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 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.     

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.     

家畜抗性等级性状的QTL定位方法

在广义线性模型的框架内模拟研究了家畜抗性等级性状的QTL定位方法,QTL参数的估计采用最大似然方法,比较了阈模型方法与一般线性方法的QTL定位效率,并对影响等级性状QTL定位效率的主要因素（QTL效应、性状的遗传力）进行了模拟研究,实验设计为多个家系的女儿设计,资源群体大小为500头。研究结果表明：在QTL位置参数估计及检验功效方面,阈模型方法具有一定的优势,对抗性等级性状QTL定位的功效也高于线性方法。另外,性状遗传力和QTL效应的大小对QTL定位的准确度也有直接的影响,随着性状遗传力QTL效应的     

Optimal sampling of a population to determine QTL location, variance, and allelic number

In a population intended for breeding and selection, questions of interest relative to a specific segregating QTL are the variance it generates in the population, and the number and effects of its alleles. One approach to address these questions is to extract several inbreds from the population and use them to generate multiple mapping families. Given random sampling of parents, sampling strategy may be an important factor determining the power of the analysis and its accuracy in estimating QTL variance and allelic number. We describe appropriate multiple-family QTL mapping methodology and apply it to simulated data sets to determine optimal sampling strategies in terms of family number versus family size. Genomes were simulated with seven chromosomes, on which 107 markers and six QTL were distributed. The total heritability was 0.60. Two to ten alleles were segregating at each QTL. Sampling strategies ranged from sampling two inbreds and generating a single family of 600 progeny to sampling 40 inbreds and generating 40 families of 15 progeny each. Strategies involving only one to five families were subject to variation due to the sampling of inbred parents. For QTL where more than two alleles were segregating, these strategies did not sample QTL alleles representative of the original population. Conversely, strategies involving 30 or more parents were subject to variation due to sampling of QTL genotypes within the small families obtained. Given these constraints, greatest QTL detection power was obtained for strategies involving five to ten mapping families. The most accurate estimation of the variance generated by the QTL, however, was obtained with strategies involving 20 or more families. Finally, strategies with an intermediate number of families best estimated the number of QTL alleles. We conclude that no overall optimal sampling strategy exists but that the strategy adopted must depend on the objective.Communicated by P. Langridge     

Combined linkage and linkage disequilibrium QTL mapping in multiple families of maize (Zea mays L.) line crosses highlights complementarities between models based on parental haplotype and single locus polymorphism

Advancements in genotyping are rapidly decreasing marker costs and increasing marker density. This opens new possibilities for mapping quantitative trait loci (QTL), in particular by combining linkage disequilibrium information and linkage analysis (LDLA). In this study, we compared different approaches to detect QTL for four traits of agronomical importance in two large multi-parental datasets of maize (Zea mays L.) of 895 and 928 testcross progenies composed of 7 and 21 biparental families, respectively, and genotyped with 491 markers. We compared to traditional linkage-based methods two LDLA models relying on the dense genotyping of parental lines with 17,728 SNP: one based on a clustering approach of parental line segments into ancestral alleles and one based on single marker information. The two LDLA models generally identified more QTL (60 and 52 QTL in total) than classical linkage models (49 and 44 QTL in total). However, they performed inconsistently over datasets and traits suggesting that a compromise must be found between the reduction of allele number for increasing statistical power and the adequacy of the model to potentially complex allelic variation. For some QTL, the model exclusively based on linkage analysis, which assumed that each parental line carried a different QTL allele, was able to capture remaining variation not explained by LDLA models. These complementarities between models clearly suggest that the different QTL mapping approaches must be considered to capture the different levels of allelic variation at QTL involved in complex traits.     

QTL detection power of multi-parental RIL populations in Arabidopsis thaliana

A major goal of today's biology is to understand the genetic basis of quantitative traits. This can be achieved by statistical methods that evaluate the association between molecular marker variation and phenotypic variation in different types of mapping populations. The objective of this work was to evaluate the statistical power of quantitative trait loci (QTL) detection of various multi-parental mating designs, as well as to assess the reasons for the observed differences. Our study was based on an empirical data of 20 Arabidopsis thaliana accessions, which have been selected to capture the maximum genetic diversity. The examined mating designs differed strongly with respect to the statistical power to detect QTL. We observed the highest power to detect QTL for the diallel cross with random mating design. The results of our study suggested that performing sibling mating within subpopulations of joint-linkage mapping populations has the potential to considerably increase the power for QTL detection. Our results, however, revealed that using designs in which more than two parental alleles segregate in each subpopulation increases the power even more.     

离散性状QTL区间定位的最大似然方法

采用最大似然区间定位法对阈模型与一般线性模型的QTL定位效率进行了比较,并对影响离散性状QTL检测效率的主要因素（QTL效应、性状的遗传力和表型发生率）进行了模拟研究,实验设计为多个家系的女儿设计．资源群体大小为500头。研究结果表明：在QTL参数估计及检验功效方面,阈模型方法具有较大的优势,对离散性状QTL定位的效率明显高于LM（Linear Model）方法,定位的准确性也较高。另外,性状遗传力、QTL效应的大小和性状表型发生率对QTL定位的准确度也有直接的影响,随着性状遗传力和表型发生率的提高,随着QTL效应的增大,QTL定位的效率也进一步提高。     

Bayesian joint mapping of quantitative trait loci for Gaussian and categorical characters in line crosses

Methodology for joint mapping of quantitative trait loci (QTL) affecting continuous and binary characters in experimental crosses is presented. The procedure consists of a Bayesian Gaussian-threshold model implemented via Markov chain Monte Carlo, which bypasses bottlenecks due to high-dimensional integrals required in maximum likelihood approaches. The method handles multiple binary traits and multiple QTL. Modeling of ordered categorical traits is discussed as well. Features of the method are illustrated using simulated datasets representing a backcross design, and the data are analyzed using mixed-trait and single-trait models. The mixed-trait analysis provides greater detection power of a QTL than a single-trait analysis when the QTL affects two or more traits. The number of QTL inferred in the mixed-trait analysis does not pertain to a specific trait, but the roles of each QTL on specific traits can be assessed from estimates of its effects. The impacts of varying incidence level and sample size on the mixed-trait QTL mapping analysis are investigated as well.     

Multiple trait multiple interval mapping of quantitative trait loci from inbred line crosses

ABSTRACT: BACKGROUND: Although many experiments have measurements on multiple traits, most studies performed the analysis of mapping of quantitative trait loci (QTL) for each trait separately using single trait analysis. Single trait analysis does not take advantage of possible genetic and environmental correlations between traits. In this paper, we propose a novel statistical method for multiple trait multiple interval mapping (MTMIM) of QTL for inbred line crosses. We also develop a novel score-based method for estimating genome-wide significance level of putative QTL effects suitable for the MTMIM model. The MTMIM method is implemented in the freely available and widely used Windows QTL Cartographer software. RESULTS: Throughout the paper, we provide compelling empirical evidences that: (1) the score-based threshold maintains proper type I error rate and tends to keep false discovery rate within an acceptable level; (2) the MTMIM method can deliver better parameter estimates and power than single trait multiple interval mapping method; (3) an analysis of Drosophila dataset illustrates how the MTMIM method can better extract information from datasets with measurements in multiple traits. CONCLUSIONS: The MTMIM method represents a convenient statistical framework to test hypotheses of pleiotropic QTL versus closely linked nonpleiotropic QTL, QTL by environment interaction, and to estimate the total genotypic variance-covariance matrix between traits and to decompose it in terms of QTL-specific variance-covariance matrices, therefore, providing more details on the genetic architecture of complex traits.     

A general mixture model approach for mapping quantitative trait loci from diverse cross designs involving multiple inbred lines

Most current statistical methods developed for mapping quantitative trait loci (QTL) based on inbred line designs apply to crosses from two inbred lines. Analysis of QTL in these crosses is restricted by the parental genetic differences between lines. Crosses from multiple inbred lines or multiple families are common in plant and animal breeding programmes, and can be used to increase the efficiency of a QTL mapping study. A general statistical method using mixture model procedures and the EM algorithm is developed for mapping QTL from various cross designs of multiple inbred lines. The general procedure features three cross design matrices, W, that define the contribution of parental lines to a particular cross and a genetic design matrix, D, that specifies the genetic model used in multiple line crosses. By appropriately specifying W matrices, the statistical method can be applied to various cross designs, such as diallel, factorial, cyclic, parallel or arbitrary-pattern cross designs with two or multiple parental lines. Also, with appropriate specification for the D matrix, the method can be used to analyse different kinds of cross populations, such as F2 backcross, four-way cross and mixed crosses (e.g. combining backcross and F2). Simulation studies were conducted to explore the properties of the method, and confirmed its applicability to diverse experimental designs.     

Linkage Disequilibrium with Linkage Analysis of Multiline Crosses Reveals Different Multiallelic QTL for Hybrid Performance in the Flint and Dent Heterotic Groups of Maize

Multiparental designs combined with dense genotyping of parents have been proposed as a way to increase the diversity and resolution of quantitative trait loci (QTL) mapping studies, using methods combining linkage disequilibrium information with linkage analysis (LDLA). Two new nested association mapping designs adapted to European conditions were derived from the complementary dent and flint heterotic groups of maize (Zea mays L.). Ten biparental dent families (N = 841) and 11 biparental flint families (N = 811) were genotyped with 56,110 single nucleotide polymorphism markers and evaluated as test crosses with the central line of the reciprocal design for biomass yield, plant height, and precocity. Alleles at candidate QTL were defined as (i) parental alleles, (ii) haplotypic identity by descent, and (iii) single-marker groupings. Between five and 16 QTL were detected depending on the model, trait, and genetic group considered. In the flint design, a major QTL (R² = 27%) with pleiotropic effects was detected on chromosome 10, whereas other QTL displayed milder effects (R² < 10%). On average, the LDLA models detected more QTL but generally explained lower percentages of variance, consistent with the fact that most QTL display complex allelic series. Only 15% of the QTL were common to the two designs. A joint analysis of the two designs detected between 15 and 21 QTL for the five traits. Of these, between 27 for silking date and 41% for tasseling date were significant in both groups. Favorable allelic effects detected in both groups open perspectives for improving biomass production.     

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.     

Joint-multiple family linkage analysis predicts within-family variation better than single-family analysis of the maize nested association mapping population

Quantitative trait locus (QTL) mapping has been used to dissect the genetic architecture of complex traits and predict phenotypes for marker-assisted selection. Many QTL mapping studies in plants have been limited to one biparental family population. Joint analysis of multiple biparental families offers an alternative approach to QTL mapping with a wider scope of inference. Joint-multiple population analysis should have higher power to detect QTL shared among multiple families, but may have lower power to detect rare QTL. We compared prediction ability of single-family and joint-family QTL analysis methods with fivefold cross-validation for 6 diverse traits using the maize nested association mapping population, which comprises 25 biparental recombinant inbred families. Joint-family QTL analysis had higher mean prediction abilities than single-family QTL analysis for all traits at most significance thresholds, and was always better at more stringent significance thresholds. Most robust QTL (detected in >50% of data samples) were restricted to one family and were often not detected at high frequency by joint-family analysis, implying substantial genetic heterogeneity among families for complex traits in maize. The superior predictive ability of joint-family QTL models despite important genetic differences among families suggests that joint-family models capture sufficient smaller effect QTL that are shared across families to compensate for missing some rare large-effect QTL.     

Multiple Trait Analysis of Genetic Mapping for Quantitative Trait Loci

We present in this paper models and statistical methods for performing multiple trait analysis on mapping quantitative trait loci (QTL) based on the composite interval mapping method. By taking into account the correlated structure of multiple traits, this joint analysis has several advantages, compared with separate analyses, for mapping QTL, including the expected improvement on the statistical power of the test for QTL and on the precision of parameter estimation. Also this joint analysis provides formal procedures to test a number of biologically interesting hypotheses concerning the nature of genetic correlations between different traits. Among the testing procedures considered are those for joint mapping, pleiotropy, QTL by environment interaction, and pleiotropy vs. close linkage. The test of pleiotropy (one pleiotropic QTL at a genome position) vs. close linkage (multiple nearby nonpleiotropic QTL) can have important implications for our understanding of the nature of genetic correlations between different traits in certain regions of a genome and also for practical applications in animal and plant breeding because one of the major goals in breeding is to break unfavorable linkage. Results of extensive simulation studies are presented to illustrate various properties of the analyses.     

Multiple-interval mapping for ordinal traits

Many statistical methods have been developed to map multiple quantitative trait loci (QTL) in experimental cross populations. Among these methods, multiple-interval mapping (MIM) can map QTL with epistasis simultaneously. However, the previous implementation of MIM is for continuously distributed traits. In this study we extend MIM to ordinal traits on the basis of a threshold model. The method inherits the properties and advantages of MIM and can fit a model of multiple QTL effects and epistasis on the underlying liability score. We study a number of statistical issues associated with the method, such as the efficiency and stability of maximization and model selection. We also use computer simulation to study the performance of the method and compare it to other alternative approaches. The method has been implemented in QTL Cartographer to facilitate its general usage for QTL mapping data analysis on binary and ordinal traits.     

MCQTL: multi-allelic QTL mapping in multi-cross design

The aim of the MCQTL software package is to perform QTL mapping in multi-cross designs. It allows the analysis of the usual populations derived from inbred lines and can link the families by assuming that the QTL locations are the same in all of them. Moreover, a diallel modelling of the QTL genotypic effects is allowed in multiple related families. The implemented model is a linear regression model. A composite interval mapping and an iterative QTL mapping are implemented to deal with multiple QTL models. Marker cofactor selections by forward or backward stepwise methods are implemented as well as computation of threshold test value by permutation. AVAILABILITY: The program is available on request after signing a licence agreement; free of charge for academic and non-profit organizations at http://www.genoplante.org (Bioinformatics products).     

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/).     

Using mating designs to uncover QTL and the genetic architecture of complex traits

Analysis of quantitative trait loci (QTL) affecting complex traits is often pursued in single-cross experiments. For most purposes, including breeding, some assessment is desired of the generalizability of the QTL findings and of the overall genetic architecture of the trait. Single-cross experiments provide a poor basis for these purposes, as comparison across experiments is hampered by segregation of different allelic combinations among different parents and by context-dependent effects of QTL. To overcome this problem, we combined the benefits of QTL analysis (to identify genomic regions affecting trait variation) and classic diallel analysis (to obtain insight into the general inheritance of the trait) by analyzing multiple mapping families that are connected via shared parents. We first provide a theoretical derivation of main (general combining ability (GCA)) and interaction (specific combining ability (SCA)) effects on F(2) family means relative to variance components in a randomly mating reference population. Then, using computer simulations to generate F(2) families derived from 10 inbred parents in different partial-diallel designs, we show that QTL can be detected and that the residual among-family variance can be analyzed. Standard diallel analysis methods are applied in order to reveal the presence and mode of action (in terms of GCA and SCA) of undetected polygenes. Given a fixed experiment size (total number of individuals), we demonstrate that QTL detection and estimation of the genetic architecture of polygenic effects are competing goals, which should be explicitly accounted for in the experimental design. Our approach provides a general strategy for exploring the genetic architecture, as well as the QTL underlying variation in quantitative traits.     

Supermodels: sorghum and maize provide mutual insight into the genetics of flowering time

Nested association mapping (NAM) offers power to dissect complex, quantitative traits. This study made use of a recently developed sorghum backcross (BC)-NAM population to dissect the genetic architecture of flowering time in sorghum; to compare the QTL identified with other genomic regions identified in previous sorghum and maize flowering time studies and to highlight the implications of our findings for plant breeding. A subset of the sorghum BC-NAM population consisting of over 1,300 individuals from 24 families was evaluated for flowering time across multiple environments. Two QTL analysis methodologies were used to identify 40 QTLs with predominately small, additive effects on flowering time; 24 of these co-located with previously identified QTL for flowering time in sorghum and 16 were novel in sorghum. Significant synteny was also detected with the QTL for flowering time detected in a comparable NAM resource recently developed for maize (Zea mays) by Buckler et al. (Science 325:714–718, 2009). The use of the sorghum BC-NAM population allowed us to catalogue allelic variants at a maximal number of QTL and understand their contribution to the flowering time phenotype and distribution across diverse germplasm. The successful demonstration of the power of the sorghum BC-NAM population is exemplified not only by correspondence of QTL previously identified in sorghum, but also by correspondence of QTL in different taxa, specifically maize in this case. The unification across taxa of the candidate genes influencing complex traits, such as flowering time can further facilitate the detailed dissection of the genetic control and causal genes.