Bayesian mapping of quantitative trait loci for multiple complex traits with the use of variance components

Complex traits important for humans are often correlated phenotypically and genetically. Joint mapping of quantitative-trait loci (QTLs) for multiple correlated traits plays an important role in unraveling the genetic architecture of complex traits. Compared with single-trait analysis, joint mapping addresses more questions and has advantages for power of QTL detection and precision of parameter estimation. Some statistical methods have been developed to map QTLs underlying multiple traits, most of which are based on maximum-likelihood methods. We develop here a multivariate version of the Bayes methodology for joint mapping of QTLs, using the Markov chain-Monte Carlo (MCMC) algorithm. We adopt a variance-components method to model complex traits in outbred populations (e.g., humans). The method is robust, can deal with an arbitrary number of alleles with arbitrary patterns of gene actions (such as additive and dominant), and allows for multiple phenotype data of various types in the joint analysis (e.g., multiple continuous traits and mixtures of continuous traits and discrete traits). Under a Bayesian framework, parameters--including the number of QTLs--are estimated on the basis of their marginal posterior samples, which are generated through two samplers, the Gibbs sampler and the reversible-jump MCMC. In addition, we calculate the Bayes factor related to each identified QTL, to test coincident linkage versus pleiotropy. The performance of our method is evaluated in simulations with full-sib families. The results show that our proposed Bayesian joint-mapping method performs well for mapping multiple QTLs in situations of either bivariate continuous traits or mixed data types. Compared with the analysis for each trait separately, Bayesian joint mapping improves statistical power, provides stronger evidence of QTL detection, and increases precision in estimation of parameter and QTL position. We also applied the proposed method to a set of real data and detected a coincident linkage responsible for determining bone mineral density and areal bone size of wrist in humans.     

Comparing linkage disequilibrium-based methods for fine mapping quantitative trait loci

Recently, a method for fine mapping quantitative trait loci (QTL) using linkage disequilibrium was proposed to map QTL by modeling covariance between individuals, due to identical-by-descent (IBD) QTL alleles, on the basis of the similarity of their marker haplotypes under an assumed population history. In the work presented here, the advantage of using marker haplotype information for fine mapping QTL was studied by comparing the IBD-based method with 10 markers to regression on a single marker, a pair of markers, or a two-locus haplotype under alternative population histories. When 10 markers were genotyped, the IBD-based method estimated the position of the QTL more accurately than did single-marker regression in all populations. When 20 markers were genotyped for regression, as single-marker methods do not require knowledge of haplotypes, the mapping accuracy of regression in all populations was similar to or greater than that of the IBD-based method using 10 markers. Thus for populations similar to those simulated here, the IBD-based method is comparable to single-marker regression analysis for fine mapping QTL.     

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.     

QTL linkage analysis of connected populations using ancestral marker and pedigree information

The common assumption in quantitative trait locus (QTL) linkage mapping studies that parents of multiple connected populations are unrelated is unrealistic for many plant breeding programs. We remove this assumption and propose a Bayesian approach that clusters the alleles of the parents of the current mapping populations from locus-specific identity by descent (IBD) matrices that capture ancestral marker and pedigree information. Moreover, we demonstrate how the parental IBD data can be incorporated into a QTL linkage analysis framework by using two approaches: a Threshold IBD model (TIBD) and a Latent Ancestral Allele Model (LAAM). The TIBD and LAAM models are empirically tested via numerical simulation based on the structure of a commercial maize breeding program. The simulations included a pilot dataset with closely linked QTL on a single linkage group and 100 replicated datasets with five linkage groups harboring four unlinked QTL. The simulation results show that including parental IBD data (similarly for TIBD and LAAM) significantly improves the power and particularly accuracy of QTL mapping, e.g., position, effect size and individuals’ genotype probability without significantly increasing computational demand.     

Bayesian multiple quantitative trait loci mapping for complex traits using markers of the entire genome

A Bayesian methodology has been developed for multiple quantitative trait loci (QTL) mapping of complex binary traits that follow liability threshold models. Unlike most QTL mapping methods where only one or a few markers are used at a time, the proposed method utilizes all markers across the genome simultaneously. The outperformance of our Bayesian method over the traditional single-marker analysis and interval mapping has been illustrated via simulations and real data analysis to identify candidate loci associated with colorectal cancer.     

A Decision Rule for Quantitative Trait Locus Detection Under the Extended Bayesian LASSO Model

Bayesian shrinkage analysis is arguably the state-of-the-art technique for large-scale multiple quantitative trait locus (QTL) mapping. However, when the shrinkage model does not involve indicator variables for marker inclusion, QTL detection remains heavily dependent on significance thresholds derived from phenotype permutation under the null hypothesis of no phenotype-to-genotype association. This approach is computationally intensive and more importantly, the hypothetical data generation at the heart of the permutation-based method violates the Bayesian philosophy. Here we propose a fully Bayesian decision rule for QTL detection under the recently introduced extended Bayesian LASSO for QTL mapping. Our new decision rule is free of any hypothetical data generation and relies on the well-established Bayes factors for evaluating the evidence for QTL presence at any locus. Simulation results demonstrate the remarkable performance of our decision rule. An application to real-world data is considered as well.     

Advances in Bayesian multiple quantitative trait loci mapping in experimental crosses

Many complex human diseases and traits of biological and/or economic importance are determined by interacting networks of multiple quantitative trait loci (QTL) and environmental factors. Mapping QTL is critical for understanding the genetic basis of complex traits, and for ultimate identification of genes responsible. A variety of sophisticated statistical methods for QTL mapping have been developed. Among these developments, the evolution of Bayesian approaches for multiple QTL mapping over the past decade has been remarkable. Bayesian methods can jointly infer the number of QTL, their genomic positions and their genetic effects. Here, we review recently developed and still developing Bayesian methods and associated computer software for mapping multiple QTL in experimental crosses. We compare and contrast these methods to clearly describe the relationships among different Bayesian methods. We conclude this review by highlighting some areas of future research.     

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.     

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.     

Bayesian shrinkage analysis of quantitative trait Loci for dynamic traits

Many quantitative traits are measured repeatedly during the life of an organism. Such traits are called dynamic traits. The pattern of the changes of a dynamic trait is called the growth trajectory. Studying the growth trajectory may enhance our understanding of the genetic architecture of the growth trajectory. Recently, we developed an interval-mapping procedure to map QTL for dynamic traits under the maximum-likelihood framework. We fit the growth trajectory by Legendre polynomials. The method intended to map one QTL at a time and the entire QTL analysis involved scanning the entire genome by fitting multiple single-QTL models. In this study, we propose a Bayesian shrinkage analysis for estimating and mapping multiple QTL in a single model. The method is a combination between the shrinkage mapping for individual quantitative traits and the Legendre polynomial analysis for dynamic traits. The multiple-QTL model is implemented in two ways: (1) a fixed-interval approach where a QTL is placed in each marker interval and (2) a moving-interval approach where the position of a QTL can be searched in a range that covers many marker intervals. Simulation study shows that the Bayesian shrinkage method generates much better signals for QTL than the interval-mapping approach. We propose several alternative methods to present the results of the Bayesian shrinkage analysis. In particular, we found that the Wald test-statistic profile can serve as a mechanism to test the significance of a putative QTL.     

Multiparent intercross populations in analysis of quantitative traits

Most traits of interest to medical, agricultural and animal scientists show continuous variation and complex mode of inheritance. DNA-based markers are being deployed to analyse such complex traits, that are known as quantitative trait loci (QTL). In conventional QTL analysis, F2, backcross populations, recombinant inbred lines, backcross inbred lines and double haploids from biparental crosses are commonly used. Introgression lines and near isogenic lines are also being used for QTL analysis. However, such populations have major limitations like predominantly relying on the recombination events taking place in the F1 generation and mapping of only the allelic pairs present in the two parents. The second generation mapping resources like association mapping, nested association mapping and multiparent intercross populations potentially address the major limitations of available mapping resources. The potential of multiparent intercross populations in gene mapping has been discussed here. In such populations both linkage and association analysis can be conductted without encountering the limitations of structured populations. In such populations, larger genetic variation in the germplasm is accessed and various allelic and cytoplasmic interactions are assessed. For all practical purposes, across crop species, use of eight founders and a fixed population of 1000 individuals are most appropriate. Limitations with multiparent intercross populations are that they require longer time and more resource to be generated and they are likely to show extensive segregation for developmental traits, limiting their use in the analysis of complex traits. However, multiparent intercross population resources are likely to bring a paradigm shift towards QTL analysis in plant species.     

Bayesian functional mapping of dynamic quantitative traits

Without consideration of other linked QTLs responsible for dynamic trait, original functional mapping based on a single QTL model is not optimal for analyzing multiple dynamic trait loci. Despite that composite functional mapping incorporates the effects of genetic background outside the tested QTL in mapping model, the arbitrary choice of background markers also impact on the power of QTL detection. In this study, we proposed Bayesian functional mapping strategy that can simultaneously identify multiple QTL controlling developmental patterns of dynamic traits over the genome. Our proposed method fits the change of each QTL effect with the time by Legendre polynomial and takes the residual covariance structure into account using the first autoregressive equation. Also, Bayesian shrinkage estimation was employed to estimate the model parameters. Especially, we specify the gamma distribution as the prior for the first-order auto-regressive coefficient, which will guarantee the convergence of Bayesian sampling. Simulations showed that the proposed method could accurately estimate the QTL parameters and had a greater statistical power of QTL detection than the composite functional mapping. A real data analysis of leaf age growth in rice is used for the demonstration of our method. It shows that our Bayesian functional mapping can detect more QTLs as compared to composite functional mapping.     

Mapping quantitative trait loci in F2 incorporating phenotypes of F3 progeny

In plants and laboratory animals, QTL mapping is commonly performed using F(2) or BC individuals derived from the cross of two inbred lines. Typical QTL mapping statistics assume that each F(2) individual is genotyped for the markers and phenotyped for the trait. For plant traits with low heritability, it has been suggested to use the average phenotypic values of F(3) progeny derived from selfing F(2) plants in place of the F(2) phenotype itself. All F(3) progeny derived from the same F(2) plant belong to the same F(2:3) family, denoted by F(2:3). If the size of each F(2:3) family (the number of F(3) progeny) is sufficiently large, the average value of the family will represent the genotypic value of the F(2) plant, and thus the power of QTL mapping may be significantly increased. The strategy of using F(2) marker genotypes and F(3) average phenotypes for QTL mapping in plants is quite similar to the daughter design of QTL mapping in dairy cattle. We study the fundamental principle of the plant version of the daughter design and develop a new statistical method to map QTL under this F(2:3) strategy. We also propose to combine both the F(2) phenotypes and the F(2:3) average phenotypes to further increase the power of QTL mapping. The statistical method developed in this study differs from published ones in that the new method fully takes advantage of the mixture distribution for F(2:3) families of heterozygous F(2) plants. Incorporation of this new information has significantly increased the statistical power of QTL detection relative to the classical F(2) design, even if only a single F(3) progeny is collected from each F(2:3) family. The mixture model is developed on the basis of a single-QTL model and implemented via the EM algorithm. Substantial computer simulation was conducted to demonstrate the improved efficiency of the mixture model. Extension of the mixture model to multiple QTL analysis is developed using a Bayesian approach. The computer program performing the Bayesian analysis of the simulated data is available to users for real data analysis.     

Bayesian shrinkage estimation of quantitative trait loci parameters

Mapping multiple QTL is a typical problem of variable selection in an oversaturated model because the potential number of QTL can be substantially larger than the sample size. Currently, model selection is still the most effective approach to mapping multiple QTL, although further research is needed. An alternative approach to analyzing an oversaturated model is the shrinkage estimation in which all candidate variables are included in the model but their estimated effects are forced to shrink toward zero. In contrast to the usual shrinkage estimation where all model effects are shrunk by the same factor, we develop a Bayesian method that allows the shrinkage factor to vary across different effects. The new shrinkage method forces marker intervals that contain no QTL to have estimated effects close to zero whereas intervals containing notable QTL have estimated effects subject to virtually no shrinkage. We demonstrate the method using both simulated and real data for QTL mapping. A simulation experiment with 500 backcross (BC) individuals showed that the method can localize closely linked QTL and QTL with effects as small as 1% of the phenotypic variance of the trait. The method was also used to map QTL responsible for wound healing in a family of a (MRL/MPJ x SJL/J) cross with 633 F(2) mice derived from two inbred lines.     

QTL mapping using a memetic algorithm with modifications of BIC as fitness function

The problem of locating quantitative trait loci (QTL) for experimental populations can be approached by multiple regression analysis. In this context variable selection using a modification of the Bayesian Information Criterion (mBIC) has been well established in the past. In this article a memetic algorithm (MA) is introduced to find the model which minimizes the selection criterion. Apart from mBIC also a second modification (mBIC2) is considered, which has the property of controlling the false discovery rate. Given the Bayesian nature of our selection criteria, we are not only interested in finding the best model, but also in computing marker posterior probabilities using all models visited by MA. In a simulation study MA (with mBIC and mBIC2) is compared with a parallel genetic algorithm (PGA) which has been previously suggested for QTL mapping. It turns out that MA in combination with mBIC2 performs best, where determining QTL positions based on marker posterior probabilities yields even better results than using the best model selected by MA. Finally we consider a real data set from the literature and show that MA can also be extended to multiple interval mapping, which potentially increases the precision with which the exact location of QTLs can be estimated.     

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.     

Bayesian inference for genomic imprinting underlying developmental characteristics

The identification of imprinted genes is becoming a standard procedure in searching for quantitative trait loci (QTL) underlying complex traits. When a developmental characteristic such as growth or drug response is observed at multiple time points, understanding the dynamics of gene function governing the underlying feature should provide more biological information regarding the genetic control of an organism. Recognizing that differential imprinting can be development-specific, mapping imprinted genes considering the dynamic imprinting effect can provide additional biological insights into the epigenetic control of a complex trait. In this study, we proposed a Bayesian imprinted QTL (iQTL) mapping framework considering the dynamics of imprinting effects and model multiple iQTLs with an efficient Bayesian model selection procedure. The method overcomes the limitation of likelihood-based mapping procedure, and can simultaneously identify multiple iQTLs with different gene action modes across the whole genome with high computational efficiency. An inference procedure using Bayes factors to distinguish different imprinting patterns of iQTL was proposed. Monte Carlo simulations were conducted to evaluate the performance of the method. The utility of the approach was illustrated through an analysis of a body weight growth data set in an F(2) family derived from LG/J and SM/J mouse stains. The proposed Bayesian mapping method provides an efficient and computationally feasible framework for genome-wide multiple iQTL inference with complex developmental traits.     

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.     

Advances in Statistical Methods to Map Quantitative Trait Loci in Outbred Populations

Statistical methods to map quantitative trait loci (QTL) in outbred populations are reviewed, extensions and applications to human and plant genetic data are indicated, and areas for further research are identified. Simple and computationally inexpensive methods include (multiple) linear regression of phenotype on marker genotypes and regression of squared phenotypic differences among relative pairs on estimated proportions of identity-by-descent at a locus. These methods are less suited for genetic parameter estimation in outbred populations but allow the determination of test statistic distributions via simulation or data permutation; however, further inferences including confidence intervals of QTL location require the use of Monte Carlo or bootstrap sampling techniques. A method which is intermediate in computational requirements is residual maximum likelihood (REML) with a covariance matrix of random QTL effects conditional on information from multiple linked markers. Testing for the number of QTLs on a chromosome is difficult in a classical framework. The computationally most demanding methods are maximum likelihood and Bayesian analysis, which take account of the distribution of multilocus marker-QTL genotypes on a pedigree and permit investigators to fit different models of variation at the QTL. The Bayesian analysis includes the number of QTLs on a chromosome as an unknown.     

Mapping quantitative trait loci with dominant markers in four-way crosses

 A common problem in mapping quantitative trait loci (QTLs) is that marker data are often incomplete. This includes missing data, dominant markers, and partially informative markers, arising in outbred populations. Here we briefly present an iteratively re-weighted least square method (IRWLS) to incorporate dominant and missing markers for mapping QTLs in four-way crosses under a heterogeneous variance model. The algorithm uses information from all markers in a linkage group to infer the QTL genotype. Monte Carlo simulations indicate that with half dominant markers, QTL detection is almost as efficient as with all co-dominant markers. However, the precision of the estimated QTL parameters generally decreases as more markers become missing or dominant. Notable differences are observed on the standard deviation of the estimated QTL position for varying levels of marker information content. The method is relatively simple so that more complex models including multiple QTLs or fixed effects can be fitted. Finally, the method can be readily extended to QTL mapping in full-sib families. Received: 16 June 1998 / Accepted: 29 September 1998