Complementing or replacing genetic markers with transcriptomic data and use of reproducing kernel Hilbert space regression based on Gaussian kernels increases hybrid predictionaccuracies for complex agronomic traits in canola.Brassica napus L./canola) employing different omics profiles, individually and in combination. To this end, a population of 950 F1 hybrids was evaluated for seed yield and six other agronomically relevant traits in commercial field trials at several locations throughout Europe. A subset of these hybrids was also evaluated in a climatized glasshouse regarding early biomass production. For each of the 477 parental rapeseed lines, 13,201 single nucleotide polymorphisms (SNPs), 154 primary metabolites, and 19,479 transcripts were determined and used as predictive variables. Both, SNP markers and transcripts, effectively predict hybrid performance using (genomic) best linear unbiased prediction models (gBLUP). Compared to models using pure genetic markers, models incorporating transcriptome data resulted in significantly higher prediction accuracies for five out of seven agronomic traits, indicating that transcripts carry important information beyond genomic data. Notably, reproducing kernel Hilbert space regression based on Gaussian kernels significantly exceeded the predictive abilities of gBLUP models for six of the seven agronomic traits, demonstrating its potential for implementation in future canola breeding programmes.In plant breeding, hybrids gained particular importance due to heterosis, the superior performance of offspring compared to their inbred parents. Since the development of new top performing hybrids requires labour-intensive and costly breeding programmes, including testing of large numbers of experimental hybrids, the prediction of hybrid performance is of utmost interest to plant breeders. In this study, we tested the effectiveness of hybrid prediction models in spring-type oilseed rape (The online version contains supplementary material available at 10.1007/s00122-020-03759-x. F1 heterosis  best linear unbiased predictions (gBLUP) and reproducing kernel Hilbert space regressions (RKHS) based on a Gaussian kernel for prediction of hybrid performance under field and glasshouse conditions. We focused on the following questions: (1) can hybrid performance in the field be effectively predicted using parental line genotype information, (2) can other parental omics data sets collected in controlled conditions be utilised to effectively predict hybrid performance and how do their prediction accuracies compare with that of genetic markers, (3) can prediction accuracy be increased by stacking multiple omics sets and which combinations are the most promising for which set of traits, (4) can per se line and hybrid performance in the glasshouse be effectively predicted using the omics data sets, and (5) can higher prediction accuracies be achieved for agronomic traits in canola by employing reproducing kernel Hilbert space regression models compared to gBLUP?B. napus (canola) genotypes with double-low seed quality (low erucic acid and low glucosinolate content) from a previously described breeding programme  from a pool of testers carrying the Male Sterility Lembke (MSL) sterility system .The experimental materials consisted of 477 spring-type Brassica Infinium 60k genotyping array . Genotyping data, including single nucleotide polymorphisms (SNPs) and copy-number variations (CNVs), were generated using the ‘gsrc’ R package . An improved version of the Brassica napus cv. Darmor v4.1 reference genome assembly  for each trait are estimated by Eq. n = 4) per genotype  and Deutsche Saatveredelung AG (DSV). Hybrids were evaluated at eight different locations in Denmark , Germany , Poland , Latvia (Jelgava) and Estonia (Viljandi), chosen according to the origin of the crosses ; thus, each hybrid was represented at least at three locations (Data S1). All hybrids were tested in one common location in Poland (Krzyżewo). Four commercial lines  were included as standards at all locations. These standards were analysed as duplicates in all but one trial. Various traits of agronomic importance were evaluated, including the content of total seed glucosinolates , the days to onset of flowering , the seed oil yield (dt/ha), seedling emergence , the seed oil content (% volume per seed dry weight), the seed protein content (% volume per seed dry weight) and seed yield (dt/ha). Best linear unbiased estimators for each agronomic trait  were calculated using a linear mixed model Eq. . In the dels Eq.  and RStuPlants were cultivated and phenotyped in the IPK phenotyping facility for large plants , between seven to nine hours after illumination, and immediately shock-frozen in liquid nitrogen. Plant material was homogenised in a scintillation vial  for 2 min at −60 °C using two 8-mm steel balls and a cryogenic plant grinding and dispensing system . The material of four individual plants per carrier was combined and equal amounts of plant material from all carriers from the different experiments were pooled. Samples from the first phenotyping experiment were omitted, as a breakdown of the cooling system in the glasshouse caused elevated temperatures during this experiment and an increased developmental speed, which might have biased the results. 15 ± 1.5 mg fresh weight were dispersed in 1.4 ml storage tubes  for metabolite profiling. For transcript profiling, 50 ± 1.5 mg fresh weight were manually weighed. Material was stored at −80 °C until use.v/v MeOH/CHCl3/H2O) and 0.250 ml H2O after incubation. Extraction was split into six batches with 96 samples each and aliquots of 50 μl of polar phase were sampled. The dried extracts were in-line derivatised directly prior to injection according to Erban et al. . Quantitative peak information was extracted with the R package ‘TargetSearch’  file using Hisat2 v2.0.4  according to the manufacturer’s protocol and eluted in 50 µl nuclease-free water. RNA quantity and purity were assessed using a NanoDrop One Microvolume UV–Vis Spectrophotometer, and RNA integrity number (RIN) was checked for each fourth RNA sample using a Bioanalyzer 2100. RNA was diluted to 100 ng/µl and 30 µl were used for sequencing in randomized order. cDNA libraries were constructed using the Lexogen SENSE mRNA-Seq Library Prep Kit V2 . Sequencing was performed using 100 bp single end (SE) reads on a HiSeq 2500 platform . In total, 27 flow cell lanes on five high-output and rapid-run flow cells were used, resulting in a total of approximately 520 Gbases/4.8 billion single-end reads, covering each genotype with on average approximately 9.5 million reads. More than 96% of all 477 samples could be covered with at least 7 million reads. Adapter-trimmed raw reads were further quality trimmed using the Trimmomatic software v0.36  times ‘number of features’ . The columns of these predictor matrices were centred and standardised to result in values between 0 and 2 (G) and 0 and 1 (T and M), respectively. The in silico generated hybrid profiles were used to calculate the realised relationship matrices  best linear unbiased prediction , n the number of hybrids, μ a vector of fixed effects that represent the overall mean. In the model n × 1 vectors of random effects, and The gBLUP as well as the RKHS models were implemented using R . The corresponding bandwidth parameters were estimated from the respective log-likelihood profile generated using the kin.blup function of the ‘rrBLUP’ R package  defined by Eq. . Here \dEndelman .3\documeŷ) and observed phenotypes (y). Differences in prediction accuracy between the data sets and combinations were analysed using an analysis of variance (ANOVA), followed by a post hoc Tukey test implemented using the R packages ‘stats’ and ‘agricolae’.Predictions were performed for the seven agronomic traits (BLUEs) of the 950 hybrids, as well as for growth-related traits and biomass of the 120 hybrids, which were phenotyped in the glasshouse experiment. Three sets of predictors  were generated for the parental lines . Predictions were performed for each set of predictors  separately, and in combinations of two or three . A cross-validation (cv) scheme with 1000 cv-cycles was applied, separating the data in a training set (75%) and a validation set (25%). The phenotypes (BLUEs) of each of the 1,000 validation sets, always containing both hybrids of randomly selected pollinators (e.g. Pol1 × MS1 and Pol1 × MS2), were masked and predicted. Prediction accuracies were obtained as average Pearson product-moment correlation coefficients between predicted  of the genetic and transcriptomic markers were sampled, respectively. These reduced predictor sets were subjected to the same procedures as described above to compare the predictive abilities of the metabolic features to genetic and transcriptomic markers based on the same number of features for each data set. In addition, the predictive abilities of copy number variation marker (n = 3110) data and time-resolved image-derived paternal phenotype data  and the mean value of the two parents [MP = (P1 + P2)/2] for end-point biomass, projected leaf area, estimated biovolume, and early plant height at all time points with available data (Eq. 1) and the better performing parent (Eq. Mid-parent heterosis (MPH in %) was calculated as difference between hybrid performance , a method for constructing a low-dimensional embedding of high-dimensional data.The analysis was performed using the panel of 13,201 SNP and 3110 CNV markers and a Barnes-Hut implementation of t-SNE (Rtsne function) provided by the ‘Rtsne’ R package. The following parameters were used: dims = 2, perplexity = 50, theta = 0, pca = T, and max_iter = 1000. Principal component analyses (PCA) were performed for metabolites and transcripts on centred and scaled data using the pca function of the ‘pcaMethods’ R package  were calculated, as in particular the raw data for seed oil yield, DTF and seedling emergence displayed substantial differences between the locations. The BLUEs of all seven traits followed a proximate normal distribution , while seed oil content and seed protein content displayed a strong negative correlation (r = −0.75). Broad sense heritability values (H2) estimated across the different trials/field locations ranged from 0.34 for the trait seedling emergence to 0.92 for total seed GSL content. Field trials of 950 FThe 477 parental lines were previously genotyped , transcripts  and metabolites (n = 154). These data sets were used individually and in all possible combinations for prediction analyses; prediction accuracies are illustrated in Fig. n = 10,000, the transcripts still yielded higher prediction accuracies than the SNP markers. However, in contrast to the initial hypothesis, no significant increase in prediction accuracies could be achieved by stacking data of all three omics data sets.Using the best linear unbiased estimators (BLUEs) of all seven agronomic traits, (genomic) best linear unbiased predictions (gBLUP) were performed with omics data sets, comprising SNP markers n = 1,201, traCombining SNP and CNV markers resulted in a significant improvement of prediction accuracies using gBLUP models for three traits: DTF, GSL, and in particular seed yield, increasing the mean prediction accuracy by 1.3% compared to SNP markers only. However, the beneficial effect of adding CNVs was negligible in the stacked models including genetic, transcriptomic, and metabolite data (Data S2).Dominance and epistasis are known to be an important source of non-additive genetic variance for many agronomic traits. Hence, we explored the effect of a dominance and a second order additive × additive epistatic genomic relationship matrix in our gBLUP models for prediction of agronomic traits.Combining dominance and/or epistatic relationship matrices, calculated based on SNP markers, with the additive genomic relationship matrix significantly increased the prediction accuracies for five traits . The highest increase in median prediction accuracy was observed for seed protein content (2.1%), seed oil yield (3.7%) and seed oil content (4.2%). The specific combination yielding the highest prediction accuracy differed between traits. A detailed comparison of the performance of the individual matrices and combinations is given in Data S2. Stacking of the additive, dominance and epistatic genomic relationship matrices with transcript and metabolite data did further improve prediction accuracies for seed protein content, DTF, seed oil content, and GSL (Data S2).In addition to gBLUP prediction  with hybrid seed yield in the field. The collection of phenotypic data for these hybrids in combination with the data of the 477 parental lines provided the basis to calculate best-parent heterosis (BPH) and mid-parent heterosis (MPH) values. Five traits with high heritability values were chosen: end-point biomass (fresh weight and dry weight), projected leaf area, estimated biovolume, and early plant height. Overall, far more positive than negative mid-parent values heterosis were detected, and even for BPH a trend towards positive heterosis was observed . For projected leaf area and estimated biovolume, calculations have been performed individually for all 21 days on the basis of BLUEs across all five phenotyping experiments. Strong positive, as well as negative MPH could be detected ranging from −29.0 to 64.6% for projected leaf area , from −40.8 to 122.5% for estimated biovolume  and from −22.9 to 43.3% for early plant height . Determined BPH values ranged from −36.4 to 47.9% for projected leaf area , from −53.3 to 63.3% for estimated biovolume  and from −27.4 to 37.3% for early plant height . MPH values ranged from −27.2 to 72.3% for FW  and from −23.2 to 73.0% for DW, respectively . BPH values ranged from −39.6 to 41.3% for FW  and from −32.7 to 45.1% for DW , respectively. Distinct differences in the value distributions were detected when the 120 lines were grouped into ‘good’ and ‘bad’ hybrids with respect to seed yield data from the field trials. The set of ‘good’ hybrids displayed significantly higher MPH , as well as BPH values  for all five traits  compared to the set of ‘bad’ hybrids. Hybrids originating from crosses with MS1 displayed significantly higher MPH values for leaf area and biovolume than crosses with MS2 (Data S1). In addition, performance differences regarding the agronomic traits were observed between the breeding pools when crossed to the two different MS lines (Data S3).Five phenotyping experiments were performed in a climatized glasshouse with the 477 parental genotypes and a selection of 120 of the 950 hybrids . Both methods showed the same trends regarding the ranking of the predictor sets and combinations. In the following, we focus on gBLUP and early biomass production.Predicting parental biomass with parental data, no increased prediction accuracy was achieved by stacking all three omics data sets. However, a clear pattern with the highest prediction accuracies for the models including the transcriptome data could be observed for parental fresh and dry weight Fig. a, b. ThePredicting hybrid biomass with parental data, it was hypothesised that the parental transcript and metabolite profiles might reflect more closely the hybrid biomass production, as parental lines and selected hybrids were grown in the same facilities under the same controlled environmental regime. Prediction analyses were performed with the same models and the predictor sets and their combinations as described above. In addition, parental phenotypic data  were also used as predictors. In contrast to the parental line performance, highest prediction accuracies (gBLUP) of 0.595 (FW) and 0.646 (DW) were achieved, using the genomic data set  was chosen for sampling as data obtained at an early growth stage harbour important information for the prediction of hybrid performance  or seed yield (H2 = 0.62), could only be predicted with low to moderate prediction accuracy. In contrast, traits with a high heritability like seed oil content (H2 = 0.90) and total seed GSL content (H2 = 0.92) were predicted with high accuracy. The median prediction accuracies, ranging from 0.25 (seedling emergence) to 0.72 , reflected the differences in genetic architecture of the traits. Seed yield is a highly polygenic trait heavily influenced by Prediction accuracies strongly correlated with trait heritability, which is consistent with previous observations that prediction accuracies depend on many factors including trait heritability, genetic complexity of the trait, density of genetic markers, and quality of the input phenotype data  did not influence the ranking of the predictors in our study; the polar leaf metabolites still show the lowest prediction accuracies. It seems likely that the analysed set of metabolites and/or their low number does not sufficiently cover the pathways associated with the traits of interest and underlying genetic factors and their interactions, or that a higher variability in the metabolite levels compared to the other omics data sets causes the overall lower prediction accuracies. These observations are in accordance with results of Westhues et al.  and the transcriptome  data showed particularly high predictive abilities. These results indicate that the number of genetic/transcriptomic markers was sufficient to cover most causative genes and their effects. This observation is supported by a recent study in maize , also reflected in the ability of the parental phenotype data to predict hybrid biomass in the glasshouse. This correlation differed substantially between hybrids with ‘good’ (r = 0.62) and ‘bad’ (r = 0.21) seed yield in the field, pointing to a link between the per se performance of parental lines and biomass for at least a subset of the hybrids. These findings and the positive correlation between hybrid seed yield and hybrid biomass (r = 0.52) indicate that biomass production is positively linked to seed yield in canola, which is in concordance with previous studies . Prediction accuracies for ‘projected leaf area’ and ‘estimated biovolume’, scored between 6 and 21 DAS, were relatively low for early time points, but increased over time and reached saturation at a value of approximately 0.6 for both traits. This observation may reflect maternal effects in the earlier phases of plant growth and/or environmental effects during seed formation. The effects diminish as soon as plants establish new leaves and shift from drawing nutrients from the storage tissue to own photosynthesis, as observed in Arabidopsis seeds and young seedlings Supplementary material 2 (XLSX 156990 kb)Supplementary material 3 (XLSX 8863 kb)Supplementary material 4 (PDF 20 kb)Below is the link to the electronic supplementary material.


Genetic and environmental covariances between pairs of complex traits are important quantitative measurements that characterize their shared genetic and environmental architectures. Accurate estimation of genetic and environmental covariances in genome-wide association studies (GWASs) can help us identify common genetic and environmental factors associated with both traits and facilitate the investigation of their causal relationship. Genetic and environmental covariances are often modeled through multivariate linear mixed models. Existing algorithms for covariance estimation include the traditional restricted maximum likelihood (REML) method and the recent method of moments (MoM). Compared to REML, MoM approaches are computationally efficient and require only GWAS summary statistics. However, MoM approaches can be statistically inefficient, often yielding inaccurate covariance estimates. In addition, existing MoM approaches have so far focused on estimating genetic covariance and have largely ignored environmental covariance estimation. Here we introduce a new computational method, GECKO, for estimating both genetic and environmental covariances, that improves the estimation accuracy of MoM while keeping computation in check. GECKO is based on composite likelihood, relies on only summary statistics for scalable computation, provides accurate genetic and environmental covariance estimates across a range of scenarios, and can accommodate SNP annotation stratified covariance estimation. We illustrate the benefits of GECKO through simulations and applications on analyzing 22 traits from five large-scale GWASs. In the real data applications, GECKO identified 50 significant genetic covariances among analyzed trait pairs, resulting in a twofold power gain compared to the previous MoM method LDSC. In addition, GECKO identified 20 significant environmental covariances. The ability of GECKO to estimate environmental covariance in addition to genetic covariance helps us reveal strong positive correlation between the genetic and environmental covariance estimates across trait pairs, suggesting that common pathways may underlie the shared genetic and environmental architectures between traits. Phenotypic covariance between pairs of traits can be partitioned into two components: genetic covariance and environmental covariance. Effective partitioning of phenotypic covariance and accurate estimation of the genetic and environmental covariances can help us understand the relationship between traits and shed light on the causal and mediation mechanisms underlying disease etiology. Here we present a new computational method, GECKO, for estimating genetic and environmental covariances using summary statistics from GWASs. GECKO improves covariance estimation accuracy upon previous methods and provides environmental covariance estimates in addition to genetic covariance estimates. We illustrate the benefits of GECKO through extensive simulations and applications to 22 traits collected from five GWASs. Phenotypic covariance between pairs of traits describes how one trait varies with respect to another in the population. Phenotypic covariance can be naturally partitioned into two components: the genetic covariance, which represents the part of the phenotypic covariance explained by common genetic factors; and the environmental covariance, which represents the part of the phenotypic covariance explained by common environmental factors . The genA standard statistical model for estimating genetic and environmental covariances in genome-wide association studies (GWASs) is the multivariate linear mixed model (mvLMM) . mvLMM eDespite the popularity of MoM algorithms for covariance estimation, however, two important limitations of MoM exist. First, MoM algorithms are often statistically inefficient and produce less accurate parameter estimates when compared to the likelihood-based REML approach . Indeed,Here, we present a new method, for estimating both genetic and environmental covariances using GWAS summary statistics, that improves estimation accuracy of MoM algorithms while keeping computation in check. Our method relies on composite likelihood, is scalable computationally, uses only summary statistics, provides accurate genetic and environmental covariance estimates across a range of scenarios, and accommodates SNP annotation stratified covariance estimation. We refer to our method as Genetic and Environmental Covariance estimation by composite-likelihood Optimization (GECKO). We illustrate the benefits of GECKO with simulations and applications to 22 traits collected from five large scale GWASs.y1 and y2 as the two phenotype vectors, measured on n1 and n2 individuals, respectively. We denote ns as the number of overlapping individuals who have both phenotypes measured. In the case of two separate studies, the sample size n1 generally does not equal to n2 and ns = 0. In the case of a single study with completely overlapping individuals, the sample size n1 = n2 = ns. Besides phenotypic measurements, all individuals have their genotypes measured on a common set of m SNPs. We denote the n1 by m matrix X1 as the genotype matrix for the n1 individuals measured with the first phenotype and denote the n2 by m matrix X2 as the genotype matrix for the n2 individuals measured with the second phenotype. To facilitate computation, we center and standardize each phenotype vector as well as each column of the genotype matrices to have a mean of zero and a standard deviation of one. We consider the following regression model to related phenotypes to genotypesβ1 and β2 are m-vectors of genotype effect sizes for the two phenotypes, respectively; and ϵ1 and ϵ2 are vectors of environmental effects/residual errors, with dimension n1 and n2, respectively.We consider an mvLMM  to modelj’th SNP on the two phenotypes follow a multivariate normal (MVN) distribution a priorig represents the genetic covariance, which characterizes the phenotypic covariance explained by genetic effects. We further define the genetic correlation as We follow the standard mvLMM assumption and assume that the genetic effect sizes of ns individuals who have both phenotypes measured, we assume that their environmental effects for the i’th individual followe represents the environmental covariance, which characterizes the phenotypic covariance explained by environmental effects. The environmental correlation is further defined as 1i for the i'th individual follows a normal distribution N is Xdj represents j’th column of the genotype matrix Xd. We derive the marginal distribution for the two z-scores of each j’th SNP based on mvLMM. Such marginal distribution is in the following form . The. The3). ρ = ρe+ρg; j’th SNP computed based on a reference panel, with rij being the correlation coefficient between j’th SNP and i’th SNP.Here wj, a j’th SNP specific weight. The weight wj is critical for ensuring estimation accuracy of the composite likelihood based algorithm in the presence of LD We are sorry that we cannot be more positive about your manuscript at this stage. Please do not hesitate to contact us if you have any concerns or questions.Yours sincerely,Michael P. EpsteinAssociate EditorPLOS GeneticsHua TangSection Editor: Natural VariationPLOS GeneticsReviewer's Responses to QuestionsComments to the Authors:Please note here if the review is uploaded as an attachment.Reviewer #1: This is a great paper with contributions to an important problem. Overall, the manuscript is well-structured and clearly written. Although I have some concerns and questions about the technical details of GECKO and some analyses presented in the paper, all my comments should be addressable in principle.Major comments:1. It was not immediately obvious to me if GECKO assumes the design matrix (X) to be fixed or random. Methods for genetic correlation estimation  have not been consistent on this issue. My recollection is that the GREML approach and GNOVA assume a fixed design while LDSC is based on a random design. While the field seems to have accepted that genetic correlation estimation is robust to these two different designs , whether this will affect the estimation of environmental covariance is unclear. Under a random design, the intercept of LDSC is the phenotypic covariance of two traits. Under a fixed design, I am under the impression that the intercept may actually be the environmental covariance instead. When the goal is to estimate the genetic covariance, such differences does not matter since the slope in LDSC is always a good estimator for rho_g. But the interpretation of intercept is completely different under two model assumptions. It will be important to clarify the assumptions in GECKO and provide some discussions on whether the environmental covariance is robust to different assumptions.2. The field has not paid as much attention to the environmental covariance compared to genetic covariance. One reason is that the environmental covariance is more susceptible to technical artifacts (such as insufficiently controlled confounding) in the GWAS summary statistics. In fact, this was a highlight in genetic covariance estimation since if there are any technical correlations between two sets of GWAS, such correlations will most likely be independent from LD. The first LDSC paper on contrasting confounding from polygenicity is a good example on this issue (in the single-trait context). Suppose there is unadjusted population stratification in a GWAS, such inflation is randomly applied to SNPs in the genome while true polygenic effects are stronger in genomic regions with stronger LD. Therefore, all the technical bias will be absorbed into the intercept term of single-trait LDSC. The multi-trait extension of this problem is that technical correlation of two traits caused by population stratification will show up in the intercept of LDSC. I have been using LDSC to illustrate the problem since it's a famous method but I don't think GECKO is immune to the exact same issue. If you run two GWASs without adjusting for PCs, and apply GECKO to their sumstats, I expect the "environmental covariance" estimates to be severely biased while the genetic covariance estimates will remain solid. If the authors can come up with an approach to address this problem, it will improve this manuscript. Otherwise, at least treat this as a serious problem and list it as a limitation.3. Only ~600k SNPs were used in the real data analyses which seems low. I wonder whether the somewhat heuristic composite likelihood approach in GECKO is still effective when there are dense SNPs with extensive LD in the analysis. This is important considering that both simulation and real data analysis in this paper were based on relatively low numbers of SNPs. LDSC also requires trimming of sumstats by taking the overlap with HAPMAP SNPs while GNOVA uses all available SNPs. After all, if there are few SNPs with minimal LD in the data, genetic and environmental covariance estimation becomes a much simpler problem. The tricky issue really is LD here.4. It is helpful that GECKO can partition genetic covariance by annotation. It will be helpful if the authors can provide some discussions on whether the procedure can be easily generalized to overlapping annotations.5. The inflated type-I error of GNOVA illustrated in Table 1 is striking and also inconsistent with the original publication of GNOVA. However, these results are also inconsistent with what is shown in Figure 2. There, when the true genetic covariance is 0, all three approaches seem to have comparable type-I error rates. Why the contradiction?Minor comments:6. Figure 2's panels are labeled differently between the main text and the actual plot. See line 403 on page 24. I think the authors meant Fig2D and Fig2E instead of panels B and D.7. In the real data analysis, is there any trait pair showing significant genetic and environmental covariance but with opposite signs? How to interest these results?8. Fig1E seems to suggest that LDSC had noisy estimates on genetic covariance but it contradicts Fig1F.9. In the simulation corresponding to Fig2C, if we apply GECKO, should we expect to see near-null results with a "power" of 5%? In practice, it may not be obvious if two GWASs have overlapping samples. Consequently, it may not be easy (or even possible) to distinguish a lack of environmental covariance and a lack of sample overlap. It may be helpful to add some discussions on this issue.https://www.nature.com/articles/s41588-020-0653-y) uses a likelihood-based approach to improve the efficiency of estimation and SUPERGNOVA (https://www.biorxiv.org/content/10.1101/2020.05.08.084475v1.abstract) de-correlates the summary statistics first and then apply standard regression analysis. In addition, trans-ethnic estimation is an important topic and a clear future direction (https://www.biorxiv.org/content/10.1101/803452v3). Since these are very recent papers, it wouldn't be fair to ask for comparisons between GECKO and these methods. But some discussions regarding these new developments will improve the manuscript.10. There have been some recent developments on genetic covariance estimation. For example, HDL . The authors performed extensive simulations to validate their approach and demonstrate its improved performance. They then followed with application to real data as an illustration. The idea is interesting, and I found the manuscript extremely well written and straightforward to follow. I detail my main comments below.Major Comments:1. The authors performed a considerable number of simulations demonstrating the mostly improved performance of GECKO with respect to LDSC and GNOVA. However, all simulations were performed essentially under the correct generative model, with no inspection of performance under model misspecification. A few scenarios come to mind:a) LD pattens match exactly between the focal population and LD score computation. To investigate this point further, can the authors showcase the performance of all three methods when 1000G LD information is used to compute LD scores?b) How does GECKO  perform under misspecification of n_s?c) How does stratification affect the performance of GECKO? A straightforward simulation would be to generate both phenotypes as a function of several leading PCs, in addition to the remaining genetic + environmental model.Minor Comments:1. MoM estimators are unbiased, and it is not unexpected to see estimates largely agree across all three methods. However, as the authors note, GECKO’s development was partly motivated by improving statistical power. The use of MSE ratios is interesting to get a relative sense of performance, but still find it difficult to judge what my expectation should be here . I appreciated the authors decision to include power calculations in the non-stratified setting given its clear interpretation but noticed their absence in the functional stratified simulations.2. Replace “*” with “x” in estimates/p-values.Reviewer #3: Gao et al. presented GECKO, a statistical method that aims to estimate both genetic and environmental covariances between different traits using GWAS summary statistics. The key idea of GECKO is to approximate complicated joint likelihood with a relatively simple composite likelihood, which holds the potential to be computationally fast if the weight parameters in the composite likelihood are correctly specified. The authors evaluated the performance of GECKO by simulations as well as the analysis of 22 traits collected from five large scale GWASs.I have the following concerns about this paper.Main comments:1. Simulations: While the authors have simulated >200 scenarios, these mostly reflect the small step size used to evaluate the impact of genetic covariance (-0.4 to 0.4) or environmental covariance (-0.4 to 0.4), whereas the other key parameter, e.g., heritability, which is also extremely important, was set at constant values. The values used in the simulations do not seem to reflect those seen in real data that the authors showed later on. Overall, I feel that the current parameter values used in simulations are too different from what one would expect from real data, which make it difficult to judge whether these simulation results are useful. To get a better sense on the performance of GECKO, I think it is important to simulate data that are more representative of real data. I would suggest the authors to do the following:a) Estimate heritability for the 22 traits collected from real GWASs. Estimate genetic covariances and environmental covariances using these real data. Use these estimates as a guideline to determine the range of parameter values for simulations.b) The data were simulated from the true model. What if the model is mis-specified? In particular, what if the weights are not optimal? The authors should illustrate that GECKO is robust to misspecification of the model and weights.c) Evaluation of type I error at 0.05 level seems to be too liberal. I am curious to see if GECKO still has controlled type I error rate at a more stringent significance level, e.g. 0.005.2. Real data analysis:a) Please show the heritability for the 22 traits. This will help readers interpret the results and connect with simulations.b) I am surprised that all estimated covariances are very small. However, even for such small covariances, they are highly significant. For example, the positive genetic covariance between BMI and WBC . What is the biological significance of such a small but highly significant covariance? Can you also show estimation error of the estimated covariances? What I am surprised is that all of the estimated covariances are extremely small. I am not sure that these results are biologically significant. The authors should provide better interpretation of the results and show that these estimates are reliable.c) Since 223 pairs of traits were tested, multiple testing issue should be considered. I am also curious to see the performance of GEKCO at a more stringent significance threshold. Considering correlation among the 223 pairs, 100 independent tests might be a good start. With this, using Bonferroni, the threshold will be 0.05/100=0.0005.3. The methods section should include more details on how the weights are estimated as this is the key of GECKO.4. In terms of comparison, shouldn’t you also compare with mvLMM?5. Can you comment on the computing time of GECKO? How does it compare with other compared methods with respect to computing speed?6. Can you comment on how to use GECKO for admixed populations?Minor comments:1. Line 404: How did you compare the power of different methods at fixed type I error rate?2. Can you comment on why the relative powers of different methods are different across study designs? What is the intuition behind this difference?3. For real data analysis, it would still be interesting to include GNOVA’s estimates for covariance.**********Have all data underlying the figures and results presented in the manuscript been provided?PLOS Geneticsdata availability policy, and numerical data that underlies graphs or summary statistics should be provided in spreadsheet form as supporting information.Large-scale datasets should be made available via a public repository as described in the Reviewer #1: YesReviewer #2: YesReviewer #3: Yes**********what does this mean?). 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The effect of heat stress on livestock production is a worldwide issue. Animal performance is influenced by exposure to harsh environmental conditions potentially causing genotype-by-environment interactions (G × E), especially in highproducing animals. In this context, the main objectives of this study were to (1) detect the time periods in which heifer fertility traits are more sensitive to the exposure to high environmental temperature and/or humidity, (2) investigate G × E due to heat stress in heifer fertility traits, and, (3) identify genomic regions associated with heifer fertility and heat tolerance in Holstein cattle.PLAG1, AMHR2, SP1, KRT8, KRT18, MLH1, and EOMES were suggested as candidate genes for heifer fertility. The genes HCRTR1, AGRP, PC, and GUCY1B1 are strong candidates for association with heat tolerance.Phenotypic records for three heifer fertility traits  were collected, from 2005 to 2018, for 56,998 Holstein heifers raised in 15 herds in the Beijing area (China). By integrating environmental data, including hourly air temperature and relative humidity, the critical periods in which the heifers are more sensitive to heat stress were located in more than 30 days before the first service for age at first calving and interval from first to last service, or 10 days before and less than 60 days after the first service for conception rate. Using reaction norm models, significant G × E was detected for all three traits regarding both environmental gradients, proportion of days exceeding heat threshold, and minimum temperature-humidity index. Through single-step genome-wide association studies, The critical periods in which the reproductive performance of heifers is more sensitive to heat stress are trait-dependent. Thus, detailed analysis should be conducted to determine this particular period for other fertility traits. The considerable magnitude of G × E and sire re-ranking indicates the necessity to consider G × E in dairy cattle breeding schemes. This will enable selection of more heat-tolerant animals with high reproductive efficiency under harsh climatic conditions. Lastly, the candidate genes identified to be linked with response to heat stress provide a better understanding of the underlying biological mechanisms of heat tolerance in dairy cattle.The online version contains supplementary material available at 10.1186/s12864-021-07496-3. In modern dairy cattle farms, female fertility is of great importance, due to its close relationship with reproductive management, veterinary treatments, involuntary culling and, consequently, the farm profitability . HoweverWith global warming and climatic change, heat stress has become an issue for livestock production in many countries around the world . The temThe average daily THI in many regions of the world exceed 72 throughout most summer period days, indicating that dairy cattle located in these regions may suffer from mild to severe heat stress . For insReaction norm models (RNM) are widely used to detect G × E when the differences in environments can be measured by a continuous environmental gradient (EG) . In RNM,The main objectives of this study were to: (1) explore the most heat-sensitive periods for three heifer fertility traits: age at first calving – AFC, interval from first to last service – IFL, and conception rate of first service – CR; (2) detect G × E for heifer fertility traits using RNM with pedigree-genomic combined relationship matrix; and (3) unravel genomic regions contributing to heat tolerance and heifer fertility traits in high-producing Holstein cattle.The summary statistics for heifer fertility traits are shown in Table Two heat related EGs were used in the current study: 1) the number of days that exceeded the THI threshold in the evaluated critical period (prop-EG); 2) the minimum THI for each day of the candidate period (mTHI-EG). To avoid the underestimation of the heat stress effect, the days in which the hourly THI was higher than 72 for six continuous hours were considered as heat-stress days for prop-EG. The Akaike Information Criterion (AIC)  was obtaThe critical periods Fig.  selectedH matrix (hybrid pedigree-genomic relationship matrix), approximately 75% of the heifers were the same in both scenarios for AFC and IFL, but relatively low (29.63% ~ 65.82%) overlap rates were observed in CR  were similar for all traits analyzed. The correlation coefficients between the intercept and slope for each trait were all negative and ranged from − 0.25 (IFL in S1 of prop-EG) to − 0.98 (CR in both S1 and S2 of mTHI-EG) when using the H matrix (Table A matrix are shown in Supp. File The estimates of (co) variance components obtained from RNMs with different kinship matrices , indicating the existence of G × E. Genetic correlations between different EGs, from RNM with the H matrix, are shown in Fig. A matrix  in Fig. Overall, similar genomic regions were detected to be associated with the same trait when using two scenarios of prop-EG, especially for CR Figs.  and 6. FThe genomic windows explaining the highest variance were not connected for AFC and IFL under S2. However, the genomic region from 26,819,709 to 26,887,021 bp that explained the highest proportion of the total additive genetic variance  for CR, was still located in BTA14. We detected 21 overlapping genomic windows for CR between two variables, which is more than detected for AFC and IFL .More shared genomic regions were detected when the same variables (the intercept or slope) of the two scenarios were tested. For AFC and IFL, more than 10 genomic areas were connected, although they did not explain the largest amount of the total additive genetic variance. However, the longest shared region in BTA14 was still detected for both the intercept  and the slope  for CR. Similarly, more than 25 overlapping genomic regions were mapped for each variable of CR.The Manhattan plots of mTHI-EG are provided in Supp. Files P < 0.05) related to heifer reproduction were: developmental process involved in reproduction, oocyte maturation, oocyte development, oocyte differentiation, oogenesis, placenta blood vessel development, and embryo development. Two identified pathways were related to stress response: regulation of response to stress and response to oxidative stress. Other pathways included muscle and body development.The mapped positional candidate genes are shown in Table Candidate genomic regions of the slope term have been previously linked to a variety of trait groups, including luteal activity, body weight, stillbirth, and many milk-related QTLs. Similarly, most of the QTLs identified were associated with milk production traits as most of the QTLs overlapped between the two genetic terms. The reproductive biological processes identified (P < 0.05) using the slope term were: reproductive processes, fertilization, sexual reproduction, granulosa cell differentiation, oocyte development, acrosome reaction, oocyte differentiation, and regulation of luteinizing hormone secretion  are shown in Fig. Approximately 50 common genes were identified in C1 and C2, whereas 10 or 17 shared genes were detected in C3. However, over 330 positional genes were detected for each trait in C3, which is nearly twice the number of genes mapped in C1 and C2 Fig. . AdditioThe heritability estimates (SE) were 0.03 (0.005) for CR, 0.06 0.007) for IFL, and 0.16 (0.011) for AFC  folThe critical periods found under S2 of CR, on the basis of first insemination day, were  and  for prop-EG and mTHI-EG, respectively, which are similar to the values reported in previous studies. Khan et al.  exploredBos taurus crossbred cows in all three breeding periods: , , and . Another study indicated that a high heat load 3–5 weeks before and 1 week after service was associated with reduced CR in cattle [To the best of our knowledge, the critical periods of heat stress for heifer reproduction traits have not been well defined, and most studies focused solely on pregnancy rate or CR. Amundson et al.  detectedn cattle . In Schün cattle , the CR n cattle  proposedThe overlapping test of the two EGs under two scenarios Table  indicateA and H  were linked with several reproductive genes. For example, the overlaps of genomic windows between the intercept of AFC and IFL contained genes such as a et al.  found thsa cells . KRT18 mithelium . Togetheal cells . SP1 hasntiation . PLAG1, n cattle –45. The le [SDC3  and FABP3 [FABP3 . One posGPER1, which has been reported to induce the non-genomic suppression luteinizing hormone secretion in cattle [RAD51 has been reported to be associated with bovine oocytes meiosis progress [LAP3, GLYCAM1, PDE1B, MICALL2, NPC1) are related with other economically-important traits in cattle such as milk production, carcass traits, body weight, body height, and body length [HCRTR1, which regulates orexin receptor type 1, has been suggested to participate in negative feedback regulation in the adrenal gland [AGRP would inhibit the cortisol production of adrenal gland [HCRTR1 and AGRP may play vital roles in the cattle stress reaction process. White et al. [PC expression in bovine primary hepatocytes and kidney epithelial was significantly higher in thermal stress conditions, which indicate that PC may contribute to the physiological response to thermal stress. PC has been reported to be associated with fatty acids regulations while feeding and thus affect the feed intake of cattle [Various candidate genes associated with heat tolerance (slope term) were also reported in the literature. Several candidate genes are related to cow reproduction such as n cattle  and was progress , 50. Somy length . Additioal gland . Severalal gland , 58. Theal gland . Thus, He et al.  reportedf cattle , 61.PLAG1. Another genomic region with high genetic variance is located in BTA29 (from 44.3 to 44.9 Mb), which includes the heat stress-related gene PC. Most of the QTLs were shared by the intercept and slope, which is consistent with the strong negative genetic correlations between them. The fertility-related QTLs would be evident in heat stress conditions as well because the change in environments may alter the genetic expression of these regions when a strong G × E exists.The biological processes identified in functional analyses were mainly reproduction- and stress response-related, which corroborates with our findings. The QTLs identified in this study are mainly related to milk production traits  genes might be compromised in heat stress conditions. Thus, these altered SNP effects were observed at higher levels of prop- and mTHI-EGs , whereas Khan et al. [HSPA13 as a differentially expressed gene in heat-stressed bovine granulosa cells. This indicates that the interaction between GUCY1B and HSP90 in cattle may be related to the heat stress response.When mTHI-EG was used, l papers , 71 haven et al.  detectedWe analyzed the impact of heat stress on dairy cattle based on three fertility traits. The critical periods, which are when heifers may be more affected by heat stress, were found to be related to the environmental gradient used, centered on the first insemination day. This indicates that detailed analysis for other traits should be applied to derive this period. Genetic parameters suggest significant and considerable magnitude of G × E for all three heifer fertility traits, indicating that breeding values may change under heat stress conditions for these traits. The re-ranking of sires between different environments further demonstrates the effects of G × E on animal breeding. Several reproduction–, growth–, production–, and resilience–related genes and QTLs were identified in the candidate genomic regions affecting fertility traits. Overall, G × E models should be integrated into current dairy cattle breeding schemes to select more climatic resilient animals. The heat stress-related genes or QTLs are important for exploring the mechanisms of heat stress response in dairy cattle.www.afimilk.com.cn). All herds were kept in a free-stall design and included 1000 to 2000 heifers and the management strategies in these farms are similar. The analyzed heifer traits are AFC , IFL , and CR (conception rate of first service). CR was coded as 1 when there was a confirmed pregnancy after the first service and 0 otherwise. The IFL was 0 when a heifer was pregnant after the first service. Further criteria for the data editing included AFC between 500 and 1100 days and IFL between 0 and 365 days. Records with values outside these ranges were dropped of further analyses. Animals which changed herds during the analyzed period were also excluded. The number of records for each trait after editing was 56,998  in Beijing, China, were collected through the herd management software AfiFarm . Individuals with genotype call rate lower than 0.90 were dropped and only autosomal markers were retained for this study. After quality control, 111,068 SNPs remained in the dataset.Phenotypes (of daughters) and genotypes were available for 3731 heifers and 537 bulls. All bulls and a subset of 2379 heifers were genotyped using the GeneSeek Genomic Profiler Bovine 50 K Chip, whereas the remaining 1352 heifers were genotyped using the GeneSeek Genomic Profiler Bovine 150 K Chip. In addition, 1769 heifers genotyped with the 150 K chip were included in the reference population to improve imputation accuracy. The animals genotyped with the 50 K chip were imputed to the 150 K using the Beagle v5.0 software [www.noaa.gov). Hourly THI was calculated using the following formula [db is the hourly dry bulb temperature (°C) and Tdp is the dew point temperature (°C). Then, the daily THI was calculated by averaging hourly values.Hourly recorded temperature data of Beijing  during the test years were obtained from the National Oceanic and Atmospheric Administration ; X is an incidence matrix connecting the vector of fixed effects b  to y; Z is an incidence matrix connecting a (additive genetic effect) to y, and e is the residual effect. The following RNM was used to investigate G × E:Z0 is an incidence matrix connecting a0 (intercept) to y, and Z1 is an incidence matrix containing EGs as covariables to connect a1 to y (slope). It was assumed that K is A (numerator relationship matrix) for pedigree-based BLUP, or H (combined pedigree-genomic relationship matrix) for ssGBLUP.Many studies have shown that the estimates from linear models for categorical fertility traits are similar to those from threshold models, while the former require less computation time , 76. FurA was constructed using pedigree data only for conventional BLUP, whereas for the ssGBLUP models, the inverse of the H matrix (H− 1), which was calculated as follows [A22 is the subset of A for genotyped individuals and G is the blended genomic relationship matrix. G was built using (1 − ω)G0 + ωA22, where ω is the assumed weight of the genetic variance not captured by markers and was set to 0.05 according in previous studies [G0 was constructed using the method proposed by [G was tuned to have the same scale as A22 [The matrix  follows , 77:\do studies . G0 was posed by . Finallye as A22 , 81.e is assumed to follow I is an identity matrix. The residual variances were assumed to be the same across different EGs, based on the results of preliminary analyses, which suggested that residual variances were similar across different levels of EG. The full data was divided into three subsets based on the percentile of EGs  to achieve similar sample sizes across subsets.The random residual vector The (co) variance components for RNMs were estimated using the average information REML method implemented in the BLUPF90 programs . The staTo evaluate the impact of heat stress on heifer reproduction, the basic assumption was that heifers experiencing higher THI in a specific period have reduced fertility performance based on critical periods, as defined in Fig. G × E exists if the variance of the slope was significantly different from zero by using a one-tailed t-test with the significance level of 0.05 , 33. Onehttp://hgdownload.soe.ucsc.edu/goldenPath/bosTau9/bigZips/genes/) and the Cattle QTL database . The biological processes of candidate genes were annotated using the PANTHER Classification System [The marker effects of the intercept and the slope for all traits were estimated using the ssGWAS method proposed by Wang et al. . The pern System .n = 555) SNPs with the highest or the lowest slope effects were selected, whereas for C3, the 0.5% SNPs with slope effects closest to zero were selected. Choosing the top 1% SNPs for trajectory analyses has been implemented in several GWAS studies based on the 50 K SNP panel [The SNPs were chosen to investigate the trajectories of their effects. Firstly, SNPs were ranked according to the magnitude of their slope effects for each trait, EG, and scenario. Then, three clusters (C) of SNPs were obtained according to the trajectory of their SNP effects across each EG: C1 = SNP effects changes in preferential ways ; C2 = SNP effects changes in opposite ways ; C3 = constant SNP effects over time. For C1 and C2, the top 0.5% (NP panel , 86. TheAdditional file 1: Figure S1. Climate conditions in Beijing, China, during experiment years.Additional file 2: Table S1. Detailed results of AICs for 19 tested combinations.Additional file 3: Table S2. The equivalents of prop and mTHI-EGs and top 1% overlapping heifers between these two kinds of EGs for each trait and scenario estimated by A matrix.Additional file 4: Table S3. The genetic parameters estimated by A matrix.Additional file 5: Figure S2. Heritabilities estimated by RNMs with the matrix A.Additional file 6: Figure S3. Genetic correlations estimated by RNMs with the matrix A.Additional file 7: Figure S4. Percentages of the intercept and slope genetic variances explained by a sliding window of 20 SNPs for three traits, which were estimated under scenario one of mTHI-EG.Additional file 8: Figure S5. Percentages of the intercept and slope genetic variances explained by a sliding window of 20 SNPs for three traits, which were estimated under scenario two of mTHI-EG.Additional file 9: Table S4. Detailed information for genomic regions, genes and QTLs detected using prop-EG.Additional file 10: Table S5. Detailed information for genomic regions, genes and QTLs detected using mTHI-EG.Additional file 11: Figure S6. Average conception rate of Holstein population in different months.


The GY genetic gains were estimated as genetic progress per se (GYP) and in terms of local checks (GYLC). In total, 239 international locations were classified into two groups: high- and low-rainfall environments based on climate variables and trial management practices. In the high-rainfall environment, the annual genetic gains for GYP and GYLC were 3.8 and 1.17 % (160 and 65.1 kg ha−1 yr−1), respectively. In the low-rainfall environment, the genetic gains were 0.93 and 0.73 % (40 and 33.1 kg ha−1 yr−1), for GYP and GYLC respectively. The GY of the lines included in each nursery showed a significant phenotypic correlation between high- and low-rainfall environments in all the examined years and several of the five best performing lines were common in both environments. The GY progress was mainly associated with increased grain weight (R2 = 0.35 p < 0.001), days to maturity  and grain filling period . These results indicate continuous GY genetic progress and yield stability in the HRWYT germplasm developed and distributed by CIMMYT.The effects of climate change together with the projected future demand represents a huge challenge for wheat production systems worldwide. Wheat breeding can contribute to global food security through the creation of genotypes exhibiting stress tolerance and higher yield potential. The objectives of our study were to ( A ME, seen also as a target population environment, is defined as a broad, not necessarily contiguous area, occurring in more than one country and frequently transcontinental, defined by climate, soil, similar biotic and abiotic stresses, cropping system requirements, etc. . There aepentis) . The ME2epentis) .In response to the needs of farmers in high rainfall areas, CIMMYT annually distributes the High Rainfall Wheat Screening Nursery (HRWYT) to more than 100 collaborators worldwide within the International Wheat Improvement Network (IWIN). The germplasm carries the required characteristics for variety release or use as progenitors by local breeding programs. Although HRWYT targets high rainfall areas, often they are grown in regions classified within other MEs as per collaborators request, since the yield stability of CIMMYT germplasm often allows them to find lines that also perform well in others MEs. Cooperators evaluate this elite germplasm at several locations across the world and return the phenotypic data to CIMMYT. Information from such trials is strategic for CIMMYT as it helps breeders to make better breeding decisions by selecting superior parents; and it allows the estimation of grain yield (GY) genetic gains achieved over a certain period, as a measure of breeding efficiency.This type of evaluations are imperative due to the significant impact of CIMMYT wheat germplasm on world wheat production, given that 70 % of cultivars grown in target countries are either direct CGIAR releases or used as parents , and sinIn addition to assessing the GY genetic gains, studying the genetic base and the changes in yield-related traits is valuable to recognize which of them are associated with GY, identify yield-limiting factors and specially to plan future effective approaches to increase the GY genetic gains in breeding programs. Historical studies have shown that traits such as grains per square meter, biomass, harvest index, and reduced plant height are positively associated with GY progress . Howeveri) estimate the annual GY genetic gain for High Rainfall Wheat Yield Trial grown from 2007 (15th HRWYT) to 2016 (24th HRWYT) across international environments, and (ii) determine the changes produced by GY genetic improvement on associated physiological traits.In general, complex traits such as GY and its components show Genotype-by-Environment interaction (GE), which may be expressed as heterogeneity in genetic variance amongst environments  and/or in the ranking of individuals (crossover interaction) . Genotyp22.1Tilletia indica) free site in Northwestern Mexico . Then, the yield trial nursery is distributed globally as requested by cooperators. The trial design is an α-lattice with two replicates. New lines are distributed each year, hence the composition of entries is different from year to year.Each HRWYT consisted of 30–50 entries, one of which is a local check (LC) included in the trial by each collaborator  and the remaining entries are new CIMMYT’s elite wheat lines and checks. The elite lines are selected after 2 years of testing under optimal irrigated conditions and an additional 1–2 years under drought and heat-stressed conditions at CIMMYT’s experimental station in Ciudad Obregon, Mexico . Germplasm selected for assembling HRWYTs each year are multiplied at Mexicali, Baja California Norte, a Karnal bunt . Briefely, days to heading (DH) was considered when 50 % of the plants in a plot had spikes completely emerged from the flag leaf. Similarly, days to maturity (DM) was considered when 50 % of the spikes in a plot had lost all green coloration. Days from heading to maturity (DHM) were calculated by subtracting the time to maturity from the time to heading. Plant height was determined by measuring the plant height in centimeter, excluding awns, on two measurements per plot. Plots are harvested soon after physiological maturity, GY was recorded in grams per square meters and then converted into tons per hectare. Grain weight was determined using two samples of 200 grains were randomly taken from each plot and weighed on a precise balance. Finally, GN was calculated as the quotient between GY and GW.The physiological yield related traits were analyzed, but only in high-rainfall environment from 2010 (18th) to 2016 (24th), because data was not available in the previous years.2.Regarding the management practices carried out and data returned by cooperators, almost all trials were fertilized (99 %), while the 96 % reported having used fungicides. In line with the reported fungicide treatments, only 8 % of the trials showed severe disease development and 29, 20, 27, and 16 % displayed moderate, slight, traces and no foliar disease development, respectively . In all reported data trials, weeds were well controlled, since 53 % did not report weed problems, and 14, 16, 15 and only 1 % reported traces, slight, moderate, and severe weed problems, respectively. Of all the sites, 30 % were reported to be irrigated and 2 % had severe lodging, while 6, 13, 22, and 57 % showed moderate, slight, traces and no lodging problems, respectively. The average experimental plot was 4.6 m2.22.2.12) using the following statistical model:jR is the fixed effects of the replicates , iG is the fixed effects of the wheat lines , and kB represents the random effects of the incomplete -blocks , assumed to be independently and identically normal distributed (iid) with mean zero and variance σ2sb(r). The term εijk is a random residual assumed to be iid with mean zero and variance σ2ε. The heritability was estimated using the variance components of each site and those sites with H2 <0.05 were excluded from further analysis. Then, a k-means clustering analysis using the POWER weather data of each remained trial was used to define the designation of the sites in each of two high- and low-rainfall environments. Once the sites were grouped, data from each HRWYT were analyzed in a combined analysis across locations and by group.The linear mixed models used in the statistical analyses are in line with those used by 2.2.21 is a column vector of ones, μ is the overall mean, sX is the incidence matrix for the fixed effects of sites, and r, Zb, Zg,Z and geZ are the design matrices for the random effects of replicates within sites, incomplete blocks within replicates and sites, genotypes, and GE, respectively. Vector s denotes the fixed effect of sites; while vectors r, b, g, ge, and e contain random effects of replicates within sites, incomplete blocks within replicates and sites, lines, GE, and residuals, respectively, and they are assumed to be iid random variables, normally distributed with zero mean vectors and variance-covariance matrices R, B, G, GE, and E, respectively.The linear mixed model used in the multi-site trial analysis can be written as:R, B, G, and E are simple variance-covariance component with R = rI, B = bI, G = gI and E = rsgI, where I. The variance-covariance matrix GE is the variance of the line effects in individual environments and can be represented as the Kronecker product between the variance-covariance matrix for the dimension of the sites (gI): GE = g⊗ I. In this study, we modeled the environmental component, gI, the genotypes are iid , in which the GY BLUPs of the five highest-yielding lines (HYL), estimated across all locations within each year, were regressed over the years of HRWYT evaluation. Here, the percentage of the genetic gain was calculated over average yield across the 10 years analyzed. In addition, we also determined the genetic progress of the CIMMYT’s line in terms of LCs as the regression slope of the five HYL GY BLUP minus the mean yield of LCs across all locations divided by the grain yield of the local check of HRWYT evaluation (GYLC),BLUP represents the average GY BLUP value of CIMMYT’s five HYL, estimated across locations within each year, and LCGY represents the predicted GY BLUP mean of the LCs.The GY genetic gains were determined as genetic progress On the other hand, the association between GY and physiological related traits was tested and illustrated using biplot, whereas multiple regression analyses were performed to determine the effects and relative contribution of each trait to GY in the high-rainfall environment.33.1Of the 360 sites, 239 (66 %) with complete data and heritability higher than 0.05 were used for further analysis. These trials were distributed across Asia (28.5 %), South America (22.2 %), Africa (19.2 %), North America (18.4 %) and Europe (11.7 %). Based on the POWER weather data and trial management information provided by cooperators, the k-means clustering analysis grouped 138 sites (57.7 %) to the high-rainfall (well-watered) environment, and 101 (42.3 %) belonged to the low-rainfall environment ; Fig. 1.3.2−1 and ranged from 3.57 (17th) to 5.97 t ha−1 (24th)  to 5.75 t ha–1 (23rd) (−1 (24th) and from 3.88 (20th) to 5.34 t ha−1 (23rd) in high- and low-rainfall environments, respectively. In addition, the GY of the lines included in each nursery, showed a significant (p < 0.01) phenotypic correlation between high- and low-rainfall environments ranging from r = 0.41 (22nd) to r = 0.83 (15th). The performance of the HYL expressed as a GY BLUP, percentage of GY increase relative to LCs, and GY rank across high- and low-rainfall environments are given in Supplementary Table 1. In general, the outstanding lines performed well in both environments, in fact, 26 % of the best performing lines in the high-rainfall environment were also amongst the best performing lines in the low-rainfall environment (Supplementary Table 1).The mean GY of the five HYL across all HRWYT in high-rainfall environment was 4.60 t ha1 (24th) a, wherea1 (23rd) d. MeanwhSignificant genetic progress for GYP and GYLC was observed in both environments by modeling the GE interaction with the FA structure .−1 yr−1 , with R2 of 0.40 and 0.19, respectively.In the high-rainfall environment, the annual rate of GYP increase was 3.8 %, which represents 160 kg ha−1 yr−1 a. When t−1 yr−1 b, the an−1 yr−1 (−1 yr−1) of yield gain .Overall, GW ranged from 30.1 (19th) to 46.9 (24th) mg, showing a strong and positive linear relationship when it was plotted against years a. Thus, −2 and presented a negative linear relationship .The GN ranged from 9279 21 st to 13922 19th grains mp = 0.55 and p = 0.93, respectively) breeding effect was detected when they were regressed over HRWYT years.Plant height ranged from 73.7 (23rd) to 96.7 (22nd) cm, while DH showed phenotypic values from 60.7 (19th) to 78.1 (22nd) days c and d. p < 0.001) and the respective annual rates of progress were 4.10 and 4.13 days yr−1, respectively  to 138.7 (24th) and from 30.2 (18th) to 72.6 (24th) days, respectively. Both trends were significant . The individual trait contributions ranged from 0.9–38.9%, being GW and DM the most significant  and genetic rates of 3.8 % and 1.17 % for GYP and LCs, respectively. Similar genetic progress per se has been observed in some countries located in high-rainfall environments. For example, in Ethiopia, the national production showed an annual yield gain of 6.4 % considering the same period analyzed in this study (−1 (Our study shows reliable genetic progress for GY over time when the data is expressed as genetic progress is study . The gentudy (−1 .p = 0.02), but not as significant as in the high-rainfall environment. This is attributed to the fact the HRWYT germplasm is particularly selected to target sites where precipitation is not a constraint for wheat production. Regarding rainfed areas, the genetic gains reported have been more variables ranging from 0.3 to more than 3 % yr−1 , artificial irrigation was used, however, many cooperators did not report the amount of water applied.2 ranging from 0.41 to 0.83. In this sense, the 26 % of the best performing lines in the high-rainfall environment were also amongst the best in the low-rainfall environment (Supplementary Table 1), which is an indicator that outstanding lines have yield stability across multiple locations. Yield stability is one of the features of CIMMYT germplasm achieved through the strategy of shuttle breeding, together with the possibility to simulate various growing conditions at the main yield-testing site in Ciudad Obregon, Sonora, Mexico  are almost fixed in the elite germplasm  due to the improvement in disease resistance along with genetic variation in the length of DM and DHM in the analyzed material. Both, the importance of genetic resistance in high rainfall zones, where the foliar diseases pressure is higher than in dryer environments, as well as a significant genotypic variation for DM and DHM in spring wheat have been reported .There is evidence that GY could be substantially improved in high-rainfall regions if early-sown, long-season wheat lines are grown to take advantage of the longer growing season compared with the traditional varieties grown in the low and medium-rainfall region .p = 0.03). As reported by Despite the positive association observed between DHM and DM with grain yield, they did not show a strong association with GW, explaining together only 15 % of its phenotypic variation  and reduced pH were positively associated with yield progress . The int5In agreement with previous reports, our results show a linear and consistent genetic progress in the wheat breeding lines from the HRWYT in both analyzed environments over the last 10 years. The best performing lines were associated with pedigrees that have been widely reported as stable and high-yielding lines across different environments. The present study also revealed that the genetic yield potential improvement of HRWYTs over time was associated with paralleled increases in GW, DM, and DHM as these traits were strongly correlated with GY genetic gains.Guillermo S. Gerard: Writing - Original Draft, Formal Analysis and Methodology. Leonardo A. Crespo-Herrera: Conceptualization, Methodology, Resources and Investigation. José Crossa: Formal Analysis and Methodology. Suchismita Mondal: Investigation and Resources. Govindan Velu: Investigation and Resources. Philomin Juliana: Investigation and Resources. Julio Huerta‑Espino: Investigation and Resources. Mateo Vargas: Formal Analysis. Mandeep S. Rhandawa: Investigation. Sridhar Bhavani: Investigation. Hans Braun: Project administration, Funding acquisition; Ravi P. Singh: Project administration, Funding acquisition, Investigation, Resources and Supervision. All authors: Writing - Review & EditingThe authors declare that there is no conflict of interest.