Chen-Hung Kao

Estimating the genetic architecture of quantitative traits.

Understanding and estimating the structure and parameters associated with the genetic architecture of quantitative traits is a major research focus in quantitative genetics. With the availability of a well-saturated genetic map of molecular markers, it is possible to identify a major part of the structure of the genetic architecture of quantitative traits and to estimate the associated parameters. Multiple interval mapping, which was recently proposed for simultaneously mapping multiple quantitative trait loci (QTL), is well suited to the identification and estimation of the genetic architecture parameters, including the number, genomic positions, effects and interactions of significant QTL and their contribution to the genetic variance. With multiple traits and multiple environments involved in a QTL mapping experiment, pleiotropic effects and QTL by environment interactions can also be estimated. We review the method and discuss issues associated with multiple interval mapping, such as likelihood analysis, model selection, stopping rules and parameter estimation. The potential power and advantages of the method for mapping multiple QTL and estimating the genetic architecture are discussed. We also point out potential problems and difficulties in resolving the details of the genetic architecture as well as other areas that require further investigation. One application of the analysis is to improve genome-wide marker-assisted selection, particularly when the information about epistasis is used for selection with mating.

General formulas for obtaining the MLEs and the asymptotic variance-covariance matrix in mapping quantitative trait loci when using the EM algorithm.

We present in this paper general formulas for deriving the maximum likelihood estimates and the asymptotic variance-covariance matrix of the positions and effects of quantitative trait loci (QTLs) in a finite normal mixture model when the EM algorithm is used for mapping QTLs. The general formulas are based on two matrices D and Q, where D is the genetic design matrix, characterizing the genetic effects of the QTLs, and Q is the conditional probability matrix of QTL genotypes given flanking marker genotypes, containing the information on QTL positions. With the general formulas, it is relatively easy to extend QTL mapping analysis to using multiple marker intervals simultaneously for mapping multiple QTLs, for analyzing QTL epistasis, and for estimating the heritability of quantitative traits. Simulations were performed to evaluate the performance of the estimates of the asymptotic variances of QTL positions and effects.

Mapping quantitative trait loci using the experimental designs of recombinant inbred populations.

In the data collection of the QTL experiments using recombinant inbred (RI) populations, when individuals are genotyped for markers in a population, the trait values (phenotypes) can be obtained from the genotyped individuals (from the same population) or from some progeny of the genotyped individuals (from the different populations). Let Fu be the genotyped population and Fv (v ≥ u) be the phenotyped population. The experimental designs that both marker genotypes and phenotypes are recorded on the same populations can be denoted as (Fu/Fv, u = v) designs and that genotypes and phenotypes are obtained from the different populations can be denoted as (Fu/Fv, v > u) designs. Although most of the QTL mapping experiments have been conducted on the backcross and F2(F2/F2) designs, the other (Fu/Fv, v ≥ u) designs are also very popular. The great benefits of using the other (Fu/Fv, v ≥ u) designs in QTL mapping include reducing cost and environmental variance by phenotyping several progeny for the genotyped individuals and taking advantages of the changes in population structures of other RI populations. Current QTL mapping methods including those for the (Fu/Fv, u = v) designs, mostly for the backcross or F2/F2 design, and for the F2/F3 design based on a one-QTL model are inadequate for the investigation of the mapping properties in the (Fu/Fv, u ≤ v) designs, and they can be problematic due to ignoring their differences in population structures. In this article, a statistical method considering the differences in population structures between different RI populations is proposed on the basis of a multiple-QTL model to map for QTL in different (Fu/Fv, v ≥ u) designs. In addition, the QTL mapping properties of the proposed and approximate methods in different designs are discussed. Simulations were performed to evaluate the performance of the proposed and approximate methods. The proposed method is proven to be able to correct the problems of the approximate and current methods for improving the resolution of genetic architecture of quantitative traits and can serve as an effective tool to explore the QTL mapping study in the system of RI populations.

An investigation of the power for separating closely linked QTL in experimental populations.

SummaryHu & Xu (2008) developed a statistical method for computing the statistical power for detecting a quantitative trait locus (QTL) located in a marker interval. Their method is based on the regression interval mapping method and allows experimenters to effectively investigate the power for detecting a QTL in a population. This paper continues to work on the power analysis of separating multiple-linked QTLs. We propose simple formulae to calculate the power of separating closely linked QTLs located in marker intervals. The proposed formulae are simple functions of information numbers, variance inflation factors and genetic parameters of a statistical model in a population. Both regression and maximum likelihood interval mappings suitable for detecting QTL in the marker intervals are considered. In addition, the issue of separating linked QTLs in the progeny populations from an F2 subject to further self and/or random mating is also touched upon. One of the primary keys to our approach is to derive the genotypic distributions of three and four loci for evaluating the correlation structures between pairwise unobservable QTLs in the model across populations. The proposed formulae allow us to predict the power of separation when several factors, such as sample sizes, sizes and directions of QTL effects, distances between QTLs, interval sizes and relative QTL positions in the intervals, are considered together at a time in different experimental populations. Numerical justifications and Monte Carlo simulations were provided for confirmation and illustration.

A study on the mapping of quantitative trait loci in advanced populations derived from two inbred lines

In genetic and biological studies, the F2 population is one of the most popular and commonly used experimental populations mainly because it can be readily produced and its genome structure possesses several niceties that allow for productive investigation. These niceties include the equivalence between the proportion of recombinants and recombination rates, the capability of providing a complete set of three genotypes for every locus and an analytically attractive first-order Markovian property. Recently, there has been growing interest in using the progeny populations from F2 (advanced populations) because their genomes can be managed to meet specific purposes or can be used to enhance investigative studies. These advanced populations include recombinant inbred populations, advanced intercrossed populations, intermated recombinant inbred populations and immortalized F2 populations. Due to an increased number of meiosis cycles, the genomes of these advanced populations no longer possess the Markovian property and are relatively more complicated and different from the F2 genomes. Although issues related to quantitative trait locus (QTL) mapping using advanced populations have been well documented, still these advanced populations are often investigated in a manner similar to the way F2 populations are studied using a first-order Markovian assumption. Therefore, more efforts are needed to address the complexities of these advanced populations in more details. In this article, we attempt to tackle these issues by first modifying current methods developed under this Markovian assumption to propose an ad hoc method (the Markovian method) and explore its possible problems. We then consider the specific genome structures present in the advanced populations without invoking this assumption to propose a more adequate method (the non-Markovian method) for QTL mapping. Further, some QTL mapping properties related to the confounding problems that result from ignoring epistasis and to mapping closely linked QTL are derived and investigated across the different populations. Simulations show that the non-Markovian method outperforms the Markovian method, especially in the advanced populations subject to selfing. The results presented here may give some clues to the use of advanced populations for more powerful and precise QTL mapping.

A New Simple Method for Improving QTL Mapping Under Selective Genotyping

The selective genotyping approach, where only individuals from the high and low extremes of the trait distribution are selected for genotyping and the remaining individuals are not genotyped, has been known as a cost-saving strategy to reduce genotyping work and can still maintain nearly equivalent efficiency to complete genotyping in QTL mapping. We propose a novel and simple statistical method based on the normal mixture model for selective genotyping when both genotyped and ungenotyped individuals are fitted in the model for QTL analysis. Compared to the existing methods, the main feature of our model is that we first provide a simple way for obtaining the distribution of QTL genotypes for the ungenotyped individuals and then use it, rather than the population distribution of QTL genotypes as in the existing methods, to fit the ungenotyped individuals in model construction. Another feature is that the proposed method is developed on the basis of a multiple-QTL model and has a simple estimation procedure similar to that for complete genotyping. As a result, the proposed method has the ability to provide better QTL resolution, analyze QTL epistasis, and tackle multiple QTL problem under selective genotyping. In addition, a truncated normal mixture model based on a multiple-QTL model is developed when only the genotyped individuals are considered in the analysis, so that the two different types of models can be compared and investigated in selective genotyping. The issue in determining threshold values for selective genotyping in QTL mapping is also discussed. Simulation studies are performed to evaluate the proposed methods, compare the different models, and study the QTL mapping properties in selective genotyping. The results show that the proposed method can provide greater QTL detection power and facilitate QTL mapping for selective genotyping. Also, selective genotyping using larger genotyping proportions may provide roughly equivalent power to complete genotyping and that using smaller genotyping proportions has difficulties doing so. The R code of our proposed method is available on http://www.stat.sinica.edu.tw/chkao/.