Categorical traits without ordinal representation of classes do not qualify for threshold models. Alternatively, the multinomial problem can be assessed by a sequence of independent binary contrasts using schemes such as one-vs-all or one-vs-one. Class probabilities can be arrived at by normalization or pair-wise coupling strategies. We assessed the predictive ability of whole-genome regression models and support vector machines for the classification of horses into four German Warmblood breeds.Prediction accuracies of leave-one-out cross-validation were high and ranged from 0.75 to 0.97 depending on the binary classifier and breeds incorporated in the training. An analysis of the population structure using eigenvectors of the genomic relationship matrix revealed clustering of individuals beyond the given breed labels. Admixture between two breeds became apparent which had substantial impact on the prediction accuracies between those two breeds and also influenced the contrasts between other breeds. Genomic prediction of unordered categorical traits was successfully applied to subpopulation assignment of German Warmblood horses. The applied methodology is a straightforward extension of existing binary threshold models for genomic prediction. Categorical traits play an important role in livestock breeding. It is often possible to arrange K distinct classes of a particular trait into a meaningful order. Ordinal threshold models can then be applied by introducing K − 1 thresholds to the liability. However, if ordering classes is doubtful or impossible, those threshold models do not apply in a meaningful manner. This is the case when treating subpopulations or breeds (both terms are used interchangeably throughout the paper) as classes of the trait “subpopulation assignment”. Partitioning individuals into groups or subpopulations is elementary in livestock breeding and can be done in various ways based on geographical, phenotypical or ideational criteria.In breeding organisations, the composition of a particular breed is subject to selection, migration and drift. Also, breed definitions and breeding goals change over time, making stringent permanent classification difficult. Different subpopulations of the German Warmblood horse have different breeding schemes, breeding value definitions and evaluations, geographical locations, ancestral breed compositions and migration policies. We expect those differences to be visible on the genomic scale using high-density single nucleotide polymorphisms (SNPs). The most fundamental metric to measure divergence between populations is the fixation index , which aAt this point, it is important to distinguish between divergence between subpopulations caused by genetic drift and gene flow (demographic history) and actual selection, both of which will be reflected by the fixation index. When sequence data is available, this can be assessed by distinguishing between SNPs located in genic and nongenic regions  . From a Genomic prediction is widely used for the prediction of quantitative and qualitative traits in livestock, plant breeding and human genetics. It has been shown, that regression on markers can pick up genomic relationships between individuals as well as linkage disequilibrium and co-segregation between markers in founders and descendants, respectively . Breed cThe aim of this study was to investigate the ability of whole-genome regression methods that are used for genomic selection, to discriminate between subpopulations. We did not seek to use genomic prediction methodology to cluster a given data set into subsets, but used predefined subpopulation assignments, which were assumed as fixed cluster indices, and assessed the predictive ability of various methods. However, as a consequence of this approach being applied to subpopulation categories, predictions of uncategorized individuals, based on the training of marker effects, can be used for subpopulation assignments or probabilistic assignments to several subpopulations. In addition, the discriminating ability of models based on genome-wide dense SNP panels can be used as a criterion for the divergence between breeds.Genomic prediction of ordinal traits has been successfully applied in animal and plant breeding using either threshold models –7 or cat® BeadChip that contains 54,602 SNPs. Genotyping results for autosomal SNPs only were filtered based on Illumina’s GenCall score applying a threshold of 0.15, which resulted in a mean call rate higher than 0.98. Furthermore, markers with a minor allele frequency less than 0.05, and a genotyping rate of at least 90 % were discarded. The final set comprised 44,159 SNPs. Before excluding SNPs based on the described criteria, we analysed each SNP for opposing fixation of certain alleles between the breeds. This should have allowed prediction of subpopulation assignment based only on those SNPs. However, none of the SNPs showed opposing fixation of alleles.The dataset included 917 stallions from four German Warmblood horse breeding populations that were part of a breed survey between 2005 and 2008, i.e. Hanoverian, Holsteiner, Oldenburger and Trakehner. The animals were genotyped with the Illumina Equine-SNPThe pedigree consisted of 270 unique sires and 876 unique dams, yielding a pedigree size of 2061 individuals that are summarized in Table k is a value between 1 and K. In that case, the use of ordinal threshold models is questionable. We have treated the multiple class prediction problem as one that can be represented by binary contrasts between two or more classes. In logistic regression, the multinomial case is dealt with by conducting K − 1 binary contrasts against a base line class. Several schemes are possible to combine a collection of binary classifiers into a single multinomial classifier. The main difficulty is to define the scheme in a consistent way and to guarantee a unique classification, since more than one binary classifier might claim the same data for different classes or none of the binary classifiers might claim the data at all. A customary scheme, also used here, is the one-vs-all (OVA), where each of the K classes is paired with the complement of samples from the K − 1 other classes. This will almost always be unbalanced, since each binary classifier usually will see a larger number of negative samples consisting of the When investigating categorical traits, for which the different classes cannot be arranged in a meaningful order, the distinct categories have to be treated as K independent classes termed or labelled K-dimensional vector of posterior predictive probability assignments:k and summing over the complementary Q, it is then shown that this yields a unique solution although Eq.  The linear predictor is mapped to the expectation of n, k, k. k for individual n averaged over the fixed effects and can be obtained as:kth binary OVA model. Deviations of the offspring from the population means can be computed as The computation of multinomial predictors based on binary contrasts can also be used to estimate multinomial breeding values expressed as expected fractions of offspring in the different categories. Considering the probit threshold model described above and a OVA scheme, a base population using the allele frequencies of the marker covariates   are a suHaving established the basic binary SVM classifier, there are two important extensions to be made: a scheme for concatenating binary classifiers, such as described above, can be set up to build a single multinomial classifier, and some (nonlinear) feature transformation can be carried out before using the basic SVM to accommodate for nonlinearities in the data. A feature transformation is an arbitrary (but somewhat 'reasonable') mapping K that takes the form:H with scalar product K on pairs of data, while, conversely, the RHS of Eq. . Hence, an sBS value of 0 indicates that the model predicts no better than random assignments of individuals to classes with uniform probabilities. A value close to 1 indicates perfect prediction. This is only slightly different from the definition given in Steyerberg et al. . The Hanoverian breeding policy can be regarded as more open than that for the aforementioned populations. The Hanoverian breed has also been influenced by several waves of Thoroughbred introgression, but also by the Trakehner horse breed, which has been extensively used after World War II, and also by the Holsteiner breed since 1993, but to a lesser extent in the context of the Hanoverian Jumper Breeding program. It has been estimated that approximately half of the Hanoverian gene pool is attributable to original Hanoverian founder animals [http://www.oldenburger-pferde.net/upload/Satzung_Stand_Juni_2014__korrigierte_Version].Actually, the Trakehner horse breed has been partially closed since 1732 and only introgression of Thoroughbreds and Arab horses are allowed. Their total gene contribution has been estimated at 34 % . The Hol animals . FinallyIn order to assess the population structure of the data set, a principal component analysis was conducted using the realized genomic relationships based on the available markers . Genomics et al.  for quanA threshold model of the form given in Eq.  was assu Mallick  and Habi Mallick   as a measurement of model complexity. Higher values of pEV indicate a more complex relationship between two breeds, hence less discriminative power of single eigenvectors. An alternative would be to treat one eigenvector at a time as the response variable and perform an analysis of variance using the known class labels as the explanatory factor variable as in Patterson et al. [We report the posterior mean of the proportion of eigenvectors with non zero effect  , with \dHowever, for the whole population, the global s et al.  based ons et al. , for whipEV using BGLR and JAGS, respectively, the number of iterations was 60,000 while discarding the first 20,000 as burn-in.All evaluations were performed in R . The Bayml-R 3.0 . The Bay in JAGS  using th in JAGS . For theAn overview of the breeds under investigation is in Table Averages over the additive genetic and genomic relationships are in Table The absolute and relative differences in the mean values of these measurements were small, while average The first two eigenvectors of the genomic relationship matrix of all individuals in the data set were used to visually inspect the clustering of individuals according to their assigned breeds Fig. a. The HopEV is a measure of model complexity using the eigenvectors of the genomic relationship matrix between two breeds , serve as a measure of divergence between the two breeds, with higher values of pEV indicating lower divergence. The posterior means of pEV ranged from 0.0061 (Holsteiner vs. Trakehner) to 0.048 (Hanoverian vs. Oldenburger), resulting in on average 2.4 and 25.15 eigenvectors included in the models, respectively. The individual inclusion probabilities of the eigenvectors in the pair-wise contrasts are in Fig. In addition to the visual inspection, we used eigenvectors to estimate two additional metrics that are in Table een them . Hence, entclass1pt{minimaIn addition to the eigenvector analysis used here, a wide range of methods exists to cluster datasets into subgroups and assign individuals to these. Besides distance measures used in k-means clustering or discriminative analysis, model-based approaches are frequently assessed. In particular, Pritchard et al.  and Raj Besides the measurements already described in Table ionships . We woulSince it may not be possible to generalize inferences without validation data , we usedThe results of the leave-one-out cross-validation are in Table In general, prediction accuracies were high and ranged from 0.59 to 0.97 across the whole data set. Using all classes for training results in intermediate overall prediction accuracies. OVO and OVA Ridge Regression yielded accuracies of 0.74 and 0.84, respectively, while the SVM reached an accuracy of 0.81. More interesting are the individual accuracies for the different subpopulations. In the OVO Ridge Regression, the Holsteiner were predicted with an accuracy of 0.99 while the Oldenburger animals were all misclassified (0.00). In the OVA scheme, the accuracy reached 0.54, which was only slightly better than random guessing. These results were also reflected by the scaled Brier scores, for which the Oldenburger animals reached a value of Generally, the prediction accuracies and scaled Brier scores were below average for the least represented breed in the data set (Trakehner), irrespective of the classifier used. Unbalanced sample sizes in the different classes can cause unreliable probability/classification estimates in weakly represented classes . This isCross-validation results, when one of the breeds was excluded, provided additional insights into the structure of the data. The schemes that achieved the highest overall and individual accuracies and sBS were those in which either the Hanoverian or Oldenburger breed was excluded. The OVA Ridge Regression model in which all but the Oldenburger breed reached the highest overall accuracies i.e., 0.97, 1.00 and 0.80 for the Hanoverian, Holsteiner and Trakehner breeds, respectively, and the average overall accuracy was equal to 0.97. The results were similar for the scheme that excludes the Hanoverian breed. This result implies that the inclusion of two breeds, even if they are strongly admixed, but given discrete class labels, significantly reduces the power of any of the binary classifiers to unambiguously predict unobserved data to the available classes. The reason is the error in the phenotype measurement, as previously discussed, which gives the class labels. One consequence of this finding is that given the present sample of individuals, the classification of the Hanoverian and Oldenburger animals into two distinct breeds is questionable. Numerous admixed individuals, resulting from stallion exchange between the two breeding organizations, have a strong impact on the results. This becomes more apparent when summarizing the predictive distributions of class label probabilities by subpopulations. This has been done in Figs. In terms of binary classification, the probability mass concentrated in the class of origin of a particular breed that is higher than 0.5 is of importance. The probabilities in the other classes clearly show a large proportion of individuals that are not unambiguously classified in one or the other class. This was especially true for the Hanoverian and Oldenburger animals. In the OVO scheme, the majority of the Oldenburger individuals had a predicted probability to belong to the Hanoverian breed higher than 0.5, which explains the 0.00 prediction accuracy for that breed and scheme Table . In the The difference in identical classifiers between the two coupling schemes (OVO vs OVA) used raises the question which one might be more appropriate. Across all scenarios, the OVA Ridge Regression scheme yielded higher accuracies within and between breeds compared to OVO. The leave-one-out scheme appears to be more sensitive to weak contrasts, which is readily explained by the fact that the classifier in that scheme never sees more than two classes. If that particular contrast is weak, the resulting prediction cannot be expected to yield high accuracies. In the OVA scheme, the classifier is always presented as the contrast between one class and all the remaining classes. Hence, even if two classes are not easily separated, the remaining classes still contribute to the binary discrimination. Although the SVM uses the OVO coupling scheme, the prediction accuracies are similar to those obtained with the threshold model using OVA. This might be due to the voting used in contrast to the estimation of posterior predictive probabilities.In addition to the admixture between the Hanoverian and Oldenburger animals, Fig. Generally, the classification of unobserved individuals to one  subpopulation(s) achieved high cross-validation accuracies. One application would be to use a reference population, that reflects subpopulations or breeds with high confidence, to estimate marker effects. To which subpopulations belong uncertain individuals can be predicted based on the training of those effects if prior breed assignment is doubtful. Probabilities for the different categories can be used to make inference about the genetic background of individuals and assign them to one or partially to several subpopulations under the restriction that the subpopulations represented in the training set are the only choices for classification.It should be noted that it is certainly possible to increase the overall prediction accuracies by alternative clustering of the data. However, the scope of our investigation was to evaluate the ability to predict unobserved individuals given the original subpopulation assignments. Migration of animals through the exchange of sires has a large impact on the results, but from a breeding organisation perspective the definition of a particular breed still makes sense, even with strong admixture with other populations. Breeding organisations define breeding goals and provide tools that might be more or less restrictive to achieve these goals. Breeding goals are an aggregate of desired phenotypic expressions that the breeders want to achieve by artificial selection and controlled mating. Given an unrestricted breeding policy, breeders would use stallions and mares, regardless of their origin, that most suitably contribute to the given breeding goal, hence having high estimated breeding values for traits of interest.A recent study investigated the impact of selection according to identical criteria in two distinct populations on the change in allele frequencies of SNPs on a genome-wide scale . The maiσG2+σGW2 . InsteadσG2+σGW2 .The described methodology for conducting multinomial prediction rests on the assumption that the multiple class problem can be represented by a number of pair-wise binary contrasts. In order to compute posterior predictive probabilities for all classes, it is sufficient to normalize the posterior predictive probabilities from the binary models in the OVA scheme and to solve a fixed point equation in OVO. Besides its appealing simplicity and the ability to make use of the variety of whole-genome regression models used in the genomic prediction framework, it also allows the use of existing software implementations without explicit multinomial extensions. The framework described is not limited to the application given here. There are numerous categorical traits of interest in livestock breeding, that cannot, or only barely, be ordered in a sensitive way to allow for ordinal threshold models to apply. Examples include coat color, functional traits or anatomical discrete measurements. Traits that are usually being ordered like calving ease might also benefit from a non-ordinal treatment of the different classes if the applied ordering scheme is questionable.The binary classification scheme used here implicitly assumes independence between any two contrasts and ignores unobserved classes . In theDefining a single additive genetic variance for unordered categorical traits is counterintuitive and, therefore treating the different categories as potentially correlated traits could provide insights about the genetic architecture of the multivariate trait aggregate. Genetic correlations could be estimated on the liability scales and the multinomial probit model would be a good candidate to investigate this topic since it does not rely on the independence of irrelevant alternatives assumption and can therefore take genetic covariances across the pair-wise binary contrasts into account .In addition, selection index theory could be used to generate a genotypic aggregate of the different categories of an unordered categorical trait. Selection could then proceed on an economically weighted index, since it is very likely that different categories have different economical impacts. The heritability for a single unordered categorical trait would then be defined by the applied index.The application given here certainly has alternatives such as substituting the marker matrix used for prediction by the lower Cholesky factor of the numerator relationship matrix  of a genomic relationship matrix in pair-wise contrasts serves as a sensitive indicator for population divergence and can be easily extended for the analysis of quantitative traits, when the number and location of relevant eigenvectors to explain variation in the response are doubtful [The estimated inclusion proportion of eigenvectors (doubtful .The reduction of breeding populations to a single class label does not allow inferences about the divergence of certain quantitative traits between populations. Further research is necessary to determine the genetic variability across the Warmblood horse populations with respect to the traits under selection.


Motivation: Expression Quantitative Trait Locus (eQTL) mapping tackles the problem of identifying variation in DNA sequence that have an effect on the transcriptional regulatory network. Major computational efforts are aimed at characterizing the joint effects of several eQTLs acting in concert to govern the expression of the same genes. Yet, progress toward a comprehensive prediction of such joint effects is limited. For example, existing eQTL methods commonly discover interacting loci affecting the expression levels of a module of co-regulated genes. Such “modularization” approaches, however, are focused on epistatic relations and thus have limited utility for the case of additive (non-epistatic) effects.Results: Here we present POEM (Pairwise effect On Expression Modules), a methodology for identifying pairwise eQTL effects on gene modules. POEM is specifically designed to achieve high performance in the case of additive joint effects. We applied POEM to transcription profiles measured in bone marrow-derived dendritic cells across a population of genotyped mice. Our study reveals widespread additive, trans-acting pairwise effects on gene modules, characterizes their organizational principles, and highlights high-order interconnections between modules within the immune signaling network. These analyses elucidate the central role of additive pairwise effect in regulatory circuits, and provide computational tools for future investigations into the interplay between eQTLs.Availability: The software described in this article is available at csgi.tau.ac.il/POEM/. The transcriptional regulatory program that controls the expression of a gene may combine the joint effect of several regulatory mechanisms that act in concert during the cellular response to internal and external signals. These regulatory programs are apparent across a variety of joint contributions, from the independent contribution of each of the regulatory mechanisms to a cooperative contribution of several mechanisms. A regulatory program may include a variety of mechanisms such as transcription factors, chromatin remodeling complexes, and promoter regulatory elements.Natural genetic variations may provide important insights into regulatory programs. In particular, transcription profiles can be integrated with genotypic data across a population to identify genomic loci that have an effect on gene expression , or alternatively, at least one of the loci acts in the proximity of the gene (termed “cis-acting effects”).Pairwise effects on gene expression may be classified on the basis of two main categorizations. One major categorization reflects the presence or absence of epistasis. In the absence of epistasis the effect of one eQTL remains the same regardless of the genotype of the other eQTL; the resulting model is therefore said to be “additive” or “non-epistatic.” In the presence of epistasis, by contrast, there is a change in magnitude or direction of one eQTL that depends on the genotype of the other eQTL. “Epistasis” therefore refers to a modification of the additive effects of two loci in a regulatory program , suggesting that the apparent scarcity of such effects in the abovementioned publications , a novel methodology that we developed on the basis of iterative refinements of a residual-based stepwise regression (RBSR) model. In our scheme, a “poeModule” is a group of genes (under particular stimulations) that are affected by the same pair of eQTLs. The aim of POEM is to identify poeModules together with their underlying pairwise effects. In particular, it is specifically designed for the case of additive relations. Our analysis of synthetic data demonstrated the superiority of POEM over existing eQTL methods. We applied POEM to a dataset of murine bone-marrow-derived dendritic cells, and found that additive n individuals at m genetic variants, as well as the levels of l traits across the same individuals. Let Y = [y1y2…yl] be a n × l matrix of trait measurements, where each column vector yj represents the measurements of a trait j across all n individuals. In addition, let G = [g1g2…gm] be a n × m matrix of measured genotypes where gx is a column vector of genotyping of variant x across all individuals. For simplicity, we assume a haploid organism or homozygous recombinant inbred strains; therefore, genotypic values for the variants are either −1 or 1. To assess the association between a given trait j and a genetic variant x, we test the significance of the genetic fixed effect βx under the following model:x is the global mean and e represents a n × 1 vector of normally distributed error terms. The significance is evaluated through a standard ANOVA F-test and the resulting P-value is referred to as an association score. A standard 1-locus scan calculates the association scores across all genetic variants. Let A be the output (l × m) association matrix where Ajx is the association score between trait j and variant x.We consider a previously measured genotyping of A. For example, let vj be the best-scoring locus of trait j where vj = argmaxxAjx. In case that Ajvj is statistically significant, we refer to vj as an eQTL. The set of significant best eQTL-trait pairs, referred to as an eQTL map, is denoted V and defined as follows:Tj is a significance cutoff. We note that the POEM algorithm does not utilize Equation (2) to construct the output eQTL map. Instead, it utilizes a revised definition of this map, as formulated in Equation (4) below.The identification of eQTLs is based on the association scores in conditioned 1-locus scan that depends on a given eQTL map V0 from a previous scan. Specifically, in this study we focus on a conditioned scan that utilizes the residuals of the eQTLs in V0 as the dependent variables, while the calculation of A and V remains unchanged. Thus, we use the following model:Importantly, it is possible to apply a n × 1 vector of residual rj instead of the trait measurements yj. For each trait j, rj is calculated as j in map V0  ∉ V0), then we set rj = yj.We note that Equation (3) is the same as Equation (1) but using a m2 pairs of variants. Stepwise regression is designed to reduce this complexity by applying two sequential 1-locus scans, with the first scan identifying the “primary” best variant and the second scan detecting the “secondary” best variant given the information about the primary one.In order to identify pairwise effects, the regression model from Equation (1) should be extended into a multiple regression model where the values of a trait are predicted based on the genotyping of two variants. However, such approach entails multiple testing of j RBSR first scans for the best (primary) eQTL and then rescans for an additional (secondary) eQTL conditioned on the primary eQTL. Assuming an additive pairwise effect, the second scan is performed on the residuals of the phenotypic measurements when the effect of the primary eQTL is removed.A common stepwise regression approach is the “RBSR,” which is equivalent to the “Either-significant” strategy proposed by Evans et al. . For a gprimary eQTLs, denoted VP. We further use the notation AP to denote the output association matrix of this primary scan. Step 2 applies a conditioned 1-locus scan that depends on the map of the primary eQTLs (using Equation 3 where V0 = VP); the map of secondary eQTLs, denoted VS, is then calculated using Equation (2). We use the notation AS to denote the resulting association matrix of the secondary scan. Based on this procedure, the pair of eQTLs affecting a given trait j is Formally, a standard RBSR is applied in two steps. Step 1 applies the standard 1-locus scan using the model and definitions in Equations (1) and (2), respectively. The output of this scan is a map of We note that the RBSR method is particularly suitable for the POEM algorithm since it relies on additive pairwise effects. An alternative partition-based approach, which is tailored for epistasis, is detailed in Section Synthetic Data Analysis.co-association groups , typically relies on the association matrix from a 1-locus scan across all traits  eQTL. The identification of such P-value of 0.01. The output grouping solution c is a set of co-association groupsc = {cv} where each group cv is associated with a single representative eQTL v. Notably, this allows revising the standard identification of eQTLs (as in Equation 2) according to the representative loci, thus increasing the overall robustness of the learned model. A revised eQTL map Vc, based on a grouping solution c, is given by the following definition:A key step of the InVamod algorithm is the filtration of non-significant groups based on an association-score cutoff. In this study, the chosen cutoff corresponds to an association expression trait is the transcriptional response of a particular gene under a certain stimulus across all individuals under study. POEM takes as input a collection of expression traits across individuals and the genotyping of the same individuals. The output is a collection of poeModules, where each poeModule is a group of traits that are associated with a specific pair of eQTLs. Similarly to existing eQTL approaches  and further extends RBSR in two ways Figure . First, VAP be the output association matrix. Next, InVamod is applied on matrix AP to generate the co-association groups. As detailed above, this provides (i) a collection of primary groups, denoted cP; and (ii) a map of primary eQTLs, denoted cP using Equation 4).In this step, the task is to learn the map of primary eQTLs assuming that the map of secondary eQTLs from stage 2  a collection of secondary groups, denoted cS; and (ii) the output map of secondary eQTLs, calculated using Equation (4) and denoted In this step, the task is to learn the secondary groups and secondary eQTLs assuming that the map of primary eQTLs from stage 1 is known. This stage is precisely the procedure we use in stage 1. The only difference is that the input cP, and the initial primary eQTL map , denoted The initialization stage involves the identification of primary eQTLs in the absence of prior loci. The procedure starts with a (non-conditioned) 1-locus scan (Equation 1), whose output association matrix is then utilized by the InVamod algorithm. As detailed above, the output of the InVamod algorithm includes the initial primary groups, denoted P-value for the overrepresentation of overlapping traits uv1v2 = cv1∩cv2 . We call these P-values the overlap P-values, and only overlaps with P-values that are lower than a certain cutoff are selected. We refer to each significant overlap poeModule and denote the entire collection of poeModules as U. Thus, the pair of eQTLs affecting a given trait j ∈ uv1v2 consists of a primary eQTL v1 and a secondary eQTL v2 , where v1 and v2 are the representative eQTLs of groups The final stage of POEM is based on the observation that large overlaps between the primary and secondary groups are unlikely to be generated at random. It is therefore possible to use significant overlaps to infer pairwise effects on groups of traits. In particular, for each pair of a primary group Permutation-based FDR for poeModules were determined by generating 100 permuted gene expression datasets (by random reshuffling of strain labels), running POEM on each of them, and then calculating the ratio between the numbers (or sizes) of the permuted vs. real poeModule.uv1v2 = cv1 ∩ cv2 where v1 and v2, respectively). For each trait j residing in the poeModule we applied a standard interaction test:As a final step we aimed to characterize epistatic vs. additive poeModules. We consider a poeModule yj is a n × 1 vector of trait measurements; μ is a global mean; gv1 and gv2 are n × 1 genotyping vectors of eQTLs v1 and v2 ; e is a n × 1 vector of normally distributed error terms; and gv1 × gv2 is a n × 1 vector of interaction terms, generated by multiplication of the genotypes from each individual. βv1 and βv2 are the additive effects of the eQTLs and βv1v2 is the interaction term. The epistasis score refers to the significance of the interaction term, calculated by testing the alternative hypothesis F-test). We define an epistatic poeModule as a poeModule consisting of at least one epistatically-affected trait (FDR < 0.01). In particular, this was performed by first calculating an epistasis score for the interaction term of each trait, and then correcting for multiple testing using the FDR method. In this study we mainly focus on the non-epistatic poeModules—which are not annotated as epistatic ones—referred to as the additive poeModules.Here, l traits that are associated with two eQTLs, denoted v1 and v2. The synthetic expression data were generated by means of a two-locus model using Equation (5). Genotypic values for the variants are either −1 or 1, as in the case of homozygous mice. We assume that the additive effect is one-half of the difference in mean trait level between the two genotypic values (with no departure from additivity due to dominance effect). We tested two models: first, an additive model with βv1 = βv2 = γ , βv1v2 = 0 , and second, an epistasis (co-adaptive) model with βv1 = βv2 = 0 , βv1v2 = γ , where γ represents the magnitude of effect, referred to as the genetic effect size. In the epistasis model, each variant alone has no significant individual effect, but the joint effect of the two variants is notable. In addition, each collection includes l1 traits that are associated with variant v1 using the model y = μ + gv1βv1 + e ; l2 traits that are associated with v2 using the model y = μ + gv2βv2 + e; and l0 traits that are not associated with any variant, based on the model y = e. A single “synthetic collection” consisted of l + l1 + l2 + l0 traits and 100 variants, which included the two eQTL variants v1 and v2. Each variant was genotyped by randomly sampling the two alleles with equal probabilities.A synthetic collection of expression traits consists of synthetic datasets of 500 collections, where each dataset was constructed for a certain (additive or epistatic) model, a certain number of individuals , and a given genetic effect size γ (ranging from 0.4 to 1.4). In all cases we used l = l1 = l2 = 10, l0 = 50, and σ2 = 0.5.Overall, we generated We compared the following four alternative methods.Residual-based stepwise regression (RBSR). A stepwise identification of pairwise effects, applied on each trait independently as proposed by Evans et al. . This method applies a stepwise identification of allele-specific interactions on each trait independently, as described by Brem et al. . A standard partition-based modularization method is the “module-network” approach . In this case, the analysis involves two sequential scans: one that does not depend on any additional locus , and one that is conditioned on the primary loci (POEM's stage 2). For a single trait, this approach is equivalent to RBSR; for multiple traits, POEM further exploits the modularization in the biological system.Non-iterative POEM. The full POEM algorithm using multiple iterations (k = 6).By testing the five alternative methods it is possible to assess the utility of a residual-based methods compared to a partition-based methods (PBMs) ; to evaluate the advantages of grouping over the analysis of each transcript independently  and to explore the contribution of the iterative approach (#4 compared to #5).mp traits associated with both x1 and x2, and the other contains the non-associated traits. Based on the predictions of the method, we further split these traits into two additional classes : a “positive” class for a transcript that is predicted to be associated with both x1 and x2, and a “negative” class for the remaining traits. For each P-value cutoff the true positive, true negative, false positive and false negative counts are then calculated. Next, the area under the receiver operating characteristic (ROC) curve is computed on the basis of the resulting counts, referred to as the accuracy score. The accuracy score ranges between 0 for a random solution and 1 for an optimal solution. Since true positive pairwise effects can rarely be detected randomly, an accuracy score of 0.5—which consists of a relatively large amount of true positives—typically reflects a highly informative, non-random solution.For determination of the ability of a method to correctly identify pairwise effects on each of the synthetic datasets, all traits in a given dataset are split into two classes: one contains P-value cutoffs, which is the main parameter of interest and serves as input in all compared methods. In all cases, these cutoffs refer to the maximum P-value attained by the primary and secondary eQTLs. In the case of RBSR and PBSR, we used the association P-values of individual traits. For the case of for PBM we used its module P-values  we used the InVamod's P-value cutoff. To assess the difference in accuracy between two different methods we compared the corresponding ROC curves using a paired t-test (comparing specificity for the same sensitivity levels).We note that the accuracy score is obtained by testing a range of in-vitro stimulation with one of three pathogenic components: lipopolysaccharide (LPS), polyinosinic polycytidylic acid (poly I:C), or Pam3CSK4 (PAM) . A gene's response to a given stimulation in a given strain was calculated as the difference between the log-transformed expression levels at 6 h after stimulation and at steady state. Overall, the data consisted of 1209 distinct response traits (403 genes × 3 stimulations), each of which is measured across the BXD strains and is referred to as an “expression trait.” Genotyping data of 3796 SNPs was downloaded from WebQTL . A trans-acting poeModule (cis-acting poeModule) is defined as a poeModule whose fraction of trans-trans-affected traits is above (below) 0.66. The TLR signaling model for poeModules M14-M18 was generated by adding all target genes whose promoters were bound to the transcription factors Nfkb1,2, Irf3,7, or Stat1,2 (1.5 standard deviations or more) in LPS-stimulated dendritic cells during at least one time point  relations among the underlying eQTLs. In particular, we used a single synthetic collection comprised of two eQTLs and 80 expression traits, where each of these eQTLs affected 20 traits (10 of them affected by both eQTLs) and additional 50 traits were not associated with any of these eQTLs. Overall, a single dataset consisted of 500 collections with three important parameters: the joint-effect model of the two eQTLs (either additive or epistatic effects); the number of individuals; and the genetic effect size. These synthetic datasets and performance evaluations based on an “accuracy score” are described in more detail in the Materials and Methods Section.k = 1, “Non-iterative POEM”; see Section Materials and Methods). The comparison provided valuable insights regarding three key components of the POEM algorithm: (i) the residual-based methodology; (ii) the grouping of traits; and (iii) the iterative learning approach. For each POEM component, we first consider the relevant comparison and then discuss the contribution of the particular component to the overall performance.We investigated the performance of POEM compared to four alternative methods: a RBSR applied to each transcript independently  or a partition-based (PBSR) approach, performs better in detecting additive effects. We found that the residual-based approach performed well in predicting additive effects . To confirm the validity of our synthetic data analysis we tested the performance of these methods in the case of a dataset with epistatic effects, where the partition-based approach is expected to perform better since it is tailored for allele-specific alterations. We found that in the case of epistasis the PBM indeed outperformed the residual-based methods . Specifically, we investigated the performance of three residual-based methods, namely “RBSR,” “POEM,” and “Non-iterative POEM,” compared to the “PBM.” We found that all three residual-based methods outperformed the partition-based approach when applied to an additive model . Notably, the performance of the non-iterative POEM approach was significantly better than that attained by the RBSR approach [P < 0.0002 (additive model) and P < 0.001 (epistasis model); paired t-test; Figure An iterative refinement of the eQTLs contributes to the performance of POEM. To evaluate the contribution of POEM's iterative approach, we applied POEM using k = 6 iterative steps compared to a single iteration (k = 1). We found that in the case of additive effects and large effect sizes, accuracy scores were further increased when the iterative approach was used  we identified 109 primary groups, 54 secondary groups, and 19 poeModules . To assess the empirical false discovery rate (FDR) generated by POEM, we repeated the analysis 100 times with permuted gene-expression data generated by randomly shuffling the labels of strains . Using the permuted data, we found an average of 0.2 poeModules, indicating significant poeModules at FDR < 0.006. Similar results were obtained when we counted the numbers of identified traits within the poeModules  for the generation of intermediate groups, while using a stringent overlap cutoff (P < 10−3–10−6) for the generation of the final poeModules . Taken together, although the intermediate groups may consist of false discoveries, POEM successfully controls the FDR in its final poeModules output.Here we focus on the poeModules that were generated after six iterative steps  of the traits in the poeModules were affected by two trans-acting eQTLs  since the remaining modules were nested within them . To build a regulatory model of this network, we focused on the particular anti-viral and inflammatory TLR signaling pathways that are triggered by the pathogenic stimulations in our dataset: poly I:C, PAM, and LPS . Interestingly, whereas a pleiotropic secondary eQTL has an effect on all 31 genes, a variety of primary effects are more specific to particular subsets of genes. For example, three components of the NFκB complex, which plays a key role in TLR signaling, have the same secondary eQTL with distinct primary eQTLs . This observation highlights the central role of combinatorial regulation in molecular processes, and emphasizes the importance of pairwise additive effects for interrogating regulatory circuits.Next, we demonstrate the multifurcating structure of poeModules M14−M18 Figure , which rS Figure . We inclS Figure . We furtIn an effort to identify co-regulated modules, most modularization methods are focused on co-expression of traits, tacitly assuming that the shared regulatory mechanisms underlie similar expression patterns [e.g., “module networks”  than those of the PCluster algorithm, with smaller standard deviation . InVamod produced best (most significant) median association P-values that are lower than 0.01 in 109 (100%) of the groups, compared to only 76 (70%) of the PCluster groups. The improved predictions of InVamod are in agreement with the absence of genotyping data as prior to the PCluster algorithm. Thus, substitution of InVamod with a co-expression-based grouping method such as PCluster resulted in reduced performance.Here we asked whether the co-association-based grouping played a key role in identifying the poeModules. To this end we compared the co-association-based InVamod algorithm to the co-expression-based PCluster algorithm. When applied to the dataset of dendritic cells, InVamod created 109 groups. PCluster was then applied using this number of groups as input. In most cases, using InVaMod increased the coherence of the association signal within the groups. For example, the results in Supplementary Figure Pairwise eQTL effects provide a model for deciphering regulatory programs that act on gene expression. In this study we presented POEM, a novel algorithm for the characterization of gene modules that are affected additively by pairs of eQTLs. POEM relies on trait grouping based on eQTLs inferred by the RBSR method, and further refines the primary and secondary eQTLs in an iterative manner. The poeModules are manifested as significant overlaps between the primary and secondary groups. Whereas, existing eQTL methods centered on joint epistatic effects  stepwise regression approach through a series of iterative refinement steps , the European Research Council (637885), and a fellowship from the Edmond J. Safra Center for Bioinformatics at Tel-Aviv University . Research in the IG. Lab is supported by the Israeli Centers of Research Excellence (I-CORE): Center No. 41/11 and the Broad-ISF program (1168/14). IG is a Faculty Fellow of the Edmond J. Safra Center for Bioinformatics at Tel Aviv University and an Alon Fellow.The authors declare that the research was conducted in the absence of any commercial or financial relationships that could be construed as a potential conflict of interest.


F) were assessed to locate genomic regions with increased inbreeding. Despite finding some decreases in fertility associated with incremental increases in F, most emerging local ROH were not significantly associated with DPR or SCS. Furthermore, the analyses of ROH could be approximated with the most frequent haplotype(s), including the associations of ROH and F or traits. The analysis of the most frequent haplotype revealed that associations of ROH and fertility could be accounted for by the additive genetic effect on the trait. Thus, we suggest that a change of autozygosity is more likely to demonstrate footprints of selected haplotypes for production rather than highlight the possible increased local autozygosity of a recessive detrimental allele resulting from the mating between closely related animals in Jersey cattle.Inbreeding is often an inevitable outcome of strong directional artificial selection but on average it reduces population fitness with increased frequency of recessive deleterious alleles. Runs of homozygosity (ROH) representing genomic autozygosity that occur from mating between selected and genomically related individuals may be able to reveal the regions affecting fitness. To examine the influence of genomic autozygosity on fitness, we used a genome-wide association test to evaluate potential negative correlations between ROH and daughter pregnancy rate (DPR) or somatic cell score (SCS) in US Jersey cattle. In addition, relationships between changes of local ROH and inbreeding coefficients ( Drosophila melanogaster is expected to be substantial (25%) [Inbreeding increases autozygosity of loci throughout the genome, some of which causes homozygosity of recessive alleles that may cause expression of an unfavorable phenotype. Although the exact number of deleterious mutations have not been estimated, it is important to note that most animals are assumed to be carriers of at least one recessively inherited disorder . Previoual (25%) . While aal (25%) .F), but in an outbred natural population correlations between F and sparse molecular markers were relatively weak [Although inbreeding on average increases the expression of recessive deleterious alleles corresponding to genetic disease , creatively weak . Recent ely weak . Varyingely weak . Moreoveely weak . Europeaely weak . In cattely weak . Becauseely weak , genomicely weak .F was examined to identify if specific genomic regions contribute more frequently to overall autozygosity. These ROH were then compared to examine if there was a genetic effect on two fitness traits. Furthermore, the whole genome was scanned using haplotype windows allowing comparison between the most frequent haplotype alleles relative to ROH. These latter analyses provided insights of inbreeding depression relative to locus-specific  and general genome-wide effects , in an attempt to elucidate possible mechanisms underlying associations between genetic diversity and fitness [The combination of intense selection for improved milk production based and artificial insemination from a limited number of related elite sires has reduced effective population size in most dairy breeds, and possibly also reduced average animal fitness. For example, a negative genetic correlation between reproduction and milk yield is currently one of the obstacles for accelerated genetic improvement of fertility . While m fitness .F) [F ~ 0.1) and outbred structure because of intensive use of a small number of influential males selected for artificial insemination (AI) and mated to cows that probably originated from common ancestors born more than three generations ago, which creates complex pedigree structures consisting of multiple inbreeding loops. The genotyped animals consisted of 1,219 male and 383 female Jersey animals that were born between the 1960s and the 2000s. Most genotyped animals  were born after 1990s, but frozen semen samples maintained for AI enabled us to genotype influential ancestors that were born before 1990s (N = 322), which may affect genetic polymorphisms of contemporary Jersey cattle. To reduce population stratification, animals were only sampled from small half-sib families (n = 3–30) without considering relationships in the extended pedigree. Then, the Illumina BovineSNP50 beadchip  was used to genotype 1,602 animals sampled from the U.S. Jersey cattle described above. The PLINK software [10p<3), genotyping rate (>0.8), and individuals with missing genotypes (<20%) and data were removed that did not conform. Finally, 36,869 SNPs were selected in autosomal chromosomes and used in all subsequent analyses. SNP genome coordinates were obtained from the bovine genome reference assembly UMD 3.1.Pedigree information obtained from the USDA-Animal Improvement Programs Laboratory  on 19,966 registered Jersey animals born in the United States between 1953 and 2008 was used to compute inbreeding coefficients (F) estimated based on pedigree (FPED). Haplotypes were obtained using fastphase [Two approaches were used to define a homozygous genomic region. First, runs of homozygosity (ROH) were determined under the following criteria using a Perl script according to the modified methods that were suggested for the analysis of data in humans . Consideastphase  and the F and ROH was assessed with or without adjusting for potential stratification. Therefore, the homozygous state of SNPs defined by ROH was used to conduct the analysis of principal components. Principal components (PCs) are obtained, which are included as covariates in a multiple regression model involving ROH and the inbreeding coefficient, which enables to adjust for underlying population structure [Principal components analysis (PCA) was performed using genotypes and ROH separately to examine and correct the potential population structure . Associatructure . To calctructure  was usedy = β0 + β1H + e where, y was the inbreeding coefficient (FPED) of an individual, β0 was the intercept of the equation, β1 was the coefficient of the predictor, H is the ROH as a particular locus (0 or 1), and e is the random error. To correct the effect of population stratification, principal components (PCn) from PCA were included as covariates in the linear model [PCi is ith principal component obtained using ROH and n is the number of PCs. Additionally, birth year of an animal was analyzed as a response (y) using the same model to assess the change of homozygosity. Statistical thresholds for a genome wide search of associations of genomic homozygosity and pedigree based inbreeding coefficient were determined empirically using permutation tests [To evaluate associations between ROH at individual loci and inbreeding coefficients, a linear model was used without transforming FPED. The regression model was ar model , y=β0+β1on tests . Inbreedon tests . The genhttps://aipl.arsusda.gov/). Daughter pregnancy rate (DPR) was calculated from days open and directly relates to the proportion of females eligible to become pregnant in a 21 day period [y = β0 + β1G + e, where y is PTA of DPR or SCS of an individual, β0 is the intercept, and β1 is a vector of recessive genetic effect. G is an indicator variable for the effect of ROH of an individual, and e is the random error. In this analysis, the effect of population structure in PTA was reduced by statistical adjustment using a linear mixed model including the effect of genetic relationships and environmental factors. The genome-wide statistical significance level was determined using 1,000 permutation tests [Association of ROH and a trait was examined across the genome. In addition, the sliding window approach partitioned the haplotype homozygosity (HH) effect of each allele contributing to overall homozygosity. The predicted transfer ability (PTA) of daughter pregnancy rate and somatic cell count (SCS) in milk records of U.S. Jersey, which are genetic values after adjusting for environmental and polygenic effects, were obtained from USDA-AIPL  using the same model for ROH-trait association. Similarly, the homozygote of the most frequent haplotype was assumed to encompass the recessive allele and other haplotypes were assumed to have no recessive allele. The genetic effect was evaluated using the same model used for association test of ROH and trait by replacement of homozygote status indicator with an allele number in the sliding window. Finally, functional annotation for genes located in the regions identified from the association test between ROH and phenotypes was performed using the Enrichr software [Since ROH reflects the sum of haplotype homozygosity, the additive effect is inestimable. Therefore, associations of the most frequent haplotype and fertility (DPR) were also estimated using the additive or recessive model for the further understanding of genetic effect of the haplotype that is highly correlated with ROH. To estimate the additive genetic effect of the most common haplotype, the number of alleles of the most frequent haplotype are considered SNP-like genotypes  with increasing FPED during the last five decades  were assessed to identify genomic regions accounting for FPED increases. This analysis revealed ROH levels had increased at 60 or more regions , which account >90% of the total variation, revealed only a few regions that were significantly associated with FPED . Three rVCAM1 and SLC35A3 are closely located between 42 and 43 Mb on BTA 3. There was an overlap between ROH-DPR and ROH-SCS associations at a region on BTA 7 but ROH was not associated with FPED in this region.Next, the association of the ROH and traits, including DPR (ROH-DPR associations) or SCS (ROH-SCS associations) were examined across the genome using linear regression, and the results identified genomic regions affecting DPR . ROH wasCOL4A2, COL4A1, GJA2, GJB3, and GJB6) are located on BTA 3 (~40 Mb). Annotation of the regions that were associated with SCS revealed several interesting genes that may be involved in biological pathways like immune response, including CBLB (BTA 1) and NCK1 (BTA 1) that are involved in T cell receptor signaling pathway.The genes identified in the regions that were significantly associated with DPR or SCS and the respective biological pathways are listed in F, ROH that affected the change of inbreeding coefficient did not greatly influence the traits known to be influenced by inbreeding depression in most genomic regions . This analysis examines the considerable change of ROH during the last five decades, which represents the regions under directional selection. Inbreeding coefficients in dairy cattle have increased mostly by selection of phenotypically superior ancestors that were frequently used as parents the next generation. When considering the annual change of ROH across the genome, several local autozygosity appeared to be involved in inbreeding and the increased level of ROH on BTA 2, 3, 5, 7, 8, 9, 16 and other regions , S3 Fig,To examine the genetic effect mode of ROH, the correlation of the most frequent haplotype and ROH was assessed using haplotype homozygosity (HH) and then an association test between the most frequent haplotype and a trait was conducted. The HH calculated with a sliding window approach was then compared with ROH based homozygosity, allowing us further understanding of ROH. With a 50-SNP sliding window, mean length of haplotype was 3.4 Mb (standard deviation = 0.52) with an average of 123.5 (standard deviation = 77.0) alleles per haplotype . The corWhen testing the association between ROH and fertility, we found that many regions with high or moderate levels of ROH were associated with fertility positively, suggesting that the genetic effect of most ROH appears to be unrelated to decreasing DPR. In A deleterious mutation may emerge randomly or one that exists may increase in frequency due to close linkage with a favorable allele under strong selection . BecauseSome inbreeding depression has been shown to be due to rare large-effect viability and fertility mutations in dairy cattle , 33. HowSpecifically, more than 60 regions showed a considerable change of local autozygosity accompanied by an elevation of inbreeding during the last few decades in Jersey cattle. On the contrary, only a small amount of ROH was associated with lower fertility in contemporary Jersey cattle. In addition, regions with high ROH (>0.3) created by recent inbreeding have not always been detrimental. In Australian Jersey cattle, only one genomic region affecting fertility was detected on the X chromosome using ROH based test . SystemaInbreeding coefficient (FPED) tends to be affected by influential and recent common ancestors, which may include stratification. According to the PCA based on genotypes, there was no obvious structure in Jersey data. However, the association test including or excluding stratification suggested to us some debatable results. After removing stratification using PCA of ROH, only three regions appeared to be bound to change of inbreeding coefficient, which deviated from our expectation. Interestingly, the correlation of the first principal component (PC1) and FPED was 0.7, which suggests that PC1 may account for the large amount of variation due to inbreeding coefficient (FPED). Therefore, we focused on the results from the analyses without PCs.F value compared to overall genomic autozygosity. In Jersey cattle, multiple regions showed substantial change of autozygosity during the last few decades, which suggested genetic forces like selection are more likely to play an important role in determining genome-wide patterns of autozygosity as well as inbreeding.The correlation between the genomic and pedigree inbreeding coefficient is less than 0.7 in Jersey cattle, which is lower than the value observed in small human families (~0.85) . The corSLC35A3 on BTA 3 causes complex vertebral malformation (CVM) [SLC35A3 or a neighboring gene such as vascular cell adhesion molecule-1 (VCAM1) that mediates leukocyte recruitment from blood into tissues and affects embryo survival rate in the mouse [The local effect hypothesis assumes that heterozygosity-fitness correlations are the results of heterozygosity of particular genes and loci . A mutaton (CVM) , which ion (CVM) . A causaon (CVM) . Anotherhe mouse . Decreashe mouse . Lethal he mouse , 44, buthe mouse . This maDrosophila [Our primary aim was to identify the regions corresponding to local autozygosity by detecting associations of ROH and DPR or SCS. Despite the fact that generally the inbreeding coefficient is the most influential factor associated with the decline in fertility, reproduction traits are generally quantitative traits and are largely affected by environment . QTL maposophila , demonstosophila . Despiteosophila  Using haosophila . It is iosophila . The powosophila . Estimatosophila . Thus, aIn humans, susceptibility of disease increased with the increase of overall autozygosity, but a local effect of autozygosity on a disease has not been clarified . In the F with large sample size  [If records are not available, ROH appears to be the most powerful method for analysis of detecting inbreeding effects from among several alternative estimates of >12,000) . However>12,000) , 51. In >12,000) , which cWe have suggested methods for assessing the relationships between inbreeding, genomic autozygosity, and trait performance using prior information from a Jersey cattle population. ROH could be considered the optimal one to measure inbreeding based on genomic information . A surveS1 Table1Definition of ROH based on the number of continuous homozygous SNP  or size (3 or 5 Mb). 2Regression coefficients of FPED on FROH are shown.(DOCX)Click here for additional data file.S2 TableCorrelation coefficient (r) are shown: r of the ROH-FPED, ROH-Year, ROH-DPR, and ROH-SCS represents associations between ROH and FPED, birth year, DPR, and SCS, respectively.(DOCX)Click here for additional data file.S3 Tablewww.kegg.org) are summarized in Excel file.The function of gene is annotated using Enrichr. Genes annotated by KEGG database ((XLSX)Click here for additional data file.S4 Tablewww.kegg.org) are summarized in Excel file.The function of gene is annotated using Enrichr. Genes annotated by KEGG database ((XLSX)Click here for additional data file.S1 FigThe y axis represents the pedigree inbreeding coefficient (FPED), and the y axis indicates the levels of genomic inbreeding based on ROH (FPED).(TIF)Click here for additional data file.S2 FigA) Significant level of associations, B) Regression coefficient of associations. A dotted line shows genome-wide significant threshold . The positive or negative effect of daughter pregnancy rate (SCS) and ROH, which are defined by the slope of regression is plotted across the genome (B). Negative effect represents the region with decreased the levels of SCS by increased levels of ROH.(TIF)Click here for additional data file.S3 FigROH-F associations are plotted with gray bars and associations of ROH and birth year are shown in black dots. The dotted line shows the genome-wide significance level of ROH-FPED associations.(TIF)Click here for additional data file.S4 FigROH (y axis) corresponding to sliding window is plotted against haplotype homozygosity . Mean ROH of in a sliding window is calculated to compare with 50-SNP haplotype homozygosity.(TIF)Click here for additional data file.S1 File(DOCX)Click here for additional data file.


Arabidopsis thaliana root length and found that the genomic narrow-sense heritability for this trait in the examined population was statistically zero. The low amount of additive genetic variance that could be captured by the genome-wide genotypes likely explains why no associations to root length could be found using standard additive-model-based genome-wide association (GWA) approaches. However, as the broad-sense heritability for root length was significantly larger, and primarily due to epistasis, we also performed an epistatic GWA analysis to map loci contributing to the epistatic genetic variance. Four interacting pairs of loci were revealed, involving seven chromosomal loci that passed a standard multiple-testing corrected significance threshold. The genotype-phenotype maps for these pairs revealed epistasis that cancelled out the additive genetic variance, explaining why these loci were not detected in the additive GWA analysis. Small population sizes, such as in our experiment, increase the risk of identifying false epistatic interactions due to testing for associations with very large numbers of multi-marker genotypes in few phenotyped individuals. Therefore, we estimated the false-positive risk using a new statistical approach that suggested half of the associated pairs to be true positive associations. Our experimental evaluation of candidate genes within the seven associated loci suggests that this estimate is conservative; we identified functional candidate genes that affected root development in four loci that were part of three of the pairs. The statistical epistatic analyses were thus indispensable for confirming known, and identifying new, candidate genes for root length in this population of wild-collected A. thaliana accessions. We also illustrate how epistatic cancellation of the additive genetic variance explains the insignificant narrow-sense and significant broad-sense heritability by using a combination of careful statistical epistatic analyses and functional genetic experiments.Efforts to identify loci underlying complex traits generally assume that most genetic variance is additive. Here, we examined the genetics of  Arabidopsis thaliana to determine the additive and non-additive genetic contributions to the phenotypic variation in root length. Surprisingly, much of the observed phenotypic variation in root length across genetically divergent strains was explained by epistasis. We mapped seven loci contributing to the epistatic genetic variance and validated four genes in these loci with mutant analysis. For three of these genes, this is their first implication in root development. Together, our results emphasize the importance of considering both non-additive and additive genetic variance when dissecting complex trait variation, in order not to lose sensitivity in genetic analyses.Complex traits, such as many human diseases or climate adaptation and production traits in crops, arise through the action and interaction of many genes and environmental factors. Classic approaches to identify contributing genes generally assume that these factors contribute mainly additive genetic variance. Recent methods, such as genome-wide association studies, often adhere to this additive genetics paradigm. However, additive models of complex traits do not reflect that genes can also contribute with non-additive genetic variance. In this study, we use  Identifying the loci underlying quantitative phenotypes is among the central challenges in genetics. The current quantitative genetics paradigm is based on the assumption of additive gene action, despite an increasing body of evidence showing that non-additive effects are crucial to many, if not most, biological systems –4. The kArabidopsis thaliana [Drosophila melanogaster [The drawbacks of studying epistasis in natural populations of higher organisms include the cost and time to collect large datasets, the lack of power due to noisy phenotypes, and the difficulties in performing the necessary follow-up studies to replicate the epistatic associations, identify the polymorphic genes, and dissect the underlying mechanisms leading to statistical epistatic associations. Efforts to utilize epistatic interactions for detection of novel loci, however, are of importance for enhancing sensitivity in statistical analyses , for selthaliana  and Drosnogaster . Many poA. thaliana RegMap [D. melanogaster Drosophila Genetic Reference Panel [A. thaliana collections and data sets [A. thaliana root phenotypes [A. thaliana has a small genome (~125 Mb), large numbers of readily available inbred accessions, a large knowledge base on how to efficiently score quantitative phenotypes in large numbers of individuals, and powerful experimental resources, such as collections of knock-out lines. This makes such populations into particularly well-suited models for performing exhaustive scans for epistasis and for evaluating whether non-additive genetic analyses are able to uncover novel contributions of genes to the studied phenotypes. However, due to the advantages of the model populations outlined above, the power of a GWA will be considerably larger than in a segregating natural population of the same size in, for example, humans. Hence, advanced analyses are still worth considering, as the power to map novel loci will be considerably higher than expected based on sample size alone.Two major challenges have often been discussed in relation with GWA analyses based on statistical epistatic models: the lack of sufficiently large experimental datasets to obtain reasonable statistical power and the computational complexity when screening the genome for multi-locus epistasis. One of the causes of low power is the need to use stringent, multiple-testing corrected significance-thresholds to account for the large number of statistical tests performed when exploring epistasis between all possible combinations of loci . As allea RegMap  or 1001 a RegMap  or the Dce Panel  collectiata sets ,28, inclenotypes –31. A. tModel organism GWA studies, however, are often based on a hundred or fewer individuals, and valid concerns have been raised regarding the challenges of testing for associations with hundreds of thousands of markers, or millions of combinations of epistatic markers in such small samples. In particular, the risk of detecting a random association will increase especially when the number of individuals in the minor two-locus genotype class is small. This situation is not accounted for by common statistical multiple-testing correction. Further, there is also a risk of statistical epistatic associations arising due to “apparent epistasis”  in whichA. thaliana root length in a population of wild-collected accessions. The broad-sense heritability for this trait was of intermediate magnitude. However, very little additive, but considerable epistatic, genetic variance could be captured by the genomic relationships between the accessions. The genetic contribution to this trait was therefore unlikely to be detected using standard additive genetic models, and it was not surprising that no locus was associated with root-length in the standard additive-model based GWA analyses. An exhaustive GWA scan for two-locus interactions, however, revealed four genome-wide significant interacting pairs. All of these displayed genotype-phenotype relationships in which epistasis cancelled the marginal additive genetic effects of the interacting loci, explaining the low levels of additive genetic variance in the population. A new statistical method was used to evaluate the robustness of the GWA associations in the small population, and it was estimated that half of these pairs are at risk of being false positives. We also found low risk that the statistical epistatic interactions were caused by “apparent epistasis” [A. thaliana populations that would have remained undetected without an epistatic GWA analysis.Here, we conduct a GWA study to dissect the genetics underlying istasis” . Our expistasis” . TogetheA. thaliana root mean lengths across 93 wild-collected accessions /VP) to be 0.24 (95% CI 0.10–0.38) for mean root length (−6) and only a small, insignificant fraction was additive . These results suggest that an epistatic GWA analysis might be needed to identify the loci explaining the heritable phenotypic variation in root length mean.In our dataset, mean root length varied significantly across accessions , this outcome was not surprising.Two GWA analysis methods were used to screen for additive loci contributing to the phenotypic variation in mean root length. The first method was a standard single-locus additive mixed-model based approach as implemented in the R-package GenABEL  that accA. thaliana genome [2 < 0.8).To identify loci that contributed to the epistatic genetic variance for mean root length, we performed an exhaustive two-locus SNP-by-SNP epistasis analysis (PLINKv1.07) . Despitea genome . In totaThe four SNP pairs that were significant in the epistatic GWA ; Fig 2a We next applied a conservative genome-wide test to estimate the risk that the epistatic GWA associations were false positives due to the small population size. A pseudo-marker was created for each pair of SNPs in the epistatic scan, resulting in an epistatic pseudo-genome with 78 million markers. Given the epistatic effect estimated for the four pairs that reached genome-wide significance in the epistatic GWA, we computed the genome-wide probability of these pairs being false-positives using the R/p.exact package , assumin2) with one of the “epistatic pseudo-markers” described above. Therefore, “apparent epistasis” involving physically unlinked markers is only expected for very strong mutations. However, if any such hidden variant exists, by definition neither its genetic effect nor its linkage disequilibrium to the “epistatic pseudo-markers” can be estimated from the GWA datasets. However, if a GWA study is performed in a species with sequenced reference populations, the genomic data from that population can be used to estimate the risk of observing a high r2 between the tested markers and other polymorphisms in the genome. Here, we used the 1001 Genomes Project A. thaliana genomes [2 (r2>0.8) between any individual SNP in the re-sequencing data and the “epistatic pseudo-markers” for the four pairs. This analysis revealed that the “epistatic pseudo-markers” corresponding to the identified pairs have very low r2 to most of the hidden variants (mean r2 = 0.01). There is a very low risk to observe a high r2 (r2 > 0.8) to any hidden variant in our population for the most strongly associated pair (p < 0.005). For the other three pairs, this risk is still low (p = 0.18–0.42) but sufficiently large to indicate that the results should be interpreted with caution  and located within 5kb upstream or downstream of the leading SNP in the epistatic GWA were considered further as candidate functional mutations. In We therefore first extracted the genotypes for 13 million genome-wide SNPs from the whole-genome re-sequencing data from the 1001 Genomes Project . SNPs thNAC6, TFL1, At2g28880), we are the first to show that these underlying genes contribute to root growth, and for one locus (ATL5), our results confirm previous findings.Further, we identified T-DNA insertion lines for 12/15 genes and transposons comprising the seven statistically epistatic loci and experimentally evaluated them for effects on root length . For thrA. thaliana root length through an epistatic interaction with a distal SNP on the same chromosome (3_66596). Twenty-one SNPs located in four genes were in high LD with the two linked SNPs (Tables Col-0) controls. The insertion mutant for At3g25520 had significantly shorter roots than wild type ; the insertion mutants for At3g25530 and At3g25540 did not differ significantly  from wild type  [A. thaliana based on these earlier studies would thus be in the range between 5–10% of the evaluated T-DNA lines.Functional dissection of epistasis, however, is a daunting task in natural populations and therefore model organisms are an indispensable resource in this work. A first step in taking on this challenge is to reduce the list of potential candidate genes and mutations for further, in-depth functional explorations. Here, we use mutational analysis of the genes that harbor SNPs in LD with the associated SNPs in the epistatic GWA analysis to identify the most likely candidate genes for the inferred associations. Detection of functional candidate genes using T-DNA analyses in regions identified using GWA analyses is also a potential way to functionally validate the associations. However, its value for this purpose depends on generally unknown mutational target size; what proportion of randomly mutated genes will affect the studied phenotype? Although the majority of evaluated single-gene mutations do not affect root length ,35,50, rmsu.edu) . Anothermsu.edu)  of glucoA. thaliana, suggesting that the current approach increases not only the efficiency to find candidate loci for root-length, but that it also is valuable for validating the importance of the detected associations in shaping natural variation in the studied population.We tested six of the seven epistatic regions with 13 available T-DNA insertion mutants . For fouA. thaliana root length represents an excellent model trait for future in-depth studies of this type of complex signals arising from GWA studies.It will be a major endeavor to functionally explore the biological mechanisms contributing to the statistical epistatic associations. None of the genes affecting root development contain non-synonymous mutations with likely phenotypic effects. Also, none of these genes have previously been described to interact with their respective epistatic locus. Future functional studies will have to explore the effects of the observed regulatory SNPs in transgenic plants in several backgrounds to study the whether these statistical interactions are the result of an underlying biological interaction or not. This endeavor will require significant time and resources, yet we believe that In order to reduce environmental variation among accessions, eighteen individuals from each of the 93 accessions studied in Atwell et al (2010)  were verThirty-one seedlings with hypocotyls less than 5mm were removed because it is likely that germination was severely delayed . Betweene are independent and normally distributed residuals. The narrow sense heritability was estimated as the ratio of the additive genetic variance to the total phenotypic variance, and the broad sense heritability was estimated as the ratio of the sum of the additive and epistatic (additive-by-additive) variance to the total phenotypic variance.Heritability of root length variance and mean were estimated using the repeated measures with genotype as a fixed effect using ANOVA. To parse the contribution of additive and epistatic effects, a linear mixed model including both additive and epistatic effects as random effects was fitted using the R/hglm package , i.e.y=Using the transformed mean and the genotypes for the 93 accessions from Atwell et al. (2010), we performed a series of analyses to detect marker-trait associations. For the standard additive GWA analysis with population structure controlled, we used the function egscore from R/GenABEL . The funbAB3.. In the analysis, all possible pairs of SNPs with a minor allele frequency > 25% were tested.To screen the genome for pairs of epistatic SNPs, we used the PLINK–epistasis procedure  that is −8 were also removed as it was considered unlikely that they would be significant after correction for population structure and applying a multiple-testing corrected significance threshold. For the remaining pairs, a linear mixed model including fixed, additive and epistatic effects, as in the PLINK based initial scan, were fitted together with a kinship correction for population stratification, as in the single locus analyses, using the package R/hglm. We derived a multiple-testing corrected significance threshold for this epistatic analysis (p = 3.2 × 10−10) by estimating the number of independent tests based on the number of estimated LD blocks in the genome. For this, we used the fact that the A. thaliana genome is approximately 125Mb with average LD-blocks of about 10kb , and th, we extracted all the sequencing variants. All four significant pseudo-markers had at least four accessions with the double-recessive genotype  in the 728 sequenced accessions. For each of the pseudo-markers, we therefore sampled 93 accessions from the 728 sequenced ones, keeping the same allele frequency as for the pseudo-marker representing the epistatic pair detected in the GWA dataset, and obtained the genome-wide, high-order LD r2 distribution across all the sequenced variants. This procedure was repeated 50 times for each pseudo-marker, and we calculated the estimates and standard errors of the r2mean, r2max and estimated the risk for “apparent epistasis” as the probability of observing an r2max > 0.8 in a particular random sample.From 728 whole-genome sequences in A. thaliana genome was used to determine the effects of the leading SNPs and the SNPs in high LD with them. Genes were considered as candidates if they were within 1kb of a variant. Expression of the candidate genes was determined using the BAR eFP Arabidopsis browser [The Ensembl Variant Effect Predictor based on the TAIR10 release of the  browser .Col-0).T-DNA lines were obtained for the candidate genes . Root leS1 DatasetRoot length was measured for 210 individuals for 93 accessions.(XLSX)Click here for additional data file.S1 Fig(A) In the GWAS based on an additive model no SNPs reached the Bonferroni threshold. (B) Using a whole-genome generalized ridge regression model, in which SNPs were modeled as random effects, no SNPs reached the Bonferroni threshold.(PDF)Click here for additional data file.S2 Fig(A-C) Mean root length is displayed for all four two-locus genotype-classes for the four pairs of significantly interacting loci. The major allele is indicated by -1 and the minor allele is indicated by 1. The combination of two minor alleles  always has the longest root length compared to the other three-allele combinations, which are more similar to one another. Pictured are mean root lengths for the epistatic pairs for SNPs (A) 3_66596 and 3_9272294/3_9273674, (B) 3_66596 and 5_18241640, and (C) 3_10891195 and 5_1027939.(PDF)Click here for additional data file.S1 Table(DOCX)Click here for additional data file.S2 Table(DOCX)Click here for additional data file.


Bos indicus (Nellore) cattle. The genotypic data comprised over 777,000 single-nucleotide polymorphism markers scored in 995 bulls, and the phenotypic data included deregressed breeding values (dEBV) for weight measurements at birth, weaning and yearling, as well visual scores taken at weaning and yearling for carcass finishing precocity, conformation and muscling. Both analyses pointed to the pleomorphic adenoma gene 1 (PLAG1) as a major pleiotropic gene. VEGAS analysis revealed 224 additional candidates. From these, 57 participated, together with PLAG1, in a network involved in the modulation of the function and expression of IGF1 (insulin like growth factor 1), IGF2 (insulin like growth factor 2), GH1 (growth hormone 1), IGF1R (insulin like growth factor 1 receptor) and GHR (growth hormone receptor), suggesting that those pleiotropic genes operate as satellite regulators of the growth pathway.Two complementary methods, namely Multi-Trait Meta-Analysis and Versatile Gene-Based Test for Genome-wide Association Studies (VEGAS), were used to identify putative pleiotropic genes affecting carcass traits in  Carcass yield plays a major economic role in beef cattle, as the carcass retail price is essentially determined by its weight. As differences in carcass yield between steers are partially heritable, selection and breeding are determinant operations in the beef cattle sector . HoweverBos indicus) cattle. These traits are inexpensive and effortless to measure, and present moderate heritability in the breed [Weight measurements and visual scores of conformation, carcass finishing precocity and muscling (CPM) have been routinely employed in selection to improve carcass yield in Brazilian Nellore . Data filtering was performed with PLINK v1.9 [A total of 995 Nellore bulls were genotyped with the Illumina® BovineHD Genotyping BeadChip assay, according to the manufacturer's protocol. The panel included 777,962 SNPs annotated in the UMD v3.1 bovine genome assembly. These bulls are part of the genomic selection reference population from a commercial breeding program, namely DeltaGen  for birth weight (BW), weaning gain (WG), conformation at weaning (CW), carcass finishing precocity at weaning (PW), muscling at weaning (MW), post-weaning gain (PG), conformation at yearling (CY), carcass finishing precocity at yearling (PY), and muscling at yearling (MY) were obtained from routine genetic evaluations. The single-trait animal models used to generate the EBVs were corrected for environmental and maternal effects, and included records from 1,278,057 animals born between 1985 and 2012, and raised in 315 grazing-based Brazilian herds. The variance ratios required to solve the mixed model equations were computed based on restricted maximum likelihood estimates of the variance components. The heritabilities obtained for BW, WG, CW, PW, MW, PG, CY, PY and MY were 0.37, 0.26, 0.25, 0.25, 0.26, 0.33, 0.31, 0.31 and 0.30, respectively. Prior to the association analysis, EBVs were deregressed following [Records for WG and PG were based on the weight gain from birth to weaning (adjusted for a period of 205 days) and from weaning to yearling (adjusted for a period of 550 days), respectively. Records for conformation, carcass finishing precocity and muscling were taken at weaning and yearling based on visual score evaluations relative to the animals of the same management group. Scores were assigned in a discrete ordered scale ranging from 1 to 5. The model used for BW included the fixed effects of contemporary group  and age of dam at calving, as well as random maternal effects . The model used for weaning traits included fixed effects of contemporary group (concatenation of BW contemporary group and herd-management group at weaning), Julian birth date within birth season, age at phenotype recording and age of dam at calving, in addition to the maternal effects described for BW. Post-weaning gain and the remaining yearling traits were corrected for the fixed effects of contemporary group (concatenation of WG contemporary group and herd-management group at yearling), age at phenotype recording and age of dam at calving.y is the n x 1 vector of dEBVs, 1n is a n x 1 vector of 1s, μ is the overall mean, g is the n x 1 vector of random polygenic effects, and e is the n x 1 vector of random residual effects. Vector g was assumed a linear combination of additive marker effects:X is a n x m matrix of allele dosages at m markers , and b is a m x 1 vector of random additive marker effects. Vector g was assumed N, where G = Kσg2, K is the n x n matrix of additive relationships between individuals, and σg2 is the variance due to genome-wide markers. Vector e was assumed N, where R = Wσe2, W is a diagonal matrix of weights accounting for heterogeneity of variance in dEBVs, and σe2 is the residual variance. The resulting variance-covariance matrix of the model was V = G + R. Notice that this is essentially the polygenic [Prior to testing SNPs for association, the same single-trait regression model was applied across traits:olygenic ,11 or tholygenic  correcteolygenic .W were computed as w = λ-1(d + c), where λ = (1-h2)/h2, h2 is the heritability of the trait before deregression, d = (1 –r2)/r2, r2 is the squared vector of accuracies  of the pseudo-phenotypes, and c is a parameter taking values between 0 and 1 controlling the relative contribution of pseudo-phenotypes on the basis of their reliabilities. As can be seen in the formula, the weights w used here are linearly related to the weights d used by Neves et al. [c. In order to achieve a balanced contrast between dEBVs with high and low accuracy, we adopted c = 0.5.Following Garrick et al., the dias et al. , except g = Xb. This introduces a bias known as 'proximal contamination' [j, we fit a modified null model where matrix K is built excluding all markers on j, guaranteeing that the null model does not contain the marker being tested or any other marker in linkage disequilibrium (LD) with it. Then, each marker on chromosome j is contrasted against the modified null model by using the test statistic t = b/SE(b), where b = (x'V-1x)-1x'V-1y* for y* = y—1μ and SE(b) is the square root of VAR(b) = (x'V-1x)-1. In this way, t is conditional on the random effects of genome-wide markers, such that the model preserves power while correcting for relatedness and population substructure. Additionally, by incorporating matrix W on V, we accounted for heterogeneity of variance in dEBVs while estimating b and SE(b). Estimates for b and SE(b) were obtained by providing V and y* to the mmscore function in GenABEL v1.8–0 [Conceptually, associations are tested by contrasting the null polygenic model against alternative models that include the fixed effect of one candidate marker at a time ,15,16. Hination' , which cination' . BrieflyL v1.8–0 . In summL v1.8–0  in GCTA L v1.8–0 , except In order to identify pleiotropic genes affecting CPM and body weight traits, we combined two distinct but complementary strategies. The Multi-Trait Meta-Analysis method described by Bolormaa et al.  was usedt as the q x 1 vector of signed t-values across q traits, and C the q x q matrix of t-values correlations across genome-wide markers. The test statistic t'C-1t is distributed as χ2 with q degrees of freedom (df) under the null hypothesis of no pleiotropic effect. One standing issue in this implementation is that assuming Ci,i = 1 when some traits present higher average correlations than others may cause highly significant composite scores even when single-trait analyses collectively present poor evidence of association. This issue was corrected here by adding the average correlation of each trait to their respective diagonal elements. The expected proportion of false discoveries among the markers declared significant was computed as f = αm/s, where m is the number of tests, α is the significance level threshold, and s is the number of tests with p < α. In order to select a value of α resulting in a false discovery rate lower than 5%, we applied the procedure described by [fs/m and defined s as the largest p-value rank position i satisfying pi ≤ fi/m for f = 0.05.For each SNP, consider ribed by . Brieflyt-values across markers. The distribution of the VEGAS test under the null hypothesis of no association is unknown but can be approximated using Monte Carlo simulations. The simulation procedure is carried out as follows: random draws from the null distribution are generated as x2 = z12 + z22 + … + zm2, where zm2 is one element of a m x 1 random vector z sampled from a multivariate normal distribution with mean 0q and covariance D, encoded by the matrix of signed genotypic correlations among the m markers within the gene. The probability of observing a VEGAS statistic value as extreme as the one obtained if the null hypothesis is true is then computed as the number of times the simulated x2 values were greater than or equal to the observed VEGAS statistic, divided by the total number of simulations. The number of simulations required to approximate the null distribution is chosen adaptively: let x2 be the vector of random draws from the null distribution. At every iteration, if the p-value is lower than the inverse of the current number of simulations k, k—length(x2) extra samples are obtained, the new p-value is computed, and the new number of simulations is set to 10k)+1log10.Interactions between protein-coding genes were predicted using the STRING database . AdditioA total of 516,740 SNPs and 995 individuals passed the filtering criteria and were retained in the dataset. After data filtering, the mean and median gap between any pair of consecutive markers on the same chromosome were approximately 5.26 kb and 3.15 kb, respectively. The resulting genotyping rate across markers and samples was approximately 0.99. The number of bulls with dEBV accuracy higher than 0.5 for BW, WG, CW, PW, MW, PG, CY, PY and MY were 837, 915, 875, 876, 875, 880, 844, 844 and 844, respectively.t-values represent marked additive genetic effects, we considered both dEBVs and t-values correlations between traits as proxies for the additive genetic correlations between traits. On average (Table 1), dEBVs were moderately correlated across traits (r = 0.442), and strikingly similar results were found for t-values across traits (r = 0.423). An exception was BW, which was only mildly correlated with WG and moderately correlated with conformation traits. These noteworthy genetic correlations suggest that pleiotropic loci may contribute to the genetic variance of these traits.As dEBVs encapsulate individual additive genetic values and association p < 9.20 x 10−5 , resulting in a false discovery rate of approximately 5%. A single large, dominant signal mapping to chromosome 14:19.46–34.92 Mb was identified. The leading SNP, namely rs136543212 (probe BovineHD1400007373), mapped to position 25,502,915, in the vicinity of the well-known PLAG1 (pleomorphic adenoma gene 1) chromosome domain.After combining the results across the nine traits with the Multi-Trait Meta-Analysis method, a total of 983 markers were declared significant at an empirical threshold of PLAG1 is well supported by its regulation of the expression of insulin-like growth factors (IGF) [PLAG1 chromosomal region [PLAG1 are also plausible functional candidates, notably MOS [CHCHD7 [XKR4 [PENK [Bos taurus origin  associated with decreased IGF1 serum concentrations and precocity, as well as with increased height and weight [This chromosomal segment has been implicated as a highly pleiotropic locus underlying genetic differences in growth, weight and reproductive traits in cattle ,23–29. Trs (IGF) . These frs (IGF) , and serrs (IGF) ,33 and p [CHCHD7 , XKR4 [3D7 [XKR4 –37 and PR4 [PENK –40. The d weight .Fig 1B), which suggests that this large segment is a single LD block. In fact, all significant markers were in moderate to high LD with the top scoring marker . However, as correcting for the leading SNP was not sufficient to completely eliminate the signal, it is hard to distinguish between the presence of more than one causal nucleotide within the LD block and residual effects captured by the remaining markers due to imperfect tagging of rs136543212.To determine if this large segment comprised a single signal driven by a large LD block or it construed a mixture of signals, we re-analyzed our data using the same GWA model conditional on the fixed effect of the top scoring SNP. Correction for the effect of rs136543212 alone was able to remove most of the signal  [GDF10) [GDF11) [GAS2L3) [FGF22) [PHLPP1) [STAT2) [SMAD4) [IGFBP5) [MUT) [TNNT1) [TNNI3) [SGCD, also known as 35kDa dystrophin-associated glycoprotein) [Besides 2 (GDF2) , 10 (GDF [GDF10)  and 11 ( [GDF11) , growth [GAS2L3) , fibrobl [FGF22) , PH doma[PHLPP1) , signal  [STAT2) , SMAD fa [SMAD4) , and ins[IGFBP5) . Genes i5) [MUT) , troponi [TNNT1) , troponi [TNNI3) , and sarprotein) .Fig 3A). The remaining 28 genes formed smaller networks ranging from two to six genes (S1 Data). Interestingly, PLAG1 was not present in any network, in spite of being the major pleiotropic gene in our study.We then used the STRING database to annotate protein-protein interactions among the candidate genes. From the initial list of 176 genes, 82 presented connections. From these, 54 genes were involved in a single network (IGF1 (insulin like growth factor 1), IGF2 (insulin like growth factor 2), GH1 (growth hormone 1), IGF1R (insulin like growth factor 1 receptor) and GHR (growth hormone receptor), were not found to shelter pleiotropic variants affecting carcass traits in the present study, we decided to add them to the list to reveal their interactions with the pleiotropic genes found here . The main motivation for this was the known function of PLAG1: by fine-tuning the expression of the IGF family, the predicted downstream consequence of mutations in PLAG1 is direct interference in the growth pathway. This could also be the case for other pleiotropic genes identified here. Another argument to support this strategy was that mutations directly occurring in the growth pathway are likely to produce extreme phenotypes, such that common variation in carcass size is expected to be explained by variation in satellite genes modulating that pathway. For instance, mutations in IGF1, IGF1R and GHR have been implicated in extreme body size reduction across dog breeds [Although key genes involved in growth, such as g breeds , as oppog breeds .IGF1, IGF2, GH1, IGF1R and GHR were added to the gene list, the pleiotropic genes formed a satellite network surrounding a central hub comprising the growth pathway. Additionally, FGF22, ITGB5, LOC506088 and PLAG1 were incorporated to the network when the genes mentioned above were included. This finding supported our hypothesis that the pleiotropic genes detected here are modulators of the growth pathway. It also opens the question whether genetic correlations among the weight and CPM traits encapsulate information about the growth pathway, in the sense that each one of these traits may be serving as a partial surrogate for the same underlying intermediate trait (i.e. growth). In this scenario, one would expect to find similar results in a direct GWA analysis on the intermediate phenotype.Surprisingly, when Bos indicus (Nellore) cattle. These pleiotropic genes formed a network modulating IGF1, IGF1R, IGF2, GH1 and GHR, which are well known major actors of the growth pathway. This finding suggests that common variation in carcass traits is not likely to be explained by mutations in essential genes controlling growth. Instead, the variation may lie in accessory genes that regulate the function and expression of essential genes. Among these accessory genes, PLAG1 seems to be the most influential. Moreover, we did not rule out the possibility that, at some extent, the genetic correlations among the nine traits studied here represent the indirect, partial measurement of a single underlying growth trait. In this case, the pleiotropic genes identified here may simply represent genes affecting the intermediate phenotypes. The future characterization of causal variants in these genes may contribute to improved prediction of carcass yield and more informed mating decisions in B. indicus cattle.We identified genes associated with multiple carcass traits in S1 Data(GZ)Click here for additional data file.