Crossbreeding schemes are widely used in animal and plant breeding for the purpose of exploiting the heterosis and breed complementarity that often occur in crosses [1]. The main goal of crossbreeding is to improve the performance of crossbred populations. Pure breed/line performance is an imperfect predictor of crossbred performance; there are two reasons that explain this incomplete correlation between purebred and crossbred populations. First, phenotypic measurements on purebred/line individuals are often recorded in only one environment (e.g., management) that differs from the environment in which the crossbred individuals are raised (genotype-by-environment interaction). Second, non-additive genetic effects, such as dominance and/or epistasis, which likely determine heterosis, may result in different breeding values between purebreds and crossbreds.

In the case of dominant inheritance, the theory of pedigree-based genetic evaluation and estimation of genetic parameters for crossbred populations was proposed by Lo et al. [2, 3]. In this model, each individual has two genetic values, one on the purebred scale and one on the crossbred scale. In the absence of inbreeding, it is necessary to estimate nine genetic variance components for an \(F_{1}\) cross between breeds/lines (A and B): additive variance for breed A, dominance variance for breed A, additive variance for breed B, dominance variance for breed B, additive variance for the \(F_{1}\) population due to the effects of the alleles inherited from breed A, additive variance for the \(F_{1}\) population due to the effects of the alleles inherited from breed B, the dominance variance for the \(F_{1}\) population, additive covariance between a parent from breed A and an \(F_{1}\) offspring, and the additive covariance between a parent from breed B and an \(F_{1}\) offspring. Although the relevance of the crossbred model has been shown [4, 5], its use in applied breeding programs is limited, because pedigree relationships between purebred and crossbred individuals are often not known, and large datasets on crosses are needed [6].

Genomic approaches offer tools that allow to perform much deeper analyses, and thus, to understand the effects and the mechanisms of the genes and their interactions that underlie complex traits and to explore new directions for their improvement [7]. In addition, genomic evaluation renews the interest in crossbred individuals because they can be used as training animals [8]. In the case of additive inheritance, a joint genomic evaluation of purebred and crossbred individuals was proposed [9]. Toosi et al. [10] and Zeng et al. [11] extended this approach in order to include dominance. All these studies focused on the selection of purebred individuals for crossbred performance. However, the formal definition of substitution effects and dominant deviations and the estimation of genetic variance components in two breeds/lines and the \(F_{1}\) population have not been revisited so far within the genomic framework. This is needed for correct genetic evaluation and for planning selection schemes. The additive variances due to the gametes from the pure lines that compose the \(F_{1}\) population are an indicator of how much can be gained by selection. Estimation of dominance variance for the \(F_{1}\) individuals can be considered as a predictor of the variability of specific combining ability, i.e. how relevant is assortative mating between purebred lines to maximize the phenotype at a trait of interest in the \(F_{1}\) population. As an example, a common procedure in maize breeding is to use “testers” to evaluate the performance of a pure line as a parent in a cross. If the level of dominance variance is high, the use of testers might severely bias selection towards those lines that combine adequately with a particular tester. In practice, the estimated variance components serve as a guide for choosing breeds/lines with good combining abilities (e.g., pigs, corn, etc.) in animal and plant breeding schemes.

The objective of this work was twofold. First, we decomposed variance components for an \(F_{1}\) population using a genomic model with additive and non-additive (dominance) inheritance. Second, we applied the approach to estimate variance components using pig data. To our knowledge, there is no published description of the theoretical variance components (additive and dominant) in terms of substitution effect across two pure populations and the crossbred population. The next section describes the theory on which the estimation of genotypic values is based using GBLUP.

An \(F_{1}\) population involves gametes from the parental populations 1 and 2. If dominance is present, and because allelic frequencies differ in each breed, the within-breed (additive) substitution effects are not equal to the substitution effects across the \(F_{1}\) population. Thus, purebred individuals have different breeding values depending on whether they are mated to individuals from the same or another breed/line. This situation is well known [3, 12, 13], and holds even if the genotype effects are constant across breeds or crossbred individuals.

\(\sigma_{{A_{1} }}^{2}\) \((\sigma_{{A_{2} }}^{2} )\) is the variance of the GCA of the alleles of individuals from population 1 crossed to individuals from population 2 (alleles of individuals from population 2 crossed with individuals from population 1) or it can also be considered as the additive variance of gametes inherited from population 1 (from population 2) in the \(F_{1}\) population as in Lo et al. [3].

The variance of the GCA (\(\sigma_{A}^{2}\)) is an important parameter to understand if selection of purebred individuals can increase crossbred performance [1]. If variance of the GCA explains a large part of the total genetic variance for the \(F_{1}\) population, it means that within-population selection will result in a large increase of the crossbred performance, without resorting to specific matings to create crossbreds with large dominance deviations.

The term \(ad\) appears in \(\sigma_{G}^{2}\) but is completely embedded in \(\sigma_{{A_{1} }}^{2}\) and \(\sigma_{{A_{2} }}^{2}\). This term differs from 0 if there is covariance between \(a\) and \(d\), i.e. if \(a\) and \(d\) are of the same magnitude and direction or if there is overdominance. This covariance between additive and dominant effects of genes implies the presence of inbreeding depression or heterosis. Different models have been proposed to take the dependency between additive and dominant effects into account [15].

If \(a\) and \(d\) effects are considered as random variables with a covariance of 0 between \(a\) and \(d\), variance components for the \(F_{1}\) population can be obtained from these expressions using markers in a GBLUP context as detailed in the next section.

In this section, we illustrate the partition of variance components (additive and dominant) across two pig lines 1 and 2 and the crossbred population. Data for this study were provided by Genus plc (Hendersonville, TN, USA). Animal Care and Use Committee approval was not obtained for this study because the data were obtained from an existing database.

Lines 1 and 2 were two unrelated lines, and population 12 consisted of both reciprocal crosses of animals from lines 1 and 2. Data on litter size (total number of piglets born per litter) were analyzed. The average litter size was equal to 12.68 ± 3.07, 13.15 ± 3.20 and 13.64 ± 3.16 for lines 1 and 2 and population 12, respectively. A total of 34,753 records were available for 8265 sows. Genotypes for all sows were generated using the Illumina PorcineSNP60 BeadChip (Illumina, San Diego, CA). After quality control, i.e. after excluding genotypes with a minor allele frequency lower than 0.05 and a SNP call rate less than 0.90 in the overall population, 40,634 SNPs remained and were used to build genomic relationship matrices. Animals with a call rate less than 0.90 were removed. Thus, the number of sows with genotypes was equal to 3509, 2706 and 2050 in lines 1 and 2 and population 12, respectively.

Phenotypes were collected for the genetic nucleus (pure lines) and commercial herds (crosses). Records were analyzed using a GBLUP (mixed) model. Fixed effects included parity order, farm, year and month of farrowing, and mating type (artificial insemination or natural service).

To estimate the variance components, lines 1 and 2 and population 12 were considered as three different traits with correlations between pure and cross lines [3]. This model is equivalent to a model where marker effects are correlated across populations [20–22] and assumes that additive and dominant effects of a gene (\(a_{1} , a_{2} , a_{12}\) and \(d_{1} , d_{2} , d_{12} )\) are not necessarily the same in the three populations. Quantitative trait loci (QTL) that segregate in different breeds are not necessarily identical. In addition, linkage disequilibrium between SNPs and QTL can differ between populations. Even with causal genes, the effects may differ, which was confirmed by experimental results. One example is the bovine myostatin gene (GDF8), i.e. both the Belgian Blue and South Devon breeds carry the same GDF8 mutation, but they have different conformation and double-muscling phenotypes [23]. Functional mutations in the GDF8 gene appear to be breed-specific [24]. Effects can be population-specific and the variation can be interpreted as a dependency of the gene effect on the environmental (GxE) and genetic (i.e. epistasis) backgrounds. Parental pure lines and the \(F_{1}\) population have only half of their genetic background in common.

SNP dominance variances \((\sigma_{{d_{1} }}^{2} , \sigma_{{d_{2} }}^{2} ,\sigma_{{d_{12} }}^{2} )\) are obtained similarly, e.g., as \(\sigma_{{d_{1} }}^{2} = \hat{\sigma }_{{D_{1}^{*} }}^{2} /\left( {\left\{ {tr\left[ {{\mathbf{WW^{\prime}}}} \right]} \right\}/n} \right)\). The covariance matrices for permanent environmental and residual effects are as follows: \( Var\left[ {\begin{array}{*{20}c} {{\mathbf{p}}_{1} } \\ {{\mathbf{p}}_{2} } \\ {{\mathbf{p}}_{12} } \\ \end{array} } \right] = \left[ {\begin{array}{*{20}c} {\sigma_{{p_{1} }}^{2} } & {\bf 0} & {\bf 0} \\ {\bf 0} & {\sigma_{{p_{2} }}^{2} } & {\bf 0} \\ {\bf 0} & {\bf 0} & {\sigma_{{p_{12} }}^{2} } \\ \end{array} } \right]\varvec{ } \otimes \;{\mathbf{I}}_{3} \), and \(Var\left[ {\begin{array}{*{20}c} {{\mathbf{e}}_{1} } \\ {{\mathbf{e}}_{2} } \\ {{\mathbf{e}}_{12} } \\ \end{array} } \right] = \left[ {\begin{array}{*{20}c} {\sigma_{{e_{1} }}^{2} } & {\bf 0} & {\bf 0} \\ {\bf 0} & {\sigma_{{e_{2} }}^{2} } & {\bf 0} \\ {\bf 0} & {\bf 0} & {\sigma_{{e_{12} }}^{2} } \\ \end{array} } \right]\varvec{ } \otimes \;{\mathbf{I}}_{3}\), respectively.

Variance components for the genomic model (GBLUP model) and for a pedigree-based model (PED model, not including dominance) were estimated by EM-REML (expectation maximization restricted maximum likelihood) using the software remlf90 ([26]; available at http://nce.ads.uga.edu/wiki/doku.php), plus an additional iteration of AIREML to obtain the average information matrix. It should be noted that estimated values of \(\sigma_{{A^{*} }}^{2}\) and \(\sigma_{{D^{*} }}^{2}\) have per se no meaningful genetic interpretation.

Additive and dominance variance components at the SNP level (\(\sigma_{{a_{1} }}^{2} , \sigma_{{a_{2} }}^{2} ,\sigma_{{a_{12} }}^{2}\) and \(\sigma_{{d_{1} }}^{2} ,\sigma_{{d_{2} }}^{2} ,\sigma_{{d_{12} }}^{2}\)) were back-solved (dividing by \(\left\{ {tr\left[ {{\mathbf{ZZ}}^{'} } \right]} \right\}/n)\) and \(\left\{ {tr\left[ {{\mathbf{WW}}\varvec{'}} \right]} \right\}/n\)) from variance component estimates (\(\sigma_{{A_{1}^{*} }}^{2} ,\sigma_{{A_{2}^{*} }}^{2} ,\sigma_{{A_{12}^{*} }}^{2}\) and \(\sigma_{{D_{1}^{*} }}^{2} ,\sigma_{{D_{2}^{*} }}^{2} ,\sigma_{{D_{12}^{*} }}^{2}\), respectively) for the three populations. Genetic variance components for the \(F_{1}\) population were obtained from Eqs. (5), (6) and (7). Asymptotic standard errors of variance component estimates were obtained as in [27].

Assuming that SNP effects are independent of the origin of alleles and that allelic frequencies differ between parental populations, we show that the genetic variance for the \(F_{1}\) population includes the biological additive and dominant effects of the gene and a covariance term. Genetic variance can be partitioned into additive variance (due to substitution effects of the parental gametes) and dominance deviations. Breeding values of crossbred individuals are generated by substitution effects, where the effects for each parental line depend on the allele frequencies from the other line. In addition, dominance variance is proportional to the product of the heterozygosities of both lines. If the biological additive and dominant effects of markers are considered random with the covariance between them equal to 0, genetic variance components for the \(F_{1}\) population can be obtained using an equivalent GBLUP model based on SNPs. The method presented here allows selection for specific combining ability, i.e. selection of a specific pair of parents to produce superior \(F_{1}\) individuals, in a GBLUP evaluation framework. The identification of superior \(F_{1}\) individuals between inbred/pure lines is an important focus of research in animals and plants [33].