We provide a theoretical framework for understanding the results of PGI analyses and a practical methodology for estimating heritability and indirect genetic effects while accounting for assortative mating.Both direct genetic effects  and indirect genetic effects — effects of alleles in an individual (e.g. parents) on another individual (e.g. offspring) — can contribute to phenotypic variation and genotype-phenotype associations. Here, we consider a phenotype affected by direct and parental indirect genetic effects under assortative mating at equilibrium. We generalize classical theory to derive a decomposition of the equilibrium phenotypic variance in terms of direct and indirect genetic effect components. We extend this theory to show that popular methods for estimating indirect genetic effects or ‘genetic nurture’ through analysis of parental and offspring polygenic predictors  are substantially biased by assortative mating. We propose an improved method for estimating indirect genetic effects while accounting for assortative mating that can also correct heritability estimates for bias due to assortative mating. We validate our method in simulations and apply it to PGIs for height and educational attainment (EA), estimating that the equilibrium heritability of height is 0.699 (S.E. = 0.075 ) and finding no evidence for indirect genetic effects on height. We estimate a very high correlation between parents’ underlying genetic components for EA, 0.755 (S.E. = 0.035), which is inconsistent with twin based estimates of the heritability of EA, possibly due to confounding in the EA PGI and/or in twin studies. We implement our method in the software package  R.A. Fisher’s foundational 1918 paper, The correlation between relatives on the supposition of Mendelian inheritance2, unified the biometrical approach to heredity — whose intellectual lineage traces back to Galton — with Mendelian inheritance3. Fisher’s primary concern in this work was to show how Mendelian genetics, which describes the inheritance of discrete entities called alleles, could explain resemblance between relatives for continuous phenotypes like height. Fisher showed that, when alleles have additive effects, and there are no other sources of correlation between relatives, the phenotypic correlations between relatives are determined by the heritability of the phenotype, 6.Since Galton’s 1886 work on the relationship between parent and offspring height9. Fisher derived the phenotypic correlations between relatives at equilibrium which Greg Clark showed can be closely fit to correlations between relatives’ social status in England in a dataset spanning 1600–2022 (ref10) — although these data do not rule out environmental effects of parents on offspring.After this groundbreaking result, Fisher’s paper goes on to address the more difficult problem of assortative mating (AM) — non-random mating leading to phenotypic correlation  between mothers and fathers, and therefore correlations between maternally and paternally inherited alleles. Fisher showed that AM induces a correlation between maternal and paternal genetic components, which increases the variance of the genetic component in the subsequent generation. The correlations between relatives’ genetic components and the variance of the genetic component increase towards equilibrium values as this process continues, requiring only a handful of generations to reach an approximate equilibrium11, also detailed in Chapter 4 of Crow and Kimura’s textbook7. They showed that, for a phenotype affected by many genetic variants spread across the genome, the genetic variance is inflated by a factor of 13. Assuming primary phenotypic assortment, and that the regression of phenotype onto genetic component is linear8, it can be shown that 11.Building on Fisher’s work, Crow and Felsenstein gave an alternative derivation of the increase in genetic variance due to AM15. The models of Cavalli-Sforza and Feldman were influential in the creation of the field of gene-culture coevolution16. In 1978, Cloninger, Rice, and Reich extended the models of Cavalli-Sforza and Feldman to include transmission of a general “cultural value” (possibly distinct from the offspring phenotype) from parent to offspring17. They extended this model to include AM due to matching on the phenotype, giving equilibrium results18. Their model makes predictions about the correlations between relatives18, and versions of their model have been used to analyse lifespan using a large pedigree from Ancestry. com19 and educational attainment (EA) using Swedish register data20. However, these analyses — like Clark’s analysis of social status in England10 — are unable to separate genetic transmission (heritability) from cultural transmission without making assumptions that are unlikely to be true.The theory described above applies to phenotypes determined by additive effects of alleles in an individual on that individual, called direct genetic effects (DGEs), and random environmental effects/noise. In the 1970s, Cavalli-Sforza and Feldman developed models where, in addition to genetic transmission from parents to offspring, parental phenotypes affect the offspring’s phenotype through an environmental process called “vertical transmission” or “cultural transmission”12. The ‘stealth model’ from Truett et al. in 199421 enabled modelling of assortative mating due to matching on the observed phenotype. Keller et al.12 introduced the ‘cascade model’ in 2009, a generalisation of the stealth model that allows the matching to take place on a latent, unobserved phenotype. Although ETFDs can separately identify genetic and cultural transmission from parents while accounting for assortative mating, obtaining precise estimates of the parameters of these models can be difficult due to limited samples of twin pairs with phenotype data on their relatives22.Building on these vertical/cultural transmission models, behaviour genetics researchers extended the classical twin design — based on estimation of heritability by comparison of monozygotic (MZ) and dizygotic (DZ) twin pairs — to ‘extended twin and family designs’ (ETFDs) that also model the phenotypes of parents and other relatives of DZ and MZ twins. These ETFDs can model vertical transmission from parents, in addition to heritability27, correlations induced by assortative mating30, and indirect genetic effects 33 — robust estimation of vertical transmission model parameters remains challenging 33.While empirical analyses of genotype-phenotype data have produced estimates of some important parameters that relate to vertical transmission models — including heritability35. This manuscript focuses on parental IGEs, effects of alleles in parents on their offspring through the rearing environment, but IGEs could come from other classes of relatives, such as siblings36, or from unrelated individuals. Vertical transmission models induce parental IGEs when the parental phenotype that affects offspring through the environment is heritable: the genetic variants that affect the parental phenotype will have IGEs on the offspring phenotype34. There is evidence that parental IGEs are important for educational outcomes32, but this evidence has been contested as confounded with the influence of population stratification and AM39.IGEs are causal effects of alleles in one individual on another individual’s phenotype, mediated through the environment. When IGEs come from genetically related individuals, they contribute to the genotype-phenotype associations estimated in genome-wide association studies (GWASs) and lead to bias in heritability estimates from many methods40, including: classical twin studies42; methods based on realized relatedness between relatives, such as siblings24 and more distant relatives, as in Relatedness Disequilibrium Regression, or RDR26; LD-score regression, or LDSC40; and genomic relatedness-matrix restricted maximum likelihood, or GREML40. While techniques for adjusting for the bias due to AM have been proposed23, they typically assume that all the correlation between maternal and paternal genetic components is explained by matching on the phenotype. This model has been shown to be inaccurate for EA, where the correlation between maternal and paternal genetic predictors of EA is far higher than can be explained by matching on the phenotype43. Factors that may contribute to maternal and paternal genetic predictors (or components) becoming more correlated than expected due to matching on the observed phenotype include: matching on a correlated phenotype that is more highly correlated with the underlying genetic predictor/component than the observed phenotype30, and matching based on the phenotypes of the mate’s family members — and/or ancestry — in addition to the phenotype of the mate. It would therefore be desirable to have a technique for adjusting for bias due to AM that does not assume the correlation between parents’ genetic components is entirely due to matching on the phenotype.AM generates bias in most methods for estimating heritability32. While this correlation captures IGEs, it also partly captures the genetic component of the phenotype with which the non-transmitted alleles are correlated due to AM39. Kong et al.31 attempted to adjust for this bias due to AM, concluding the bias was small. However, they did not measure the uncertainty in their adjustment, and the adjustment relied upon assuming that there had been only one generation of AM and that DGEs and IGEs are perfectly correlated. Balbona et al. proposed a structural equation model (SEM) to adjust for the bias due to AM in estimates of IGEs but this method assumes correlations between parents’ genetic components are due entirely to matching on the phenotype. The method of Balbona et al. additionally requires observations on the parental phenotype through which IGEs/vertical transmission operate, or knowledge of the true heritability of the phenotype and an assumption that the parental phenotype through which IGEs/vertical transmission operates is the same as the offspring phenotype.Most studies examining evidence for IGEs have proceeded by correlating parental alleles not transmitted to offspring with offspring phenotypes11 approach (which considered only DGEs) to also include parental IGEs. We derive a decomposition of the equilibrium phenotypic variance in terms of DGE and IGE components that does not assume a particular model of assortative mating. We connect our results to estimation of heritability using random-variation in realized relatedness due to Mendelian segregations26, deriving the bias in these methods due to AM. We extend our approach to derive results on analysis of polygenic predictors . We show how to assess evidence for IGEs from PGI analysis when there is AM and to adjust for bias in heritability estimates due to AM. We apply these results to PGIs for height and EA, and we correct for the bias in the height heritability estimate due to AM.In this paper, we generalize Crow and Felsenstein’sWe model the phenotype of sibling  ignored .AM induces a correlation between parents’ DGE components, which we define to be:We decompose the equilibrium phenotypic variance in terms of the random mating variance decomposition and the equilibrium correlations between parents’ DGE and IGE components. 31:The random-mating variance decomposition is the same as given in Young et al. 2018, who derived it for the RDR method for estimating heritabilityIn The equilibrium phenotypic variance due to covariance between DGE and IGE components, 34, such asymmetries do not affect the PGI analysis results contained in this manuscript.In theory, it would be possible to estimate this variance decomposition given estimates of the random-mating variance components and correlations between parents’ DGE and IGE components . In SuppIf 7) when there is  correlation between DGE and IGE components. When DGE and IGE components are positively correlated, the equilibrium correlation between parents’ genetic components will thus be higher than would be predicted using the classic result, If we assume primary phenotypic assortment, then it can be shown that :rδ=heq26). By examining how the phenotypic correlation between siblings changes with realized relatedness, an estimate of heritability can be obtained that is robust to population stratification26. We call this method ‘sib-regression’.While siblings have a relatedness coefficient of ½ in expectation — based on the expected proportions of the genome shared identical-by-descent (IBD) from each parent — there is variation around this expectation due to random segregation of genetic material in the parents during meiosis. The realized relatedness between siblings is computed from the proportions of the genome shared IBD from each parent, and therefore captures the random variation in relatedness around the expectation. In outbred samples, the realized relatedness between siblings has an approximate normal distribution with mean close to 0.5 and a standard deviation around 0.04  can be used to estimate 31, many studies have examined both offspring and parental PGIs as predictors of offspring phenotypes 43. In the following sections, we show how to interpret the results of these studies in our model.Building on Kong et al.A PGI is a weighted sum of genotypes  across genetic variants:Given parental genotypes, offspring genotypes vary due to random segregations during meiosis in the mother and father. Furthermore, Mendelian inheritance induces an important relationship between parent and offspring PGIs. Letting The most common type of PGI analysis is a regression of phenotype onto PGI without controlling for parental PGIs :43:34. Often, 39. Since offspring PGI is conditionally independent of environment given parental genotypes, and 34.Most of the evidence for parental IGEs has derived from fitting a version of the following regression equation par(i)/2 , δPGI re26 . By inputting an estimate of snipar will estimate We implemented the estimation procedure  in snipeariances .simulate.py module in sinipar , ≈δl/1-rδ  with k=0To investigate how PGIs constructed from standard GWAS perform in our procedure, we constructed ‘population effect PGls’ as such:We compared inference of I cannot . This isrrelated . A limit43) , we used an estimate from a meta-analysis of twin studies53. We did not use an RDR or sib-regression estimate as the only available estimates26 lacked sufficient precision for reliable inference  . Okbay enference .26, which leads to underestimation of Our results using the RDR estimate of 26, and confounding in the EA GWAS summary statistics34 that inflates the correlation between parents’ PGI values. The second phenomenon implies that part of the correlation between parents’ PGIs is due to confounding factors that are correlated between parents — such as social class — in addition to DGEs, leading to a correlation that is higher than would be obtained from a DGE PGI that explains the same fraction of heritability.For EA, we estimated that 11 to a phenotype affected by both DGEs and IGEs. The equilibrium variance is expressed in terms of the random-mating variance components as defined by Young et al. in relation to the RDR method of estimating heritability26  and parental PGIsmponents , with αPpulation .51, i.e. how the coefficient on the proband PGI changes when controlling for parental or sibling PGIs. In our framework, this ‘shrinkage’ corresponds to the ratio between direct and population effects, We derived theoretical results for PGI analysis under AM at equilibrium. We showed that, when assortative mating is strong and  of IGEs . Many st of IGEs . These rIn 2018, Kong et al. presented analyses of an educational attainment (EA) PGI in Icelandic data, and estimated that snipar, that accounts for AM by combining the results of two-generation PGI analysis with a family-based heritability estimate We develop a procedure , implemerecisely –7. Unlik51. The advantage of such PGIs over PGIs constructed from DGE estimates from family-based GWAS is greater statistical power due to larger sample sizes and the fact that ‘population effects’ capture both DGEs and IGEs. Under a ‘null model’ of AM but no IGEs, DGEs and population effects of variants should be approximately perfectly correlated, implying that population effect PGIs can be used in our procedure , which estimated Our procedure for estimating the correlation between parents’ underlying DGE components, While the simulation results indicated that using a population effect PGI in our procedure produces reasonably accurate results, the simulations did not consider the impact of population stratification confounding in population effect estimates. We hypothesize that confounding factors in the EA GWAS have inflated the correlation between parents’ EA PGIs beyond that which would be expected for a DGE PGI (without confounding) that explains the same amount of heritability. This would have the effect of overestimating the correlation between parents’ DGE components and therefore the equilibrium heritability.18, extended twin and family designs (ETDFs) such as the ‘stealth’ and ‘cascade’ models, and the structural equation model of Balbona et al.33. Unlike previous research, our results do not rely upon a particular model of assortative mating. A difference from previous models is that we do not model the total contribution of vertical/cultural transmission from parents to offspring, only the heritable component of the parental phenotype that influences offspring through the environment. While modelling the total contribution of vertical/cultural transmission is important for predicting phenotypic correlations between relatives, it is less important for PGI analyses and estimation of IGEs.This manuscript has focused on estimating a consequence of cultural/vertical transmission from parents to offspring, the IGEs that are induced when the parental phenotypes affecting the offspring through the environment are themselves heritable. Previous research on estimation of cultural/vertical transmission includes the cultural transmission model of Cloninger, Rice, and Reich19. ETFDs can separately identify genetic and cultural transmission, but these models generally assume vertical/cultural transmission operates through an observed phenotype. The same is true of the structural equation model of Balbona et al., which is the only other method that uses PGIs.Our method does not make any assumptions about the parental phenotype through which cultural/vertical transmission from parents to offspring operates and can model arbitrary correlations between DGEs and IGEs. This is in contrast to most previous research - except for the model from Cloninger, Rice, and Reich, which, while flexible, cannot separately identify genetic and cultural transmission when fit to phenotypic correlations between relatives24 and RDR54. Together, these will enable application of the theoretical results and methodology developed here to quantify the impact of IGEs and AM on phenotype variation.This manuscript provides a general theoretical framework for understanding the joint impact of AM and IGEs on phenotypic variance components, heritability estimation, and PGI analyses. As more genetic data on families becomes available, we will obtain more powerful family-based GWAS summary statistics, enabling us to apply our estimation procedure to DGE PGIs. We will also obtain more precise heritability estimates from sib-regressionIn real-world applications, we do not know To estimate In order to make inferences about the impact of assortative mating on the DGE component, we need to estimatete rδ by :Equati to zero .As shown above, To estimate what the ratio between direct and population effects of a PGIs would be under a model with AM at equilibrium but without IGEs, one can combine ed above  and 21:Since the parameters are estimated using a series of non-linear equations, we expect the estimators to be biased even when the input parameter estimates are themselves unbiased. Furthermore, standard errors are approximated using the Delta method , which chf2 . We founhf2 , i.e. thsimulate.py module in snipar (55. For each SNP We simulated 16 phenotypes with varying parameters using the n snipar . Each phThis implies that the heritability in the random-mating population was 0.5 for all phenotypes. For phenotypes with IGEs, the proportion of variance explained by parental IGEs was 0.125, and the proportion of variance explained by covariance between DGEs and IGEs was Subsequent generations were produced by random-mating or AM. AM was simulated by matching males and females into parent-pairs according to their rank on a noisy observation of their phenotype value:Supplement 1Supplement 2


Growth traits, such as body weight and height, are essential in the design of genetic improvement programs of dairy cattle due to their relationship with feeding efficiency, longevity, and health. We investigated genomic regions influencing height across growth stages in Japanese Holstein cattle using a single-step random regression model. We used 72,921 records from birth to 60 mo of age with 4,111 animals born between 2000 and 2016. The analysis included 1,410 genotyped animals with 35,319 single nucleotide polymorphisms, consisting of 883 females with records and 527 bulls, and 30,745 animals with pedigree information. A single genomic region at the 58.4 megabase pair on chromosome 18 was consistently identified across 6 age points of 10, 20, 30, 40, 50, and 60 mo after multiple testing corrections for the significance threshold. Twelve candidate genes, previously reported for longevity and gestation length, were found near the identified genomic region. Another location near the identified region was also previously associated with body conformation, fertility, and calving difficulty. Functional Gene Ontology enrichment analysis suggested that the candidate genes regulate dephosphorylation and phosphatase activity. Our findings show that further study of the identified candidate genes will contribute to a better understanding of the genetic basis of height in Japanese Holstein cattle. Growth traits such as BW and height play an economically important role in reducing feeding costs and enhancing the profit of dairy farms. Poor growth during the rearing period leads to a delay in age at first calving and increases culling risk at an earlier age. In addition, heifer growth rates are related to milk production during the first lactation . PreviouRRM) is applied to the genetic analysis of growth traits because it captures genetic changes  across time by fitting the additive genetic effect as a covariance function  method is an extension of the single-step genomic best linear unbiased prediction to the GWAS framework , the Hokkaido Agricultural Research Center in National Agriculture and Food Research Organization , the Agricultural Research Department in Hokkaido Research Organization , the Yamagata Integrated Agricultural Research Center , the Gunma Prefectural Livestock Experiment Station , the Chiba Prefectural Livestock Experiment Station , the Gifu Prefectural Livestock Research Institute , the Awaji Agricultural Technology Center , and the Nagasaki Agricultural and Forestry Technical Development Center  in Japan. We included individuals born between 2000 and 2016 with at least 3 records from birth to 60 mo of age. The animals were required to have records before 10 and after 30 mo of age. All animals included in this study were registered by the Holstein Cattle Association of Japan . Phenotypic records that deviated from the mean by more than 5 times the standard deviation at each age were excluded as outliers. After data editing, 72,921 records from 4,111 animals were retained for subsequent analyses. The average (± SD) number of records per animal was 17.7 ± 8.3.We used 883 females with records and 527 bulls, which were either their sires or maternal grand-sires. These individuals were genotyped using the Illumina Bovine50 BeadChip or BovineLD BeadChip (Illumina), designed based on the UMD3.1 genome assembly, and missing genotypes were imputed using findhap.f90 software . A totalGEBV) of height for genotyped animals were estimated using RRM-ssGWAS using the following equation:ijlty is the observation of the lth animal at age t, iHY is the fixed effect of ith herd-birth-year combination, jM is the fixed effect of jth birth-month, kb is the kth fixed regression coefficient explaining the overall mean, lku is the kth random regression coefficient of additive genetic effect, lkpe is the kth random regression coefficient of permanent environment effect, ijlte is the residual, andkth Legendre orthogonal polynomial of animal l evaluated at age t. In matrix notation, y = Xb + Z1a + Z2p + e, where y is the vector of phenotype, b is the vector of fixed effects, a is the vector of random additive genetic effect, p is the vector of random permanent effect, e is the vector of residuals, and X, Z1, and Z2 are incidence matrices. The corresponding mixed model equations (MME) can be shown as follows:H is the pedigree-based relationship matrix augmented with genomic information, I is the identity matrix, R is the diagonal matrix of heterogeneous residuals ranging from birth to 60 mo of age, Vg and Vp are 3 × 3 variance-covariance matrices for additive genetic and permanent environment effect, respectively, andH was defined asA−1 is the inverted numerator relationship matrix, G−1 is the inverted genomic relationship matrix, andG matrix was calculated by the first method of A22 as (1 − α)G + αA22, where α = 0.05, to avoid a singularity problem. The means of diagonal and off-diagonal elements of G were adjusted to be equal to those of A22  by calculatingt  enrichment analysis. The nearby genes were queried from the Genome Ensemble database (release version 94) using the biomaRt Bioconductor package  were investigated further using Gene Ontology ( package  coupled  package .10[1 − (1 − 0.05)1/n] = 5.24, where n = 8,876 is the effective number of SNP. The RRM-ssGWAS analysis identified persistent SNP across age groups. However, transient SNP whose effects vary over time were not detected. One SNP, Hapmap45189-BTA-43948 (rs41636786), on BTA 18 was consistently significant across all ages . Similarly, Hapmap40537-BTA-43945 (rs41582522), a neighbor of Hapmap45189-BTA-43948, was identified as a significant SNP at 40, 50, and 60 mo of age . We also investigated the change of proportion of the total additive genetic variance explained by each marker over time. The results showed a similar trend with the P-value-based Manhattan plots. Namely, the peak was observed on BTA18, and Hapmap45189-BTA-43948 (rs41636786) always had the largest value in the range of 0.12 to 0.16% across the 6 time periods. Several markers on BTA18 close to the 2 identified SNP have also been reported in recent studies using Canadian Holstein bulls  gene, which is one of the candidate genes identified in this study, has been reported to be associated with longevity (SIGLEC5) gene located at 57.9 Mbp (based on the UMD 3.1 assembly) may increase calf size in association with longer gestation length. Height is known to have a strong genetic correlation with BW and body conformation traits, including body depth and rump width (A total of 12 candidate genes were found in the vicinity of 2 significant SNP at 58.4 to 58.6 Mbp on BTA18 . The ideongevity  and gestongevity . Within mp width , 2014. Imp width . The resmp width . OverallP-values as shown in this study. However, computation of P-values in RRM-ssGWAS may be challenging when using a larger left-hand side matrix coupled with higher-order random regression coefficients and more genotyped animals because obtaining the inverse matrix is difficult. Therefore, in such a situation, different longitudinal GWAS approaches (In this study, we applied the RRM-ssGWAS to longitudinal height data. The ssGWAS approach enables the simultaneous inclusion of data from phenotyped, but nongenotyped, animals in the analyses. In addition, it can identify significant SNP based on their proaches  may be mTo the best of our knowledge, this is the first longitudinal study to investigate the changes in genomic marker effects over time using longitudinal height data in dairy cattle. Although the genetic correlations of height across the 6 age points were not constant and ranged between 0.39 and 0.99, the ssGWAS results did not identify significant transient genes affecting height at only one of these periods. The results of this study suggest that many genes with minor effects influence height over time. For the growth trait of BW, This study investigated the locations of QTL regions for dairy cattle height using time-series data by integrating ssGWAS and RRM. The longitudinal ssGWAS estimates SNP effects while accounting for covariance across age groups. In conclusion, we detected a single persistent QTL region on BTA18, likely to continuously influence height across growth stages and have a pleiotropic effect on various traits in dairy cattle. Therefore, further analysis is warranted to understand the association with important traits, such as calving, reproduction, and body conformation traits in Japanese Holsteins. In addition, investigating the effect of the identified candidate genes on heifer growth rate is expected to offer further insights into gene function. This study serves as the first step toward understanding the genetic basis of height in Japanese Holstein dairy cattle.


A comparison of genetic trends by source indicated that CIMMYT entries had a gain of 1.98% year-1 or 106 kg ha-1 year-1. In contrast, NARO and private sector maize entries recorded genetic gains of 1.30% year-1 (59 kg ha-1 year-1) and 1.71% year-1 (79 kg ha-1 year-1), respectively. Varieties from NARO and private sector showed comparable mean yields of 4.56 t ha-1 and 4.62 t ha-1, respectively, while hybrids from CIMMYT had a mean of 5.37 t ha-1. Era analysis indicated significant genetic gain of 1.69% year-1 or 55 kg ha-1 year-1, while a significant national productivity gain of 1.48% year-1 (37 kg ha-1 year-1) was obtained. The study, thus, demonstrated the importance of public-private partnerships in development and delivery of new genetics to farmers in Uganda.Estimating genetic gains is vital to optimize breeding programs for increased efficiency. Genetic gains should translate into productivity gains if returns to investments in breeding and impact are to be realized. The objective of this study was to estimate genetic gain for grain yield and key agronomic traits in pre-commercial and commercial maize varieties from public and private breeding programs tested in (i) national performance trials (NPT), (ii) era trial and, (iii) compare the trends with the national average. The study used (i) historical NPT data on 419 improved maize varieties evaluated in 23 trials at 6-8 locations each between 2008 and 2020, and (ii) data from an era trial of 54 maize hybrids released between 1999 and 2020. The NPT data was first analyzed using a mixed model and resulting estimate for each entry was regressed onto its first year of testing. Analysis was done over all entries, only entries from National Agricultural Research Organization (NARO), International Maize and Wheat Improvement Center (CIMMYT), or private seed companies. Estimated genetic gain was 2.25% or 81 kg ha It occupies the largest land area of all staples, with more than 35 million hectares (M ha) harvested annually . The data was categorized as follows: 1961 – 2020, 1999 – 2020 and 2008 – 2020 for analysis.We obtained estimates of maize grain yield in Uganda from 1961 to 2020 from FAO statistics  was performed within each experiment using the model:ijy is the mean phenotypic value of the trait within an experiment, ig is the random effect of the ith genotype with jr is the random effect of the jth replication with ije is the residual error with 2H) within an experiment with r replications was calculated aswhere ANOVA was performed within each trial using the model:ijky is the phenotype of the ith genotype in the jth location, in the kth replication. µ is the overall mean; gi is the random effect of the ith genotype with jl is the random effect of the jth location with r(l)jk is a random effect of the kth replication nested in the jth location with ijgl is the interaction of the ith genotype and jth location with ϵijk is the error with Where 2H > 0.20 were used in the subsequent analysis.where l is the number of locations for that trial and r the number of replications. Only data from experiments with 2 value greater than 0.05. The genetic value of an entry was estimated using the mixed model:Genetic gain in the NPT was estimated by regressing the estimated genetic value of each entry on the first year the entry was tested (FYT) in 2008 to 2020 data set. The FYT was the base year the entry was entered into the NPT. An experiment is a year/season/test/location combination, and an experiment is nested in a trial. We modeled the genotype x experiment interaction which encompasses genotype x location interaction. We declared a genetic gain to be significant if the probability of the slope was less than 0.05 and it was accompanied by an Rijklmy is the phenotypic value for genotype i tested in control group j, trial k, experiment l, and rep m. µ is the overall mean; ig is the fixed effect of the ith genotype, jc is the fixed effect of the thj control group were the checks, where j=1 for the control population consisting of the two checks and j=2 for the tested group; kt is a random effect for the thk trial with x(t)kl is a random effect for the lth experiment nested in the kth trial with ikgtis the random effect of the interaction of the ith genotype and kth trial with iklgx(t) is the random effect of the interaction of the ith genotype and lth experiment nested in the kth trial with r(x)lm is a random effect of the mth replication nested in the lth experiment with ϵijklm is the residual error with where u) *100, where u is the intercept in equation 5. For GY, the analysis was run across trials, and separately for trials with mean GY > 3.3 t/ha (high), and for trials with mean GY< 3.3 t/ha (low). We assessed the significance of the source of the NPT entries using the model shown in equation 5 by adding a term for the nth source (ns) as a fixed effect and changing gi to be the effect of the ith genotype nested in the nth source, ing(s).The model in equation 5 was run separately for all entries: only entries from private sources, only entries from CIMMYT sources, and only entries from NARO. The same controls were used in each analysis. The Best Linear Unbiased Estimator (BLUE) for each entry from equation 5 was then regressed onto the FYT value for that entry. Data from the checks were excluded from the regression analysis. The regression was done separately for each source of entries. The slope of this regression was used as the estimate of genetic gain. The percent genetic gain per FYT was calculated as the  effect was assessed in two ways. First, we analyzed variance using FYT as a covariate using the model:ijkly is the phenotypic value for genotype j, from FYT i, tested in environment k, and rep l. µ is the overall mean; ifis the fixed effect of the ith FYT, jg is the fixed effect of the jth genotype, kv is the random effect of the kth environment with jkgvis the random effect of the interaction of the jth genotype and kth environment with r(v)kl is a random effect of the lth replication nested in the kth environment with ϵijkl is the residual error with fi to test its significance and then once without fi to obtain BLUEs for each entry. These BLUEs were then regressed onto their FYT values to estimate the effect of increasing FYT. A genetic gain was declared significant if the FYT covariate test had a probability of P< 0.05, and the slope was significant at P< 0.05.where 2.4.3Analysis of the was done across all 60 years of average maize yield estimates, for 22 years (1999–2020) to reflect the range of years of entries in the Era trial, and for 13 years (2008–2020) to reflect the range of years for entries in the NPT. The national average maize yield was regressed over the baseline years.33.1There was considerable variation among the experiment means and heritability for the traits examined in this study Table 3.We obtained BLUEs using a model that used the checks as a fixed effect to adjust for the effect of experiments. Correlation analysis of the entry means with control/check means within experiments and within trials indicated that control means were a suitable surrogate for experiment effect for most traits with correlations greater than 0.70 for 5 of the 7 traits Table 5.3.1.1P< 0.05) in genetic gain for hybrid yields from all sources entered into the NPT (-1 or 106 kg ha-1 year-1). Entries from NARO and private sector sources had lower mean GY of 4.56 t ha-1 and 4.62 t ha-1, respectively. Entries from NARO had the lowest estimated genetic gain (1.30% year-1 or 59 kg ha-1 year-1).There was a significant difference ( the NPT Table 6.-1) and low GY experiments (mean GY< 3.3 t ha-1). The overall genetic gains in the high mean GY experiments were lower compared to that estimated across all environments. The genetic gain estimates for high mean GY experiments varied by source relative to estimates from all experiments with some increasing and others reducing (-1 year-1) compared to 106 kg ha-1 year-1 across all environments. The genetic gains were lower in the low GY experiments for the different germplasm sources (-1 year-1) in lower mean GY experiments.We also analyzed across the different sources of germplasm for high GY experiments (mean GY >3.3 t hareducing Table 6.3.1.2P< 0.05) genetic gain for decreasing days to anthesis when analyzed over all entry sources or within just the CIMMYT and commercial entries  of any trait/source combination. Reduced days to anthesis was observed in all the entries though the change could not always be declared significant. Entries from NARO had a lower AD mean compared with CIMMYT entries, which had the highest AD mean among all sources. There was evidence of increasing PH in the entries in the NPT though the only significant gain was for the NARO entries, where the increase was 1.58 cm year-1, or a gain of 0.89% year-1 (-1 at a rate of 0.93% year-1 (P< 0.05) genetic gain for EPP was noted among the CIMMYT and NARO entries, where the decrease was -1.05% and -0.72% year-1, respectively  and the two lowest yielding sites  (-1 year-1) although this gain was greatest on a percentage basis (2.52% year-1). The annual gains (43 to 61 kg ha-1 year-1) in the era trial were lower than those estimated in the NPT trial (59 to 81 kg ha-1 year-1) , accounting for more than 72% of the variation (-1 year -1 (2.85% per year).All three regressions of grain yield onto year were significant  of the inbred lines that are used to develop new products that are allocated and released for commercial production  and the era trials (1.52%). This shows that either of these datasets can be used to obtain reliable estimates of genetic gain, but NPTs offer a cheaper option for the estimates. The high overall genetic gain (2.25%) in NPTs obtained from all collaborators indicates that a combined effort of public-private partnerships is critical in delivering improved genetics to the farmers. The results showed that CIMMYT hybrids were significantly higher yielding than the NARO and private sector entries in the NPTs. This highlights the need for continuous improvement in breeding, for integrating new tools and methods to drive rates of genetic gain with sustainable funding for the public sector. We also noted significantly lower rates of genetic gain in the low GY experiments.In broad terms, currently grown maize varieties in Uganda have higher GY, have an earlier anthesis date, and are taller than those cultivated in the past, as these traits showed the same trend in each source of NPT entries and in the era trials 8. The tThe genetic trends indicated that grain texture significantly decreased over time in the CIMMYT entries tending to be more flint-type, but we observed a significant increase in the grain texture of the NARO entries, towards selection for dent type. The trends for PH and EH were interesting because while PH increased in all sources, EH was unchanged in the CIMMYT and private sector sources but increased in the NARO entries. Increasing PH over time probably indicates selection for more vigorous plant types. With product profile-based breeding and foreseen mechanized operations among emerging commercial farmers, there is a need to breed and select for shorter and more modern plant types for increased plant population density and higher harvest index.Exserohilum turcicum (Pass.) K. J. Leonard & Suggs) and gray leaf spot (caused by Cercospora zeae-maydis Tehon & E. Y. Daniels). This genetic improvement program of maize in Uganda, supported initially by USAID and later by the Rockefeller Foundation, partly explains the highest national productivity gains (62 kg ha-1 year-1) starting from 1999. The other major contributing factor was the advent of the private seed sector that took over from government managed seed industry in the mid-1990s.Maize productivity in Uganda increased for the three truncated periods analyzed . This significant upward trend of the national average maize yields in Uganda could be partly due to improved genetics and production practices. The highest national maize productivity was achieved between 1999 and 2020, consistent with a systematic and sustained maize breeding program that resulted in the release of the first suite of improved stress-tolerant hybrids in 1999. The first formal release of maize varieties in Uganda was in 1960 followed by another release of Kawanda Composite A in 1971. These varieties were grown for a long time until the late 1980s when they succumbed to the maize streak virus (MSV) during the outbreak of the disease in 1987. These two open-pollinated varieties (OPVs) were lost due to a combination of lack of maintenance breeding and civil strife between 1980 and 1986. The varieties developed later took into account breeding for host plant resistance for MSV and other major foliar diseases like Turcicum leaf blight (caused by -1 year-1) for 2008 – 2020, compared to 59 to 106 kg ha-1 year-1  using datasets from the NPTs (2008 – 2020) and the era trials on varieties released between 1999 – 2020 in Uganda. The results revealed significant annual genetic gains for grain yield and agronomic traits over the years. The study indicates that breeding for the target maize-growing environments in Uganda has been successful, with significant contribution to maize improvement from the private sector, CIMMYT and NARO supported through increased collaboration and expanded testing networks. The rates of gains estimated from the national maize yield average were lower than those obtained from other estimates but these would be higher if farmers use appropriate crop management practices, including the use of fertilizers, which is still very low in Uganda (less than 6 kg/ha on average) . The susThe finding from this study demonstrated continuous improvement in terms of realized genetic gains and productivity gains under research and on-farm conditions, respectively. To achieve larger genetic gains there is a need to modernize the breeding programs by incorporating use of molecular markers and doubled haploid technologies, faster recycling of elite lines, and switch to product profile-based breeding with optimized breeding schemes to improve selection efficiency. In the last three years, NARO’s breeding program has been assessed, and improvement plans and investment cases have been identified under the CGIAR Excellence in Breeding (EiB) platform and Accelerating Genetic Gains for Maize and Wheat Improvement (AGG) Project. These new breeding strategies now inform the product-profile based variety development and replacement. Gaps in seed systems still need to be addressed to effectively deliver improved stress-resilient varieties to the smallholder farmers for greater impact.The raw data supporting the conclusions of this article will be made available by the authors, without undue reservation.GA, DK, and CK – conceptualization, field experimentation, data analysis and manuscript development. CS – conceptualization, data analysis and manuscript development and review. BD and LM – conceptualization, manuscript review and funding. DM and YB – germplasm development, experimentation and manuscript review and, BP – manuscript review, editing, and funding. All authors contributed to the article and approved the submitted version.


BW) in dogs has been associated with developmental as well as degenerative diseases, but the heritability of BW in dog breeds is largely unknown. The aim of the current study was to estimate heritability and genetic change (genetic trend) for BW in a range of dog breeds in Sweden. Body weight registrations from 19 dog breeds  of varying body size, type and usage were collected from 2007 to 2016. The average BW of the breeds was 8 to 56 kg. The BW registrations were performed when the dogs were 12 to 24 mo of age (18 to 30 mo for one large-sized breed) in connection with an official radiographic screening program for hip dysplasia. Collected weight records were used to estimate heritability and genetic trends for BW. Several statistical models were used. The preliminary model included the fixed effects of breed (P < 0.001), sex (P < 0.001), year of screening (P < 0.001), litter size (P = 0.06), parity of the dam (P = 0.03) and linear regression on age at screening (P < 0.001), the latter five effects all nested within breed, and the random effects of litter and dam. Season of birth and the quadratic effect of age were also tested, but were not significant (P > 0.10). For the genetic analysis, various mixed linear models were tested within breed with different combinations of random effects; the most complex model included random effects of litter, direct additive, and maternal genetic effects, and maternal permanent environmental effects. The average heritability for BW over all 19 breeds was 51%, with a range of 35% to 70%, and the additive genetic coefficient of variance was around 9%. Maternal heritability was 5% to 9% and litter variance was below 10% with one exception (15% in Shetland Sheepdogs). For nine breeds, there was a genetic trend of increasing BW, whereas seven breeds had a genetic trend of decreasing BW. The largest absolute genetic change over a 10-yr period was around 0.6 kg or about 2% of the mean. In conclusion, given the small genetic changes in spite of the high heritability, it seems that there is generally a very weak selection, if any, for BW in the included dog breeds.High body weight ( Body weight in dogs is highly heritable, but there is very little selection on it. BW), due to large body size or excessive body fat content, predisposes for various diseases in dogs  is directly proportional to heritability. If there is indirect selection on BW, by selection on a highly correlated trait, e.g., height, then the selection response in BW is dependent on the heritability of height, the genetic correlation between height and BW, and the genetic standard deviation for BW, given a certain selection intensity. Thus, also in this case, heritability, as a measure of genetic variation, is an important parameter. With higher heritability, we can thus expect a higher genetic change, whether by intent or inadvertently.BMI), length, and height are known to be highly heritable in most mammals. Heritability (h2) of height in humans is a classic example, with heritabilities usually being very high, around 0.8 (2) has a high h2, with reported median 0.75 from twin studies and 0.46 from family-based studies . In 2005, registration of the dog’s BW was made mandatory in the screening program. According to screening guidelines, official screening results can be obtained from 12 mo of age for most breeds (or 18 mo for certain large breeds). In the present study, screening data registered between January 1, 2007 and December 31, 2016 was used, for dogs born during the years 2005 to 2015 and screened at 12 to 24 mo of age (or 18 to 30 mo for large breeds). Individual dogs without a Swedish registration number at screening, dogs screened prior to the official age and dogs with extreme BW registrations, indicating misreporting, were excluded. If a dog had more than one screening result, the first result was used. Only breeds with a minimum of 15 BW recordings each individual year and a minimum of 400 registrations for the 10-yr period were eligible for inclusion in the study. From breeds meeting these inclusion criteria, 19 breeds were selected to represent a broad range of body size, type and usage in the current study. The breeds represented 8 of 10 breed groups as defined by Fédération Cynologique Internationale , sexj(i) is the fixed effect of sex j  within breed i; yeark(i) is the fixed effect of year of screening k  within breed i; seasoni)l; littersizem(i) is the fixed effect of litter size class m within breed i (approximately divided into tertiles within breed); parityn(i) is the fixed effect of parity n of the dam  within breed i; bi)1( and bi)2 ~IND as additive genetic standard deviation (SDA) divided by the mean BW of the breed, where SDA is the additive genetic standard deviation AIC) and the model with the lowest AIC was considered the optimal model. All genetic models ([1] to [5]) were analyzed using the DMU package  was calculated within breed.P < 0.001) in the preliminary model. Season and the quadratic term for age were nonsignificant , while parity of dam was significant (P = 0.03). Litter size was almost significant (P = 0.06) and we decided to include it in the genetic models [1] to [5].Breed, sex, year, and the linear regression on age were highly significant  for 10 of the breeds. This was not surprising, as this type of model needs large amounts of data in which both mother and offspring have observed phenotypes. Therefore, model [4], which did not include that correlation, was chosen as the most complex model.The optimal model (based on AIC) differed between breeds . There wHeritabilities from the four models [1] to [4] are summarized in h2. On the other hand, the Shetland Sheepdog, which is one of the most light-weight breeds in this study, and is screened from 12 mo, had the lowest heritability of all breeds, at 35%. The reason for this disparity is unknown, but it should be noted that the standard error for the heritability estimate is quite large for both breeds.The optimal model for each breed is presented in A  and a permanent nongenetic  effect of the mother (pe in model [4]). In model [3], dam includes both these components.The total maternal influence was generally small, between 5% and 12% . The matMaternal heritabilities  were between 5% and 9%. The litter effect, i.e, the effect of the common environment particular for this litter, was generally small (below 10%) and decreased when the dam effect was introduced to the model (results not shown), except for Shetland Sheepdog, for which the effect was stable at ~15%. For several breeds, litter variance became zero when dam was introduced into the model. This could be explained by difficulties in separating these two effects, obviously so for mothers with only one litter, which was between 54% and 82% of all dams, depending on breed. The variance proportion attributable to the permanent environmental effect of dam also decreased when the maternal genetic effect was introduced (results not shown). In several cases, the estimates of litter and dam permanent environmental variance were close to zero, even though the model including that or those effect(s) was the best according to AIC. Thus, for practical purposes, some of these optimal models could be simplified.A/yr or 0.12 to 0.15 percentage points of the mean per year. The largest annual decreases were for Newfoundland Dog, Swedish Elkhound, and Standard Poodle with absolute values of 0.041 to 0.059 kg/yr, corresponding to 0.014 to 0.023 SDA/yr or 0.10 to 0.19 percentage points of the mean per year. The genetic trend for the two breeds with the largest changes in kg is shown in A/yr for weaning (200 d) weight in beef cattle in USA. However, in dog breeds, even if there would be an intention to select for larger  dogs, there are generally upper and lower limits, usually on height, that may restrict the change also in BW.The genetic trend of BW was calculated between the birth years 2005 and 2015 . For ninIn a previous study, partly using the same dataset, the average breed BW for 72 dog breeds was calculated, including estimates of breed BW changes over the 10-yr period . In thatThe BW of an individual is affected by genes as well as environmental factors, such as nutritional status . NutritiIn the present data set, we had access to BW information, while information on size and body composition was lacking. It is therefore difficult to discern what constitutes the BW of included dogs. The study population consisted of young adult dogs, with data collected in conjunction with the official screening program for hip dysplasia. In Sweden, dogs traditionally undergo screening at the allowed age for the specific breed or shortly thereafter. Hence, the inclusion criteria for age in our study were decided with the intention to include dogs from young adult age within an age interval when a majority of dogs are examined. Because overweight due to excess body fat seems to be less prevalent in young adult dogs compared to middle-aged or old dogs , the obsIt might be hypothesized that there could be a negative assortative mating for BW (or for correlated size traits), i.e., that heavier males would be mated to lighter females and vice versa, because there are often breed requirements where there are limits on, e.g., wither height. However, the correlations between male and female BW in mating pairs were mostly nonsignificant and those significantly different from zero were positive. The highest values were found for Kleiner Münsterländer (0.34) and other significant values ranged between 0.07 and 0.14, i.e., a sign of positive assortative mating, although not very strong.i that would be needed to result in the observed genetic trend (based on breeder’s equation: Genetic change in SDA/yr = (i h)/L, where h is L is the generation interval), assuming direct selection on phenotypic value of BW only. For the highest genetic response in SDA (Bernese Mountain Dog), the selection intensity was 0.144 which translates to a proportion selected of 93% . Such small changes, in spite of the high heritability, would indicate a weak selection, if any, for BW in these studied dog breeds.Given the estimates of heritabilities, it is possible to calculate a selection intensity A was around 9%. These values are high enough that any selection, either directly on the phenotype of BW or on a strongly correlated trait, such as height, could result in a large genetic change. Previous selection has at most resulted in a genetic change of 0.6 kg over a 10-yr period or about 2% of the average BW. Given the small genetic changes in spite of the high heritability, it seems that there is generally a very weak selection, if any, for BW in the included dog breeds.The average heritability for BW over all 19 breeds was 51%, with a range from 35% to 70%, and the CV


Lacerta agilis). The badge functions as a cue to male fighting ability in this species, and male–male interactions determine mate acquisition. Using animal models on a pedigree including three generations of males measured over an extensive 9-year field study, we partition phenotypic variance in both badge size and body condition into additive genetic, maternal, and permanent environmental effects experienced by an individual over its lifespan. Heritability of badge size was 0.33 with a significant estimate of underlying additive genetic variance. Body condition was strongly environmentally determined in this species and did not show either significant additive genetic variance or heritability. Neither badge size nor body condition was responsive to maternal effects. We propose that the lack of additive genetic variance and heritability of body condition makes it unlikely that genic capture mechanisms maintain additive genetic variance for badge size. That said, genic capture was originally proposed for male traits under female choice, not agonistic selection. If developmental pathways generating variance in body condition, and/or the covarying secondary sex trait, differ between inter- and intrasexual selection, or the rate at which their additive genetic variance or covariance is depleted, future work may show whether genic capture is largely restricted to intersexual selection processes.Sexual selection on fitness-determining traits should theoretically erode genetic variance and lead to low heritability. However, many sexually selected traits maintain significant phenotypic and additive genetic variance, with explanations for this “lek paradox” including genic capture due to condition-dependence, and breaks on directional selection due to environmental sources of variance including maternal effects. Here we investigate genetic and environmental sources of variance in the intrasexually selected green badge of the sand lizard ( Strong directional selection on sexually selected signals should erode trait variance and result in a loss of underlying additive genetic variance and low heritability , for example, sexually selected badge size of cross-fostered offspring more closely resembles the badge size of their foster fathers than biological fathers, a result that demonstrates the strong contribution of environmental and parental rearing conditions on sexually selected trait development . Badge size is an intrasexually selected signal of male fighting ability in this species (Olsson Eulamprus quoyii), for example, fitness-determining motor behaviors of offspring are more strongly influenced by maternal effects than additive genetic variance , sexually dimorphic ground-dwelling lizards distributed primarily across central Europe but also extending into the UK, Sweden and central Asia  and body mass to the nearest 0.001 g. We measured badge size by scanning each male on a flatbed scanner, and calculating the proportion (%) of green coloration versus the remaining side of the body  and calculated estimates of heritability (h2) for badge size and body condition using animal models , nongenetic effects due to maternal identity (VMA), and effects due to permanent environment experienced over a lifespan by an individual (VPE). Our models represent the total phenotypic variance as VP = VA + VMA + VPE + VR, where VR encompasses residual variance. In each model, we also include fixed effects and covariates known to influence trait variance . We tested for the significance of random effects components  using log-likelihood ratio test comparisons of models with and without each random effect, and a significance threshold of p < 0.05. When the random effect for VMA was not significant following log-likelihood ratio testing, it was excluded from the final model . Maternal ID (VMA) was non-significant  and was dropped from the final model.We found a significant additive genetic variance component to badge size with a heritability estimate of 0.33 Table . Other fVMA) was non-significant  and excluded from the final model, although its inclusion did not alter any other variance estimates.Variance in body condition was significantly affected by the permanent environment and the fixed effects of year and capture date Table . We founDuring these analyses, we also made attempts at quantifying the genetic correlation between body condition and badge size, but unfortunately, these models did not converge and will not be further reported on.We found significant additive genetic variance underlying the sexually selected sand lizard badge. Genic capture is an unlikely resolution to this maintenance of genetic variance in the face of selection , since body conditions show low and non-significant additive genetic variance, with variance significantly predicted by the environmental conditions experienced over an individual’s lifetime. We propose that complex genetic, structural and pigmentary mechanisms involved in the production of the multicomponent badge, along with age-dependent and sex-specific variance in selection on badge size, act as a check on loss of genetic variance for this fitness-determining trait. Moreover, intrasexual selection on the male badge, which drives access to mates , the strength of selection on fitness-linked physical endurance capacity varies across age classes, with positive viability selection apparent in juveniles and yearlings only and weak sexual selection in adult males. These relationships were non-linear, and likely permissive to the high heritability of locomotor performance in this species  were negligible and therefore unlikely to influence our estimate of badge size heritability  and physiological response to parasite load through the production of corticosteroid stress hormones , condition metrics that reflect body composition showed higher estimates of heritability than those that described variance in metabolic responses to feeding vs. fasting. It was for these body composition metrics of condition that genetic correlations with sexually selected singing behavior indicated that heritability in song rate might be “captured” by condition (Burkhard et al. The condition can be measured in a variety of ways, and while some metrics of condition may be more responsive to environmental conditions, others may be more affected by genetic traits (e.g., Merilä et al. Differences in estimates of heritability may exist between the size and pattern of a color badge and the reflectance parameters of that patch (Karino and Haijima Electronic supplement


In addition, k-mer-based analyses can be used in species that lack a reference genome. However, the use of k-mers for GWAS presents challenges such as data size and complexity, lack of standard tools, and potential detection of false associations. Nevertheless, efforts are being made to overcome these challenges and a general analysis workflow has started to emerge. We identify the priorities for k-mer-based GWAS in years to come, notably in the development of user-friendly programs for their analysis and approaches for linking significant k-mers to sequence variation.Genome-wide association studies (GWAS) have allowed the discovery of marker–trait associations in crops over recent decades. However, their power is hampered by a number of limitations, with the key one among them being an overreliance on single-nucleotide polymorphisms (SNPs) as molecular markers. Indeed, SNPs represent only one type of genetic variation and are usually derived from alignment to a single genome assembly that may be poorly representative of the population under study. To overcome this,  Genome-wide association studies (GWAS) have routinely been used in plant science for the discovery of significant markers, candidate genes, and beneficial alleles, as well as understanding the genetic architecture of traits of economic importance ,2,3. PutSW9-1 (ss246792949T/C) locus associated with soybean seed size, and the dCAPS marker was able to discriminate between accessions with small and large seeds.The GWAS provides several advantages over its sister mapping strategy, linkage mapping. Notably, the GWAS requires less time to assemble the mapping population, provides higher resolution and more recombination events, facilitates the identification of candidate genes, and generates results that can be transferred more easily to marker-assisted breeding ,9. For eThe most commonly used markers in GWAS are single-nucleotide polymorphisms (SNPs), which can be easily discovered and genotyped on the genome scale using various array- or sequencing-based genotyping platforms . SNP arrAlthough the GWAS has proved useful for quantitative trait loci (QTL) discovery in crop species, the task of identifying candidate genes and causal variants from the output of GWAS is still difficult ,17. As aBar gene is linked to small eyes in fruit flies [cis-regulatory sequences [Some limitations of SNP-based GWAS could be overcome using different types of genotypic data . For exait flies . Over thit flies ,37,38,39equences . The potequences , soybeanequences , cucumbeequences , and tomequences , an apprDespite the relevance of SVs, the identification of SVs from short-read sequencing is difficult and unreliable, leaving the majority of SVs poorly resolved and their molecular as well as phenotypic impacts largely overlooked ,44,45. Ak-mer-based GWAS approaches that are agnostic to variant types and reference-free have been developed [k-mer-based GWAS approaches, examples of use in crops, and specific challenges, as well as future developments and perspectives.In summary, SNP datasets are not sufficient to describe the whole spectrum of sequence variation and SVs are difficult to accurately discover and genotype on the population scale. Therefore, methods that enable researchers to encompass all types of variation without the pitfalls of genome-wide SV discovery are much needed. To overcome these limitations, eveloped ,53,54. Tk-mers are subsequences of a fixed length k which can be obtained from sequencing data or genome assemblies by extracting all such subsequences found in the input dataset. For example, from the sequence ACCGTCG, the following k-mers of length four (4-mers) can be observed: ACCG, CCGT, CGTC, and GTCG. k-mers have been used for various applications in genomics, notably genome assembly [k-mers can be used to identify statistical associations between the occurrence of k-mers in a dataset and traits of interest. In its simplest implementation, the k-mer-based GWAS relies on a presence/absence table of k-mers found across various accessions . In . In k-mes . In allk-mer-based GWAS analyses work because variation in genomic sequences between individuals ultimately results in differences in the occurrence of k-mers in sequencing data , variation in repetitive regions may not result in k-mer patterns that can be observed in typical short-read WGS data. Also, although k-mer-based GWAS could theoretically be applied to reduced-representation sequencing methods such as GBS, their full potential to pinpoint causal variants will only be reaped using WGS data.ing data . For exaariation . In prack-mer-based GWAS provide several advantages over conventional SNP-based GWAS. First, k-mers are agnostic to the nature of the sequence variation underlying differences in their occurrence [k-mer-based analyses do not depend on a reference genome and can therefore be used in species that lack a reference genome . Even fs . Third,S . Finalle, e.g., ).k-mer-based GWAS approaches is the lack of standardized methods for conducting such analyses. Indeed, these methods are relatively recent and therefore have not yet matured to the point where state-of-the-art analysis pipelines have emerged. As a result, user-friendly programs such as TASSEL [k-mer-based GWAS programs user-friendly for wider application in the plant genomics community. Nevertheless, a general workflow for conducting k-mer-based GWAS has started to emerge  for GWAS, but the thorough testing of a wide range of k-mer lengths has yet to be performed. Shorter k-mers reduce the probability of including sequencing errors and result in more k-mers being tallied for a given dataset, therefore providing higher counts for resolving copy number variants based on k-mer counts. On the other hand, longer k-mers would provide a larger number of unique k-mers and more power to resolve sequences stemming from repetitive regions. In addition to properly choosing the k-mer length, raw k-mer counts must be filtered prior to downstream analyses. Typically, k-mers that are only observed once in a sample are discarded as they might be due to sequencing errors  may indicate contamination, e.g., due to sequencing adapters. k-mers may be further filtered based on minor allele frequency, as in standard GWAS methods.The first step in conducting ing data a. This s , their approach first computes an approximate score for each k-mer in the dataset. The most significant k-mers identified using the approximate scores are then used to compute the exact p-values in GEMMA. Gaurav et al. [k-mers by keeping only those whose presence/absence pattern was correlated to phenotypic data above some preset threshold; exact p-values for the filtered k-mers were then computed using linear regression models, accounting for population structure. Presence/absence-based approaches are convenient because the minimum average sequencing depth required to attain low error rates in determining presence/absence is rather low. Using empirical data [k-mer-based GWAS relying on presence/absence . Count-based methods will therefore need to rely on deeper sequencing to accurately resolve the statistical relationship between k-mer copy number and phenotypic variation.More elaborate for GWAS ,53. Thesfor GWAS , as theye et al.  proposedk-mer-based GWAS analyses truly differ from conventional GWAS when the time comes to analyze significant signals  to extract all of the sequence reads containing significant k-mers. This approach is more exhaustive than simply mapping k-mers to a reference because it considers the entire read sequence for mapping purposes. signals c. While r e.g., ,54). Onc. Onck-mey et al. , who devk-mers lies in clustering them into coherent signals  of our summarized framework are truly mandatory; steps 3 to 5  are optional to some extent. For example, in species that lack a reference genome, step 3 is not possible, but significant k-mers can nevertheless be clustered and/or assembled to gather meaningful information. For example, in a study of resistance to wheat stem rust in Aegilops tauschii, for which no reference genome was available, Arora et al. [k-mers were subsequently mapped to local assemblies of NLR genes generated from the sequencing data themselves. For such analyses involving species without a reference genome, the local assembly of significant k-mers or reads containing them will likely be necessary to obtain meaningful results. Step 5 (linking k-mers to sequence variation) may however not be necessary at all to obtain meaningful results from k-mer-based GWAS. For example, in a set of studies involving Ae. tauschii [It is important to note that only steps 1 and 2 . Given the expectation that genes associated with resistance to this pathogen would be NLR genes, they performed their analysis on sequencing data enriched for NLR genes, an approach that they named AgRenSeq. In a presence/absence-based strategy that involved pre-filtering based on a correlation with the trait and linear regression models accounting for population structure, 51-mers were used to derive p-values. Candidate genes were identified by mapping significant k-mers to locally assembled NLR gene sequences. This analysis readily identified four NLRs that were consistent with the results published in previous studies [k-mers for association mapping in plants.In the first study reporting a et al.  phenotyp studies ,79,80,81 studies  was not Arabidopsis, tomato, and maize, respectively, to conduct k-mer- and SNP-based GWAS with several hundreds of traits , yellow endosperm 1, carotenoid cleavage dioxygenase 1, and zeaxanthin epoxidase 1 (kernel color) [k-mer analysis with gene expression data and used k-mers to predict phenotypes.He et al.  used the content . A numbel color)  were detAe. Tauschii accessions. In their study, the significantly associated k-mers for stem rust (caused by P. graminis f. sp. tritici) were mapped to both the Ae. tauschii AL8/78 reference genome and a de novo assembly of a relevant accession (with two cloned stem rust resistance genes), which was anchored to the reference genome. In addition, they identified a number of candidate genes which had previously been demonstrated to influence the studied traits. These include stem rust (SrTA1662) [FLOWERING LOCUS T1) [RAMOSA3 and TPP4) [RAMOSA3) [Cmc3 and Cmc4) [Yr15) [Rpg1 and Sr60) [Gaurav et al.  extendedrTA1662) ,79, flowOCUS T1) ,84,85, lnd TPP4) , spikeleRAMOSA3) , and whend Cmc4) ,89. Inte) [Yr15) , stem rund Sr60) ,92, and nd Sr60) .Pyricularia oryzae , transformed with either PWT3 or PWT4 from a previous study [k-mer-based association mapping based on NLR-enriched sequencing data, from which they identified two candidate genes, TraesCS1D02G029900 and TraesJAG1D03G00423690, for PWT3 and PWT4, respectively. These genes, respectively, overlapped with Rwt3 [and Rwt4 [TraesCS1D02G029900 (Rwt3) in a TILLING population of Jagger [TraesCS1D02G029900 (Rwt3), which showed that this gene is required for resistance to P. oryzae, expressing the PWT3 effector. They subsequently developed kompetitive allele-specific PCR markers (for both Rwt3 and Rwt4), which they tested in the landrace subpopulation, and obtained a validation rate of 97 and 99%, respectively. The functional validation results from this study provide compelling evidence for the value of k-mer application in GWAS mapping.Arora et al.  used 320us study . They coith Rwt3  and Rwt4and Rwt4  from a bf Jagger . A virusk-mer-based GWAS with 363 Glycine max accessions for 13 traits including flower, pubescence, seed coat, and hilum color, as well as stem termination type, pubescence density, and seed coat luster , and warrant application in plant science to discover novel loci/candidate genes for traits of economic importance.So far, these seven k-mer-based GWAS papers published to date [k-mers from 261 inbred lines in their study, with the majority of lines having 0.35–0.55 billion non-redundant k-mers. This can lead to challenges in data storage, processing, and analysis. Therefore, large k-mer datasets can be computationally intensive and require significant computational resources, such as dedicated high-performance computing servers. Efficient algorithms and programs designed specifically for the analysis of k-mer-based GWAS have therefore been developed  prohibiiFi data , which ciFi data  may be ak-mer-based association methods in years to come will prove to be transformative for the discovery of new gene–trait associations. As discussed in this review, k-mer-based methods have already provided novel insights into crops such as soybean, maize, and wheat. Going forward, we believe that such approaches will greatly enhance our understanding of crop genetics and ultimately our ability to develop crop varieties that will tackle the challenges associated with population growth and climate change.Beyond methodological improvements in GWAS analysis, we believe that the use of