Improving genomic selection in hexaploid wheat with sub-genome additive and epistatic models

Abstract

The goal of wheat breeding is the development of superior cultivars tailored to specific environments, and the identification of promising crosses is crucial for the success of breeding programs. Although genomic estimated breeding values were developed to estimate additive effects of genotypes before testing as parents, application has focused on predicting performance of candidate lines, ignoring nonadditive genetic effects. However, nonadditive genetic effects are hypothesized to be especially important in allopolyploid species due to the interaction between homeologous genes. The objectives of this study were to model additive and additive-by-additive epistatic effects to better delineate the genetic architecture of grain yield in wheat and to improve the accuracy of genome-wide predictions. The data set utilized consisted of 3,740 F_5:6 experimental lines tested in the K-State wheat breeding program across the years 2016 and 2018. Covariance matrices were calculated based on whole- and sub-genome marker data, and the natural and orthogonal interaction approach was used to estimate variance components for additive and additive-by-additive epistatic effects. Incorporating epistatic effects in additive models resulted in nonorthogonal partitioning of genetic effects but increased total genetic variance and reduced deviance information criteria. Estimation of sub-genome effects indicated that genotypes with the greatest whole-genome effects often combine sub-genomes with intermediate to high effects, suggesting potential for crossing parental lines that have complementary sub-genome effects. Modeling epistasis in either whole-genome or sub-genome models led to a marginal (3%) improvement in genomic prediction accuracy, which could result in significant genetic gains across multiple cycles of breeding.

Introduction

In wheat breeding, selection of experimental lines has historically been based on phenotypic selection, which encapsulates all forms of genetic effects. However, utilization of genomic selection (Meuwissen et al. 2001) to predict the performance of candidate lines overlooks the potential contribution of nonadditive effects, especially epistasis, which potentially could play an important role in the genetic expression of agronomic traits (Chapman and McNeal 1971; Sun et al. 1972; Goldringer et al. 1997; Zhang et al. 2008; Jiang et al. 2017). Epistasis is defined as the interaction between alleles of different genes and is normally classified into additive by additive, additive by dominance, and dominance by dominance, with higher-order interactions normally ignored.

Historically, complex mating designs were used to create the progeny that could be used to estimate epistatic effects, e.g. the triple test cross (Bauman 1959). Current breeding programs have an abundance of experimental lines genotyped with phenotypic records that could be leveraged to estimate genetic effects. With large numbers of markers covering the genome and quantitative trait locus effects assumed to have a normal distribution, genome-wide markers can be used to estimate the additive covariance between individuals and predict their additive genetic value (Nejati-Javaremi et al. 1997; VanRaden 2008). To obtain an approximation of the additive-by-additive relationship matrix between individuals, Henderson (1985) proposed the use of the Hadamard product of the additive covariance matrix based on pedigree information, which was later expanded to accommodate genomic-based additive relationship matrices (Su et al. 2012; Jiang and Reif 2015). The natural and orthogonal interaction (NOIA) approach (Alvarez-Castro and Carlborg 2007) was expanded to build genomic relationship matrices using genotypic frequencies and to include genome-wide epistasis (Vitezica et al. 2017), resulting in an orthogonal partitioning of genetic variance components, even if not in Hardy–Weinberg equilibrium (HWE). The main feature of the NOIA approach is to account for Hardy–Weinberg disequilibrium, which is quite common in agriculture or livestock, e.g. inbred line populations, F₁ crosses, 3-way crosses, or backcrosses. However, most of the previous studies on estimation of genomic variance have ignored this by assuming HWE. If genomic relationship matrices are built incorrectly assuming HWE, inclusion of more genetic effects in the model can dramatically change estimates (Vitezica et al. 2017). In the absence of HWE, the genomic relationship matrices do not have a diagonal mean of 1 and overall sum to 0. To account for deviation from HWE, Legarra (2016) proposed a correction for estimated variances, assuming HWE, to transform them to a proper scale. In the NOIA approach, the genomic relationship matrices are standardized prior to fitting the model to ensure an average diagonal of 1, which approximates Legarra's method (Vitezica et al. 2017). This ensures that estimates of genetic variance correspond to the reference population under evaluation.

Moreover, to address the challenge of partitioning estimates of genomic variance in the presence of extensive linkage disequilibrium (LD), as is the case of common wheat, Sorensen et al. (2001) and Lehermeier et al. 2017 developed methods to account for the contribution of the covariance between loci, i.e. LD and covariance between genetic effects, when estimating genomic variances. This approach allows variance and covariance to contribute to genetic effects, thereby preventing the overestimation or underestimation of these effects. Thus, there is the potential to better delineate the genetic architecture of important agronomic traits while leveraging existing data sets in crop species, as demonstrated on wheat grain yield using historical CIMMYT and Cornell wheat breeding data (Santantonio et al. 2019) and on important wheat diseases using synthetic wheat data (Cuevas et al. 2024).

Common wheat (Triticum aestivum spp aestivum) is a hexaploid species originated from the interspecific hybridization between tetraploid wheat, Triticum turgidum (Sax 1922; Kihara 1924), the source of the AABB genome, and Aegilops tauschii (Kihara 1944; McFadden and Sears 1946), the source of the DD genome. The presence of homeologous genomes could create positive or negative epistasis through subfunctionalization (Lynch and Force 2000) or substrate competition (Qian et al. 2010), while also impacting gene expression patterns (Leach et al. 2014; Akhunova et al. 2010). Intra-genomic interactions have also been reported in wheat (Tranquilli and Dubcovsky 2000; Reif et al. 2011; Sehgal et al. 2020), underscoring the potential importance of epistasis underpinning agronomic traits. In addition, the inbreeding nature of wheat leads to a fast fixation of alleles (Charlesworth 2003), revealing epistatic interactions whose favorable combinations are perpetuated with selection of superior genotypes in breeding programs. Although hexaploid, the 3 sub-genomes of common wheat undergo disomic segregation (Feldman and Levy 2012) and are normally treated as diploids in breeding programs. For that reason, partitioning whole-genome genetic effects into sub-genome opens up the possibility to attribute biological importance to whole-genome effects and variance estimates, as was previously done by Santantonio et al. (2019), Cuevas et al. (2024), and Bernardo (2021).

One of the challenges to accurately estimate the extent of epistatic effects in wheat lines is the conversion of epistatic variance into additive variance, which reaches its maximum when an allele becomes fixed (Whitlock et al. 1995; Cheverud and Routman 1996; Holland 2001; Technow et al. 2021). The Kansas State wheat breeding program is notable for maintaining a large genetic base through consistent crossings with external germplasm and wild accessions. This breeding strategy enhances the probability of observing epistatic interactions segregating while increasing the minor allele frequency of such interactions. Consequently, the statistical power to detect epistatic effects is higher compared to studies that relied in less diverse breeding data (Santantonio et al. 2019). Therefore, the objectives of this study were to (i) partition the genetic variance of wheat grain yield into additive and epistatic effects; (ii) partition whole-genome effects into sub-genome effects; and (iii) incorporate epistasis into the genomic selection model. We used a population of hexaploid wheat genotypes from the Kansas State wheat breeding program and implemented recent developments and methods for estimating genetic variance components as the NOIA approach and correction for LD and covariance between genetic effects.

Materials and methods

Estimation of epistatic variance

Phenotypic data

Phenotypic data consisted of grain yield records for 3,740 F_5:6 wheat genotypes grown in the early yield trail stage, comprised of individual plant short rows (IPSRs), of the K-State hard red winter wheat breeding program across the years of 2016 and 2018. The IPSR is the first stage of the breeding program where selection is based on grain yield data (previous stages consisted of visual single plant selection). The IPRS represents the foundational genetic diversity within each breeding cycle. The experimental design used for the phenotypic evaluation consisted of unreplicated trials and was a modified augmented design (Federer and Raghavarao 1975) type II with 1 replication of each experimental line. However, to quantify and remove spatial variability of these large experiments, we employed a 2-stage analyses (Smith et al. 2001; Welham et al. 2010) and included fitting a spatial model to account for field patterns associated with environmental factors (Cullis and Gleeson 1991; Cullis et al. 1998; Smith et al. 2001). Due to the limited number of replications in the trials, an extra step was added to the first-stage analysis, as reported in Arief et al. (2019). In the first step, the genotypes were fitted as random effects to provide an estimate of the spatial trend, which permits utilizing all genotypes to estimate the trend. In the second step, the genotypes were fitted as fixed effects to calculate the best linear unbiased estimates (BLUEs) necessary for the second-stage analysis (Smith et al. 2001). The statistical model used in both steps of the first-stage analysis was as follows:

$${\mathbf{y}}_{\mathbf{i}\mathbf{p}\mathbf{q}}=\mathbf{\mu }+{\mathbf{g}}_{\mathbf{i}}+{\mathbf{r}}_{\mathbf{p}}+{\mathbf{c}}_{\mathbf{q}}+{\epsilon }_{\mathbf{i}\mathbf{p}\mathbf{q}}:\mathbf{A}\mathbf{R}\mathbf{1}\mathbf{(}\mathbf{R}\mathbf{)}\otimes \mathbf{A}\mathbf{R}\mathbf{1}\mathbf{(}\mathbf{C}\mathbf{)}\mathbf{,}$$

where ${\mathbf{y}}_{\mathbf{i}\mathbf{p}\mathbf{q}}$ is the grain yield of genotype i in row p and column q; $\mathbf{\mu }$ is the intercept; ${\mathbf{g}}_{\mathbf{i}}$ is the effect of genotype modeled as random at first, then as fixed to obtain BLUEs; ${\mathbf{r}}_{\mathbf{p}}$ is the random effect of row p; ${\mathbf{c}}_{\mathbf{q}}$ is the random effect of column q; and ${\epsilon }_{\mathbf{i}\mathbf{p}\mathbf{q}}$ is the residual effect modeled using first-order autoregression (AR1) for row q and column q. As previously described, the initial phase of our analysis employed the first-step approach to calculate variance component estimates for the random spatial terms, while fitting the genotype as a random effect. In the second step, these variance component estimates were utilized to derive the BLUEs for the genotypes and their corresponding weights, where genotype was treated as a fixed effect. The weights were computed as the inverse of the residual variance for each field, multiplied by the replication count for each genotype, and the pooled residual variance across all fields (Cullis et al. 1996).

The second-stage analysis was conducted using the BLUEs and weights for genotypes within each location in order to obtain the across-location BLUEs according to the following statistical model:

$${\mathrm{Z}}_{\mathrm{ij}}\odot {\mathrm{w}}_{\mathrm{ij}}=\mathbf{\mu }+{\mathbf{g}}_{\mathbf{i}}+{\mathbf{l}}_{\mathbf{j}}+{\epsilon }_{\mathbf{i}\mathbf{j}},$$

where ${\mathrm{Z}}_{\mathrm{ij}}\odot {\mathrm{w}}_{\mathrm{ij}}$ is the weighted BLUE from the first-stage analysis for genotype i in location j; ${\mathbf{l}}_{\mathbf{j}}$ is the effect of location j; ${\mathbf{g}}_{\mathbf{i}}$ is the fixed effect of the genotypes; and ${\epsilon }_{\mathbf{i}\mathbf{j}}$ is the residual. Genotype-by-environment interaction was not modeled because the small number of genotypes replicated across all environments was insufficient for estimating interaction effects. The across-location BLUEs obtained from the second-stage analysis were used in the downstream analyses of this study. The 2-stage analysis was conducted using the Echidna Mixed Model Software (Gilmour 2018).

Genotypic data

The lines analyzed were in the F_5:6 generation and were genotyped using genotype by sequencing (Poland et al. 2012). To ensure data quality, markers with a minor allele frequency lower than 0.01, more than 20% missing values, and more than 20% heterozygotes were filtered out. The marker imputation process was carried out using the “rrBLUP” package (Endelman 2011), where missing markers were imputed as the mean value among all lines for that marker, resulting in a data set containing a total of 55,148 single nucleotide polymorphisms (21,338 SNPs on sub-genome A, 26,996 on sub-genome B, and 6,844 for sub-genome D).

Statistical models

Four models were used to quantify additive and additive-by-additive epistatic (herein described as epistasis) effects across sub-genomes. The first 2 models consisted of analyses at the whole-genome level while the last 2 models partitioned genetic effects, both additive and epistatic, by sub-genome. Variance components, for all models, were estimated using the NOIA approach parametrizations outlined by Alvarez-Castro and Carlborg (2007) and Vitezica et al. (2017).

Whole-genome models

Model 1: additive effects

The linear model for grain yield that includes only an additive term is given by the following:

$$\mathbf{y}=1\mathrm{\mu }+\mathbf{Z}\mathbf{g}+\epsilon ,$$ (1)

where $\mathbf{y}$ is the response vector (adjusted phenotypic data); $\mathit{\mu }$ is the intercept; $\mathit{\epsilon }$ is a vector of random residuals with $\epsilon$ ∼ N (0, ${\mathrm{\sigma }}_{\mathrm{e}}^2$), where ${\mathrm{\sigma }}_{\mathrm{e}}^2$ is the residual variance; $\mathbf{Z}$ is a design matrix of random effects; and $\mathbf{g}$ is the vector of additive genomic breeding values with $\mathbf{g}$ ∼ N (0, ${\mathrm{\sigma }}_{\mathrm{G}}^2\mathbf{G}$), where ${\mathrm{\sigma }}_{\mathrm{G}}^2$ is the genomic additive genetic variance and the $\mathbf{G}$ is an additive genomic relationship constructed according to Vitezica et al. (2017) (see Raffo et al. 2022). The main difference between the NOIA approach and VanRaden (2008) to compute the additive genomic relationship is the scaling of the matrix (see Joshi et al. 2020). In NOIA, the mean trace of the matrix is used for standardization and to have an average diagonal equal to 1. Thereby, the amount of the estimated genetic variances refers to the genotyped population, i.e. F_5:6 wheat genotypes. Note that this means that the proportion of additive variance in the F_5:6 generation is approximately double that expected in the F₂ generation.

Model 2: additive and epistatic effects

Following Henderson (1985), the epistatic covariance of individuals can be calculated as the Hadamard product of the component covariance matrices. Hence, the second model used in this study extends the first model by including an additive-by-additive epistatic term:

$$\mathbf{y}=1\mathrm{\mu }+\mathbf{Z}\mathbf{g}+\mathbf{Z}\mathbf{i}+\epsilon ,$$ (2)

where $\mathbf{y},\mathbf{\mu },\mathbf{Z},\mathbf{g}$, and $\epsilon$ are the same as in the first model and $\mathbf{i}$ is a vector of additive-by-additive epistatic genomic effects with $\mathbf{i}$ ∼ N (0, ${\mathrm{\sigma }}_{\mathrm{I}}^2\mathbf{I}$), where ${\mathrm{\sigma }}_{\mathrm{I}}^2$ is the epistatic variance and $\mathbf{I}$ is the additive-by-additive relationship matrix, built following Vitezica et al. (2017). This method consists of calculating the Hadamard product of the additive genomic relationship matrix, followed by standardization using the mean trace of the resulting matrix to have an average diagonal equal to 1. As a result, the amount of the estimated genetic variances refers to the genotyped population, i.e. F_5:6 wheat genotypes. This matrix is designed to capture deviations from additivity due to epistatic interactions across loci, thereby accounting for nonadditive genetic variance arising from epistatic deviation effects. Note that the additive-by-additive epistatic variance in the F_5:6 generation is approximately 4 times the one expected in the F₂ generation.

Sub-genome models

Model 3: additive effects

As described in Santantonio et al (2019), ${\mathrm{G}}_{\mathrm{j}}$ can be decomposed into individual additive effects for each sub-genome, such that ${\mathrm{G}}_{\mathrm{j}}={\mathrm{A}}_{\mathrm{j}}+{\mathrm{B}}_{\mathrm{j}}+{\mathrm{D}}_{\mathrm{j}}$, resulting in the following model:

$$\mathit{y}=\mathbf{1}\mu +{\mathit{Z}}_A{\mathit{g}}_A+{\mathit{Z}}_B{\mathit{g}}_B+{\mathit{Z}}_D{\mathit{g}}_D+\mathit{\epsilon }\mathbf{.}$$ (3)

Decomposition of the whole genome into sub-genomes permits each sub-genome to have an individual additive genetic variance and covariance among individuals, such that ${\mathbf{g}}_{\mathrm{A}}\sim \mathrm{N}(\mathbf{0},{\mathrm{\sigma }}_{\mathrm{A}}^2\mathbf{A})$, ${\mathbf{g}}_{\mathrm{B}}\sim \mathrm{N}(\mathbf{0},{\mathrm{\sigma }}_{\mathrm{B}}^2\mathbf{B})$, and ${\mathbf{g}}_{\mathrm{D}}\sim \mathrm{N}(\mathbf{0},{\mathrm{\sigma }}_{\mathrm{D}}^2\mathbf{D})$. The matrices $\mathbf{A}$, $\mathbf{B}$, and $\mathbf{D}$ are the realized additive genetic variance–covariance for each sub-genome and are calculated using the method used for calculating the whole-genome additive variance–covariance matrix.

Model 4: additive and epistatic effects

The whole-genome epistatic component, $\mathbf{I}$, can also be decomposed into intra- and inter-sub-genome epistatic interactions, such that ${\mathrm{I}}_{\mathrm{i}}$ = $\mathrm{A}{\mathrm{A}}_{\mathrm{i}}+\mathrm{B}{\mathrm{B}}_{\mathrm{i}}+\mathrm{D}{\mathrm{D}}_{\mathrm{i}}+\mathrm{A}{\mathrm{B}}_{\mathrm{i}}+\mathrm{A}{\mathrm{D}}_{\mathrm{i}}+\mathrm{B}{\mathrm{D}}_{\mathrm{i}}$. The new additive plus epistatic sub-genome model is then expressed as follows:

$$\mathit{y}=\mathbf{1}\mu +{\mathit{Z}}_A{\mathit{g}}_A+{\mathit{Z}}_B{\mathit{g}}_B+{\mathit{Z}}_D{\mathit{g}}_D+{\mathit{Z}}_{AA}{\mathit{g}}_{AA}+{\mathit{Z}}_{BB}{\mathit{g}}_{BB}+{\mathit{Z}}_{DD}{\mathit{g}}_{DD}+$$

$${\mathit{Z}}_{AB}{\mathit{g}}_{AB}+{\mathit{Z}}_{AD}{\mathit{g}}_{AD}+{\mathit{Z}}_{BD}{\mathit{g}}_{BD}+\mathit{\epsilon }\mathbf{.}$$ (4)

Decomposition of the whole-genome epistatic term into sub-genomes results in expression of additive-by-additive genetic variance–covariance among individuals for each pair of sub-genomes. This decomposition is generalized as ${\mathbf{g}}_{\mathrm{AA}}\sim \mathrm{N}(\mathbf{0},{\mathrm{\sigma }}_{\mathrm{AA}}^2\mathbf{A}\mathbf{A})$ and ${\mathbf{g}}_{\mathbf{A}\mathbf{B}}\sim \mathbf{N}(\mathbf{0},{\mathbf{\sigma }}_{\mathbf{A}\mathbf{B}}^2\mathbf{A}\mathbf{B})$, where the matrices $\mathbf{A}\mathbf{A}$ and $\mathbf{A}\mathbf{B}$ are the realized intra- and inter-sub-genome epistatic variance–covariance matrices, respectively, calculated using the previously described procedures.

Variance components, covariance, and heritability

Variance components were estimated using the Bayesian BGLR R-package (Pérez and de los Campos 2014). To enhance computational efficiency, eigenvalue decomposition of the variance–covariance matrices described by Acosta-Pech et al. (2017) was used and modeled as Bayesian ridge regression. For each model, a total of 50,000 iterations were executed, with the first 5,000 iterations discarded as burn-in. Subsequent iterations were thinned by selecting one of every 10 samples, resulting in 4,500 samples used for estimating the variance components. Based upon a Bayesian Markov chain Monte Carlo (MCMC) framework, genetic variance components were calculated using the genetic values derived from each MCMC sample, per the methods of Sorensen, Fernando, and Gianola (2001). By following this method, we account for the contribution of the covariance between loci, i.e. LD when estimating genomic variances (Lehermeier et al. 2017). As described by Lehermeier et al. (2017), within a Bayesian framework, samples from the posterior distribution of the total genetic variance, incorporating covariances among effects, can be derived from posterior samples of the effects themselves. In this study, following ideas presented by Lehermeier et al. (2017), variance components from each MCMC sample were used to calculate the correlation between additive and epistatic terms in both whole-genome (model 2) and sub-genome models (models 3 and 4). Broad-sense heritability was calculated for each model as the ratio of total genetic variance divided by total phenotypic variance.

Genomic prediction

Five-fold cross-validation, with 10 replicates, was used to test predictive ability of the models. For each replicate, the set of 3,740 lines was randomly divided into 5 groups, irrespective of which year they were tested, with 4 used to train the model and the remaining used to predict genetic values. Using that structure, all 4 above-described models were fitted, resulting in predicted genetic values for each line. Prediction of the 5-folds was correlated with the phenotypic value, with the resulting value representing the prediction accuracy of the model. Correlation between predicted genetic values and phenotypic values provided comparisons of accuracy among models. To test statistical significance between the prediction accuracies of the models fitted, Tukey's honestly significant difference was calculated using the “agricolae” package on R (De Mendiburu and De Mendiburu 2019).

Results

Variance components and broad-sense heritability

Whole-genome models 1 and 2

Estimated posterior means of the whole-genome additive model 1 resulted in additive genetic effects accounting for 28% of the observed phenotypic variance for wheat grain yield. However, in whole-genome additive plus epistatic model 2, the contribution of additive variance decreased to only 12%, while epistatic variance became prominent, explaining 29% of the observed phenotypic variance. By including an epistatic term in the whole-genome additive plus epistatic model 2, total genetic variance increased while residual variance decreased compared to whole-genome additive model 1 (Fig. 1). These results suggest that the epistatic term, although not orthogonal to additive effects, captures a portion of the genetic variance that is not captured by the additive term alone. Consequently, broad-sense heritability estimated using the whole-genome additive plus epistatic model 2 (0.40) exceeded the estimate of the fully additive model (0.28; see Supplementary Table 1). Interestingly, while epistatic effects did not result in orthogonal partitioning of genetic effects, the correlation between additive and epistatic terms was virtually 0 (Supplementary Table 2). The deviance information criteria value was smaller for whole-genome additive plus epistatic model 2 than for whole-genome additive model 1 (Supplementary Table 1), suggesting that the incorporation of an epistatic term captured epistatic interactions and provided a better fit of data.

Fig. 1.

Substantial additive-by-additive epistatic contribution to variance of grain yield. Estimated posterior means and standard deviations of genetic variance components of model 1 a) and model 2 b). ${\mathrm{\sigma }}_{\mathrm{G}}^2$ refers to additive variance, ${\mathrm{\sigma }}_{\mathrm{I}}^2$ to additive-by-additive epistatic variance, ${\mathrm{\sigma }}_{\mathrm{T}}^2$ to total genetic variance, and ${\mathrm{\sigma }}_{\mathrm{E}}^2$ is the residual variance of the model.

Sub-genome models

The sum of additive genetic variances across genomes for the sub-genome additive model 3 (35.84) was similar to the total genetic variance estimated using the whole-genome additive model 1, which was expected since breaking the genome into sub-genomes should not imply different additive effects (Figs. 1 and 2). For the sub-genome additive plus epistatic model 4, the sum of sub-genome additive added variance to 16.13 and the epistatic to 40.63, indicating a slight increase in epistatic variance with little change in additive variance compared to corresponding values from the whole-genome additive plus epistatic model 2 (Supplementary Table 3). The sum of the additive and epistatic variances deviated from the observed total genotypic variance because of a negative covariance between the terms in the sub-genome additive plus epistatic model 4 (Supplementary Table 6). Using sub-genome additive model 3, the largest additive variance was estimated for the D sub-genome, followed by the corresponding estimates for the A and B sub-genomes (Fig. 2). Inclusion of the intra- and inter-sub-genome interaction terms reduced the amount of additive variance associated with each sub-genome, and the D sub-genome displayed the highest additive effects among the 3 sub-genomes. As with whole-genome additive plus epistatic model 2, the addition of intra- and inter-sub-genome epistatic terms resulted in nonorthogonal partitioning of genetic effects. However, the covariance within and between additive and epistatic terms was small, such that the correlations were all about 0 in all instances (Supplementary Table 6).

Fig. 2.

Wheat sub-genomes additive and epistatic variance components in grain yield. Estimated posterior means and standard deviations of genetic variance components of model 3 a) and model 4 b). ${\mathrm{\sigma }}_{\mathrm{A}}^2$ refers to additive variance in sub-genome A, ${\mathrm{\sigma }}_{\mathrm{B}}^2$ to sub-genome B, and ${\mathrm{\sigma }}_{\mathrm{D}}^2$ to sub-genome D. Intra- and inter-sub-genome additive-by-additive epistatic variances are depicted by ${\mathrm{\sigma }}_{\mathrm{AA}}^2$, ${\mathrm{\sigma }}_{\mathrm{BB}}^2$, and ${\mathrm{\sigma }}_{\mathrm{DD}}^2$ and ${\mathrm{\sigma }}_{\mathrm{AB}}^2$, ${\mathrm{\sigma }}_{\mathrm{AD}}^2$, and ${\mathrm{\sigma }}_{\mathrm{BD}}^2$, respectively. ${\mathrm{\sigma }}_{\mathrm{T}}^2$ is the total genetic variance, and ${\mathrm{\sigma }}_{\mathrm{E}}^2$ is the residual variance of the model.

The larger total genetic variance estimated using model 4 increased the magnitude of estimated broad-sense heritability compared to estimates from the whole-genome models 1 and 2 (Supplementary Table 3). In addition, the deviance information criteria values were smaller than those obtained using the whole-genome models, indicating a better fit of data.

Correlation between whole- and sub-genome breeding values

Correlations between whole- and sub-genome effects were estimated to characterize the relationships between sub-genome additive effects and whole-genome effects (Fig. 3). The smallest correlation was observed between the effects of sub-genome D and the whole-genome effects (0.64; Supplementary Table 4), while the correlations between the effects of sub-genome A and B with the whole-genome effects were higher (0.71 and 0.68, respectively). The correlations among sub-genome effects were very low. Interestingly, when analyzing the sub-genome additive effects from model 4, which includes epistasis, the correlations between sub-genome effects and whole-genome effects, as well as between sub-genomes, generally decreased, except for the correlation between sub-genomes A and D. Although correlations between sub-genomes and whole genome varied, all 3 sub-genomes resulted in similar maximum and minimum additive effects and standard deviations (Supplementary Table 5).

Fig. 3.

Sub-genome additive effects correlate with whole-genome effects for grain yield in wheat. The dotted lines indicate the 95 percentile for whole- or sub-genome additive effects. The solid black lines indicate the correlation between whole-genome and sub-genome effects. Upper-left dots indicate genotypes with whole-genome additive effects above the 95 percentile. Bottom-right dots indicate genotypes with sub genome additive effects above the 95 percentile. Top-right dots indicate genotypes above the 95 percentile for both thresholds.

In total, 92 lines were found above the 95-percentile threshold for both whole- and sub-genome A additive effects (Fig. 3). Of those, 41 and 20 genotypes were also above the 95-percentile threshold for sub-genomes B and D, respectively. In sub-genomes B and D, 74 and 54 lines were found above the 95-percentile threshold for both sub-genomes and whole-genome additive effects, with 15 lines above the 95-percentile threshold for both sub-genomes B and D. In total, 10 genotypes were above the 95-percentile for all sub-genomes, 9 of which were sister lines. Hence, in general, genotypes with the highest whole-genome effects tended to also exhibit intermediate to high additive effects in more than 1 sub-genome. Of the top-100 genotypes with the highest whole-genome additive effects, 53 were advanced to the preliminary yield trials. Among these, 5 were selected for the advanced yield trials, and only one of these inbreds was advanced to the elite yield trials (EYTs). One possible reason for the discarding of genotypes with high whole-genome additive effects could be selection for or against other important agronomic traits not considered in this prediction model, traits such as disease resistance, maturity, and lodging. For example, from this pool of 3,740 genotypes evaluated across 3 breeding cycles, only 8 reached the EYT stage, 5 of which were above the top-90 percentile. It is necessary to acknowledge that the total genetic value of each genotype across many traits is used to make advancement decisions, with those genetic values being a function of both additive and epistatic effects. In that regard, the sub-genome additive plus epistatic model 4 estimated higher genotypic values for the 8 lines that reached the EYT stage, with 6 of those 8 falling within the top-10 percentile. Compared to the sub-genome additive model 3, these 8 lines had an average rank change of 145 positions toward the top when epistasis was modeled. This analysis of past selection decisions highlights the advantage of modeling epistasis to attain a better estimation of the true genotypic values of inbred lines that have not yet been extensively field tested.

Genomic predictive ability

The prediction accuracy between the predicted total genetic values and the phenotypic values was evaluated using the 5-fold cross-validation method (Fig. 4). For both whole- and sub-genome models, inclusion of epistatic genetic effects significantly improved the prediction accuracy compared to fully additive models, based upon Tukey's honestly significant difference test. In contrast, partitioning whole-genome effects into sub-genome effects did not improve the accuracy of predictions. The correlations among total genetic values of the 4 models fitted showed very high values (Supplementary Table 7). As expected, the smallest correlation was observed between whole-genome additive model 1 and sub-genome additive plus epistatic model 4. Similarly, correlations among genomic estimated breeding values (sum of additive effects only) included a smaller correlation between the purely additive and the additive plus epistatic models (Supplementary Table 8).

Fig. 4.

Increased prediction accuracy using models that include epistasis. Bar plot of predictive abilities of additive whole-genome a) and sub-genome b) models with the inclusion or not of an epistatic term using a 5-fold cross validation approach with 10 replications.

Discussion

Modeling epistasis improves the total genetic variance and broad-sense heritability

The role of epistasis is expected to be especially relevant in polyploid species due to the interaction between homologous genes, but epistasis has been commonly ignored in wheat. In this study, we rescaled the relationship matrices according to Vitezica et al. (2017) to obtain genetic variance estimates that properly reflect the generation of the reference population (F_5:6), which implies increasing the amount of additive variance nearly 2 times and the additive-by-additive variance by 4times relative to the base F₂ population, should we had followed VanRaden (2008) method. However, based upon modeling variance components, inclusion of epistatic effects in whole- and sub-genome additive models reduced the additive variance, indicating a lack of orthogonality or statistical independence between additive and epistatic effects. Hence, although the 2 whole-genome duplication events that led to the formation of hexaploid wheat suggest more prominent inter-genomic epistatic effects compared to intra-genomic, the lack of orthogonality prevents precise partitioning of inter- and intra-genomic epistatic variance components and the validation of this hypothesis. Despite the implementation of recent developments for orthogonal estimations, i.e. NOIA and correction for LD and covariance between genetic effects, the lack of independence due to nonorthogonal partitioning of genetic effects could be attributed to the strong LD, which introduces nonindependence between loci (Vitezica et al. 2017). Under the presence of LD, covariance between genetic effects can be introduced and exact partitioning genetic effects is not possible (Hill and Mäki-Tanila 2015; Zeng et al. 2005). In simulated and experimental studies, lack of orthogonality was evidenced through a high correlation between additive and epistatic variance component estimates (Vitezica et al. 2017; Raffo et al. 2022). Lack of orthogonality has also been observed in other studies within the context of hybrid breeding (González-Diéguez et al. 2021; Bernardo 1995). González-Diéguez et al. (2021) reported that additive genomic relationship tends to capture additive-by-additive effects when there is a significant correlation between additive and additive-by-additive genomic relationships. This finding aligns with the explanation given by Raffo et al. (2022) and the observations from our study, which reported a correlation of 0.55 between G and I. Furthermore, epistasis itself also contributes to additive genetic variance (Hill et al. 2008; Mäki-Tanila and Hill 2014; Vitezica et al. 2017). Although we did not obtain empirically orthogonal estimates, using methods from Sorensen, Fernando, and Gianola (2001) and Lehermeier et al. (2017), we derived genomic variance estimates that, when combined with error variance estimates, match the sample variance of phenotypes. Therefore, even if the partitioning into additive and epistatic genetic effects is biased due to strong LD, the total genetic variance estimates remain accurate. This checkpoint—summing the estimated genomic variance and error variance—is often overlooked but is crucial to ensure the correctness of genetic variance component estimates.

In this study, the inclusion of epistatic effects in additive models increased the total genetic variance and reduced the deviance information criteria. In contrast, a study of the Nordic Seed A/S program reported a slight reduction in total genetic variance when epistasis was modeled (Raffo et al. 2022), while a study of Cornell's and CIMMYT's breeding populations found that partitioning whole-genome effects into sub-genome additive and epistatic effects only marginally increased the AIC for wheat grain yield (Santantonio et al. 2019). One possible explanation for differing results could relate to the large genetic diversity of the K-State wheat breeding program. The introduction and maintenance of diverse germplasm in the K-State program is expected to increase the allele frequency of genomic regions that would otherwise be (nearly) fixed in programs with limited genetic diversity, as may be the case for CIMMYT, Cornell, and Nordic Seed A/S programs. A small allele frequency can ultimately convert epistatic variance into additive variance (Whitlock et al. 1995; Cheverud and Routman 1996; Holland 2001; Technow et al. 2021), while the segregation of epistatic interactions found at relatively high minor allele frequencies can increase the statistical power to capture epistatic interactions.

Sub-genome additive effects as a tool for parental selection if epistasis is irrelevant

Estimation of sub-genome additive effects has the potential of attributing biological importance to genetic effects commonly assigned to the whole genome (Santantonio et al 2019) and could be leveraged for selecting parental lines with complementary sub-genome effects when designing breeding crosses. A study of Cornell's and CIMMYT's wheat breeding populations found that genotypes with the highest sub-genome additive effects were normally not among the top genotypes having high whole-genome effects, and a low correlation between sub-genomes was regarded as a potential indicator of independence between sub-genomes (Santantonio et al. 2019). In this study, making breeding crosses using parental lines (genotypes) having the highest additive effects within each sub-genome could produce theoretical maximum whole-genome effects of 13.7, 21% higher than the observed highest whole-genome effect (11.3). The success of this approach is tightly linked to the interplay of epistatic effects between and within sub-genomes. Although low pairwise correlations among sub-genomes were observed, high epistatic variance resulted from both intra- and inter-sub-genome interactions, variances that were nearly zero in Cornell's and CIMMYT's breeding populations (Santantonio et al. 2019). In addition, estimation of sub-genome additive effects was less accurate than sub-genome additive plus epistatic effects in predicting which lines reached the elite stages of evaluation in the wheat breeding pipeline. Although the magnitude of epistasis is hard to estimate, epistasis has been reported to affect several traits in wheat, including stem rust (Singh et al. 2013; Rouse et al. 2014), stripe rust (Vazquez et al. 2015), plant height (Zhang et al. 2008; Sannemann et al. 2018), photoperiod response (Shaw et al. 2020), vernalization (Kippes et al. 2014), and grain yield heterosis (Boeven et al. 2020; Jiang et al. 2017). These reports suggest epistatic effects should not be overlooked when estimating or predicting the genetic value of wheat genotypes.

In the germplasm used in this study, all sub-genomes exhibited similar additive effects on grain yield in wheat, despite differences in the number of markers within each sub-genome. While the D sub-genome is typically associated with limited genetic variability and fewer markers, one might expect a lower contribution from it. However, the results indicate that the D sub-genome contributed similarly to sub-genomes A and B. Bernardo (2021) also observed equal contributions from each sub-genome for grain yield, although contributions varied for other traits, suggesting that the impact may depend largely on the specific trait being analyzed. The high additive effects could be in part attributed to major QTLs conferring resistance to barley yellow dwarf virus (BYD). In 2017, a severe BYD infection affected the experimental farm, but a large proportion of the advanced genotypes had parental lines carrying Bdv1 or Bdv2 resistance genes. Bdv1 reportedly confers partial tolerance to BYD and is located on chromosome 7D (Singh et al. 1993; Ayala et al. 2002), while Bdv2, an introgression from Thinopyrum intermedium, is also found on chromosome 7D in hexaploid wheat (Banks et al. 1995; Hohmann et al. 1996; Larkin et al. 2002). Hence, the presence of Bdv1 and Bdv2 alleles in many advanced families could have drastically increased the genetic effects associated with the D sub-genome, as the high disease pressure greatly affected grain yield performance.

Prediction accuracy improves when epistasis is modeled

As epistasis is potentially an important component of genetic effects impacting grain yield, inclusion of epistatic effects in genomic prediction models could significantly improve prediction accuracies. In this study, modeling epistasis in whole- or sub-genome models marginally but significantly improved prediction accuracies, contrasting with the results from Cornell's and CIMMYT's wheat breeding populations, where no improvement in prediction accuracy was observed for grain yield, even though inclusion of epistatic effects improved predictability of other traits (Santantonio et al. 2019). In a study of CIMMYT's Global Wheat and Semiarid wheat breeding populations, an average improvement of 6% in prediction accuracy was observed when modeling additive-by-additive epistatic interactions (Jiang and Reif 2015), while an increase of 5% in prediction accuracy was achieved in KWS European wheat breeding program (He et al. 2016). These consistent but small improvements in prediction accuracy reported do not support the expectation that epistasis has a major role in wheat given the allopolyploid nature of wheat. In allopolyploid species, subfunctionalization could create positive or negative epistasis (Lynch and Force 2000), competition for the same substrate (Qian et al. 2010), or affect the patterns of gene expression (Leach et al. 2014; Akhunova et al. 2010), and directly influence the expression of agronomic traits. Furthermore, homeologous interaction effects could be minor relative to the total epistatic genetic effect in wheat, since epistasis has been reported in diploid wheat, Triticum monccum (Tranquilli and Dubcovsky 2000), and also within sub-genomes in hexaploid wheat (Reif et al. 2011; Sehgal et al. 2020). In this study, although portioning of variance component estimates suggests that epistasis may have a prominent role in grain yield, the marginal improvement in prediction accuracy when modeling epistasis suggests that drawing any conclusion from nonorthogonal variance component estimates requires caution, as estimates may be biased. Another potential reason for the marginal improvement in prediction accuracy when modeling epistasis could relate to overfitting epistatic effects when training the GS model, because, even though the genomic estimates more closely approximate the observed phenotypic values in the training population, that does not translate into equally better prediction accuracies in the validation population. This observation implies that modeling epistasis could be highly useful for selecting the best genotypes in a breeding cycle but that may not deliver commensurate accuracies when selecting new parents for the development of new breeding populations.

Conclusion

To delineate the genetic architecture of wheat, as expressed for grain yield, whole-genome additive and epistatic effects were estimated at the whole-genome level and at the sub-genome level. We implemented recent developments and methods for estimating genetic variance components including the NOIA approach to account for deviation from HWE and correction for LD and covariance between genetic effects. Although estimated phenotypic variance matched the observed phenotypic variance, partitioning of genetic variance components into additive and epistatic variance components was not orthogonal, which can be attributed to the LD. Incorporating epistasis in additive models did augment the total genetic variance and reduced the DIC, indicating existence of epistatic effects controlling the expression of the trait grain yield. Using a fully additive model, additive effects were estimated for each sub-genome with results indicating that genotypes with the largest whole-genome additive effects also tended toward intermediate to large sub-genome additive effects. Estimating sub-genome additive effects has potential as a tool during parental selection, as it could permit pairing lines with complementary sub-genome effects. However, the success of this approach is tightly linked to the extent of inter- and intra-sub-genome epistatic effects. Modeling epistasis in genomic selection models marginally but significantly improved prediction accuracy for both whole-genome and sub-genome models, which can potentially contribute to significant improvements in genetic gain over multiple cycles of selection and advancement.

Data availability

Marker information is included in files “Sub-Genome-A.hmp.txt,” “Sub-Genome-B.hmp.txt,” and “Sub-Genome-D.hmp.txt,” for sub-genomes A, B, and D, respectively. Raw phenotypic data are included in file “Raw Phenotypes.txt” and across-location BLUEs are included in file “BLUEs.txt.” The Var.R, GS.R, and Rank.R are the codes used to estimate the variance components, to run the genomic prediction models, and to rank lines according to sub-genome additive effects, respectively. Supplementary material available at G3 FigShare: https://doi.org/10.25387/g3.25655385. The raw genotypic data can be accessed at NCBI PRJNA1221944.

Supplemental material available at G3 online.

Funding

We acknowledge the Kansas Wheat Alliance and the Kansas Wheat Commission for their essential financial support, which made this research possible.