Mapping quantitative trait loci in complex pedigrees: a two-step variance component approach

There is a growing need for the development of statistical techniques capable of mapping quantitative trait loci (QTL) in general outbred animal populations. Presently used variance component methods, which correctly account for the complex relationships that may exist between individuals, are challenged by the difficulties incurred through unknown marker genotypes, inbred individuals, partially or unknown marker phases, and multigenerational data. In this article, a two-step variance component approach that enables practitioners to routinely map QTL in populations with the aforementioned difficulties is explored. The performance of the QTL mapping methodology is assessed via its application to simulated data. The capacity of the technique to accurately estimate parameters is examined for a range of scenarios.     

Bayesian QTL analyses using pedigreed families of an outcrossing species,with application to fruit firmness in apple

Key message

Proof of concept of Bayesian integrated QTL analyses across pedigree-related families from breeding programs of an outbreeding species. Results include QTL confidence intervals, individuals’ genotype probabilities and genomic breeding values.

Abstract

Bayesian QTL linkage mapping approaches offer the flexibility to study multiple full sib families with known pedigrees simultaneously. Such a joint analysis increases the probability of detecting these quantitative trait loci (QTL) and provide insight of the magnitude of QTL across different genetic backgrounds. Here, we present an improved Bayesian multi-QTL pedigree-based approach on an outcrossing species using progenies with different (complex) genetic relationships. Different modeling assumptions were studied in the QTL analyses, i.e., the a priori expected number of QTL varied and polygenic effects were considered. The inferences include number of QTL, additive QTL effect sizes and supporting credible intervals, posterior probabilities of QTL genotypes for all individuals in the dataset, and QTL-based as well as genome-wide breeding values. All these features have been implemented in the FlexQTL^? software. We analyzed fruit firmness in a large apple dataset that comprised 1,347 individuals forming 27 full sib families and their known ancestral pedigrees, with genotypes for 87 SSR markers on 17 chromosomes. We report strong or positive evidence for 14 QTL for fruit firmness on eight chromosomes, validating our approach as several of these QTL were reported previously, though dispersed over a series of studies based on single mapping populations. Interpretation of linked QTL was possible via individuals’ QTL genotypes. The correlation between the genomic breeding values and phenotypes was on average 90 %, but varied with the number of detected QTL in a family. The detailed posterior knowledge on QTL of potential parents is critical for the efficiency of marker-assisted breeding.     

QTL analysis in arbitrary pedigrees with incomplete marker information

Mapping quantitative trait loci (QTL) in arbitrary outbred pedigrees is complicated by the combinatorial possibilities of allele flow relationships and of the founder allelic configurations. Exact methods are only available for rather short and simple pedigrees. Stochastic simulation using Markov chain Monte Carlo (MCMC) integration offers more flexibility. MCMC methods are less natural in a frequentist than in a Bayesian context, which we therefore adopt. Among the MCMC algorithms for updating marker locus genotypes, we implement the descent-graph algorithm. It can be used to update marker locus allele flow relationships and can handle arbitrarily complex pedigrees and missing marker information. Compared with updating marker genotypic information, updating QTL parameters, such as position, effects, and the allele flow relationships is relatively easy with MCMC. We treat the effect of each diploid combination of founder alleles as a random variable and only estimate the variance of these effects, ie, we model diploid genotypic effects instead of the usual partition in additive and dominance effects. This is a variant of the random model approach. The number of QTL alleles is generally unknown. In the Bayesian context, the number of QTL present on a linkage group can be treated as variable. Computer simulations suggest that the algorithm can indeed handle complex pedigrees and detect two QTL on a linkage group, but that the number of individuals in a single extended family is limited to about 50 to 100 individuals.     

Limitations of a two-step moment method for mapping linked multiple QTL

As an alternative to multiple-interval mapping a two-step moment method was recently proposed to map linked multiple quantitative trait loci (QTLs). The advantage of this moment method was supposed to be its simplicity and computational efficiency, especially in detecting closely linked QTLs within a marker bracket, but also in mapping QTLs in different marker intervals. Using simulations it is shown that the two-step moment method may give poor results compared with multiple-interval mapping, irrespective of whether the QTLs are in the same or in different marker intervals, especially if linked QTLs are in repulsion. The criteria of comparison are number of identified QTLs, likelihood ratio test statistics, means and empirical standard errors of the QTL position and QTL effects estimates, and the accuracy of the residual variance estimates. Further, the joint conditional probabilities of QTL genotypes for two putative QTLs within a marker interval were derived and compared with the unmodified approach ignoring the non-independence of the conditional probabilities.     

Family-based association tests for genomewide association scans

With millions of single-nucleotide polymorphisms (SNPs) identified and characterized, genomewide association studies have begun to identify susceptibility genes for complex traits and diseases. These studies involve the characterization and analysis of very-high-resolution SNP genotype data for hundreds or thousands of individuals. We describe a computationally efficient approach to testing association between SNPs and quantitative phenotypes, which can be applied to whole-genome association scans. In addition to observed genotypes, our approach allows estimation of missing genotypes, resulting in substantial increases in power when genotyping resources are limited. We estimate missing genotypes probabilistically using the Lander-Green or Elston-Stewart algorithms and combine high-resolution SNP genotypes for a subset of individuals in each pedigree with sparser marker data for the remaining individuals. We show that power is increased whenever phenotype information for ungenotyped individuals is included in analyses and that high-density genotyping of just three carefully selected individuals in a nuclear family can recover >90% of the information available if every individual were genotyped, for a fraction of the cost and experimental effort. To aid in study design, we evaluate the power of strategies that genotype different subsets of individuals in each pedigree and make recommendations about which individuals should be genotyped at a high density. To illustrate our method, we performed genomewide association analysis for 27 gene-expression phenotypes in 3-generation families (Centre d'Etude du Polymorphisme Humain pedigrees), in which genotypes for ~860,000 SNPs in 90 grandparents and parents are complemented by genotypes for ~6,700 SNPs in a total of 168 individuals. In addition to increasing the evidence of association at 15 previously identified cis-acting associated alleles, our genotype-inference algorithm allowed us to identify associated alleles at 4 cis-acting loci that were missed when analysis was restricted to individuals with the high-density SNP data. Our genotype-inference algorithm and the proposed association tests are implemented in software that is available for free.     

A Bayesian approach for constructing genetic maps when markers are miscoded

The advent of molecular markers has created opportunities for a better understanding of quantitative inheritance and for developing novel strategies for genetic improvement of agricultural species, using information on quantitative trait loci (QTL). A QTL analysis relies on accurate genetic marker maps. At present, most statistical methods used for map construction ignore the fact that molecular data may be read with error. Often, however, there is ambiguity about some marker genotypes. A Bayesian MCMC approach for inferences about a genetic marker map when random miscoding of genotypes occurs is presented, and simulated and real data sets are analyzed. The results suggest that unless there is strong reason to believe that genotypes are ascertained without error, the proposed approach provides more reliable inference on the genetic map.     

The effect of missing marker genotypes on the accuracy of gene-assisted breeding value estimation: a comparison of methods

In livestock populations, missing genotypes on a large proportion of the animals is a major problem when implementing gene-assisted breeding value estimation for genes with known effect. The objective of this study was to compare different methods to deal with missing genotypes on accuracy of gene-assisted breeding value estimation for identified bi-allelic genes using Monte Carlo simulation. A nested full-sib half-sib structure was simulated with a mixed inheritance model with one bi-allelic quantitative trait loci (QTL) and a polygenic effect due to infinite number of polygenes. The effect of the QTL was included in gene-assisted BLUP either by random regression on predicted gene content, i.e. the number of positive alleles, or including haplotype effects in the model with an inverse IBD matrix to account for identity-by-descent relationships between haplotypes using linkage analysis information (IBD-LA). The inverse IBD matrix was constructed using segregation indicator probabilities obtained from multiple marker iterative peeling. Gene contents for unknown genotypes were predicted using either multiple marker iterative peeling or mixed model methodology. For both methods, gene-assisted breeding value estimation increased accuracies of total estimated breeding value (EBV) with 0% to 22% for genotyped animals in comparison to conventional breeding value estimation. For animals that were not genotyped, the increase in accuracy was much lower (0% to 5%), but still substantial when the heritability was 0.1 and when the QTL explained at least 15% of the genetic variance. Regression on predicted gene content yielded higher accuracies than IBD-LA. Allele substitution effects were, however, overestimated, especially when only sires and males in the last generation were genotyped. For juveniles without phenotypic records and traits measured only on females, the superiority of regression on gene content over IBD-LA was larger than when all animals had phenotypes. Missing gene contents were predicted with higher accuracy using multiple-marker iterative peeling than with using mixed model methodology, but the difference in accuracy of total EBV was negligible and mixed model methodology was computationally much faster than multiple iterative peeling. For large livestock populations it can be concluded that gene-assisted breeding value estimation can be practically best performed by regression on gene contents, using mixed model methodology to predict missing marker genotypes, combining phenotypic information of genotyped and ungenotyped animals in one evaluation. This technique would be, in principle, also feasible for genomic selection. It is expected that genomic selection for ungenotyped animals using predicted single nucleotide polymorphism gene contents might be beneficial especially for low heritable traits.     

Markov chain Monte Carlo for mapping a quantitative trait locus in outbred populations

A Bayesian approach is presented for mapping a quantitative trait locus (QTL) using the 'Fernando and Grossman' multivariate Normal approximation to QTL inheritance. For this model, a Bayesian implementation that includes QTL position is problematic because standard Markov chain Monte Carlo (MCMC) algorithms do not mix, i.e. the QTL position gets stuck in one marker interval. This is because of the dependence of the covariance structure for the QTL effects on the adjacent markers and may be typical of the 'Fernando and Grossman' model. A relatively new MCMC technique, simulated tempering, allows mixing and so makes possible inferences about QTL position based on marginal posterior probabilities. The model was implemented for estimating variance ratios and QTL position using a continuous grid of allowed positions and was applied to simulated data of a standard granddaughter design. The results showed a smooth mixing of QTL position after implementation of the simulated tempering sampler. In this implementation, map distance between QTL and its flanking markers was artificially stretched to reduce the dependence of markers and covariance. The method generalizes easily to more complicated applications and can ultimately contribute to QTL mapping in complex, heterogeneous, human, animal or plant populations.     

An Index of Information Content for Genotype Probabilities Derived from Segregation Analysis

A genotype probability index (GPI) is proposed to indicate the information content of genotype probabilities derived from a segregation analysis. Typically, some individuals are genotyped at a marker locus or a quantitative trait locus, and segregation analysis is used to make genotype inferences about ungenotyped relatives. Genotype probabilities for a two-allele autosomal locus are plotted on a triangular surface. The GPI has a value of zero at the point corresponding to Hardy-Weinberg frequencies, and a value of 100% at the vertices of the triangle. Trigonometric functions are used to help calculate intermediate index values. It is proposed that such an index can be useful to help identify which ungenotyped individuals or loci should be genotyped to maximize the benefit/cost of genotyping operations.     

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.     

Estimation of Effects of Quantitative Trait Loci in Large Complex Pedigrees

A method was derived to estimate effects of quantitative trait loci (QTL) using incomplete genotype information in large outbreeding populations with complex pedigrees. The method accounts for background genes by estimating polygenic effects. The basic equations used are very similar to the usual linear mixed model equations for polygenic models, and segregation analysis was used to estimate the probabilities of the QTL genotypes for each animal. Method R was used to estimate the polygenic heritability simultaneously with the QTL effects. Also, initial allele frequencies were estimated. The method was tested in a simulated data set of 10,000 animals evenly distributed over 10 generations, where 0, 400 or 10,000 animals were genotyped for a candidate gene. In the absence of selection, the bias of the QTL estimates was <2%. Selection biased the estimate of the Aa genotype slightly, when zero animals were genotyped. Estimates of the polygenic heritability were 0.251 and 0.257, in absence and presence of selection, respectively, while the simulated value was 0.25. Although not tested in this study, marker information could be accommodated by adjusting the transmission probabilities of the genotypes from parent to offspring according to the marker information. This renders a QTL mapping study in large multi-generation pedigrees possible.     

Fruit size QTL identification and the prediction of parental QTL genotypes and breeding values in multiple pedigreed populations of sweet cherry

Large fruit size is a critical trait for any new sweet cherry (Prunus avium L.) cultivar, as it is directly related to grower profitability. Therefore, determining the genetic control of fruit size in relevant breeding germplasm is a high priority. The objectives of this study were (1) to determine the number and positions of quantitative trait loci (QTL) for sweet cherry fruit size utilizing data simultaneously from multiple families and their pedigreed ancestors, and (2) to estimate fruit size QTL genotype probabilities and genomic breeding values for the plant materials. The sweet cherry material used was a five-generation pedigree consisting of 23 founders and parents and 424 progeny individuals from four full-sib families, which were phenotyped for fruit size and genotyped with 78 RosCOS single nucleotide polymorphism and 86 simple sequence repeat markers. These data were analyzed by a Bayesian approach implemented in FlexQTL? software. Six QTL were identified: three on linkage group (G) 2 with one each on groups 1, 3, and 6. Of these QTL, the second G2 QTL and the G6 QTL were previously discovered while other QTL were novel. The predicted QTL genotypes show that some QTL were segregating in all families while other QTL were segregating in a subset of the families. The progeny varied for breeding value, with some progeny having higher breeding values than their parents. The results illustrate the use of multiple pedigree-linked families for integrated QTL mapping in an outbred crop to discover novel QTL and predict QTL genotypes and breeding values.     

Mapping of epistatic quantitative trait loci in four-way crosses

Four-way crosses (4WC) involving four different inbred lines often appear in plant and animal commercial breeding programs. Direct mapping of quantitative trait loci (QTL) in these commercial populations is both economical and practical. However, the existing statistical methods for mapping QTL in a 4WC population are built on the single-QTL genetic model. This simple genetic model fails to take into account QTL interactions, which play an important role in the genetic architecture of complex traits. In this paper, therefore, we attempted to develop a statistical method to detect epistatic QTL in 4WC population. Conditional probabilities of QTL genotypes, computed by the multi-point single locus method, were used to sample the genotypes of all putative QTL in the entire genome. The sampled genotypes were used to construct the design matrix for QTL effects. All QTL effects, including main and epistatic effects, were simultaneously estimated by the penalized maximum likelihood method. The proposed method was confirmed by a series of Monte Carlo simulation studies and real data analysis of cotton. The new method will provide novel tools for the genetic dissection of complex traits, construction of QTL networks, and analysis of heterosis.     

A Monte Carlo method for Bayesian analysis of linkage between single markers and quantitative trait loci. I. Methodology

A Bayesian approach to the statistical mapping of Quantitative Trait Loci (QTLs) using single markers was implemented via Markov Chain Monte Carlo (MCMC) algorithms for parameter estimation and hypothesis testing. Parameter estimators were marginal posterior means computed using a Gibbs sampler with data augmentation. Variables sampled included the augmented data (marker-QTL genotypes, polygenic effects), an indicator variable for linkage, and the parameters (allele frequency, QTL substitution effect, recombination rate, polygenic and residual variances). Several MCMC algorithms were derived for computing Bayesian tests of linkage, which consisted of the marginal posterior probability of linkage and the marginal likelihood of the QTL variance associated with the marker.     

Quantifying evidence for candidate gene polymorphisms: Bayesian analysis combining sequence-specific and quantitative trait loci colocation information

We calculate posterior probabilities for candidate genes as a function of genomic location. Posterior probabilities for quantitative trait loci (QTL) presence in a small interval are calculated using a Bayesian model-selection approach based on the Bayesian information criterion (BIC) and used to combine QTL colocation information with sequence-specific evidence, e.g., from differential expression and/or association studies. Our method takes into account uncertainty in estimation of number and locations of QTL and estimated map position. Posterior probabilities for QTL presence were calculated for simulated data with n = 100, 300, and 1200 QTL progeny and compared with interval mapping and composite-interval mapping. Candidate genes that mapped to QTL regions had substantially larger posterior probabilities. Among candidates with a given Bayes factor, those that map near a QTL are more promising for further investigation with association studies and functional testing or for use in marker-aided selection. The BIC is shown to correspond very closely to Bayes factors for linear models with a nearly noninformative Zellner prior for the simulated QTL data with n > or = 100. It is shown how to modify the BIC to use a subjective prior for the QTL effects.     

Multi-QTL mapping for quantitative traits using distorted markers

Marker segregation distortion is a common natural phenomenon. However, relatively little is known about utilizing distorted markers for detecting quantitative trait loci (QTL). Therefore, in this study we proposed a multi-QTL mapping approach that uses distorted markers. First, the information from all markers, including distorted markers, was used to detect segregation distortion loci (SDL). Second, the information from the detected SDL was used to correct the conditional probabilities of the QTL genotypes conditional on marker information, and these corrected probabilities were then incorporated into a multi-QTL mapping methodology. Finally, the proposed approach was validated by both Monte Carlo simulation studies and real data analysis. The results from the simulation studies show that as long as one or two SDL are placed around the simulated QTL, there are no differences between the new method and the ordinary interval mapping method in terms of the power of QTL detection or the estimates of the position and dominant effects of the QTL. However, the power of QTL detection is higher under the dominant genetic model of SDL than under the additive genetic model, and the estimate for the additive effect of QTL using the new method is significantly different from the estimate obtained using ordinary interval mapping. The above results were further confirmed by the detection of QTL for dried soymilk in 222 F_2:4 families in soybean.     

A Bayesian Approach to Detect Quantitative Trait Loci Using Markov Chain Monte Carlo

Markov chain Monte Carlo (MCMC) techniques are applied to simultaneously identify multiple quantitative trait loci (QTL) and the magnitude of their effects. Using a Bayesian approach a multi-locus model is fit to quantitative trait and molecular marker data, instead of fitting one locus at a time. The phenotypic trait is modeled as a linear function of the additive and dominance effects of the unknown QTL genotypes. Inference summaries for the locations of the QTL and their effects are derived from the corresponding marginal posterior densities obtained by integrating the likelihood, rather than by optimizing the joint likelihood surface. This is done using MCMC by treating the unknown QTL genotypes, and any missing marker genotypes, as augmented data and then by including these unknowns in the Markov chain cycle along with the unknown parameters. Parameter estimates are obtained as means of the corresponding marginal posterior densities. High posterior density regions of the marginal densities are obtained as confidence regions. We examine flowering time data from double haploid progeny of Brassica napus to illustrate the proposed method.     

Bayesian analysis of linkage between genetic markers and quantitative trait loci. II. Combining prior knowledge with experimental evidence

Summary A Bayesian method was developed for identifying genetic markers linked to quantitative trait loci (QTL) by analyzing data from daughter or granddaughter designs and single markers or marker pairs. Traditional methods may yield unrealistic results because linkage tests depend on number of markers and QTL gene effects associated with selected markers are overestimated. The Bayesian or posterior probability of linkage combines information from a daughter or granddaughter design with the prior probability of linkage between a marker locus and a QTL. If the posterior probability exceeds a certain quantity, linkage is declared. Upon linkage acceptance, Bayesian estimates of marker-QTL recombination rate and QTL gene effects and frequencies are obtained. The Bayesian estimates of QTL gene effects account for different amounts of information by shrinking information from data toward the mean or mode of a prior exponential distribution of gene effects. Computation of the Bayesian analysis is feasible. Exact results are given for biallelic QTL, and extensions to multiallelic QTL are suggested.     

Bayesian methods for quantitative trait loci mapping based on model selection: approximate analysis using the Bayesian information criterion.

We describe an approximate method for the analysis of quantitative trait loci (QTL) based on model selection from multiple regression models with trait values regressed on marker genotypes, using a modification of the easily calculated Bayesian information criterion to estimate the posterior probability of models with various subsets of markers as variables. The BIC-delta criterion, with the parameter delta increasing the penalty for additional variables in a model, is further modified to incorporate prior information, and missing values are handled by multiple imputation. Marginal probabilities for model sizes are calculated, and the posterior probability of nonzero model size is interpreted as the posterior probability of existence of a QTL linked to one or more markers. The method is demonstrated on analysis of associations between wood density and markers on two linkage groups in Pinus radiata. Selection bias, which is the bias that results from using the same data to both select the variables in a model and estimate the coefficients, is shown to be a problem for commonly used non-Bayesian methods for QTL mapping, which do not average over alternative possible models that are consistent with the data.     

Changes in Variance Components of Flanking Marker Genotypes Under Varying Selection Intensities

Selection is practically ubiquitous during marker-QTL linkage analysis with an experimental population. Thus, it is necessary to investigate the impacts of selection upon linkage analyses in order to obtain unbiased estimates of QTL position and effect. In this article, by exploiting flanking markers through the widely applied half-sib design, we have developed the structures of three variance components, i.e., variance component between marker genotypes, polygenic variance component and recombinant variance component within marker genotypes. Changes in these variance components under varying selection intensities were investigated in this study to formulate the effects of selection on various variance components. Results showed clearly that all variance components presented were quite sensitive to changes in selection intensity. As selection intensity increased, all variance components declined by differing extents in a quadratic fashion. Comparatively speaking, the variance between marker genotypes decreased most drastically, followed by the polygenic variance within marker genotypes and then the recombinant variance within marker genotypes, which suggested a decrease of power for QTL linkage analysis. Therefore, steps should be taken to avoid as much as possible the presence of selection in real populations, so as to further eliminate the negative effects of selection on QTL linkage analysis.