominance by dominance, and additive by dominance), using either pedigree- or marker-based information. We show that, compared with the pedigree, use of realized genomic relationships in marker-based models yields a substantially more precise separation of additive and nonadditive components of genetic variance. We conclude that the marker-based relationship matrices in a model including additive and nonadditive effects performed better, improving breeding value prediction. Moreover, our results suggest that, for tree height in this population, the additive and nonadditive components of genetic variance are similar in magnitude. This novel result improves our current understanding of the genetic control and architecture of a quantitative trait and should be considered when developing breeding strategies.     

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.     

The mapping of epistatic effects onto a genealogical tree in haploid populations

In this paper we present a model that maps epistatic effects onto a genealogical tree for a haploid population. Prior work has demonstrated that genealogical structure causes the genotypic values of individuals to covary. Our results indicate that epistasis can reduce genotypic covariance that is caused by genealogical structure. Genotypic effects (both additive and epistatic) occur along the branches of a genealogical tree, from the base of the tree to its tips. Epistasis reduces genotypic covariance because there is a reweighting of the contribution of branches to the states of genotypes compared to the additive case. Branches near the tips of a genealogical tree contribute proportionally more genetic effects with epistasis than without epistasis. Epistatic effects are most numerous at basal positions in a genealogical tree when a population is constant in size and experiencing no selection, optimizing selection, diversifying selection or directional selection, indicating that epistatic effects are typically old. For a population that is growing in size, epistatic effects are most numerous at midpoints in a genealogical tree, indicating epistatic effects are of moderate age. Our results are important in that they suggest epistatic effects may typically explain deep (old) divergences and broad patterns of divergence that exist in populations, except in growing populations. In a growing population, epistatic effects may cause more within group divergence higher up in a tree and less between group divergence that is deep in a tree. The distribution of the number of epistatic effects and the expected variance and covariance in the number of epistatic effects is also provided assuming neutrality.     

Evolution of genetic architecture under directional selection

We investigate the multilinear epistatic model under mutation-limited directional selection. We confirm previous results that only directional epistasis, in which genes on average reinforce or diminish each other's effects, contribute to the initial evolution of mutational effects. Thus, either canalization or decanalization can occur under directional selection, depending on whether positive or negative epistasis is prevalent. We then focus on the evolution of the epistatic coefficients themselves. In the absence of higher-order epistasis, positive pairwise epistasis will tend to weaken relative to additive effects, while negative pairwise epistasis will tend to become strengthened. Positive third-order epistasis will counteract these effects, while negative third-order epistasis will reinforce them. More generally, gene interactions of all orders have an inherent tendency for negative changes under directional selection, which can only be modified by higher-order directional epistasis. We identify three types of nonadditive quasi-equilibrium architectures that, although not strictly stable, can be maintained for an extended time: (1) nondirectional epistatic architectures; (2) canalized architectures with strong epistasis; and (3) near-additive architectures in which additive effects keep increasing relative to epistasis.     

Nonparametric Method for Genomics-Based Prediction of Performance of Quantitative Traits Involving Epistasis in Plant Breeding

Genomic selection (GS) procedures have proven useful in estimating breeding value and predicting phenotype with genome-wide molecular marker information. However, issues of high dimensionality, multicollinearity, and the inability to deal effectively with epistasis can jeopardize accuracy and predictive ability. We, therefore, propose a new nonparametric method, pRKHS, which combines the features of supervised principal component analysis (SPCA) and reproducing kernel Hilbert spaces (RKHS) regression, with versions for traits with no/low epistasis, pRKHS-NE, to high epistasis, pRKHS-E. Instead of assigning a specific relationship to represent the underlying epistasis, the method maps genotype to phenotype in a nonparametric way, thus requiring fewer genetic assumptions. SPCA decreases the number of markers needed for prediction by filtering out low-signal markers with the optimal marker set determined by cross-validation. Principal components are computed from reduced marker matrix (called supervised principal components, SPC) and included in the smoothing spline ANOVA model as independent variables to fit the data. The new method was evaluated in comparison with current popular methods for practicing GS, specifically RR-BLUP, BayesA, BayesB, as well as a newer method by Crossa et al., RKHS-M, using both simulated and real data. Results demonstrate that pRKHS generally delivers greater predictive ability, particularly when epistasis impacts trait expression. Beyond prediction, the new method also facilitates inferences about the extent to which epistasis influences trait expression.     

Effect of varying epistasis on the evolution of recombination

Whether recombination decelerates or accelerates a population's response to selection depends, at least in part, on how fitness-determining loci interact. Realistically, all genomes likely contain fitness interactions both with positive and with negative epistasis. Therefore, it is crucial to determine the conditions under which the potential beneficial effects of recombination with negative epistasis prevail over the detrimental effects of recombination with positive epistasis. Here, we examine the simultaneous effects of diverse epistatic interactions with different strengths and signs in a simplified model system with independent pairs of interacting loci and selection acting only on the haploid phase. We find that the average form of epistasis does not predict the average amount of linkage disequilibrium generated or the impact on a recombination modifier when compared to results using the entire distribution of epistatic effects and associated single-mutant effects. Moreover, we show that epistatic interactions of a given strength can produce very different effects, having the greatest impact when selection is weak. In summary, we observe that the evolution of recombination at mutation-selection balance might be driven by a small number of interactions with weak selection rather than by the average epistasis of all interactions. We illustrate this effect with an analysis of published data of Saccharomyces cerevisiae. Thus to draw conclusions on the evolution of recombination from experimental data, it is necessary to consider the distribution of epistatic interactions together with the associated selection coefficients.     

Genome-enabled methods for predicting litter size in pigs: a comparison

Predictive ability of models for litter size in swine on the basis of different sources of genetic information was investigated. Data represented average litter size on 2598, 1604 and 1897 60K genotyped sows from two purebred and one crossbred line, respectively. The average correlation (r) between observed and predicted phenotypes in a 10-fold cross-validation was used to assess predictive ability. Models were: pedigree-based mixed-effects model (PED), Bayesian ridge regression (BRR), Bayesian LASSO (BL), genomic BLUP (GBLUP), reproducing kernel Hilbert spaces regression (RKHS), Bayesian regularized neural networks (BRNN) and radial basis function neural networks (RBFNN). BRR and BL used the marker matrix or its principal component scores matrix (UD) as covariates; RKHS employed a Gaussian kernel with additive codes for markers whereas neural networks employed the additive genomic relationship matrix (G) or UD as inputs. The non-parametric models (RKHS, BRNN, RNFNN) gave similar predictions to the parametric counterparts (average r ranged from 0.15 to 0.23); most of the genome-based models outperformed PED (r = 0.16). Predictive abilities of linear models and RKHS were similar over lines, but BRNN varied markedly, giving the best prediction (r = 0.31) when G was used in crossbreds, but the worst (r = 0.02) when the G matrix was used in one of the purebred lines. The r values for RBFNN ranged from 0.16 to 0.23. Predictive ability was better in crossbreds (0.26) than in purebreds (0.15 to 0.22). This may be related to family structure in the purebred lines.     

Epistasis can increase multivariate trait diversity in haploid non-recombining populations

We evaluate the effect of epistasis on genetically-based multivariate trait variation in haploid non-recombining populations. In a univariate setting, past work has shown that epistasis reduces genetic variance (additive plus epistatic) in a population experiencing stabilizing selection. Here we show that in a multivariate setting, epistasis also reduces total genetic variation across the entire multivariate trait in a population experiencing stabilizing selection. But, we also show that the pattern of variation across the multivariate trait can be more even when epistasis occurs compared to when epistasis is absent, such that some character combinations will have more genetic variance when epistasis occurs compared to when epistasis is absent. In fact, a measure of generalized multivariate trait variation can be increased by epistasis under weak to moderate stabilizing selection conditions, as well as neutral conditions. Likewise, a measure of conditional evolvability can be increased by epistasis under weak to moderate stabilizing selection and neutral conditions. We investigate the nature of epistasis assuming a multivariate-normal model genetic effects and investigate the nature of epistasis underlying the biophysical properties of RNA. Increased multivariate diversity occurs for populations that are infinite in size, as well as populations that are finite in size. Our model of finite populations is explicitly genealogical and we link our findings about the evenness of eigenvalues with epistasis to prior work on the genealogical mapping of epistatic effects.     

Bridging the gap between marker-assisted and genomic selection of heading time and plant height in hybrid wheat

Based on data from field trials with a large collection of 135 elite winter wheat inbred lines and 1604 F₁ hybrids derived from them, we compared the accuracy of prediction of marker-assisted selection and current genomic selection approaches for the model traits heading time and plant height in a cross-validation approach. For heading time, the high accuracy seen with marker-assisted selection severely dropped with genomic selection approaches RR-BLUP (ridge regression best linear unbiased prediction) and BayesCπ, whereas for plant height, accuracy was low with marker-assisted selection as well as RR-BLUP and BayesCπ. Differences in the linkage disequilibrium structure of the functional and single-nucleotide polymorphism markers relevant for the two traits were identified in a simulation study as a likely explanation for the different trends in accuracies of prediction. A new genomic selection approach, weighted best linear unbiased prediction (W-BLUP), designed to treat the effects of known functional markers more appropriately, proved to increase the accuracy of prediction for both traits and thus closes the gap between marker-assisted and genomic selection.     

The Causal Meaning of Genomic Predictors and How It Affects Construction and Comparison of Genome-Enabled Selection Models

The term “effect” in additive genetic effect suggests a causal meaning. However, inferences of such quantities for selection purposes are typically viewed and conducted as a prediction task. Predictive ability as tested by cross-validation is currently the most acceptable criterion for comparing models and evaluating new methodologies. Nevertheless, it does not directly indicate if predictors reflect causal effects. Such evaluations would require causal inference methods that are not typical in genomic prediction for selection. This suggests that the usual approach to infer genetic effects contradicts the label of the quantity inferred. Here we investigate if genomic predictors for selection should be treated as standard predictors or if they must reflect a causal effect to be useful, requiring causal inference methods. Conducting the analysis as a prediction or as a causal inference task affects, for example, how covariates of the regression model are chosen, which may heavily affect the magnitude of genomic predictors and therefore selection decisions. We demonstrate that selection requires learning causal genetic effects. However, genomic predictors from some models might capture noncausal signal, providing good predictive ability but poorly representing true genetic effects. Simulated examples are used to show that aiming for predictive ability may lead to poor modeling decisions, while causal inference approaches may guide the construction of regression models that better infer the target genetic effect even when they underperform in cross-validation tests. In conclusion, genomic selection models should be constructed to aim primarily for identifiability of causal genetic effects, not for predictive ability.     

Predicting Growth Traits with Genomic Selection Methods in Zhikong Scallop (<Emphasis Type="Italic">Chlamys farreri</Emphasis>)

Selective breeding is a common and effective approach for genetic improvement of aquaculture stocks with parental selection as the key factor. Genomic selection (GS) has been proposed as a promising tool to facilitate selective breeding. Here, we evaluated the predictability of four GS methods in Zhikong scallop (Chlamys farreri) through real dataset analyses of four economical traits (e.g., shell length, shell height, shell width, and whole weight). Our analysis revealed that different GS models exhibited variable performance in prediction accuracy depending on genetic and statistical factors, but non-parametric method, including reproducing kernel Hilbert spaces regression (RKHS) and sparse neural networks (SNN), generally outperformed parametric linear method, such as genomic best linear unbiased prediction (GBLUP) and BayesB. Furthermore, we demonstrated that the predictability relied mainly on the heritability regardless of GS methods. The size of training population and marker density also had considerable effects on the predictive performance. In practice, increasing the training population size could better improve the genomic prediction than raising the marker density. This study is the first to apply non-linear model and neural networks for GS in scallop and should be valuable to help develop strategies for aquaculture breeding programs.     

Genomic selection for grain yield and quality traits in durum wheat

The prediction accuracies of genomic selection depend on several factors, including the genetic architecture of target traits, the number of traits considered at a given time, and the statistical models. Here, we assessed the potential of single-trait (ST) and multi-trait (MT) genomic prediction models for durum wheat on yield and quality traits using a breeding panel (BP) of 170 varieties and advanced breeding lines, and a doubled-haploid (DH) population of 154 lines. The two populations were genotyped with the Infinium iSelect 90K SNP assay and phenotyped for various traits. Six ST-GS models (RR-BLUP, G-BLUP, BayesA, BayesB, Bayesian LASSO, and RKHS) and three MT prediction approaches (MT-BayesA, MT-Matrix, and MT-SI approaches which use economic selection index as a trait value) were applied for predicting yield, protein content, gluten index, and alveograph measures. The ST prediction accuracies ranged from 0.5 to 0.8 for the various traits and models and revealed comparable prediction accuracies for most of the traits in both populations, except BayesA and BayesB, which better predicted gluten index, tenacity, and strength in the DH population. The MT-GS models were more accurate than the ST-GS models only for grain yield in the BP. Using BP as a training set to predict the DH population resulted in poor predictions. Overall, all the six ST-GS models appear to be applicable for GS of yield and gluten strength traits in durum wheat, but we recommend the simple computational models RR-BLUP or G-BLUP for predicating single trait and MT-SI for predicting yield and protein simultaneously.     

Genome-wide mapping and prediction suggests presence of local epistasis in a vast elite winter wheat populations adapted to Central Europe

Key message

Genome-wide association mapping as well as marker- and haplotype-based genome-wide selection unraveled a complex genetic architecture of grain yield with absence of large effect QTL and presence of local epistatic effects.

Abstract

The genetic architecture of grain yield determines to a large extent the optimum design of genomic-assisted wheat breeding programs. The main goal of our study was to examine the potential and limitations to dissect the genetic architecture of grain yield in wheat using a large experimental data set. Our study was based on phenotypic information and genomic data of 13,901 SNPs of a diverse set of 3816 elite wheat lines adapted to Central Europe. We applied genome-wide association mapping based on experimental and simulated data sets and performed marker- and haplotype-based genomic prediction. Computer simulations revealed for our mapping population a high power to detect QTL, even if they individually explained only 2.5% of the genetic variation. Despite this, we found no stable marker–trait associations when validating in independent subsets. A two-dimensional scan for marker–marker interactions indicated presence of local epistasis which was further supported by improved prediction abilities when shifting from marker- to haplotype-based genome-wide prediction approaches. We observed that marker effects estimated using genome-wide prediction approaches strongly varied across years albeit resulting in high prediction abilities. Thus, our results suggested that the prediction accuracy of genomic selection in wheat is mainly driven by relatedness rather than by exploiting knowledge of the genetic architecture.