Marker pair selection for mapping quantitative trait loci

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

A review of statistical methods for expression quantitative trait loci mapping

With high-throughput technologies now widely available, investigators can easily measure thousands of phenotypes for quantitative trait loci (QTL) mapping. Microarray measurements are particularly amenable to QTL mapping, as evidenced by a number of recent studies demonstrating utility across a broad range of biological endeavors. The early success stories have impelled a rapid increase in both the number and complexity of expression QTL (eQTL) experiments. Consequently, there is a need to consider the statistical principles involved in the design and analysis of these experiments and the methods currently being used. In this article we review these principles and methods and discuss the open questions most likely to yield significant progress toward increasing the amount of meaningful information obtained from eQTL mapping experiments.     

Mixed linear model approach for mapping quantitative trait loci underlying crop seed traits

The crop seed is a complex organ that may be composed of the diploid embryo, the triploid endosperm and the diploid maternal tissues. According to the genetic features of seed characters, two genetic models for mapping quantitative trait loci (QTLs) of crop seed traits are proposed, with inclusion of maternal effects, embryo or endosperm effects of QTL, environmental effects and QTL-by-environment (QE) interactions. The mapping population can be generated either from double back-cross of immortalized F₂ (IF₂) to the two parents, from random-cross of IF₂ or from selfing of IF₂ population. Candidate marker intervals potentially harboring QTLs are first selected through one-dimensional scanning across the whole genome. The selected candidate marker intervals are then included in the model as cofactors to control background genetic effects on the putative QTL(s). Finally, a QTL full model is constructed and model selection is conducted to eliminate false positive QTLs. The genetic main effects of QTLs, QE interaction effects and the corresponding P-values are computed by Markov chain Monte Carlo algorithm for Gaussian mixed linear model via Gibbs sampling. Monte Carlo simulations were performed to investigate the reliability and efficiency of the proposed method. The simulation results showed that the proposed method had higher power to accurately detect simulated QTLs and properly estimated effect of these QTLs. To demonstrate the usefulness, the proposed method was used to identify the QTLs underlying fiber percentage in an upland cotton IF₂ population. A computer software, QTLNetwork-Seed, was developed for QTL analysis of seed traits.     

Model selection for quantitative trait loci mapping in a full-sib family

Statistical methods for mapping quantitative trait loci (QTLs) in full-sib forest trees, in which the number of alleles and linkage phase can vary from locus to locus, are still not well established. Previous studies assumed that the QTL segregation pattern was fixed throughout the genome in a full-sib family, despite the fact that this pattern can vary among regions of the genome. In this paper, we propose a method for selecting the appropriate model for QTL mapping based on the segregation of different types of markers and QTLs in a full-sib family. The QTL segregation patterns were classified into three types: test cross (1:1 segregation), F₂ cross (1:2:1 segregation) and full cross (1:1:1:1 segregation). Akaike’s information criterion (AIC), the Bayesian information criterion (BIC) and the Laplace-empirical criterion (LEC) were used to select the most likely QTL segregation pattern. Simulations were used to evaluate the power of these criteria and the precision of parameter estimates. A Windows-based software was developed to run the selected QTL mapping method. A real example is presented to illustrate QTL mapping in forest trees based on an integrated linkage map with various segregation markers. The implications of this method for accurate QTL mapping in outbred species are discussed.     

A bivalent polyploid model for mapping quantitative trait loci in outcrossing tetraploids

Two major aspects have made the genetic and genomic study of polyploids extremely difficult. First, increased allelic or nonallelic combinations due to multiple alleles result in complex gene actions and interactions for quantitative trait loci (QTL) in polyploids. Second, meiotic configurations in polyploids undergo a complex biological process including either bivalent or multivalent formation, or both. For bivalent polyploids, different degrees of preferential chromosome pairings may occur during meiosis. In this article, we develop a maximum-likelihood-based model for mapping QTL in tetraploids by considering the quantitative inheritance and meiotic mechanism of bivalent polyploids. This bivalent polyploid model is implemented with the EM algorithm to simultaneously estimate QTL position, QTL effects, and QTL-marker linkage phases by incorporating the impact of a cytological parameter determining bivalent chromosome pairings (the preferential pairing factor). Simulation studies are performed to investigate the performance and robustness of our statistical method for parameter estimation. The implication and extension of the bivalent polyploid model are discussed.     

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 sib-pair approach to interval mapping of quantitative trait loci.

An interval mapping procedure based on the sib-pair method of Haseman and Elston is developed, and simulation studies are carried out to explore its properties. The procedure is analogous to other interval mapping procedures used with experimental material, such as plants and animals, and yields very similar results in terms of the location and effect size of a quantitative trait locus (QTL). The procedure offers an advantage over the conventional Haseman and Elston approach, in terms of power, and provides useful information concerning the location of a QTL. Because of its simplicity, the method readily lends itself to the analysis of selected samples for increased power and the evaluation of multilocus models of complex phenotypes.     

Linkage disequilibrium mapping of quantitative trait loci under truncation selection

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

Bayesian model selection for genome-wide epistatic quantitative trait loci analysis

The problem of identifying complex epistatic quantitative trait loci (QTL) across the entire genome continues to be a formidable challenge for geneticists. The complexity of genome-wide epistatic analysis results mainly from the number of QTL being unknown and the number of possible epistatic effects being huge. In this article, we use a composite model space approach to develop a Bayesian model selection framework for identifying epistatic QTL for complex traits in experimental crosses from two inbred lines. By placing a liberal constraint on the upper bound of the number of detectable QTL we restrict attention to models of fixed dimension, greatly simplifying calculations. Indicators specify which main and epistatic effects of putative QTL are included. We detail how to use prior knowledge to bound the number of detectable QTL and to specify prior distributions for indicators of genetic effects. We develop a computationally efficient Markov chain Monte Carlo (MCMC) algorithm using the Gibbs sampler and Metropolis-Hastings algorithm to explore the posterior distribution. We illustrate the proposed method by detecting new epistatic QTL for obesity in a backcross of CAST/Ei mice onto M16i.     

Bayesian LASSO for quantitative trait loci mapping

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

A 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.     

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

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

An efficient Bayesian model selection approach for interacting quantitative trait loci models with many effects

We extend our Bayesian model selection framework for mapping epistatic QTL in experimental crosses to include environmental effects and gene-environment interactions. We propose a new, fast Markov chain Monte Carlo algorithm to explore the posterior distribution of unknowns. In addition, we take advantage of any prior knowledge about genetic architecture to increase posterior probability on more probable models. These enhancements have significant computational advantages in models with many effects. We illustrate the proposed method by detecting new epistatic and gene-sex interactions for obesity-related traits in two real data sets of mice. Our method has been implemented in the freely available package R/qtlbim (http://www.qtlbim.org) to facilitate the general usage of the Bayesian methodology for genomewide interacting QTL analysis.     

A new approach for mapping quantitative trait loci using complete genetic marker linkage maps

A new approach based on nonlinear regression for the mapping of quantitative trait loci (QTLs) using complete genetic marker linkage maps is advanced in this paper. We call the approach joint mapping as it makes comprehensive use of the information from every marker locus on a chromosome. With this approach, both the detection of the existence of QTLs and the estimation of their positions, with corresponding confidence intervals, and effects can be realized simultaneously. This approach is widely applicable because only moments are used. It is simple and can save considerable computer time. It is especially useful when there are multiple QTLs and/or interactions between them on a chromosome.     

Lineage-specific mapping of quantitative trait loci

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

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.