Optimal haplotype structure for linkage disequilibrium-based fine mapping of quantitative trait loci using identity by descent

A linkage disequilibrium-based method for fine mapping quantitative trait loci (QTL) has been described that uses similarity between individuals' marker haplotypes to determine if QTL alleles are identical by descent (IBD) to model covariances among individuals' QTL alleles for a mixed linear model. Mapping accuracy with this method was found to be sensitive to the number of linked markers that was included in the haplotype when fitting the model at a putative position of the QTL. The objective of this study was to determine the optimal haplotype structure for this IBD-based method for fine mapping a QTL in a previously identified QTL region. Haplotypes consisting of 1, 2, 4, 6, or all 10 available markers were fit as a "sliding window" across the QTL region under ideal and nonideal simulated population conditions. It was found that using haplotypes of 4 or 6 markers as a sliding "window" resulted in the greatest mapping accuracy under nearly all conditions, although the true IBD state at a putative QTL position was most accurately predicted by IBD probabilities obtained using all markers. Using 4 or 6 markers resulted in greater discrimination of IBD probabilities between positions while maintaining sufficient accuracy of IBD probabilities to detect the QTL. Fitting IBD probabilities on the basis of a single marker resulted in the worst mapping accuracy under all conditions because it resulted in poor accuracy of IBD probabilities. In conclusion, for fine mapping using IBD methods, marker information must be used in a manner that results in sensitivity of IBD probabilities to the putative position of the QTL while maintaining sufficient accuracy of IBD probabilities to detect the QTL. Contrary to expectation, use of haplotypes of 4-6 markers to derive IBD probabilities, rather than all available markers, best fits these criteria. Thus for populations similar to those simulated here, optimal mapping accuracy for this IBD-based fine-mapping method is obtained with a haplotype structure including a subset of all available markers.     

A method for computing identity by descent probabilities and quantitative trait loci mapping with dominant (AFLP) markers

Amplified fragment length polymorphisms (AFLPs) are a widely used marker system: the technique is very cost-effective, easy and rapid, and reproducibly generates hundreds of markers. Unfortunately, AFLP alleles are typically scored as the presence or absence of a band and, thus, heterozygous and dominant homozygous genotypes cannot be distinguished. This results in a significant loss of information, especially as regards mapping of quantitative trait loci (QTLs). We present a Monte Carlo Markov Chain method that allows us to compute the identity by descent probabilities (IBD) in a general pedigree whose individuals have been typed for dominant markers. The method allows us to include the information provided by the fluorescent band intensities of the markers, the rationale being that homozygous individuals have on average higher band intensities than heterozygous individuals, as well as information from linked markers in each individual and its relatives. Once IBD probabilities are obtained, they can be combined into the QTL mapping strategy of choice. We illustrate the method with two simulated populations: an outbred population consisting of full sib families, and an F2 cross between inbred lines. Two marker spacings were considered, 5 or 20 cM, in the outbred population. There was almost no difference, for the practical purpose of QTL estimation, between AFLPs and biallelic codominant markers when the band density is taken into account, especially at the 5 cM spacing. The performance of AFLPs every 5 cM was also comparable to that of highly polymorphic markers (microsatellites) spaced every 20 cM. In economic terms, QTL mapping with a dense map of AFLPs is clearly better than microsatellite QTL mapping and little is lost in terms of accuracy of position. Nevertheless, at low marker densities, AFLPs or other biallelic markers result in very inaccurate estimates of QTL position.     

Bayesian mapping of quantitative trait loci under the identity-by-descent-based variance component model

Variance component analysis of quantitative trait loci (QTL) is an important strategy of genetic mapping for complex traits in humans. The method is robust because it can handle an arbitrary number of alleles with arbitrary modes of gene actions. The variance component method is usually implemented using the proportion of alleles with identity-by-descent (IBD) shared by relatives. As a result, information about marker linkage phases in the parents is not required. The method has been studied extensively under either the maximum-likelihood framework or the sib-pair regression paradigm. However, virtually all investigations are limited to normally distributed traits under a single QTL model. In this study, we develop a Bayes method to map multiple QTL. We also extend the Bayesian mapping procedure to identify QTL responsible for the variation of complex binary diseases in humans under a threshold model. The method can also treat the number of QTL as a parameter and infer its posterior distribution. We use the reversible jump Markov chain Monte Carlo method to infer the posterior distributions of parameters of interest. The Bayesian mapping procedure ends with an estimation of the joint posterior distribution of the number of QTL and the locations and variances of the identified QTL. Utilities of the method are demonstrated using a simulated population consisting of multiple full-sib families.     

Multiple interval mapping for quantitative trait loci.

A new statistical method for mapping quantitative trait loci (QTL), called multiple interval mapping (MIM), is presented. It uses multiple marker intervals simultaneously to fit multiple putative QTL directly in the model for mapping QTL. The MIM model is based on Cockerham's model for interpreting genetic parameters and the method of maximum likelihood for estimating genetic parameters. With the MIM approach, the precision and power of QTL mapping could be improved. Also, epistasis between QTL, genotypic values of individuals, and heritabilities of quantitative traits can be readily estimated and analyzed. Using the MIM model, a stepwise selection procedure with likelihood ratio test statistic as a criterion is proposed to identify QTL. This MIM method was applied to a mapping data set of radiata pine on three traits: brown cone number, tree diameter, and branch quality scores. Based on the MIM result, seven, six, and five QTL were detected for the three traits, respectively. The detected QTL individually contributed from approximately 1 to 27% of the total genetic variation. Significant epistasis between four pairs of QTL in two traits was detected, and the four pairs of QTL contributed approximately 10.38 and 14.14% of the total genetic variation. The asymptotic variances of QTL positions and effects were also provided to construct the confidence intervals. The estimated heritabilities were 0.5606, 0.5226, and 0. 3630 for the three traits, respectively. With the estimated QTL effects and positions, the best strategy of marker-assisted selection for trait improvement for a specific purpose and requirement can be explored. The MIM FORTRAN program is available on the worldwide web (http://www.stat.sinica.edu.tw/chkao/).     

Generalized linear model for interval mapping of quantitative trait loci

We developed a generalized linear model of QTL mapping for discrete traits in line crossing experiments. Parameter estimation was achieved using two different algorithms, a mixture model-based EM (expectation–maximization) algorithm and a GEE (generalized estimating equation) algorithm under a heterogeneous residual variance model. The methods were developed using ordinal data, binary data, binomial data and Poisson data as examples. Applications of the methods to simulated as well as real data are presented. The two different algorithms were compared in the data analyses. In most situations, the two algorithms were indistinguishable, but when large QTL are located in large marker intervals, the mixture model-based EM algorithm can fail to converge to the correct solutions. Both algorithms were coded in C++ and interfaced with SAS as a user-defined SAS procedure called PROC QTL.     

A random model approach to mapping quantitative trait loci for complex binary traits in outbred populations.

Mapping quantitative trait loci (QTL) for complex binary traits is more challenging than for normally distributed traits due to the nonlinear relationship between the observed phenotype and unobservable genetic effects, especially when the mapping population contains multiple outbred families. Because the number of alleles of a QTL depends on the number of founders in an outbred population, it is more appropriate to treat the effect of each allele as a random variable so that a single variance rather than individual allelic effects is estimated and tested. Such a method is called the random model approach. In this study, we develop the random model approach of QTL mapping for binary traits in outbred populations. An EM-algorithm with a Fisher-scoring algorithm embedded in each E-step is adopted here to estimate the genetic variances. A simple Monte Carlo integration technique is used here to calculate the likelihood-ratio test statistic. For the first time we show that QTL of complex binary traits in an outbred population can be scanned along a chromosome for their positions, estimated for their explained variances, and tested for their statistical significance. Application of the method is illustrated using a set of simulated data.     

Using the mixed model for interval mapping of quantitative trait loci in outbred line crosses

Interval mapping by simple regression is a powerful method for the detection of quantitative trait loci (QTLs) in line crosses such as F2 populations. Due to the ease of computation of the regression approach, relatively complex models with multiple fixed effects, interactions between QTLs or between QTLs and fixed effects can easily be accommodated. However, polygenic effects, which are not targeted in QTL analysis, cannot be treated as random effects in a least squares analysis. In a cross between true inbred lines this is of no consequence, as the polygenic effect contributes just to the residual variance. In a cross between outbred lines, however, if a trait has high polygenic heritability, the additive polygenic effect has a large influence on variation in the population. Here we extend the fixed model for the regression interval mapping method to a mixed model using an animal model. This makes it possible to use not only the observations from progeny (e.g. F2), but also those from the parents (F1) to evaluate QTLs and polygenic effects. We show how the animal model using parental observations can be applied to an outbred cross and so increase the power and accuracy of QTL analysis. Three estimation methods, i.e. regression and an animal model either with or without parental observations, are applied to simulated data. The animal model using parental observations is shown to have advantages in estimating QTL position and additive genotypic value, especially when the polygenic heritability is large and the number of progeny per parent is small.     

Efficient algorithms for quantitative trait loci mapping problems.

Rapid advances in molecular genetics push the need for efficient data analysis. Advanced algorithms are necessary for extracting all possible information from large experimental data sets. We present a general linear algebra framework for quantitative trait loci (QTL) mapping, using both linear regression and maximum likelihood estimation. The formulation simplifies future comparisons between and theoretical analyses of the methods. We show how the common structure of QTL analysis models can be used to improve the kernel algorithms, drastically reducing the computational effort while retaining the original analysis results. We have evaluated our new algorithms on data sets originating from two large F(2) populations of domestic animals. Using an updating approach, we show that 1-3 orders of magnitude reduction in computational demand can be achieved for matrix factorizations. For interval-mapping/composite-interval-mapping settings using a maximum likelihood model, we also show how to use the original EM algorithm instead of the ECM approximation, significantly improving the convergence and further reducing the computational time. The algorithmic improvements makes it feasible to perform analyses which have previously been deemed impractical or even impossible. For example, using the new algorithms, it is reasonable to perform permutation testing using exhaustive search on populations of 200 individuals using an epistatic two-QTL model.     

Bayesian LASSO for quantitative trait loci mapping

The mapping of quantitative trait loci (QTL) is to identify molecular markers or genomic loci that influence the variation of complex traits. The problem is complicated by the facts that QTL data usually contain a large number of markers across the entire genome and most of them have little or no effect on the phenotype. In this article, we propose several Bayesian hierarchical models for mapping multiple QTL that simultaneously fit and estimate all possible genetic effects associated with all markers. The proposed models use prior distributions for the genetic effects that are scale mixtures of normal distributions with mean zero and variances distributed to give each effect a high probability of being near zero. We consider two types of priors for the variances, exponential and scaled inverse-chi(2) distributions, which result in a Bayesian version of the popular least absolute shrinkage and selection operator (LASSO) model and the well-known Student's t model, respectively. Unlike most applications where fixed values are preset for hyperparameters in the priors, we treat all hyperparameters as unknowns and estimate them along with other parameters. Markov chain Monte Carlo (MCMC) algorithms are developed to simulate the parameters from the posteriors. The methods are illustrated using well-known barley data.     

A general mixture model for mapping quantitative trait loci by using molecular markers

Summary In a segregating population a quantitative trait may be considered to follow a mixture of (normal) distributions, the mixing proportions being based on Mendelian segregation rules. A general and flexible mixture model is proposed for mapping quantitative trait loci (QTLs) by using molecular markers. A method is discribed to fit the model to data. The model makes it possible to (1) analyse non-normally distributed traits such as lifetimes, counts or percentages in addition to normally distributed traits, (2) reduce environmental variation by taking into account the effects of experimental design factors and interaction between genotype and environment, (3) reduce genotypic variation by taking into account the effects of two or more QTLs simultaneously, (4) carry out a (combined) analysis of different population types, (5) estimate recombination frequencies between markers or use known marker distances, (6) cope with missing marker observations, (7) use markers as covariables in detection and mapping of QTLs, and finally to (8) implement the mapping in standard statistical packages.     

Lineage-specific mapping of quantitative trait loci

We present an approach for quantitative trait locus (QTL) mapping, termed as ‘lineage-specific QTL mapping'', for inferring allelic changes of QTL evolution along with branches in a phylogeny. We describe and analyze the simplest case: by adding a third taxon into the normal procedure of QTL mapping between pairs of taxa, such inferences can be made along lineages to a presumed common ancestor. Although comparisons of QTL maps among species can identify homology of QTLs by apparent co-location, lineage-specific mapping of QTL can classify homology into (1) orthology (shared origin of QTL) versus (2) paralogy (independent origin of QTL within resolution of map distance). In this light, we present a graphical method that identifies six modes of QTL evolution in a three taxon comparison. We then apply our model to map lineage-specific QTLs for inbreeding among three taxa of yellow monkey-flower: Mimulus guttatus and two inbreeders M. platycalyx and M. micranthus, but critically assuming outcrossing was the ancestral state. The two most common modes of homology across traits were orthologous (shared ancestry of mutation for QTL alleles). The outbreeder M. guttatus had the fewest lineage-specific QTL, in accordance with the presumed ancestry of outbreeding. Extensions of lineage-specific QTL mapping to other types of data and crosses, and to inference of ancestral QTL state, are discussed.     

Robust Bayesian mapping of quantitative trait loci using Student-t distribution for residual

In most quantitative trait loci (QTL) mapping studies, phenotypes are assumed to follow normal distributions. Deviations from this assumption may affect the accuracy of QTL detection, leading to detection of false positive QTL. To improve the robustness of QTL mapping methods, we replace the normal distribution assumption for residuals in a multiple QTL model with a Student-t distribution that is able to accommodate residual outliers. A Robust Bayesian mapping strategy is proposed on the basis of the Bayesian shrinkage analysis for QTL effects. The simulations show that Robust Bayesian mapping approach can substantially increase the power of QTL detection when the normality assumption does not hold and applying it to data already normally distributed does not influence the result. The proposed QTL mapping method is applied to mapping QTL for the traits associated with physics–chemical characters and quality in rice. Similarly to the simulation study in the real data case the robust approach was able to detect additional QTLs when compared to the traditional approach. The program to implement the method is available on request from the first or the corresponding author. Xin Wang and Zhongze Piao contributed equally to this study.     

Bayesian mapping of quantitative trait loci under complicated mating designs

Quantitative trait loci (QTL) are easily studied in a biallelic system. Such a system requires the cross of two inbred lines presumably fixed for alternative alleles of the QTL. However, development of inbred lines can be time consuming and cost ineffective for species with long generation intervals and severe inbreeding depression. In addition, restriction of the investigation to a biallelic system can sometimes be misleading because many potentially important allelic interactions do not have a chance to express and thus fail to be detected. A complicated mating design involving multiple alleles mimics the actual breeding system. However, it is difficult to develop the statistical model and algorithm using the classical maximum-likelihood method. In this study, we investigate the application of a Bayesian method implemented via the Markov chain Monte Carlo (MCMC) algorithm to QTL mapping under arbitrarily complicated mating designs. We develop the method under a mixed-model framework where the genetic values of founder alleles are treated as random and the nongenetic effects are treated as fixed. With the MCMC algorithm, we first draw the gene flows from the founders to the descendants for each QTL and then draw samples of the genetic parameters. Finally, we are able to simultaneously infer the posterior distribution of the number, the additive and dominance variances, and the chromosomal locations of all identified QTL.     

Linkage disequilibrium mapping of quantitative trait loci under truncation selection

As a dense map of single nucleotide polymorphism (SNP) markers are available, population-based linkage disequilibrium (LD) mapping or association study is becoming one of the major tools for identifying quantitative trait loci (QTL) and for fine gene mapping. However, in many cases, LD between the marker and trait locus is not very strong. Approaches that maximize the potential of detecting LD will be essential for the success of LD mapping of QTL. In this paper, we propose two strategies for increasing the probability of detecting LD: (1) phenotypic selection and (2) haplotype LD mapping. To provide the foundations for LD mapping of QTL under selection, we develop analytic tools for assessing the impact of phenotypic selection on allele and haplotype frequencies, and LD under three trait models: single trait locus, two unlinked trait loci, and two linked trait loci with or without epistasis. In addition to a traditional chi(2) test, which compares the difference in allele or haplotype frequencies in the selected sample and population sample, we present multiple regression methods for LD mapping of QTL, and investigate which methods are effective in employing phenotypic selection for QTL mapping. We also develop a statistical framework for investigating and comparing the power of the single marker and multilocus haplotype test for LD mapping of QTL. Finally, the proposed methods are applied to mapping QTL influencing variation in systolic blood pressure in an isolated Chinese population.     

Least squares interval mapping of quantitative trait loci under the infinitesimal genetic model in outbred populations.

Genetic marker and phenotypic data for a quantitative trait were simulated on 20 paternal half-sib families with 100 progeny to investigate properties of within-family-regression interval mapping of a postulated single quantitative trait locus (QTL) in a marker interval under the infinitesimal genetic model, which has been the basis of the application of quantitative genetics to genetic improvement programs, and to investigate use of the infinitesimal model as null hypothesis in testing for presence of a major QTL. Genetic effects on the marked chromosome were generated based on a major gene model, which simulated a central biallelic QTL, or based on 101 biallelic QTL of equal effect, which approximated the infinitesimal model. The marked chromosome contained 0, 3.3%, 13.3%, or 33.3% of genetic variance and heritability was 0.25 or 0.70. Under the polygenic model with 3.3% of genetic variance on the marked chromosome, which corresponds to the infinitesimal model for the bovine, significant QTL effects were found for individual families. Correlations between estimates of QTL effects and true chromosome substitution effects were 0.29 and 0.47 for heritabilities of 0.25 and 0.70 but up to 0.85 with 33.3% of polygenic variance on the marked chromosome. These results illustrate the potential of marker-assisted selection even under the infinitesimal genetic model. Power of tests for presence of QTL was substantially reduced when the polygenic model with 3.3% of genetic variance on the chromosome was used as a null hypothesis. The ability to determine whether genetic variance on a chromosome was contributed by a single QTL of major effect or a large number of QTL with minor effects, corresponding to the infinitesimal model, was limited.     

Marker pair selection for mapping quantitative trait loci

Mapping of quantitative trait loci (QTL) for backcross and F(2) populations may be set up as a multiple linear regression problem, where marker types are the regressor variables. It has been shown previously that flanking markers absorb all information on isolated QTL. Therefore, selection of pairs of markers flanking QTL is useful as a direct approach to QTL detection. Alternatively, selected pairs of flanking markers can be used as cofactors in composite interval mapping (CIM). Overfitting is a serious problem, especially if the number of regressor variables is large. We suggest a procedure denoted as marker pair selection (MPS) that uses model selection criteria for multiple linear regression. Markers enter the model in pairs, which reduces the number of models to be considered, thus alleviating the problem of overfitting and increasing the chances of detecting QTL. MPS entails an exhaustive search per chromosome to maximize the chance of finding the best-fitting models. A simulation study is conducted to study the merits of different model selection criteria for MPS. On the basis of our results, we recommend the Schwarz Bayesian criterion (SBC) for use in practice.     

Sib mating designs for mapping quantitative trait loci

The power to separate the variance of a quantitative trait locus (QTL) from the polygenic variance is determined by the variability of genes identical by descent (IBD) at the QTL. This variability may increase with inbreeding. Selfing, the most extreme form of inbreeding, increases the variability of the IBD value shared by siblings, and thus has a higher efficiency for QTL mapping than random mating. In self-incompatible organisms, sib mating is the closest form of inbreeding. Similar to selfing, sib mating may also increase the power of QTL detection relative to random mating. In this study, we develop an IBD-based method under sib mating designs for QTL mapping. The efficiency of sib mating is then compared with random mating. Monte Carlo simulations show that sib mating designs notably increase the power for QTL detection. When power is intermediate, the power to detect a QTL using full-sib mating is, on average, 7% higher than under random mating. In addition, the IBD-based method proposed in this paper can be used to combine data from multiple families. As a result, the estimated QTL parameters can be applied to a wide statistical inference space relating to the entire reference population. This revised version was published online in July 2006 with corrections to the Cover Date.     

Bayesian quantitative trait loci mapping for multiple traits

Most quantitative trait loci (QTL) mapping experiments typically collect phenotypic data on multiple correlated complex traits. However, there is a lack of a comprehensive genomewide mapping strategy for correlated traits in the literature. We develop Bayesian multiple-QTL mapping methods for correlated continuous traits using two multivariate models: one that assumes the same genetic model for all traits, the traditional multivariate model, and the other known as the seemingly unrelated regression (SUR) model that allows different genetic models for different traits. We develop computationally efficient Markov chain Monte Carlo (MCMC) algorithms for performing joint analysis. We conduct extensive simulation studies to assess the performance of the proposed methods and to compare with the conventional single-trait model. Our methods have been implemented in the freely available package R/qtlbim (http://www.qtlbim.org), which greatly facilitates the general usage of the Bayesian methodology for unraveling the genetic architecture of complex traits.     

A random model for mapping imprinted quantitative trait loci in a structured pedigree: an implication for mapping canine hip dysplasia

Genetic imprinting may have played a more notable role in shaping embryonic development of plants, animals, and humans than previously appreciated. Quantitative trait loci that are imprinted (iQTL) exert monoallelic effects, depending on the parent of origin, which is an exception to the laws of Mendelian genetics. In this article, we present a modified random effect-based mapping model to use in a genome-wide scan for the distribution of iQTL that contribute to genetic variance for a complex trait in a structured pedigree. This model, implemented with the maximum likelihood method, capitalizes on a network of relatedness for maternally and paternally derived alleles through identical-by-descent sharing, thus allowing for the discrimination of the genetic variances due to alleles derived from maternal and paternal parents. The model was employed to map iQTL responsible for canine hip dysplasia in a multihierarchical canine pedigree, founded with seven greyhounds and six Labrador retrievers. Of eight significant QTL detected, three, located on CFA1, CFA8, and CF28, were found to trigger significant parent-of-origin effects on the age of femoral capital ossification measured at the left and right hips of a canine. The detected iQTL provide important candidate regions for fine-mapping of imprinted genes and for studying their structure and function in the control of complex traits.     

Genome-wide evaluation for quantitative trait loci under the variance component model

The identity-by-descent (IBD) based variance component analysis is an important method for mapping quantitative trait loci (QTL) in outbred populations. The interval-mapping approach and various modified versions of it may have limited use in evaluating the genetic variances of the entire genome because they require evaluation of multiple models and model selection. In this study, we developed a multiple variance component model for genome-wide evaluation using both the maximum likelihood (ML) method and the MCMC implemented Bayesian method. We placed one QTL in every few cM on the entire genome and estimated the QTL variances and positions simultaneously in a single model. Genomic regions that have no QTL usually showed no evidence of QTL while regions with large QTL always showed strong evidence of QTL. While the Bayesian method produced the optimal result, the ML method is computationally more efficient than the Bayesian method. Simulation experiments were conducted to demonstrate the efficacy of the new methods.