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ABSTRACT: The single-step genomic model has become the golden standard for routine evaluation in livestock species, such as Holstein dairy cattle. The single-step genomic model with direct estimation of marker effects has been proven to be efficient in accurately accounting for millions of genotype records. For diverse applications including frequent genomic evaluation updates on a weekly basis, estimates of the marker effects from the single-step evaluations play a central role in genomic prediction. In this study we focused on exploring the marker effect estimates from the single-step evaluation. Phenotypic, genotypic, and pedigree data were taken from the official evaluation for German dairy breeds in April 2021. A multilactation random regression test-day model was applied to more than 242 million test-day records separately for 4 traits: milk, fat, and protein yields, and somatic cell scores (SCS). Approximately one million genotyped Holstein animals were considered in the single-step genomic evaluations including ∼21 million animals in pedigree. Deregressed multiple across-country breeding values of Holstein bulls having daughters outside Germany were integrated into the national test-day data to increase the reliability of genomic breeding values. To assess the stability and bias of the marker effects of the single-step model, test-day records of the last 4 yr were deleted, and the integrated bulls born in the last 4 yr were truncated from the complete phenotypic dataset. Estimates of the marker effects were shown to be highly correlated, with correlations ∼0.9, between the full and truncated evaluations. Regression slope values of the marker-effect estimates from the full on the truncated evaluations were all close to their expected value, being ∼1.03. Calculated using random regression coefficients of the marker effect estimates, drastically different shapes of the genetic lactation curve were seen for 2 markers on chromosome 14 for the 4 test-day traits. The contribution of individual chromosomes to the total additive genetic variances seemed to follow the polygenic inheritance mode for protein yield and SCS. However, chromosome 14 was found to make an exceptionally large contribution to the total additive genetic variance for milk and fat yields because of markers near the major gene DGAT1. For the first lactation test-day traits, we obtained ∼0 correlations of chromosomal direct genomic values between any pair of the chromosomes; no spurious correlations were found in our analysis, thanks to the large reference population. For trait milk yield, chromosomal direct genomic values appeared to have a large variation in the between-lactation correlations among the chromosomes, especially between first and second or third lactations. The optimal features of the random regression test-day model and the single-step marker model allowed us to track the differences in the shapes of genetic lactation curves down to the individual markers. Furthermore, the single-step random regression test-day model enabled us to better understand the inheritance mode of the yield traits and SCS (e.g., variable chromosomal contributions to the total additive genetic variance and to the genetic correlations between lactations).

The implementation of genomic selection for six German beef cattle populations was evaluated. Although the multiple‐step implementation of genomic selection is the status quo in most national dairy cattle evaluations, the breeding structure of German beef cattle, coupled with the shortcoming and complexity of the multiple‐step method, makes single step a more attractive option to implement genomic selection in German beef cattle populations. Our objective was to develop a national beef cattle single‐step genomic evaluation in five economically important traits in six German beef cattle populations and investigate its impact on the accuracy and bias of genomic evaluations relative to the current pedigree‐based evaluation. Across the six breeds in our study, 461,929 phenotyped and 14,321 genotyped animals were evaluated with a multi‐trait single‐step model. To validate the single‐step model, phenotype data in the last 2 years were removed in a forward validation study. For the conventional and single‐step approaches, the genomic estimated breeding values of validation animals and other animals were compared between the truncated and the full evaluations. The correlation of the GEBVs between the full and truncated evaluations in the validation animals was slightly higher in the single‐step evaluation. The regression of the full GEBVs on truncated GEBVs was close to the optimal value of 1 for both the pedigree‐based and the single‐step evaluations. The SNP effect estimates from the truncated evaluation were highly correlated with those from the full evaluation, with values ranging from 0.79 to 0.94. The correlation of the SNP effect was influenced by the number of genotyped animals shared between the full and truncated evaluations. The regression coefficients of the SNP effect of the full evaluation on the truncated evaluation were all close to the expected value of 1, indicating unbiased estimates of the SNP markers for the production traits. The Manhattan plot of the SNP effect estimates identified chromosomal regions harbouring major genes for muscling and body weight in breeds of French origin. Based on the regression intercept and slope of the GEBVs of validation animals, the single‐step evaluation was neither inflated nor deflated across the six breeds. Overall, the single‐step model resulted in a more accurate and stable evaluation. However, due to the small number of genotyped individuals, the single‐step method only provided slightly better results when compared to the pedigree‐based method. [ABSTRACT FROM AUTHOR]

In the last decade, a number of methods have been suggested to deal with large amounts of genetic data in genomic predictions. Yet, steadily growing population sizes and the suboptimal use of computational resources are pushing the practical application of these approaches to their limits. As an extension to the C/CUDA library miraculix, we have developed tailored solutions for the computation of genotype matrix multiplications which is a critical bottleneck in the empirical evaluation of many statistical models. We demonstrate the benefits of our solutions at the example of single-step models which make repeated use of this kind of multiplication. Targeting modern Nvidia® GPUs as well as a broad range of CPU architectures, our implementation significantly reduces the time required for the estimation of breeding values in large population sizes. miraculix is released under the Apache 2.0 license and is freely available at https://github.com/alexfreudenberg/miraculix. [ABSTRACT FROM AUTHOR]

In the last decade, a number of methods have been suggested to deal with large amounts of genetic data in genomic predictions. Yet, steadily growing population sizes and the suboptimal use of computational resources are pushing the practical application of these approaches to their limits. As an extension to the C/CUDA library miraculix, we have developed tailored solutions for the computation of genotype matrix multiplications which is a critical bottleneck in the empirical evaluation of many statistical models. We demonstrate the benefits of our solutions at the example of single-step models which make repeated use of this kind of multiplication. Targeting modern Nvidia® GPUs as well as a broad range of CPU architectures, our implementation significantly reduces the time required for the estimation of breeding values in large population sizes. miraculix is released under the Apache 2.0 license and is freely available at https://github.com/alexfreudenberg/miraculix.

ABSTRACT: Genomic evaluation based on a single-step model uses all available data of phenotype, genotype, and pedigree; therefore, it should provide unbiased genomic breeding values with a higher correlation of prediction than the current multistep genomic model. Since 2019, a mixed reference population of cows and bulls has been applied to the routine multistep genomic evaluation in German Holsteins. For a fair comparison between the single-step and multistep genomic models, the same phenotype, genotype, and pedigree data were used. Because of its simple structure of the standard multitrait animal model used for German Holstein conventional evaluation, conformation traits were chosen as the first trait group to test a single-step SNP BLUP model for the large, genotyped population of German Holsteins. Genotype, phenotype, and pedigree data were taken from the official August 2020 conventional and genomic evaluation. Because of the same trait definition in national and multiple across-country evaluation for the conformation traits, deregressed multiple across-country evaluation estimated breeding value (EBV) of foreign bulls were treated as a new source of data for the same trait in the genomic evaluations. Due to a short history of female genotyping in Germany, the last 3 yr of youngest cows and bulls were deleted, instead of 4 yr, to perform a genomic validation. In comparison to the multistep genomic model, the single-step SNP BLUP model resulted in a higher correlation and greater variance of genomic EBV according to 798 national validation bulls. The regression of genomic prediction of the current, full evaluation on the earlier, truncated evaluation was slightly closer to 1 than the multistep model. For the validation bulls or youngest genomic artificial insemination bulls, correlation of genomic EBV between the 2 models was, on average, 0.95 across all the conformation traits. We did not find overprediction of young animals by the single-step SNP BLUP model for the conformation traits in German Holsteins.

Hanwoo beef cattle are well known for the flavor and tenderness of their meat. Genetic improvement programs have been extremely successful over the last 40 yr. Recently, genomic selection was initiated in Hanwoo to enhance genetic progress. Routine genomic evaluation based on the single-step breeding value model was implemented in 2020 for all economically important traits. In this study, we tested a single-step marker effect model for the genomic evaluation of four carcass traits, namely, carcass weight (CW), eye muscle area, backfat thickness, and marbling score. In total, 8,023,666 animals with carcass records were jointly evaluated, including 29,965 genotyped animals. To assess the prediction stability of the single-step model, carcass data from the last 4 yr were removed in a forward validation study. The estimated genomic breeding values (GEBV) of the validation animals and other animals were compared between the truncated and full evaluations. A parallel conventional best linear unbiased prediction (BLUP) evaluation with either the full or the truncated dataset was also conducted for comparison with the single-step model. The estimates of the marker effect from the truncated evaluation were highly correlated with those from the full evaluation, ranging from 0.88 to 0.92. The regression coefficients of the estimates of the marker effect for the full and truncated evaluations were close to their expected value of 1, indicating unbiased estimates for all carcass traits. Estimates of the marker effect revealed three chromosomal regions (chromosomes 4, 6, and 14) harboring the major genes for CW in Hanwoo. For validation of cows or steers, the single-step model had a much higher R2 value for the linear regression model than the conventional BLUP model. Based on the regression intercept and slope of the validation, the single-step evaluation was neither inflated nor deflated. For genotyped animals, the estimated GEBV from the full and truncated evaluations were more correlated than the estimated breeding values from the two conventional BLUP evaluations. The single-step model provided a more accurate and stable evaluation over time. [ABSTRACT FROM AUTHOR]

The success and sustainability of a breeding program incorporating genomic information is largely dependent on the accuracy of predictions. For low heritability traits, large training populations are required to achieve high accuracies of genomic estimated breeding values (GEBV). By including genotyped and nongenotyped animals simultaneously in the evaluation, the singlestep genomic BLUP (ssGBLUP) approach has the potential to deliver more accurate and less biased genomic evaluations. The aim of this study was to compare the accuracy and bias of genomic predictions for various traits in Canadian Holstein cattle using ssGBLUP and multi-step genomic BLUP (msGBLUP) under different strategies, such as (1) adding genomic information of cows in the analysis, (2) testing different adjustments of the genomic relationship matrix, and (3) using a blending approach to obtain GEBV from msGBLUP. The following genomic predictions were evaluated regarding accuracy and bias: (1) GEBV estimated by ssGBLUP; (2) direct genomic value estimated by msGBLUP with polygenic effects of 5 and 20%; and (3) GEBV calculated by a blending approach of direct genomic value with estimated breeding values using polygenic effects of 5 and 20%. The effect of adding genomic information of cows in the evaluation was also assessed for each approach. When genomic information was included in the analyses, the average improvement in observed reliability of predictions was observed to be 7 and 13 percentage points for reproductive and workability traits, respectively, compared with traditional BLUP. Absolute deviation from 1 of the regression coefficient of the linear regression of de-regressed estimated breeding values on genomic predictions went from 0.19 when using traditional BLUP to 0.22 when using the msGBLUP method, and to 0.14 when using the ssGBLUP method. The use of polygenic weight of 20% in the msGBLUP slightly improved the reliability of predictions, while reducing the bias. A similar trend was observed when a blending approach was used. Adding genomic information of cows increased reliabilities, while decreasing bias of genomic predictions when using the ssGBLUP method. Differences between using a training population with cows and bulls or with only bulls for the msGBLUP method were small, likely due to the small number of cows included in the analysis. Predictions for lowly heritable traits benefit greatly from genomic information, especially when all phenotypes, pedigrees, and genotypes are used in a single-step approach. [ABSTRACT FROM AUTHOR]

In the future an approach incorporating cows' measured phenotypes and marker genotypes of cows and bulls within a single model can be applied. The most important advantage of such a model is the simultaneous use of pedigree and marker-based genomic relationship data. Such a solution allows the use of both genotyped and non-genotyped animals in the prediction procedure. This pilot study is aimed towards implementation of a one-step approach in a random regression test day model context for the Polish Holstein Friesian population, considering various ways of adjusting the relationship matrix. Data consisted of 890 animals (10 genotyped bulls, 100 cows with phenotypic data and 780 ancestors without genotypes or phenotypes). Random regression test day models with a polygenic effect on milk yield modeled by second order Legendre polynomials for the estimation of variance-covariance parameters and were used for prediction of genomically enhanced breeding values (GEBV). In this model, a matrix combining pedigree and marker-based information was used instead of a traditional numerator relationship matrix. In this matrix the proportions of information coming from pedigree and markers were defined by weighting parameters w and 1-w for pedigree and marker-based information matrices, respectively. Various weights of the two sources of information were considered. The accuracy of GEBV both for genotyped bulls and for cows with phenotypes was highest for weighting parameter w=0 and lowest for w=1. Incorporating genomic information into a conventional genetic evaluation improves reliabilities of breeding value prediction, however, pedigree information is important to maintain the stability of evaluation for non-genotyped animals. Implementation of the single-step approach in a random regression test day model framework is very attractive for genomic prediction in dairy cattle, since it allows to incorporate genomic information directly into a conventional genetic evaluation. However, for accurate predictions it is essential to achieve the right balance between the numerator relationship and markers-based relationship information. [ABSTRACT FROM AUTHOR]

We study (2+1)-dimensional single step model for crystal growth including both deposition and evaporation processes parametrized by a single control parameter p . Using extensive numerical simulations with a relatively high statistics, we estimate various interface exponents such as roughness, growth and dynamic exponents as well as various geometric and distribution exponents of height clusters and their boundaries (or iso-height lines) as function of p . We find that, in contrary to the general belief, there exists a critical value ${p}_{c}\approx 0.25$ at which the model undergoes a roughening transition from a rough phase with $p\lt {p}_{c}$ in the Kardar–Parisi–Zhang universality to a smooth phase with $p\gt {p}_{c}$ , asymptotically in the Edwards–Wilkinson class. We validate our conclusion by estimating the effective roughness exponents and their extrapolation to the infinite-size limit.