Shizhong Xu

Quantitative Trait Locus Mapping Can Benefit From Segregation Distortion

Segregation distortion is a phenomenon that has been observed in many experimental systems. How segregation distortion among markers arises and its impact on mapping studies are the focus of this work. Segregation distortion of markers can be considered to arise from segregation distortion loci (SDL). I develop a theory of segregation distortion and show that the presence of only a few SDL can cause the entire chromosome to distort from Mendelian segregation. Segregation distortion is detrimental to the power of detecting quantitative trait loci (QTL) with dominance effects, but it is not always a detriment to QTL mapping for additive effects. When segregation distortion of a locus is a random event, the SDL is beneficial to QTL mapping ∼44% of the time. If SDL are present and ignored, power loss can be substantial. A dense marker map can be used to ameliorate the situation, and if dense marker information is incorporated, power loss is minimal. However, other situations are less benign. A method that can simultaneously map QTL and SDL is discussed, maximizing both use of mapping resources and use by agricultural and evolutionary biologists.

Supervised cluster analysis for microarray data based on multivariate Gaussian mixture

MOTIVATION Grouping genes having similar expression patterns is called gene clustering, which has been proved to be a useful tool for extracting underlying biological information of gene expression data. Many clustering procedures have shown success in microarray gene clustering; most of them belong to the family of heuristic clustering algorithms. Model-based algorithms are alternative clustering algorithms, which are based on the assumption that the whole set of microarray data is a finite mixture of a certain type of distributions with different parameters. Application of the model-based algorithms to unsupervised clustering has been reported. Here, for the first time, we demonstrated the use of the model-based algorithm in supervised clustering of microarray data. RESULTS We applied the proposed methods to real gene expression data and simulated data. We showed that the supervised model-based algorithm is superior over the unsupervised method and the support vector machines (SVM) method. AVAILABILITY The program written in the SAS language implementing methods I-III in this report is available upon request. The software of SVMs is available in the website http://svm.sdsc.edu/cgi-bin/nph-SVMsubmit.cgi

Mapping quantitative trait loci with epistatic effects.

Epistatic variance can be an important source of variation for complex traits. However, detecting epistatic effects is difficult primarily due to insufficient sample sizes and lack of robust statistical methods. In this paper, we develop a Bayesian method to map multiple quantitative trait loci (QTLs) with epistatic effects. The method can map QTLs in complicated mating designs derived from the cross of two inbred lines. In addition to mapping QTLs for quantitative traits, the proposed method can even map genes underlying binary traits such as disease susceptibility using the threshold model. The parameters of interest are various QTL effects, including additive, dominance and epistatic effects of QTLs, the locations of identified QTLs and even the number of QTLs. When the number of QTLs is treated as an unknown parameter, the dimension of the model becomes a variable. This requires the reversible jump Markov chain Monte Carlo algorithm. The utility of the proposed method is demonstrated through analysis of simulation data.

Genetic control of the rate of wound healing in mice

There have been few studies of the inheritance of wound healing in mammals. In this study, we demonstrate that inbred strains of mice differ significantly in the rate of wound healing. Of the 20 strains tested, fast healers (MRL/MpJ-Faslpr and LG/J) healed wounds four times faster than slow healers (Balb/cByJ and SJL/J). The genetic basis underlying the difference in the healing capacity was analysed using F2 populations of two different crosses. We show that the wound healing is a polygenically determined quantitative trait with an average estimated heritability of 86%. The modes of gene action in these two crosses are different. In the (MRL/MpJ × SJL/J) cross, genes regulating fast healing in MRL/MpJ mice exhibited additive effects, whereas these effects were suppressed by a dominant repressor gene in CBA/J mice in the (MRL/MpJ-Faslpr × CBA/J) cross. Information gained from this investigation provides insight into further study of molecular mechanisms underlying the rate of wound healing in mammals.

A quantitative genetics model for viability selection

Viability selection will change gene frequencies of loci controlling fitness. Consequently, the frequencies of marker loci linked to the viability loci will also change. In genetic mapping, the change of marker allelic frequencies is reflected by the departure from Mendelian segregation ratio. The non-Mendelian segregation of markers has been used to map viability loci along the genome. However, current methods have not been able to detect the amount of selection (s) and the degree of dominance (h) simultaneously. We developed a method to detect both s and h using an F2 mating design under the classical fitness model. We also developed a quantitative genetics model for viability selection by proposing a continuous liability controlling the viability of individuals. With the liability model, mapping viability loci has been formulated as mapping quantitative trait loci. As a result, nongenetic systematic environmental effects can be easily incorporated into the model and subsequently separated from the genetic effects of the viability loci. The quantitative genetic model has been verified with a series of Monte Carlo simulation experiments.

Mapping Quantitative Trait Loci for Expression Abundance

Mendelian loci that control the expression levels of transcripts are called expression quantitative trait loci (eQTL). When mapping eQTL, we often deal with thousands of expression traits simultaneously, which complicates the statistical model and data analysis. Two simple approaches may be taken in eQTL analysis: (1) individual transcript analysis in which a single expression trait is mapped at a time and the entire eQTL mapping involves separate analysis of thousands of traits and (2) individual marker analysis where differentially expressed transcripts are detected on the basis of their association with the segregation pattern of an individual marker and the entire analysis requires scanning markers of the entire genome. Neither approach is optimal because data are not analyzed jointly. We develop a Bayesian clustering method that analyzes all expressed transcripts and markers jointly in a single model. A transcript may be simultaneously associated with multiple markers. Additionally, a marker may simultaneously alter the expression of multiple transcripts. This is a model-based method that combines a Gaussian mixture of expression data with segregation of multiple linked marker loci. Parameter estimation for each variable is obtained via the posterior mean drawn from a Markov chain Monte Carlo sample. The method allows a regular quantitative trait to be included as an expression trait and subject to the same clustering assignment. If an expression trait links to a locus where a quantitative trait also links, the expressed transcript is considered to be associated with the quantitative trait. The method is applied to a microarray experiment with 60 F2 mice measured for 25 different obesity-related quantitative traits. In the experiment, ∼40,000 transcripts and 145 codominant markers are investigated for their associations. A program written in SAS/IML is available from the authors on request.

Mapping viability loci using molecular markers

In genetic mapping experiments, some molecular markers often show distorted segregation ratios. We hypothesize that these markers are linked to some viability loci that cause the observed segregation ratios to deviate from Mendelian expectations. Although statistical methods for mapping viability loci have been developed for line-crossing experiments, methods for viability mapping in outbred populations have not been developed yet. In this study, we develop a method for mapping viability loci in outbred populations using a full-sib family as an example. We develop a maximum likelihood (ML) method that uses the observed marker genotypes as data and the proportions of the genotypes of the viability locus as parameters. The ML solutions are obtained via the expectation–maximization algorithm. Application and efficiencies of the method are demonstrated and tested using a set of simulated data. We conclude that mapping viability loci can be accomplished using similar statistical techniques used in quantitative trait locus mapping for quantitative traits.

Iteratively Reweighted Least Squares Mapping of Quantitative Trait Loci

Mapping quantitative trait loci (QTL) is a typical problem of regression with uncertain independent variables because the genotype of a putative QTL is not observed. Rather, the genotype is inferred from marker information. The method of maximum likelihood (ML) methods is considered to be the optimal solution for this problem because the distribution of the unobserved QTL genotype is fully taken into account. The simple linear regression method (REG) is a first-order approximation to ML and usually performs very well. In this study, an iteratively reweighted least squares method (IRWLS) is proposed. The new method is a second-order approximation to ML because both the expectation and the variance of the unobserved QTL genotype are taken into consideration. The IRWLS is developed in the context of a single large outbred family. The properties of IRWLS are demonstrated and compared with REG and ML via replicated Monte Carlo simulations. The conclusions are: (1) when marker information content is high, the three methods perform equally well, but ML and IRWLS outperform REG when marker information content is low and the variance explained by the QTL is high; (2) when the residual distribution is not normal, ML can fail or have low power to detect small QTLs, but REG and IRWLS are robust to non-normality; and (3) when the residual distribution is normal, the performance of IRWLS is almost identical to ML, but the computational speed of IRWLS is many times faster than that of ML.

PROC QTL—A SAS Procedure for Mapping Quantitative Trait Loci

Statistical analysis system (SAS) is the most comprehensive statistical analysis software package in the world. It offers data analysis for almost all experiments under various statistical models. Each analysis is performed using a particular subroutine, called a procedure (PROC). For example, PROC ANOVA performs analysis of variances. PROC QTL is a user-defined SAS procedure for mapping quantitative trait loci (QTL). It allows users to perform QTL mapping for continuous and discrete traits within the SAS platform. Users of PROC QTL are able to take advantage of all existing features offered by the general SAS software, for example, data management and graphical treatment. The current version of PROC QTL can perform QTL mapping for all line crossing experiments using maximum likelihood (ML), least square (LS), iteratively reweighted least square (IRLS), Fisher scoring (FISHER), Bayesian (BAYES), and empirical Bayes (EBAYES) methods.

Mapping quantitative trait loci for ordered categorical traits in four-way crosses.

Many quantitative traits of economical importance are ordinal in nature. Although methods of mapping quantitative trait loci (QTLs) for continuous quantitative characters are well developed, such methods for ordinal characters are generally lacking. In this paper, we develop a method based on the framework of a generalized linear model using four-way cross populations. The method estimates and tests the average effects of a gene substitution in the parents. All markers in the same linkage group are simultaneously used to infer the allelic transmission of a putative QTL. General results of the method are demonstrated by a few simulation experiments. We discuss extensions of the method to QTL mapping in full-sib families.