A Random Model Approach to Interval Mapping of Quantitative Trait Loci

Mapping quantitative trait loci in outbred populations is important because many populations of organisms are noninbred. Unfortunately, information about the genetic architecture of the trait may not be available in outbred populations. Thus, the allelic effects of genes can not be estimated with ease. In addition, under linkage equilibrium, marker genotypes provide no information about the genotype of a QTL (our terminology for a single quantitative trait locus is QTL while multiple loci are referred to as QTLs). To circumvent this problem, an interval mapping procedure based on a random model approach is described. Under a random model, instead of estimating the effects, segregating variances of QTLs are estimated by a maximum likelihood method. Estimation of the variance component of a QTL depends on the proportion of genes identical-by-descent (IBD) shared by relatives at the locus, which is predicted by the IBD of two markers flanking the QTL. The marker IBD shared by two relatives are inferred from the observed marker genotypes. The procedure offers an advantage over the regression interval mapping in terms of high power and small estimation errors and provides flexibility for large sibships, irregular pedigree relationships and incorporation of common environmental and fixed effects.     

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 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.     

植物QTL分析的理论研究进展

数量性状的表型是由数量性状基因座 ( Quantitative trait locus,QTL)和环境效应共同作用的结果。传统的数量遗传学采用统计学的方法由一级统计量和二级统计量描述处理 QTL的复合作用 ,估计各种遗传参数 (例如遗传力、遗传相关、遗传进度、有效因子数等 ) ,用于指导遗传育种实践。然而 ,在传统的数量遗传学分析中 ,往往假设数量性状受微效多基因控制 ,这些基因具有相同的并且是较微小的效应 ,所估计的遗传参数反映的是数量性状多基因系统的整体特征 ,其理论方法不能用于追踪研究和描述单个数量性状基因的作用。近年来 ,由于分子生物学技…     

Bayesian mapping of genome-wide epistatic imprinted loci for quantitative traits

Genomic imprinting, an epigenetic phenomenon of parent-of-origin-specific gene expression, has been widely observed in plants, animals, and humans. To detect imprinting genes influencing quantitative traits, the least squares and maximum likelihood approaches for fitting a single quantitative trait locus (QTL) and Bayesian methods for simultaneously modeling multiple QTL have been adopted, respectively, in various studies. However, most of these studies have only estimated imprinting main effects and thus ignored imprinting epistatic effects. In the presence of extremely complex genomic imprinting architectures, we introduce a Bayesian model selection method to analyze the multiple interacting imprinted QTL (iQTL) model. This approach will greatly enhance the computational efficiency through setting the upper bound of the number of QTLs and performing selective sampling for QTL parameters. The imprinting types of detected main-effect QTLs can be estimated from the Bayes factor statistic formulated by the posterior probabilities for the genetic effects being compared. The performance of the proposed method is demonstrated by several simulation experiments. Moreover, this method is applied to dissect the imprinting genetic architecture for body weight in mouse and fruit weight in tomato. Matlab code for implementing this approach will be available from the authors upon request.     

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.     

Epistatic interaction between two nonstructural loci on chromosomes 7 and 3 influences hepatic lipase activity in BSB mice

BSB mice exhibit a wide range of obesity despite being produced by a backcross of lean C57BL/6J (B) x lean Mus spretus (SPRET/Pt) F1 animals x B. Previous linkage studies identified a quantitative trait locus (QTL) on mouse chromosome 7 with coincident peaks for hepatic lipase activity, obesity, and plasma cholesterol. However, these mice were not analyzed for gene x gene epistasis. Hepatic lipase activity is correlated with obesity and plasma cholesterol levels. In this study, we identified QTLs for plasma hepatic lipase activity with three statistical mapping methods: maximum likelihood interval mapping, Bayesian nonepistatic mapping, and Bayesian epistatic mapping. Bayesian epistatic mapping detected not only the QTL on chromosome 7 but also an additional QTL on chromosome 3, which has a weak main effect but a strong interaction with chromosome 7. SPRET/Pt alleles of the QTL on each chromosome promote hepatic lipase activity. The proportion of phenotypic variance explained by the epistatic effect is higher than that explained by the main effect of the QTL on chromosome 7.     

An EM algorithm for mapping binary disease loci: application to fibrosarcoma in a four-way cross mouse family

Many diseases show dichotomous phenotypic variation but do not follow a simple Mendelian pattern of inheritance. Variances of these binary diseases are presumably controlled by multiple loci and environmental variants. A least-squares method has been developed for mapping such complex disease loci by treating the binary phenotypes (0 and 1) as if they were continuous. However, the least-squares method is not recommended because of its ad hoc nature. Maximum Likelihood (ML) and Bayesian methods have also been developed for binary disease mapping by incorporating the discrete nature of the phenotypic distribution. In the ML analysis, the likelihood function is usually maximized using some complicated maximization algorithms (e.g. the Newton-Raphson or the simplex algorithm). Under the threshold model of binary disease, we develop an Expectation Maximization (EM) algorithm to solve for the maximum likelihood estimates (MLEs). The new EM algorithm is developed by treating both the unobserved genotype and the disease liability as missing values. As a result, the EM iteration equations have the same form as the normal equation system in linear regression. The EM algorithm is further modified to take into account sexual dimorphism in the linkage maps. Applying the EM-implemented ML method to a four-way-cross mouse family, we detected two regions on the fourth chromosome that have evidence of QTLs controlling the segregation of fibrosarcoma, a form of connective tissue cancer. The two QTLs explain 50-60% of the variance in the disease liability. We also applied a Bayesian method previously developed (modified to take into account sex-specific maps) to this data set and detected one additional QTL on chromosome 13 that explains another 26% of the variance of the disease liability. All the QTLs detected primarily show dominance effects.     

Genetic architecture of variation in sex-comb tooth number in Drosophila simulans

The sex comb on the forelegs of Drosophila males is a secondary sexual trait, and the number of teeth on these combs varies greatly within and between species. To understand the relationship between the intra- and interspecific variation, we performed quantitative trait locus (QTL) analyses of the intraspecific variation in sex-comb tooth number. We used five mapping populations derived from two inbred Drosophila simulans strains that were divergent in the number of sex-comb teeth. Although no QTLs were detected on the X chromosome, we identified four QTLs on the second chromosome and three QTLs on the third chromosome. While identification and estimated effects of the second-chromosome QTLs depend on genetic backgrounds, significant and consistent effects of the two third-chromosome QTLs were found in two genetic backgrounds. There were significant epistatic interactions between a second-chromosome QTL and a third-chromosome QTL, as well as between two second-chromosome QTLs. The third-chromosome QTLs are concordant with the locations of the QTLs responsible for the previously observed differences in sex-comb tooth number between D. simulans and D. mauritiana.     

Empirical Bayesian elastic net for multiple quantitative trait locus mapping

In multiple quantitative trait locus (QTL) mapping, a high-dimensional sparse regression model is usually employed to account for possible multiple linked QTLs. The QTL model may include closely linked and thus highly correlated genetic markers, especially when high-density marker maps are used in QTL mapping because of the advancement in sequencing technology. Although existing algorithms, such as Lasso, empirical Bayesian Lasso (EBlasso) and elastic net (EN) are available to infer such QTL models, more powerful methods are highly desirable to detect more QTLs in the presence of correlated QTLs. We developed a novel empirical Bayesian EN (EBEN) algorithm for multiple QTL mapping that inherits the efficiency of our previously developed EBlasso algorithm. Simulation results demonstrated that EBEN provided higher power of detection and almost the same false discovery rate compared with EN and EBlasso. Particularly, EBEN can identify correlated QTLs that the other two algorithms may fail to identify. When analyzing a real dataset, EBEN detected more effects than EN and EBlasso. EBEN provides a useful tool for inferring high-dimensional sparse model in multiple QTL mapping and other applications. An R software package ‘EBEN'' implementing the EBEN algorithm is available on the Comprehensive R Archive Network (CRAN).     

Development and high-throughput genotyping of substitution lines carring the chromosome segments of indica 9311 in the background of japonica Nipponbare

Chromosome segment substitution lines (CSSLs) are useful for the precise mapping of quartitative trait loci (QTLs) and dissection of the genetic basis of complex traits.In this study,two whole-genome sequenced rice cultivars,the japonica Nipponbare and indica 9311 were used as recipient and dtonor,respectively.A population with 57 CSSLs was developed after crossing and back-crossing assisted by mo lecular rnarkers,and genotypes were identified using a high-throughput resequencing strategy,Detailed graphical genotypes of 38 lines were constructed based on resequencing data.These CSSLs had a total of 95 substituted segments derived from indica 9311,with an average of about 2.5 segments pet CSSL and eight segments per chromosome,and covered about 87.4％ of the rice whole genome.A multiple linear regression QTL analysis mapped four QTLs for 1000-grain weight.The largest-effect QTL was located in a region on chromosome 5 that contained a cloned major QTL GW5/qSW5 for grain size in rice.These CSSLs with a background of Nipponbare may provide powerful tools for future whole-genome discovery and functional study of essential genes/QTLs in rice,and offer ideal materials and foundations for japonica breeding.     

An empirical Bayes method for estimating epistatic effects of quantitative trait loci

Summary .   The genetic variance of a quantitative trait is often controlled by the segregation of multiple interacting loci. Linear model regression analysis is usually applied to estimating and testing effects of these quantitative trait loci (QTL). Including all the main effects and the effects of interaction (epistatic effects), the dimension of the linear model can be extremely high. Variable selection via stepwise regression or stochastic search variable selection (SSVS) is the common procedure for epistatic effect QTL analysis. These methods are computationally intensive, yet they may not be optimal. The LASSO (least absolute shrinkage and selection operator) method is computationally more efficient than the above methods. As a result, it has been widely used in regression analysis for large models. However, LASSO has never been applied to genetic mapping for epistatic QTL, where the number of model effects is typically many times larger than the sample size. In this study, we developed an empirical Bayes method (E-BAYES) to map epistatic QTL under the mixed model framework. We also tested the feasibility of using LASSO to estimate epistatic effects, examined the fully Bayesian SSVS, and reevaluated the penalized likelihood (PENAL) methods in mapping epistatic QTL. Simulation studies showed that all the above methods performed satisfactorily well. However, E-BAYES appears to outperform all other methods in terms of minimizing the mean-squared error (MSE) with relatively short computing time. Application of the new method to real data was demonstrated using a barley dataset.     

Identification of QTLs influencing wood property traits in loblolly pine ( Pinus taeda L.). II. Chemical wood properties

Chemical wood property traits were analyzed for the presence of quantitative trait loci (QTLs) in a three-generation outbred pedigree of loblolly pine (Pinus taeda L.). These traits were assayed using pyrolysis molecular beam mass spectrometry and include mass spectrum peak intensities associated with carbohydrates, α-cellulose and hemicellulose sugars, and lignin. Models for projection to latent structures (PLS) were used to also estimate the chemical composition of cell walls (i.e., α-cellulose, galactan and lignin) from mass spectrum data using multivariate regression. Both earlywood and latewood fractions from the fifth annual ring were analyzed for each trait. An interval mapping approach designed for an outbred pedigree was used to estimate the number of QTLs, the magnitude of QTL effects, and their genomic position. Eight unique QTLs influencing cell wall chemistry were detected from multiple peak intensities and/or PLS estimates using the one- and two-QTL models. Significant differences in chemical contents were observed among the populations from North Carolina vs Oklahoma, and results from QTL×environment analyses suggest that QTLs interact with environmental location. QTLs should be verified in larger experiments and in different genetic and environmental backgrounds. QTL mapping will help towards eventually identifying genes having a major effect on chemical wood properties. Received: 19 January 2001 / Accepted: 31 May 2001     

Stability over genetic backgrounds, generations and years of quantitative trait locus (QTLs) for organoleptic quality in tomato

The efficiency of marker-assisted backcross for the introgression of a quantitative trait locus (QTL) from a donor line into a recipient line depends on the stability of QTL expression. QTLs for six quality traits in tomato (fruit weight, firmness, locule number, soluble solid content, sugar content and titratable acidity) were studied in order to investigate their individual effect and their stability over years, generations and genetic backgrounds. Five chromosome regions carrying fruit quality QTLs were transferred following a marker-assisted backcross scheme from a cherry tomato line into three modern lines with larger fruits. Three sets of genotypes corresponding to three generations were compared: (1) an RIL population, which contained 50% of each parental genome, (2) three BC3S1 populations which segregated simultaneously for the five regions of interest but were almost fully homozygous for the recipient genome on the eight chromosomes carrying no QTL and (3) three sets of QTL-NILs (BC3S3 lines) which differed from the recipient line only in one of the five regions. QTL detection was performed in each generation, in each genetic background and during 2 successive years for QTL-NILs. About half of the QTLs detected in QTL-NILs were detected in both years. Eight of the ten QTLs detected in RILs were recovered in the QTL-NILs with the genetic background used for the initial QTL mapping experiment, with the exception of two QTLs for fruit firmness. Several new QTLs were detected. In the two other genetic backgrounds, the number of QTLs in common with the RILs was lower, but several new QTLs were also detected in advanced generations.     

QTL Express: mapping quantitative trait loci in simple and complex pedigrees

QTL Express is the first application for Quantitative Trait Locus (QTL) mapping in outbred populations with a web-based user interface. User input of three files containing a marker map, trait data and marker genotypes allows mapping of single or multiple QTL by the regression approach, with the option to perform permutation or bootstrap tests.     

Detection of multiple quantitative trait loci and their pleiotropic effects in outbred pig populations

Background

Simultaneous detection of multiple QTLs (quantitative trait loci) may allow more accurate estimation of genetic effects. We have analyzed outbred commercial pig populations with different single and multiple models to clarify their genetic properties and in addition, we have investigated pleiotropy among growth and obesity traits based on allelic correlation within a gamete.

Methods

Three closed populations, (A) 427 individuals from a Yorkshire and Large White synthetic breed, (B) 547 Large White individuals and (C) 531 Large White individuals, were analyzed using a variance component method with one-QTL and two-QTL models. Six markers on chromosome 4 and five to seven markers on chromosome 7 were used.

Results

Population A displayed a high test statistic for the fat trait when applying the two-QTL model with two positions on two chromosomes. The estimated heritabilities for polygenic effects and for the first and second QTL were 19%, 17% and 21%, respectively. The high correlation of the estimated allelic effect on the same gamete and QTL test statistics suggested that the two separate QTL which were detected on different chromosomes both have pleiotropic effects on the two fat traits. Analysis of population B using the one-QTL model for three fat traits found a similar peak position on chromosome 7. Allelic effects of three fat traits from the same gamete were highly correlated suggesting the presence of a pleiotropic QTL. In population C, three growth traits also displayed similar peak positions on chromosome 7 and allelic effects from the same gamete were correlated.

Conclusion

Detection of the second QTL in a model reduced the polygenic heritability and should improve accuracy of estimated heritabilities for both QTLs.     

Bayesian mapping of quantitative trait loci for complex binary traits

A complex binary trait is a character that has a dichotomous expression but with a polygenic genetic background. Mapping quantitative trait loci (QTL) for such traits is difficult because of the discrete nature and the reduced variation in the phenotypic distribution. Bayesian statistics are proved to be a powerful tool for solving complicated genetic problems, such as multiple QTL with nonadditive effects, and have been successfully applied to QTL mapping for continuous traits. In this study, we show that Bayesian statistics are particularly useful for mapping QTL for complex binary traits. We model the binary trait under the classical threshold model of quantitative genetics. The Bayesian mapping statistics are developed on the basis of the idea of data augmentation. This treatment allows an easy way to generate the value of a hypothetical underlying variable (called the liability) and a threshold, which in turn allow the use of existing Bayesian statistics. The reversible jump Markov chain Monte Carlo algorithm is used to simulate the posterior samples of all unknowns, including the number of QTL, the locations and effects of identified QTL, genotypes of each individual at both the QTL and markers, and eventually the liability of each individual. The Bayesian mapping ends with an estimation of the joint posterior distribution of the number of QTL and the locations and effects of the identified QTL. Utilities of the method are demonstrated using a simulated outbred full-sib family. A computer program written in FORTRAN language is freely available on request.     

Mapping multiple QTL of different effects: comparison of a simple sequential testing strategy and multiple QTL mapping

The aim of this study was to explore, by computer simulation, the mapping of QTLs in a realistic but complex situation of many (linked) QTLs with different effects, and to compare two QTL mapping methods. A novel method to dissect genetic variation on multiple chromosomes using molecular markers in backcross and F₂ populations derived from inbred lines was suggested, and its properties tested using simulations. The rationale for this sequential testing method was to explicitly test for alternative genetic models. The method consists of a series of four basic statistical tests to decide whether variance was due to a single QTL, two QTLs, multiple QTLs, or polygenes, starting with a test to detect genetic variance associated with a particular chromosome. The method was able to distinguish between different QTL configurations, in that the probability to `detect' the correct model was high, varying from 0.75 to 1. For example, for a backcross population of 200 and an overall heritability of 50%, in 78% of replicates a polygenic model was detected when that was the underlying true model. To test the method for multiple chromosomes, QTLs were simulated on 10 chromosomes, following a geometric series of allele effects, assuming positive alleles were in coupling in the founder lines For these simulations, the sequential testing method was compared to the established Multiple QTL Mapping (MQM) method. For a backcross population of 400 individuals, power to detect genetic variance was low with both methods when the heritability was 0.40. For example, the power to detect genetic variation on a chromosome on which 6 QTLs explained 12.6% of the genetic variance, was less than 60% for both methods. For a large heritability (0.90), the power of MQM to detect genetic variance and to dissect QTL configurations was generally better, due to the simultaneous fitting of markers on all chromosomes. It is concluded that when testing different QTL configurations on a single chromosome using the sequential testing procedure, regions of other chromosomes which explain a significant amount of variation should be fitted in the model of analysis. This study reinforces the need for large experiments in plants and other species if the aim of a genome scan is to dissect quantitative genetic variation.     

Interval Mapping of Multiple Quantitative Trait Loci

The interval mapping method is widely used for the mapping of quantitative trait loci (QTLs) in segregating generations derived from crosses between inbred lines. The efficiency of detecting and the accuracy of mapping multiple QTLs by using genetic markers are much increased by employing multiple QTL models instead of the single QTL models (and no QTL models) used in interval mapping. However, the computational work involved with multiple QTL models is considerable when the number of QTLs is large. In this paper it is proposed to combine multiple linear regression methods with conventional interval mapping. This is achieved by fitting one QTL at a time in a given interval and simultaneously using (part of) the markers as cofactors to eliminate the effects of additional QTLs. It is shown that the proposed method combines the easy computation of the single QTL interval mapping method with much of the efficiency and accuracy of multiple QTL models.     

Methods for the detection of multiple linked QTL applied to a mixture of full and half sib families

A new multiple trait strategy based on discriminant analysis was studied for efficient detection of linked QTL in outbred sib families, in comparison with a multivariate likelihood technique. The discriminant analysis technique describes the segregation of a linear combination of the traits in a univariate likelihood. This combination is calculated for each pair of positions depending on the inheritance of the pairs of QTL haplotypes in the progeny. The gains in power and accuracy for position estimations of multiple trait methods in grid searches were evaluated in reference to single trait detections of linked QTL. The methods were applied to simulated designs with two correlated traits submitted to various effects from the linked QTL. Multiple trait strategies were generally more powerful and accurate than the single trait technique. Linked QTL were distinguished when they were separated enough to identify informative recombinations: at least two genetic markers and 25 cM between the QTL under the simulated conditions. Except in a particular case, discriminant analysis was at least as powerful as the multivariate technique and its implementation was five times faster. Combining the advantages from both methodologies, we finally propose a complete strategy for rapid and efficient systematic multivariate detections in outbred populations.