Prudul26A013473 as a candidate gene responsible for the main QTL found in crack-out percentage, Prudul26A012082 and Prudul26A017782 as candidate genes for the QTLs found in double kernels percentage, and Prudul26A000954 as a candidate gene for the QTL found in blooming time. Our study enhances our knowledge of almond dissemination history and will have a great impact on almond breeding.Domestication drastically changed crop genomes, fixing alleles of interest and creating different genetic populations. Genome-wide association studies (GWASs) are a powerful tool to detect these alleles of interest (and so QTLs). In this study, we explored the genetic structure as well as additive and non-additive genotype–phenotype associations in a collection of 243 almond accessions. Our genetic structure analysis strongly supported the subdivision of the accessions into five ancestral groups, all formed by accessions with a common origin. One of these groups was formed exclusively by Spanish accessions, while the rest were mainly formed by accessions from China, Italy, France, and the USA. These results agree with archaeological and historical evidence that separate modern almond dissemination into four phases: Asiatic, Mediterranean, Californian, and southern hemisphere. In total, we found 13 independent QTLs for nut weight, crack-out percentage, double kernels percentage, and blooming time. Of the 13 QTLs found, only one had an additive effect. Through candidate gene analysis, we proposed  One of the landmarks of human history was the transition from nomadic hunter–gatherer societies to settled agriculture-based societies. This transition, known as the Neolithic Revolution, marked the beginning of the domestication of wild plant species as cultivated crops . DomestiNowadays, the main actor changing crop genomes is modern breeding. The efficient implementation of breeding strategies requires the correct management of germplasms, optimized genotyping and phenotyping methods, concise knowledge of crop genetic structure, and the study of genetic determinism behind traits of interest , 5. GenoPrunus dulcis Miller (D.A. Webb)] is the most economically important temperate nut tree worldwide. In the period 2011 to 2021, its production increased 54%, reaching 1 684 395 metric tons of kernel [Rosaceae family and the Prunus genus with other important crops, including peach, plum, apricot, and cherry.Almond [While research on the almond domestication process is in the early stages, some important insights have been made. The most accepted theory to date is that almond originated from hybridizations with several wild relatives somewhere between the Eastern Mediterranean and Southwest Asia, expanding rapidly to Central Asia and the Western Mediterranean. Many studies using different approaches support this theory, from analyses based on morphology, habitat, and/or coexistence in cultivated areas , throughThe recent publication of three almond reference genomes  and the In this study, we explored the genetic structure and genotype–phenotype associations in a collection of 243 almond accessions from different origins. For this purpose, we first characterized the genetic diversity of the collection using the almond 60 K SNP array. Then we carried out a GWAS using additive and non-additive models for different traits, including kernel and nut weight, crack-out percentage, double kernels percentage, and blooming time. As far as we know, this is the first non-additive GWAS in Rosaceae species. Using candidate gene analysis, we also proposed candidate genes responsible for the main QTLs found in this analysis.The results from the population structure analysis, additive kinship, phylogenetic tree, and PCA indicated that the most acceptable genetic structure of the 152 landraces included in this analysis is a model with five ancestral populations . AccordiAdditive kinship results showed five dense clusters . These cThe phylogenetic tree had three main clades: C1, C2, and C3 . C1, maiBased on the Principal Compound Analysis, the ancestral groups were well separated according to the first two principal components, explaining 9.18% of the overall variation . Admixtu2F or 0.25F values  to 6904 bp (PD02) .A significant negative correlation was found between NW and CRO (−0.77) . A weaker correlation was also found between NW and KW (0.49). Genetic variance and the variance due to the environment  were extracted from the linear mixed models for each trait . HeritabIn total, 19 associations were detected for the traits under study . Of thesWithin these 13 QTLs, only qP-CRO6 had an additive effect, the rest had non-additive effects on the phenotype. By trait, we found three, seven, two and one QTL for NW, CRO, DK and BLO, respectively. No QTL was found for KW. NW and CRO had QTLs located at the same position: qP-NW2 and qP-CRO2 had SPD02_19131932 as top SNP with a recessive effect, while qP-NW3 and qP-CRO3 had SPD03_1356626 as top SNP with a overdominant effect. The variance explained by the QTLs ranged from 9% (qP-NW6) to 49% (qP-CRO2). The combined variance explained for all the QTLs was 45.74% for NW, 71.78% for CRO, 65.87% for DK, and the only QTL found for BLO explained 16.44% of the variance.Prudul26A013473, was annotated as a NAC domain-containing protein.The length of the qP-CRO2 region was 13.8 kb, containing just one gene . This geThe length of the qP-DK7.1 region was 11.5 kb, containing two genes  encodingThe length of qP-DK7.2 was 11.5 kb, containing four genes  encodingThe length of qP-BLO2 was 13.8 kb, containing three genes  encodingPrupe.2G196600, the homologous gene of Prudul26A013473, indicated 25 enriched terms, with nine classified as top enriched terms (TET)  analysis of Prupe.7G052700, the homologous gene of Prudul26A012082, indicated 192 enriched terms, and 28 were classified as TETs ’ within group G4. This genetic relatedness between French and Californian accessions is explained by the introduction of French commercial cultivars in California from 1850 to 1900 . In thisG5 formation also included three Tunisian accessions. This close genetic relatedness may be explained by the exchange of material between the two countries during the French occupation of Tunisia from 1881 to 1956.Finally, Californian cultivars were introduced to adapted regions of the southern hemisphere, including Chile, Argentina, South Africa, and Australia. This explains the genetic relatedness between Californian and Australian accessions.ROH lengths allowed us to detect modern (2ROH) and ancient (0.25ROH) inbreeding events. In general, all accessions had a low 0.25F and 2F value, regardless of whether they were classified as landraces or breeding cultivars. Nevertheless, there were some breeding cultivars with an 2F value over 0.4, such as ‘Ayles’, ‘Amandier rose’, and ‘Garfi’. These results are in agreement with [Homozygosity analysis was used to study the inbreeding levels in the studied germplasm. Different ent with , who con0.25Freq and 2Freq, we found no region with high 2Freq. On the other hand, there were several regions with a high 0.25Freq, in chromosomes 1, 4, and 5. This may respond to selection pressure during the almond domestication process, fixing in the genome alleles of interest. In the case of chromosome 4, the peak of high 0.25Freq was formed only by one gene, Prudul26A014015. According to phylomeDB, Prudul26A014015 had two homologous genes in Arabidopsis, AT1G21390 and AT1G76980. AT1G21390 was annotated as EMBRYO DEFECTIVE (EMB) 2170, while AT1G76980 was annotated as patatin-like phospholipase domain protein. EMB genes are related with embryo development in Arabidopsis [Looking at bidopsis . Since tPrunus genus genomes, this phenomenon could be repeated in other cross-pollinating Prunus species. In peach, a self-compatible Prunus species that still maintains an important level of heterozygosity, this may have predominantly led to the fixation of dominant/recessive QTLs and to the selection for heterozygosis in those that are overdominant.Among the 13 QTLs detected in this study, only one had an additive effect. This indicates that non-additive effects could be the main source of genotype–phenotype interactions in almond. Furthermore, due to the similarity between Another remarkable aspect was that most of the QTLs detected in this study were defined by only one SNP (the top SNP). This is caused by the short LD decay found in the population, affecting this study in two different ways: first, it has facilitated the search for candidate genes, since the genomic regions associated with the traits of interest were small and only a few genes were found in these regions . Second, this could have limited the detection of some regions of interest. In genomic regions with a lower concentration of SNPs, some regions of interest might have been lost because the distance between SNPs was greater than the LD decay. This could be the case for KW, were no QTL was found.For the rest of the traits under study, we found several QTLs that could be used in marker assisted selection. However, the mathematical models used to calculate the combined variance explained by these QTLs did not take into account possible epistatic interactions between the QTLs. More research is needed before these QTLs can be used in marker assisted selection.Prudul26A013473. This gene, annotated as a NAC TF, was homologous to NST1 in Arabidopsis. NST1 has been reported as a key regulator of the formation of secondary cell walls in woody tissues [Prupe.2G196600, the homologous gene of Prudul26A013473, gave several enriched terms related to secondary cell wall biogenesis and organization. All this evidence indicates that Prudul26A013473 is the gene responsible for qP-CRO2, having a major role in the transcriptional regulation of the almond endocarp lignification.Among the 13 QTLs detected in this study, only qP-CRO2 has been reported in other studies , 30, 31. tissues  and spec tissues . FurtherPrudul26A012082 and Prudul26A002885. These genes were annotated as a eukaryotic translation initiation factor 3 subunit E (eIF3e) and alpha-carbonic anhydrase domain-containing protein, respectively. eIF3e has been reported as essential for embryo development and normal plant cell growth in Arabidopsis [Prupe.7G052700, the homologous gene of Prudul26A012082, gave several terms related to embryo development . This indicates that Prudul26A012082 is the gene responsible for qP-DK7.1, having a major role in almond ovule and embryo development.For double kernels percentage, we found two different QTLs, qP-DK7.1, and qP-DK7.2. In almond, the presence of double kernels in a shell is due to the development and fertilization of two ovules in the ovary when the secondary ovule does not degenerate . In the bidopsis , 36. ThePrudul26A029836, Prudul26A017782, Prudul26A005959, and Prudul26A008330. These genes were annotated as a ML domain-containing protein, a ribosomal large subunit 6 (RL6) and two uncharacterized proteins, respectively. It has been suggested that RL6, among other ribosomal subunits, is essential for embryogenesis in Arabidopsis [Prupe.7G052700, homologous gene of Prudul26A012082, showed several terms related to cell cycle regulation and cell development : Prudul26A017782 is most likely the gene responsible for qP-DK7.2, having a major role in almond ovule and embryo development.There were four candidate genes in qP-DK7.2 region, bidopsis . The GCNPrudul26A000954, Prudul26A028547, and Prudul26A019171. These genes were annotated as a cyclic nucleotide-binding domain-containing protein, an uncharacterized protein and a beta-amylase. Both AT1G01710 and AT4G00520, the homologous genes of Prudul26A000954 in Arabidopsis, were annotated as Acyl-CoA thioesterases. These proteins catalyze the hydrolysis of acyl-CoAs to free fatty acids and coenzyme A. During dormancy breaking in perennial fruit trees, reactive oxygen species (ROS) are produced. One of the pathways that produces these ROS starts from fatty acids, beta-oxidated to monosaccharides, and these monosaccharides produce ROS via mitochondrial respiration or are oxidized via the Pentose Phosphate Pathway [Prupe.2G169700, the homologous gene of Prudul26A000954, gave several terms related with SWI/SNF complex and BAF60 . SWI/SNF complexes have been shown to participate in the control of flower development and blooming time [Prudul26A000954 therefore appears to be the gene responsible for qP-BLO2, having a major role in blooming time through fatty acids metabolism.Within qP-BLO2, three genes were found,  Pathway . The GCNing time . BAF60 iing time . Prudul2Prunus species. Based on the fast LD decay and the use of the peach GCN we propose four candidate genes for the main QTLs found in this study.In this study, we carried out genetic structure analysis and non-additive GWAS in a set of 243 almond accessions. Our genetic results agreed with the archaeological and historical evidence that separate modern almond dissemination into four phases: Asiatic, Mediterranean, Californian, and southern hemisphere. Of the 13 QTLs found for the traits of interest, only one had an additive effect, suggesting that non-additive effects could be the major source of genotype–phenotype interactions in almond and other We used a diversity panel of 243 accessions from 21 countries and five continents . Of the 243 accessions, 161 were maintained in the INRAE collection  and 97 in the IRTA collection , with 78 accessions in common at the two locations. DNA of the 180 accessions from the INRAE and IRTA collections was extracted from leaves according to [For the remaining 63 accessions, genotype information was obtained from two different sources: 45 resequences were from a previous study  and reseSNP calling of samples from the DNA libraries was according to Duval et al. 2023. Only SNPs present in the 60 K almond SNP array were selected (60 581 SNPs). All the samples were merged, giving a dataset with 243 accessions and 60 581 SNPs. These SNPs were filtered following these criteria: i) call rate per sample higher than 82%, ii) call rate per SNP higher than 90%, and iii) minimum allele frequency (MAF) higher than 5%.Using this initial dataset , we calcFrom the initial dataset, three more datasets were created for each analysis in this study. For the genetic structure analysis, only the 152 accessions classified as landraces were selected. We also created two more data sets for GWAS, including only phenotyped individuals for nut traits and blooming time, with 79 and 167 accessions respectively. After selecting the accessions, the datasets were filtered again following the same criteria described above. For the datasets used in GWAS, we included two more criteria: (iv) SNPs with three genotypic classes, (v) minimum genotypic class frequency higher than 5%. The four datasets used in this study are presented in A population structure analysis, an additive kinship, a phylogenetic tree and a principal component analysis (PCA) were used to determine the genetic structure of the 152 accessions classified as landraces. The population structure analysis was performed using the LEA R package . The numROHs) were analyzed in all 243 accessions using the detectRUNS R package (https://cran.r-project.org/package=detectRUNS). Two lengths of ROHs were analyzed: 2ROH higher than 4 163 686 bp and 0.25ROH higher than 520 461 bp , kernel weight (KW), crack-out percentage (CRO) and double kernels percentage (DK). Each accession was phenotyped between 9 and 12 years in the IRTA collection. From each accession, at least 100 mature fruits were randomly collected. The fruit was considered mature when the mesocarp was fully dry and split along the fruit suture and the peduncle was near to complete abscission. Samples were stored at room temperature for at least 2 weeks. After measuring NW, the shells were cracked to measure the weight of the kernels. All weights were measured using an electronic balance. DK was measuring by counting the number of shells containing double kernels. CRO was calculated according to Equation Blooming time (BLO) was phenotyped for three consecutive years (2020–22) as Julian days when about 5% of flower buds were fully open for each tree. This trait was measured in the INRAE collection.Best Linear Unbiased Prediction (BLUP) for NW, KW, CRO, and BLO was estimated for each genotype using a linear mixed model according to Equation Where BLUP for DK was estimated for each genotype using a linear mixed model according to Equation entclass1pt{minimaFor every trait, broad-sense heritability (a1) and an alternative allele (a2), were written in numeric representation as 1 , 0  and −1 . For dominant effects, genotypes a1a1 and a1a2 have the same effect in the phenotype, so a1a1 and a1a2 were codified as 1 and a2a2 as −1. For recessive effects, genotypes a1a2 and a2a2 have the same effect in the phenotype, so a1a2 and a2a2 were codified as −1 and a1a1 as 1. Note that the dominant and recessive transformations correspond to a dominant-recessive genotype–phenotype interaction, but we had to differentiate the effects of a dominant reference allele or a dominant alternative allele. For overdominant effects, genotypes a1a1 and a2a2 have the same effect in the phenotype, so genotypes a1a1 and a2a2 were codified as 1 and a1a2 as 0  [We explored additive and non-additive genotype–phenotype associations in two different datasets: nut traits and blooming time datasets, with 79 and 167 accessions respectively. For this purpose, we transformed these genotypic datasets as follow. For additive effects, the three possible genotypes of a biallelic marker with a reference allele . We considered any significant genotype–phenotype association matching its phenotypic data distribution as a true positive QTL.2 as the variance explained for those QTL. We also calculated the combined variance of the QTLs detected for every trait. In this case, we assumed as the combined variance explained the R2 of a linear regression using all the top SNPs detected for a trait.For the QTLs considered as true positives, we assumed the Simple model’s RFor every trait, we selected the QTL with the highest -log10. If the QTL selected had a -log10 higher than 6.5, we performed a candidate gene analysis. In the case of DK, we used candidate gene analysis on the two QTLs we found, as both had a -log10 higher than 6.5.Prunus persica) and Arabidopsis .Every QTL region was defined using the position of the top SNP and the estimated LD decay for every chromosome. The beginning of the QTL was defined as the top SNP position minus the estimated LD decay and the end of the QTL was defined as the top SNP position plus the estimated LD decay. We determined the number of genes located in the QTL region using the ‘Texas’ reference genome v2.0 . Any genPrudul26A013473, Prudul26A012082, Prudul26A017782, and Prudul26A000954, we used the PeachGCN v1 [To obtain more information on the function of the most suitable candidate genes, chGCN v1 , 49. TheWeb_Material_uhad193Click here for additional data file.


Polygenic scores have become an important tool in human genetics, enabling the prediction of individuals’ phenotypes from their genotypes. Understanding how the pattern of differences in polygenic score predictions across individuals intersects with variation in ancestry can provide insights into the evolutionary forces acting on the trait in question, and is important for understanding health disparities. However, because most polygenic scores are computed using effect estimates from population samples, they are susceptible to confounding by both genetic and environmental effects that are correlated with ancestry. The extent to which this confounding drives patterns in the distribution of polygenic scores depends on patterns of population structure in both the original estimation panel and in the prediction/test panel. Here, we use theory from population and statistical genetics, together with simulations, to study the procedure of testing for an association between polygenic scores and axes of ancestry variation in the presence of confounding. We use a general model of genetic relatedness to describe how confounding in the estimation panel biases the distribution of polygenic scores in a way that depends on the degree of overlap in population structure between panels. We then show how this confounding can bias tests for associations between polygenic scores and important axes of ancestry variation in the test panel. Finally, we use the understanding gained from this analysis to develop a method that uses patterns of genetic similarity between the two panels to guard against these biases, and show that this method can provide better protection against confounding than the standard PCA-based approach. Complex traits are influenced by both genetics and the environment. Human geneticists increasingly use polygenic scores, calculated as the weighted sum of trait-associated alleles, to predict genetic effects on a phenotype. Differences in polygenic scores across groups would therefore seem to indicate differences in the genetic basis of the trait, which are of interest to researchers across disciplines. However, because polygenic scores are usually computed using effect sizes estimated using population samples, they are susceptible to confounding due to both the genetic background and the environment. Here, we use theory from population and statistical genetics, together with simulations, to study how environmental and background genetic effects can confound tests for association between polygenic scores and axes of ancestry variation. We then develop a simple method to protect these tests from confounding and show that our approach succeeds in situations that are difficult to protect with standard methods. Our work helps clarify how bias in the distribution of polygenic scores is produced and provides a tool to researchers wishing to protect their analyses from confounding. The effects of individual variants are typically estimated from population samples in which the environments individuals experience vary as a function of their social, cultural, economic, and political contexts. Differences in these factors are often correlated with differences in ancestry within population samples, and these ancestry-environment correlations induce systematic biases in the estimated effects of individual variants. Similar biases can also arise if genetic effects on the phenotype vary as a function of ancestry within the GWAS panel. Ancestry stratification is a long recognized problem in the GWAS study design , and manSeparation of direct genetic effects from correlations between ancestry and environment or genetic background is important to all applications of polygenic scores. Empirically, polygenic scores exhibit geographic clustering even in relatively homogeneous samples and after strict control for population stratification , 10, 11.This issue has been particularly apparent in the detection of directional selection acting on complex traits. Polygenic scores are an ideal tool for this task, as studying the distribution of scores among individuals who differ in ancestry allows us to aggregate the small changes in allele frequency induced by selection on a polygenic trait into a detectable signal , 16, 17.Recently, in the context of selection tests, Chen et al. 34] proposed a strategy to mitigate the impact of stratification by carefully choosing the GWAS panel so that even if residual stratification biases in effect size estimates exist, they will be unlikely to confound the test . We do To test for association of polygenic scores with the test vector, we consider the linear modeloses see . We assuUnder the null model, We can also re-frame this test as a statement about the association between the effect sizes and a set of genotype contrasts, ach site . SimilarThe equivalence between Thus far we have assumed that causal effects are known and therefore the only source of randomness in 3.Now, given these modeling assumptions, we describe how the relationship between the GWAS and test panels impacts the distribution of polygenic scores and the association between the polygenic scores and the test vector. We first consider the case where no attempt is made to correct for population structure. Motivated by these results, we propose a new correction procedure to remove bias associated with the test vector, and consider several details associated with the implementation of our method.3.1.Conditional on the genetic and environmental effects, and genotypes at the focal site, the marginal effect size estimate is given by. (2015)  supplemeects see .3.2.The deviation of an allele’s estimated effect size from its expectation depends on ˜GXsee  where E[VAAX see , μS=αVAAHere This result provides explicit theoretical justification for the procedure of choosing GWAS and test panels that do not overlap in population structure as a means to avoiding bias in polygenic score analyses , 34, 35,3.3.The bias in the polygenic score association test statistic Therefore, we expect the polygenic score association test to be biased when the test vector We can gain further intuition by recalling that testing for an association between the polygenic scores and the test vector is equivalent to testing for an association between the effect size estimates and the genotype contrasts see . This alWe gain two main insights here. First, AS panel . Thus, q3.4.3.4.1.Given the above results, how can we ensure that patterns we observe in the distribution of polygenic scores are not the result of stratification bias? As discussed above, a conservative solution is to prevent bias by choosing a GWAS panel that does not have any overlap in population structure with the test panel, but this is not ideal due to the well documented portability issues that plague polygenic scores , 47, 40,Alternately, our analysis above suggests that including ased see  and S5.3.4.2.A standard approach to controlling for population stratification in polygenic scores is to include the top lue of J . How doeIn our model, ith PC of To see how our approach relates to the PCA correction approach, we can write As a corollary, if we perform a polygenic score association test using GWAS effect size estimates in which the top In practice, however, we do not directly observe ype data . The accgenetics , PCA exhgenetics , 50. ThuA second interpretation of the PC correction approach is that it operates on a hypothesis that the major axes of confounding in a given GWAS panel . Both our predictor 3.4.5.Our approach proposes to use the test panel genotype data twice: once when controlling for stratification in the GWAS panel, and a second time when testing for an association between the polygenic scores and the test vector. Shouldn’t this remove the signal we are trying to detect? While the answer is yes, at least for naive applications, the effect will be small so long as the number of SNPs used to compute the correction is large relative to the number included in the polygenic score.To see why, we can rewrite the regression ion data , and linion data , where iNotably, controlling for PCs of the GWAS panel genotype matrix will induce a similar effect if the axis of structure corresponding to the test is also present in the GWAS panel and gets included in the GWAS model as a covariate. In either case, the downward bias should be small so long as sites used to compute the polygenic score are only a small subset of those used to estimate  million , suggest million , 5 or vi3.5.In this section, using both theory and simulations, we consider a number of concrete examples with varying degree of alignment between the axis of stratification and axis of population structure relevant to the polygenic association test, demonstrating how these biases play out in practice, and how our proposed correction reduces false positives to nominal levels.3.5.1.We first consider a toy model with four populations , which are related to one another by an evenly balanced population phylogeny . The GWAFor simplicity, we consider a purely environmental phenotype  . Followiement of ).Then, using these effect sizes to test for a difference in polygenic score between populations C and D, the bias in our association test statistic is,tatistic , 56, staTo test this prediction, we simulated 100 replicates of four populations related by this topology where the B;C,D)>0 , but unbB;C,D)=0 .Given the population model, B;C,D)=0 . When we removed  and fals removed . For the removed .Next, we wanted to confirm that including When the identities of the causal sites are known and there is no environmental stratification, all three schemes for estimating effects perform similarly, and equally as well as when the causal effects themselves are known exactly  . We sampFor the first example, the confounder, nal axis . When weIn the second example, we simulate confounding along the diagonal, resulting in uncorrected polygenic scores that are correlated with both latitude and longitude in the test panel and an association test that is biased along both axes . When weJ = 10 principal components of the GWAS panel genotype matrix as covariates in the GWAS successfully control for confounding along the latitudinal axis, albeit for potentially different reasons. As outlined in Section 3.4.2, PCA is successful when either the confounder or In the first two examples, the standard approach of including the top p 10 PCs . This paIn the third example, we simulate an increased environmental effect in a single deme, which induces a more complex spatial pattern in the uncorrected polygenic scores . We thenon model . Includic scores . We reviHere, PCA alone fails because neither the confounder nor uded PCs . In factuded PCs , and doiFinally, we tested our approach for each of these for scenarios over a range of magnitudes for the confounding effect. In our simulations, g effect .4.Interpreting patterns in the distribution of polygenic scores is difficult, especially when confounding cannot be ruled out. Because most well-powered GWAS are conducted on population samples where the relationship between genetic background, ancestry, and the environment is not well controlled, stratification bias remains a significant concern , 40, 61.Another advantage of our approach is related to the tradeoff between statistical power and robustness in polygenic selection analyses. Researchers motivated by the difficulties with stratification bias and height , 33 haveOur results are also relevant for analyses that use GWAS summary statistics and coalescent methods to test for signals of directional polygenic selection , 24, 64.A related point is that for both the coalescent approaches as well as methods relying on direct comparison of polygenic scores, both the evolutionary hypothesis being tested and the degree of susceptibility to bias follow directly from the set of genotype contrasts used in the test. Some authors have suggested that certain coalescent methods of testing for polygenic selection are more robust to stratification bias than others [We frame our model in terms of detecting an association between polygenic scores and an ancestry gradient of interest due selection acting on genotypes in the test panel. We focus on this scenario because there is a well-defined null hypothesis and residual stratification is known problem. However, none of our results about the source of bias rely on the signal being generated by selection. Other non-neutral process, such as assortative mating along ancestry gradients, phenotype dependent migration, or phenotype dependent ascertainment could generate signals similar to those generated by natural selection. In other settings, researchers may simply be interested in accurately estimating the association between a polygenic score and a given ancestry gradient, whether or not such an association could plausibly have been generated by neutral evolutionary processes or not. Our results apply in this scenario as well.One disadvantage of our approach is that it requires that the test panel genotypes are available and the test axis specified before carrying out the GWAS. Thus, the benefits cannot immediately be extended to situations where the individual level data for both panels is not available. Tests for association between polygenic scores and axes of ancestry variation are closely related to bivariate LD score regression as applied to a combination of effect estimates for one trait and frequency/genotype contrasts from an independent dataset . A more We also note several additional limitations of our approach. First, our model assumes that all loci share the same underlying genetic relatedness matrix. For simple demographic histories like that of our toy model, this is unlikely to be an issue, but could be an issue for more complex demographic histories experienced by real human populations, as mutations of different ages experience the demographic history differently. One solution to this problem would be to bin variants by allele frequency or estimated age  and compAnother limitation is that in real human datasets with complex patterns of genetic variation and confounding, our approach alone will not be sufficient to mitigate all of the negative consequences of confounding. For example, while a GWAS conducted using only our covariate to control for stratification biases would result in effect size estimates that are uncorrelated with the target axis of population structure under the null, stratification along other axes would still function as additional noise in the effect size estimates. This would result in the ascertainment of tag SNPs that are less tightly coupled to their causal variants, and less accurate polygenic scores, reducing power to detect real signals. Therefore, control of population structure via inclusion of top PCs and/or linear mixed models, in addition to the inclusion of our projected test vector, is still desirable.Finally, we note that the interpretation of results derived from applying our method will be limited by standard assumptions made in many polygenic score association tests and caution interpretation in light of these assumptions . Finally5.5.1.msprime [−8. We then sample 20,000 individuals per population and create two configurations of GWAS and test panels based on the diagrams in We used msprime  to simulster i.e  the averm = 0.01. We fix the effective population size to 1,000 diploid individuals and simulate 200 chromosomes per individual. To reduce computational burden, we scale down to 10kb chromosomes with a per base mutation and recombination rates equal to 1 × 10−7. We then sample 20 individuals per deme to create a test panel (N = 720) and 60 individuals per deme to create the GWAS panel (M = 2160). Again we filter out SNPs with MAF < 0.05 in either panel.The second model, modified from [5.1.1.To study the effect of environmental stratification on association tests, we first simulated nongenetic phenotypes for an individual on model  we then gradient  we addedviations  .To study the effect of controlling for stratification in cases where there is a true signal of association between polygenic scores and the test vector , we usedTo simulate a signal of true difference in polygenic score in the test panel, we calculate the frequency difference 5.1.2.glm. Finally we compute sscore in plink2, taking care to standardize by the variance in the GWAS panel genotypes.We computed a single covariate e plink2  function5.1.3.where glm.For each set of phenotypes, we first carried out four separate marginal association GWASs using the regression equations below,e plink2  functionlm under the four models above, including all 200 SNPs along with the covariates.We then ascertain 200 SNPs to do our association test by picking the SNP with the lowest p-value per chromosome. Additionally, for heritable trait simulations, we know the identity of the underlying causal sites and can choose to use those sites . For eacFinally, we construct polygenic scores for the individuals in the test panel as 5.1.4.For each replicate we first compute the test statistic To evaluate the significance of our association statistic, we compute p-values using an empirical null approach. Specifically, we obtain psuedosamples of Supplement 1Supplement 2


For centuries, morphology has been the most commonly selected trait in horses. A 3D video recording enabled us to obtain the coordinates of 43 anatomical landmarks of 2089 jumping horses. Generalized Procrustes analysis provided centered and scaled coordinates that were independent of volume, i.e., centroid size. Genetic analysis of these coordinates  allowed us to estimate a variance–covariance matrix. New phenotypes were then defined: the “summarized shapes”. They were obtained by linear combinations of Procrustes coordinates with, as coefficients, the eigenvectors of the genetic variance–covariance matrix. These new phenotypes were used in genome-wide association analyses (GWAS) and multitrait genetic analysis that included judges’ scores and competition results of the horses.g = − 0.12, standard error = 0.07). Positive and negative genetic correlations between judges’ scores and shapes were found. This means that the breeding objective defined by judges involves improvement of anatomical parts of the body that are negatively correlated with each other. Known single nucleotide polymorphisms (SNPs) on chromosomes 1 and 3 for height at withers were significant for centroid size but not for any of the shapes. As these SNPs were not associated with the shape that distinguished rectangular horses from square horses (with height at withers greater than body length), we hypothesize that these SNPs play a role in the overall development of horses, i.e. in height, width, and length but not in height at withers when standardized to unit centroid size. Several other SNPs were found significant for other shapes.We defined ten shapes that represented 86% of the variance, with heritabilities ranging from 0.14 to 0.42. Only one of the shapes was found to be genetically correlated with competition success  of a sire based on the shape of its potential progeny, which is easier for breeders to visualize in a single synthetic image than a full description based on linear profiling. However, the acceptance of these new phenotypes by breeders and the complex nature of summarized shapes may be challenging. Due to the low genetic correlations of the summarized shapes with jumping performance, the methodology did not allow indirect performance selection criteria to be defined.The online version contains supplementary material available at 10.1186/s12711-023-00837-8. Horse breeding has been based on the selection for conformation traits since the nineteenth century. Conformation traits were part of the definition of the breed standard when studbooks were created. The traditional assessment of conformation was based on subjective evaluation in relation to the breeding objective . CurrentMorphology is the study of the form and structure of organisms and their specific structural features. In 2002, Pourcelot et al.  developeMorphological data were recorded on 2053 young jumping horses, at the end of a show jumping competition, and on 36 of their sires. A first sample of 784 horses was recorded in 2002 and subsequent samples in 2015 (734) and in 2016 (571). Recording was performed at 24 different events, with between 15 and 201 horses per event (average: 87.0). Gender was evenly distributed: 49% females and 51% males (including 26% geldings). Age was also evenly distributed among the young horses: 52% were 4 years old and 48% were 5 or 6 years old (with only four horses aged 6). The 36 sires were between 9 and 25 years old. The breed of the horses was mostly Selle Français  with a few foreign sport horses (5%), Anglo Arabs (2%), and crossed sport horses (2%). Among those that were not SF, 39% had at least one SF parent.The 3D morphometric method was developed by , 37, 38 Analysis of the jumping competition data was performed with the complete dataset of official jumping competitions from 2001 to 2019 since the birth year 1997, including 150,817 horses and 654,340 annual performances. Horses born abroad with a partial career in France were excluded from this dataset. Success in competition was measured by the repeated observations of an annual performance, which was the logarithm of an annual sum of points. Points were distributed in each competition according to rank and technical difficulty. Among the young horses that were measured for morphometry, nearly all (95%) performed in jumping competitions.Morphology was judged during young horse breeding shows organized by the French stud book of Selle Français. From 2005 to 2019, 28,412 horses aged 2 or 3 were recorded, from one to nine times per horse, with a mean of 1.6, for a total of 46,349 records. Recorded horses were 46% female. Horses were scored on a scale from 1 to 20 for nine criteria: neck, body forehand , withers-back-loins, body hindquarters , joints, stance of forelegs, stance of hindlegs, impression, and chic. Only 19% of the horses that were measured for morphometry were also judged in breeding shows but 92% of them had paternal half-sibs that were judged in breeding shows (mean number of siblings: 188.5).–6, a call rate < 90%, no valid position on an autosome, and P-value for the test of a difference in MAF between the three chips < 10–5. The EquCab 3.0 reference sequence was used to specify SNP positions on the reference map; EquCab 2.0 used as alternative reference when this information was not available (N = 2127 SNPs). After quality control, 375,687 SNPs were retained, which represented 73.2, 73.2, and 55.7% of the SNPs on, respectively, the Illumina Equine SNP50 chip, the Equine SNP70 BeadChip, and the Affymetrix Axiom Equine chip. Imputation was then performed using the Fimpute 3.0 software [In total, 3658 jumping horses, including jumping horses and stallions from previous studies –42, weresoftware , and by Pedigree data were provided by the Institut Français du Cheval et de l’Equitation (IFCE), on behalf of the breeding organizations. Depending on the analysis, the number of horses in the pedigrees over six generations varied: 17,994, 28,270, 90,507, 331,062 horses for analyses that, respectively, involved only the morphometric data, the morphometric and the genomic data, the morphometric data and judges’ scores, and joint analysis with competition data.Coordinates of the forehand and of the hindquarters reference images were combined to constitute the set of coordinates for a horse. These coordinates do not correspond to a real position of the horse because they were taken at different moments of the walking cycle, according to the position of the fore and hind cannon bone. Nevertheless, they allow a shape that is specific to the horse to be synthesized. Among the three redundant points between the two images, we chose to take two on the forelimb image: the top of the withers and the tuber sacral; and one on the hindlimb image: the tuber coxae. To obtain a three-dimensional shape, the landmarks collected from the right side were mirrored on the left side. For the neck and head landmarks, we retained the base of the neck, but the position of the top of the head and nostrils was standardized with a fixed angle between them and with the base of the neck. Because of differences in the walking phase when the image was selected, the actual position of the neck was not relevant for the morphological description. Each horse was therefore finally characterized by the three coordinates for 43 points  between all available pairs of SNPs by chromosome [A mixed linear animal model similar to Eq. software . The relsoftware –52, as wsoftware  to constsoftware . Genomicsoftware  was applromosome . LinkageBivariate models for each of the ten initial summarized shapes software . The modmentclass2pt{minimElementary statistics on raw coordinates, judges’ scores, and competition performances are in Additional file The ten shapes were estimated to be moderately to highly heritable Table , especiaThe estimate of heritability of jumping performance was 0.30 (se 0.005) and its repeatability between years was equal to 0.51. Only one genetic correlation between summarized shapes and performance was estimated to be marginally significantly different from 0, i.e. + 0.12 for shape 2  = 5.46. Genomic control was applied to ensure a regression coefficient close to 1 between ranked observed log on expected log in the QQ plots  for stallions and mares were calculated for the ten summarized shapes. Then, the ten EBV were converted to the 63 coordinates, which were used to create a single image as the expected shape of progeny, which is an attractive way to present EBV for breeders. An example for the famous stallion APACHE D’ADRIERS is in Fig. Morphology of an organism must be studied in the form of synthetic criteria, which are the combinations of the coordinates of the elementary points placed anatomically on the organism. However, the possible number of anatomical landmarks used to describe morphology is necessarily large. In our case, we recorded the position in space (three coordinates) of 25 points, i.e., 75 variables. Since the position of some was fixed beforehand, 63 variables remained. Estimating the variance–covariance matrix of 63 variables is a challenge since it involves 2016 genetic (co-)variances and as many residuals (co-)variances. In order to estimate these variance components while reducing the size of the system, Meyer  developeCervantes et al.  were theMany studies have reported genetic parameters on conformation traits for horses. Traits were either based on subjective judgments, linear profiling, or objective measurements and horses were from different breeds and types . This variety can easily explain the variability in genetic parameters reported in the meta-analysis of . GeometrOur estimates of genetic correlations between judges’ scores and shapes indicate that the conformation desired by judges does not correspond to conformation determined by genetic correlations between anatomical landmarks. This is in agreement with the findings of , who faiPrevious estimates of genetic correlations of conformation linear profiling traits or judges’ scores with jumping competition results are low. In the Netherlands, Rovere et al.  estimateLCORL gene, which is known to be linked to height at withers [We did not find any significant SNPs in the regions that were reported in  for conf withers –80. Thes withers , who perLCORL gene is not associated with shape of the horse (greater than long) but with its overall development. We propose to routinely use ten summarized shapes defined as linear combinations of Procrustes coordinates using the ten first eigenvectors of genetic (co-)variance matrix of coordinates. We found no significant genetic correlation of shapes with jumping performance but conformation remains a selection objective in itself, which will be facilitated by estimated breeding values presented as a single attractive image of the shape of future progeny. An image is easy to understand but the synthetic and complex aspect of the summarized shape may be difficult to handle for breeders, thus its analytical translation into the language traditionally used in breeding may be necessary.Conformation of the horse has been studied for a very long time. Genetic analysis of generalized Procrustes coordinates avoids the challenging problem of describing and judging conformation. It also allowed us to eliminate the volume effect, which impacts any other subjective or objective conformation traits. In particular, we found that the known quantitative trait locus linked to the Additional file 1: Table S1. Elementary statistics of raw coordinates, judges’ scores and competition performances. This table contains three sheets for raw coordinates, judges’ scores and competition performance, respectively, and provides the number of records, mean, standard deviation, minimum and maximum of all variables used in the analysis.Additional file 2: Table S2. Heritability and genetic correlations between judges’ scores.Additional file 3: Figure S1. QQ plot and Manhattan plot of GWAS for summarized shapes and centroid size.


For theoretical studies, reaction norm evolution in a changing environment can be modeled by means of the multivariate breeder's equation, with the reaction norm parameters treated as traits in their own right. This is, however, not a feasible approach for the use of field data, where the intercept and slope values are not available. An alternative approach is to use infinite‐dimensional characters and smooth covariance function estimates found by, e.g., random regression. This is difficult because of the need to find, for example, polynomial basis functions that fit the data reasonably well over time, and because reaction norms in multivariate cases are correlated, such that they cannot be modeled independently. Here, I present an alternative approach based on a multivariate linear mixed model of any order, with dynamical incidence and residual covariance matrices that reflect the changing environment. From such a mixed model follows a dynamical BLUP model for the estimation of the individual reaction norm parameter values at any given parent generation, and for updating of the mean reaction norm parameter values from generation to generation by means of Robertson's secondary theorem of natural selection. This will, for example, make it possible to disentangle the microevolutionary and plasticity components in climate change responses. The BLUP model incorporates the additive genetic relationship matrix in the usual way, and overlapping generations can easily be accommodated. Additive genetic and environmental model parameters are assumed to be known and constant, but it is discussed how they can be estimated by means of a prediction error method. The identifiability by the use of field or laboratory data containing environmental, phenotypic, fitness, and additive genetic relationship data is an important feature of the proposed model. A general reaction norm model of any order can be formulated as a linear mixed model with dynamical incidence and residual covariance matrices. The evolution of mean reaction norms follows from a dynamical BLUP model. This leads to a straightforward and multivariate alternative to infinite‐dimensional and random regression modeling of reaction norms, accommodating changing environments, overlapping generations, and multivariate data. Mean reaction norms in a population can evolve, and this evolution can be modeled by the use of several methods. In its simplest form, a mean reaction norm is characterized by an intercept value and a plasticity slope value, but it may also be natural to use models with a multiple of reaction norms, and they may be nonlinear.A common model is the multivariate breeder's equation Lande, , where tAn alternative approach for reaction norm modeling is to use infinite‐dimensional characters, as in a method introduced by Kirkpatrick and Heckman , and as In the present article, I introduce a multivariate modeling approach based on best linear unbiased predictions (BLUP), allowing for multiple  traits to have joint norms of reaction, which can be linear or approximated by power series , utilizing the information contained in environmental cues and individual phenotypic trait values over many generations, as well as fitness information  based on the multivariate breeder's equation Ergon, , 2022b. After this introduction, Theory and Methods follow in Section 22.1Mathematical symbols with descriptions in the order they appear in equations are shown in Table 2.2yi,t measured on a single individual. In this case, the BLUP estimate of the true additive genetic value for that individual is a′^i,t=h2yi,t−mean, where h2 is the heritability, while mean denotes any fixed effects adjustment. If we substitute this estimate into Robertson's secondary theorem of natural selection  and use BLUP to estimate the vector of additive effects associated with the norm of reaction functions, with these BLUPs then substituted into the expression for Robertson's secondary theorem of natural selection. The incremental changes in the mean reaction norm parameter values thus follow from covwi,t,z′^i,t, where z′^i,t stands for the BLUP estimates of the true additive genetic values involved (Equation (What is done below is to consider a much more complicated phenotype (an observed vector Equation  below).At=In, the BLUP estimates yield a matrix‐based inheritance expression (using the correlated nature of the random effects) to replace h2, and a Robertson‐Price term covwi,tyi,t to measure phenotypic selection . Equation  is unnec2.3n individuals says that the evolution of the mean trait of an n×1 vector zt of individual quantitative traits is described by  of individual i in generation t. The trait zi,t may be any property we can assign a numerical value to, not necessarily biological. In a biological context, the trait may be a behavioral, morphological, or physiological characteristic, but it may also be a parameter in a reaction norm model that describes a plastic organism. Disregarding the second term on the righthand side of Equation  random variables.The nonadditive effects xi,t from one generation to the next besides selection, that is, Ewi,txi,tdescendants−xi,t=0.There are no expected fitness‐weighted changes in the individual additive genetic effects xi,t, and the environmental effects ei,t and ei,tdescendants, are multivariate normal.The additive genetic effects xi,t and nonadditive effects ei,t and ei,tdescendants influence individual fitness only through zi,t.The additive genetic effects At is a unity matrix.All individuals in the population are genetically unrelated, which means that the additive genetic relationship matrix Second, we need to see how the multivariate breeder's equation Lande, ,2)∆z¯∆z¯(2)Equation , 2022c:At=In are compared with results based on the multivariate breeder's equation.In what follows, we will make the use of Assumptions 1, 2, and 3, while Assumptions 4, 5, and 6 will be used only indirectly when the BLUP results with ut is an environmental cue, and where the mean reaction norm is y¯t=a¯t+b¯tut. Here, ai,t=a¯t+ai,t′+υi,t and bi,t=b¯t+bi,t′+ηi,tut are the individual intercept and slope parameters, where υi,t and ηi,t are iid and zero mean random variables, with variances συ2 and ση2, respectively. We thus use ai,t′+υi,t and bi,t′+ηi,t to denote individual deviations from mean values a¯t and b¯t, respectively, where ai,t′ and bi,t′ are the additive genetic components of these deviations. Such additive genetic deviations will be the random effects in the linear mixed model developed below, and thus the random effects that are estimated by the use of the BLUP equations.In order to see how Equation  can be aG and P given by G=Eai,t′bi,t′ai,t′bi,t′=GaaGabGabGbb and P=Gaa+συ2GabGabGbb+ση2. The additive genetic and phenotypic covariance matrices G and P may be time‐varying, but for simplicity, we will here assume that they are constant. As discussed in Ergon  form,ai,t and bi,t, as in Equation   ai,tbi,t2.4p=2 phenotypic traits, and q=2 environmental cues, and a similar simplified system will also be used in the simulations. The theory will, however, be developed in such a way that extensions to higher dimensions are obvious.For clarity of presentation, some details will here be limited to a system with j, the individual reaction norm model with q=2 environmental cues isa¯j,t+aj,i,t′+vj,i,t, b¯j1,t+bj1,i,t′+ηj1,i,tu1,t and b¯j2,t+bj2,i,t′+ηj2,i,tu2,t are the individual parameter values, whilen individuals, we may collect yj,i,t, aj,i,t′, etc., in n×1 vectors and obtain the individual trait vector for trait j,For a specific trait p=2 traits and q=2 environmental cues, we thus obtain the linear mixed model in general form yt=y1,tTy2,tTT, y¯t=y¯1,ty¯2,tT, xt=a1,t′Ta2,t′Tb11,t′Tb12,t′Tb21,t′Tb22,t′TT, and et=e1,tTe2,tTT, where e1,t=v1,t+η11,tu1,t+η12,tu2,t and e2,t=v2,t+η21,tu1,t+η22,tu2,t. Here, X=1n001n, with dimension pn×p=2n×2, while Z~t has dimension pn×p1+qn=2n×6n. In more detail, we have Z~t=Zt⨂In, where Zt=10u1,t010u2,t000u1,tu2,t, and where ⨂ is the Kronecker product operator, which means that all elements in Zt should be multiplied by In. For use in Appendix Zt=IpUtT, with Ut=u1,tu2,t0000u1,tu2,tT. It is an essential feature of the model in Equation yt=Xy¯t+G=ExtxtT=GaaGabGabTGbb. The residuals e1,t and e2,t in Equation  , as in Equation , we alsop=2 and q=2, y¯^t=y¯^1,ty¯^2,tT and x^t=a′^1,tTa′^2,tTb′^11,tTb′^12,tTb′^21,tTb′^22,tTT. Here, Ey¯^j,i,t=y¯j,t and Ea′^1,i,t=Ea′^2,i,t=Eb′^11,i,t=Eb′^12,i,t=Eb′^21,i,t=Eb′^22,i,t=0. Note that the derivation of Equation  among the individuals in the parent generation, the corresponding mean values in the offspring generation before new reproduction will be different from a′^¯1,t, etc., and these within‐generation differences may be used for updating of the mean reaction norm parameters a¯1,t, a¯2,t, b¯11,t, b¯12,t, b¯21,t, and b¯22,t in Equations x^t will z′^i,t is any estimated individual value a′^1,i,t, a′^2,i,t, b′^11,i,t, b′^12,i,t, b′^21,i,t, or b′^22,i,t in the random effects vector x^t in Equation z¯tEquation , while zz¯tEquatEquation , that isAssumption 7. There are no expected fitness‐weighted changes in estimated random effects from individual parents to their descendants after selection but before reproduction, that is, Ewi,tz′^i,tdescendants−z′^i,t=0.z¯toffspring−z¯tparents will be entirely determined by the additive genetic values z′^i,t and individual fitness, and when these changes are used for updating we thus obtain between‐generation changes in the mean values as given by Robertson's secondary theorem of natural selection  give identical results.With individual phenotypic traits according to Equation , and witFor Equation  with x^tA proof of Theorem 2.7The fact that updated mean reaction norm parameter values, and thus also updated mean phenotypic trait values, are found by the use of Robertson's secondary theorem of natural selection applied on estimated random effects, as stated in Theorem 5a∆y¯1,tGjj=Eaj,i,t′2 becomes especially transparent with diagonal G and P matrices, where we from Equations  fitness values are integers from 0 to 4, with number of descendants as unit, and cases with both nonoverlapping and overlapping generations are simulated. The population size is assumed to be constant, which implies that not all descendants survive until reproduction. A constant population size is not essential for the principal results, but it simplifies the simulations.In the simulations, the two environmental reference values are assumed to be known from historical data, that is, it is assumed that the population was fully adapted to the stationary stochastic environment before the onset of anthropogenic and global climate change around 1970.3.3u1,ref= 10°C , and mean spring rainfall u2,ref=2 mm/day . Also assume phenotypic scales such that the phenotypic values that maximize fitness in the reference environment are given by θ1,ref=θ2,ref=0. Further assume environmental cues u1,t=μU1,t−10+u1,s,t and u2,t=μU2,t−2+u2,s,t, where the mean values μU1,t and μU2,t are ramp functions as shown in Figure u1,s,t and u2,s,t are zero mean and white random variables, that is, without autocorrelation. In a corresponding way assume that θ1,t=μΘ1,t+θ1,s,t and θ2,t=μΘ2,t+θ2,s,t, where μΘ1,t and μΘ2,t are ramp functions as shown in Figure θ1,s,t and θ2,s,t are zero mean and white random variables.Assume a population that is fully adapted to a stationary stochastic environment with mean spring temperature u1,s,t, u2,s,t,θ1,s,t, and θ2,s,t have a joint normal distribution with variances σU1,s2, σU2,s2, σΘ1,s2, and σΘ2,s2, and covariances σU1,sU2,s, σU1,sΘ1,s=−2ρτσU1,s2, and σU2,sΘ2,s=−2ρτσU2,s2, where ρτ is the autocorrelation of background environmental fluctuations, as described in more detail in Lande  are thus integers from 0 to 4.The individual fitness function is assumed to be rounded values oft=1. We will assume that the phenotypic values are scaled such that the initial mean intercept values in a stationary stochastic environment are a¯1,t=a¯2,t=0, and that the initial mean reaction norm slope values are the optimal values in a stationary stochastic environment. These optimal values are the ones that maximize the expected individual fitness according to Equation  (dotted magenta lines), are somewhat different from the BLUP results. These differences are clearly smaller for a population size of n=1000. Note that y¯1,t and y¯2,t lag behind θ1,t and θ2,t, as shown in Figure Simulation results with population size Equation  A=zt=AMz0,t, where zt stands for a1,t′, a2,t′, b11,t′, b12,t′, b21,t′, or b22,t′, and where the different data vectors z0,t were drawn from normal distributions in accordance with the given G matrix. Here, AM is the matrix square root, that is, AMAM=A. This reduced the variances of the random effects to approximately 75% of the nominal values. The random effects were also mean centred.As shown in Figure ft=0.5 of any given generation are offspring from the previous generation, that is, with the use of Equation  plots in panels (b) and (d).The simulations with results as in Figure Equation . As seena¯1,t over 60 generations in a stationary stochastic environment as before t=10 in Figure n=10to400, and based on 100 repeated simulations this change had a mean value of approximately zero, and a standard error of approximately 0.1, that is, around 10% of the corresponding changes in panel (a) in Figures n=2, this standard error due to drift increased to 15%. There were no noticeable differences between cases with At≠In and At=In. In order to find an explanation of the drift, the environmental cue u1,t over 60 generations in a stationary environment was approximated by a least squares straight line, denoted u^1,t. Based on 100 repeated simulations with n=100, the resulting change in u^1,t over 60 generations had a mean value of approximately zero, and a standard error of approximately 0.30, that is, around 12% of the corresponding changes of u1,t in Figure The changes in mean values over 60 generations in Figures 4p phenotypic traits and q environmental cues, that is, with p1+q mean reaction norm parameters, can be formulated as a linear mixed model with fixed and random effects. In this model, the fixed effects will be the mean trait values, while the random effects will be the additive genetic components of individual deviations from mean reaction norm parameter values. From this follows that for any given parent generation, estimates of the mean trait values and the additive genetic components of individual deviations from mean reaction norm parameter values, can be found from BLUP equations. Because the incidence and residual covariance matrices are functions of the environmental cues, and thus of time, I have introduced the concept of dynamical BLUP. Note that the elements in the random effects vector in Equation , this leads to ∆z¯t=covwi,txi,t+ei,t. Robertson's secondary theorem of natural selection, on the other hand, states that ∆z¯t=covwi,txi,t where xi,t is the breeding value. For the special application of the Price equation in Equation  where data from all generations are used. Note that this confounding is not a problem in the modeling process, where a¯t and b¯t in ai,t′=ai,t−a¯t and bi,t′=bi,t−b¯t, respectively, are found from Equation  applied on data from a single generation is not possible for plastic organisms. A simple reason for this is that the variance components Equation , all theEquation , and wheft of the population will be offspring from the previous generation, and as shown in Equation ; formal analysis ; investigation ; methodology .There are no competing interests.


Furthermore, we revealed that widespread genetic complementarity contributed to indica–japonica intersubspecific heterosis in yield traits, with dominance effect loci making a greater contribution to phenotypic variance than overdominance effect loci. On the basis of the comprehensive dataset, a genomic model applicable to diverse rice varieties was developed and optimized to predict the performance of hybrid combinations. Our data offer a valuable resource for advancing the understanding and facilitating the utilization of heterosis in rice.Exploitation of crop heterosis is crucial for increasing global agriculture production. However, the quantitative genomic analysis of heterosis was lacking, and there is currently no effective prediction tool to optimize cross-combinations. Here 2,839 rice hybrid cultivars and 9,839 segregation individuals were resequenced and phenotyped. Our findings demonstrated that  2 individuals from 18 elite crosses is used to characterize the genetics underlying a range of grain yield-related traits, providing insights into heterosis during breeding and a predictive model.Resequencing of 2,839 hybrid rice cultivars and 9,839 F With the development of the first commercial hybrid maize variety in the 1930s (ref. 1), and the development of rice hybrid varieties in the early 1970s in China2, exploitation of heterosis in crop plants has achieved remarkable yield advantages over inbred lines, and remains a crucial approach to increase agricultural production for global food demand in response to rapidly increasing global population and changing climate4.Heterosis or hybrid vigor refers to the phenomenon that the heterozygous first filial generation  is observed to contribute to heterosis in Arabidopsis, maize and rice13. Large-scale genomic analysis based on commercial rice hybrids and associated populations, provides the support for partial dominance of heterozygous loci for yield-related traits, and the phenomenon of heterosis in rice agricultural production emerges from the nonlinear effects of multiple heterozygous genomic loci combined15. The observation provides evidence that there is a dosage-sensitive component to heterosis16. Genomic studies have identified a number of quantitative trait loci (QTLs) that contribute to rice heterosis18. However, comprehensive analysis that scans genome-wide breeding footprints in rice hybrids to identify and evaluate heterotic loci in a dynamic perspective is lacking.Three non-mutually exclusive hypotheses—dominance, overdominance and epistasis—have been proposed to explain the genetic basis of crop heterosis2 individuals derived from 18 elite hybrids, which represents the majority of commercially used rice hybrids in China over past 50 years. We systematically investigated the genetic basis of phenotypic changes for grain yield-related traits, flowering time (heading date) and grain quality, and explored how breeding history applied heterotic loci and shaped the genomic architecture in rice hybrids. We further investigated the genetic basis underlying strong heterosis in intersubspecific rice hybrids. Moreover, a genomic selection model was developed on the basis of the comprehensive dataset, to facilitate selection of hybrid combinations with high yield potential and design of crossing plans. The model has been validated with respect to its accuracy and broad applicability, including testing in intersubspecific rice hybrids.In this Article, we performed a comprehensive genome analysis of 2,839 rice hybrid cultivars and 9,839 FOryzasativa cv. Nipponbare IRGSP 1.0 reference genome19 to identify polymorphisms. When compared with 100-fold sequencing data, 35-fold sequencing data could capture 96.70% of high-quality single-nucleotide polymorphisms (SNPs), with accuracy estimated at 95.21% and 94.72% for all and heterozygous loci  for 964 representative hybrids.The experimental design of this study is shown in Extended Data Fig. 2 progenies . Users can rapidly access the sample, phenotype, variant and genome-wide association analysis (GWAS) information of interest. The tool also demonstrates the allele frequency and potential impact of variants (according to annotation by SnpEff22), and provides haplotype analysis.To facilitate access to the resource, we constructed a web-based tool , Boro II (BT) and Honglian (HL) cytoplasms from three-line breeding system, as well as Twoline-Jap (TJ) and Twoline-Ind (TI) cytoplasms from two-line breeding system. Phylogenetic analysis based on nuclear polymorphisms implied that WA, TJ and BT hybrids were clearly separated with only a few exceptions and that TI hybrids were genetically mixed with WA and TJ hybrids .According to previous report, hybrids hold five types of cytoplasmPhenotypic comparison of hybrids from three breeding periods was conducted. Of involved traits, heading date, source and sink organ traits, and grain quality changed significantly Fig. . CorrelaPhenotypic changes demonstrated that the achievements of hybrid breeding in the past five decades were reflected in the substantial enlargement of source and sink organs, slight reduction of flowering time and improvement of grain quality.indica–indica hybrids but that TJ hybrids possessed japonica cytoplasms23, we further surveyed the japonica introgression level. Across the whole genome, unequal levels of both heterozygous and homozygous introgression between WA and TJ hybrids were observed (the details for definition of heterozygous and homozygous introgression in NAL1 (Os04g0615000) controlling leaf size and spikelet numbers25, Waxy (Os06g0133000) associated with grain quality28, GW3p6 (Os03g0215400) (refs. 30) and GS6 (Os06g0127800) (ref. 31) controlling grain size and weight  or large (≥100) count of alleles leading to short heading date was less than that in Y1 and Y2 periods  (ref. 32), Hd1 (Os06g0275000) (ref. 33) and Ghd7 (Os07g0261200) (ref. 34), increased from periods Y1 to Y3   respectively explained 12.71% and 8.11% phenotypic variance for leaf width and grain number per panicle in hybrids, with d/a value estimated at 0.61 and 0.79. The number of WA and TI hybrids holding heterozygous genotypes slightly increased during improvement breeding. A newly identified locus GNP3p1 explained 1.89% of phenotypic variance for full grain number per panicle in hybrids, and had its d/a value estimated at −0.14, indicating a negative dominance effect   , ALK (Os06g0229800) (ref. 26), GW5 (Os05g0187500) (refs. 38), GS3 (Os03g0407400) (ref. 39) and GW7 (Os07g0603300) (refs. 41), were identified as master loci controlling grain quality by GWAS. For those genes, frequencies of the breeding-favorable alleles increased during improvement breeding , Ghd7, Ghd8 (Os08g0174500) and Hd1, which were differentiated between indica and japonica subspecies according to previous report46. These results indicated genetic complementation was prevalent in intersubspecific hybrids. We further evaluated the effects of genetic complementation by F2 individuals derived from an indica–japonica hybrid Quanjingyou No.1. In the hybrid, 0.98%, 59.22% and 39.80% genomic segments were indica/indica, indica/japonica and japonica/japonica genotypes, respectively  located nearby) and bin1687 (Hd1 located nearby), breeding-favorable alleles were contributed by maternal line and their d/a were evaluated at 1.40 and 0.24, respectively. Loci at bin0796 and bin2547 (DEP1 (Os09g0441900) located nearby) were from paternal line, with d/a evaluated at 0.09 and 0.01  method. The prediction accuracies for yield per plant, valid panicle number, full grain number per panicle, seed setting rate, plant height, heading date and grain shape were 0.518, 0.559, 0.689, 0.728, 0.790, 0.799 and 0.945, respectively  (ref. 49) and rf4 (Os10g0495200) (ref. 50), both of which were responsible for wild-abortive sterility. Moreover, there was no high-scoring TJ combination containing the homozygous genotype of tms5 (Os02g0214300) (refs. 52), which contributed to environment-sensitive genic male sterility  the magnitude of dominance effects varied across different loci, posing a challenge for reliably estimating the dominance effects; (2) including dominance in model elevated complexity, hindering precise estimation of marker weights. Current models were inadequate in handling the non-additive effect, and the community will need more time to construct a new model to address this issue. The breeding potential predicted by our selection model could help identify core collections from extensive germplasm resources, thereby serving as backbone materials for further improvement. In the future, with the characterization of more quantitative trait genes and better understanding of genetic frameworks, molecular design can be implemented to fully develop the potential of heterosis.The genomic selection model developed in this study presented a valuable tool for breeders to search efficiently for optimal crossbreeding combinations and accelerate breeding cycles while reducing costs. Given the significance of non-additive effects in rice hybrid heterosisEthics approval was not required in this article.13. According to information recorded by the China Rice Data Center (https://www.ricedata.cn/), 2,715 out of 2,839 hybrids had registration information and were released from year 1985 to 2017. We performed Y1 to Y3 classification for 2,715 hybrids based on the following: (1) Policy steering in late 1990s and 2000s. China Ministry of Agriculture established a national mega project entitled ‘Breeding and cultivation system of super rice in China’ in 1996. The project encouraged combined use of heterosis and ideal plant type to increase yield, and they set standards for ideal plant type of super rice: erect panicle model, long erect leaves with delayed senescence model, early vigorous growth and heavy panicle model4. (2) Boom of two-line hybrids in 2000s . With the reference sequence available and development of high-throughput sequencing and genotyping methods, rice molecular biology and functional genomics made great progress, which further accelerated the rational design and molecular breeding of hybrid super rice by marker assisted selection in the past decade62.All 2,839 rice hybrids were from the collections preserved at the China National Rice Research Institute in Hangzhou, China, comprising 1,495 rice hybrid accessions reported previously00s Fig. . (3) Pro1 lines were grown in the consecutive farmland with well-distributed soil status. For each accession, 15 samples were germinated and planted in three rows with row and column spacing setting to 20 and 26.7 cm, respectively. Plants in the middle of the plot were selected to investigate phenotype, involving morphological characteristics , yield components and relevant factors , grain quality-related traits  and physiological feature (heading date). Among them, heading date and morphological characteristics were measured in the field. The heading date was recorded as the duration in days from the date of sowing to the emergence of first inflorescences above flag leaf sheath of five plants for each accession. The other field traits were investigated after heading, and three individuals were selected for phenotyping: the tallest panicle was selected to investigate plant height, flag leaf length and width; tiller angle was measured between tiller and the ground level. The collection of observations for grain-related traits was carried out in the laboratory after harvest, and they were investigated with three biological replicates for each sample. Full/total grain number per plant was counted manually; valid panicle number was directly counted after removing ineffective panicles; full grain number per panicle was estimated by dividing the full grain number per plant by the valid panicle number; seed setting rate was estimated by calculating the ratio of full grain number to total grain number per plant; kilo-grain weight was obtained by weighing 400 fully filled grains and then converting the value to 1,000-grain weight; yield per plant was recorded as the weight of all full grains per plant. Grain quality traits were investigated according to the Chinese national standard . Grain shape was measured as length-to-width ratio of polished grain, which was the grain having its husk and outer brown layers removed. Grains from mixed harvest of the same accession were randomly selected to investigate grain quality-related traits, with two replicates.All hybrid accessions were germinated and planted in an experimental field in Hangzhou, China in summer of 2020. All the F2 populations, generating a total of 9,839 F2 lines. The 18 hybrids were carefully selected to encompass a comprehensive range of major hybrid rice types. Specifically, they consisted of ten intrasubspecific and eight intersubspecific combinations; additionally, they were also categorized into five WA, three TJ, three TI and seven BT hybrids according to cytoplasmic types, providing a diverse representation within the chosen set. The planting standards utilized for F2 individuals were as those employed for F1s. Similar to the procedures employed for F1s, the investigation of physiological feature, morphological characteristics, yield components and relevant factors was also carried out for the F2 lines. No replication was prepared for F2 lines because F2 individuals have different genotypes.Eighteen hybrid accessions were chosen to construct FLeaves were collected from plants in vegetative growth phase in the field for each accession, and genomic DNA was then extracted from the fresh leaf tissue using DNeasy Plant Mini Kit (Qiagen). A sequencing library with ~400 bp insert size was constructed by Trueprep Tagment 3 Enzyme . The amplification-free method was used for library construction to accurately judge heterozygous sites by reducing duplicate sequences. Sequencing was performed on the NovaSeq 6000 platform, generating 150 bp paired-end reads.63), and bases of low quality or from adapter contamination were removed by the tools with parameters ‘ILLUMINACLIP:TruSeq3-PE.fa:2:30:10:2:true MAXINFO:50:0.6’. The clean reads were then aligned against the rice reference genome IRGSP 1.0 by software BWA (version 0.7.1) (ref. 64) with parameters of ‘-R ‘@RG\tID:SampleID\tPL:Illumina\tSM:SampleName’ -M’. Read alignment results were sorted according to their coordination by ‘SortSam’ functions in Genome Analysis Toolkit v4.1.4.1 (GATK) (ref. 65) with default parameters. Variations were called using ‘HaplotypeCaller’, ‘GenomicsDBImport’ and ‘GenotypeGVCFs’ functions in GATK. High-quality variations were contained by the ‘VariantFiltration’ function in GATK with parameters of ‘–cluster-size 3 –cluster-window-size 10 QD < 10.00 FS > 15.000 AC < 3 DP > 200||DP < 5’ for SNPs and ‘QD < 10.00 FS > 30.000 DP > 200||DP < 5’ for InDels.Quality control was conducted using the software Trimmomatic (version 0.38) (ref. 66) using 5,222,902 high-quality SNPs. Kinship analysis was conducted to identify hybrid pairs sharing close relationships in the same breeding period, allowing for further assessment of genomic diversity of rice hybrids. Kinship coefficients were estimated using the ‘emmax-kin’ program in EMMAX software (version emmaxbeta-07Mar2010) (ref. 67) with default parameters. The cutoff value of kinship coefficient was set as 0.9. The result was visualized using Cytoscape desktop software version 3.8.2 (ref. 68). A dot represented a hybrid accession and was colored according to its cytoplasm type. Pairs of hybrids with kinship coefficient greater than 0.9 were connected by a beeline. Principal component analysis (PCA) and admixture analysis were based on 223,533 fourfold degenerate (4DTv) sites. SNPs were annotated by a gene model (snpEff_v4_3_ENSEMBL_BFMPP_32_268.zip) in software SnpEff (v 4.3t) (ref. 22), and the 4DTv sites were identified using customized Python script from the annotated SNPs. PCA was performed by software GCTA  (ref. 69) with default parameters. The highest-ranked three principal components were chosen for visualizing the population structure. Ancestral components for hybrids were inferred by the ADMIXTURE program (version 1.3.0) (ref. 70) with K values from 2 to 5, and the population admixture result was visualized using the R package pophelper (version 2.3.1) (ref. 71).Nucleotide diversity was calculated with VCFtools (version 0.1.15)  ≥21 japonica varieties held the same genotype; (3) the major alleles in indica and japonica populations were different. We identified 830,245 indica–japonica differential SNPs in total.Zhao et al. have reported sequence variants for 66 divergent rice accessions that included 19 japonica fragments were detected across the whole genome: First, we analyzed the genotypes  at the 830,245 differentiated loci in the samples to be tested. Then, a 199-SNP sliding window with 1-SNP step was applied to identify introgressive fragments. Within the 199-SNP length fragment, if there were at least 120 SNPs with homozygous genotypes of japonica origin, the segment was labeled as homozygous japonica introgression; if there were at least 120 SNPs with heterozygous genotypes (indica/japonica), the segment was deemed as heterozygous japonica-introgressive fragments. Adjacent fragments in the same state were then combined for further analysis.On the basis of the differentiated SNPs, introgressive indica- or japonica-origin sequences, representative O. sativa japonica cv. Nipponbare and O. sativa indica cv. Shuhui498 (ref. 73) were chosen to scan for the distribution of indica- or japonica-origin segments across the whole genome, respectively. Out of 830,245 loci, 99.97% SNPs from cv. Nipponbare were judged as of japonica origin and all the 199-SNP length fragments across the whole genome were considered to be of japonica lineage. For cv. Shuhui498, 95.39% SNPs were judged as of indica origin and all the 199-SNP length fragments were identified as of indica lineage 74 for all three cohorts. For rice hybrid cohort, the high-quality SNP data were further filtered by software PLINK (v.1.90b6.12 64-bit)75 to keep variants with a missing rate ≤10% and minor allele frequency ≥5%. For F2 progenies, every individual was genotyped on the basis of high-density SNPs of both parental lines, as described previously21, and further imputed by the genotyping results and SNP information in both parental lines using a customized Perl script. In total, 1,296,386 and 1,059,427 SNPs were supplied to perform GWAS analysis for indica–indica and indica–japonica F2 progenies, respectively. PCA was performed using the input genetic markers by the software GCTA69, and the first two principal components were incorporated as the covariates to effectively account for population structure. Kinship matrix was generated based on the input genetic markers by ‘Kinship’ function in TASSEL. The significant threshold was set as 1 × 10−6 for all three cohorts.GWAS was performed for three cohorts separately, including 2,716 d/a) for association signal was calculated based on genotype effect estimated by TASSEL. The genotype effects of peak SNP in target association signal were chosen to calculate d/a index:A and C respectively represented the genotype effects of two homozygous genotypes, and M represented the genotype effect of heterozygous genotype. For traits with high observation preferred by breeding practice , the degree of dominance was quantified by calculating the ratio of d/a; for traits with low observation preferred by breeding practice , the degree of dominance was quantified by calculating the ratio of −d/a. If the number of observations of any homozygous genotype was less than 5, the estimated effect could be influenced by outliers and might be unreliable. Thus, such locus was excluded in the calculation of d/a index.The dominance-effect/additive-effect  (ref. 76).Furthermore, for QTL mapping by F77. The candidate regions were the 1 Mb region surrounding the genome-wide association signals. A mixed linear model with multiple random effect was applied to estimate the variance components using the R package sommer (version 4.2.0.1) (refs. 79). The model can be written as:y is a vector of the phenotypic value, b is a vector of fixed effects, and ua1, ud1, ua2 and ud2 are vectors of random effect. L, T1, T2, T3 and T4 are incidence matrices for b, ua1, ud1, ua2 and ud2, respectively. e represents a matrix of residual effects and e ~ N and outside (i = 2) the candidate region, and udi ~ N and outside (i = 2) the candidate region, and The PVE by the candidate region surrounding the association signal was estimated according to previously report80. With respect to others, the genotype effects of leading SNP in significant association signal were used to define the breeding-favorable genotype. If a locus associated with multiple phenotypes, only the leading SNP in the association signal with the largest −log10(P) value was selected for analysis. Genotyping was conducted by the leading SNP, and the average phenotypes of both homozygous genotypes were calculated to help determine the breeding-favorable genotype. For grain yield-related traits, the genotype associated with higher grain yield was considered to be the breeding-favorable genotype. For morphological traits, the genotype associated with larger leaf size or tiller angle (erect plant architecture) was defined as the breeding-favorable genotype. With respect to heading date, short heading date resulted in a yield penalty but long heading date was impractical for agricultural production in most rice planting areas; thus, an appropriate heading date was chosen for hybrid breeding. According to the performance of hybrid cultivars in our collection, almost all accessions completed their life cycle when planting in Hangzhou, and a slightly shortened heading date was observed in Y3 hybrids. Thus, the genotype with shorter heading date was defined as the breeding-favorable genotype in our work. Definition of the breeding-favorable genotype for grain quality referred to the Chinese national standard . According to the standard, the value of chalkiness and grain translucency (level) were lower for O. sativa varieties with better grain quality, and the value of gel consistency was higher for O. sativa varieties with better grain quality. Thus, the genotype leading to lower chalkiness, lower grain translucency (level) or higher gel consistency was deemed as the breeding-favorable genotype in our work. With respect to amylose content, the value range was 13.0–18.0 for first-class O. sativa indica and japonica rice varieties, was 13.0–20.0 and 13.0–19.0 for second-class O. sativa indica and japonica rice varieties, and was 13.0–22.0 and 13.0–20.0 for third-class O. sativa indica and japonica rice varieties, respectively. The value range was narrower for first-class varieties, with a lower upper limit. Thus, for convenience, the genotype associated with lower amylose content within limit (13.0–22.0) was determined as the breeding-favorable genotype.For loci with definite QTNs reported previously, the definition of breeding-favorable genotype was in accord with previous reporty is a vector of the phenotypic value, β is the vector of fixed effects, ua and ud are vectors for additive and dominant effects, X, Z1 and Z2 are incidence matrices for fixed and random effects, and ε is the residual error. The additive relationship matrix A was constructed with the A.mat function in sommer, and markers were coded −1, 0 and 1 with respect to the homozygous reference genotype, heterozygous genotype and homozygous alternative genotype, respectively. The dominance relationship matrix D was calculated with the D.mat function in sommer, and markers were coded as 0 and 1 for homozygotes and heterozygote respectively. The equations for the additive relationship matrix A and the dominance relationship matrix D are as follows:W and H being the scaled marker matrix for A and D, respectively, and p and q being the allelic frequencies for the jth marker (j = 1…m).We used the GBLUP method in R package sommer to train the model. The basic equation for our model was as follows:We used sixfold cross-validation to test the accuracy of the model. The accuracy was evaluated through calculating the Pearson correlation coefficient between the GEBV and observed value. Three iterations were performed and averaged to determine the final accuracy.All individuals were sorted by their GEBV, and a high score was given to individuals in the best rank. The score of seven traits was comprehensively incorporated by calculating the selection index. For yield per plant, seed setting rate, full grain number per panicle and valid panicle number, the following percentile GEBV ranges and associated scores were respectively assigned: >80%: 40; 60–80%: 30; 40–60%: 20; 20–40%: 10; <20%: 0. With respect to the rest of three traits, the optimal value ranges were 80–90 days for heading date, 115–125 cm for plant height, and 3.0–3.4 for grain shape. A score of 40 was assigned to the GEBV falling within the optimal range, while a score of 20 was allocated to the GEBV falling outside the optimal range.A validation population was derived from a random combination of 58 inbred lines and 19 commercial male sterile lines. Among all possible combinations, we randomly selected 67 combinations and planted them in the field conditions of Shanghai, China in the summer of 2021. On the basis of the selection index, 13 and 10 individuals were scored in the top 20% and bottom 20%, respectively. We investigated grain yield-related traits . Two-sided Wilcoxon test was conducted to discern significant differences between high-scoring and low-scoring subsets.indica–indica hybrids, to better identify loci associated with phenotypic variation in indica–indica rice hybrids. The field work and phenotypic investigation of randomly selected 67 pseudo-combinations out of all 1,102 combinations were parallel to the model construction and selection index calculation, and the analysis was under double-blind experimental control.No statistical method was used to predetermine sample size. No data were excluded from the phenotypic comparison analysis. Considering the TASSEL software was sensitive to outliers, the outliers were identified by ‘boxplot$out’ script in R-4.1.0 and removed for yield per plant, full grain number per plant, seed setting rate, valid panicle number and chalkiness in P ≤ 0.05) for data with distributions conforming to the assumption of homoscedasticity and data with a heteroscedastic distribution. ‘LSD.test’ in R package agricolae (version 1.3–5) and ‘kwAllPairsNemenyiTest’ in R package PMCMRplus (version 1.9.6) with parameter of ‘dist = Chisquare’ were further used to determine multiple comparison in data with distributions of homoscedasticity and heteroscedasticity, respectively. Box plot was used to demonstrate the distribution of data. For each box plot, the upper and lower boundaries represent the 25th and 75th percentile, respectively; the middle horizontal lines represent the median; the whiskers represent 1.5× the interquartile range; and the dots beyond the whiskers represent outliers.The function of ‘aov’ and ‘Kruskal.test’ in R 4.1.0 were used to identify significant phenotypic change (Further information on research design is available in the Any methods, additional references, Nature Portfolio reporting summaries, source data, extended data, supplementary information, acknowledgements, peer review information; details of author contributions and competing interests; and statements of data and code availability are available at 10.1038/s41588-023-01495-8.Supplementary InformationSupplementary Fig. 1.Reporting SummaryPeer Review FileSupplementary DataSupplementary Table 1. Information of 2,839 rice hybrids. Supplementary Table 2. Resequencing and alignment information of 2,839 rice hybrids. Supplementary Table 3. Estimation of SNP capture rate and calling accuracy for resequencing data with graded depth. Supplementary Table 4. Phenotype of 2,839 rice hybrids. Supplementary Table 5. Association signals underlying grain quality and explaining phenotypic variance larger than 1%. Supplementary Table 6. The list of 58 inbred lines and 19 commercial male sterile lines applied for model verification.