Efficient computation of the inverse of gametic relationship matrix for a marked QTL

Best linear unbiased prediction of genetic merits for a marked quantitative trait locus (QTL) using mixed model methodology includes the inverse of conditional gametic relationship matrix (G^-1) for a marked QTL. When accounting for inbreeding, the conditional gametic relationships between two parents of individuals for a marked QTL are necessary to build G^-1 directly. Up to now, the tabular method and its adaptations have been used to compute these relationships. In the present paper, an indirect method was implemented at the gametic level to compute these few relationships. Simulation results showed that the indirect method can perform faster with significantly less storage requirements than adaptation of the tabular method. The efficiency of the indirect method was mainly due to the use of the sparseness of G^-1. The indirect method can also be applied to construct an approximate G^-1 for populations with incomplete marker data, providing approximate probabilities of descent for QTL alleles for individuals with incomplete marker data.     

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.     

Power of tests for QTL detection using replicated progenies derived from a diallel cross

In crop species, most QTL (quantitative trait loci) mapping strategies use segregating populations derived from an initial cross between two lines. However, schemes including more than two parents could also be used. We propose an approach using a high-density restriction fragment length polymorphism (RFLP) map established on six F ₂ populations derived from diallel crosses among four inbred lines and the phenotypic performances of two types of replicated progenies (F ₃ and topcross). The QTL is supposed to be on the marker locus considered. Three linear model tests for the detection of QTL effects (T ₁, T ₂ and T ₃) are described and their power studied for the two types of progeny. T ₁ tests the global genetic effects of the QTL (additivity and dominance) and T ₂ tests only additive effects assuming dominance is absent when it could exist. The models of these two tests assume that the main effects of QTL alleles are constant in different genetic backgrounds. The additive model of test T ₃ considers the six F ₂ populations independently, and T ₃ is the equivalent of the classical mean comparison test if we neglect dominance; it uses only contrasts between the homozygote marker classes. The results show that T ₂ is much more powerful than T ₃. The power of T ₁ and T ₂ depends on the relative sizes of the additive and dominance effects, and their comparison is not easy to establish. Nevertheless, T ₂ seems to be the more powerful in most situations, indicating that it is often more interesting to ignore dominance when testing for a QTL effect. For a given size of genetic effects, the power is affected by the total number of individuals genotyped in F ₂ and the recombination rate between the marker locus and the putative QTL. The approach presented in this paper has some drawbacks but could be easily generalized to other sizes of diallels and different progeny types.     

Genomewide predictions from maize single-cross data

Maize (Zea mays L.) breeders evaluate many single-cross hybrids each year in multiple environments. Our objective was to determine the usefulness of genomewide predictions, based on marker effects from maize single-cross data, for identifying the best untested single crosses and the best inbreds within a biparental cross. We considered 479 experimental maize single crosses between 59 Iowa Stiff Stalk Synthetic (BSSS) inbreds and 44 non-BSSS inbreds. The single crosses were evaluated in multilocation experiments from 2001 to 2009 and the BSSS and non-BSSS inbreds had genotypic data for 669 single nucleotide polymorphism (SNP) markers. Single-cross performance was predicted by a previous best linear unbiased prediction (BLUP) approach that utilized marker-based relatedness and information on relatives, and from genomewide marker effects calculated by ridge-regression BLUP (RR-BLUP). With BLUP, the mean prediction accuracy (r _MG) of single-cross performance was 0.87 for grain yield, 0.90 for grain moisture, 0.69 for stalk lodging, and 0.84 for root lodging. The BLUP and RR-BLUP models did not lead to r _MG values that differed significantly. We then used the RR-BLUP model, developed from single-cross data, to predict the performance of testcrosses within 14 biparental populations. The r _MG values within each testcross population were generally low and were often negative. These results were obtained despite the above-average level of linkage disequilibrium, i.e., r ² between adjacent markers of 0.35 in the BSSS inbreds and 0.26 in the non-BSSS inbreds. Overall, our results suggested that genomewide marker effects estimated from maize single crosses are not advantageous (compared with BLUP) for predicting single-cross performance and have erratic usefulness for predicting testcross performance within a biparental cross.     

Mapping quantitative trait loci using molecular marker linkage maps

Summary High-density restriction fragment length polymorphism (RFLP) and allozyme linkage maps have been developed in several plant species. These maps make it technically feasible to map quantitative trait loci (QTL) using methods based on flanking marker genetic models. In this paper, we describe flanking marker models for doubled haploid (DH), recombinant inbred (RI), backcross (BC), F₁ testcross (F₁TC), DH testcross (DHTC), recombinant inbred testcross (RITC), F₂, and F₃ progeny. These models are functions of the means of quantitative trait locus genotypes and recombination frequencies between marker and quantitative trait loci. In addition to the genetic models, we describe maximum likelihood methods for estimating these parameters using linear, nonlinear, and univariate or multivariate normal distribution mixture models. We defined recombination frequency estimators for backcross and F₂ progeny group genetic models using the parameters of linear models. In addition, we found a genetically unbiased estimator of the QTL heterozygote mean using a linear function of marker means. In nonlinear models, recombination frequencies are estimated less efficiently than the means of quantitative trait locus genotypes. Recombination frequency estimation efficiency decreases as the distance between markers decreases, because the number of progeny in recombinant marker classes decreases. Mean estimation efficiency is nearly equal for these methods.     

The covariance between relatives conditional on genetic markers

The development of molecular genotyping techniques makes it possible to analyze quantitative traits on the basis of individual loci. With marker information, the classical theory of estimating the genetic covariance between relatives can be reformulated to improve the accuracy of estimation. In this study, an algorithm was derived for computing the conditional covariance between relatives given genetic markers. Procedures for calculating the conditional relationship coefficients for additive, dominance, additive by additive, additive by dominance, dominance by additive and dominance by dominance effects were developed. The relationship coefficients were computed based on conditional QTL allelic transmission probabilities, which were inferred from the marker allelic transmission probabilities. An example data set with pedigree and linked markers was used to demonstrate the methods developed. Although this study dealt with two QTLs coupled with linked markers, the same principle can be readily extended to the situation of multiple QTL. The treatment of missing marker information and unknown linkage phase between markers for calculating the covariance between relatives was discussed.     

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.     

Theory for identification of marker locus-QTL associations in population by line crosses

The objective of this paper is to present genetic theory demonstrating the conditions under which it should be possible to identify molecular marker-quantitative trait locus (QTL) associations in crosses of random-mating populations to inbreds. Using as an example the cross of a corn (Zea mays L.) population to an inbred, the expected disequilibrium for testcross and per se performance of F₂, F₃, BC₁ (to the inbred) and recombinant inbred generations was derived for cases where a marker allele is linked to an unfavorable QTL allele in the inbred and where the marker allele is linked to a favorable QTL allele in the inbred. Disequilibrium in segregating generations was shown to be a function of disequilibrium in the parent population, the frequency of marker and QTL alleles in the parent population, and the recombination distance between the marker and the QTL. To maximize the opportunity to identify a favorable QTL the following procedures are suggested:

Confirmation of QTL controlling seed yield in spring canola (Brassica napus L.) hybrids

Allelic effects observed in QTL discovery experiments must be confirmed to be useful in subsequent breeding efforts. Two QTL affecting seed yield of spring hybrid canola (Brassica napus L.) were previously identified in two populations of inbred backcross lines (IBLs) containing germplasm introgressed from a winter cultivar. The effects of favorable alleles at these QTL were retested by crossing two selected IBLs (M5 and M31) to three spring canola lines having different genetic backgrounds. Doubled haploid (DH) lines derived from each F₁ were genotyped with RFLP markers flanking the QTL and grouped into the four possible QTL genotypes. For the first field experiment, DH lines derived by crossing the M5 line to one spring line were crossed to two female testers and evaluated as individual testcross progenies in one environment. QTL genotypes had large variances and were not significantly different. A second field experiment was conducted using the DH lines from the first experiment and two other sets of DH lines derived from the M31 line crossed to two different spring canola lines. Individual lines within each QTL genotype of each set were bulked and crossed to the same testers used in Experiment 1. Bulked hybrid seeds of each QTL genotype were planted in a split-split plot randomized block design and 12 replicates. QTL genotypes had smaller variances in this experiment, and the effects of one QTL were confirmed in some genetic backgrounds. These results suggest that bulking of QTL genotypes and use of an appropriate experimental design with many replicates are needed to detect small differences between QTL genotypes.     

Testcross additive and dominance effects in best linear unbiased prediction of maize single-cross performance

Best linear unbiased prediction (BLUP) has been found to be useful in maize (Zea mays L.) breeding. The advantage of including both testcross additive and dominance effects (Intralocus Model) in BLUP, rather than only testcross additive effects (Additive Model), has not been clearly demonstrated. The objective of this study was to compare the usefulness of Intralocus and Additive Models for BLUP of maize single-cross performance. Multilocation data from 1990 to 1995 were obtained from the hybrid testing program of Limagrain Genetics. Grain yield, moisture, stalk lodging, and root lodging of untested single crosses were predicted from (1) the performance of tested single crosses and (2) known genetic relationships among the parental inbreds. Correlations between predicted and observed performance were obtained with a delete-one cross-validation procedure. For the Intralocus Model, the correlations ranged from 0.50 to 0.66 for yield, 0.88 to 0.94 for moisture, 0.47 to 0.69 for stalk lodging, and 0.31 to 0.45 for root lodging. The BLUP procedure was consistently more effective with the Intralocus Model than with the Additive Model. When the Additive Model was used instead of the Intralocus Model, the reductions in the correlation were largest for root lodging (0.06–0.35), smallest for moisture (0.00–0.02), and intermediate for yield (0.02–0.06) and stalk lodging (0.02–0.08). The ratio of dominance variance (v _D) to total genetic variance (v _G) was highest for root lodging (0.47) and lowest for moisture (0.10). The Additive Model may be used if prior information indicates that V_D for a given trait has little contribution to V_G. Otherwise, the continued use of the Intralocus Model for BLUP of single-cross performance is recommended.     

Mapping quantitative trait loci by genotyping haploid tissues.

Mapping strategies based on a half- or full-sib family design have been developed to map quantitative trait loci (QTL) for outcrossing species. However, these strategies are dependent on controlled crosses where marker-allelic frequency and linkage disequilibrium between the marker and QTL may limit their application. In this article, a maximum-likelihood method is developed to map QTL segregating in an open-pollinated progeny population using dominant markers derived from haploid tissues from single meiotic events. Results from the haploid-based mapping strategy are not influenced by the allelic frequencies of markers and their linkage disequilibria with QTL, because the probabilities of QTL genotypes conditional on marker genotypes of haploid tissues are independent of these population parameters. Parameter estimation and hypothesis testing are implemented via expectation/conditional maximization algorithm. Parameters estimated include the additive effect, the dominant effect, the population mean, the chromosomal location of the QTL in the interval, and the residual variance within the QTL genotypes, plus two population parameters, outcrossing rate and QTL-allelic frequency. Simulation experiments show that the accuracy and power of parameter estimates are affected by the magnitude of QTL effects, heritability levels of a trait, and sample sizes used. The application and limitation of the method are discussed.     

Hybrid maize breeding with doubled haploids: V. Selection strategies for testcross performance with variable sizes of crosses and S1 families

In hybrid maize (Zea mays L.) breeding, doubled haploids (DH) are increasingly replacing inbreds developed by recurrent selfing. Doubled haploids may be developed directly from S₀ plants in the parental cross or via S₁ families. In both these breeding schemes, we examined 2 two-stage selecting strategies, i.e., considering or ignoring cross and family structure while selection among and within parental crosses and S₁ families. We examined the optimum allocation of resources to maximize the selection gain ΔG and the probability P(q) of identifying the q% best genotypes. Our specific objectives were to (1) determine the optimum number and size of crosses and S₁ families, as well as the optimum number of test environments and (2) identify the superior selection strategy. Selection was based on the evaluation of testcross progenies of (1) DH lines in both stages (DHTC) and (2) S₁ families in the first stage and of DH lines within S₁ families in the second stage (S₁TC-DHTC) with uniform and variable sizes of crosses and S₁ families. We developed and employed simulation programs for selection with variable sizes of crosses and S₁ families within crosses. The breeding schemes and selection strategies showed similar relative efficiency for both optimization criteria ΔG and P (0.1%). As compared with DHTC, S₁TC-DHTC had larger ΔG and P (0.1%), but a higher standard deviation of ΔG. The superiority of S₁TC-DHTC was increased when the selection was done among all DH lines ignoring their cross and family structure and using variable sizes of crosses and S₁ families. In DHTC, the best selection strategy was to ignore cross structures and use uniform size of crosses.     

Inheritance of grain filling duration in spring wheat (<Emphasis Type="Italic">Triticum aestivum</Emphasis> L. em thell)

To understand the genetic control of grain filling duration (GFD), i.e., the number of days from anthesis to physiological maturity, we studied the F₁, F₂, BC1 and BC₂ generations of six spring wheat crosses from nine varieties/genotypes. Generation mean analysis for gene effects indicated that one or more types of epistasis were significant in all crosses. In each pairing, the F₁ and F₂ means were either intermediate or closer to the mean of the parent having the longer GFD. Our narrow-sense heritability estimate was reasonably high, at 47.67 (based on diallel analysis). This demonstrated that progress could be made from the selection in these crosses for either long or short GFD. The two early varieties that had identical maturity durations differed in their GFD values, indicating that maturity dates are not good criteria when choosing parents for modifying GFD. To utilize favorable additive × additive effects during this selection, we suggest that a single seed descent (SSD) or bulk popula-tion approach be adopted. In comparison, dominance effects would prove quite useful in hybrid wheat breeding programs.     

Interactions between markers can be caused by the dominance effect of quantitative trait loci

F₂ populations are commonly used in genetic studies of animals and plants. For simplicity, most quantitative trait locus or loci (QTL) mapping methods have been developed on the basis of populations having two distinct genotypes at each polymorphic marker or gene locus. In this study, we demonstrate that dominance can cause the interactions between markers and propose an inclusive linear model that includes marker variables and marker interactions so as to completely control both additive and dominance effects of QTL. The proposed linear model is the theoretical basis for inclusive composite-interval QTL mapping (ICIM) for F₂ populations, which consists of two steps: first, the best regression model is selected by stepwise regression, which approximately identifies markers and marker interactions explaining both additive and dominance variations; second, the interval mapping approach is applied to the phenotypic values adjusted by the regression model selected in the first step. Due to the limited mapping population size, the large number of variables, and multicollinearity between variables, coefficients in the inclusive linear model cannot be accurately determined in the first step. Interval mapping is necessary in the second step to fine tune the QTL to their true positions. The efficiency of including marker interactions in mapping additive and dominance QTL was demonstrated by extensive simulations using three QTL distribution models with two population sizes and an actual rice F₂ population.     

A two‐step approach to map quantitative trait loci for meat quality in connected porcine F2 crosses considering main and epistatic effects

The aim of this study was to map QTL for meat quality traits in three connected porcine F₂ crosses comprising around 1000 individuals. The three crosses were derived from the founder breeds Chinese Meishan, European Wild Boar and Pietrain. The animals were genotyped genomewide for approximately 250 genetic markers, mostly microsatellites. They were phenotyped for seven meat quality traits (pH at 45 min and 24 h after slaughter, conductivity at 45 min and 24 h after slaughter, meat colour, drip loss and rigour). QTL mapping was conducted using a two‐step procedure. In the first step, the QTL were mapped using a multi‐QTL multi‐allele model that was tailored to analyse multiple connected F₂ crosses. It considered additive, dominance and imprinting effects. The major gene RYR1:g.1843C>T affecting the meat quality on SSC6 was included as a cofactor in the model. The mapped QTL were tested for pairwise epistatic effects in the second step. All possible epistatic effects between additive, dominant and imprinting effects were considered, leading to nine orthogonal forms of epistasis. Numerous QTL were found. The most interesting chromosome was SSC6. Not all genetic variance of meat quality was explained by RYR1:g.1843C>T. A small confidence interval was obtained, which facilitated the identification of candidate genes underlying the QTL. Epistasis was significant for the pairwise QTL on SSC12 and SSC14 for pH24 and for the QTL on SSC2 and SSC5 for rigour. Some evidence for additional pairwise epistatic effects was found, although not significant. Imprinting was involved in epistasis.     

Development of diagnostic markers for use in breeding potatoes resistant to Globodera pallida pathotype Pa2/3 using germplasm derived from Solanum tuberosum ssp. andigena CPC 2802

Quantitative resistance to Globodera pallida pathotype Pa2/3, originally derived from Solanum tuberosum ssp. andigena Commonwealth Potato Collection (CPC) accession 2802, is present in several potato cultivars and advanced breeding lines. One genetic component of this resistance, a large effect quantitative trait locus (QTL) on linkage group IV (which we have renamed GpaIV _adg ^s ) has previously been mapped in the tetraploid breeding line 12601ab1. In this study, we show that GpaIV _adg ^s is also present in a breeding line called C1992/31 via genetic mapping in an F₁ population produced by crossing C1992/31 with the G. pallida susceptible cultivar Record. C1992/31 is relatively divergent from 12601ab1, confirming that GpaIV _adg ^s is an ideal target for marker-assisted selection in currently available germplasm. To generate markers exhibiting diagnostic potential for GpaIV _adg ^s , three bacterial artificial chromosome clones were isolated from the QTL region, sequenced, and used to develop 15 primer sets generating single-copy amplicons, which were examined for polymorphisms exhibiting linkage to GpaIV _adg ^s in C1992/31. Eight such polymorphisms were found. Subsequently, one insertion/deletion polymorphism, three single nucleotide polymorphisms and a specific allele of the microsatellite marker STM3016 were shown to exhibit diagnostic potential for the QTL in a panel of 37 potato genotypes, 12 with and 25 without accession CPC2082 in their pedigrees. STM3016 and one of the SNP polymorphisms, C237(119), were assayed in 178 potato genotypes, arising from crosses between C1992/31 and 16 G. pallida susceptible genotypes, undergoing selection in a commercial breeding programme. The results suggest that the diagnostic markers would most effectively be employed in MAS-based approaches to pyramid different resistance loci to develop cultivars exhibiting strong, durable resistance to G. pallida pathotype Pa2/3.     

QTL analysis: a simple ‘marker-regression’ approach

A method to locate quantitative trait loci (QTL) on a chromosome and to estimate their additive and dominance effects is described. It applies to generations derived from an F₁ by selfing or backcrossing and to doubled haploid lines, given that marker genotype information (RFLP, RAPD, etc.) and quantitative trait data are available. The method involves regressing the additive difference between marker genotype means at a locus against a function of the recombination frequency between that locus and a putative QTL. A QTL is located, as by other regression methods, at that point where the residual mean square is minimised. The estimates of location and gene effects are consistent and as reliable as conventional flanking-marker methods. Further applications include the ability to test for the presence of two, or more, linked QTL and to compare different crosses for the presence of common QTL. Furthermore, the technique is straightforward and may be programmed using standard pc-based statistical software.     

Most significant genome regions involved in the control of earliness traits in bread wheat, as revealed by QTL meta-analysis

Earliness is one of the most important adaptation traits in plant breeding. Our purpose was to identify the genome regions of bread wheat involved in the control of earliness and its three components: photoperiod sensitivity (PS), vernalization requirement (VR) and intrinsic earliness (IE). A QTL meta-analysis was carried out to examine the replicability of QTL across 13 independent studies and to propose meta-QTL (MQTL). Initial QTL were projected on a recent consensus map (2004). Quality criteria were proposed to assess the reliability of this projection. These criteria were based on the distances between markers in the QTL regions. Chromosomes of groups 2 and 5 had a greater incidence on earliness control as they carry the known, major genes Ppd and Vrn. Other chromosome regions played an intermediate role in earliness control: 4A [heading date (HD) Meta-QTL], 4B (HD MQTL), 2B (VR MQTL) and 5B (IE MQTL). Markers at this four MQTL should prove helpful in marker-assisted selection, to better control earliness.     

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.     

Some practical considerations for using RFLP markers to aid in selection during inbreeding of maize

Summary If molecular markers are to be routinely used in maize (Zea mays L.) breeding for selection of quantitative trait loci (QTL), then consistent marker-trait associations across breeding populations are needed, as are efficient methods for weighting information from different markers. Given 15 restriction fragment length polymorphism (RFLP) markers associated with grain yield in testcrosses of 220 [BS11(FR)C7 x FRMol7] F₂ individuals to FRB73, separate weighting schemes were attempted in order to maximize the frequency of favorable marker genotypes associated with increased grain yield in selected F₂ individuals and F₂:S₄ Unes. The following principles were apparent: (1) Differential weighting among markers, in addition to weighting individual marker genotypes on the basis of associated mean effects, should be emphasized when using markers to select in breeding populations. This is due to limited population sizes that can readily be handled. (2) Relatively few markers may need to be used to screen segregating populations (e.g., F₂) of limited size for loci affecting complex traits, such as combining ability for grain yield, assuming prior knowledge of marker-QTL associations. Markers given greatest weight (largest estimates of associated effects) will determine most selections. (3) When marker-based selection is among individuals at higher levels of inbreeding (e.g., S₄) within selected families, more markers need to be used in screening because those associated with relatively small effects have an increased chance of affecting selection.These results suggest a qualitative approach for utilizing RFLP markers to aid in selection of complex traits in commercial hybrid maize breeding programs. Commercial research programs produce thousands of crosses each year aimed at inbred line development. Discovery of molecular markers with consistent QTL associations across breeding populations and close QTL linkages would allow for rapid screening of new F₂ populations at a few key markers. Early elimination of individuals with undesirable genotypes would reduce the extent of hybrid performance testing necessary during later stages of inbreeding.