A rapid method for computing the inverse of the gametic covariance matrix between relatives for a marked Quantitative Trait Locus

The inverse of the gametic covariance matrix between relatives, G^-1, for a marked quantitative trait locus (QTL) is required in best linear unbiased prediction (BLUP) of breeding values if marker data are available on a QTL. A rapid method for computing the inverse of a gametic relationship matrix for a marked QTL without building G itself is presented. The algorithm is particularly useful due to the approach taken in computing inbreeding coefficients by having to compute only few elements of G. Numerical techniques for determining, storing, and computing the required elements of G and the nonzero elements of the inverse are discussed. We show that the subset of G required for computing the inbreeding coefficients and hence the inverse is a tiny proportion of the whole matrix and can be easily stored in computer memory using sparse matrix storage techniques. We also introduce an algorithm to determine the maximum set of nonzero elements that can be found in G^-1 and a strategy to efficiently store and access them. Finally, we demonstrate that the inverse can be efficiently built using the present techniques for very large and inbred populations.     

A model for marker-assisted selection among single crosses with multiple genetic markers

 Trait means of marker genotypes are often inconsistent across experiments, thereby hindering the use of regression techniques in marker-assisted selection. Best linear unbiased prediction based on trait and marker data (TM-BLUP) does not require prior information on the mean effects associated with specific marker genotypes and, consequently, may be useful in applied breeding programs. The objective of this paper is to present a flanking-marker, TM-BLUP model that is applicable to interpopulation single crosses that characterize maize (Zea mays L.) breeding programs. The performance of a single cross is modeled as the sum of testcross additive and dominance effects at unmarked quantitative trait loci (QTL) and at marked QTL (MQTL). The TM-BLUP model requires information on the recombination frequencies between flanking markers and the MQTL and on MQTL variances. A tabular method is presented for calculating the conditional probability that MQTL alleles in two inbreds are identical by descent given the observed marker genotypes (G ^k _obs) at the kth MQTL. Information on identity by descent of MQTL alleles can then be used to calculate the conditional covariance of MQTL effects between single crosses given G ^k _obs. The inverse of the covariance matrix for dominance effects at unmarked QTL and MQTL can be written directly from the inverse of the covariance matrices of the corresponding testcross additive effects. In practice, the computations required in TM-BLUP may be prohibitive. The computational requirements may be reduced with simplified TM-BLUP models wherein dominance effects at MQTL are excluded, only the single crosses that have been tested are included, or information is pooled across several MQTL. Received: 22 June 1997 / Accepted: 25 February 1998     

The effect of using approximate gametic variance covariance matrices on marker assisted selection by BLUP

Under additive inheritance, the Henderson mixed model equations (HMME) provide an efficient approach to obtaining genetic evaluations by marker assisted best linear unbiased prediction (MABLUP) given pedigree relationships, trait and marker data. For large pedigrees with many missing markers, however, it is not feasible to calculate the exact gametic variance covariance matrix required to construct HMME. The objective of this study was to investigate the consequences of using approximate gametic variance covariance matrices on response to selection by MABLUP. Two methods were used to generate approximate variance covariance matrices. The first method (Method A) completely discards the marker information for individuals with an unknown linkage phase between two flanking markers. The second method (Method B) makes use of the marker information at only the most polymorphic marker locus for individuals with an unknown linkage phase. Data sets were simulated with and without missing marker data for flanking markers with 2, 4, 6, 8 or 12 alleles. Several missing marker data patterns were considered. The genetic variability explained by marked quantitative trait loci (MQTL) was modeled with one or two MQTL of equal effect. Response to selection by MABLUP using Method A or Method B were compared with that obtained by MABLUP using the exact genetic variance covariance matrix, which was estimated using 15 000 samples from the conditional distribution of genotypic values given the observed marker data. For the simulated conditions, the superiority of MABLUP over BLUP based only on pedigree relationships and trait data varied between 0.1% and 13.5% for Method A, between 1.7% and 23.8% for Method B, and between 7.6% and 28.9% for the exact method. The relative performance of the methods under investigation was not affected by the number of MQTL in the model.     

Derivation of single-locus relationship coefficients conditional on marker information

The coefficient of relationship is defined as the correlation between the additive genetic values of two individuals. This coefficient can be defined specifically for a single quantitative trait locus (QTL) and may deviate considerably from the overall expectation if it is taken conditional on information from linked marker loci. Conditional halfsib correlations are derived under a simple genetic model with a biallelic QTL linked to a biallelic marker locus. The conditional relationship coefficients are shown to depend on the recombination rate between the marker and the QTL and the population frequency of the marker alleles, but not on parameters of the QTL, i.e. number and frequency of QTL alleles, degree of dominance etc., nor on the (usually unknown) QTL genotype of the sire. Extensions to less simplified cases (multiple alleles at the marker locus and the QTL, two marker loci flanking the QTL) are given. For arbitrary pedigrees, conditional relationship coefficients can also be derived from the conditional gametic covariance matrix suggested by Fernando and Grossman (1989). The connection of these two approaches is discussed. The conditional relationship coefficient can be used for marker-assisted genetic evaluation as well as for the detection of QTL and the estimation of their effects.     

Detection of quantitative trait loci in outbred populations with incomplete marker data.

Augmentation of marker genotypes for ungenotyped individuals is implemented in a Bayesian approach via the use of Markov chain Monte Carlo techniques. Marker data on relatives and phenotypes are combined to compute conditional posterior probabilities for marker genotypes of ungenotyped individuals. The presented procedure allows the analysis of complex pedigrees with ungenotyped individuals to detect segregating quantitative trait loci (QTL). Allelic effects at the QTL were assumed to follow a normal distribution with a covariance matrix based on known QTL position and identity by descent probabilities derived from flanking markers. The Bayesian approach estimates variance due to the single QTL, together with polygenic and residual variance. The method was empirically tested through analyzing simulated data from a complex granddaughter design. Ungenotyped dams were related to one or more sons or grandsires in the design. Heterozygosity of the marker loci and size of QTL were varied. Simulation results indicated a significant increase in power when ungenotyped dams were included in the analysis.     

Identification of gametes and treatment of linear dependencies in the gametic QTL-relationship matrix and its inverse

The estimation of gametic effects via marker-assisted BLUP requires the inverse of the conditional gametic relationship matrix G. Both gametes of each animal can either be identified (distinguished) by markers or by parental origin. By example, it was shown that the conditional gametic relationship matrix is not unique but depends on the mode of gamete identification. The sum of both gametic effects of each animal – and therefore its estimated breeding value – remains however unaffected. A previously known algorithm for setting up the inverse of G was generalized in order to eliminate the dependencies between columns and rows of G. In the presence of dependencies the rank of G also depends on the mode of gamete identification. A unique transformation of estimates of QTL genotypic effects into QTL gametic effects was proven to be impossible. The properties of both modes of gamete identification in the fields of application are discussed.     

Prediction of Complex Human Traits Using the Genomic Best Linear Unbiased Predictor

Despite important advances from Genome Wide Association Studies (GWAS), for most complex human traits and diseases, a sizable proportion of genetic variance remains unexplained and prediction accuracy (PA) is usually low. Evidence suggests that PA can be improved using Whole-Genome Regression (WGR) models where phenotypes are regressed on hundreds of thousands of variants simultaneously. The Genomic Best Linear Unbiased Prediction (G-BLUP, a ridge-regression type method) is a commonly used WGR method and has shown good predictive performance when applied to plant and animal breeding populations. However, breeding and human populations differ greatly in a number of factors that can affect the predictive performance of G-BLUP. Using theory, simulations, and real data analysis, we study the performance of G-BLUP when applied to data from related and unrelated human subjects. Under perfect linkage disequilibrium (LD) between markers and QTL, the prediction R-squared (R²) of G-BLUP reaches trait-heritability, asymptotically. However, under imperfect LD between markers and QTL, prediction R² based on G-BLUP has a much lower upper bound. We show that the minimum decrease in prediction accuracy caused by imperfect LD between markers and QTL is given by (1−b)², where b is the regression of marker-derived genomic relationships on those realized at causal loci. For pairs of related individuals, due to within-family disequilibrium, the patterns of realized genomic similarity are similar across the genome; therefore b is close to one inducing small decrease in R². However, with distantly related individuals b reaches very low values imposing a very low upper bound on prediction R². Our simulations suggest that for the analysis of data from unrelated individuals, the asymptotic upper bound on R² may be of the order of 20% of the trait heritability. We show how PA can be enhanced with use of variable selection or differential shrinkage of estimates of marker effects.     

Computation of the Full Likelihood Function for Estimating Variance at a Quantitative Trait Locus

The proportion of alleles identical by descent (IBD) determines the genetic covariance between relatives, and thus is crucial in estimating genetic variances of quantitative trait loci (QTL). However, IBD proportions at QTL are unobservable and must be inferred from marker information. The conventional method of QTL variance analysis maximizes the likelihood function by replacing the missing IBDs by their conditional expectations (the expectation method), while in fact the full likelihood function should take into account the conditional distribution of IBDs (the distribution method). The distribution method for families of more than two sibs has not been obvious because there are n(n - 1)/2 IBD variables in a family of size n, forming an n X n symmetrical matrix. In this paper, I use four binary variables, where each indicates the event that an allele from one of the four grandparents has passed to the individual. The IBD proportion between any two sibs is then expressed as a function of the indicators. Subsequently, the joint distribution of the IBD matrix is derived from the distribution of the indicator variables. Given the joint distribution of the unknown IBDs, a method to compute the full likelihood function is developed for families of arbitrary sizes.     

Estimation of the position and effect of a lethal factor locus on a molecular marker linkage map

In the mapping of DNA markers the distortion of segregation of marker genotypes is often observed, which may be caused by a lethal factor acting in filial generations derived from distant crosses. A method is presented for estimating the recombination values between a lethal factor locus and neighboring molecular markers, and the relative viability or fertilization ability of gametes or zygotes affected by the lethal factor in an F₂ population using the maximum likelihood method and the expectation conditional maximization (ECM) algorithm. Three selection models of gamete or zygote were considered, and the most likely one was determined by goodness of fit of the observed frequency of the phenotypes to the expected ones under the models. The method was applied to segregation data of molecular markers of an F₂ population consisting of 144 individuals derived from a cross between an Indica and a Japonica rice variety. The presence of a lethal factor locus (L) located on chromosome III that caused partial gametic selection in both the male and female sides was suggested. The locus L was tightly linked to RFLP marker number 23 of the RFLP linkage map of Saito et al. (1991a), and the fertilization chance of a male or female gamete possessing the lethal factor was, on average, 41.5% of that of the normal gamete.     

A gene frequency model for QTL mapping using Bayesian inference

Background

Information for mapping of quantitative trait loci (QTL) comes from two sources: linkage disequilibrium (non-random association of allele states) and cosegregation (non-random association of allele origin). Information from LD can be captured by modeling conditional means and variances at the QTL given marker information. Similarly, information from cosegregation can be captured by modeling conditional covariances. Here, we consider a Bayesian model based on gene frequency (BGF) where both conditional means and variances are modeled as a function of the conditional gene frequencies at the QTL. The parameters in this model include these gene frequencies, additive effect of the QTL, its location, and the residual variance. Bayesian methodology was used to estimate these parameters. The priors used were: logit-normal for gene frequencies, normal for the additive effect, uniform for location, and inverse chi-square for the residual variance. Computer simulation was used to compare the power to detect and accuracy to map QTL by this method with those from least squares analysis using a regression model (LSR).

Results

To simplify the analysis, data from unrelated individuals in a purebred population were simulated, where only LD information contributes to map the QTL. LD was simulated in a chromosomal segment of 1 cM with one QTL by random mating in a population of size 500 for 1000 generations and in a population of size 100 for 50 generations. The comparison was studied under a range of conditions, which included SNP density of 0.1, 0.05 or 0.02 cM, sample size of 500 or 1000, and phenotypic variance explained by QTL of 2 or 5%. Both 1 and 2-SNP models were considered. Power to detect the QTL for the BGF, ranged from 0.4 to 0.99, and close or equal to the power of the regression using least squares (LSR). Precision to map QTL position of BGF, quantified by the mean absolute error, ranged from 0.11 to 0.21 cM for BGF, and was better than the precision of LSR, which ranged from 0.12 to 0.25 cM.

Conclusions

In conclusion given a high SNP density, the gene frequency model can be used to map QTL with considerable accuracy even within a 1 cM region.     

A Two-Stage Approximation for Analysis of Mixture Genetic Models in Large Pedigrees

Information from cosegregation of marker and QTL alleles, in addition to linkage disequilibrium (LD), can improve genomic selection. Variance components linear models have been proposed for this purpose, but accommodating dominance and epistasis is not straightforward with them. A full-Bayesian analysis of a mixture genetic model is favorable in this respect, but is computationally infeasible for whole-genome analyses. Thus, we propose an approximate two-step approach that neglects information from trait phenotypes in inferring ordered genotypes and segregation indicators of markers. Quantitative trait loci (QTL) fine-mapping scenarios, using high-density markers and pedigrees of five generations without genotyped females, were simulated to test this strategy against an exact full-Bayesian approach. The latter performed better in estimating QTL genotypes, but precision of QTL location and accuracy of genomic breeding values (GEBVs) did not differ for the two methods at realistically low LD. If, however, LD was higher, the exact approach resulted in a slightly higher accuracy of GEBVs. In conclusion, the two-step approach makes mixture genetic models computationally feasible for high-density markers and large pedigrees. Furthermore, markers need to be sampled only once and results can be used for the analysis of all traits. Further research is needed to evaluate the two-step approach for complex pedigrees and to analyze alternative strategies for modeling LD between QTL and markers.DUE to advances in molecular genetics, high-density single-nucleotide polymorphisms (SNPs) are becoming available in animal and plant breeding. These can be used for whole-genome analyses such as prediction of genomic breeding values (GEBVs) and fine mapping of quantitative trait loci (QTL). Genomic selection (GS) (Meuwissen et al. 2001) is promising to improve response to selection by exploiting linkage disequilibrium (LD) between SNPs and QTL (Hayes et al. 2009; Vanraden et al. 2009), but the accuracy of GEBVs depends on additive-genetic relationships between the individuals used to estimate SNP effects and selection candidates (Habier et al. 2007, 2010). Use of cosegregation information, in addition to LD, may reduce this dependency and improve GS. Calus et al. (2008) used a variance components linear model for this purpose in which random QTL effects are modeled conditional on marker haplotypes. The covariance between founder haplotypes allows one to include LD (Meuwissen and Goddard 2000), and the covariance between nonfounder haplotypes computed as in Fernando and Grossman (1989) allows one to include cosegregation. The resulting covariance matrices, however, can be nonpositive definite, which necessitates bending with the effect that information can be lost (Legarra and Fernando 2009). Furthermore, accommodating dominance and epistasis is not straightforward with linear models, especially for crossbred data. In contrast with mixture genetic models, genetic covariance matrices do not enter into the analysis, and accommodating dominance and epistasis is more straightforward (Goddard 1998; Pong-Wong et al. 1998; Stricker and Fernando 1998; Du et al. 1999; Du and Hoeschele 2000; Hoeschele 2001; Yi and Xu 2002; Pérez-Enciso 2003; Yi et al. 2003, 2005).Mixture model analyses, however, are more computationally demanding because the unknowns to be estimated in these analyses include the effects of unobservable QTL genotypes. In linear model analyses, in contrast, it is effects of observable marker genotypes that are estimated. The mixture model analysis can be thought of as a weighted sum of linear model analyses corresponding to each possible state for the unobservable QTL genotypes, where the weights are the probabilities of the QTL genotype states conditional on the observed marker genotypes and trait phenotypes. In practice, the analysis needs to consider all possible haplotypes at the markers also because even when all marker genotypes are observed, some of the marker haplotypes may not be known. As a result, the computational burden of these analyses stems from the number of unknown genotype and haplotype states that need to be summed over being exponentially related to the number of individuals in the pedigree and the number of loci.It can be shown that conditional on the genotypes of their parents, genotypes of offspring are independent of the genotypes of all their ancestors. This conditional independence can be exploited to efficiently compute the weighted summation in the mixture model analysis, provided the pedigree is not too complex (Lauritzen and Sheehan 2003). In genetics, this strategy is called peeling (Elston and Stewart 1971; Cannings et al. 1978) and is equivalent to variable elimination in graphical models (Lauritzen and Sheehan 2003). This approach, however, becomes infeasible when the pedigree is complex and the number of loci is large. Another strategy for analysis of mixture models is based on using Markov chain Monte Carlo (MCMC) theory to draw samples of QTL genotypes and marker haplotypes conditional on the observed marker genotypes and trait phenotypes. Pérez-Enciso (2003) developed an MCMC-based Bayesian analysis for a mixture genetic model that uses information from both LD and cosegregation to fine map a single QTL, but this approach becomes computationally infeasible for whole-genome analyses without approximations.In this article, we investigate a two-stage, approximate analysis that uses information from both LD and cosegregation. In the first stage, ordered genotypes of markers are sampled conditional only on the observed, unordered marker genotypes, ignoring information from the trait phenotypes. These samples are drawn using a Gibbs sampler with overlapping blocks (Thomas et al. 2000; Abraham et al. 2007) in which peeling is performed within a block while conditioning on variables outside the block. From these samples, founder haplotype probabilities and segregation probabilities for the QTL, also called probabilities of descent of QTL (PDQs) alleles, are calculated. In the second stage, these probabilities are used to sample QTL genotypes conditional on the trait phenotypes. In this analysis, information from LD is incorporated by allowing the QTL allele frequencies in founders to be dependent on the marker haplotypes, and information from cosegregation is incorporated by using the PDQs from the first stage to sample QTL alleles in nonfounders. The approximation comes from ignoring trait phenotypes in sampling ordered marker genotypes. A major advantage of the two-step approach is that markers have to be sampled only once and can then be used to analyze all quantitative traits with a mixture model.The objective of this study is to test the hypothesis that this approximation is negligible given high-density SNPs. To test this hypothesis, results from the two-stage, approximate analysis are compared to a full-Bayesian analysis that does not ignore the information from the trait phenotypes in sampling the ordered marker genotypes. The full-Bayesian approach was selected, because it is considered to be the ideal statistical model as it accounts for all uncertainties (Hoeschele 2001). Because the full-Bayesian approach is computationally too demanding for application to GS, the approximate and full-Bayesian analyses are used to fine map within a simulated chromosomal region that is known to contain a QTL to make the comparison computationally feasible. If the consequences of ignoring trait phenotypes to sample ordered marker genotypes are negligible, further research to apply mixture genetic models to GS and comparisons with linear models are justifiable.     

标记基因型中QTL基因型条件概率分布

随着分子数量遗传学的发展,人们提出了很多统计模型用于QTL定位分析。在这些模型中,首先得确定QTL在标记基因型中的条件概率分布,然后利用适当的统计方法对QTL在基因组中所处的位置进行估计。本文讨论了常见作图群体（如F2和回交群体）中在给定标记基因型下QTL的条件概率分布,提出了用Mathematics软件推导QTL基因型条件概率分布的方法。用该方法能够快速地、准确地推导出比较复杂的标记基因型中QTL基因型的条件概率分布。     

Conditional QTL mapping for plant height with respect to the length of the spike and internode in two mapping populations of wheat

Plant height (PH) in wheat is a complex trait; its components include spike length (SL) and internode lengths. To precisely analyze the factors affecting PH, two F_8:9 recombinant inbred line (RIL) populations comprising 485 and 229 lines were generated. Crosses were performed between Weimai 8 and Jimai 20 (WJ) and between Weimai 8 and Yannong 19 (WY). Possible genetic relationships between PH and PH components (PHC) were evaluated at the quantitative trait locus (QTL) level. PH and PHC (including SL and internode lengths from the first to the fourth counted from the top, abbreviated as FIITL, SITL, TITL, and FOITL, respectively) were measured in four environments. Individual and the pooled values from four trials were used in the present analysis. A QTL for PH was mapped using data on PH and on PH conditioned by PHC using IciMapping V2.2. All 21 chromosomes in wheat were shown to harbor factors affecting PH in two populations, by both conditional and unconditional QTL mapping methods. At least 11 pairwise congruent QTL were identified in the two populations. In total, ten unconditional QTL and five conditional QTL that could be detected in the conditional analysis only have been verified in no less than three trials in WJ and WY. In addition, three QTL on the short arms of chromosomes 4B, 4D, and 7B were mapped to positions similar to those of the semi-dwarfing genes Rht-B1, Rht-D1 and Rht13, respectively. Conditional QTL mapping analysis in WJ and WY proved that, at the QTL level, SL contributed the least to PH, followed by FIITL; TITL had the strongest influence on PH, followed by SITL and FOITL. The results above indicated that the conditional QTL mapping method can be used to evaluate possible genetic relationships between PH and PHC, and it can efficiently and precisely reveal counteracting QTL, which will enhance the understanding of the genetic basis of PH in wheat. The combination of two related populations with a large/moderate population size made the results authentic and accurate.     

Polyploidy with an arbitrary mixture of chromosome- and chromatid segregation

Summary The gametic algebraG is constructed for a random mating population of 2r-ploid individuals which differ in a single locus with the allelesA anda. It is assumed that every kind of segregation between chromosome- and chromatid segregation occurs with a given probability. This algebraG is a convex combination of 2r+1 genetic algebras which have a common canonical basis. The train roots of these algebras are calculated and shown to be monotonically descending. The algebraG possesses a one-dimensional manifold of idempotents. With a generalization of Gonshor's theorem on the convergence of the sequence of plenary powers of an element of unit weight it is shown that for every initial gametic distribution the distribution in the following generations converges towards an equilibrium state whose coordinates are polynomials in the frequency of the alleleA in the initial generation.     

A recursive procedure to compute the gametic relationship matrix and its inverse for marked QTL clusters

Recently, a variety of mixed linear models have been proposed for marker-assisted prediction of the effects of quantitative trait loci (QTLs) in outbred populations of animals. One of them addresses the effects of a cluster of linked QTLs, or those of a particular chromosomal segment, marked by DNA marker(s) and requires that the inverse of the corresponding gametic relationship matrix whose elements are the conditional expected values of the identity-by-descent (IBD) proportions between gametes for individuals be evaluated. Here, for a model of this type, utilizing the property of the IBD set and using the information on the joint gametogenesis processes at the flanking marker loci, we present a recursive method to systematically calculate the elements of the gametic relationship matrix and its inverse. A numerical example is given to illustrate the proposed computing procedure.     

Genetic models to estimate additive and non-additive effects of marker-associated QTL using multiple regression techniques

Summary The development of molecular markers has recently raised expectations for their application in selection programs. However, some questions related to quantitative trait loci (QTL) identification are still unanswered. The objectives of this paper are (1) to develop statistical genetic models for detecting and locating on the genome multi-QTL with additive, dominance and epistatic effects using multiple linear regression analysis in the backcross and F_n generations from the cross of two inbred lines; and (2) to discuss the bias caused by linked and unlinked QTL on the genetic estimates. Non-linear models were developed for different backcross and F_n generations when both epistasis and no epistasis were assumed. Generation analysis of marked progenies is suggested as a way of increasing the number of observations for the estimates without additional cost for molecular scoring. Some groups of progenies can be created in different generations from the same scored individuals. The non-linear models were transformed into approximate multivariate linear models to which combined stepwise and standard regression analysis could be applied. Expressions for the biases of the marker classes from linked QTL were obtained when no epistasis was assumed. When epistasis was assumed, these expressions increased in complexity, and the biases were caused by both linked and unlinked QTL.     

Kernel weight per spike: what contributes to it at the individual QTL level?

Spike length (SL), spikelet number (SPN) per spike, kernel number per spike (KNPS), and thousand-kernel weight (TKW) have strong genetic associations with kernel weight per spike (KWPS) in wheat. To investigate their genetic relationships at the individual quantitative trait locus (QTL) level, both unconditional and conditional QTL mapping for KWPS with respect to SL, SPN, KNPS, and TKW were conducted. Two related F_8:9 recombinant inbred line populations, comprising 485 and 229 lines, respectively, were used. The trait phenotypic performances of each population were evaluated in four different environments. Unconditional QTL mapping analysis identified 22 putative additive QTL for KWPS, five of which were stable QTL, and only QKwps-WJ-1B.2 showed significant additive-by-environment interaction effects. In comparison with unconditional QTL mapping analysis, conditional QTL mapping analysis indicated that, at the QTL level, KNPS and TKW contributed more to KWPS than did SL and SPN. Any unconditional QTL for KWPS detected in this study was associated with at least one of its four related traits. The present study will provide assistance in the understanding of the genetic relationships between KWPS and its related traits.     

QTL consistency and meta-analysis for grain yield components in three generations in maize

Grain yield is the most important and complex trait in maize. In this study, a total of 258 F₉ recombinant inbred lines (RIL), derived from a cross between dent corn inbred Dan232 and popcorn inbred N04, were evaluated for eight grain yield components under four environments. Quantitative trait loci (QTL) and their epistatic interactions were detected for all traits under each environment and in combined analysis. Meta-analysis was used to integrate genetic maps and detected QTL across three generations (RIL, F_2:3 and BC₂F₂) derived from the same cross. In total, 103 QTL, 42 pairs of epistatic interactions and 16 meta-QTL (mQTL) were detected. Twelve out of 13 QTL with contributions (R ²) over 15% were consistently detected in 3–4 environments (or in combined analysis) and integrated in mQTL. Only q100GW-7-1 was detected in all four environments and in combined analysis. 100qGW-1-1 had the largest R ² (19.3–24.6%) in three environments and in combined analysis. In contrast, 35 QTL for 6 grain yield components were detected in the BC₂F₂ and F_2:3 generations, no common QTL across three generations were located in the same marker intervals. Only 100 grain weight (100GW) QTL on chromosome 5 were located in adjacent marker intervals. Four common QTL were detected across the RIL and F_2:3 generations, and two between the RIL and BC₂F₂ generations. Each of five important mQTL (mQTL7-1, mQTL10-2, mQTL4-1, mQTL5-1 and mQTL1-3) included 7–12 QTL associated with 2–6 traits. In conclusion, we found evidence of strong influence of genetic structure and environment on QTL detection, high consistency of major QTL across environments and generations, and remarkable QTL co-location for grain yield components. Fine mapping for five major QTL (q100GW-1-1, q100GW-7-1, qGWP-4-1, qERN-4-1 and qKR-4-1) and construction of single chromosome segment lines for genetic regions of five mQTL merit further studies and could be put into use in marker-assisted breeding.     

Wheat kernel dimensions: how do they contribute to kernel weight at an individual QTL level?

Kernel dimensions (KD) contribute greatly to thousand-kernel weight (TKW) in wheat. In the present study, quantitative trait loci (QTL) for TKW, kernel length (KL), kernel width (KW) and kernel diameter ratio (KDR) were detected by both conditional and unconditional QTL mapping methods. Two related F(8:9) recombinant inbred line (RIL) populations, comprising 485 and 229 lines, respectively, were used in this study, and the trait phenotypes were evaluated in four environments. Unconditional QTL mapping analysis detected 77 additive QTL for four traits in two populations. Of these, 24 QTL were verified in at least three trials, and five of them were major QTL, thus being of great value for marker assisted selection in breeding programmes. Conditional QTL mapping analysis, compared with unconditional QTL mapping analysis, resulted in reduction in the number of QTL for TKW due to the elimination of TKW variations caused by its conditional traits; based on which we first dissected genetic control system involved in the synthetic process between TKW and KD at an individual QTL level. Results indicated that, at the QTL level, KW had the strongest influence on TKW, followed by KL, and KDR had the lowest level contribution to TKW. In addition, the present study proved that it is not all-inclusive to determine genetic relationships of a pairwise QTL for two related/causal traits based on whether they were co-located. Thus, conditional QTL mapping method should be used to evaluate possible genetic relationships of two related/causal traits.     

Different models of genetic variation and their effect on genomic evaluation

Background

The theory of genomic selection is based on the prediction of the effects of quantitative trait loci (QTL) in linkage disequilibrium (LD) with markers. However, there is increasing evidence that genomic selection also relies on "relationships" between individuals to accurately predict genetic values. Therefore, a better understanding of what genomic selection actually predicts is relevant so that appropriate methods of analysis are used in genomic evaluations.

Methods

Simulation was used to compare the performance of estimates of breeding values based on pedigree relationships (Best Linear Unbiased Prediction, BLUP), genomic relationships (gBLUP), and based on a Bayesian variable selection model (Bayes B) to estimate breeding values under a range of different underlying models of genetic variation. The effects of different marker densities and varying animal relationships were also examined.

Results

This study shows that genomic selection methods can predict a proportion of the additive genetic value when genetic variation is controlled by common quantitative trait loci (QTL model), rare loci (rare variant model), all loci (infinitesimal model) and a random association (a polygenic model). The Bayes B method was able to estimate breeding values more accurately than gBLUP under the QTL and rare variant models, for the alternative marker densities and reference populations. The Bayes B and gBLUP methods had similar accuracies under the infinitesimal model.

Conclusions

Our results suggest that Bayes B is superior to gBLUP to estimate breeding values from genomic data. The underlying model of genetic variation greatly affects the predictive ability of genomic selection methods, and the superiority of Bayes B over gBLUP is highly dependent on the presence of large QTL effects. The use of SNP sequence data will outperform the less dense marker panels. However, the size and distribution of QTL effects and the size of reference populations still greatly influence the effectiveness of using sequence data for genomic prediction.