Simulation-based establishment of base pools for a hybrid breeding program in winter rapeseed.

KEY MESSAGE: Simulation planned pre-breeding can increase the efficiency of starting a hybrid breeding program. Starting a hybrid breeding program commonly comprises a grouping of the initial germplasm in two pools and subsequent selection on general combining ability. Investigations on pre-breeding steps before starting the selection on general combining ability are not available. Our goals were (1) to use computer simulations on the basis of DNA markers and testcross data to plan crosses that separate genetically two initial germplasm pools of rapeseed, (2) to carry out the planned crosses, and (3) to verify experimentally the pool separation as well as the increase in testcross performance. We designed a crossing program consisting of four cycles of recombination. In each cycle, the experimentally generated material was used to plan the subsequent crossing cycle with computer simulations. After finishing the crossing program, the initially overlapping pools were clearly separated in principal coordinate plots. Doubled haploid lines derived from the material of crossing cycles 1 and 2 showed an increase in relative testcross performance for yield of about 5% per cycle. We conclude that simulation-designed pre-breeding crossing schemes, that were carried out before the general combining ability-based selection of a newly started hybrid breeding program, can save time and resources, and in addition conserve more of the initial genetic variation than a direct start of a hybrid breeding program with general combining ability-based selection.

Accurate prediction of quantitative traits with failed SNP calls in canola and maize.

In modern plant breeding, genomic selection is becoming the gold standard to select superior genotypes in large breeding populations that are only partially phenotyped. Many breeding programs commonly rely on single-nucleotide polymorphism (SNP) markers to capture genome-wide data for selection candidates. For this purpose, SNP arrays with moderate to high marker density represent a robust and cost-effective tool to generate reproducible, easy-to-handle, high-throughput genotype data from large-scale breeding populations. However, SNP arrays are prone to technical errors that lead to failed allele calls. To overcome this problem, failed calls are often imputed, based on the assumption that failed SNP calls are purely technical. However, this ignores the biological causes for failed calls-for example: deletions-and there is increasing evidence that gene presence-absence and other kinds of genome structural variants can play a role in phenotypic expression. Because deletions are frequently not in linkage disequilibrium with their flanking SNPs, permutation of missing SNP calls can potentially obscure valuable marker-trait associations. In this study, we analyze published datasets for canola and maize using four parametric and two machine learning models and demonstrate that failed allele calls in genomic prediction are highly predictive for important agronomic traits. We present two statistical pipelines, based on population structure and linkage disequilibrium, that enable the filtering of failed SNP calls that are likely caused by biological reasons. For the population and trait examined, prediction accuracy based on these filtered failed allele calls was competitive to standard SNP-based prediction, underlying the potential value of missing data in genomic prediction approaches. The combination of SNPs with all failed allele calls or the filtered allele calls did not outperform predictions with only SNP-based prediction due to redundancy in genomic relationship estimates.

Haplotype blocks for genomic prediction: a comparative evaluation in multiple crop datasets.

In modern plant breeding, genomic selection is becoming the gold standard for selection of superior genotypes. The basis for genomic prediction models is a set of phenotyped lines along with their genotypic profile. With high marker density and linkage disequilibrium (LD) between markers, genotype data in breeding populations tends to exhibit considerable redundancy. Therefore, interest is growing in the use of haplotype blocks to overcome redundancy by summarizing co-inherited features. Moreover, haplotype blocks can help to capture local epistasis caused by interacting loci. Here, we compared genomic prediction methods that either used single SNPs or haplotype blocks with regards to their prediction accuracy for important traits in crop datasets. We used four published datasets from canola, maize, wheat and soybean. Different approaches to construct haplotype blocks were compared, including blocks based on LD, physical distance, number of adjacent markers and the algorithms implemented in the software "Haploview" and "HaploBlocker". The tested prediction methods included Genomic Best Linear Unbiased Prediction (GBLUP), Extended GBLUP to account for additive by additive epistasis (EGBLUP), Bayesian LASSO and Reproducing Kernel Hilbert Space (RKHS) regression. We found improved prediction accuracy in some traits when using haplotype blocks compared to SNP-based predictions, however the magnitude of improvement was very trait- and model-specific. Especially in settings with low marker density, haplotype blocks can improve genomic prediction accuracy. In most cases, physically large haplotype blocks yielded a strong decrease in prediction accuracy. Especially when prediction accuracy varies greatly across different prediction models, prediction based on haplotype blocks can improve prediction accuracy of underperforming models. However, there is no "best" method to build haplotype blocks, since prediction accuracy varied considerably across methods and traits. Hence, criteria used to define haplotype blocks should not be viewed as fixed biological parameters, but rather as hyperparameters that need to be adjusted for every dataset.

Stacked ensembles on basis of parentage information can predict hybrid performance with an accuracy comparable to marker-based GBLUP.

Testcross factorials in newly established hybrid breeding programs are often highly unbalanced, incomplete, and characterized by predominance of special combining ability (SCA) over general combining ability (GCA). This results in a low efficiency of GCA-based selection. Machine learning algorithms might improve prediction of hybrid performance in such testcross factorials, as they have been successfully applied to find complex underlying patterns in sparse data. Our objective was to compare the prediction accuracy of machine learning algorithms to that of GCA-based prediction and genomic best linear unbiased prediction (GBLUP) in six unbalanced incomplete factorials from hybrid breeding programs of rapeseed, wheat, and corn. We investigated a range of machine learning algorithms with three different types of predictor variables: (a) information on parentage of hybrids, (b) in addition hybrid performance of crosses of the parental lines with other crossing partners, and (c) genotypic marker data. In two highly incomplete and unbalanced factorials from rapeseed, in which the SCA variance contributed considerably to the genetic variance, stacked ensembles of gradient boosting machines based on parentage information outperformed GCA prediction. The stacked ensembles increased prediction accuracy from 0.39 to 0.45, and from 0.48 to 0.54 compared to GCA prediction. The prediction accuracy reached by stacked ensembles without marker data reached values comparable to those of GBLUP that requires marker data. We conclude that hybrid prediction with stacked ensembles of gradient boosting machines based on parentage information is a promising approach that is worth further investigations with other data sets in which SCA variance is high.

Genomic prediction in hybrid breeding: II. Reciprocal recurrent genomic selection with full-sib and half-sib families.

KEY MESSAGE: Genomic prediction of GCA effects based on model training with full-sib rather than half-sib families yields higher short- and long-term selection gain in reciprocal recurrent genomic selection for hybrid breeding, if SCA effects are important. Reciprocal recurrent genomic selection (RRGS) is a powerful tool for ensuring sustainable selection progress in hybrid breeding. For training the statistical model, one can use half-sib (HS) or full-sib (FS) families produced by inter-population crosses of candidates from the two parent populations. Our objective was to compare HS-RRGS and FS-RRGS for the cumulative selection gain ([Formula: see text]), the genetic, GCA and SCA variances ([Formula: see text],[Formula: see text], [Formula: see text]) of the hybrid population, and prediction accuracy ([Formula: see text]) for GCA effects across cycles. Using SNP data from maize and wheat, we simulated RRGS programs over 10 cycles, each consisting of four sub-cycles with genomic selection of [Formula: see text] out of 950 candidates in each parent population. Scenarios differed for heritability [Formula: see text] and the proportion [Formula: see text] of traits, training set (TS) size ([Formula: see text]), and maize vs. wheat. Curves of [Formula: see text] over selection cycles showed no crossing of both methods. If [Formula: see text] was high, [Formula: see text] was generally higher for FS-RRGS than HS-RRGS due to higher [Formula: see text]. In contrast, HS-RRGS was superior or on par with FS-RRGS, if [Formula: see text] or [Formula: see text] and [Formula: see text] were low. [Formula: see text] showed a steeper increase and higher selection limit for scenarios with low [Formula: see text], high [Formula: see text] and large [Formula: see text]. [Formula: see text] and even more so [Formula: see text] decreased rapidly over cycles for both methods due to the high selection intensity and the role of the Bulmer effect for reducing [Formula: see text]. Since the TS for FS-RRGS can additionally be used for hybrid prediction, we recommend this method for achieving simultaneously the two major goals in hybrid breeding: population improvement and cultivar development.

Genomic prediction with haplotype blocks in wheat.

Haplotype blocks might carry additional information compared to single SNPs and have therefore been suggested for use as independent variables in genomic prediction. Studies in different species resulted in more accurate predictions than with single SNPs in some traits but not in others. In addition, it remains unclear how the blocks should be built to obtain the greatest prediction accuracies. Our objective was to compare the results of genomic prediction with different types of haplotype blocks to prediction with single SNPs in 11 traits in winter wheat. We built haplotype blocks from marker data from 361 winter wheat lines based on linkage disequilibrium, fixed SNP numbers, fixed lengths in cM and with the R package HaploBlocker. We used these blocks together with data from single-year field trials in a cross-validation study for predictions with RR-BLUP, an alternative method (RMLA) that allows for heterogeneous marker variances, and GBLUP performed with the software GVCHAP. The greatest prediction accuracies for resistance scores for B. graminis, P. triticina, and F. graminearum were obtained with LD-based haplotype blocks while blocks with fixed marker numbers and fixed lengths in cM resulted in the greatest prediction accuracies for plant height. Prediction accuracies of haplotype blocks built with HaploBlocker were greater than those of the other methods for protein concentration and resistances scores for S. tritici, B. graminis, and P. striiformis. We hypothesize that the trait-dependence is caused by properties of the haplotype blocks that have overlapping and contrasting effects on the prediction accuracy. While they might be able to capture local epistatic effects and to detect ancestral relationships better than single SNPs, prediction accuracy might be reduced by unfavorable characteristics of the design matrices in the models that are due to their multi-allelic nature.

Training set designs for prediction of yield and moisture of maize test cross hybrids with unreplicated trials.

Unreplicated field trials and genomic prediction are both used to enhance the efficiency in early selection stages of a hybrid maize breeding program. No results are available on the optimal experimental design when combining both approaches. Our objectives were to investigate the effect of the training set design on the accuracy of genomic prediction in unreplicated maize test crosses. We carried out a cross validation study on basis of an experimental data set consisting of 1436 hybrids evaluated for yield and moisture for which genotyping information of 461 SNP markers were available. Training set designs of different size, implementing within environment prediction, within year prediction, across year prediction, and combinations of data sources across years and environments were compared with respect to their prediction accuracy. Across year prediction did not reach prediction accuracies that are useful for genomic selection. Within year prediction across environments provided useful correlations between observed and predicted breeding values. The prediction accuracies did not improve when adding to the training set data from previous years. We conclude that using all data available from unreplicated tests of the current breeding cycle provides a good accuracy of predicting test crosses, whereas adding data from previous breeding cycles, in which the genotypes are less related to the tested material, has only limited value for increasing the prediction accuracy.

Genomic Prediction Can Provide Precise Estimates of the Genotypic Value of Barley Lines Evaluated in Unreplicated Trials.

Genomic prediction has been established in breeding programs to predict the genotypic values of selection candidates without phenotypic data. First results in wheat showed that genomic predictions can also prove useful to select among material for which phenotypic data are available. In such a scenario, the selection candidates are evaluated with low intensity in the field. Genome-wide effects are estimated from the field data and are then used to predict the genotypic values of the selection candidates. The objectives of our simulation study were to investigate the correlations r(y, g) between genomic predictions y and genotypic values g and to compare these with the correlations r(p, g) between phenotypic values p and genotypic values g. We used data from a yield trial of 250 barley lines to estimate variance components and genome-wide effects. These parameters were used as basis for simulations. The simulations included multiple crossing schemes, population sizes, and varying sizes of the components of the masking variance. The genotypic values g of the selection candidates were obtained by genetic simulations, the phenotypic values p by simulating evaluation in the field, and the genomic predictions y by RR-BLUP effect estimation from the phenotypic values. The correlations r(y, g) were greater than the correlations r(p, g) for all investigated scenarios. We conclude that using genomic predictions for selection among candidates tested with low intensity in the field can proof useful for increasing the efficiency of barley breeding programs.

Strategies and considerations for implementing genomic selection to improve traits with additive and non-additive genetic architectures in sugarcane breeding.

KEY MESSAGE: Simulations highlight the potential of genomic selection to substantially increase genetic gain for complex traits in sugarcane. The success rate depends on the trait genetic architecture and the implementation strategy. Genomic selection (GS) has the potential to increase the rate of genetic gain in sugarcane beyond the levels achieved by conventional phenotypic selection (PS). To assess different implementation strategies, we simulated two different GS-based breeding strategies and compared genetic gain and genetic variance over five breeding cycles to standard PS. GS scheme 1 followed similar routines like conventional PS but included three rapid recurrent genomic selection (RRGS) steps. GS scheme 2 also included three RRGS steps but did not include a progeny assessment stage and therefore differed more fundamentally from PS. Under an additive trait model, both simulated GS schemes achieved annual genetic gains of 2.6-2.7% which were 1.9 times higher compared to standard phenotypic selection (1.4%). For a complex non-additive trait model, the expected annual rates of genetic gain were lower for all breeding schemes; however, the rates for the GS schemes (1.5-1.6%) were still greater than PS (1.1%). Investigating cost-benefit ratios with regard to numbers of genotyped clones showed that substantial benefits could be achieved when only 1500 clones were genotyped per 10-year breeding cycle for the additive genetic model. Our results show that under a complex non-additive genetic model, the success rate of GS depends on the implementation strategy, the number of genotyped clones and the stage of the breeding program, likely reflecting how changes in QTL allele frequencies change additive genetic variance and therefore the efficiency of selection. These results are encouraging and motivate further work to facilitate the adoption of GS in sugarcane breeding.

Breeding improves wheat productivity under contrasting agrochemical input levels.

The world cropping area for wheat exceeds that of any other crop, and high grain yields in intensive wheat cropping systems are essential for global food security. Breeding has raised yields dramatically in high-input production systems; however, selection under optimal growth conditions is widely believed to diminish the adaptive capacity of cultivars to less optimal cropping environments. Here, we demonstrate, in a large-scale study spanning five decades of wheat breeding progress in western Europe, where grain yields are among the highest worldwide, that breeding for high performance in fact enhances cultivar performance not only under optimal production conditions but also in production systems with reduced agrochemical inputs. New cultivars incrementally accumulated genetic variants conferring favourable effects on key yield parameters, disease resistance, nutrient use efficiency, photosynthetic efficiency and grain quality. Combining beneficial, genome-wide haplotypes could help breeders to more efficiently exploit available genetic variation, optimizing future yield potential in more sustainable production systems.

Effect of breeding on nitrogen use efficiency-associated traits in oilseed rape.

Oilseed rape is one of the most important dicotyledonous field crops in the world, where it plays a key role in productive cereal crop rotations. However, its production requires high nitrogen fertilization and its nitrogen footprint exceeds that of most other globally important crops. Hence, increased nitrogen use efficiency (NUE) in this crop is of high priority for sustainable agriculture. We report a comprehensive study of macrophysiological characteristics associated with breeding progress, conducted under contrasting nitrogen fertilization levels in a large panel of elite oilseed rape varieties representing breeding progress over the past 20 years. The results indicate that increased plant biomass at flowering, along with increases in primary yield components, have increased NUE in modern varieties. Nitrogen uptake efficiency has improved through breeding, particularly at high nitrogen. Despite low heritability, the number of seeds per silique is associated positively with increased yield in modern varieties. Seed weight remains unaffected by breeding progress; however, recent selection for high seed oil content and for high seed yields appears to have promoted a negative correlation (r= -0.39 at high and r= -0.49 at low nitrogen) between seed weight and seed oil concentration. Overall, our results reveal valuable breeding targets to improve NUE in oilseed rape.

Genetic insights into underground responses to Fusarium graminearum infection in wheat.

The ongoing global intensification of wheat production will likely be accompanied by a rising pressure of Fusarium diseases. While utmost attention was given to Fusarium head blight (FHB) belowground plant infections of the pathogen have largely been ignored. The current knowledge about the impact of soil borne Fusarium infection on plant performance and the underlying genetic mechanisms for resistance remain very limited. Here, we present the first large-scale investigation of Fusarium root rot (FRR) resistance using a diverse panel of 215 international wheat lines. We obtained data for a total of 21 resistance-related traits, including large-scale Real-time PCR experiments to quantify fungal spread. Association mapping and subsequent haplotype analyses discovered a number of highly conserved genomic regions associated with resistance, and revealed a significant effect of allele stacking on the stembase discoloration. Resistance alleles were accumulated in European winter wheat germplasm, implying indirect prior selection for improved FRR resistance in elite breeding programs. Our results give first insights into the genetic basis of FRR resistance in wheat and demonstrate how molecular parameters can successfully be explored in genomic prediction. Ongoing work will help to further improve our understanding of the complex interactions of genetic factors influencing FRR resistance.

Effects of different management regimes on microbial biodiversity in vineyard soils.

An active and diverse soil biota is important for maintaining crop productivity and quality, and preservation of these traits is a major goal of sustainable farming. This study aimed at unravelling the impact of different management practices on soil fungal and bacterial biodiversity in vineyards as a model for permanent crops. Species diversity was assessed using an amplicon sequencing approach in a long-term field experiment in the Rheingau wine region of Germany where integrated, organic and biodynamic management practices had been in place for 10 years. Fungal community composition under integrated management differed significantly from organic and biodynamic management, whereas fungal species richness remained unaffected. Soil under integrated management had a significantly reduced bacterial species richness compared to organic, but community composition was similar to organically and biodynamically managed soils. Highest fungal richness was obtained under cover crop between rows in topsoil, arising from cover cropping and organic carbon supply.

Small RNA-based prediction of hybrid performance in maize.

BACKGROUND: Small RNA (sRNA) sequences are known to have a broad impact on gene regulation by various mechanisms. Their performance for the prediction of hybrid traits has not yet been analyzed. Our objective was to analyze the relation of parental sRNA expression with the performance of their hybrids, to develop a sRNA-based prediction approach, and to compare it to more common SNP and mRNA transcript based predictions using a factorial mating scheme of a maize hybrid breeding program. RESULTS: Correlation of genomic differences and messenger RNA (mRNA) or sRNA expression differences between parental lines with hybrid performance of their hybrids revealed that sRNAs showed an inverse relationship in contrast to the other two data types. We associated differences for SNPs, mRNA and sRNA expression between parental inbred lines with the performance of their hybrid combinations and developed two prediction approaches using distance measures based on associated markers. Cross-validations revealed parental differences in sRNA expression to be strong predictors for hybrid performance for grain yield in maize, comparable to genomic and mRNA data. The integration of both positively and negatively associated markers in the prediction approaches enhanced the prediction accurary. The associated sRNAs belong predominantly to the canonical size classes of 22- and 24-nt that show specific genomic mapping characteristics. CONCLUSION: Expression profiles of sRNA are a promising alternative to SNPs or mRNA expression profiles for hybrid prediction, especially for plant species without reference genome or transcriptome information. The characteristics of the sRNAs we identified suggest that association studies based on breeding populations facilitate the identification of sRNAs involved in hybrid performance.

Parental Expression Variation of Small RNAs Is Negatively Correlated with Grain Yield Heterosis in a Maize Breeding Population.

Heterosis refers to a quantitative phenomenon in which F1 hybrid trait values exceed the mean of the parental values in a positive direction. Generally, it is dependent on a high degree of heterozygosity, which is maintained in hybrid breeding by developing parental lines in separate, genetically distinct heterotic groups. The mobility of small RNAs (sRNAs) that mediate epigenetic regulation of gene expression renders them promising candidates for modulating the action of combined diverse genomes in trans-and evidence already indicates their contribution to transgressive phenotypes. By sequencing small RNA libraries of a panel of 21 maize parental inbred lines we found a low overlap of 35% between the sRNA populations from both distinct heterotic groups. Surprisingly, in contrast to genetic or gene expression variation, parental sRNA expression variation is negatively correlated with grain yield (GY) heterosis. Among 0.595 million expressed sRNAs, we identified 9,767, predominantly 22- and 24-nt long sRNAs, which showed an association of their differential expression between parental lines and GY heterosis of the respective hybrids. Of these sRNAs, 3,485 or 6,282 showed an association with high or low GY heterosis, respectively, thus the low heterosis associated group prevailing at 64%. The heterosis associated sRNAs map more frequently to genes that show differential expression between parental lines than reference sets. Together these findings suggest that trans-chromosomal actions of sRNAs in hybrids might add up to a negative contribution in heterosis formation, mediated by unfavorable gene expression regulation. We further revealed an exclusive accumulation of 22-nt sRNAs that are associated with low GY heterosis in pericentromeric genomic regions. That recombinational suppression led to this enrichment is indicated by its close correlation with low recombination rates. The existence of this enrichment, which we hypothesize resulted from the separated breeding of inbred lines within heterotic groups, may have implications for hybrid breeding strategies addressing the recombinational constraints characteristic of complex crop genomes.

Prediction of Means and Variances of Crosses With Genome-Wide Marker Effects in Barley.

Background: The expected genetic variance is an important criterion for the selection of crossing partners which will produce superior combinations of genotypes in their progeny. The advent of molecular markers has opened up new vistas for obtaining precise predictors for the genetic variance of a cross, but fast prediction methods that allow plant breeders to select crossing partners based on already available data from their breeding programs without complicated calculations or simulation of breeding populations are still lacking. The main objective of the present study was to demonstrate the practical applicability of an analytical approach for the selection of superior cross combinations with experimental data from a barley breeding program. We used genome-wide marker effects to predict the yield means and genetic variances of 14 DH families resulting from crosses of four donor lines with five registered elite varieties with the genotypic information of the parental lines. For the validation of the predicted parameters, the analytical approach was extended by the masking variance as a major component of phenotypic variance. The predicted parameters were used to fit normal distribution curves of the phenotypic values and to conduct an Anderson-Darling goodness-of-fit test for the observed phenotypic data of the 14 DH families from the field trial. Results: There was no evidence that the observed phenotypic values deviated from the predicted phenotypic normal distributions in 13 out of 14 crosses. The correlations between the observed and the predicted means and the observed and predicted variances were r = 0.95 and r = 0.34, respectively. After removing two crosses with downward outliers in the phenotypic data, the correlation between the observed and predicted variances increased to r = 0.76. A ranking of the 14 crosses based on the sum of predicted mean and genetic variance identified the 50% best crosses from the field trial correctly. Conclusions: We conclude that the prediction accuracy of the presented approach is sufficiently high to identify superior crosses even with limited phenotypic data. We therefore expect that the analytical approach based on genome-wide marker effects is applicable in a wide range of breeding programs.

Genome-wide regression models considering general and specific combining ability predict hybrid performance in oilseed rape with similar accuracy regardless of trait architecture.

KEY MESSAGE: Genomic prediction using the Brassica 60 k genotyping array is efficient in oilseed rape hybrids. Prediction accuracy is more dependent on trait complexity than on the prediction model. In oilseed rape breeding programs, performance prediction of parental combinations is of fundamental importance. Due to the phenomenon of heterosis, per se performance is not a reliable indicator for F1-hybrid performance, and selection of well-paired parents requires the testing of large quantities of hybrid combinations in extensive field trials. However, the number of potential hybrids, in general, dramatically exceeds breeding capacity and budget. Integration of genomic selection (GS) could substantially increase the number of potential combinations that can be evaluated. GS models can be used to predict the performance of untested individuals based only on their genotypic profiles, using marker effects previously predicted in a training population. This allows for a preselection of promising genotypes, enabling a more efficient allocation of resources. In this study, we evaluated the usefulness of the Illumina Brassica 60 k SNP array for genomic prediction and compared three alternative approaches based on a homoscedastic ridge regression BLUP and three Bayesian prediction models that considered general and specific combining ability (GCA and SCA, respectively). A total of 448 hybrids were produced in a commercial breeding program from unbalanced crosses between 220 paternal doubled haploid lines and five male-sterile testers. Predictive ability was evaluated for seven agronomic traits. We demonstrate that the Brassica 60 k genotyping array is an adequate and highly valuable platform to implement genomic prediction of hybrid performance in oilseed rape. Furthermore, we present first insights into the application of established statistical models for prediction of important agronomical traits with contrasting patterns of polygenic control.

Genomic selection of crossing partners on basis of the expected mean and variance of their derived lines.

In a line or a hybrid breeding program superior lines are selected from a breeding pool as parental lines for the next breeding cycle. From a cross of two parental lines, new lines are derived by single-seed descent (SSD) or doubled haploid (DH) technology. However, not all possible crosses between the parental lines can be carried out due to limited resources. Our objectives were to present formulas to characterize a cross by the mean and variance of the genotypic values of the lines derived from the cross, and to apply the formulas to predict means and variances of flowering time traits in recombinant inbred line families of a publicly available data set in maize. We derived formulas which are based on the expected linkage disequilibrium (LD) between two loci and which can be used for arbitrary mating systems. Results were worked out for SSD and DH lines derived from a cross after an arbitrary number of intermating generations. The means and variances were highly correlated with results obtained by the simulation software PopVar. Compared with these simulations, computation time for our closed formulas was about ten times faster. The means and variances for flowering time traits observed in the recombinant inbred line families of the investigated data set showed correlations of around 0.9 for the means and of 0.46 and 0.65 for the standard deviations with the estimated values. We conclude that our results provide a framework that can be exploited to increase the efficiency of hybrid and line breeding programs by extending genomic selection approaches to the selection of crossing partners.