The one-century-old theory of orthogonal genetic variance decomposition originated the field of quantitative genetics and has kept on being improved ever since. Recently, serious concerns about the possibility of attaining a satisfactory implementation of genetic variance decomposition with linkage disequilibrium (LD) and epistasis have been raised. In this paper we dissipate such doubts by completing the classical theory of variance decomposition into additive, dominance and epistasis components with LD. We apply that theory to the analysis of the genotype-to-phenotype maps of two cases of particular evolutionary interest—Bateson-Dobzhansky-Müller incompatibilities and sign epistasis. For the first case we show how negative LD and reduction of heterozygotes may contribute to maintain genetic variability after secondary contact. For the second case we show that LD transforms the set of frequencies leading to an evolutionary plateau into a ridge. Our theoretical developments reassuringly reflect the complexity LD conveys to genetic systems throughout novel properties—as compared with systems under linkage equilibrium. We argue that such particularities might have actually contributed to cause confusion about the feasibility of developing this methodology. In any case, the theory we provide in this paper enables new perspectives in both evolutionary and quantitative genetics studies. Genetic variance decomposition has gained increased scientific attention one century after it was first developed by Fisher . At thatWith time, models of genetic effects enabling variance decomposition (and thus disentangling resemblance between relatives) in the face of increasingly complex genetic systems and/or population facts were developed. A historical, key step forward was given by Kempthorne  and CockMore recently, the development of molecular genetics has made it possible to obtain datasets large enough to undertake realistic strategies of genetic mapping and genomic prediction. Consequently, along the latest quarter century models of genetic effects and variance decomposition have been thoroughly revisited  and from linkage equilibrium (LE). We also provide applications of our methodology to two cases of special evolutionary interest—Bateson-Dobzhansky-Müller (BDM) incompatibilities and sign epistasis—and review and discuss arguments on which doubts about the feasibility of a genetic decomposition with LD and epistasis were based.Following Fisher's , 1930 scA with two alleles A1 and A2 decomposes the genotypic values—i.e., the expected phenotype of each genotype, G = —into additive and interaction (dominance) components using the linear regression framework G = 1μ+Nα+δ  genetic model for a locus N indicates the number of alleles of each type in each genotype, αi, i = 1,2 (the explanatory variables), are the average (additive) effects of the alleles and δij, j = 1,2, i ≤ j, (the error terms) are the dominance deviations.where μ is the population mean phenotype, The regression model in Equation (1) is solved by first clearing away the term with the mean phenotype using the mean-corrected genotypic values asNα +δ. The weighted least-squares (WLS) solution and error terms for this regression may be obtained from its normal equations, N'PNα = N'PN is called the design matrix of the regression, N' stands for its transpose and P  contains the population genotypic frequencies (f (AiAj) = pij, j = 1,2, i ≤ j) in its diagonal, i.e., P = diag(pij)  is the annihilation matrix. Using the latest matrix the error terms may be equated asMatrix G = (1|N)(μ|α)'+δ expanding toIt is worth noting that the regression model in Equation (1), as solved above, is not equivalent to G = 1μ+η. In this expression, vector 1 = ′ is the design matrix, 11′P is the hat matrix, μ = 1′PG is the explanatory variable, I–11′P is the annihilation matrix and η = (I–11′P)G are the error terms. Hence, the mean-corrected vector of genotypic values G–1μ in Equation (2) can actually be interpreted as the error terms of regression G = 1μ+η, since the WLS solution of that regression provides the error terms as η = (I–11′P)G = G–1μ.In fact, the WLS solution of the regression in Equation (5) can easily be found to be different from the results obtained within Equations (2–4). More to the point, Equation (2) can actually be obtained from the WLS solution of regression G = 1μ+η, evidently leading to η = Nα+δ. Below it will be made clear that the methodology provided in this paper for decomposing the genotypic values and the genetic variance in the face of LD and epistasis fits squarely with this sequential procedure.Overall, for obtaining the biologically meaningful parameters aimed in the regression model in Equation (1), the solution is not achieved in a single step—i.e., is not achieved as by means of Equation (5). It is instead achieved in two regression steps, the first of which is Using the solution to Equation (1) obtained above within Equations (2–4), the genotypic values may be decomposed as:Gij, its additive component is αij = αi+αj—cf Equation (1). These αij are actually the breeding values, at least under HWE. In any case, the decomposition of the genotypic values in Equation (6) directly provides the corresponding decomposition of the genetic variance, which can be given by the variance of the values obtained for each set of components  = VA+VD, and hold their biological interpretations. Orthogonality is also an extremely useful statistical property for the development of appropriate model selection strategies in genetic mapping studies, as resumed in the discussion.Since the above decomposition of the genotypic values (Equation 6) is orthogonal by construction, the variance components in Equation (7) provide an accurate decomposition of the genetic variance, VThe mathematical model in Equation (1) has recently been extended to an arbitrary number of alleles with arbitrary HWD  above. Therefore, we consider regression G = 1μ+Nαα+ηα, expanding toWe now consider an additional biallelic locus, Nα are just all possible combinations of two rows of the one-locus matrix N in Equation (1). In any case, in a first step we obtain Ḡ = G–1μ, as in Equation (2) for the one-locus case. Then, the second step consists in obtaining the additive component as the WLS solution of regression Ḡ = Nαα+ηα. Equations  and related text would typically provide the WLS solution for this case as well, did our new matrix Nα in the regression model in Equation (8) not lead to a singular matrix Nα′PNα. However, it actually does. Conveniently, this issue may be overcome by performing the regression outside the kernel, as follows.The rows of Nα′PNα are computed—in practice, this is done simply using e.g., the appropriate built-in commands of R Core Team , using in this case the matrices obtained just above, asFirst, the eigenvalues and eigenvectors of singular matrix ore Team . Next, aThe error term is then obtained, as in Equation (4), asHα = NαMα = I–Hα, being understood that the identity matrix I is used in its appropriate dimension, which for this case is 9 × 9.using the hat matrix l multiple biallelic loci is straightforward, by just enlarging the rows of regression matrix Nα to accommodate all combinations of l repetitions of the rows of regression matrix N in Equation (1). Complexity may also be increased straightforwardly in what regards the numbers of alleles, by just building Nα from single-locus design matrices that, as opposed to N in Equation (1), are appropriate for the number of alleles, nj, of each locus, j. As mentioned above, such matrices have already been provided  to Nα in Equation (8) that the additive components of the genotypic values are αijkl = j = 1,2, i ≤ j, l = 1,2, k ≤ l. In the general case  we would have αG = G indicates the genotype. Then, the additive component of the genetic variance is, analogous to the one-locus case above (Equation 7), VA = V(αijkl) for the two-locus case and VA = V(αG) in the general case.For the case of two biallelic loci, it is easy to derive from the design matrix pijkl, i ≤ j, k ≤ l  ensures by construction that the additive variance computed from Equation (9) as explained just above is accurate regardless both any dominance and/or epistatic interactions involved in the genetic system and any departures from equilibrium—from HWE and/or from LE—affecting the population frequencies. Indeed, such additive variance accounts for any possible departures from equilibrium frequencies because it is obtained using expressions that involve the genotypic frequencies α, entail all possible interactions together—including also epistasis. Thus, further regression steps are still required for detaching the remaining terms needed for completing the genetic decomposition, leading to a full orthogonal partition of both the genotypic values and the genetic variance. Specifically, the next step consists in detaching the dominance component, δ, from the error terms of Equation (8), ηα. This can be done with regression ηα = Nδδ+ηδ, expanding toAnalogous to the one-locus case (Equation 1), the mean and the additive components of the two-locus regression model (Equation 8) have been detached above in two steps. However, as opposed to the error terms in the one-locus case, which accounted for only dominance interactions, the error terms of the regression model in Equation (8), ηNδ just indicates the dominance deviations associated to each genotype, at each locus. The rows of Nδ can be obtained with all possible repetitions of the rows of the identity matrix of the appropriate dimension, 3 × 3, in the same way as the two-locus design matrix Nα in the regression model in Equation (8) are repetitions of rows of the single locus design matrix N in the regression model in Equation (1). This analogy makes special sense by rewriting the one-locus regression model (Equation 1) as G = 1μ+Nα+Iδ. In the general case, with multiple multiallelic loci, the number of rows combined would equal the number of loci and the dimension of each identity matrix would equal the number of marginal genotypes of the corresponding locus.Design matrix In any case, the WLS solution to Equation (11) may be obtained with the method used for Equation (9) above, leading toDδ and Uδ are built with the non-nil eigenvalues of matrix Nδ′PNδ and with their corresponding eigenvectors, respectively. The error term is obtained once again as in Equations  aswhere Hδ = NδMδ = I–Hδ, I being the identity matrix with the appropriate dimension, 9 × 9.with ijkl = i ≤ j, k ≤ l. In the general case  they would be δG = D = V(δijkl) or, in general, VD = V(δG). This dominance variance is by construction orthogonal to the additive variance provided above through Equation (9), since it comes from the error terms of Equation (8). Furthermore, this dominance component is also accurate regardless both dominance, epistasis and departures from HWE and from LE that may occur, in the same way as justified in relation to the additive variance above.As pointed out above for the multilocus additive components of the genotypic values, having a look at Equation (11) makes it evident that the dominance components of the genotypic values are δδ, account only for the between-locus interactions  or for also higher order interactions in the general case. Hence, the epistasis variance is just the variance of those error terms. We may thus call ε = ηδ and express the epistatic variance as VI = V(εijkl) for two loci or VI = V(εG) in the general case. Once again, orthogonality holds by construction since the epistasis terms so obtained are necessarily orthogonal both to the dominance component, from which they have been detached as error terms in Equation (11), and to the additive component, from which they had previously been detached within the error terms of Equation (8). As a consequence of orthogonality, the sum of the three variance components  equals the genetic variance and, hence, the epistatic variance can also be expressed as VI = VG − (VA+VD).After having detached both the additive and the dominance components, the remaining error terms in Equation (11), ηαα, asWith this, we have achieved the main goal of this paper—to overcome the established misconception that LD precludes orthogonal genetic variance decomposition into additive, dominance and epistasis components. In what follows we nevertheless briefly describe how to further split the epistasis terms, ε, into their natural components (arising as the different groups of interactions of the previous variables) in the face of LD. To do so, additional regressions must keep on being solved sequentially, the first of which detaches the additive-by-additive (AA) component, εwhere ⊗ stands for the Kronecker product—the operator providing interaction terms in regressions  and the additive-by-dominance (AD) components, εδα and εαδ, respectively, aswith Nδα|Nαδ), which for this particular case works the same way as splitting it in two steps—one for obtaining DA and another one for obtaining AD. The WLS solution of the regression in Equation (17) requires again, analogous to Equations , to compute the corresponding eigenvectors and eigenvalues 'P(Nδα|Nαδ)) to obtainIn this expression, the DA and AD components are obtained simultaneously by concatenating their design matrices into one as (The error terms are the dominance-by-dominance (DD) interactions,Hααδδ = Mααδδ = I–Hααδδ.with δα using Nδα and a second one for obtaining εαδ using Nαδ. Each of these two steps can be solved using the common WLS method (Equation 3) and the results so obtained are the same as in Equations . Thus, the regression method we are using in this paper when the regressions lead to non-singular matrices—e.g., when using design-matrix (Nδα|Nαδ)—works in the same way as alternative formulations of the regression that do not lead to non-singular matrices—using Nδα and Nαδ sequentially. In any case, the pairwise epistasis components AA, DA, AD and DD of the genotypic values and of the genetic variance may be obtained from Equations (14–19) in a way analogous to the additive and the dominance components obtained from Equations , respectively.It is worth noting at this point that the regression in Expression 17 may as well be performed in two sequential steps—one for obtaining εl loci, all l(l−1)/2 pairwise AA components must be detached first using the design matrices of to the two-locus case and concatenating them as done for the DA and AD components above (Equation 17). All DA and AD components must be detached afterwards in a similar way, followed by all DD components, for which a design matrix built as a concatenation of l(l−1)/2 identity matrices must be used. Next, all third order components AAA, DAA, ADA, AAD, DDA, DAD, ADD, and DDD must be detached sequentially in a way analogous to the pairwise components—with design matrices built as concatenations of Kronecker products of three marginal effects design matrices. The same process must then be repeated up to the lth order, at which the Dl interactions are the last error terms (as the DD interactions were in the two-locus case).In the completely general case , the occurrence of higher order interactions shall accordingly increase the number of regression steps required for completing the decomposition of the epistasis term. With The theory developed above for the decomposition of the genotypic values and of the genetic variance under arbitrary departures from both additivity, HWE and LE gets summarized byThis regression model may be derived in particular from Equations  and is meant to be solved in three sequential steps, the first of which is trivial—it consists in just computing the mean-corrected vector of genotypic values. Thus, the solution of the regression model in Equation (20) is very similar to that of the classical one-locus case (Equation 1), which requires one fewer regression step, as well as shorter design matrices.If the decomposition of the epistasis term is also required, the regression model in Equation (20) must be extended. For a genetic system with two biallelic loci, by just considering also Equations  it is easy to derive that extension asto be solved using two additional regression steps . In the general case, the regression model in Equation (21) must be further extended to accommodate all levels of higher order interactions as explained above.A and B, with pairwise epistasis—BDM incompatibilities and sign epistasis.Hereafter we consider two cases of particular evolutionary interest that may be addressed using a genetic system with two biallelic loci, A1 and B1 are fixed, which splits into two isolated populations that are in their turn invaded by initially neutral mutations A2 and B2, respectively. However, as soon as the two populations enter into secondary contact the simultaneous occurrence of alleles A2 and B2 in individuals causes a fitness decline. The left-hand side of A1A1B1B1–R = G1111—all marginal effects are nil. Nevertheless, it is well-known that the presence of all kinds of epistatsis components  implies that non-nil marginal effects shall arise both when representing the GP map from different individual-reference points and when analyzing it at the population level. Hence, it is expected that additive, dominance and epistasis variance components are non-nil under many conditions.First, we focus on the BDM case  = f (B2), which is expected to occur at the beginning of the secondary contact described above. In one case (gray lines) the individuals are assumed to come in equal numbers from the two populations and therefore f (A1) = f (B2) = 1/2, whereas in the other case (black lines) the number of individuals coming from one of the populations doubles that of the other one and therefore f (A1) = f (B2) = 1/3. Alvarez-Castro and Le Rouzic , but also to the right of that point  and to the left of it .Such variance components can be observed in e Rouzic  have obsA1B2 or A2B1, depending upon its population of origin. Starting with LE  = f (B1) = 4/5 (gray lines) and another one with f (A1) = f (B1) = 9/10 (black lines). As opposed to We have also inspected the scenario close to fixation of genotype D' = −0.6. A decrease in additive variance around the fixation of A1A1B1B1 can be perceived as an incipient plateau toward the left corner of the additive variance surface in That slow down of the selection speed toward fixation due to departures from equilibrium frequencies can also be visualized in F = 0.3 at each of the two loci. That incidence is enough to cause an evident additive variance plateau around the fixation of A1A1B1B1 .The right-hand side of A1 and B1. Negative LD causes the same effect but in the perpendicular direction (The drop of additive variance at intermediate frequencies in irection .f (A1) = f (B1) = 1/3 the additive variance (black solid line) increases with negative LD and decreases with positive LD, which reflects the formation of a ridge in the additive variance surface—growing either in the direction of equal allele frequencies in the two loci (with positive LD) or in the perpendicular direction (with negative LD) as mentioned above. D' values.The aforementioned drop of additive variance of the sign epistasis case at intermediate frequencies is also shown in A1A2B1B2 and also in an F2 population but not when frequencies are not intermediate at least at one locus. However, AA effects cannot typically (under LE) generate  marginal dominance effects. This is why the dominance variance is nil in D' = 0. However, the combined effect of AA effects and LD makes dominance variance to arise in both cases considered in Beyond what LD implies when sign epistasis occurs, this case also enables us to describe a general property of epistasis with LD, which entails a remarkable difference with systems under LE. Such property can be revealed by focusing on the dominance variances of A1B1|A1B1, A1B1|A2B2, and A2B2|A2B2. For the specific pattern of sign epistasis we are considering here, the genotypic values of these genotypes are those of a single locus with overdominance—in particular, 0, 1, and 0, respectively, as A1B2|A1B2, A1B2|A2B1, and A2B1|A2B1 would remain, with genotypic values 2, 1, and 2, respectively). Thus, regardless its sign, LD gradually transforms a two-locus genetic system with sign epistasis (with only AA epistasis and no dominance) into a single-locus system with dominance. That is exactly what the decomposition of the genetic variance with LD (performed as developed above) reflects in D' = 0 and with increasing dominance variances (and decreasing epistasis variances) toward both sides. Overall, as opposed to what occurs under LE, LD makes AA epistasis to influence marginal dominance effects. Besides, by generating dominance and dominance variance, the combination of AA effects plus LD also generates all three remaining kinds of epistatic interactions—DA, AD, and DD. Hence, in particular, the epistasis variances shown in Let us begin by considering the most extreme cases of LD. With complete positive association of alleles, only three out of the nine genotypes remain—D' , from LE genotypic frequencies.Throughout one century, the theory for the orthogonal decomposition of the genetic variance into additive, dominance and epistasis components remained unfinished. Several recent works have considered LD when performing genetic variance decomposition, although not providing an orthogonal decomposition of the genetic variance with LD  step by step. Finally, we have shown that the method used to overcome non-singular matrices in WLS regression works as desired since it provides the same solutions as alternative ways of setting out the regression that enable a conventional WLS solution .In general, the theory developed in this paper illustrates the potential of matrix algebra applied to regression analysis in the context of the orthogonal decomposition of the genotypic values and the genetic variance. With two biallelic loci, the regression model providing an orthogonal decomposition into additive, dominance and epistasis components takes a rather simple form even when expanded to show all scalars within each matrix. Then, by just describing the way in which the design matrices must be modified , the same regression model is straightforwardly extended to arbitrary numbers of alleles and loci. Also by virtue of matrix algebra, the WLS solutions of the subsequent steps of the regression model take the form of manageable expressions even when the conventional WLS solutions involve non-singular matrices.Some concerns about the feasibility of a fully orthogonal decomposition of the genetic variance with LD come from realizing that, by definition, LD generates non-independence at the among-locus level  epistasis may condition the evolutionary outcome of a genetic system while the epistatic variance remains low.D'.In the sign epistasis case, we have kept things even simpler by sticking to a GP map that can be built with just AA epistatic effects. It has been by using the simplicity of that genetic system that we have been able to show how LD may (in combination with just AA epistasis) generate dominance variance. More in particular, we have also shown how LD turns the evolutionary plateau that has been described for this system under LE into a ridge, whose orientation depends upon the sign of the standardized disequilibrium index The potential applications of the orthogonal decomposition of the genetic variance under LD certainly go beyond the applications we have here addressed. Álvarez-Castro and Yang  describeWe have already mentioned above that, in genetic mapping studies, orthogonal variance decomposition is a key ingredient of model selection strategies. Indeed, the need to overcome difficulties arising in genetic mapping studies and the lack of a satisfactory extension of the classical decomposition of the genetic variance justified the development of alternative orthogonal parameterizations , extensions of the classical models enabling orthogonal decompositions under most of the genetic phenomena involved in the data  provide estimates that are substantially more consistent  than models that enable orthogonal decompositions only under more restricted conditions (accounting for arbitrary allele frequencies alone; Vitezica et al., Overall, orthogonal genetic variance decomposition is nuclear in evolutionary and quantitative genetics, its usefulness goes nowadays far beyond what Fisher  could poJÁ-C established the research aims, conceived the regression models and design matrices, performed the analyses, wrote the paper and obtained funding. RC solved the normal equations, revised the manuscript and obtained funding.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.


ROH. We demonstrate a statistically significant negative correlation between fecundity and FROH. This work sets the stage for larger scale analyses to investigate genomic regions associated with fecundity and other measures of fitness.Inbreeding depression has been demonstrated to impact vital rates, productivity, and performance in human populations, wild and endangered species, and in recent years, the domestic species. In all cases, standardized, high-quality phenotype data on all individuals are invaluable for longitudinal analyses such as those required to evaluate vital rates of a study cohort. Further, many investigators agree upon the preference for and utility of genomic measures of inbreeding in lieu of pedigree-based estimates of inbreeding. We evaluated the association of measures of reproductive fitness in 93 Golden Retrievers enrolled in the Golden Retriever Lifetime Study with a genomic measurement of inbreeding, FThe online version of this article (10.1007/s00335-019-09805-4) contains supplementary material, which is available to authorized users. The term “inbreeding depression” encompasses a reduction of a trait, often associated with lifetime fitness, as a sequela to a sustained rate of breeding of closely related individuals (reviewed in Charlesworth and Willis PED), is used in lieu of measurement of true autozygosity (Wright ROH) preclude the need for pedigree-based COIs, which depend heavily on pedigree depth and accuracy , estimated from pedigree-based relationships between ancestors  Golden Retriever Lifetime Study (GRLS), seeks to identify genetic and environmental variables that impact longevity in the Golden Retriever . 93 dogs were successfully genotyped, ranging from 1 to 7 years of age. A total of 407 heats were recorded; heat frequency ranged from 0 to 4 heats per dog per year. Recorded heats for dogs over the age of 5 years decreased dramatically, likely reflecting the relative youth of the GRLS cohort as well as increased likelihood for elective spay in older bitches. 66 dogs had produced at least one litter, with a total of 99 litters observed. FROH on body size .Many have demonstrated a negative impact of FP = 0.19).Body size has been observed to impact both age at first estrus, ovulation frequency, and parity across dog breeds .Finally, age at time of parturition has been shown to impact litter size ; however, we did note that dogs who had more than 1 heat per year were likely to maintain this higher than average heat frequency over all years recorded.The canine interestrus cycle is roughly 7 months with high variation across breeds; bitches can also vary individually in their interestrus cycle depending on age and season  may not be reflective of a dog’s potential for SCR. We found that, while dogs with lower FROH had subjectively higher SCR, this result was not statistically significant .We next measured the association of successful conception rate (SCR) versus FROH against the number of live puppies born per litter using a mixed-effects linear model, considering FROH, median height, and age at time of litter log as fixed effect variables and dam ID as a random effect variable. We found a statistically significant negative correlation between FROH and number of live puppies ; binning dams by FROH into lower, middle, and upper thirds demonstrates appreciably lower recorded litter sizes in the uppermost or most inbred third .An alternative mixed-effects linear model was performed using FWhile other measures of reproductive success could include variables for parturition and post-natal care, our dataset included just five reported cases of dystocia and one case of mastitis; data on puppy survival and progress post-partum were not available in all cases. However, post-natal measurements for reproductive success are likely to be much more complex in nature, and will likely require a much larger dataset to inform them.ROH and litter size.We and others have already demonstrated the potential of direct-to-consumer genomics to discover novel genetic variants affecting coloration .Phenotype information was compiled and provided by the MAF; information was gathered via veterinary- and owner-submitted questionnaire annually and at each veterinary visit per MAF guidelines. Participants’ date of birth, physical exam findings, most recent estrous (heat) cycle and duration, date and method of last breeding and litter, litter size , and reproductive complications  were included.ROH was calculated using runs of homozygosity ⋧ 500 kb as described in Sams and Boyko -derived gDNA for each dog was provided by the MAF. gDNA was diluted to roughly 200 ng/µL; 50 µL of each sample as submitted for genotyping using on the Embark 220K SNP array platform as previously described , median withers height (median_height), and age in years at the time of litter recording (age_at_visit_year) as fixed effect variables and unique dam ID (barcode) as a random effect variable , data = all_data_for_kinship)Litter size was calculated as the variable livepup, number of live puppies born, compiled from MAF records. A linear mixed model (coi.with.barcode) was generated with the lmer function , data = all_data_for_kinship, varlist = kinship.matrix.pdv)A second linear mixed-effects model (coi.with.kinship) was performed using lmekin function in coxme Therneau :coi.withUsing a standardized kinship matrix (kinship.matrix.pdv) generated with GEMMA (version 0.97) as a random effective variable Below is the link to the electronic supplementary material.


Solanum melongena) breeding. A set of 10 morphologically highly diverse eggplant parents, including nine inbred S. melongena and one weedy S. insanum accessions, were intercrossed according to a half-diallel mating design without reciprocals to obtain 45 hybrids. Parents and hybrids were evaluated for 14 morphological and agronomic conventional descriptors and 14 fruit morphometric traits using Tomato Analyzer. Genetic distances among parents were estimated with 7,335 polymorphic SNP markers. Wide ranges of variation and significant differences were observed in the set of 55 genotypes for all traits, although the hybrids group had significantly higher vigour and yield than parents. General and specific combining abilities (GCA and SCA) were significant for most (GCA) or all (SCA) traits, although a wide variation was obtained for GCA/SCA ratios. Many relevant traits associated to vigour and yield had low GCA/SCA ratios and narrow-sense heritability (h2) values, while the reverse occurred for most fruit shape descriptors. Broad-sense heritability (H2) values were generally high, irrespective of GCA/SCA ratios. Significant correlations were found between traits related to size of leaf, flower and fruit, as well as among many fruit morphometric traits. Genetic distances (GD) among parents were coherent with their phylogenetic relationships, but few significant and generally low correlations were found between GD and hybrid means, heterosis or SCA. The results provide relevant information for developing appropriate strategies for parent selection and hybrid development in eggplant and suggest that GD among parents have limited value to predict hybrid performance in this crop.Evaluation and prediction of the performance of hybrids is important in eggplant ( Solanum melongena L.) is an important vegetable crop of tropical and subtropical regions of the world, being cultivated in more than 1.79 million ha and having a global production of 51.28 million tons [Eggplant  and broad-sense (H2) heritabilities, as well as correlations among traits [Several studies have used a variable number of parents (four to 10) for half-diallel analysis to evaluate GCA and SCA for yield and several traits of agronomic interest in different eggplant parents and their hybrids from the Mediterranean region , southeag traits , and in g traits  found thg traits ,31,32,40S. insanum L.) [S. insanum and S. melongena have been found to be intermediate between parents in characteristics [In this work we evaluate a large number of traits (28) of interest for eggplant breeding, including conventional descriptors –43 and fanum L.) . Experimeristics ,49, and S. melongena) accessions plus one weedy accession of S. insanum were used as parents for the present study (S. insanum accession (INS2) originating from Sri Lanka  distributed through the entire period of cultivation. Plants were trained with bamboo canes. Weeds were removed manually, and no phytosanitary treatments were performed throughout the cultivation period (May-October 2016), as pest levels were below treatment limits.Seeds of the parents and hybrids were germinated and transplanted into an open field plot situated in the campus of the Universitat Politècnica de València  on May 2016. A randomized block-design with three replications and three plants was established. Plants were grown using a spacing of 1.2 m between rows and 1.0 m within the row. Irrigation was applied using drip irrigation; fertilization was provided through the irrigation system and consisted of 80 g·plant2), Width Mid-height (cm), Maximum Width (cm), Height Mid-width (cm), Maximum Height (cm), Curved Height (cm), Fruit Shape Index External I, Fruit Shape Index External II, Curved Fruit Shape Index, Proximal Fruit Blockiness, Distal Fruit Blockiness, Fruit Shape Triangle, and Fruit Shape Index Internal. A full description of the Tomato Analyzer traits measured can be found in Kaushik et al. [Plants were characterized using 14 morphological and agronomic descriptors based on EGGNET ,43 and  descriptk et al. .t-tests to detect differences among the two groups. The significance of differences among group means was evaluated using at p < 0.05 using the Statgraphics Centurion XVI software . The general combining ability (GCA) of parents and specific combining ability (SCA) of individual hybrids, along with the variance components, and narrow (h2) and broad (H2) sense heritabilities were estimated based on the Griffing’s [2GCA / ((2 × s2GCA) + s2SGA) [r) were calculated using the Statgraphics Centurion XVI software, and significance of correlations was evaluated using the Bonferroni test [Het; %) was calculated using formula Het = 100 × ((F1 − MP)/MP), where F1 = hybrid mean, and MP = mean of the parents.For each trait measured, the mean and range were calculated for the parental (n = 10) and hybrid (n = 45) groups. Mean values of parents and hybrids were compared with iffing’s  Method 2iffing’s . The rel+ s2SGA) , Relativoni test . The F1 https://cran.r-project.org/web/packages/poppr/index.html) using a UPGMA hierarchical clustering method and Hamming distance (bitwise distance). The relationship among GD of parents of individual hybrids was used to estimate pairwise Pearson linear correlations between GD and hybrid trait values, heterosis, and SCA.Polymorphism information for the parental accessions used in this study was retrieved from genotyping data obtained by Acquadro et al. , where tp < 0.05) were found only for Plant Height, Stem Diameter, and Yield, with higher values in hybrids  block effects except for Leaf Pedicel Length, Leaf Blade Length, Leaf Blade Width, Corolla Diameter, and Yield (p < 0.001) were found among genotypes for all traits. Similarly, highly significant (p < 0.001) effects were found for general combining ability (GCA) and specific combining ability (SCA) for all the traits evaluated, although higher values for the mean squares were observed for GCA than for SCA (h2) values ranged between 0.11 for Proximal Fruit Blockiness and 0.83 for three fruit shape descriptors . Traits related to plant vigour , Fruit Calyx Prickles, and Yield had low h2 values, while most fruit size and shape traits, either using conventional or Tomato Analyzer descriptors, had h2 values above 0.5 (H2) had values above 0.5 for all traits, except for Proximal Fruit Blockiness (0.42) and Fruit Shape Triangle (0.35). Most traits related to vigour, yield and fruit size and shape had values above 0.85 for H2  detected no significant   among traits were significant according to the Bonferroni test ( 0.4928) . Only 16t Weight . Many sit Weight .S. insanum INS2 and all the cultivated accessions, and also between the cultivated A0416 and MEL1 accessions  based on 7,335 coding SNPs ranged from GD = 0.0094 between DH621 and IVIA-371 to a maximum value of GD = 0.0389 between MEL1 and INS2 . The higcessions . The lowcessions . These rtogether .p < 0.05) with hybrid’s trait values for 14 traits out of a total of 28  generally were non-significant, and only significant positive correlations with GD were found for Leaf Blade Width and Fruit Pedicel Diameter, while negative correlations were found for Proximal Fruit Blockiness and Fruit Shape Triangle. Regarding the correlations between GD and SCA the only significant correlation (negative) was observed for Fruit Weight  between parents genetic distance and yield or fruit weight of hybrids, although their results were based on just 10 hybrids. Our work tried to provide an integrated perspective for obtaining relevant information for the eggplant breeding. We used a half-diallel analysis, which gives information on the magnitude of general and specific combining abilities and trait heritabilities [In several crops, genetic distances among parents have been proved useful to predict the performance of hybrids, although the results depend on the crop, the diversity present in the parents, and the markers used ,40,58,59bilities –27, usinWe found that parents and hybrids displayed wide and overlapping ranges of variation, as compared with other works evaluating the diversity for morphological and agronomic traits of eggplant ,45,60,61The high diversity in the parental and hybrid materials for the traits evaluated was matched by significant GCA and SCA values for all traits, revealing the presence of significant additive and non-additive effects in all traits ,27. ThisH2) values were generally high, indicating that most of the variation observed is genetically determined and that selection among varieties or hybrids will be efficient [H2 values. This is in agreement with Hurtado et al. [H2 values for eggplant fruit shape traits. However, when considering narrow-sense heritability (h2), which only takes into account additive variance, values were lower, especially for traits with lowest GCA/SCA ratios. In this way, h2 values for important agronomic traits, like those related to vigour, prickliness or yield were relatively low, difficulting genetic advances in breeding programmes [h2 values) in hybrids of Spanish local varieties. This may be an indication that heritability values in eggplant largely depend on the population evaluated. On the contrary, fruit shape traits generally had high values for h2, suggesting a high selection efficiency in breeding programmes.Broad-sense heritability  are the ones with smallest leaves and fruits. Also, a negative correlation of genetic distance with yield when hybrids with all parental accessions are included is probably a consequence of the low yield of INS2. In fact, when hybrids with this parent are excluded this correlation is not significant. Traits for which genetic distances may have some predicting value are stem diameter, where a positive correlation is obtained with heterosis when all accessions are included, and with SCA when only S. melongena accessions are included. Overall, our results indicate that genetic distances based on coding SNP markers in the materials studied are of little predictive value. This is in contrast with a previous study of Rodríguez-Burruezo et al. [Correlations between genetic distances and hybrid trait values or parameters like heterosis or SCA can be very useful for selecting hybrids . In our o et al. , who fouo et al.  might aco et al. ,31,32,40o et al. ,81.Overall our study provides relevant information for eggplant breeding, in particular for the development of improved F1 hybrids. The highly significant differences observed for GCA and SCA for all traits indicates that there is large genetic and gene action diversity in the set of parents and hybrids that can be exploited for breeding. The differences in GCA/SCA ratios and heritabilities, as well as correlations among traits also will condition the breeding strategy to be followed in order to maximize genetic gains in eggplant . With ouS1 Table(XLSX)Click here for additional data file.S2 Tablep < 0.05; r ≥ 0.4928) are included.Only those traits for which at least one correlation was significant  according to the Bonferroni test ((XLSX)Click here for additional data file.S3 TableGD values are based on identity by state (IBS) and calculated as GD = 1-IBS.(XLSX)Click here for additional data file.


The six traits were studied under two environments [Campos dos Goytacazes-RJ (ENV1) and Itaocara-RJ (ENV2)] in an incomplete block design. Based on the phenotypic data of the S1 progenies and on the genetic characteristics of the parents, the MLMA was performed. Thereafter, genes annotated in the MaizeGDB platform were screened for potential linkage disequilibrium with the SNPs associated to the six evaluated traits. Overall, seven and eight genes were identified as associated with the traits in ENV1 and ENV2, respectively, and proteins encoded by these genes were evaluated for their function. The results obtained here contribute to increase knowledge on the genetic architecture of the six evaluated traits and might be used for marker-assisted selection in breeding programs.The identification of the genes responsible for complex traits is highly promising to accelerate crop breeding, but such information is still limited for popcorn. Thus, in the present study, a mixed linear model-based association analysis (MLMA) was applied for six important popcorn traits: plant and ear height, 100-grain weight, popping expansion, grain yield and expanded popcorn volume per hectare. To this end, 196 plants of the open-pollinated popcorn population UENF-14 were sampled, selfed (S Zea mays L.), the economically most important traits are usually quantitatively inherited, and their genetic basis is attributed to polygenes, as well as to the interaction effects between genes and/or between genes and environment [Z. mays has been used as a model in classical genetics and cytogenetics, it has significantly contributed to the understanding of fundamental processes such as reproduction and photosynthesis, among others [In maize , but also the identification of candidate genes based on statistical significant differences between markers and phenotypes . Single Fusarium graminearum Schwabe [Association mapping was successfully applied in maize to identify genomic regions related to plant height , root de Schwabe , and oil Schwabe . Mapping Schwabe . HoweverSince its inclusion in the germplasm bank of the State University of Northern Rio de Janeiro (UENF), the UENF-14 popcorn population has been subject to a series of recurrent intrapopulational selection cycles targeting the improvement of economically important traits and the maintenance of the population variability. Until the eighth cycle, only phenotypic evaluations were performed –20. The Thus, in the present study, a mixed linear model-based association analysis (MLMA) approach was applied to six agronomic traits of popcorn under different environments to identify potential genes in LD with associated SNPs that might be responsible for these agronomic traits.The selected study population belongs to the open-pollinated UENF-14 variety . The ori1 progenies derived from the UENF-14 population (see below) were subject to two experiments in August 2016: one in Campos dos Goytacazes (ENV1) (northern region of the State of Rio de Janeiro—21°43'15.5"S 41°20'38.3"W), and the other in Itaocara (ENV2) (northwestern region of the State of Rio de Janeiro—21°38'45.2"S 42°03'16.0"W). Each experiment comprised 98 S1 progenies, whose parents produced enough seeds to establish these experiments. Both experiments were arranged in incomplete blocks with three orthogonal replications, in which the progenies and eight controls were set. At both environments, the main traits of agronomic interest for this crop were measured: plant height , ear height , 100-grain weight , grain yield , popping expansion of the grain , and expanded popcorn volume per hectare .The S1 progenies referred in the previous sub-section were derived from 98 of these plants.Genetic polymorphisms were characterized throughout the genome of 200 plants the open-pollinated UENF-14 in the C-9 cycle. Genomic DNA was extracted from the young leaves of these plants, using the standard CTAB method with modifications  and the Capture Seq method  was appl2 statistic calculated for all marker pairs of the same chromosome using PLINK [Based on the 10,507 SNPs and 196 plants, a Genetic Relationship Matrix (GRM) was calculated using the rrBLUP package  and the ng PLINK . The LD ng PLINK , adjusteng PLINK .2 >0.1, removing one marker and maintaining the other. Both filters were applied using PLINK, resulting in 739 SNPs. The application of the Bayesian model of Pritchard et al. [Based on the SNP data for the 196 plants, the absence of population structure was confirmed in STRUCTURE . The algd et al.  was evald et al.  using thd et al. .y is the vector of the phenotypes of a given trait; β1 the vector of fixed effects including intercept, replication, and co-variables such as the number of plants per plot, counted immediately after thinning, and grain moisture for the traits 100GW, GY, PE, and PV; βii is the effect of the ith SNP candidate (regression coefficient); b is the vector of block effects within replications; u is the vector of polygenic effects; e is the vector of residual effects of the model; X is the incidence matrix of systematic fixed effects; SNPi is the vector of the number of copies of a given allele of the i-th SNP candidate randomly taken as a reference, SNPi = {0, 1 or 2}; and Z1-2 are incidence matrices of random effects.The MLMA  was perfG is GRM calculated using package rrBLUP [Ib and In are the identity matrices in the order equal to the number of incomplete blocks and the number of observations, respectively; b, u, and e, respectively. These components were estimated by the Restricted Maximum Likelihood (REML) method, using the Average Information (AI) algorithm. The MLMA was adjusted using the ASReml-R package [It was also assumed that:e rrBLUP , based oe rrBLUP ; Ib and  package  as avail package .p is the progeny effect, taken as fixed for the estimation of the adjusted means (LSMeans); the other terms were as in the previous model. This model was adjusted with package lme4 [To compute the corrected means for the systematic effects, the following model was used:age lme4  and the age lme4 , both asage lme4 .yadj corresponds to the adjusted means for a given trait; μ is the intercept of the model; g is the genetic component explained by the SNPs; and e is the random error. This model assumed: cov = 0, where the ratio of the variance explained by the markers  were defined as candidate genes for the agronomic traits evaluated. The public dataset of the maize genome and the B73 version 3 reference genome .The frequency distribution of the SNPs throughout the genome was calculated to quantify marker coverage . This ev2 among all SNPs belonging to the same chromosome was estimated. The highest LD was observed on chromosome 8, and half of the decay occurred at ~ 151 Kb, while the lowest LD was observed on chromosome 4, where half of the decay was observed at ~ 76 Kb. For all chromosomes, half of the LD was observed at ~ 110 ± 6.82 Kb, on average.To measure the magnitude of the LD and define the windows at which polymorphic locations are expected to be in high LD, the rAs expected, the absence of population structure was confirmed by the criterion of Evanno et al.   than in ENV2.The adjusted phenotypic means, as well as the heritability estimates for the SNPs of the six traits evaluated in ENV1 and ENV2, are listed in The proportion of markers explaining phenotypic traits (heritability) was not identical between environments. In ENV1, PH was the trait with the highest value (≅ 53%), while in ENV2, the proportion was highest for PE, with approximately 74% of the SNPs explaining the trait, suggesting that both traits would respond well to SNP marker-assisted selection.The correlations between the traits in ENV1 were generally very low, ranging from -0.08 (PH vs. PE) to 0.9 (PV vs. GY). In this environment, PV and GY presented the highest correlation (0.9), but a high value was also found for the correlation between PH and EH - 0.82; . Both ofIn ENV2, trait correlations ranged from -0.08 (PE vs. 100GW) to 0.93 (GY vs. PV). Although the highest values were observed GY and PV, the correlation between EH and PH was also high and significant  to 0.83 (EH), and all associations were significant (P <0.01).To find the genomic regions associated with traits of interest in popcorn, MLMA was performed. The Manhattan plot obtained  revealedBy comparing the physical locations of significant SNPs in the B73 reference genome, the candidate genes were identified. In ENV1, the genes were distributed on chromosomes 1, 2, 4, 5, and 10. For EH, the gene GRMZM2G002959  coding fArabidopsis thaliana L., and responsible for limiting the extension of the meristem and/or the development of the cotyledon domains [In ENV2, the genes were distributed on chromosomes 3, 5, 6, 7, 9, and 10 . Gene GR domains , as well domains . The sec domains . No geneThe LD decay varied between the chromosomes 76–151 Kb), which is in agreement with the mean LD decay range observed for 64 maize lines (80–100 kb)  Kb, whic was sensSeveral genes might be present in the range of an associated region , and these can be identified by MLMA. This analysis identified seven and eight genes related to the studied traits in ENV1 and ENV2, respectively. Gene GRMZM2G002959, identified for EH in both environments encodes glutaryl-CoA dehydrogenase but its function in plant species has not been identified.Gene GRMZM2G089995, identified for the 100GW in ENV1, encodes the transcription factor 209, a member of the AP2/EREBP protein family, which is directly involved in the response to biotic and abiotic stresses –54. For In ENV2, the candidate gene GRMZM2G087032, identified for 100GW, corresponds to the putative bifunctional protein C3H transcription factor 313, which may be important during the initial stage of maize grain filling . HoweverIn ENV1, the candidate gene GRMZM2G069618, related to GY, was found on chromosome 1. It encodes a protein containing tetratricopeptide repeats (TPR) that is involved in plant stress and hormone signaling , and thaA. thaliana plants [In ENV1, the candidate gene GRMZM2G461936, which encodes AGO108, was identified as being related to PE. Argonaute proteins are fundamental in the regulation of gene expression and are essential for several developmental processes [a plants . Gene GRa plants . This suIn ENV1, the gene GRMZM2G118950 identified for PH encodes AMT3. Ammonium transporters are responsible for ammonium uptake from the soil solution ,64. Two A. thaliana, RbohC-deficient mutants had short root hair on stunted roots, suggesting that RbohC regulates plant cell expansion [In ENV2, gene GRMZM2G043435, encoding the respiratory burst oxidase-like protein C (RbohC), was associated with PH. In xpansion  and mighRSR1), a transcription factor of the APT2 family, negatively regulates the expression of type I starch synthesis genes as RSR1 deficiency resulted in the increased expression of starch synthesis genes in rice seeds [One gene (GRMZM2G086573) was identified as related to PV in ENV1. This gene encodes the AP2/EREBP transcription factor 24. The distinctive feature of AP2/EREBP proteins is that they contains one or two AP2 domains, which are binding elements to the response element ethylene. The rice starch regulator gene , no SNP was common to both traits. However, these traits are directly related because PV is calculated based on GY and PE. Thus, PV should be used as a "super trait" in recurrent popcorn selection programs, as its inclusion results in optimized simultaneous gains for GY and PE ,70. AlthThe increased number of functional assays and the higher number of evaluated traits will produce more available data for the different populations, which will be extremely useful to address important biological issues. The MLMA strategy used in the present study proved useful and robust, complementary to biparental cross-mapping, and can map multiple genetic traits simultaneously . The maiThe initial stimulus for this study was the fact that, to date, few studies have addressed the genetic architecture and mechanisms that control the natural variation in the development and productivity of popcorn genotypes. The results obtained here showed that some morphological traits are moderately inheritable, such as PH and EH, with wide variation in a population containing different lines genotyped with 10,507 SNPs. The candidate genes associated to these loci are an inestimable resource for gene function analyses and for dissecting the molecular network regulating the development of popcorn traits, apart from identifying genome polymorphisms, which are essential in marker-assisted selection of agronomic traits in breeding programs. Twelve proteins were identified as being SNP-associated, providing new information useful for accelerating popcorn breeding programs, as this crop accounts for an annual turnover of about one billion dollars in the United States.S1 Table(DOCX)Click here for additional data file.S2 Table(DOCX)Click here for additional data file.S3 Table(XLS)Click here for additional data file.S4 Table(XLS)Click here for additional data file.


The seven GWAS models detected a total of 228 quantitative trait nucleotides (QTNs) for 16 traits. Most loci, defined by an interval of 100 kb up and downstream of the QTNs, harbored genes known to play role(s) in root and shoot development, suggesting them as candidates. Examples of candidate genes linked to root network QTNs included genes encoding GRAS transcription factors, mitogen-activated protein kinases, and auxin related lateral organ boundary proteins while QTN loci for shoot dry weight harbored genes involved in photomorphogenesis and plant immunity. These results provide insights into the genetic bases of early shoot and root development traits in flax that could be capitalized upon to improve its root architecture, particularly in view of better withstanding water limiting conditions during the cropping season.Roots are fundamental organs for water and nutrient uptake as well as for signal transduction in response to biotic and abiotic stresses. Flax has a shallow tap root system that relies mostly on top soil nutrient and moisture resources. The crop can easily be outcompeted by weeds or other crops in intercropping systems, especially in moisture deficit conditions. However, there is a wide range of variation among genotypes in terms of performance under scarce resources such as moisture limitation. Here we phenotyped 15 root, two shoot traits and shoot to root dry weight ratio on 115 flax accessions grown in a hydroponic pouch system and performed a genome-wide association study (GWAS) based on seven different models to identify quantitative trait loci underlying these traits. Significant variation among genotypes was observed for the two shoot and 12 of the 14 root traits. Shoot dry weight was correlated with root network volume, length, surface area, and root dry weight ( Roots are vital organs in terrestrial higher plants for acquisition of essential nutrients and water. Because roots function in a bio-physico-chemically dynamic rhizosphere, they play an important role in controlling and regulating the impacts of various edaphic factors through internal physiological adjustments  and signhttp://www.fao.org/faostat/en/#data). High and stable yielding cultivars are urgently needed to rekindle growers' interest and meet market potential.Flax, one of the founder crops of agriculture , has beeFlax, like other oilseed crops, is a tap-rooted plant. Compared to canola, sunflower, and safflower, flax has a shallower root system and, as such, it mainly relies on moisture and nutrient resources available in the soil's top layers  mainly wPrompted by the current large sets of genetic data from genome-wide association studies (GWAS) and advancements in imaging and data processing, high throughput digital root phenotyping techniques have become attractive. Here we applied this technique to study the early root and shoot development of a flax mini-core collection (n = 115) that comprises representative genotypes from all major flax growing regions of the world. A GWAS was performed using a set of single nucleotide polymorphism (SNP) markers obtained from shotgun short-read re-sequencing data of the germplasm collection. The objectives of this research were 1) to assess the extent of the variation in root traits among genotypes, 2) to identify quantitative trait nucleotides (QTNs) associated with the genetic architecture of selected root and shoot traits, and, 3) to identify candidate genes for the traits harbored at the QTN loci.A flax mini-core collection (n = 115) that comprised >95% of the genetic diversity  of the fEarly development root and shoot phenotyping was performed in a hydroponic pouch system modified from Sterile 24x30 cm blue germination blotting papers  were inserted in Ziploc bags. The assemblies were attached to rust-proofed rods using fold back clips that were suspended in the bins with the open side of the Ziploc bags down to allow blotting up of the nutrient solution. Assemblies were labeled and randomized for each of the three consecutive biological replicates performed  was made at 25% strength with deionized water and adjusted to pH∼6.3. The pouches were randomly distributed in the two bins. After complete moistening of the blotting paper, three seeds were inserted directly in the paper using tweezers, ∼2 cm below the top edge and slightly spaced out near the center.The siphons that transferred the nutrient solution were set ∼3 cm from the bottom of the bins, i.e., touching the bottom of the blotting papers, in order to keep them moist at all time. Five days after seeding, germinated seeds were thinned out to one seedling per pouch. The retained seedlings were allowed to grow for 18 days.th day, plants were carefully removed from their assemblies for imaging and measurements. Shoots, cut at the shoot and root junction, were measured with a ruler to estimate shoot length prior to being transferred to labeled envelopes and dried for 3 days at 60°C to measure dry weight. The root system of each plant was imaged using a Canon camera (EOS Rebel T5i) mounted on a custom stand to ensure a consistent 40 cm distance between lens and roots. After imaging, the roots were gently pealed from the paper and processed as the shoots to measure their dry weight. Three biological replicates were thus performed consecutively.On the 19http://www.rootnet.biology.gatech.edu/giaroots/download/recent/gia_roots_manual.pdf). Root images were processed using the General Image Analysis of Roots (GIA Roots) software . Each imShoot dry weight, shoot length, root dry weight, shoot to root dry weight ratio, and the 14 root traits measured by the GIA software were analyzed. Basic statistics of the 16 traits were computed for 111 genotypes Table 1.SNP data of the entire flax core collection (n = 407) was previously generated after resequencing each genotypes using Illumina HiSeq 2000 platform in 100 bp paired-end mode to at an average coverage of 17X. The alignment, SNP call, and quality control such as removal of SNPs in long terminal repeat region were performed as previously described . The SNPper se were removed. The two datasets were analyzed separately and results were compared.For this study, only SNPs with no missing data were used. Because the minor allele frequency (MAF) of an SNP could differ between the core and the mini-core collections, we used two datasets for GWAS. The first dataset (7K) included all SNPs with no missing data regardless of their MAF in the mini-core collection because these had already previously been included based on MAF > 5% criteria in the core collection. The second dataset was smaller (3K) because SNPs with MAF < 5% in the mini-core collection To estimate the possible number of ancestral populations (K), a cross-validation technique  was appln = the number of total SNPs) was also used as a comparative methods. Quantitative trait loci (QTL) regions spanning 100 Kb up and downstream of all associated QTNs were examined for the predicted coding genes they harbored using the flax reference genome (Arabidopsis orthologues (www.arabidopsis.org). Strongly associated SNPs and their putative underlying genes were illustrated on the flax pseudomolecules (R2 > 5%), was illustrated using box plot based on mean phenotypic value of genotypes with each allele.The use of multi-locus methods that capture small effect loci in complex polygenic traits such as in plant roots and shoots has recently become a feasible approach. To benefit the algorithmic merits of different models and support results of one by an other, it is also advantageous to apply multiple methods . To assee genome . Predictolecules  using Maolecules . The favhttps://string-db.org) (Linum usitatissimum) with a minimum regulatory confidence of 0.95.For genes previously identified to encode for interacting proteins, protein interaction networks were constructed using the tool STRING V11 (-db.org) . The intP < 0.05) among genotypes was detected for all except for root dry weight, network depth, and network distribution  at Chr9:19061342 using the multilocus FASTmrMLM  and the LFMM methods identified a grand total of 228 QTNs associated with at least one of the 16 traits, of which 33 QTNs were detected in both datasets  and sucrose synthase-6 (SUS6), respectively.Most loci within 100 kb up and downstream of the detected QTNs harbored genes that had previously been reported to play role(s) in root and/or shoot development in plants . This lovia COP1 (Most QTNs associated with SDWt were at loci containing genes predicted to encode pentatricopeptide repeat (PPR), photomorphogenesis, ubiquitin-related and plant defense proteins such as multi and toxic compound extrusion (MATE) , was dupRoot trait measurements provide essential information to facilitate varietal improvement in breeding programs to select superior genotypes especially in nutrient and moisture deficit areas . Root trThe significant variations observed among genotypes for most of the early root system and shoot development traits evaluated points to the genetic diversity of flax for such traits and the potential for genetic improvement. The highest and lowest morphometric values reflect the level of diversity within the gene pool, promising useful materials for improvement through breeding. The wide range of variation for most agronomic traits among flax genotypes  is well Correlations between root and shoot traits reflect the notion of balance between roots and shoots referring to plants partitioning their resource allocations between the two plant parts . A high Arabidopsis thaliana ; the latter also possesses E3 ubiquitin ligase activity  are not available.DS and SC conceived the project. DS, SC, and SR participated in designing and implementing the phenotyping experiment. SC and FY made the genetic data available. DS performed the phenotypic and GWAS data analysis. DS prepared the manuscript. SC, FY, and SR worked on revision of the manuscript.This Agri-Innovation Project (J-000279) was funded by the Flax Council of Canada and Agriculture and Agri-Food Canada.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.


Brief Infant-Toddler Social and Emotional Assessment , and did not meaningfully predict internalizing or externalizing scores on the Achenbach Scales of Empirically Based Assessment at 36 months. In this general population infant twin study, variation in social communication was independent from variation in other domains of general psychopathology, and exhibited a distinct genetic structure. The commonly-observed comorbidity of specific psychiatric syndromes with autism may arise from subsequent interactions between autistic liability and independent susceptibilities to other psychopathologic traits, suggesting opportunities for preventive amelioration of outcomes of these interactions over the course of development.Previous research has suggested that behavioral comorbidity is the rule rather than the exception in autism. The present study aimed to trace the respective origins of autistic and general psychopathologic traits—and their association—to infancy. Measurements of autistic traits and early liability for general psychopathology were assessed in 314 twins at 18 months, ascertained from the general population using birth records. 222 twins were re-evaluated at 36 months. Standardized ratings of variation in social communication at 18 months were highly heritable and strongly predicted autistic trait scores at 36 months. These early indices of autistic liability were independent from contemporaneous ratings of behavior problems on the The online version of this article (10.1007/s10802-018-0410-1) contains supplementary material, which is available to authorized users. Ever since it was established that the characterizing features of the autistic syndrome were continuously distributed in the human population  and shared genetic influences (genetic correlation = 0.27). The utility of the CBCL for ascertaining the characterizing traits and features of autism is somewhat limited, however, because it only captures a fraction of the variance captured by more specific assessment of autistic traits , and 3) utilizing a comprehensive analytic strategy to evaluate previously uncharacterized associations between QATs and traits of general psychopathology.To better characterize the role of autistic traits in human behavioral development, the present study was designed to prospectively examine relationships among early precursors of social behavioral outcome 1) beginning in infancy (18 months), 2) using specific developmental measures of quantitative autistic traits , a longitudinal study characterizing the development of QATs from infancy through the toddler years. Autistic trait data related to this study have been published previously  and the State of Missouri Department of Health and Senior Services Institutional Review Board  approved all study procedures. Informed consent was obtained from parents of twins who participated in the study.Measures were completed by the consenting individual. Descriptive statistics for all measures are reported in Table SCI) and restricted interests and repetitive behavior  and consists of SCI and RRB subscales. Higher RSB, SCI, and RRB scores indicate greater impairment. Importantly, RSB and SCI scores are continuously distributed, capturing trait variation in the general population  a Competence score measuring key features of social adaptation, including social-emotional skills and aspects of social relatedness. Specifically, the BITSEA Competence score is comprised of 11 items assessing rule-abiding behavior, play behavior, attention, empathy, motivation to master new skills, and quality of peer interactions. Qualitatively, these items exhibit substantial overlap with the vrRSB SCI scale; however, SCI includes more items assessing communicative competency and social engagement.The Brief Infant-Toddler Social and Emotional Assessment . Higher Total scores indicate greater impairment. In the present study, SRS-2 Total served as the primary outcome measure in prospective longitudinal predictions of QATs.The Social Responsiveness Scale, second edition  and five DSM-oriented scores . Emotionally Reactive, Anxious/Depressed, Somatic Complaints, and Withdrawn syndrome scores can be combined to create an Internalizing composite, and Aggressive Behavior and Attention Problems syndrome scores can be combined to create an Externalizing composite. In the present study, Internalizing and Externalizing composites served as the primary outcome measures in prospective longitudinal predictions of psychopathology. Although there is evidence to support their use as autism screeners, Withdrawn and Autism Spectrum Problems scores were not included in the present analyses due to low discriminative validity  if zygosity could not be determined by questionnaire.The Goldsmith Child Zygosity Questionnaire  and vrRSB  subscales. Non-independence of twin observations was mitigated by randomly selecting one twin per family for inclusion in the primary set of analyses, with co-twins used for the purpose of quasi-replication. Next, we implemented exploratory factor analysis (EFA) as an initial step in empirically-deriving independent and overlapping domains of behavioral variation from assessment data collected at baseline. Subscales rather than individual items were entered into factor analyses to increase the ratio of number of observations to number of items. These subscales included Externalizing, Internalizing, Dysregulation, Competence, SCI , and RRB (restricted interests and repetitive behavior) indices. Composite indices  were omitted to avoid item redundancy. As above, dependencies in twin data were accounted for by randomly selecting one twin per family to be included in analyses. Observations were evaluated via principal component and minimum residual factoring methods, followed by oblimin rotation. Tucker Lewis Index (TLI) values 0.90–0.92 were judged adequate fit, 0.92–0.95 good fit, and > 0.95 excellent fit : Heritability = 2(rMZ– rDZ). Within domains, heritability was estimated separately for male/male and female/female twin pairs, as well as for the entire sample. Although Falconer’s formula is commonly used to estimate heritability  was used to 1) investigate domains of overlap during early development and 2) predict QATs, internalizing, and externalizing behaviors at 36 months. To account for twin dependency, family was modeled as a random effect.Competence and Behavior Problem indices, wherein the Behavior Problem index consisted of Externalizing, Internalizing, and Dysregulation subscales) and autistic traits . As expected, Competence was not significantly correlated with Behavior Problem subscales, and both the total vrRSB score and SCI exhibited only weak correlations with Behavior Problem subscales  and weakly onto problem behaviors , factor loadings for RRB were equivocal . In this sense, RRB diverged from SCI despite strong correlations between the two traits within individuals. In conjunction with the limited number of (n = 11) and variation in (SD = 2.8) RRB items in typically-developing children, this divergence informed our decision to use SCI as the primary index of autistic traits. The heritability of RRB is provided in Online Resource To evaluate whether correlational associations reflected an underlying factor structure, vrRSB and BITSEA subscales were submitted to exploratory factor analyses (EFA). Results of EFA identified a two-factor solution wherein Competence, Behavior Problem, and SCI indices were examined at baseline. Falconer’s estimates of heritability are reported in Table SCI and Competence indices compared to the Behavior Problem index. The magnitude of these estimates warrants mention. Across all participants, the estimation of heritability for SCI was 0.90 and the estimation of heritability for Competence was 0.89, indicating that a large majority of variance in these traits can be attributed to genetic factors. In contrast, the estimation of heritability for the Behavior Problem index was 0.30, indicating that environmental factors account for a greater proportion of variance in this trait than genetic factors.Having characterized factor structure, we aimed to determine whether the distinction between social adaptation and behavior problems may have emerged, in part, due to differences in patterns of genetic transmission. Thus, the heritabilities of Behavior Problem indices and SCI and Competence, we next investigated the extent to which quantitative risk factors for autism and general psychopathology are shared in infancy. To this end, we evaluated construct overlap among SCI, Competence, and Behavior Problems at baseline using hierarchical linear modeling  and Competence . Results indicated that Behavior Problems accounted for 4% of unique variance in SCI, whereas Competence accounted for 48% of unique variance in SCI.Given the above results, suggesting a dissociation between Competence and Behavior Problem indices as our dependent variables (Online Resource 10). Importantly, differences in unique R2 values were smaller when comparing incremental analyses within a given domain  relative to across domains . This suggested that our estimates of overlap were reliable. To visualize subscale relationships, we created a Venn diagram depicting average estimates of overlap among SCI, Competence, and Behavior Problem indices  would predict the severity of autistic traits during the toddler years  above and beyond infant measures of social communicative autistic traits . Conversely, we were interested whether infant measures of social communicative autistic traits would predict the severity of internalizing and/or externalizing behavior during the toddler years  above and beyond infant measures of general psychopathology. Finally, we were interested whether Competence would significantly improve predictive capacities above and beyond within-trait predictors. To this end, we ran incremental regressions using HLM. Within-trait predictors were entered into incremental models at Step 1, Competence was entered at Step 2, and cross-trait predictors were entered at Step 3. Independent variables were assessed at 18 months and dependent variables were assessed at 36 months.Despite relative non-overlap between SCI accounting for 26% of the variance in QATs and the Behavior Problem index accounting for 15% and 9% of the variances in Internalizing and Externalizing behavior, respectively. In Step 2, Competence accounted for a small albeit significant proportion of additional variance in QATs (unique R2 = 3%); Competence did not account for significant additional variance in either Internalizing or Externalizing behavior. Finally, in Step 3 (cross-trait prediction), the Behavior Problem index accounted for a small albeit significant proportion of additional variance in QATs (unique R2 = 2%); SCI did not account for significant additional variance in either Internalizing or Externalizing behavior. Given Behavior Problem and RRB indices clustered together in factor analyses, it remained unclear to what extent the Behavior Problem index improved prediction of QATs above and beyond comprehensive measures of autistic traits indexing both social communication and restricted, repetitive behaviors. To this end, a fourth incremental regression was run substituting RSB, a composite scale derived from SCI and RRB, for SCI. Under these conditions, Behavior Problems ceased to explain significant additional variance in QATs .Results are detailed in Online Resource Competence is shared with that of SCI  and externalizing  traits were roughly two times larger at 36 compared to 18 months, suggesting a strengthening relationship among these constructs over the course of early development.Considering the above factor analyses, heritability analyses, and patterns of cross-trait overlap, these findings suggest that half of the causal influence on SCI Fig. , and thaths Fig. . MeanwhiCompetence, a well-known index of social adaptation measuring early developmental changes in social-emotional skills and social relatedness , exhibited substantial factorial and genetic overlap with variation in SCI , which represents a core DSM5 criterion domain for autism. SCI accounted for almost half of variance in Competence at 18 months, and SCI and Competence demonstrated strong heritability. Importantly, we can be confident that rater bias was minimal given observations of high heritability, which are predicated on a single parent’s ability to provide discrepant data regarding non-identical siblings  during the first two years of life and 2) in a prospective, genetically-informative sample, epidemiologically ascertained from the general population through birth records. Doing so allowed us to investigate the extent to which autistic and general psychopathologic traits—for which extreme variation constitutes clinical abnormality—overlap early in development. We observed a general pattern of independence of these constructs, which manifested highly disparate genetic and environmental structures. Furthermore, we observed that Behavior Problem indices) were predominantly influenced by environmental rather than genetic factors in infancy and were only weakly correlated with SCI. The superordinate construct comprised of SCI and Competence explained roughly one-third of the variation in quantitative autistic traits at age 3 but did not improve explanatory power for internalizing or externalizing traits above and beyond Behavior Problem indices, reflecting its role as a major parameter of adaptation that is distinct and dissociable from general psychopathology. Interestingly, RRB (restricted interests and repetitive behavior) was more strongly correlated with Behavior Problem than Competence and SCI indices and segregated with general psychopathologic traits in exploratory factor analyses. Although such results are consistent with a theory of fractionation of autistic symptomatology  associated with the measurement of RRBs in the present normative sample. Further work with more comprehensive, RRB-specific measures—which may prove most informative in a sample enriched for children with elevations in autistic trait burden—is warranted to determine whether and how such behaviors influence the developmental course of trait overlap between risk factors for autism and general psychopathology.In contrast, early developmental precursors of internalizing and externalizing behaviors  psychopathologic symptoms can affect the severity of autistic syndromes (and vice versa) over the course of development—this would reflect the same phenomenon of phenotypic pleiotropy that was recently observed in a sibling study of autism recurrence  with longitudinal, interactive effects , studies in infant samples are well-positioned to disentangle these competing hypotheses. Integrating the present findings with extant literature, a pattern emerges wherein shared genetic variance is commonly observed during childhood but not infancy, suggesting a mediation model wherein genetic influences on autistic traits may exacerbate symptoms of general psychopathology over the course of development. Consistent with this notion, abnormalities in Competence and SCI subscales. Such analyses could provide more fine-grained information about the specific elements of SCI and Competence that contribute most strongly to their correlation, thereby designating effective targets for behavioral intervention. It bears mention, however, that there is heterogeneity in autistic syndromes. While this work supports measurable trait-level stability, heterogeneity implies the potential for different developmental mechanisms underlying autistic traits. For example, there are some autistic syndromes whose emergence is preceded by relatively normative development through the age of two years followed by severe regression. In such circumstances, the observed relationships may not apply. Finally, it remains unknown at what age interactive effects between social communicative autistic traits and traits of general psychopathology emerge. Future studies examining the relationship between these constructs over the course of early development, spanning infancy to early school age, are needed to fully characterize the time course and behavioral corollaries of trait overlap. This knowledge, in turn, may pinpoint critical periods of intervention during which negative interactions between autistic traits and traits of general psychopathology can be mitigated.Although the size of our twin sample was substantial and more than adequate to detect the associations described in this report, the derivation of more precise estimates of heritability using structural equation modeling was not possible at this sample size. Future studies of larger samples would be expected to replicate the heritability estimates obtained by Falconer’s formula. Sample size limitations also precluded item-level factor analyses on contribute to the causation of autism itself , particularly in cases in which symptomatic features of joint (comorbid) liability remain appreciable over the course of later development. At a minimum, results of the present study indicate that the clinical comorbidity of specific psychiatric syndromes with autism may arise from interactions between autistic liability and independent susceptibilities to other psychopathologic traits. This suggests opportunities for preventive amelioration of outcomes of these interactions in infancy, perhaps via interventions to target developmentally-independent, behaviorally-modifiable traits  were independent and exhibited separable mechanisms of causation . Given that autistic trait variation in the general population is related to the genetic causes of autism itself ESM 2(PDF 70.3 kb)ESM 3(PDF 96.6 kb)ESM 4(PDF 56.9 kb)ESM 5(PDF 60.7 kb)ESM 6(PDF 71.2 kb)ESM 7(PDF 67.1 kb)ESM 8(PDF 74.6 kb)ESM 9(PDF 66.4 kb)ESM 10(PDF 84.2 kb)ESM 11(PDF 96.9 kb)