GENOMIC SELECTION On the Additive and Dominant Variance and Covariance of Individuals Within the Genomic Selection Scope

Genomic evaluation models can fit additive and dominant SNP effects. Under quantitative genetics theory, additive or “breeding” values of individuals are generated by substitution effects, which involve both “biological” additive and dominant effects of the markers. Dominance deviations include only a portion of the biological dominant effects of the markers. Additive variance includes variation due to the additive and dominant effects of the markers. We describe a matrix of dominant genomic relationships across individuals, D, which is similar to the G matrix used in genomic best linear unbiased prediction. This matrix can be used in a mixed-model context for genomic evaluations or to estimate dominant and additive variances in the population. From the “genotypic” value of individuals, an alternative parameterization defines additive and dominance as the parts attributable to the additive and dominant effect of the markers. This approach underestimates the additive genetic variance and overestimates the dominance variance. Transforming the variances from one model into the other is trivial if the distribution of allelic frequencies is known. We illustrate these results with mouse data (four traits, 1884 mice, and 10,946 markers) and simulated data (2100 individuals and 10,000 markers). Variance components were estimated correctly in the model, considering breeding values and dominance deviations. For the model considering genotypic values, the inclusion of dominant effects biased the estimate of additive variance. Genomic models were more accurate for the estimation of variance components than their pedigree-based counterparts. GENOMIC evaluation models typically fit only marker or haplotypic additive effects, either explicitly, estimating the effect of each marker (Meuwissen et al. 2001; VanRaden 2008; De Los Campos et al. 2009), or implicitly through the so-called “genomic” relationship matrix (VanRaden 2008; Goddard 2009), which uses an equivalent model from which the marker effects can be inferred by backsolving. The additive substitution effect represents the average change in genotypic value that results when an A1 allele is randomly substituted for an A2 allele in a population (Falconer 1981; Lynch and Walsh 1998). The substitution effects in quantitative genetics are of primal interest, as they capture a large part of dominant and higher-order interactions across genes and alleles (i.e., epistasis) (e.g., Cockerham 1954; Kempthorn 1954; Falconer 1981). In addition, selection acts on additive substitution effects because alleles, not genotypes, are passed from parents to offspring. However, dominance is of theoretical and practical interest, because it is heavily used in crosses of animal breeds and plant lines (e.g., in pigs, poultry, or corn). In principle, assortative mating or mate allocation (when the breeder chooses specific couples of individuals as parents for the next generation, for instance, favorable dominant combinations) can boost the field performances of livestock and crops (Varona and Misztal 1999; Toro and Varona 2010). In livestock populations, one of the main reasons why dominance effects have not been widely used or estimated per individual is that pedigree relationships are not enough informative, as large full-sib families are typically needed for any accurate estimate. In prolific species such as chickens and Copyright © 2013 by the Genetics Society of America doi: 10.1534/genetics.113.155176 Manuscript received July 21, 2013; accepted for publication October 4, 2013; published Early Online October 11, 2013. Supporting information is available online at http://www.genetics.org/lookup/suppl/ doi:10.1534/genetics.113.155176/-/DC1. Corresponding author: Unité Mixte de Recherche 1289 TANDEM, ENSAT, Avenue de l’Agrobiopole, Postal Box 32607, 31326 Auzeville Tolosane, France. E-mail: [email protected] Genetics, Vol. 195, 1223–1230 December 2013 1223 pigs, the litter effect is highly confounded with family. In addition, the prediction of dominant values is typically very cumbersome because it involves complex computations (e.g., Misztal et al. 1998; Mrode and Thompson 2005). Recently, genomic evaluations have renewed the interest in the prediction of nonadditive genetic effects (e.g., Toro and Varona 2010; Su et al. 2012; Wellmann and Bennewitz 2012). One of the reasons is that it is much easier to work with dominance, knowing for each evaluated locus which animals are heterozygotes, but also that prediction of the genotypic value of future matings is straightforward (Toro and Varona 2010). An example of the richness of parametric genomic prediction methods for the estimation of marker effects is to have equivalent models and interpretations through the genetic covariance between individuals [i.e., genomic relationships (VanRaden 2008; Goddard 2009; Yang et al. 2010)] and estimation of the base population variances (Forni et al. 2011; Legarra et al. 2011b; Sillanpaa 2011). These equivalences are not quite completely described for the case of dominance effects in genomic evaluations (e.g., Toro and Varona 2010; Su et al. 2012; Wellmann and Bennewitz 2012). These works are either incomplete or, in part, induce erroneous interpretations of correct models. The aim of this study is to show the equivalences between additive and dominant effects at the marker and the population levels and to present how to compute from genotypes the covariances between individuals due to dominant deviations, e.g., “dominant genomic relationships”. Real data and a simulated example are used to illustrate the principles.