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Additional file 1: Table S1. Phenotypic distribution of AWW per country for males and females. Table S2. List of environmental effects in each national model. Table S3. Direct and maternal genetic covariances (below diagonal), genetic variances (diagonal) and genetic correlations (above diagonal) within and across countries. Table S4. National genetic, environmental, and residual variances. Table S5. Number of phenotypes and genotypes per country used in the implemented scenarios. Table S6. Distribution of adjusted de-regressed effective record contribution (dERC*) for direct and maternal EBV of publishable sires, from either pedigree-based or single-step international evaluations. Table S7. Comparison of direct and maternal EBV for all animals in the Italian pseudo-national pedigree between NATAPR and BLENDAPR integrating EBV from pedigree-based international evaluations. Table S8. Comparison of direct and maternal EBV for all animals in the Italian pseudo-national pedigree between NATAPR and BLENDAPR integrating EBV from single-step international evaluations. Table S9. Validation of the scenarios’ adequacy for direct and maternal EBV of genotyped and non-genotyped foreign publishable sires when EBVINT are computed using single-step international evaluations. Table S10. Validation of the scenarios’ predictivity for direct EBV of offspring of publishable sires and for maternal EBV of daughters of MGS with publishable EBV when international information are integrated on Scenario NATJAN and EBVINT are computed using pedigree-based international evaluations. Table S11. Validation of the scenarios’ predictivity for direct EBV of offspring of publishable sires and for maternal EBV of daughters of MGS with publishable EBV when international information are integrated on Scenario NATJAN and EBVINT are computed using single-step international evaluations. Table S12. Validation of the GOLD scenario’s adequacy for direct and maternal EBV of publishable sires when EBVINT are computed using pedigree-based international evaluations. Table S13. Validation of the GOLD scenario’s predictivity for direct EBV of offspring of publishable sires and for maternal EBV of daughters of MGS with publishable EBV when EBVINT are computed using pedigree-based international evaluations. Table S14. Validation of the GOLD scenario’s adequacy for direct and maternal EBV of publishable sires when EBVINT are computed using single-step international evaluations. Table S15. Validation of the GOLD scenario’s predictivity for direct EBV of offspring of publishable sires and for maternal EBV of daughters of MGS with publishable EBV when EBVINT are computed using single-step international evaluations.

Additional file 1: Table S1. Pearson correlations, regression coefficients, and level bias for direct and maternal GEBV computed from the full ssSNPBLUP versus from the indirect prediction approaches for 251,332 genotyped selection candidates with both parents genotyped. Table S2. Pearson correlations, regression coefficients, and level bias for direct and maternal GEBV computed from the full ssSNPBLUP versus from the indirect prediction approaches for 155,675 genotyped selection candidates with only one genotyped parent. Table S3. Pearson correlations, regression coefficients, and level bias for direct and maternal GEBV computed from the full ssSNPBLUP versus from the indirect prediction approaches for 50,164 genotyped selection candidates with no genotyped parents. Table S4. Pearson correlations, regression coefficients, and level bias for direct and maternal GEBV computed from the full ssGTABLUP versus from the indirect prediction approaches for 251,332 genotyped selection candidates with both parents genotyped. Table S5. Pearson correlations, regression coefficients, and level bias for direct and maternal GEBV computed from the full ssGTABLUP versus from the indirect prediction approaches for 155,675 genotyped selection candidates with only one genotyped parent. Table S6. Pearson correlations, regression coefficients, and level bias for direct and maternal GEBV computed from the full ssGTABLUP versus from the indirect prediction approaches for 50,164 genotyped selection candidates with no genotyped parents.

Additional file 2. R functions to compute ERC, DRP, dERC* and DRP* (also available at https://github.com/bonifazi/Integration_EBV_and_GEBV ).

Additional file 5: Figure S1. Scatterplot of statistical additive effects in different generations for the genetic model with additive and dominance effects (Model AD) under RANDOM selection. Figure S2. Scatterplot of statistical additive effects in different generations for the genetic model with additive and dominance effects (Model AD) under MASS selection. Figure S3. Scatterplot of statistical additive effects in different generations for the genetic model with additive and dominance effects (Model AD) under PBLUP selection with own performance (PBLUP_OP). Figure S4. Scatterplot of statistical additive effects in different generations for the genetic model with additive and dominance effects (Model AD) under GBLUP selection without own performance (GBLUP_NoOP). Figure S5. Scatterplot of statistical additive effects in different generations for the genetic model with additive and dominance effects (Model AD) under GBLUP selection with own performance (GBLUP_OP). Figure S6. Scatterplot of statistical additive effects in different generations for the genetic model with additive, dominance and epistatic effects (Model ADE) under RANDOM selection. Figure S7. Scatterplot of statistical additive effects in different generations for the genetic model with additive, dominance and epistatic effects (Model ADE) under MASS selection. Figure S8. Scatterplot of statistical additive effects in different generations for the genetic model with additive, dominance and epistatic effects (Model ADE) under PBLUP selection with own performance (PBLUP_OP). Figure S9. Scatterplot of statistical additive effects in different generations for the genetic model with additive, dominance and epistatic effects (Model ADE) under GBLUP selection without own performance (GBLUP_NoOP). Figure S10. Scatterplot of statistical additive effects in different generations for the genetic model with additive, dominance and epistatic effects (Model ADE) under GBLUP selection with own performance (GBLUP_OP).

Additional file 4: Figure S1. Scatterplot of allele frequencies of all causal variants in different generations for the genetic model with additive effects (Model A) under RANDOM selection. Figure S2. Scatterplot of allele frequencies of all causal variants in different generations for the genetic model with additive effects (Model A) under MASS selection. Figure S3. Scatterplot of allele frequencies of all causal variants in different generations for the genetic model with additive effects (Model A) under PBLUP selection with own performance (PBLUP_OP). Figure S4. Scatterplot of allele frequencies of all causal variants in different generations for the genetic model with additive effects (Model A) under GBLUP selection without own performance (GBLUP_NoOP). Figure S5. Scatterplot of allele frequencies of all causal variants in different generations for the genetic model with additive effects (Model A) under GBLUP selection with own performance (GBLUP_OP). Figure S6. Scatterplot of allele frequencies of all causal variants in different generations for the genetic model with additive and dominance effects (Model AD) under RANDOM selection. Figure S7. Scatterplot of allele frequencies of all causal variants in different generations for the genetic model with additive and dominance effects (Model AD) under MASS selection. Figure S8. Scatterplot of allele frequencies of all causal variants in different generations for the genetic model with additive and dominance effects (Model AD) under PBLUP selection with own performance (PBLUP_OP). Figure S9. Scatterplot of allele frequencies of all causal variants in different generations for the genetic model with additive and dominance effects (Model AD) under GBLUP selection without own performance (GBLUP_NoOP). Figure S10. Scatterplot of allele frequencies of all causal variants in different generations for the genetic model with additive and dominance effects (Model AD) under GBLUP selection with own performance (GBLUP_OP). Figure S11. Scatterplot of allele frequencies of all causal variants in different generations for the genetic model with additive, dominance and epistatic effects (Model ADE) under RANDOM selection. Figure S12. Scatterplot of allele frequencies of all causal variants in different generations for the genetic model with additive, dominance and epistatic effects (Model ADE) under MASS selection. Figure S13. Scatterplot of allele frequencies of all causal variants in different generations for the genetic model with additive, dominance and epistatic effects (Model ADE) under PBLUP selection with own performance (PBLUP_OP). Figure S14. Scatterplot of allele frequencies of all causal variants in different generations for the genetic model with additive, dominance and epistatic effects (Model ADE) under GBLUP selection without own performance (GBLUP_NoOP). Figure S15. Scatterplot of allele frequencies of all causal variants in different generations for the genetic model with additive, dominance and epistatic effects (Model ADE) under GBLUP selection with own performance (GBLUP_OP).

Additional file 2: Table S2. Estimated additive genetic and residual variances (standard errors in bracket) in the sire line.

Additional file 2: Figure S1. Allele frequency distribution of segregating causal loci over 50 generations without selection. Figure S2. Extent of LD (r2) in the simulated population before selection as a function of distance in one random replicate. Figure S3. Trend in the variation in minor allele frequency (MAF) of segregating causal loci for the five selection methods and three genetic models. The five selection methods were: RANDOM selection, MASS selection, PBLUP selection with own performance (PBLUP_OP), GBLUP selection without own performance (GBLUP_NoOP) or with own performance (GBLUP_OP). The three genetic models were a model with only additive effects (A), with additive and dominance effects (AD), or with additive, dominance and epistatic effects (ADE). Results are shown as averages of 20 replicates and the width of the lines represents the average plus and minus one standard error. Figure S4. The difference between additive genic and additive genetic variance which represents a transient loss in genetic variance for the five selection methods and three genetic models. The five selection methods were: RANDOM selection, MASS selection, PBLUP selection with own performance (PBLUP_OP), GBLUP selection without own performance (GBLUP_NoOP) or with own performance (GBLUP_OP). The three genetic models were a model with only additive effects (A), with additive and dominance effects (AD), or with additive, dominance and epistatic effects (ADE). Results are shown as averages of 20 replicates and the width of the lines represents the average plus and minus one standard error. Figure S5. Phenotypic trend for the GBLUP model with own performance records and with and without a dominance effect for the genetic models with non-additive effects. The phenotypic trend is scaled by the additive genetic standard deviation in the generation before selection in order to make the results comparable across the genetic models. The two genetic models were a model with additive and dominance effects (AD), or with additive, dominance and epistatic effects (ADE). Results are shown as averages of 20 replicates and the width of the lines represents the average plus and minus one standard error. Figure S6. Trend in additive genetic (A, B) and additive genic (C, D) variance for the GBLUP model with own performance and with and without a dominance effect for the genetic models with non-additive effects. The trend is scaled by the additive genetic or additive genic variance in the generation before selection in order to make the results comparable across the genetic models. The two genetic models were a model with additive and dominance effects (AD), or with additive, dominance and epistatic effects (ADE). Results are shown as averages of 20 replicates and the width of the lines represents the average plus and minus one standard error.

Additional file 1: Table S1. Number of phenotypes (N), minimum, mean, maximum, and phenotypic standard deviation ( $${\sigma }_{P}$$ σ P ) of males and females per country. Table S2. List of environmental effects in the national model of each country. Table S3. Direct and maternal genetic covariances (below diagonal), genetic variances (diagonal) and genetic correlations (above diagonal) within and across countries. Table S4. National genetic, environmental and residual (co)variances. Table S5. Level bias ( $${\widehat{\Delta }}_{p}$$ Δ ^ p ), dispersion bias ( $${\widehat{b}}_{p}$$ b ^ p ) and accuracy of partial EBV ( $${\widehat{acc}}_{p}$$ acc ^ p ) of direct and maternal EBV for animals with phenotypes and no genotypes born from 2014 onwards, in each scenario and for each country. Table S6. Increases in population accuracy ( $$inc\_acc$$ i n c _ a c c ) of moving from each scenario to ssSNPBLUPINT for direct and maternal EBV for animals with phenotypes and no genotypes born from 2014 onwards for each country. Table S7. Countries sending the genotypes for Common Bulls (CB) and Common Maternal Grand-Sires (CMGS). Table S8. Level bias ( $${\widehat{\Delta }}_{p}$$ Δ ^ p ), dispersion bias ( $${\widehat{b}}_{p}$$ b ^ p ) and accuracy of partial EBV ( $${\widehat{acc}}_{p}$$ acc ^ p ) of direct and maternal EBV for the focal group, in each scenario and for each country. Standard errors reported within parenthesis.

Additional file 2: Figure S1. Observed line B proportions in the simulated data (histogram), versus the theoretically expected distribution (red line).

Additional file 3: Table S3. Estimated additive genetic and residual variances (standard errors in bracket) in the dam line.

Additional file 4: Figure S3. Direct and maternal EBV realized Mendelian sampling (RMS) trends in each countrya expressed in genetic standard deviation (GSD) for genotyped and non-genotyped animalsb computed with pedigree-based BLUP international (PBLUPINT) and single-step SNP-BLUP international (ssSNPBLUPINT) models. aCZE Czech Republic, DFS Denmark, Finland and Sweden, IRL Ireland, DEU Germany, CHE Switzerland. bGenotyped animals: animals with genotype that appear in the pseudo-national pedigree (with or without phenotype in the country). Non-genotyped animals: animals that appear in the pseudo-national pedigree without genotype and with phenotype in the country.

Additional file 2: File S1. Findhap instruction file. File S2. International pedigree-based BLUP MiXBLUP instruction file. File S3. International single-step SNPBLUP MiXBLUP instruction file.

Additional file 3. Decomposition of additive genetic variance. This file provides a theoretical decomposition of the additive genetic variance.

Additional file 6: Table S1. Maximum genetic gain1 that is still possible after 50 generations of selection for the five selection methods and three genetic models2. The five selection methods were: RANDOM selection, MASS selection, PBLUP selection with own performance (PBLUP_OP), GBLUP selection without own performance (GBLUP_NoOP) or with own performance (GBLUP_OP). The three genetic models were a model with only additive effects (A), with additive and dominance effects (AD), or with additive, dominance and epistatic effects (ADE). 1The maximum genetic gain in generation 50 is estimated as the genetic gain when all loci would be fixed for the favourable allele, using the statistical additive effects of generation 50 and neglecting mutations. 2Results are shown as averages across the 20 replicates with their corresponding standard errors of the mean between brackets. Table S2. Percentual change in the components of the genetic variance after 10 and 50 generations of selection for the five selection methods and three genetic models1. The five selection methods were: RANDOM selection, MASS selection, PBLUP selection with own performance (PBLUP_OP), GBLUP selection without own performance (GBLUP_NoOP) or with own performance (GBLUP_OP). The three genetic models were a model with only additive effects (A), with additive and dominance effects (AD), or with additive, dominance and epistatic effects (ADE). 1Results are shown as averages of 20 replicates with their corresponding standard errors of the mean between brackets. Increases in the value of a component are represented in bold. Table S3. Average pedigree inbreeding coefficient after 50 generations of selection for the five selection methods and three genetic models1. The five selection methods were: RANDOM selection, MASS selection, PBLUP selection with own performance (PBLUP_OP), GBLUP selection without own performance (GBLUP_NoOP) or with own performance (GBLUP_OP). The three genetic models were a model with only additive effects (A), with additive and dominance effects (AD), or with additive, dominance and epistatic effects (ADE). 1Results are shown as averages of 20 replicates with their corresponding standard errors of the mean between brackets. Table S4. Average and variance of change in allele frequency of causal loci1 across 50 generations of selection for the five selection methods and three genetic models2. The five selection methods were: RANDOM selection, MASS selection, PBLUP selection with own performance (PBLUP_OP), GBLUP selection without own performance (GBLUP_NoOP) or with own performance (GBLUP_OP). The three genetic models were a model with only additive effects (A), with additive and dominance effects (AD), or with additive, dominance and epistatic effects (ADE). 1Causal loci included only the causal loci segregating in generation 0. 2Results are shown as averages of 20 replicates with their corresponding standard errors of the mean between brackets.

Additional file 3: Figure S1. Plot of pedigree-based (x-axis) and genomic-based (y-axis) relationships between genotyped animals. The red dots indicate the relationships of the 41 genotypes removed due to pedigree incompatibilities. Figure S2. Differences in genetic trends between ssSNPBLUPINT and PBLUPINT per country for sires with at least 10 recorded offspring in the country (only birth years with at least 5 sires are included).

Additional file 1: Table S1. Estimated heritabilities (diagonal), genetic correlations (below the diagonal) and phenotypic correlations (above the diagonal) for the traits analysed in this study, using the full data from the sire line. Standard errors are in brackets.

Additional file 1. Derivation of the termination criterion CR.

Additional file 1: Table S1. Minimum, mean, and maximum values of realized r_pc for crosses between lines that were divergently selected for 10 (P10-N10), 25 (P25-N25), or 50 generations of selection (P50-N50).

Additional file 2. Derivation of the termination criterion CM.

Additional file 3. Relationship between the residual of the system of equations of ssGBLUPand ssSNPBLUP.

Additional file 1. Simulation of the TGV coming from the sire for four-way crossbreds. Explanation of how the TGV coming from the sire was simulated for the four-way crossbreds.