Simultaneous fine mapping of multiple closely linked quantitative trait Loci using combined linkage disequilibrium and linkage with a general pedigree

Within a small region (e.g., <10 cM), there can be multiple quantitative trait loci (QTL) underlying phenotypes of a trait. Simultaneous fine mapping of closely linked QTL needs an efficient tool to remove confounded shade effects among QTL within such a small region. We propose a variance component method using combined linkage disequilibrium (LD) and linkage information and a reversible jump Markov chain Monte Carlo (MCMC) sampling for model selection. QTL identity-by-descent (IBD) coefficients between individuals are estimated by a hybrid MCMC combining the random walk and the meiosis Gibbs sampler. These coefficients are used in a mixed linear model and an empirical Bayesian procedure combines residual maximum likelihood (REML) to estimate QTL effects and a reversible jump MCMC that samples the number of QTL and the posterior QTL intensities across the tested region. Note that two MCMC processes are used, i.e., an (internal) MCMC for IBD estimation and an (external) MCMC for model selection. In a simulation study, the use of the multiple-QTL model clearly removes the shade effects between three closely linked QTL located at 1.125, 3.875, and 7.875 cM across the region of 10 cM, using 40 markers at 0.25-cM intervals. It is shown that the use of combined LD and linkage information gives much more useful information compared to using linkage information alone for both single- and multiple-QTL analyses. When using a lower marker density (11 markers at 1-cM intervals), the signal of the second QTL can disappear. Extreme values of past effective size (resulting in extreme levels of LD) decrease the mapping accuracy.     

Linkage analysis of quantitative trait loci in multiple line crosses

Simple line crosses, for example, backcross and F₂, are commonly used in mapping quantitative trait loci (QTL). However, these simple crosses are rarely used alone in commercial plant breeding; rather, crosses involving multiple inbred lines or several simple crosses but connected by shared inbred lines may be common in plant breeding. Mapping QTL using crosses of multiple lines is more relevant to plant breeding. Unfortunately, current statistical methods and computer programs of QTL mapping are all designed for simple line crosses or multiple line crosses but under a regular mating system. It is not straightforward to extend the existing methods to handle multiple line crosses under irregular and complicated mating designs. The major hurdle comes from irregular inbreeding, multiple generations, and multiple alleles. In this study, we develop a Bayesian method implemented via the Markov chain Monte Carlo (MCMC) algorithm for mapping QTL using complicated multiple line crosses. With the MCMC algorithm, we are able to draw a complete path of the gene flow from founder alleles to their descendents via a recursive process. This has greatly simplified the problem caused by irregular mating and inbreeding in the mapping population. Adopting the reversible jump MCMC algorithm, we are able to simultaneously search for multiple QTL along the genome. We can even infer the posterior distribution of the number of QTL, one of the most important parameters in QTL study. Application of the new MCMC based QTL mapping procedure is demonstrated using two different mating designs. Design I involves two inbred lines and their derived F₁, F₂, and BC populations. Design II is a half-diallel cross involving three inbred lines. The two designs appear different, but can be handled with the same robust computer program.     

Mapping-Linked Quantitative Trait Loci Using Bayesian Analysis and Markov Chain Monte Carlo Algorithms

A Bayesian method for mapping linked quantitative trait loci (QTL) using multiple linked genetic markers is presented. Parameter estimation and hypothesis testing was implemented via Markov chain Monte Carlo (MCMC) algorithms. Parameters included were allele frequencies and substitution effects for two biallelic QTL, map positions of the QTL and markers, allele frequencies of the markers, and polygenic and residual variances. Missing data were polygenic effects and multi-locus marker-QTL genotypes. Three different MCMC schemes for testing the presence of a single or two linked QTL on the chromosome were compared. The first approach includes a model indicator variable representing two unlinked QTL affecting the trait, one linked and one unlinked QTL, or both QTL linked with the markers. The second approach incorporates an indicator variable for each QTL into the model for phenotype, allowing or not allowing for a substitution effect of a QTL on phenotype, and the third approach is based on model determination by reversible jump MCMC. Methods were evaluated empirically by analyzing simulated granddaughter designs. All methods identified correctly a second, linked QTL and did not reject the one-QTL model when there was only a single QTL and no additional or an unlinked QTL.     

Bayes factors for detection of Quantitative Trait Loci

A fundamental issue in quantitative trait locus (QTL) mapping is to determine the plausibility of the presence of a QTL at a given genome location. Bayesian analysis offers an attractive way of testing alternative models (here, QTL vs. no-QTL) via the Bayes factor. There have been several numerical approaches to computing the Bayes factor, mostly based on Markov Chain Monte Carlo (MCMC), but these strategies are subject to numerical or stability problems. We propose a simple and stable approach to calculating the Bayes factor between nested models. The procedure is based on a reparameterization of a variance component model in terms of intra-class correlation. The Bayes factor can then be easily calculated from the output of a MCMC scheme by averaging conditional densities at the null intra-class correlation. We studied the performance of the method using simulation. We applied this approach to QTL analysis in an outbred population. We also compared it with the Likelihood Ratio Test and we analyzed its stability. Simulation results were very similar to the simulated parameters. The posterior probability of the QTL model increases as the QTL effect does. The location of the QTL was also correctly obtained. The use of meta-analysis is suggested from the properties of the Bayes factor.     

A unified Markov chain Monte Carlo framework for mapping multiple quantitative trait loci

In this article, a unified Markov chain Monte Carlo (MCMC) framework is proposed to identify multiple quantitative trait loci (QTL) for complex traits in experimental designs, based on a composite space representation of the problem that has fixed dimension. The proposed unified approach includes the existing Bayesian QTL mapping methods using reversible jump MCMC algorithm as special cases. We also show that a variety of Bayesian variable selection methods using Gibbs sampling can be applied to the composite model space for mapping multiple QTL. The unified framework not only results in some new algorithms, but also gives useful insight into some of the important factors governing the performance of Gibbs sampling and reversible jump for mapping multiple QTL. Finally, we develop strategies to improve the performance of MCMC algorithms.     

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.     

Study on mapping Quantitative Trait Loci for animal complex binary traits using Bayesian-Markov chain Monte Carlo approach

It is a challenging issue to map Quantitative Trait Loci (QTL) underlying complex discrete traits, which usually show discontinuous distribution and less information, using conventional statistical methods. Bayesian-Markov chain Monte Carlo (Bayesian-MCMC) approach is the key procedure in mapping QTL for complex binary traits, which provides a complete posterior distribution for QTL parameters using all prior information. As a consequence, Bayesian estimates of all interested variables can be obtained straightforwardly basing on their posterior samples simulated by the MCMC algorithm. In our study, utilities of Bayesian-MCMC are demonstrated using simulated several animal outbred full-sib families with different family structures for a complex binary trait underlied by both a QTL and polygene. Under the Identity-by-Descent-Based variance component random model, three samplers basing on MCMC, including Gibbs sampling, Metropolis algorithm and reversible jump MCMC, were implemented to generate the joint posterior distribution of all unknowns so that the QTL parameters were obtained by Bayesian statistical inferring. The results showed that Bayesian-MCMC approach could work well and robust under different family structures and QTL effects. As family size increases and the number of family decreases, the accuracy of the parameter estimates will be improved. When the true QTL has a small effect, using outbred population experiment design with large family size is the optimal mapping strategy.     

Study on mapping Quantitative Trait Loci for animal complex binary traits using Bayesian-Markov chain Monte Carlo approach

It is a challenging issue to map Quantitative Trait Loci (QTL) underlying complex discrete traits,which usually show discontinuous distribution and less information,using conventional statisti-cal methods. Bayesian-Markov chain Monte Carlo (Bayesian-MCMC) approach is the key procedure in mapping QTL for complex binary traits,which provides a complete posterior distribution for QTL parameters using all prior information. As a consequence,Bayesian estimates of all interested vari-ables can be obtained straightforwardly basing on their posterior samples simulated by the MCMC algorithm. In our study,utilities of Bayesian-MCMC are demonstrated using simulated several ani-mal outbred full-sib families with different family structures for a complex binary trait underlied by both a QTL and polygene. Under the Identity-by-Descent-Based variance component random model,three samplers basing on MCMC,including Gibbs sampling,Metropolis algorithm and reversible jump MCMC,were implemented to generate the joint posterior distribution of all unknowns so that the QTL parameters were obtained by Bayesian statistical inferring. The results showed that Bayesian-MCMC approach could work well and robust under different family structures and QTL effects. As family size increases and the number of family decreases,the accuracy of the parameter estimates will be im-proved. When the true QTL has a small effect,using outbred population experiment design with large family size is the optimal mapping strategy.     

On the generalized poisson regression mixture model for mapping quantitative trait loci with count data

Statistical methods for mapping quantitative trait loci (QTL) have been extensively studied. While most existing methods assume normal distribution of the phenotype, the normality assumption could be easily violated when phenotypes are measured in counts. One natural choice to deal with count traits is to apply the classical Poisson regression model. However, conditional on covariates, the Poisson assumption of mean-variance equality may not be valid when data are potentially under- or overdispersed. In this article, we propose an interval-mapping approach for phenotypes measured in counts. We model the effects of QTL through a generalized Poisson regression model and develop efficient likelihood-based inference procedures. This approach, implemented with the EM algorithm, allows for a genomewide scan for the existence of QTL throughout the entire genome. The performance of the proposed method is evaluated through extensive simulation studies along with comparisons with existing approaches such as the Poisson regression and the generalized estimating equation approach. An application to a rice tiller number data set is given. Our approach provides a standard procedure for mapping QTL involved in the genetic control of complex traits measured in counts.     

Mapping Quantitative Trait Loci for Complex Binary Diseases Using Line Crosses

A composite interval gene mapping procedure for complex binary disease traits is proposed in this paper. The binary trait of interest is assumed to be controlled by an underlying liability that is normally distributed. The liability is treated as a typical quantitative character and thus described by the usual quantitative genetics model. Translation from the liability into a binary (disease) phenotype is through the physiological threshold model. Logistic regression analysis is employed to estimate the effects and locations of putative quantitative trait loci (our terminology for a single quantitative trait locus is QTL while multiple loci are referred to as QTLs). Simulation studies show that properties of this mapping procedure mimic those of the composite interval mapping for normally distributed data. Potential utilization of the QTL mapping procedure for resolving alternative genetic models (e.g., single- or two-trait-locus model) is discussed.     

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.     

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 statistical model for testing the pleiotropic control of phenotypic plasticity for a count trait

The differences of a phenotypic trait produced by a genotype in response to changes in the environment are referred to as phenotypic plasticity. Despite its importance in the maintenance of genetic diversity via genotype-by-environment interactions, little is known about the detailed genetic architecture of this phenomenon, thus limiting our ability to predict the pattern and process of microevolutionary responses to changing environments. In this article, we develop a statistical model for mapping quantitative trait loci (QTL) that control the phenotypic plasticity of a complex trait through differentiated expressions of pleiotropic QTL in different environments. In particular, our model focuses on count traits that represent an important aspect of biological systems, controlled by a network of multiple genes and environmental factors. The model was derived within a multivariate mixture model framework in which QTL genotype-specific mixture components are modeled by a multivariate Poisson distribution for a count trait expressed in multiple clonal replicates. A two-stage hierarchic EM algorithm is implemented to obtain the maximum-likelihood estimates of the Poisson parameters that specify environment-specific genetic effects of a QTL and residual errors. By approximating the number of sylleptic branches on the main stems of poplar hybrids by a Poisson distribution, the new model was applied to map QTL that contribute to the phenotypic plasticity of a count trait. The statistical behavior of the model and its utilization were investigated through simulation studies that mimic the poplar example used. This model will provide insights into how genomes and environments interact to determine the phenotypes of complex count traits.     

Zero-inflated generalized Poisson regression mixture model for mapping quantitative trait loci underlying count trait with many zeros

Phenotypes measured in counts are commonly observed in nature. Statistical methods for mapping quantitative trait loci (QTL) underlying count traits are documented in the literature. The majority of them assume that the count phenotype follows a Poisson distribution with appropriate techniques being applied to handle data dispersion. When a count trait has a genetic basis, “naturally occurring” zero status also reflects the underlying gene effects. Simply ignoring or miss-handling the zero data may lead to wrong QTL inference. In this article, we propose an interval mapping approach for mapping QTL underlying count phenotypes containing many zeros. The effects of QTLs on the zero-inflated count trait are modelled through the zero-inflated generalized Poisson regression mixture model, which can handle the zero inflation and Poisson dispersion in the same distribution. We implement the approach using the EM algorithm with the Newton-Raphson algorithm embedded in the M-step, and provide a genome-wide scan for testing and estimating the QTL effects. The performance of the proposed method is evaluated through extensive simulation studies. Extensions to composite and multiple interval mapping are discussed. The utility of the developed approach is illustrated through a mouse F₂ intercross data set. Significant QTLs are detected to control mouse cholesterol gallstone formation.     

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.     

Mixed model approaches for the identification of QTLs within a maize hybrid breeding program

Two outlines for mixed model based approaches to quantitative trait locus (QTL) mapping in existing maize hybrid selection programs are presented: a restricted maximum likelihood (REML) and a Bayesian Markov Chain Monte Carlo (MCMC) approach. The methods use the in-silico-mapping procedure developed by Parisseaux and Bernardo (2004) as a starting point. The original single-point approach is extended to a multi-point approach that facilitates interval mapping procedures. For computational and conceptual reasons, we partition the full set of relationships from founders to parents of hybrids into two types of relations by defining so-called intermediate founders. QTL effects are defined in terms of those intermediate founders. Marker based identity by descent relationships between intermediate founders define structuring matrices for the QTL effects that change along the genome. The dimension of the vector of QTL effects is reduced by the fact that there are fewer intermediate founders than parents. Furthermore, additional reduction in the number of QTL effects follows from the identification of founder groups by various algorithms. As a result, we obtain a powerful mixed model based statistical framework to identify QTLs in genetic backgrounds relevant to the elite germplasm of a commercial breeding program. The identification of such QTLs will provide the foundation for effective marker assisted and genome wide selection strategies. Analyses of an example data set show that QTLs are primarily identified in different heterotic groups and point to complementation of additive QTL effects as an important factor in hybrid performance.     

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/).     

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.     

Quantitative trait locus mapping with background control in genetic populations of clonal F1 and double cross

In this study, we considered five categories of molecular markers in clonal F₁ and double cross populations, based on the number of distinguishable alleles and the number of distinguishable genotypes at the marker locus. Using the completed linkage maps, incomplete and missing markers were imputed as fully informative markers in order to simplify the linkage mapping approaches of quantitative trait genes. Under the condition of fully informative markers, we demonstrated that dominance effect between the female and male parents in clonal F₁ and double cross populations can cause the interactions between markers. We then developed an inclusive linear model that includes marker variables and marker interactions so as to completely control additive effects of the female and male parents, as well as the dominance effect between the female and male parents. The linear model was finally used for background control in inclusive composite interval mapping (ICIM) of quantitative trait locus (QTL). The efficiency of ICIM was demonstrated by extensive simulations and by comparisons with simple interval mapping, multiple‐QTL models and composite interval mapping. Finally, ICIM was applied in one actual double cross population to identify QTL on days to silking in maize.     

Approximating identity-by-descent matrices using multiple haplotype configurations on pedigrees

Identity-by-descent (IBD) matrix calculation is an important step in quantitative trait loci (QTL) analysis using variance component models. To calculate IBD matrices efficiently for large pedigrees with large numbers of loci, an approximation method based on the reconstruction of haplotype configurations for the pedigrees is proposed. The method uses a subset of haplotype configurations with high likelihoods identified by a haplotyping method. The new method is compared with a Markov chain Monte Carlo (MCMC) method (Loki) in terms of QTL mapping performance on simulated pedigrees. Both methods yield almost identical results for the estimation of QTL positions and variance parameters, while the new method is much more computationally efficient than the MCMC approach for large pedigrees and large numbers of loci. The proposed method is also compared with an exact method (Merlin) in small simulated pedigrees, where both methods produce nearly identical estimates of position-specific kinship coefficients. The new method can be used for fine mapping with joint linkage disequilibrium and linkage analysis, which improves the power and accuracy of QTL mapping.