Our findings support a minor genetic overlap between these two disorders.Parkinson's disease and Alzheimer's disease show overlapping features both clinically and neuropathologically and elucidating shared mechanisms could have important implications for therapeutic strategies. Evidence for genetic overlap is limited, although enrichment of heritability in genomic regions relevant to microglia has been demonstrated in both disorders. Using summary statistics from genome‐wide association studies, we assessed genetic covariance stratified by cell types and local genetic correlation between Parkinson's and Alzheimer's disease. Significant covariance was observed for neurons only ( Exploring the genetic relationship between complex disorders can shed light on shared pathogenic mechanisms and help generate hypotheses for novel therapies, such as drug repurposing.Parkinson's disease (PD) and Alzheimer's disease (AD) are common neurodegenerative disorders that show overlapping features both clinically and neuropathologically.Summary statistics from GWAS may be combined with tissue‐ or cell type‐specific genomic annotations to assess enrichment of common variant heritability.To further dissect shared genetic mechanisms across PD and AD, we took advantage of summary statistics from large‐scale GWAS to assess both annotation‐stratified genetic covarianceWe used the full summary statistics (including 23andMe data) from the 2019 Nalls et al. meta‐analysis of PD GWAS, which included a total of 37,688 cases, 18,618 proxy‐cases, and 1,417,791 controls.p‐values were calculated based on the coefficient z‐score. To define open chromatin in individual brain and immune cell types, we used assay for transposase‐accessible chromatin sequencing (ATAC‐seq) peak data from neurons, oligodendrocytes, astroglia, and microglia+, and CD8+ T‐cells.Linkage disequilibrium score regression (LDSC) is a statistical framework that examines the relationship between GWAS test statistics and LD patterns in a reference sample in order to estimate common variant heritability. In a recently published work, we applied stratified LDSC (s‐LDSC) to partition heritability for PD and assess enrichment across specific cell types.p‐values corrected for sample overlap by GNOVA, Bonferroni‐correcting for eight independent cell types.GNOVA  is a framework to estimate annotation‐stratified genetic covariance using GWAS summary statistics and dissect patterns of shared risk that may not be captured by a global test.Whereas GNOVA estimates correlation within a functionally defined set of regions spread out across the genome, ρ‐HESS (Heritability Estimator from Summary Statistics) is a method to partition the genome into consecutive LD blocks and assess genetic correlation between two traits locally within each of these blocks.z‐scores from summary statistics, excluding APOE. We calculated the Pearson correlation and plotted the results using R.4.0.3.To compare the magnitude and direction of effect of significant SNPs reported in PD and AD GWAS we generated p‐value = 4.0e‐5), LDSC , and HESS . When genetic covariance was stratified by open chromatin in the brain and immune cell types, significant covariance was observed for neurons only  when correcting for multiple testing , entailing the HLA association signals reported as significant in both PD and AD  was the only significant finding .Similarly, the lot Fig.  shows thus Table , whereasPrevious studies have reported limited genetic overlap between PD and AD, but a significant enrichment of heritability in microglia‐related genomic regions has been evident in both disorders. Hypothesizing that PD and AD may show genetic sharing in specific subsets or regions of the genome we analyzed annotation‐stratified genetic covariance and local genetic correlation.We note as an important caveat that clinical misdiagnosis may drive false positive signals of shared heritability when phenotypes are overlapping,Analyzing local genetic correlation using ρ‐HESS, the HLA locus was the only region where PD and AD correlated significantly. This result must be interpreted with caution, in particular because correlation on the SNP level does not necessarily imply that the same HLA alleles are involved. Consequently, ρ‐HESS results from this particular region may potentially be non‐specific, as the negative local correlation observed in the HLA region for PD and MS also suggests. GWAS signals from the HLA region have been reported in both PD and AD, and follow‐up studies have performed fine‐mapping, HLA imputation and stepwise conditional regression analyses. A 2017 AD HLA study highlighted the DR15 risk haplotype (DRB1*15:01‐ DQA1*01:02‐ DQB1*06:02), proposing that components of this haplotype could contribute to risk across multiple neurological disorders, including PD.GRN association signal reported in both PD and AD. GRN encodes progranulin and is implicated in frontotemporal lobar degeneration. Further research will be needed to clarify whether these are all genuine signals or if misdiagnosis may contribute to the apparent association of this locus with a broad range of neurodegenerative disorders.We also highlight concordant effect direction of the In conclusion, we have detected a significant genetic covariance between AD and PD in neuronal, but not microglial open chromatin regions. We also highlight the common HLA association. Still, the evidence for shared genetic architecture across the two most common neurodegenerative disorders remains limited, underlining that neuroinflammation and neurodegeneration are complex and heterogeneous phenomena following diverse pathways in different disorders.L. P. contributed to conception and design of the study; M. S. A., M. T., I. H., J. H., L. P., and IPDGC contributed to acquisition, analysis and/or interpretation of the data; M. S. A. and L. P. contributed to the drafting of the manuscript and figures; All authors contributed to critical revision of the manuscript.Nothing to report.https://research.23andme.com/dataset‐access/. AD summary statistics: https://ctg.cncr.nl/software/summary_statistics. GWAS Catalog: https://www.ebi.ac.uk/gwas/. Alkes group repository: https://alkesgroup.broadinstitute.org/LDSCORE/. Linkage disequilibrium score regression: https://github.com/bulik/ldsc. GNOVA: https://github.com/xtonyjiang/GNOVA. HESS: https://huwenboshi.github.io/hess/.23andMe: Data S1. Supplementary methods.Figure S1.Z scores of local genetic correlations.Table S1. SNPs in shared GWAS loci across PD and AD.Click here for additional data file.


Despite the importance of flowering for fruit formation, it has been considered very little in breeding programs involving fruit species, including olives. We evaluated the principal morphological flower-quality components in the olive cultivars, ‘Arbequina’ and ‘Picual’, and in the progenies of their crosses. Wide ranges of variation were obtained for all the inflorescence traits and ovary tissue sizes. An analysis of variance indicated that the residual error was the main contributor to the inflorescence traits, except for the number of perfect flowers, underlining the need to evaluate adequate numbers of inflorescences for accurate measurements of these traits. However, the high repeatability obtained for the inflorescence traits suggests that simple evaluation procedures could be accurate enough for genotype characterization. The average values for ‘Arbequina’ were in the upper range for all the traits; the opposite occurred for ‘Picual’, and the values for most of the progenies were intermediate. No significant differences between the maternal and paternal effect on inheritance were found. Some interesting transgressive segregants showed a higher flower number, greater ovary and mesocarp size, or percentage of ovaries with all four fully developed ovules. The correlations among the parameters may have reflected a relatively consistent distribution of the ovaries’ structural components and a close relationship between the ovaries and their mesocarp growth. Flowering is the critical first step in fruit formation; however, it has been evaluated very rarely in breeding programs for olive or other fruit tree species . Some ofFruit-tree flower quality encompasses the multiple aspects of floral structure and physiology that potentially affect fruit number and/or size . In olivOnce a hermaphrodite flower with a well-formed ovary is present, the fruit number is determined by its capacity for successful fertilization. In olive trees, fertilization can be limited by incomplete or anomalous ovule development, in which the embryonic sac fails to fully differentiate ,23. As wA third relevant morphological component of olive flower quality, which potentially influences fruit size rather than number, is ovary size. Among a range of olive cultivars, Rosati et al.  found thEven though a strong genetic (cultivar) influence has been reported for the above- described floral morphogenetic components, the inheritance of these traits has been seldom studied in the progenies of olive crosses. In the new olive cultivar, ‘Sikitita’ (known in USA as ‘Chiquitita’) , flower In this study, we evaluated the principal morphological components of flower quality in the olive cultivars ‘Arbequina’ and ‘Picual’, and in the progenies of their reciprocal crosses, in order to observe indications of their inheritance and determine whether any particular maternal or paternal effect could be identified. In relation to the flower-quality factors determining the final fruit number, we examined the inflorescence flower number, pistil abortion, and incomplete ovule development. In relation to the determination of the fruit size, we evaluated the ovary and ovary tissue sizes, and tested their correlation with the mature fruit tissues.Wide ranges of variation were obtained for all the inflorescence traits and ovary tissue sizes, with the greatest coefficients of variation found in the numbers and percentages of perfect flowers . The valThe ‘Arbequina’ consistently showed higher values than the ‘Picual’ for all the inflorescence and ovary parameters , as was The variance components showed different contributions of the sources of variation for the different evaluated traits . The resThese results provided repeatability estimates from 0.63 to 0.76 for the inflorescence traits, and slightly higher values (0.82–0.85) for the ovary tissue sizes . The higSignificant differences  were obtAll the parameters evaluated showed higher values in 2013 compared to 2014, especially the number and percentage of perfect flowers . This coThe similar pattern of variability inferred for both inflorescence and ovary parameters in No significant correlation, however, was found between the ovary tissue size parameters and the corresponding mature fruit tissues  (data not shown). By contrast, Rosati et al.  found a 2 = 0.09, p = 0.9929), and no significant differences obtained among the cross combinations . Significant differences were found, however, among the genotypes  at the Alameda del Obispo Center in Córdoba, Spain. Five-year-old adult seedling trees from reciprocal crosses of cultivars ‘Arbequina’ and ‘Picual’  were grown together with vegetatively cloned parent cultivars of the same age. Only one plant per genotype was evaluated per parent cultivar in order to be comparable with the single-plant genotypes of the crosses. Trees were drip-irrigated with 1500 mv/v/v). A mixture of open and closed flowers in the sampled inflorescences guaranteed the presence of recently opened flowers containing mature ovaries just prior to fertilization. Following transfer to 70% ethanol and step-wise rehydration to 30% ethanol, the number of nodes, number of flowers, and number and percentage of perfect (hermaphrodite) flowers were determined.Inflorescences were sampled and evaluated in two successive years (2013 and 2014), following the procedures in . During ® embedding paraffin with a melting point 56–58 °C , according to standard paraffin procedures [Ten pistils per genotype and year were selected from perfect flowers interpreted as having recently opened on the sampled inflorescences, using a maximum of two pistils per inflorescence. The pistils were processed in Histosecocedures ; the ovaocedures .The ovules of the ten processed pistils were observed in successive transverse ovary sections by optical microscope . Each ovary was rated 0–4, according to the number of fully developed ovules  found among the total of four ovules normally present in the olive ,27 Figua. The peOvary and ovary tissue sizes were determined in central transverse sections of the ovaries previously used for ovule analysis. Only ovaries with three or four developed ovules were used for the measurements, to eliminate any link with poor or anomalous ovule development and reduced ovary growth . In ordeIn the second year (2014) of the experiment, mature green fruits were also sampled, fixed, and later rehydrated for measurement, as described above for the inflorescences. For ten fruits per tree, the central transverse diameters of fruit and pit were measured and the relations of fruit, pulp, and pit to ovary, ovary mesocarp, and ovary endocarp size were evaluated.p = 0.05. The correlations between ovary and fruit tissues were tested by determining the Pearson coefficient (p = 0.05).Descriptive statistics of the different traits were obtained for the whole dataset. ANOVA was applied according to the following statistical model: Pijk = µ+Gi + Yj + (G × Y)ij + εijk, where Pijk is the phenotypic value of the k sample of the i genotype in the j year, µ is the overall mean of the progeny, Gi is a random effect contributed by the i genotype, Yj is a random effect of the j year, (G × Y)ij is the interaction between the i genotype and the j year, and εijkl is the random residual error effect for the k measured samples. ANOVA provided the variance among genotypes (σ²G), among years (σ²Y), associated with the genotype × year interaction (σ²GY), and residual error effect for the measured samples (σ²ε). From this model, repeatability was estimated as r = σ²G/ (σ²G + σ²GY/y + σ²ε/ys), where y is the number of years (2) and s the number of samples (25 for inflorescence traits and 10 for ovary tissue size). Comparison of means was carried out according to Tukey’s HSD test at The evaluation of genotypes from the reciprocal crosses between ‘Arbequina’ and ‘Picual’ showed mostly intermediate values for the studied flower quality traits in the progenies, with no significant differences between the maternal and the paternal effect on the inheritance. Furthermore, some interesting transgressive segregants showed higher flower numbers, greater ovary and mesocarp tissue sizes, or higher percentages of ovaries with four fully developed ovules. It is remarkable that among the descendants of the two cultivars used as parents, ‘Arbequina’ and ‘Picual’, only a few genotypes should be potentially discarded on the basis of very low values in these traits. Despite the importance of flower quality for final fruit production, its evaluation is time-consuming, particularly when histological procedures are involved, and requires careful sampling and large sample numbers. The large differences between the two parents for the evaluated flower quality parameters produced a high level of variability among the descendants. This high variability allows us to conclude that the flower-quality parameters that influence olive fruit number, such as flower number, perfect flower number, and ovule development, appear to be more pertinent for breeding programs due to the high genetic influence, whereas the link between ovary quality and fruit quality is still unclear. These assumptions, however, should be further confirmed with studies in other progenies involving different cultivars as genitors.


Genetic variation is the basis of selection, evolution and breeding. Maize landraces represent a rich source of allelic diversity, but their efficient utilization in breeding and research has been hampered by their heterogeneous and heterozygous nature and insufficient information about most accessions. While molecular inventories of germplasm repositories are growing steadily, linking these data to meaningful phenotypes for quantitative traits is challenging.ad libitum and represent a powerful biological resource available to the community. The DH lines allow high-precision phenotyping in repeated experiments and reveal the full additive genetic variance of the population. The DH lines were evaluated for nine agronomically important, quantitative traits in multi-environment field trials comprising seven locations and two years. The DH populations revealed high genetic variance and high heritability for the analysed traits. The DH lines were genotyped with 600k SNP markers. After stringent quality filtering 500k markers remained for further analyses.Here, we present comprehensive molecular and phenotypic data for ∼1,000 doubled-haploid (DH) lines derived from three pre-selected European maize landraces. Due to their full homozygosity, the DH lines can be multiplied This is the largest resource of landrace derived DH material in maize, unprecedented in its structure and dimension. The presented data are ideal for linking molecular variation to meaningful phenotypes. They can be used for genome-wide association studies, genomic prediction, and population genetic analyses as well as for developing and testing statistical methods. All plant material is available to the community for conducting additional experiments, extending the panel of traits and environments, and for testing the landrace-derived lines in combination with other genetic material. It is unique in its structure and dimension.•The biological material is available to the community and can serve as starting point for further research, extending the panel of traits and environments, testing the landrace-derived lines in combination with other genetic material and developing novel populations for (pre-)breeding and fine-mapping of candidate genes.•The data presented here, together with the available underlying biological material and efficient analysis tools, open new avenues for the efficient utilization of the genetic diversity of maize landraces for enhancing elite germplasm with novel beneficial alleles.•These data will benefit geneticists studying the genetic architecture of complex traits, gene bank curators aiming at making their inventories accessible for crop improvement, and breeders interested in harnessing native diversity for improving their breeding germplasm.•These data are ideal for developing and testing novel statistical methods for genome-wide association studies, genomic prediction, haplotype construction, and population genetic analyses.•The data can further be used to gain insights into maize history and the evolutionary consequences of maize expansion, environmental adaptation and breeding.1Zea mays L. ssp. mays) landraces: Kemater Landmais Gelb (KE), Petkuser Ferdinand Rot (PE), and Lalin (LL). In total, 1,015 DH lines were derived from the three landraces . The DH lines represent samples of gametes segregating in the original landraces. Their full homozygosity allows ad libitum seed multiplication through selfing and therefore an evaluation in repeated experiments with any degree of precision desired. The DH lines reveal the full additive genetic variance of a population, without hidden allelic effects in heterozygous states. The three landraces can be assumed to be closest related to the European flint heterotic breeding pool, which therefore should be the target germplasm for introgression of novel beneficial alleles found in the DH populations. Accordingly, the hybrid performance of the DH lines should be evaluated in testcrosses with a complementary dent breeding line. All DH lines are available to the community through standard material transfer agreements upon request, no patents or plant breeders’ rights apply. The genotypes can be evaluated for additional traits for per se and testcross performance, expanding the dataset presented here according to particular research and breeding goals. The data and material presented here, support the research article “Discovery of beneficial haplotypes for complex traits in maize landraces” by Mayer et al. This article presents high-dimensional genotypic and phenotypic data of three pre-selected European flint maize (http://www.ncbi.nlm.nih.gov/geo/query/acc.cgi?acc=GPL18778). The landrace membership of individual lines is indicated by the respective abbreviation  included in the column headers. The filtered data file contains only markers with the best quality class, “Poly High Resolution” The DH lines were genotyped with the 600k Affymetrix® Axiom® Maize Array The field experiments comprised 958 landrace-derived DH lines as well as 15 breeding lines and samples of the original landraces. Thereof, 899 DH lines  passed the quality filtering based on genotypic data. The lines were evaluated for various traits in six and five locations in 2017 and 2018, respectively, resulting in up to eleven location-year combinations per DH line and trait , Table 122.1The flint maize landraces KE, LL and PE, originating from Austria, Spain, and Germany, respectively, were pre-selected from a broad panel of 35 European landraces based on population genetic analyses 2.2DNA was extracted from leaf material of seedlings of each DH line following the protocol of Saghai-Maroof et al. For each DH line, the proportion of ancestry attributable to each of the three landraces was estimated with the software ADMIXTURE 2.3−2.The performance of the DH lines was evaluated in six and five locations in 2017 and 2018, respectively, resulting in eleven different environments (location-year combinations; Traits were measured/scored manually in the field. Experimental units were associated with plot numbers only and not with genotype names, i.e. investigators were blinded to group allocations during data collection. Early vigour was scored in growth stages V4 and V6 i; j; i and environment j; k nested in environment j; o nested in lattice k and environment j; p nested in replicate o, lattice k and environment j; and For each trait, BLUEs of genotype means across environments were calculated using the modelManfred Mayer: Conceptualization, Investigation, Data curation, Formal analysis, Visualization, Writing – original draft. Armin C. Hölker: Investigation, Data curation, Formal analysis. Thomas Presterl: Investigation. Milena Ouzunova: Conceptualization, Funding acquisition, Investigation. Albrecht E. Melchinger: Funding acquisition, Investigation. Chris-Carolin Schön: Conceptualization, Funding acquisition, Investigation, Writing – review & editing.The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.


LTBP2) has been found to have some functions in the development of thoracolumbar vertebrae. The relationship between LTBP2 and TLN (the number of thoracolumbar vertebrae) of Dezhou donkeys is yet to be reported. The purposes of this study are as follows: 1) to quantify the effect of thoracolumbar vertebrae number variation of Dezhou donkeys on body size and carcass trait; 2) to study the distribution of single nucleotide variants (SNVs) in the LTBP2 gene of Dezhou donkeys; and 3) to explore whether these SNVs can be used as candidate sites to study the mechanism of Dezhou donkey muti-thoracolumbar vertebrae development. The TLN, body size, and carcass traits of 392 individuals from a Dezhou donkey breed were recorded. All animals were sequenced for LTBP2 using GBTS liquid chip and 16 SNVs were used for further analysis. We then analyzed the relationship between these SNVs with TLN, body size, and carcass traits. The results showed that: 1) c.5547 + 860 C > T, c.5251 + 281 A > C, c.3769 + 40 C > T, and c.2782 + 3975 A > G were complete genetic linkages and significantly associated with thoracic vertebrae number (TN) (p < 0.05) (wild-type homozygotes had more TN than heterozygotes); 2) c.1381 + 768 T > G and c.1381 + 763 G > T were significantly associated with lumber vertebrae number (LN) (p < 0.05); 3) c.1003 + 704 C > T, c.1003 + 651 C > T, c.1003 + 626 A > G, and c.812 + 22526 T > G were significantly associated with chest circumference (CHC), front carcass weight (CWF), after carcass weight (CWA), and carcass weight (CW) (p < 0.05) ; and 4) the effect of variation is not consistent in c.565 + 11921 A > G, c.565 + 6840 A > G, c.565 + 3453 C > T, and c.494 + 5808 C > T. These results provide useful information that the polymorphism of LTBP2 is significantly associated with TLN, body size, and carcass traits in Dezhou donkeys, which can serve as a molecule marker to improve donkey production performance.The number of thoracolumbar vertebrae in Dezhou donkeys varies from 22 to 24 and is associated with body size and carcass traits. In mammals, the latent transforming growth factor beta binding protein 2 ( In our previous studies, through the genome-wide association study (GWAS), we predicted that LTBP2 may be associated with the vertebrae number in Dezhou donkeys , the vertical distance from the highest point of bun nail to the ground; body length (BL), the linear distance from the fore-end of humeral carina to the last internal carina of ischial tubercle; and chest circumference (CHC), make a vertical line from the back of the scapula and measure the circumference of its chest] of these 392 Dezhou donkeys was measured using The study collected 392 Dezhou donkeys in Dezhou Town, Shandong Province, China, from 2019 to 2021 (the average body size data are shown in LTBP2 gene with GBTS (genotyping by targeted sequencing) liquid chip (using GenoBaits and GenoPlexs technology), which was developed by our laboratory and the Shijiazhuang Breeding Biotechnology Co., Ltd. In order to confirm the accuracy of the chip sequencing results, eight pairs of specific primer sequences were designed using Primer Premier 5.0 software to amplify some polymorphic loci of the LTBP2 gene (50%). The primers are listed in LTBP2. The PCR system (25 μL) included: 2 μL genomic DNA (25 ng/μL); 1 μL of each primer (10 μM); 12.5 μL 2 × MasterMix ; and 8.5 μL double-distilled H2O. The PCR protocol was as follows: 95°C for 10 min; 35 cycles of denaturing at 95°C for 30 s; annealing at Tm°C  and , including observed heterozygosity (Obs-Het), predicted heterozygosity (Pred-Het), homozygosity (Ho), effective number of alleles (Ne) (r2 (correlation coefficient) based on the alleles of each site was calculated using Haploview software  and haplotype frequency was estimated between the loci  was used for correlation analysis:ij is the measured value of the related trait, μ is the population mean, ai is the genotype effect, and eij is the random residual.The genotype frequencies of all target loci (the frequency of each genotype is greater than 5%) were calculated. Using these SNVs, haplotypes were constructed, and haplotype frequency was calculated; haplotypes with frequencies greater than 5% were used for subsequent analysis. The genotype frequencies, allelic frequencies, and Hardy–Weinberg equilibrium partial values were determined by direct counting. At the same time, the population genetic parameters were estimated using online software (les (Ne) , and polles (Ne) . The linthe loci . In ordeThe variation in vertebral number among 392 Dezhou donkeys is shown in vice versa . From the perspective of TLN, adding a thoracolumbar vertebra means the total length of thoracolumbar vertebrae (TLL) will increase significantly . Individuals with 23 or 24 TLN were about 10 kg heavier than 22 TLN individuals in carcass weight, but no significant difference was detected. Theoretically, there should be a T19L6 combinational type, but this has not been observed in our long-term observations; maybe we need a larger sample size.The number of vertebrae is associated with body length, carcass weight, and other economical traits in livestock. However, for Dezhou donkeys, the effect of the vertebrae number increase on economic traits is still unclear. To evaluate their relationship, the effects of single TL, LN, TLN, and thoracic-lumbar vertebrae combinational types on carcass weight were analyzed successfully. In order to accurately calculate the effect of adding one type of vertebrae on body length and carcass weight, we studied the effect of variations in the number of thoracic vertebrae in the same population of lumbar vertebrae and ce versa . When LNA total of 16 mutations  were identified as candidate sites for multi-vertebrae and carcass traits . In ordeLTBP2 gene are shown. As shown in In LTBP2 SNVs on the number of vertebrae, body size, and carcass traits were evaluated and are presented in p < 0.05) in c.5547 + 860 C > T, c.5251 + 281 A > C, c.3769 + 40 C > T, and c.2782 + 3975 A > G loci, which means the main effect of these four loci are significant in TN. The lumbar vertebrae number in Dezhou donkeys with wild-type homozygous and mutant-type homozygous genotypes of c.1381 + 768 T > G and c.1381 + 763 G > T was 5.05 ± 0.05 and 5.25 ± 0.03, respectively, and the difference was significant (p < 0.05). The other two mutant loci significantly related to the number of lumbar vertebrae are c.812 + 10937 C > T and c.812 + 6591 T > C. In c.812 + 10937 C > T, individuals with mutant-type homozygous T/T and heterozygous C/T genotypes had more LN than wild-type homozygotes, while in c.812 + 6591 T > C, individuals with mutant-type homozygous C/C had significantly fewer LN than wild-type homozygotes and heterozygotes. As described in p < 0.05), while others  were not significant. There are five mutational sites significantly associated with TLN . The effect of c.812 + 10937 C > T in TLN was consistent with that in LN. In c.565 + 11921 A > G, c.565 + 6840 A > G, c.565 + 3453 C > T, and c.494 + 5808 C > T mutational sites, individuals with the heterozygous genotype had the most TLN (23.12 ± 0.03) compared to other genotypes (significantly).The effects of the LTBP2 gene, the linkage disequilibrium between these loci was estimated. The analysis results showed that three blocks exist between these 16 SNVs (r2 > 0.33) (r2 < 0.33).In order to reveal the linkage relationship between the 16 mutational sites of the  > 0.33) . Block1 Haplotype analysis was performed for the three blocks, and three haplotypes with a frequency greater than 0.05 were identified and labeled as Hap1 (-CACAGTTTGGAATT-), Hap2 (-CACAGTTTGGGGCC-), and Hap3 (-CACATGCCATGGCC-). The haplotype frequencies of Hap1, Hap2, and Hap3 in Dezhou donkeys account for 37.6%, 28.3%, and 10.2%, respectively.In order to explore the association between SNV polymorphism and the traits of Dezhou donkeys, haplotype combination was carried out on the basis of constructed haplotypes. Six haplotype combinations  were constructed, respectively. The results are shown in p < 0.05). Interestingly, the effect of one more TN in different LN populations is unequal, such as in the 5 lumber vertebrae number population, the CW of individuals with 18 TN is 11.1 kg more than individuals with 17 TN, while in 6 lumbar vertebrae number population, the CW of individuals with 18 TN is 3 kg more than individuals with 17 TN; the disparity between the increase between the two is 8.1 kg. Similarly, the effect of one more LN in different TN populations is also unequal.With the development of agricultural mechanization, the causative function of donkeys is gradually weakening, which is the reason why the stock of donkeys in China decreased rapidly from 1990 to 2016 . In receThe heritability of the vertebrae is high; for example, the heritability of vertebrae in pigs is 0.62 . TherefoLTBP2 have been proved to have some functions in the development of primary congenital glaucoma  , to amplify the eight mutational loci  , cardiomrgfs*35) , and lunrgfs*35) . LTBP2 h in pigs . In this studies . The liqnal loci . The aliLTBP2 gene of Dezhou donkeys; however, most of the loci cannot meet the requirements of correlation analysis, so we finally screened out 16 SNVs for subsequent analysis. Among these loci, the adjacent loci are significantly related to a certain trait (p < 0.05), centrally. For example, for c.5547 + 860 C > T, c.5251 + 281 A > C, c.3769 + 40 C > T, and c.2782 + 3975 A > G, the distance between them was 8,247 bp, 6,018 bp, and 9,253 bp, respectively, genome wide; they were all significantly associated with TN (p < 0.05). The analysis results from the Haploview software showed a strong linkage disequilibrium between them, which means when the C allele is mutated to the T allele at c.5547 + 860 C > T, the A allele at c.5251 + 281 A > C will mutate to C allele, the C allele at c.3769 + 40 C > T will mutate to T allele, and the A allele at c.2782 + 3975 A > G will mutate to G allele. In this block, individuals with wild-type homozygous had more 0.16 pieces of TN than heterozygotes, so we speculate that the mutation here has a negative effect on TN. The same situation also occurs in block2 , but the function of block2 is aimed at LN, CHC, CWF, CWA, and CW. For the development of TN, mutations at c.1381 + 768 T > G and c.1381 + 763 G > T sites are beneficial, while for CHC, CWF, CWA, and CW, mutations at c.1003 + 704 C > T, c.1003 + 651 C > T, c.1003 + 626 A > G, and c.812 + 22526 T > G are negative. In block3, four SNVs  were all significantly associated with TLN; individuals with the heterozygous genotype had higher TLN than others (p < 0.05), but the difference between wild-type homozygotes and mutant-type homozygotes was different.We actually detected a total of 544 mutational sites in the whole LTBP2 gene, a total of three haplotype blocks among 16 SNVs were identified, examining haplotypes above 5%, spanning approximately 30 kb. The most frequent haplotypes within the blocks comprised the linkage of nucleotides CACA (96.4%) for block1, GTTTGG (72.8%) for block2, and AATT (53.6%) for block3. To demonstrate the total effect of a single locus, haplotype combinations were combined for further study. The haplotype combination types with a sample size less than 5% of the total sample were removed. Our results showed no dominant type in the four haplotype combinations; the reason may be the effect of these three blocks counteracting each other. These results indicate that the LTBP2 gene is a potential functional gene for regulating the development of vertebrae in Dezhou donkeys, and this finding may provide important implications for further studies on the regulation of the TLN in Dezhou donkeys. It must be noted that our research has not explored the molecular mechanism of SNPs in LTBP2 regarding how they effect TLN development.In order to determine the linkage disequilibrium of 16 SNV markers located on the Dezhou donkey LTBP2 gene of Dezhou donkeys. These loci have low-to-high polymorphism and there are significant differences between different genotypes among different traits. The results of this study will play an important role in the further study of the molecular mechanism of Dezhou donkey molecular breeding for muti-vertebrae breeds.In summary, the effects of TLN variation on body size and carcass traits of Dezhou donkeys were quantified, and 16 trait-related SNVs have been identified in the


We investigated prediction of malting quality (MQ) phenotypes in different locations using metabolomic spectra, and compared the prediction ability of different models, and training population (TP) sizes. Data of five MQ traits was measured on 2667 individual plots of 564 malting spring barley lines from three years and two locations. A total of 24,018 metabolomic features (MFs) were measured on each wort sample. Two statistical models were used, a metabolomic best linear unbiased prediction (MBLUP) and a partial least squares regression (PLSR). Predictive ability within location and across locations were compared using cross-validation methods. For all traits, more than 90% of the total variance in MQ traits could be explained by MFs. The prediction accuracy increased with increasing TP size and stabilized when the TP size reached 1000. The optimal number of components considered in the PLSR models was 20. The accuracy using leave-one-line-out cross-validation ranged from 0.722 to 0.865 and using leave-one-location-out cross-validation from 0.517 to 0.817. In conclusion, the prediction accuracy of metabolomic prediction of MQ traits using MFs was high and MBLUP is better than PLSR if the training population is larger than 100. The results have significant implications for practical barley breeding for malting quality. In the process of brewing, the cereal needs to be malted. Under specific controlled conditions, cereal grains are sprouted and the young seedlings are grown for four to six days in order to produce malt2. This process ensure a physical and biochemical transformation within the grain that is defined as malting1. During the process of malting, the cell-wall is degraded and protein is broken-down by hydrolytic enzymes such that the physical and the biochemical structure of the barley grain is modified in order to allow malt to be used in the subsequent stages of brewing3.Brewing for alcohol production is the major end use of malt, and barley is the primary cereal used in production of malt because of optimal content of carbohydrates, dietary fibers, protein, vitamins and minerals4. Measurement of MQ traits is expensive and labor-intensive and the MQ traits have been demonstrated to have complex inheritance6. A detailed analysis of genetic variation in MQ traits in spring barley was provided by a previous study6, where a population of 1329 spring barley lines from four breeding cycles were investigated and medium to high narrow sense heritabilities were found for the MQ traits included in this study.The quality of malt can directly affect the quality and quantity of brewed beer, thus malting quality (MQ) traits are important traits in breeding of barley to be used for malting. A series of MQ traits are defined in the malting industry. These include traits as extract yield and grain protein; alpha-amylase, beta-glucanase, beta-glucan, soluble protein, and free amino nitrogen in wort; and some physical properties such as diastatic power, viscosity, taste, flavor, haze and foam head retention7. Metabolites are typical intermediates of biochemical reactions during the growth and development at all stages of plant life8. A comprehensive view of cellular metabolites can be provided by metabolomics, which is an approach to quantify the endogenous metabolites in cells and organisms. The development of metabolomics has contributed to the molecular and biological characterization of various organisms. Especially in the area of crops, compared with animals and microorganisms, metabolomics is of great importance since the crops produce very large array of metabolites collectively9. Omics technologies like genomics, transcriptomics, and metabolomics can be used in the investigation for the biological background in different organisms10.The organic compounds in a plant are mostly produced by the plant itself so that the photosynthetic and metabolic capacity of a plant is the primary factor determining its growth potential11. NMR can produce signal intensities, which can be treated as an indicator of metabolites in a biological sample, were defined as metabolomic features (MFs)12. A total of 24,018 MFs from barley wort were investigated in a previous study where the genetic variation in the MFs was investigated using a univariate model and 8,604 MFs were found to be significantly heritable13. NMR has been recognized as one of the most powerful analytical techniques which allows detailed investigation of qualitative and quantitative characteristics of complex chemical and biological samples14.Nuclear magnetic resonance (NMR) spectroscopy is one of the technologies used to analyze many metabolites simultaneously16. PLSR was first developed for the modelling of information-scarce situations in social sciences by Wold17. It is a latent variable approach which has been used to find fundamental relationships between two matrices by modelling the inner covariance structures18. Similar with traditional regression, PLSR relates two matrices by a linear multivariate model, but compared with traditional regression, the structure of two matrices can also be modelled when using PLSR19. The use of PLSR in chemistry first started in 1980s and has increased steadily for about 40 years, due to its appealing mathematical properties20. PLSR is able to analyze data sets with a large number of explanatory variables compared to the number of observations, in cases of noisy data, multi-collinearity, and incomplete variables in both the matrix of dependent variables and the matrix of predictor variables21.The most popular regression method in the field of chemometrics is partial least squares regression (PLSR)22. In the area of animal breeding, the selection of animals with highest BV was usually based on predicted/expected BVs (EBVs) derived from the records on the animals themselves and their relatives using BLUP. The use of BLUP is also widely studied in many other areas of research where the use of mixed linear models are relevant such as plant breeding23.Best linear unbiased prediction (BLUP), which is a method allowing prediction of random effects in a mixed model, was originally developed in animal breeding for prediction of breeding values (BVs) 23. Compared with a model based on individual performance, pedigree based BLUP leads to more accurate predictions and result in larger genetic gain because it efficiently uses information from all relatives by constructing an additive genetic relationship matrix (A), under the circumstances where genetic relationships between relatives exists24. The higher the additive genetic relationship between the genotype of interest with its relatives, the more information can be gained from records of these related genotypes23. With the rapid development of biochip technology, genomic BLUP (GBLUP) has been developed and widely applied because it is easy and straightforward to be implemented since technically it just needed the replacement of the A matrix in pedigreed based BLUP by a genomic matrix (G)25. More recently, metabolomic BLUP (MBLUP) has been proposed by replacing the A or G matrix by a metabolomic similarity matrix (M) and MBLUP has been shown as an promising method26.BLUP can be used in general linear mixed models that include both fixed and random effects. The simplest case is BLUP without pedigree, where genotypic effect is treated as an independent unobservable normally distributed random variable and no relationships between individuals are considered13. With this information, it is worthwhile to investigate the role of MFs involved in the prediction of phenotypes for MQ traits. PLSR is a popular method used in the studies of metabolic profiles27, and using metabolomic BLUP (MBLUP) model gave better prediction accuracies than the BLUP model using genomic information for four of five quantitative traits investigated28.In our previous study, around 36% of MFs were found having significant heritability and among which many were found to be correlated with MQ traits in spring barleyThe objectives of this study were to: (1) investigate the possibility of prediction of phenotypes of malting quality traits using metabolomic information; (2) compare the ability of predictions using PLSR and MBLUP models; (3) study the effect of different training population size on the accuracy of prediction; (4) explore the possibility of metabolomic prediction within and across location; and (5) compare different number of components considered in the PLSR model.29.All the data used are available in a public accessible repository2). Each trial was designed as a randomized complete block comprising 20–45 lines with three replicates of each line30. The capacity of trials varied so that 564 lines were not exactly equally distributed into all the trials. The breeders reference of the lines involved in this study was not provided in the published repository29 in order to comply with business rules. Each trial included two control lines in three replications. As a consequence, testing was conducted in a number of trials within each year-location combination. In total there were 139, 214 and 215 lines tested in 2014, 2015, and 2016, respectively. Two lines were tested in all three years as standards and the 2667 plots were distributed on 564 inbred lines.In this study, a total of 2667 plots of 564 spring barley malting lines were included. These lines were part of the standard breeding program from Nordic Seed A/S. All experiments in this study were conducted on land owned by Nordic Seed. There were no animal or human experiments conducted for this research, the study also did not contain any GMO. Standard farm operating procedures were used and therefore no ethical approval was needed for this study. All the experiments involving plants adhered to plant ethics guidelines. Samples from two locations in Denmark were used, and samples were taken from each plot individually and the data covered three years from 2014 to 2016. The two locations are Dyngby (55° 56′ 57.2″ N 10° 15′ 13.8″ E) and Holeby (54° 42′ 03.1″ N 11° 27′ 07.6″ E). In both locations, the fields were divided into trials, which included 52–106 plots (8.25 m13. The wort was used to measure five MQ traits which included filtering speed (FS), extract yield (EY), wort color (WC), beta glucan content (BG), and wort viscosity (WV). The wort samples needed to be filtered first, and 20 min after filtering begun, FS was scored by measuring the height of the liquid surface in the glass (cm flow-through in 20 min). EY was the percentage of dry matter in the filtered wort. Spectrophotometer was used to determine WC following the method of European Brewery Convention (EBC)31. After the filtration, the wort samples were separated in two parts and all wort phenotypes were obtained according to the Analytica-EBC 2004 manual. Briefly, one sample of 25 ml of wort was used for WV  and EY . A second sample of 3–4 ml of wort was used for BG  and WC . Detailed description of MQ traits also can be found in a previous study by Sarup, et al.6, where the heritability estimates for MQ traits were reported as 0.51 for EY, 0.31 for FS, 0.64 for WC, 0.55 for BG, 0.49 for WV.Malt sample from each plot was milled and extracted in water in order to produce a wort as described in the previous study13. MFs used in this study were 24,018 NMR intensities which obtained from one-dimensional (1D) 1H NMR spectra. The NMR intensities were integrated over small chemical shift (δ) intervals and expressed in parts per million (ppm) in the frequency range of 0.00–11.00 ppm. An in-house custom Matlab script was used to process the spectra32. First an exponential apodization function equivalent to 0.5 Hz line-broadening was used and then Fourier transformation was applied. Afterwards, all spectra were referenced to the DSS-d6 signal, automatically phased, and baseline corrected. After visual inspection data below 0.70 ppm and above 9.00 ppm was removed as it did not contain any signal. The water peak which was in the range of 4.7–4.9 ppm, and the region of the added standard which was − 0.2 ppm to 0.2 ppm, were also excluded. The raw data was then normalized using the probabilistic quotient method33, and the spectra were aligned using icoshift35. Finally, the MFs were centered and standardized to a mean of 0 and standard deviation as 1 in order to equalize the contribution from each observation independent of signal intensity12.The preparation of NMR analysis is described in detail in Guo, et al.Two models were involved in the statistical analyses. The models were a metabolomic best linear unbiased prediction (MBLUP) model and a partial least squares regression (PLSR) model.y referred to the vector of each MQ trait, m was the vector metabolomic effects, and e was a vector of residual terms that could not be explained by the other effects in the model. In this model, m was a random vector with e was a random vector with 13, was because metabolomic information include environmental factors in addition to genomic information. M denoted the metabolomic similarity matrix built from MFs using the method as for building a genomic relationship matrix (G) computed using VanRaden method 125. Specifically, Q is a n × m matrix of adjusted, centered and scaled NMR intensities with m = 24,018  and n = 2667 . Both the build of M and the MBLUP analysis were carried out by using the “qgg” R-package36.MBLUP model was as follows:36.The (co)variance components in the MBLUP model described in the previous section were estimated by restricted maximum likelihood using “qgg” R-packageM, which equal to 1. The relative variance component due to effects of m was The total variation of each MQ trait was calculated as the sum of variance components in MBLUP model:Q, the matrix of MFs, into orthogonal scores T and loadings P:y not on Q itself but on the first t columns of the scores T, t is the number of components fitted in the model when using the package mentioned below. The number of principal component (PCs) were 2667 which was the number of observations in the metabolomic data. PC1 explained large proportion of variance in MFs and 99.99% of variance in MFs is explained by first 20 PCs as shown in Figure 21, which was used in the current study to carry out the PLS analysis.The PLSR model decompose Three different leave-set-out (LSO) cross-validation strategies, in which the whole dataset was divided into a training population (TP) and a validation population (VP), were investigated in this study based on three different hypotheses regarding factors that influence prediction accuracies. The first strategy, named SIZE, was to randomly leave out VP in order to create TP of different size; the second strategy, named LINE, was to leave out VP according to line i.e. all observations of a specific line; the third strategy, named LOC, was to leave out VP according to location.In the SIZE strategy, since the TP samples and VP samples were randomly selected, the TP contained observations on the same lines, locations and years as in VP—although not the same combinations of lines, locations and years. In the LINE strategy, one out of 564 lines was left out so that the accuracy of predicting one line from all the other lines could be investigated, this strategy is similar to prediction of new lines. In the LOC strategy, one out of two locations was left out, which means the accuracy of predicting one location based on data from the other one location could be investigated.For each strategy, cross-validation was carried out to evaluate the accuracy of metabolomic prediction of five MQ traits using MFs. In order to study the effect of different size of TP in SIZE strategy, eight scenarios were investigated in this study. These scenarios varied on TP having the size of 50, 100, 200, 500, 1000, 1500, 2000, and 2500. Since the selection of TP was random, 15 replicates were carried out when selecting the TP. For the strategy of LINE, data from 564 lines were left out separately so that 564 replicates were carried out in this strategy. For the strategy of LOC, data from one of the two locations were left in turn so each location were predicted based on the other location.In each round of the cross-validation, according to the setup of TP size, a certain number of the phenotypes were selected and then the rest of phenotypes were masked. The phenotypes of the masked samples were predicted based on the TP together with the metabolomic information. Thereafter, the correlation between phenotypes and the predicted values was calculated as the accuracy of prediction. The accuracies obtained from strategy LINE and LOC were computed based on both plot level and line mean level where means were computed both for MQ and predicted MQ. This means, two accuracies were obtained for each model in LINE and LOC strategies. In each round of the prediction in LINE and LOC strategies, the predicted values for VP in this round were collected, and then when all the rounds of prediction completed, predicted values were collected for the whole population. Afterwards, the accuracy on plot level in LINE and LOC strategies was calculated as the correlation between observed phenotypes and the predicted values of each plot, and the accuracy of line mean was calculated as correlation between average observed phenotypes and the average predicted values of each line.When applying PLSR model, compared with MBLUP model, a leave-one-sample-out (LOO) cross-validation was carried out within the TP to train the model first, and afterwards the LSO cross-validation was processed using the trained model from the preliminary LOO cross-validation. In this study, different number of components considered in the PLSR model was also compared, and the number of components were 5, 10, 20 and 50. The cumulated proportion of MF variance explained by first 60 components are shown in Figure 36 and the procedure using PLSR model was carried out by “pls” package21. In all three strategies, both MBLUP and PLSR models were applied, and the datasets utilized in each round of the cross-validation were same for MBLUP and PLSR models to make sure they were comparable.The cross-validation procedure using MBLUP was carried out by “qgg” packageIn this study, the proportion of total phenotypic variance across fixed effects in malting quality (MQ) traits that can be explained by effects of metabolomic features (MFs) was evaluated. Then the phenotype of MQ traits was predicted by using MFs through MBLUP and/or PLSR models. Different size of TP, and the different number of components considered in the PLSR models, prediction across line and location were also investigated.Table A univariate MBLUP model was applied to estimate the proportion of the total variance including potential fixed effects in each MQ trait that can be explained by the MFs. The estimates were indicators for the proportion of total variance in MQ traits that is associated with metabolites.Figure For all the MQ traits, the effect of MFs explained very large proportions of the total variance. RVCm in WC, BG and WV were similar and larger than RVCm in FS and EY.A univariate MBLUP model was used for metabolomic prediction of the five MQ traits. The cross-validation results from MBLUP model at each SIZE scenario are shown in Fig. There were four PLSR models compared regarding the number of components utilized in the model. The number of components considered were 5, 10, 20 and 50. As shown in Fig. When the PLSR model considered 5 components, the prediction accuracy was low for all the MQ traits. With increase in the number of components considered in the PLSR model, the accuracy also generally increased. The accuracy kept increasing until the number of components reached 20. However, when the number of components increased further to 50, the accuracy decreased and in some cases even smaller than the accuracy from 5 components.The accuracy from MBLUP and the maximum accuracy among four PLSR models are plotted for each SIZE scenario in Fig. A total of eight sizes, including 50, 100, 200, 500, 1000, 1500, 2000, and 2500, of TP were compared. With the increase of TP size, the prediction accuracy increased regardless of the model used. When the TP size was small, PLSR could provide better predictions than MBLUP model, while along with the increase of TP size, the MBLUP outperformed PLSR models as soon as the data size reached 500 samples. As can be observed from Fig. The second cross-validation strategy investigated in this study was LINE, in which the data from one line were masked as VP and the data from the other lines were treated as TP to predict the VP. This corresponds to predicting a new line based on metabolomic information only. There were 564 lines in the whole dataset, one line was left out and then predicted based on all other lines. This process was repeated until all lines were predicted. Since PLSR model with 20 components generally yielded highest accuracy, only this PLSR model was conducted and compared with MBLUP model in this strategy. As shown in Fig. The third cross-validation strategy investigated in this study was LOC, in which the data from one location were masked as VP and the data from the other location were treated as TP to predict the VP. There were two locations in the whole dataset, therefore, this strategy had two rounds of prediction by treating each location as VP in each round. Same with LINE strategy, both MBLUP and the PLSR with 20 components were carried out in this strategy. As shown in Fig. Metabolomic prediction using 24,018 metabolomic features (MFs) were carried for a total of 2667 plots of 564 spring barley malting lines each phenotyped for five malting quality (MQ) traits. MBLUP and PLSR models were compared. Accuracy of cross-validation was investigated by varying size of training population, also using leave-one-line-out and leave-one-location-out strategies. In addition, the number of components in the PLSR model was also studied.6. The standard deviation of most of the MQ traits in the current study were smaller than the previous study and it is expected because in the previous study, 1,329 spring malting barley lines were involved and the harvest was done in four different years and three locations6, so that the samples from the current study was a subset of the previous study. The more variation in the year and location compared with the current led to the larger variation in phenotypes.The descriptive statistics for MQ traits in the current study were similar with the previous study though the number of observations in the previous study was around three times larger than in the current studyThe variance of five MQ traits explained by MFs was explored by using a univariate model integrating a metabolomic similarity matrix and the proportion of metabolomic effects were larger than 90% for all the five traits.13, similarly to a large extent, their total variance could be explained by MFs. Potentially MQ traits of WC, WV, and BG can be predicted from the MFs because MFs explains almost all the variance in these MQ traits. While among the two traits (FS and EY) having relative lower correlation with MFs, they could not be explained to the same very high degree by variation in metabolites. One of the reasons that there was a very large proportion of total variance in BG could be explained by MFs, can be due to that BG itself is a metabolite included in the NMR peaks.The utilization of metabolomic similarity matrix in the model aimed at dissection of the total variance into a metabolomic part and a random error. The proportion of the variance of MFs shows the extent that MFs can be used to predict total variance in MQ traits. In our previous study, WC, BG, and WV were found to have significant phenotypic and genetic correlation to a large proportion of the MFs37. The fact that almost all the variation in MQ can be explained by the MFs confirmed that the MFs could be used as the objective proxy for phenotype of interest and even more valuable and meaningful when the phenotype of interest is difficult or expensive to be obtain.The direct link between metabolites and phenotypic records in biological systems provide the potential of utilizing metabolomic features as an objective proxy for phenotype data41. The higher prediction accuracy in this study is probably due to a larger number of unique genotypes in the study and the fact that the NMR was performed directly on wort and not on, for example, leaves of the developing plant. When utilizing genomic information and fitting a genomic BLUP (GBLUP) model, the accuracies of genomic prediction for MQ traits were reported as from 0.28 to 0.686. The accuracy of GBLUP in the previous study was lower than the accuracy of MBLUP in the current study can be due to metabolomic data included information on both genetic factors as well as environmental factors. Thus the metabolomic information was closer related with phenotype observations than the genomic information. Though the spring barley lines involved in the current study were from the same breeding program as the lines in the previous study6, the number of lines studied in the current study was a subset from the previous one, which can also lead to the difference in prediction accuracy. However the accuracy of GBLUP is expected to be even lower if using the same dataset as in the current study, which is smaller than and be a subset of the dataset in the previous study6.The MBLUP model used for estimation of variance components was then used for metabolomic prediction of five MQ traits. The prediction accuracies using MBLUP model were quite promising as the maximum accuracy were all above 0.7. This is higher than the previous reported prediction accuracies for metabolomic prediction of plant phenotypes13. A subset of MFs were detected as significantly heritable, and a further subset of these had significant genetic correlation with MQ traits in our previous study13.Therefore, we carried out extra analysis in order to compare the performance of MBLUP using different subsets of MFs. Three matrices were built regarding to MFs included, the matrix using all the 24,018 MFs was M, the matrix using significant heritable MFs was Ms, and the matrix using MFs which were significant heritable and also had significant genetic correlation with each trait was Mgs (varied across traits). The estimation of variance due to MFs were quite similar among the MBLUP models using M, Ms and Mgs. The accuracy of prediction for using these three MBLUP models with different training population size are shown in the Supplementary Figure One of the reasons for the high prediction accuracy using MBLUP could be because the total variance explained by MFs were large in all the five MQ traits. The phenotypes of MQ traits can be predicted very well and better than when using GBLUP, because most variation in MQ is expected to be reflected in the NMR spectra. The high accuracy also shows that there is no overfitting and MBLUP can explain and predict large part of the variation in MQ. In addition, our previous study on the genetic and phenotypic correlation between MFs and MQ traits also showed a significant correlation between them, which can also be the reason for the high prediction accuracy using metabolomic informationDrosophila, where the prediction accuracy for two behavioral traits was below 0.1 when based on GBLUP and then increased to above 0.4 when using MBLUP. Such an increase have also been found for two environmental stress resistance traits in Drosophila28. In the plant field, metabolic information was introduced into prediction of complex traits by Riedelsheimer, et al.38, where the authors presented a complementary approach to exploit large-scale genomic and metabolic information in hybrid testcrosses. The MBLUP was also investigated in a previous study42, where metabolomics data were used to predict the performance agronomic traits in wheat, and metabolomic information were found as providing strong predictive power for number of grains per spike and plant height42.The performance of GBLUP and MBLUP was investigated in 43.Four PLSR models were compared with different the number of components considered in the model. The maximum accuracies provided by PLSR models were smaller than for the MBLUP model. Increasing the number of components considered in the PLSR models generally led to the increase of prediction accuracy while when the number of components reached 50, the prediction accuracy was not larger than the one provided by the models only considered 20 components. These results indicated that the non-linear relationship between the number of components and the performance of the prediction. For all the MQ traits investigated in the current study, 20 components were already enough to provide good prediction accuracy though the exact number of components varied a bit from trait to trait. In a study of genomic selection for pork pH traits, 30 was found as the optimal number of components considered in the PLSR analysis44. The accuracy of prediction for yield traits in French dairy cattle were similar between PLSR and GBLUP models but in no case PLSR provided higher accuracy than GBLUP44. It was also reported that an increase in the number of relevant variables and observations contributed to the improvement in the precision of the model parameters, which was one desirable property of PLSR model19.PLSR has also been suggested as an efficient method to analyze genomic data, because of its ability to handle large data sets and its prediction ability, and the PLSR approach is particularly suitable to predict dependent variables from a very large number of predictors and especially the predictors might be highly correlated with each other26 analyzed a hybrid population of rice, and showed that the MBLUP model was superior to PLSR model26. A similar situation was also found when utilizing genomic information instead of metabolomic information. For example, a study on rice also investigated the GBLUP model and PLSR using genomic information, showed that the GBLUP outperformed PLSR26. The superiority of BLUP model was also found in the study on genomic prediction in French Holstein and Montbéliarde breeds45. In addition to the better performance of MBLUP, it is also easy to implement, needs low demands regarding computation power, time and skill for the breeder, which makes MBLUP is more attractive in the practical breeding. For example, the time spent on the analysis of MBLUP when predict phenotypes of one year from the other two years was around 20 s per trait, while was around 5 min when using PLSR model with 20 components. The analysis of MBLUP model was realized by applying public available R package to our data, which can be easily found online together with example codes, which do not need very complicated skills from the breeders.The comparison of MBLUP and PLSR models showed that MBLUP generally outperformed PLSR for all traits, when the TP size larger than 500. PLSR could be a better choice than MBLUP only when the TP was small. In a previous study Xu et al.47. For example, the accuracy of genomic prediction in wheat has been investigated regarding to different population sizes and the results indicated that TP of around 700 lines were enough to yield the highest prediction accuracy48.A total of eight TP sizes from 50 to 2500 were compared in this study. The results showed that the prediction accuracy generally increased with increasing TP size. Though the accuracy increased all the way from smallest TP until the largest dataset, the increase in accuracy was much smaller when the TP were larger than 1000. The impact of TP size on the prediction accuracy had been demonstrated in the genomic prediction while rare in the metabolomic prediction using metabolomic informationIn addition to the first cross-validation strategy which selecting TP randomly within the whole population, two more strategies were investigated either predict the VP from different lines or growing in different locations. The accuracy of predicting plot MQ from these two strategies were smaller than when the TP randomly selected from the whole population. The reason is because when selecting TP from the whole population randomly, the observations on the same lines and/or locations were involved in TP and VP, which increased the degree of the metabolomic similarity between TP and VP.In this study, 564 lines were harvested in three years separately, which means there was almost no lines involved in two or three years. This design created difficulty in the metabolomic prediction across year based on the current dataset. The across year metabolomic prediction could be better investigated when a dataset including overlap of lines been planted in different years is available.Records of five malting quality (MQ) traits and metabolomic features (MFs) for 2667 plots of 564 spring malting barley lines that were grown in two locations were studied. The ability of prediction based on metabolomic information was investigated.The proportion of variance in MQ traits that can be explained by effects of MFs was above 0.9 for all traits when using all the records. The phenotype of MQ traits could be predicted by MFs through MBLUP and/or PLSR models. The prediction accuracy when using MBLUP was larger than 0.7 and generally surpassed PLSR models when size of training population (TP) larger than 500. When the size of TP smaller than 500, PLSR provided better accuracy than MBLUP. The prediction accuracy increased along with increasing TP size but when the population size reached 1000, the rate of increase was very small. The number of components considered in the PLSR models can affect the performance of PLSR models and 20 was the optimal number. In addition, the prediction accuracy was also explored regarding to using the TP to predict the validation population (VP) in a different year or location. The results showed that it was possible carry out the prediction across line/location with the accuracy of plot ranged from 0.5 to 0.8, and the accuracy of line mean ranged from 0.7 to 0.9.In conclusion, it is possible to carry out prediction of phenotypes of malting quality traits using metabolomic information. MBLUP is an ideal model for the prediction when TP size larger than 500. The results from the current study indicate that barley breeders can predict MQ based on MFs from the wort and have significant implications for the practical barley breeding.Supplementary Figures.


Female fertility has gained significant attention and is incorporated in dairy cattle breeding objectives worldwide. Currently, age at first calving (AFC) and calving interval (CI) are the only traits used as indicators of fertility in the genetic evaluations of South African dairy cattle, but these traits are greatly affected by the breeder’s decisions and calving interval is available late in an animal’s life which delays selection decisions. This study investigated the possible use of additional fertility traits derived from artificial insemination records. The fertility traits examined in this study could be used in addition to age at first calving and calving interval to select for improved fertility in dairy cattle.n = 64,464) were collected from 18 South African Holstein herds using on-farm automated milk recording systems. Using a multivariate model, the data were used to estimate heritabilities and correlations among several fertility traits. The pedigree data consisted of information on 18,592 animals born between 1981 and 2013. Heritability estimates observed were low to moderate (0.02 ± 0.00 to 0.24 ± 0.00), indicating that there is some genetic basis for the explored fertility traits to warrant selection. The genetic correlations observed between fertility traits were generally favorable, with some high correlations between age at first service (AFS) and services per conception for heifers (SPCh) (0.73 ± 0.00) and between days from calving to first service (CFS) and services per conception for cows (SPC) (0.90 ± 0.01). Positive genetic correlations indicate that improvement in one trait is coupled with a correlated genetic increase in another trait. The studied fertility traits could be used in addition to AFC and CI to serve as a basis for the selection of reproduction in dairy cattle to minimize selection bias.This study aimed to derive additional fertility traits from service data and estimate their genetic parameters for their possible inclusion in the South African Holstein cattle breeding programs. Service records ( Dairy cattle farming aims to increase milk production at the lowest possible input cost. Profitability of dairy cattle not only depends on increased milk production but also on non-production characteristics such as improved fertility and health traits . High-yiHeritabilities for fertility traits are relatively low, indicating strong influence by environmental factors, which largely discouraged efforts for their genetic improvement through selection ,6. HowevIn South Africa, functional traits such as fertility are currently included in breeding programs. However, the improvement of fertility is based on AFC and CI, which are readily available from calving dates . The selThe data for this study were collected from 18 South African Holstein herds. The data were collected from herds that use the Digital Information Management System South Africa (DIMSSA) program for performance recording. The records are not available on the National Milk Recording Scheme. They are available on-farm from the recording program and are used for herd management purposes. The herds used in this study are located across the country. South Africa’s climatic conditions generally range from the Mediterranean in the southwestern corner of South Africa to temperate in the interior plateau and subtropical in the northeast. These data stored in the farmer’s database included AI service records for 15,466 cows, where an outcome of each AI event was recorded, and the veterinarian-based pregnancy diagnosis using rectal palpation. Additional information included birth dates, service and calving dates of each animal, lactation number, and dam and sire identification numbers from which heifer and cow fertility traits were calculated. Data editing was carried out using the R-CRAN program . Two subsets of data were extracted from the original dataset for heifer (1st parity only) and cow (2nd parity and above). The datasets were edited to remove outliers for each trait. Removing outliers from the dataset included deleting records below 21 days or above 250 days for CFS and below 21 days or above 435 days for DO, as these biological anomalies could have been wrongly recorded. The biggest proportion of the dataset was lost during editing mainly due to missing pedigree records, as animals without parents were excluded. Furthermore, the animals that did not meet a biologically reasonable threshold for the traits were excluded from the final analysis. Two seasons of birth or calving were defined as summer (October–March) and winter (April–September). Contemporary groups were defined as the herd-year-season of birth or calving depending on the specific trait . After editing, the final data available for statistical analysis consisted of 10,019 and 24,909 records for heifers and cows, respectively. The pedigree data were edited to remove animals with unknown birth dates, with information remaining on 14,323 animals. The structure of the data before and after editing is shown in p < 0.05) and were included in the final models for estimation of genetic parameters.Descriptive statistics were computed using the core R functions and histograms were plotted using the Hist function in R . The lmey was the vector of observations; b was the vector of fixed effects which consists of herd-year-season of birth (for heifer traits), herd-year-season of calving (for cow traits), parity fitted only (for cow traits), age at insemination fitted only for (SPCh) and age at calving fitted (for cow traits) as a covariate; a was the vector of additive genetic effects; pe was the vector of random permanent environmental effects (fitted only for cow traits); e was the vector of residual effects; X, Z, and W were the corresponding incidence matrices. The expectations of E and variances were assumed as follows:The analysis for variance component estimation was first run using the AI REML of the BLUPF90 family of programs , which mX, Z, and WcdFGR were the corresponding incidence matrices. The three-trait animal model was fitted as follows:Single chains of 200,000 cycles were launched in the THRGIBBS1F90, with the first 50,000 cycles used as the burn-in period during which the sampling process moves from the initial values of the parameters to those from the joint posterior distribution. This was followed by POSTGIBBS analysis to verify the burn-in period, determine the convergence by visual examination of plots of covariance components within each iteration and finally, the use of the posterior means to calculate heritabilities and genetic correlations. The variance components estimates from the single trait analysis were no different from the multi-trait analysis.Descriptive statistics of the data are presented in Heritability estimates for all traits ranged from 0.02 ± 0.00 to 0.08 ± 0.01 in heifers and 0.04 ± 0.00 to 0.24 ± 0.00 in cows . As expeThe descriptive statistics in The heritability estimates ranged from low to moderate. Previously reported heritability estimates for AFS were 0.24 in Dutch Friesians , 0.22 inThe SPC was defined for both heifers and cows, and the variance components differed considerably. The heritability estimate of SPC in heifers was slightly lower (0.02) than for the corresponding cow trait (0.04). These estimates were consistent with previous observations in the US Holsteins, Canadian Holsteins, Brown Swiss, and Chinese Holsteins ,33,34. TThe current estimate for DO was similar to the values of 0.05 and 0.07 reported for Chinese  and DaniThe traits included in this study, excluding AFC, are not included in South African breeding programs due to a lack of information on the traits and because their estimated breeding values are not available on a nationwide basis. However, this study shows that the traits have the potential for inclusion in breeding programs. Several studies reported that DO should be preferred over CI in breeding programs for higher genetic progress because DO has higher heritability and genetic variance than CI . This apFor success traits, heritability estimates ranged from low to moderate. The heritability estimate for the first service within 80 days post-partum was 0.07 and for the two binary traits indicating whether cows were confirmed pregnant within 100 or 200 days post-partum were 0.13 and 0.24, respectively. However, Potgieter et al. (2011)  observedResults from this study were generally in agreement with previously reported estimates for fertility traits of relatively low heritability at 10%. The exception was cow success traits. Although we observed relatively low heritability estimates in this study, there is evidence of genetic basis in the analyzed fertility traits. Therefore, improvements in nutrition and reproductive management could be coupled with genetic selection for improved fertility in dairy herds. The heifer traits AFS and SPCh are available early in the animal’s life and could be used in addition to age at first calving as fertility indicators in genetic evaluations of the South African Holstein cattle population. High positive genetic relationships observed between SPCh, AFS, and AFC indicate that younger cows conceive from fewer inseminations. Lower AFS and AFC positively affect genetic progress as generation interval decreases and it allows early progeny testing of sampled bulls. Selection for decreased AFS and AFC may be an efficient strategy for dairy farmers to reduce costs ,45. HeifGenetic and phenotypic correlations were estimated amongst the defined AI service-based fertility traits . The genPositive genetic correlations of 0.90, 0.19, and 0.70 were observed between cow fertility traits SPC and CFS, SPC and DO, CFS and DO, respectively. Although the correlation between SPC and DO was low (0.19), it was similar to the estimate of 0.21 reported by Zaabza et al. (2016) . The estGenetic correlations between heifer and cow fertility traits were positive, but the scale was wide (0.05–0.95). SPCh for heifers had the lowest correlation with SPC for cows (0.05), indicating that the number of inseminations required for a heifer to conceive is not related to the number of inseminations required for conception in cows. These results were in agreement with Raheja et al. (1989) , who repIn this study, the genetic correlation of CFS with all the fertility traits was generally higher than the genetic correlations of DO with other fertility traits. The genetic relationship of heifer traits with cow traits indicates that heifer traits could be used in selection for improved fertility because these traits are available early in an animal’s life. Jansen 1987)  pointed   pointedEstimates between cow interval and success traits were negative, ranging from −0.20 to −0.89 for genetic and −0.07 to −0.80 for phenotypic correlations, excluding a phenotypic correlation of 0.19 between SPC and FS80d. These correlations were generally favorable amongst the traits. The number of days open had the lowest genetic correlation with the number of cows being inseminated within 80 days post-partum (−0.20), although the relationship was favorable. Calving to the first service had the highest favorable genetic relationship with FS80d (−0.89), indicating that selection for a shorter interval from calving to the first service would result in a greater number of cows being inseminated within 80 days post-partum and more cows conceiving from fewer inseminations as the genetic correlation between FS80d and SPC is also favorable (−0.46).The traits CFS, DO, and SPC had favorable relationships with PD100d and PD200d, suggesting that decreasing the number of inseminations and shortening CFS and DO will result in more cows confirmed pregnant within 100 or 200 days post-partum, which means a high percentage of cows will not exceed 200 days open. The genetic correlations of FS80d with PD100d and PD200d were positive and moderate (0.39 and 0.47), demonstrating that when more inseminations occur within 80 days post-partum, more cows will be confirmed pregnant within 100 days or 200 days post-partum. This study shows that fertility traits generally exhibit low heritabilities. Although heritabilities are low, they are non-zero, indicating that there is some genetic basis to warrant genetic improvement through selection. Heifer and cow fertility traits should be treated as separate but genetically correlated traits in multi-trait genetic evaluations. The high genetic correlations among the different fertility traits reveal the relationships amongst these traits, as predictions can be made on several fertility traits after performing selection on either one of the traits. Early availability of service data on fertility traits of heifers and the desirable genetic correlation with cow fertility traits presents an opportunity for these traits to be included in national genetic evaluations for South African Holstein cattle. The parameters estimated in the present study will facilitate the development of a genetic evaluation system for female fertility traits to improve the reproduction efficiency of South African Holsteins. Artificial insemination records provide an opportunity for the inclusion of additional fertility traits in genetic evaluations and potentially in the breeding objectives of the South African Holstein dairy cattle population. Therefore, farmers are encouraged to record such information and make it available for genetic evaluations in the quest to improve the reproductive performance of South African dairy herds.


Fast, memory-efficient, and reliable algorithms for estimating genomic estimated breeding values (GEBV) for multiple traits and environments are needed to make timely decisions in breeding. Multivariate genomic prediction exploits genetic correlations between traits and environments to increase accuracy of GEBV compared to univariate methods. These genetic correlations are estimated simultaneously with GEBV, because they are specific to year, environment, and management. However, estimating genetic parameters is computationally demanding with restricted maximum likelihood (REML) and Bayesian samplers, and canonical transformations or orthogonalizations cannot be used for unbalanced experimental designs.Tilde-Hat-GS (THGS) and Pseudo-Expectation-GS (PEGS). Balanced and unbalanced experimental designs were simulated to compare runtime, bias and accuracy of GEBV, and bias and standard errors of estimates of heritabilities and genetic correlations of THGS, PEGS, and REML. Models with 10 to 400 response variables, 1279 to 42,034 genetic markers, and 5990 to 1.85 million observations were fitted.We propose a multivariate randomized Gauss–Seidel algorithm for simultaneous estimation of model effects and genetic parameters. Two previously proposed methods for estimating genetic parameters were combined with a Gauss–Seidel (GS) solver, and were called Runtime of PEGS and THGS was a fraction of REML. Accuracies of GEBV were slightly lower than those from REML, but higher than those from the univariate approach, hence THGS and PEGS exploited genetic correlations. For 500 to 600 observations per response variable, biases of estimates of genetic parameters of THGS and PEGS were small, but standard errors of estimates of genetic correlations were higher than for REML. Bias and standard errors decreased as sample size increased. For balanced designs, GEBV and estimates of genetic correlations from THGS were unbiased when only an intercept and eigenvectors of genotype scores were fitted.THGS and PEGS are fast and memory-efficient algorithms for multivariate genomic prediction for balanced and unbalanced experimental designs. They are scalable for increasing numbers of environments and genetic markers. Accuracy of GEBV was comparable to REML. Estimates of genetic parameters had little bias, but their standard errors were larger than for REML. More studies are needed to evaluate the proposed methods for datasets that contain selection.The online version contains supplementary material available at 10.1186/s12711-022-00730-w. Genomic prediction  uses genEstimation of variances and covariances can be computationally demanding with standard multivariate approaches for trials with multiple quantitative traits and environments. In restricted maximum likelihood (REML) analyses, large and dense mixed-model equations need to be stored in memory and inverted repeatedly. In Bayesian analyses, model effects need to be sampled for thousands of Markov chain Monte Carlo (MCMC) iterations. This becomes time-consuming with an increasing number of response variables, because increasingly large matrices need to be inverted and factorized in each iteration. Canonical transformation  or diagoFast and reliable algorithms are economically important in plant breeding enterprises to make timely decisions and advance the breeding pipeline. With any kind of delays during harvest season, e.g., due to weather, only a few hours may be available for selection decisions. If a breeder misses a deadline to request either new breeding crosses from nurseries or seed of selected individuals or seed of test-crosses, the generation interval increases, genetic gain per year decreases, and product launches are delayed.Tilde-Hat approach to the methods of Schaeffer ,,13], calk were initialized to m columns of k was updated by:k, k and Estimates of genetic and residual variances for environment mentclasspt{minimaentclass1pt{minimaposed by  and deri-rkas in , where \Bending of ribed in  was usedGR (2.0) , functioFor balanced experimental designs, when the intercept is the only fixed effect, and either a principal components  or eigenmentclass2pt{minimk in Eq.  becomes  in Eqs.  and 6)  \documenAs a gold standard for low biases and standard errors of both GEBV and variance components, empirical genomic best linear unbiased predictions (GBLUP)  were obtcument}G  , which analyzes phenotypes of only one environment at a time with the randomised Gauss–Seidel solver and TH, was run to evaluate the increase in accuracy of GEBV with multivariate THGS over univariate THGS. Table Phenotypic data for five scenarios were simulated to evaluate bias and accuracy of GEBV within environments, runtime, and biases and standard errors of estimates of heritabilities and genetic correlations Table . The genScenario 1 contained simulated phenotypes from inbred lines that were grown in the same ten environments, using 599 inbred lines from CIMMYT’s Global Wheat Program [Scenario 2 contained simulated phenotypes from different inbred lines grown in ten different environments, using 5142 recombinant inbred lines from the SoyNAM project [Scenario 3 was used to study the evaluation statistics for an increasing number of soy inbred lines in each of the ten environments. Thus, each line could be present in multiple environments. Scenario 4 was used to study runtime of PEGS and THGS for an increasing number of environments (response variables), i.e., 10, 50, 100, 200 and 400, using the SoyNAM dataset with a random 10% of lines missing in each environment. Scenario 5 was used to study runtime with a higher marker density, using the SoyNAM dataset and genotyped at 42,034 SNPs that were obtained from the original SNPs plus from a linkage disequilibrium-based imputation of SNPs, as described in [ Program , 29 that Program . Scenari project , 36 genok were sampled as j, and Phenotypes were simulated by summing true genomic breeding values (TBV) and residuals. TBV for environment k were calculated as Biases and standard errors of estimates of heritabilities and genetic correlations were calculated as the average and standard deviation, respectively, of estimated minus true simulated values across replicates. GEBV for environment The average runtimes for the different methods used in scenario 1 are presented in Table Figure Table Accuracy of GEBV increased with increasing heritability and genetic correlation, as expected Fig. . It was Regression coefficients of TBV on GEBV are shown in Fig. Figure Figure Figures Standard errors of estimates of the genetic correlations decreased with increasing heritability and genetic correlations Fig. . StandarAs the number of observations per environment increased in scenario 3, standard errors of estimates of genetic parameters decreased, bias of estimates of genetic correlations decreased, but bias of estimates of heritabilities did not approach zero even with 3000 observations per environment Table . AdditioTable Our main goal was to develop an algorithm for multivariate genomic prediction that is efficient in runtime and memory, applicable to unbalanced experimental designs, and exploits genetic correlations between environments to increase the accuracy of GEBV compared to univariate analyses. We proposed two algorithms, PEGS and THGS, that use randomized Gauss–Seidel to estimate marker effects and simultaneously estimate variance components, based on methods developed by , 13, resPEGS and THGS were shown to be fast and memory-efficient algorithms for both balanced and unbalanced experimental designs, and had a much shorter runtime than REML using standard software implementations Tables  and 4. Mentclass1pt{minimaThe number of iterations to convergence Fig.  and runtFor balanced designs, the number of iterations to convergence can be further reduced by modeling the eigenvectors of genotype scores, which completely removes dependencies among model effects. In addition, THGS becomes an exact method that yields unbiased estimates of genetic correlations and GEBV (section Exact THGS), and reduces the bias of estimates of heritabilities, as can be demonstrated for scenario 1 Table . Matrix pistasis , and Gaupistasis , 38. TheThe trade-off for higher speed with PEGS and THGS is a slightly lower accuracy of GEBV of 0.01 compared to REML under realistic conditions when heritability was low and genetic correlations between environments were medium to high Fig. a. PEGS aResiduals were treated as uncorrelated between environments for three reasons. First, the phenotypes come from different individuals that are assumed to have uncorrelated environmental effects. Second, epistatic effects, which are not captured by the marker effects in the model of Eq. , are ass in Eqs.  to 7), , \documeentclass1pt{minimaEstimates of variances and covariances obtained by the methods PE and TH are unbiased when the mixed-model equations are weighted by the true variances and covariances as shown in Additional file from Eq. . For simformulas , the quaPEGS and THGS should be evaluated against alternative methods for modeling phenotypes from multiple environments. These are compound symmetry and extended factor analytic (XFA) models . CompounThe iterative algorithm of PEGS and THGS differs from that of REML and Bayesian Gibbs sampling. In each iteration of REML, the mixed-model equations are fully solved to obtain estimates of the model effects conditional on the current variance components of that iteration. The estimated model effects are then used to update the variance components and a new iteration begins, unless the change in variance components is small. In PEGS and THGS, in contrast, the model effects are merely updated, not solved, before variance components are updated and a new iteration begins. In Bayesian Gibbs sampling, similar computations are conducted in each iteration as in PEGS and THGS. However, rather than converging directly to a solution within a small number of iterations, the Gibbs algorithm samples from the posterior for thousands of iterations and, therefore, must have longer runtimes.PEGS and THGS are fast, memory-efficient, and reliable algorithms for genomic prediction for both balanced and unbalanced experimental designs. They are scalable with an increasing number of response variables and markers. Their runtime is much shorter than for REML and Gibbs sampling. For balanced designs, THGS provides unbiased GEBV and estimates of genetic correlations if only an intercept is modeled, and eigenvalue decomposition is feasible. Without eigenvalue decomposition, the accuracy of GEBV obtained using PEGS and THGS is slightly lower than of GEBV obtained using REML, but higher than that of univariate THGS under realistic genetic correlations between environments. Estimates of genetic parameters obtained using PEGS and THGS have little bias, but their standard errors are larger than for REML. More studies are needed to evaluate the PEGS and THGS algorithms for unbalanced datasets with selection.Additional file 1. An R implementation of PEGS.Additional file 2. Deterministic calculations to study estimators of variance components using Tilde-Hat or Pseudo-Expectation.Additional file 3. Scenario 1 with more environments.Additional file 4. Estimated covariances for different degrees of balanced data.Additional file 5. Summary of scenario 2 and balanced case.Additional file 6. R code demonstrating unbiasedness of PEGS.