Rapid inversion of dominance relationship matrices for noninbred populations by including sire by dam subclass effects

Progress in Genomic Mating in Domestic Animals

Simple Summary

Since animal domestication, breeders have been selecting candidates for breeding based on phenotypic performance. Estimating breeding values through the best linear unbiased prediction method represents a revolutionary shift in animal breeding. On this basis, selection and mating are utilized to improve the production level of animals. The application of genomic selection has once again revolutionized animal breeding methods. However, although this kind of truncated selection based on breeding values can significantly improve genetic gain, the genetic relationship between individuals with a high breeding value is usually closed, and the probability of being co-selected is greater, which will lead to a rapid increase in the rate of inbreeding in the population. Reduced genetic variation is not conducive to long-term sustainable breeding, so a trade-off between genetic gain and inbreeding is required. Genomic mating is the use of candidate individuals’ genomic information to implement optimized breeding and mating, which can effectively control the rate of inbreeding in the population and achieve long-term and sustainable genetic gain. It is more suitable for modern animal breeding, especially for conservation and genetic improvement of local domestic animal breeds.

Abstract

Selection is a continuous process that can influence the distribution of target traits in a population. From the perspective of breeding, elite individuals are selected for breeding, which is called truncated selection. With the introduction and application of the best linear unbiased prediction (BLUP) method, breeders began to use pedigree-based estimated breeding values (EBV) to select candidates for the genetic improvement of complex traits. Although truncated selection based on EBV can significantly improve the genetic progress, the genetic relationships between individuals with a high breeding value are usually closed, and the probability of being co-selected is greater, which will lead to a rapid increase in the level of inbreeding in the population. Reduced genetic variation is not conducive to long-term sustainable breeding, so a trade-off between genetic progress and inbreeding is required. As livestock and poultry breeding enters the genomic era, using genomic information to obtain optimal mating plans has formally been proposed by Akdemir et al., a method called genomic mating (GM). GM is more accurate and reliable than using pedigree information. Moreover, it can effectively control the inbreeding level of the population and achieve long-term and sustainable genetic gain. Hence, GM is more suitable for modern animal breeding, especially for local livestock and poultry breed conservation and genetic improvement. This review mainly summarized the principle of genomic mating, the methodology and usage of genomic mating, and the progress of its application in livestock and poultry.

Genome-wide interaction study reveals epistatic interactions for beef lipid-related traits in Nellore cattle

Gene-gene interactions cause hidden genetic variation in natural populations and could be responsible for the lack of replication that is typically observed in complex traits studies. This study aimed to identify gene-gene interactions using the empirical Hilbert-Schmidt Independence Criterion method to test for epistasis in beef fatty acid profile traits of Nellore cattle. The dataset contained records from 963 bulls, genotyped using a 777 962k SNP chip. Meat samples of Longissimus muscle, were taken to measure fatty acid composition, which was quantified by gas chromatography. We chose to work with the sums of saturated (SFA), monounsaturated (MUFA), polyunsaturated (PUFA), omega-3 (OM3), omega-6 (OM6), SFA:PUFA and OM3:OM6 fatty acid ratios. The SNPs in the interactions where P < 10 - 8 were mapped individually and used to search for candidate genes. Totals of 602, 3, 13, 23, 13, 215 and 169 candidate genes for SFAs, MUFAs, PUFAs, OM3s, OM6s and SFA:PUFA and OM3:OM6 ratios were identified respectively. The candidate genes found were associated with cholesterol, lipid regulation, low-density lipoprotein receptors, feed efficiency and inflammatory response. Enrichment analysis revealed 57 significant GO and 18 KEGG terms ( P < 0.05), most of them related to meat quality and complementary terms. Our results showed substantial genetic interactions associated with lipid profile, meat quality, carcass and feed efficiency traits for the first time in Nellore cattle. The knowledge of these SNP-SNP interactions could improve understanding of the genetic and physiological mechanisms that contribute to lipid-related traits and improve human health by the selection of healthier meat products.

Estimation of additive genetic, dominance, and mate sire variances for fertility traits in Thai native (Pradu Hang Dam) chickens

The objective of this study was to compare the appropriate models used to estimate the value of genetic parameters in fertility traits: fertility (FER), hatchability of fertile eggs (HOF), and hatchability of eggs set (HOS) in Thai native (Pradu Hang Dam) chickens. Data were collected for each fertility trait from 3435 test-week records from 715 hens, 158 mate sires, and 972 pedigree animals. Three random regression models were analyzed: model 1 (M1: A + PE) was adjusted by using additive genetic and permanent environmental effects. Model 2 (M2: A + PE + D) was adjusted by using the dominance effect. Finally, model 3 (M3: A + MS + PE + D) was adjusted by using the mate sire effect. The results found the low heritability of FER (M1 to M3), HOF (M1 to M3), and HOS (M1 to M3) ranged from 0.031-0.040, 0.037-0.066, and 0.040-0.059, respectively. Adjustment for the dominance and mate sire effects in M3 reduced the upward bias in heritability and improved the accuracy of variance component estimates compared to M1 and M2. In conclusion, the genetic evaluation for FER, HOF, and HOS can include the dominance and MS effects to increase the accuracy of evaluation of breeding values and plan for mate selection in breeding programs.

ggroups: an R package for pedigree and genetic groups data

BACKGROUND

R is a multi-platform statistical software and an object oriented programming language. The package archive network for R provides CRAN repository that features over 15,000 free open source packages, at the time of writing this article (https://cran.r-project.org/web/packages, accessed in October 2019). The package ggroups is introduced in this article. The purpose of this package is providing functions for checking and processing the pedigree, calculation of the additive genetic relationship matrix and its inverse, which are used to study the population structure and predicting the genetic merit of animals. Calculation of the dominance relationship matrix and its inverse are also covered. A concept in animal breeding is genetic groups, which is about the inequality of the average genetic merits for groups of unknown parents. The package provides functions for the calculation of the matrix of genetic group contributions (Q). Calculating Q is computationally demanding, and depending on the size of the pedigree and the number of genetic groups, it might not be feasible using personal computers. Therefore, a computationally optimised function and its parallel processing alternative are provided in the package.

RESULTS

Using sample data, outputs from different functions of the package were presented to illustrate a real experience of working with the package.

CONCLUSIONS

The presented R package is a free and open source tool mainly for quantitative geneticists and ecologists, who deal with pedigree data. It provides numerous functions for handling pedigree data, and calculating various pedigree-based matrices. Some of the functions are computationally optimised for large-scale data.

Genetic evaluation of eggshell color based on additive and dominance models in laying hens

Objective

Eggshells with a uniform color and intensity are important for egg production because many consumers assess the quality of an egg according to the shell color. In the present study, we evaluated the influence of dominant effects on the variations in eggshell color after 32 weeks in a crossbred population.

Methods

This study was conducted using 7,878 eggshell records from 2,626 hens. Heritability was estimated using a univariate animal model, which included inbreeding coefficients as a fixed effect and animal additive genetic, dominant genetic, and residuals as random effects. Genetic correlations were obtained using a bivariate animal model. The optimal diagnostic criteria identified in this study were: L* value (lightness) using a dominance model, and a* (redness), and b* (yellowness) value using an additive model.

Results

The estimated heritabilities were 0.65 for shell lightness, 0.42 for redness, and 0.60 for yellowness. The dominance heritability was 0.23 for lightness. The estimated genetic correlations were 0.61 between lightness and redness, −0.84 between lightness and yellowness, and −0.39 between redness and yellowness.

Conclusion

These results indicate that dominant genetic effects could help to explain the phenotypic variance in eggshell color, especially based on data from blue-shelled chickens. Considering the dominant genetic variation identified for shell color, this variation should be employed to produce blue eggs for commercial purposes using a planned mating system.

Additional considerations to the use of single-step genomic predictions in a dominance setting

Recent publications indicate that single-step models are suitable to estimate breeding values, dominance deviations and total genetic values with acceptable quality. Additive single-step methods implicitly extend known number of allele information from genotyped to non-genotyped animals. This theory is well derived in an additive setting. It was recently shown, at least empirically, that this basic strategy can be extended to dominance with reasonable prediction quality. Our study addressed two additional issues. It illustrated the theoretical basis for extension and validated genomic predictions to dominance based on single-step genomic best linear unbiased prediction theory. This development was then extended to include inbreeding into dominance relationships, which is a currently not yet solved issue. Different parametrizations of dominance relationship matrices were proposed. Five dominance single-step inverse matrices were tested and described as C¹ , C² , C³ , C⁴ and C⁵ . Genotypes were simulated for a real pig population (n = 11,943 animals). In order to avoid any confounding issues with additive effects, pseudo-records including only dominance deviations and residuals were simulated. SNP effects of heterozygous genotypes were summed up to generate true dominance deviations. We added random noise to those values and used them as phenotypes. Accuracy was defined as correlation between true and predicted dominance deviations. We conducted five replicates and estimated accuracies in three sets: between all (S₁ ), non-genotyped (S₂ ) and inbred non-genotyped (S₃ ) animals. Potential bias was assessed by regressing true dominance deviations on predicted values. Matrices accounting for inbreeding (C³ , C⁴ and C⁵ ) best fit. Accuracies were on average 0.77, 0.40 and 0.46 in S₁ , S₂ and S₃ , respectively. In addition, C³ , C⁴ and C⁵ scenarios have shown better accuracies than C¹ and C² , and dominance deviations were less biased. Better matrix compatibility (accuracy and bias) was observed by re-scaling diagonal elements to 1 minus the inbreeding coefficient (C⁵ ).

Predicting segregation of multiple fruit-quality traits by using accumulated phenotypic records in citrus breeding

In the breeding of citrus (Citrus spp.), suitable fruit quality is essential for consumer acceptance of new cultivars. To identify parental combinations producing F₁ progeny with fruit-quality traits exceeding certain selection criteria, we developed a simple and practical method for predicting multiple-trait segregation in an F₁ progeny population. This method uses breeding values of parental genotypes and an additive genetic (co)variance matrix calculated by the best linear unbiased prediction method to construct a model for trait segregation in F₁ progeny. To confirm the validity of our proposed method, we calculated the breeding values and additive genetic (co)variances based on phenotypic records on nine fruit-quality traits in 2122 genotypes, and constructed a trait segregation model. Subsequently, we applied the trait segregation model to all pairs of the 2122 genotypes (i.e., 2,252,503 combinations), and predicted the most promising combinations and evaluated their probabilities of producing superior genotypes exceeding the nine fruit-quality traits of satsuma mandarin (Citrus unshiu Marcow.) or ‘Shiranuhi’ (‘Kiyomi’ × ‘Nakano No. 3’ ponkan), two popular citrus cultivars in Japan. We consider these results to be useful not only for selecting good parental combinations for fruit quality or other important traits but also for determining the scale of breeding programs required to achieve specific breeding goals.

Considering dominance in reduced single-step genomic evaluations

Single-step models including dominance can be an enormous computational task and can even be prohibitive for practical application. In this study, we try to answer the question whether a reduced single-step model is able to estimate breeding values of bulls and breeding values, dominance deviations and total genetic values of cows with acceptable quality. Genetic values and phenotypes were simulated (500 repetitions) for a small Fleckvieh pedigree consisting of 371 bulls (180 thereof genotyped) and 553 cows (40 thereof genotyped). This pedigree was virtually extended for 2,407 non-genotyped daughters. Genetic values were estimated with the single-step model and with different reduced single-step models. Including more relatives of genotyped cows in the reduced single-step model resulted in a better agreement of results with the single-step model. Accuracies of genetic values were largest with single-step and smallest with reduced single-step when only the cows genotyped were modelled. The results indicate that a reduced single-step model is suitable to estimate breeding values of bulls and breeding values, dominance deviations and total genetic values of cows with acceptable quality.

Dominance genetic and maternal effects for genetic evaluation of egg production traits in dual-purpose chickens

1. A study was conducted to study direct dominance genetic and maternal effects on genetic evaluation of production traits in dual-purpose chickens. The data set consisted of records of body weight and egg production of 49 749 Mazandaran fowls from 19 consecutive generations. Based on combinations of different random effects, including direct additive and dominance genetic and maternal additive genetic and environmental effects, 8 different models were compared. 2. Inclusion of a maternal genetic effect in the models noticeably improved goodness of fit for all traits. Direct dominance genetic effect did not have noticeable effects on goodness of fit but simultaneous inclusion of both direct dominance and maternal additive genetic effects improved fitting criteria and accuracies of genetic parameter estimates for hatching body weight and egg production traits. 3. Estimates of heritability (h²) for body weights at hatch, 8 weeks and 12 weeks of age (BW0, BW8 and BW12, respectively), age at sexual maturity (ASM), average egg weights at 28-32 weeks of laying period (AEW), egg number (EN) and egg production intensity (EI) were 0.08, 0.21, 0.22, 0.22, 0.21, 0.09 and 0.10, respectively. For BW0, BW8, BW12, ASM, AEW, EN and EI, proportion of dominance genetic to total phenotypic variance (d²) were 0.06, 0.08, 0.01, 0.06, 0.06, 0.08 and 0.07 and maternal heritability estimates (m²) were 0.05, 0.04, 0.03, 0.13, 0.21, 0.07 and 0.03, respectively. Negligible coefficients of maternal environmental effect (c²) from 0.01 to 0.08 were estimated for all traits, other than BW0, which had an estimate of 0.30. 4. Breeding values (BVs) estimated for body weights at early ages (BW0 and BW8) were considerably affected by components of the models, but almost similar BVs were estimated by different models for higher age body weight (BW12) and egg production traits (ASM, AEW, EN and EI). Generally, it could be concluded that inclusion of maternal effects (both genetic and environmental) and, to a lesser extent, direct dominance genetic effect would improve the accuracy of genetic evaluation for early age body weights in dual-purpose chickens.

Estimates of parental-dominance and full-sib permanent environment variances in laying hens

Estimates of variance components for five egg traits on 26265 laying hens were obtained by restricted maximum likelihood (REML) using several models. In the DOMFS model, the effects included hatch group, additive genetic, full-sib, parental dominance and inbreeding depression. In the DOM model, the full-sib effect was eliminated. In the FS model, the parental dominance effect was eliminated. In the ADD model, both the full-sib and the dominance effects were eliminated. In the DOMFS model, the estimates of the full-sib variance were generally higher for egg production traits and lower for egg characteristics than those of the parental dominance variance. However, this model has partially failed in separating these two components. When the full-sib effect was removed from the model, almost all of its estimated variance moved to the estimated parental dominance variance. When the parental dominance effect was removed from the model, almost all of its estimated variance moved to the estimated full-sib variance. The estimates obtained with REML and the DOM model were compared with those obtained by method R and tilde-hat methodologies. The d2 (ratio of dominance variance to total variance) differed by up to 86% for method R and up to 225% for tilde-hat. The h2 differed by up to 26 and 28%, respectively. For data sets that are too large to be analysed with REML, method R is a feasible alternative. A model for estimation of dominance variance should also include the full-sib or a similar effect, provided the data set is large. Similarly, to analyse egg production traits, the model should include at least the full-sib effect.

Mother-offspring and nest-mate resemblance but no heritability in early-life telomere length in white-throated dippers

Telomeres are protective DNA-protein complexes located at the ends of eukaryotic chromosomes, whose length has been shown to predict life-history parameters in various species. Although this suggests that telomere length is subject to natural selection, its evolutionary dynamics crucially depends on its heritability. Using pedigree data for a population of white-throated dippers (Cinclus cinclus), we test whether and how variation in early-life relative telomere length (RTL, measured as the number of telomeric repeats relative to a control gene using qPCR) is transmitted across generations. We disentangle the relative effects of genes and environment and test for sex-specific patterns of inheritance. There was strong and significant resemblance among offspring sharing the same nest and offspring of the same cohort. Furthermore, although offspring resemble their mother, and there is some indication for an effect of inbreeding, additive genetic variance and heritability are close to zero. We find no evidence for a role of either maternal imprinting or Z-linked inheritance in generating these patterns, suggesting they are due to non-genetic maternal and common environment effects instead. We conclude that in this wild bird population, environmental factors are the main drivers of variation in early-life RTL, which will severely bias estimates of heritability when not modelled explicitly.

Accounting for dominance to improve genomic evaluations of dairy cows for fertility and milk production traits

Background

Dominance effects may contribute to genetic variation of complex traits in dairy cattle, especially for traits closely related to fitness such as fertility. However, traditional genetic evaluations generally ignore dominance effects and consider additive genetic effects only. Availability of dense single nucleotide polymorphisms (SNPs) panels provides the opportunity to investigate the role of dominance in quantitative variation of complex traits at both the SNP and animal levels. Including dominance effects in the genomic evaluation of animals could also help to increase the accuracy of prediction of future phenotypes. In this study, we estimated additive and dominance variance components for fertility and milk production traits of genotyped Holstein and Jersey cows in Australia. The predictive abilities of a model that accounts for additive effects only (additive), and a model that accounts for both additive and dominance effects (additive + dominance) were compared in a fivefold cross-validation.

Results

Estimates of the proportion of dominance variation relative to phenotypic variation that is captured by SNPs, for production traits, were up to 3.8 and 7.1 % in Holstein and Jersey cows, respectively, whereas, for fertility, they were equal to 1.2 % in Holstein and very close to zero in Jersey cows. We found that including dominance in the model was not consistently advantageous. Based on maximum likelihood ratio tests, the additive + dominance model fitted the data better than the additive model, for milk, fat and protein yields in both breeds. However, regarding the prediction of phenotypes assessed with fivefold cross-validation, including dominance effects in the model improved accuracy only for fat yield in Holstein cows. Regression coefficients of phenotypes on genetic values and mean squared errors of predictions showed that the predictive ability of the additive + dominance model was superior to that of the additive model for some of the traits.

Conclusions

In both breeds, dominance effects were significant (P < 0.01) for all milk production traits but not for fertility. Accuracy of prediction of phenotypes was slightly increased by including dominance effects in the genomic evaluation model. Thus, it can help to better identify highly performing individuals and be useful for culling decisions.

Phenotype-associated inbreeding biases estimates of inbreeding depression in a wild bird population

Inbreeding depression is usually quantified by regressing individual phenotypic values on inbreeding coefficients, implicitly assuming there is no correlation between an individual's phenotype and the kinship coefficient to its mate. If such an association between parental phenotype and parental kinship exists, and if the trait of interest is heritable, estimates of inbreeding depression can be biased. Here we first derive the expected bias as a function of the covariance between mean parental breeding value and parental kinship. Subsequently, we use simulated data to confirm the existence of this bias, and show that it can be accounted for in a quantitative genetic animal model. Finally, we use long-term individual-based data for white-throated dippers (Cinclus cinclus), a bird species in which inbreeding is relatively common, to obtain an empirical estimate of this bias. We show that during part of the study period, parents of inbred birds had shorter wings than those of outbred birds, and as wing length is heritable, inbred individuals were smaller, independent of any inbreeding effects. This resulted in the overestimation of inbreeding effects. Similarly, during a period when parents of inbred birds had longer wings, we found that inbreeding effects were underestimated. We discuss how such associations may have arisen in this system, and why they are likely to occur in others, too. Overall, we demonstrate how less biased estimates of inbreeding depression can be obtained within a quantitative genetic framework, and suggest that inbreeding and additive genetic effects should be accounted for simultaneously whenever possible.

Estimation of dominance variance for live body weight in a crossbred population of pigs

The objective of this study was to estimate the dominance variance for repeated live BW records in a crossbred population of pigs. Data were provided by the Walloon Pig Breeding Association and included 22,197 BW records of 2,999 crossbred Piétrain × Landrace K+ pigs from 50 to 210 d of age. The BW records were standardized and adjusted to 210 d of age for analysis. Three single-trait random regression animal models were used: Model 1 without parental subclass effect, Model 2 with parental subclasses considered unrelated, and Model 3 with the complete parental dominance relationship matrix. Each model included sex, contemporary group, and heterosis as fixed effects as well as additive genetic, permanent environment, and residual as random effects. Variance components and their SE were estimated using a Gibbs sampling algorithm. Heritability tended to increase with age: from 0.50 to 0.64 for Model 1, from 0.19 to 0.42 for Model 2, and from 0.31 to 0.53 for Model 3. Permanent environmental variance tended to decrease with age and accounted for 29 to 44% of total variance for Model 1, 29 to 37% of total variance for Model 2, and 34 to 51% of total variance for Model 3. Residual variance explained <10% of total variance for the 3 models. Dominance variance was computed as 4 times the estimated parental subclass variance. Dominance variance accounted for 22 to 40% of total variance for Model 2 and 5 to 11% of total variance for Model 3, with a decrease with age for both models. Results showed that dominance effects exist for growth traits in pigs and may be reasonably large. The use of the complete dominance relationship matrix may improve the estimation of additive genetic variances and breeding values. Moreover, a dominance effect could be especially useful in selection programs for individual matings through the use of specific combining ability to maximize growth potential of crossbred progeny.

Inversion of a part of the numerator relationship matrix using pedigree information

Background

In recent theoretical developments, the information available (e.g. genotypes) divides the original population into two groups: animals with this information (selected animals) and animals without this information (excluded animals). These developments require inversion of the part of the pedigree-based numerator relationship matrix that describes the genetic covariance between selected animals (A₂₂). Our main objective was to propose and evaluate methodology that takes advantage of any potential sparsity in the inverse of A₂₂ in order to reduce the computing time required for its inversion. This potential sparsity is brought out by searching the pedigree for dependencies between the selected animals. Jointly, we expected distant ancestors to provide relationship ties that increase the density of matrix A₂₂ but that their effect on A22-1

might be minor. This hypothesis was also tested.

Methods

The inverse of A₂₂ can be computed from the inverse of the triangular factor (T^-1) obtained by Cholesky root-free decomposition of A₂₂. We propose an algorithm that sets up the sparsity pattern of T^-1 using pedigree information. This algorithm provides positions of the elements of T^-1 worth to be computed (i.e. different from zero). A recursive computation of A22-1 is then achieved with or without information on the sparsity pattern and time required for each computation was recorded. For three numbers of selected animals (4000; 8000 and 12 000), A₂₂ was computed using different pedigree extractions and the closeness of the resulting A22-1 to the inverse computed using the fully extracted pedigree was measured by an appropriate norm.

Results

The use of prior information on the sparsity of T^-1 decreased the computing time for inversion by a factor of 1.73 on average. Computational issues and practical uses of the different algorithms were discussed. Cases involving more than 12 000 selected animals were considered. Inclusion of 10 generations was determined to be sufficient when computing A₂₂.

Conclusions

Depending on the size and structure of the selected sub-population, gains in time to compute A22-1 are possible and these gains may increase as the number of selected animals increases. Given the sequential nature of most computational steps, the proposed algorithm can benefit from optimization and may be convenient for genomic evaluations.

The contribution of dominance and inbreeding depression in estimating variance components for litter size in Pannon White rabbits

In a synthetic closed population of Pannon White rabbits, additive (VA ), dominance (VD ) and permanent environmental (VPe ) variance components as well as doe (bF d ) and litter (bF l ) inbreeding depression were estimated for the number of kits born alive (NBA), number of kits born dead (NBD) and total number of kits born (TNB). The data set consisted of 18,398 kindling records of 3883 does collected from 1992 to 2009. Six models were used to estimate dominance and inbreeding effects. The most complete model estimated VA and VD to contribute 5.5 ± 1.1% and 4.8 ± 2.4%, respectively, to total phenotypic variance (VP ) for NBA; the corresponding values for NBD were 1.9 ± 0.6% and 5.3 ± 2.4%, for TNB, 6.2 ± 1.0% and 8.1 ± 3.2% respectively. These results indicate the presence of considerable VD . Including dominance in the model generally reduced VA and VPe estimates, and had only a very small effect on inbreeding depression estimates. Including inbreeding covariates did not affect estimates of any variance component. A 10% increase in doe inbreeding significantly increased NBD (bF d  = 0.18 ± 0.07), while a 10% increase in litter inbreeding significantly reduced NBA (bF l  = -0.41 ± 0.11) and TNB (bF l  = -0.34 ± 0.10). These findings argue for including dominance effects in models of litter size traits in populations that exhibit significant dominance relationships.

Bayesian adaptive Markov chain Monte Carlo estimation of genetic parameters

Accurate and fast estimation of genetic parameters that underlie quantitative traits using mixed linear models with additive and dominance effects is of great importance in both natural and breeding populations. Here, we propose a new fast adaptive Markov chain Monte Carlo (MCMC) sampling algorithm for the estimation of genetic parameters in the linear mixed model with several random effects. In the learning phase of our algorithm, we use the hybrid Gibbs sampler to learn the covariance structure of the variance components. In the second phase of the algorithm, we use this covariance structure to formulate an effective proposal distribution for a Metropolis-Hastings algorithm, which uses a likelihood function in which the random effects have been integrated out. Compared with the hybrid Gibbs sampler, the new algorithm had better mixing properties and was approximately twice as fast to run. Our new algorithm was able to detect different modes in the posterior distribution. In addition, the posterior mode estimates from the adaptive MCMC method were close to the REML (residual maximum likelihood) estimates. Moreover, our exponential prior for inverse variance components was vague and enabled the estimated mode of the posterior variance to be practically zero, which was in agreement with the support from the likelihood (in the case of no dominance). The method performance is illustrated using simulated data sets with replicates and field data in barley.

Combining the genetic and clonal responses in a closed dairy cattle nucleus scheme

Designs testing clones in a closed nucleus, in which 1024 cows are tested each year, were compared for their additive genetic response to selection (genetic response) and their genetic superiority of female genotype(s) selected for commercial cloning (clonal response), using stochastic simulation. Clones were tested at the expense of dam or sire families, matings per dam (sire), or full-sibs per family. The reference design maximized the genetic response corrected for inbreeding in the absence of cloning. The trait considered was overall economic merit for milk production, which was simulated assuming an approximate infinitesimal model with both additive and dominant gene action. Bulls and cows eligible for breeding were selected on their animal model estimated additive genetic effect at either 15 or 27 months of age. Female genotypes eligible for commercial cloning were selected on their estimated total genetic effect at 27 months of age. All (fe)male full-sibs were available for selection. With only additive gene action, testing clones at the expense of sire families, matings per dam or full-sibs per family reduced genetic response, while it increased clonal response and inbreeding. Testing clones at the expense of dam families, however, added to both the genetic and clonal response without increasing inbreeding. When eight clones were tested at the expense of dam families, the genetic response and the final genetic level of commercially available cloned embryos were maximal. Accuracy of clonal selection equalled 0·83. With dominant gene action, however, testing two clones at the expense of dam families maximized the final genetic level of cloned embryos, irrespective of the level of inbreeding depression (accuracy of 0·72). Reliable commercial clone lines can be produced now and in future generations by testing clones at the expense of dam families.

Accounting for additive genetic mutations on litter size in Ripollesa sheep

Little is known about mutational variability in livestock, among which only a few mutations with relatively large effects have been reported. In this manuscript, mutational variability was analyzed in 1,765 litter size records from 404 Ripollesa ewes to characterize the magnitude of this genetic source of variation and check the suitability of including mutational effects in genetic evaluations of this breed. Threshold animal models accounting for additive genetic mutations were preferred to models without mutational contributions, with an average difference in the deviance information criterion of more than 5 units. Moreover, the statistical relevance of the additive genetic mutation term was checked through a Bayes factor approach, which showed that the models with mutational variability were 8.5 to 22.7 times more probable than the others. The mutational heritability (percentage of the phenotypic variance accounted for by mutational variance) was 0.6 or 0.9%, depending on whether genetic dominance effects were accounted for by the analytical model. The inclusion of mutational effects in the genetic model for evaluating litter size in Ripollesa ewes called for some minor modifications in the genetic merit order of the individuals evaluated, which suggested that the continuous uploading of new additive mutations could be taken into account to optimize the selection scheme. This study is the first attempt to estimate mutational variances in a livestock species and thereby contribute to better characterization of the genetic background of productive traits of interest.

Efficient Markov chain Monte Carlo implementation of Bayesian analysis of additive and dominance genetic variances in noninbred pedigrees

Accurate and fast computation of quantitative genetic variance parameters is of great importance in both natural and breeding populations. For experimental designs with complex relationship structures it can be important to include both additive and dominance variance components in the statistical model. In this study, we introduce a Bayesian Gibbs sampling approach for estimation of additive and dominance genetic variances in the traditional infinitesimal model. The method can handle general pedigrees without inbreeding. To optimize between computational time and good mixing of the Markov chain Monte Carlo (MCMC) chains, we used a hybrid Gibbs sampler that combines a single site and a blocked Gibbs sampler. The speed of the hybrid sampler and the mixing of the single-site sampler were further improved by the use of pretransformed variables. Two traits (height and trunk diameter) from a previously published diallel progeny test of Scots pine (Pinus sylvestris L.) and two large simulated data sets with different levels of dominance variance were analyzed. We also performed Bayesian model comparison on the basis of the posterior predictive loss approach. Results showed that models with both additive and dominance components had the best fit for both height and diameter and for the simulated data with high dominance. For the simulated data with low dominance, we needed an informative prior to avoid the dominance variance component becoming overestimated. The narrow-sense heritability estimates in the Scots pine data were lower compared to the earlier results, which is not surprising because the level of dominance variance was rather high, especially for diameter. In general, the hybrid sampler was considerably faster than the blocked sampler and displayed better mixing properties than the single-site sampler.

Joint modeling of additive and non-additive (genetic line) effects in multi-environment trials

A statistical approach for the analysis of multi-environment trials (METs) is presented, in which selection of best performing lines, best parents, and best combination of parents can be determined. The genetic effect of a line is partitioned into additive, dominance and residual non-additive effects. The dominance effects are estimated through the incorporation of the dominance relationship matrix, which is presented under varying levels of inbreeding. A computationally efficient way of fitting dominance effects is presented which partitions dominance effects into between family dominance and within family dominance line effects. The overall approach is applicable to inbred lines, hybrid lines and other general population structures where pedigree information is available.

Impact of dominance effects on sow longevity

The purpose of the current study was to estimate variance components, especially dominance genetic variation, for overall leg action, length of productive life and sow stayability until third and fifth parity in the Finnish pig populations. The variance components were estimated in two purebred [Landrace (LR), n = 23 602 and Large White (LW), n =22 984] and crossbred (LR x LW, n = 17 440) data sets. Five different analyses were carried out for all the traits to compare the effect of sows' inbreeding, common litter environment and parental dominance in the statistical model when determining the genetic correlations of the traits for the two purebred and crossbred populations. Estimated heritabilities for the traits ranged from 0.04 to 0.06. The estimates for the proportion of dominance variance of phenotypic variance (d(2)) varied between 0.01 and 0.17, and was highest in the crossbred dataset. The genetic correlations of the same traits in purebred and crossbred were all high (>0.75). Based on current results, the effect of dominance should be accounted for in the breeding value estimation of sow longevity, especially when data from crossbred animals are included in the analyses. Because dominance genetic variation for sow longevity exists that variation should be utilized through planned matings in producing sows for commercial production.

Method R estimates of additive genetic, dominance genetic, and permanent environmental fraction of variance for yield and health traits of Holsteins

Fractions of variance accounted for by additive genetic, dominance genetic, and permanent environmental effects for milk, fat, and protein yields; somatic cell score; and productive life were estimated from Holstein data used for national genetic evaluations. Contemporary group assignments were determined using the national procedure. Data included 1,973,317 milk and fat records for 812,659 cows, 1,019,421 protein records for 462,067 cows, 468,374 lactation average somatic cell score (SCS) records for 232,909 cows, and 735,256 cows with productive-life records. Variance components were estimated with the JAADOM program, which uses iteration on data and second-order Jacobi iteration for obtaining solutions to the mixed-model equations and Method R for estimation of variance components. Ten different random data subsets were used to estimate parameters for each trait. Estimated additive genetic, dominance genetic, and permanent environmental fractions of variance were 0.34, 0.05, and 0.10 for milk yield; 0.34, 0.05, and 0.11 for fat yield; 0.31, 0.05, and 0.10 for protein yield; and 0.17, 0.01, and 0.16 for lactation average SCS. Estimated additive genetic and dominance genetic fractions of variance were 0.12 and 0.06 for productive life. Mean empirical standard errors of additive genetic, dominance genetic, and permanent environmental variance fractions were 0.003, 0.006, and 0.006.

Prediction of parental dominance combinations for planned matings, methodology, and simulation results

Optimal use of dominance information requires a mating system and predictions of specific combining abilities for each set of prospective parents. Current evaluation procedures provide such predictions only for a limited number of parents. A procedure is described that predicts the specific combining ability for any parents. In this procedure, for each set of parents and their ancestors, the additive relationship matrix is created as a dense matrix. This matrix is then used to create a parental dominance matrix in a sparse matrix form, in which the rows of the matrix correspond to all parental combinations for which predictions are already available. Each new prediction requires a solution of the system of equations with the parental dominance matrix as the left-hand side. The efficiency of the mating system that accounts for dominance was evaluated in a simulation study. The simulated data files varied with respect to proportion of males and females selected, proportion of cattle born through embryo transfer, and additive and dominance variance. Sires and dams were preselected based on the additive merits only, but specific matings were arranged based on the combined additive plus dominance merit. The response to selection with consideration of dominance increased from 3.8 to 16.6% of the response from one generation of additive selection. The response was greater when the additive variance was smaller, the dominance variance was larger, the intensity of additive selection was lower, and the proportion of full sibs was greater. Use of dominance in the mating system is feasible and results in an additional genetic response to selection.

Supply of genetic information--amount, format, and frequency

The volume and complexity of genetic information is increasing because of new traits and better models. New traits may include reproduction, health, and carcass. More comprehensive models include the test day model in dairy cattle or a growth model in beef cattle. More complex models, which may include nonadditive effects such as inbreeding and dominance, also provide additional information. The amount of information per animal may increase drastically if DNA marker typing becomes routine and quantitative trait loci information is utilized. In many industries, evaluations are run more frequently. They result in faster genetic progress and improved management and marketing opportunities but also in extra costs and information overload. Adopting new technology and making some organizational changes can help realize all the added benefits of the improvements to the genetic evaluation systems at an acceptable cost. Continuous genetic evaluation, in which new records are accepted and breeding values are updated continuously, will relieve time pressures. An online mating system with access to both genetic and marketing information can result in mating recommendations customized for each user. Such a system could utilize inbreeding and dominance information that cannot efficiently be accommodated in the current sire summaries or off-line mating programs. The new systems will require a new organizational approach in which the task of scientists and technicians will not be simply running the evaluations but also providing the research, design, supervision, and maintenance required in the entire system of evaluation, decision making, and distribution.

Detection and utilization of single genes without DNA assays

Quantitative traits are often assumed to be controlled by a large number of loci that each have a small effect. Under this assumption, the distribution of genotypic and phenotypic values can be adequately modeled by a multivariate normal distribution. Thus, most genetic analyses are based on mixed linear models. Evidence is accumulating, however, for the presence of loci that have large effects on traits of economic importance. If the genotypes for such loci can be observed without error, then--conditional on these observed genotypes--genotypic and phenotypic values follow a multivariate normal distribution, and data from very large pedigrees can be analyzed using a mixed linear model that includes the genotypic effects for these loci as fixed effects. However, when the major genotype is not observed, the genotypic and phenotypic values follow a mixture of multivariate normal distributions, and analyses based on fitting a mixed linear model may not be optimum, especially for populations undergoing selection and nonrandom mating. Several approaches are discussed for the genetic analysis of data when the major genotypes are not known.

Effect of full sibs on additive breeding values under the dominance model for stature in United States Holsteins

Differences in breeding values between dominance and additive models were examined theoretically and with field data. Data included 5.2 million records on stature from 3.0 million US Holsteins. The largest full-sib family had 29 animals, and 7% of all animals had at least one full sib. The dominance model, which accounted for dominance covariances, included the following effects: management, age, stage of lactation, permanent environment, animal additive, and parental dominance (one-quarter of dominance variance) as well as a regression coefficient for inbreeding percentage. Two reduced models were also assumed; in the first, the parental dominance effect was removed, and, in the second, the inbreeding regression coefficient was also removed. The correlations between breeding values in the three models were > 0.999, but breeding values of some animals from full-sib families changed > 5 standard deviations of parental dominance. The largest changes were observed for parents with large numbers of full-sib progeny, with limited information from parents, and without individual performance records. On average, the differences were up to four times larger for cows than for bulls and up to five times larger for dams than for sires. The greatest differences in breeding values between the dominance and the additive models were observed for dams with full-sib progeny, female full sibs, and low reliability bulls with full sibs in the extended family. Animals with large amounts of additive information as progeny-tested bulls were influenced little by the inclusion of dominance. Animals with a large proportion of information coming from animals with dominance relationships, such as cows originating via embryo transfer changed the most.

Dominance models with method R for stature of Holsteins

Estimates of variance components were obtained with method R for several additive and dominance models. The data included 301,960 records for first parity and 280,040 records for later parities of Holsteins. The single-record model included effects of management, regression on inbreeding percentage, age at calving, stage of lactation, and additive and dominance effects. The repeatability model included these effects in addition to permanent environment. For the single-record model, estimates were 46% of the total variance for additive variance, 12% of total variance for dominance variance, and -0.06 for the regression coefficient on inbreeding. In the repeatability model, the variance for permanent environment was estimated at 5%; other estimates were similar. When the dominance effect was eliminated, the estimate of the variance for permanent environment increased to 17% for the repeatability model. Elimination of stage of lactation increased regression on inbreeding to 0.09 and the estimate of dominance variance to 17% in the single-record model. The same change increased the estimate of additive variance to 64% for the repeatability model. Elimination of regression on inbreeding or stage of lactation had a small effect on the estimates. The presence or absence of the dominance effect had little influence on additive variance. In the absence of dominance, the estimate of the permanent environment effect included the dominance effect. Estimates of variances with method R are very sensitive to age adjustments. With the adjustments, the estimates of the dominance and additive variances are consistent.

Production traits of Holstein cattle: estimation of nonadditive genetic variance components and inbreeding depression

Additive, dominance, and additive by additive components of genetic variance and inbreeding depression were estimated for production traits from a group of daughters of young sires from the Canadian Holstein population. First lactations of 92,838 cows were analyzed. Three sire and dam models (additive, additive plus dominance, additive plus dominance plus additive by additive genetic effects), all including regression of the trait on inbreeding coefficient of the cow, were used to estimate the effect of inbreeding on production traits. For all production traits, heritability in the narrow sense was overestimated with the simplest model, in which only the additive effect was fitted. Estimates of dominance variance were low for all traits, .9 to 3%. Additive by additive components were low for milk, 2.8%, and fat yield, 2.8%, but higher for protein yield, 6.8%, and for fat, 9%, and protein percentages, 8.9%. Estimates of inbreeding depression for the five traits were similar across all models (-25, -.9, and -.8 kg; .05% and .05% per 1% increase in inbreeding for milk, fat, and protein production and fat and protein percentages, respectively). More accurate estimates of additive effects might be obtained with the inclusion of nonadditive effects for genetic evaluation. If the estimation of inbreeding depression is the only objective, simple models and small random samples of the population may be adequate.

Nonadditive genetic effects and inbreeding depression for somatic cell counts of Holstein cattle

A total of 65,491 lactation means of log2-transformed SCC measures were analyzed from first lactation Holstein cows in Ontario. Effects of inbreeding on SCC were estimated by a nonadditive sire and dam model that included additive, dominance, and additive by additive genetic effects and regression of lactation somatic cell score on inbreeding coefficients of the cows. Variance components were estimated using the tildehat approximation to REML. Solutions were by iteration on data. Estimates of heritability for lactation somatic cell score in the narrow sense were .165 and in the broad sense were .203. The additive by additive component (2.5% of the total phenotypic variance) was almost twice as large as the dominance component (1.3%). The regression coefficient of lactation somatic cell score per 1% increase of inbreeding was .012. The average increase of the population mean of lactation somatic cell score caused by a 10% increase of inbreeding coefficient was estimated to be 10.5% of the original phenotypic standard deviation of 1.153. The inbreeding depression was thus relatively low, but, on average, inbred animals tended to have higher lactation somatic cell score. This study provides preliminary evidence that inbreeding is related to disease prevalence in large purebred dairy populations.

Theory for modelling means and covariances in a two-breed population with dominance inheritance

This paper presents theory and methods to compute genotypic means and covariances in a two breed population under dominance inheritance, assuming multiple unlinked loci. It is shown that the genotypic mean is a linear function of five location parameters and that the genotypic covariance between relatives is a linear function of 25 dispersion parameters. Recursive procedures are given to compute the necessary identity coefficients. In the absence of inbreeding, the number of parameters for the mean is reduced from five to three and the number for the covariance is reduced from 25 to 12. In a two-breed population, for traits exhibiting dominance, the theory presented here can be used to obtain genetic evaluations by best linear unbiased prediction and to estimate genetic parameters by maximum likelihood.

Additive and nonadditive genetic variances for milk yield, fertility, and lifetime performance traits of dairy cattle

Additive and nonadditive genetic variances were estimated for yield traits and fertility for three subsequent lactations and for lifetime performance traits of purebred and crossbred dairy cattle populations. Traits were milk yield, energy-corrected milk yield, fat percentage, protein percentage, calving interval, length of productive life, and lifetime FCM of purebred Simmental, Simmental including crossbreds, and Braunvieh crossed with Brown Swiss. Data files ranged from 66,740 to 375,093 records. An approach based on pedigree information for sire and maternal grandsire was used and included additive, dominance, and additive by additive genetic effects. Variances were estimated using the tildehat approximation to REML. Heritability estimated without nonadditive effects in the model was overestimated, particularly in presence of additive by additive variance. Dominance variance was important for most traits; for the lifetime performance traits, dominance was clearly higher than additive variance. Additive by additive variance was very high for milk yield and energy-corrected milk yield, especially for data including crossbreds. Effect of inbreeding was low in most cases. Inclusion of nonadditive effects in genetic evaluation models might improve estimation of additive effects and may require consideration for dairy cattle breeding programs.

Use of reproductive technology to estimate variances and predict effects of gene interactions

Advanced reproductive techniques are creating the large numbers of close relatives needed to study gene interactions. Identical triplets, a set of 26 full sisters, a family of 4215 three-quarter sisters (same sire and maternal grandsire), a family of 76,698 half sisters, and 1.6 million granddaughters of Round Oak Rag Apple Elevation now have lactation records. Similarity of closest relatives might be explained by similar nonadditive as well as additive genetic merit. The 23,015 families of full sisters with mean family size of 3 provide nearly as much information about dominance variation as do the 55,779 families of three-quarter sisters with mean family size of 13; the 79 families of clones provide little information by comparison. Hypothetically, REML analysis of all US Holstein data could provide estimates of dominance and additive x additive variance with standard errors approximately 1% of phenotypic variance, but estimates of any higher order interactions would have standard errors greater than 10%. The tilde-hat approximation proved to be incompatible with animal models but was used for sire-maternal grandsire analysis of 765,868 first lactation records. Dominance variance was estimated as 3.5% of phenotypic variance for milk and 3.3% for fat with standard error of 4.2%. With constant data set size, variances are estimated most precisely if family sizes equal 1 plus ratio of within-family to between-family variance. An animal model evaluation including dominance relationships for 581,670 animals was computed, but gene interactions from distant ancestor pairs were ignored. Mating advice and improved additive predictions, especially for clones, could be obtained by including dominance in models.

Additive and nonadditive genetic variance in female fertility of Holsteins

Additive and nonadditive genetic variances were estimated for cow fertility of Holsteins. Measures of fertility were first lactation days open and service period as recorded and with upper bounds of 150 and 91 d, respectively. Six million inseminations from the Raleigh, North Carolina Processing Center were used to form fertility records of 379,009 cows. Data were analyzed with a model accounting for all additive, dominance, and additive by additive covariances traced through sires and maternal grandsires. Variance components were estimated by the tilde-hat approximation to REML. Heritability in the narrow sense was 2% for days open and .8% for service period. Dominance and additive by additive variance as a percentage of phenotypic variation strongly depended on imposition of upper bounds. Heritabilities in the broad sense ranged from 2.2 to 6.6% and were at least twice as large as heritabilities in the narrow sense. Effect of 25% inbreeding was only around an additional 3 d open. Specific combining abilities among bulls were estimated as sums of dominance and additive by additive interactions removing effect of inbreeding depression. Differences between maximum and minimum estimates were in the order of twice the estimated standard deviation, ranging from 1.5 to 6.7 d. Effects of inbreeding and specific combining ability could be jointly considered in mating programs following sire selection.

Rapid inversion of additive by additive relationship matrices by including sire-dam combination effects

Inverses of relationship matrices are useful for prediction of individual additive or nonadditive genetic merits and for estimation of variance components. An algorithm to form inverses of additive by additive relationship matrices rapidly from lists of individuals and their parents was developed. The algorithm uses simple recurrences among additive by additive and sire-dam combination effects to construct inverses for noninbred or inbred populations. Dimensions of matrices produced may be several times the number of individuals in the population because combination effects for sire-dam subclasses must be included in matrices. Rules of inheritance of sire-dam combination effects are the same as for dominance combination effects. Cost of forming inverses increases linearly with number of individuals. Each individual contributes 36 or fewer nonzero coefficients, and each sire-dam subclass contributes an additional 81 or fewer nonzero coefficients to the matrix. Computation of inverse of the relationship matrix due to 1003 sires and maternal grandsires of 765,868 cows required forming a matrix of order 137,830 and 4 Mbytes of memory.