Inferring linkage disequilibrium between a polymorphic marker locus and a trait locus in natural populations

Three approaches are proposed in this study for detecting or estimating linkage disequilibrium between a polymorphic marker locus and a locus affecting quantitative genetic variation using the sample from random mating populations. It is shown that the disequilibrium over a wide range of circumstances may be detected with a power of 80% by using phenotypic records and marker genotypes of a few hundred individuals. Comparison of ANOVA and regression methods in this article to the transmission disequilibrium test (TDT) shows that, given the genetic variance explained by the trait locus, the power of TDT depends on the trait allele frequency, whereas the power of ANOVA and regression analyses is relatively independent from the allelic frequency. The TDT method is more powerful when the trait allele frequency is low, but much less powerful when it is high. The likelihood analysis provides reliable estimation of the model parameters when the QTL variance is at least 10% of the phenotypic variance and the sample size of a few hundred is used. Potential use of these estimates in mapping the trait locus is also discussed.     

Methodology and Accuracy of Estimation of Quantitative Trait Loci Parameters in a Half-Sib Design Using Maximum Likelihood

Maximum likelihood methods were developed for estimation of the six parameters relating to a marker-linked quantitative trait locus (QTL) segregating in a half-sib design, namely the QTL additive effect, the QTL dominance effect, the population mean, recombination between the marker and the QTL, the population frequency of the QTL alleles, and the within-family residual variance. The method was tested on simulated stochastic data with various family structures under two genetic models. A method for predicting the expected value of the likelihood was also derived and used to predict the lower bound sampling errors of the parameter estimates and the correlations between them. It was found that standard errors and confidence intervals were smallest for the population mean and variance, intermediate for QTL effects and allele frequency, and highest for recombination rate. Correlations among standard errors of the parameter estimates were generally low except for a strong negative correlation (r = -0.9) between the QTL's dominance effect and the population mean, and medium positive and negative correlations between the QTL's additive effect and, respectively, recombination rate (r = 0.5) and residual variance (r = -0.6). The implications for experimental design and method of analysis on power and accuracy of marker-QTL linkage experiments were discussed.     

Models and tests of linkage and association studies of quantitative trait locus for multi-allele marker Loci

In this paper, we explore models and tests for association and linkage studies of a quantitative trait locus (QTL) linked to a multi-allele marker locus. Based on the difference between an offspring's conditional trait means of receiving and not receiving an allele from a parent at marker locus, we propose three statistics T(m), T(m,row) and T(m,col) to test association or linkage disequilibrium between the marker locus and the QTL. These tests are composite tests, and use the offspring marginal sample means including offspring data of both homozygous and heterozygous parents. For the linkage study, we calculate the offspring's conditional trait mean given the allele transmission status of a heterozygous parent at the marker locus. Based on the difference between the conditional means of a transmitted and a nontransmitted allele from a heterozygous parent, we propose statistics T(parsi), T(satur), T(gen) and T(m,het) to perform composite tests of linkage between the marker locus and the quantitative trait locus in the presence of association. These tests only use the offspring data that are related to the heterozygous parents at the marker locus. T(parsi) is a parsimonious or allele-wise statistic, T(satur) and T(gen )are satured or genotype-wise statistics, and T(m,het) compares the row and column sample means for offspring data of heterozygous parents. After comparing the powers and the sample sizes, we conclude that T(parsi) has higher power than those of the bi-allele tests, T(satur), T(gen), and T(m,het). If there is tight linkage between the marker and the trait locus, T(parsi) is powerful in detecting linkage between the marker and the trait locus in the presence of association. By investigating the goodness-of-fit of T(parsi), we find that T(satur) does not gain much power compared to that of T(parsi). Moreover, T(parsi) takes into account the pattern of the data that is consistent with linkage and linkage disequilibrium. As the number of alleles at the marker locus increases, T(parsi) is very conservative, and can be useful even for sparse data. To illustrate the usefulness and the power of the methods proposed in this paper, we analyze the chromosome 6 data of the Oxford asthma data, Genetic Analysis Workshop 12.     

Modeling linkage disequilibrium between a polymorphic marker locus and a locus affecting complex dichotomous traits in natural populations

Linkage disequilibrium is an important topic in evolutionary and population genetics. An issue yet to be settled is the theory required to extend the linkage disequilibrium analysis to complex traits. In this study, we present theoretical analysis and methods for detecting or estimating linkage disequilibrium (LD) between a polymorphic marker locus and any one of the loci affecting a complex dichotomous trait on the basis of samples randomly or selectively collected from natural populations. Statistical properties of these methods were investigated and their powers were compared analytically or by use of Monte Carlo simulations. The results show that the disequilibrium may be detected with a power of 80% by using phenotypic records and marker genotype when both the trait and marker variants are common (30%) and the LD is relatively high (40-100% of the theoretical maximum). The maximum-likelihood approach provides accurate estimates of the model parameters as well as detection of linkage disequilibrium. The likelihood method is preferred for its higher power and reliability in parameter estimation. The approaches developed in this article are also compared to those for analyzing a continuously distributed quantitative trait. It is shown that a larger sample size is required for the dichotomous trait model to obtain the same level of power in detecting linkage disequilibrium as the continuous trait analysis. Potential use of these estimates in mapping the trait locus is also discussed.     

The Effects of Selection on Linkage Analysis for Quantitative Traits

The effects of within-sample selection on the outcome of analyses detecting linkage between genetic markers and quantitative traits were studied. It was found that selection by truncation for the trait of interest significantly reduces the differences between marker genotype means thus reducing the power to detect linked quantitative trait loci (QTL). The size of this reduction is a function of proportion selected, the magnitude of the QTL effect, recombination rate between the marker locus and the QTL, and the allele frequency of the QTL. Proportion selected was the most influential of these factors on bias, e.g., for an allele substitution effect of one standard deviation unit, selecting the top 80%, 50% or 20% of the population required 2, 6 or 24 times the number of progeny, respectively, to offset the loss of power caused by this selection. The effect on power was approximately linear with respect to the size of gene effect, almost invariant to recombination rate, and a complex function of QTL allele frequency. It was concluded that experimental samples from animal populations which have been subjected to even minor amounts of selection will be inefficient in yielding information on linkage between markers and loci influencing the quantitative trait under selection.     

Estimating linkage disequilibrium between a polymorphic marker locus and a trait locus in natural populations.

Positional cloning of gene(s) underlying a complex trait requires a high-resolution linkage map between the trait locus and genetic marker loci. Recent research has shown that this may be achieved through appropriately modeling and screening linkage disequilibrium between the candidate marker locus and the major trait locus. A quantitative genetics model was developed in the present study to estimate the coefficient of linkage disequilibrium between a polymorphic genetic marker locus and a locus underlying a quantitative trait as well as the relevant genetic parameters using the sample from randomly mating populations. Asymptotic covariances of the maximum-likelihood estimates of the parameters were formulated. Convergence of the EM-based statistical algorithm for calculating the maximum-likelihood estimates was confirmed and its utility to analyze practical data was exploited by use of extensive Monte-Carlo simulations. Appropriateness of calculating the asymptotic covariance matrix in the present model was investigated for three different approaches. Numerical analyses based on simulation data indicated that accurate estimation of the genetic parameters may be achieved if a sample size of 500 is used and if segregation at the trait locus explains not less than a quarter of phenotypic variation of the trait, but the study reveals difficulties in predicting the asymptotic variances of these maximum-likelihood estimates. A comparison was made between the statistical powers of the maximum-likelihood analysis and the previously proposed regression analysis for detecting the disequilibrium.     

Bayesian analysis of linkage between genetic markers and quantitative trait loci. I. Prior knowledge

Summary Prior information on gene effects at individual quantitative trait loci (QTL) and on recombination rates between marker loci and QTL is derived. The prior distribution of QTL gene effects is assumed to be exponential with major effects less likely than minor ones. The prior probability of linkage between a marker and another single locus is a function of the number and length of chromosomes, and of the map function relating recombination rate to genetic distance among loci. The prior probability of linkage between a marker locus and a quantitative trait depends additionally on the number of detectable QTL, which may be determined from total additive genetic variance and minimum detectable QTL effect. The use of this prior information should improve linkage tests and estimates of QTL effects.     

Bayesian analysis of linkage between genetic markers and quantitative trait loci. II. Combining prior knowledge with experimental evidence

Summary A Bayesian method was developed for identifying genetic markers linked to quantitative trait loci (QTL) by analyzing data from daughter or granddaughter designs and single markers or marker pairs. Traditional methods may yield unrealistic results because linkage tests depend on number of markers and QTL gene effects associated with selected markers are overestimated. The Bayesian or posterior probability of linkage combines information from a daughter or granddaughter design with the prior probability of linkage between a marker locus and a QTL. If the posterior probability exceeds a certain quantity, linkage is declared. Upon linkage acceptance, Bayesian estimates of marker-QTL recombination rate and QTL gene effects and frequencies are obtained. The Bayesian estimates of QTL gene effects account for different amounts of information by shrinking information from data toward the mean or mode of a prior exponential distribution of gene effects. Computation of the Bayesian analysis is feasible. Exact results are given for biallelic QTL, and extensions to multiallelic QTL are suggested.     

METHODS FOR DETECTION AND ESTIMATION OF LINKAGE BETWEEN A MARKER LOCUS AND QUANTITATIVE TRAIT LOCI Ⅰ. USING F_2 FAMILY SUBPOPULATION

Several methods heve been proposed over the years to detect linkage between a marker gene and a quantitative trait locus (QTL). Use of isozymes and restriction fragment length polymorphisms (RFLPs) as genetic markers has encouraged the development of new methods to detect linkage. In the paper authors present three methods to detect linkage and two methods to measure recombination frequency (r). The three methods that detect linkage are fit for the test of one or several QTLs of quantitative trait considered as long as the net gene effect vaIue of all loci belonging to a linkage cluster is greater significantly than zero, irrespective of their linkage relationship among the several QTLs.     

Fine mapping of a quantitative trait locus for twinning rate using combined linkage and linkage disequilibrium mapping

A novel and robust method for the fine-scale mapping of genes affecting complex traits, which combines linkage and linkage-disequilibrium information, is proposed. Linkage information refers to recombinations within the marker-genotyped generations and linkage disequilibrium to historical recombinations before genotyping started. The identity-by-descent (IBD) probabilities at the quantitative trait locus (QTL) between first generation haplotypes were obtained from the similarity of the marker alleles surrounding the QTL, whereas IBD probabilities at the QTL between later generation haplotypes were obtained by using the markers to trace the inheritance of the QTL. The variance explained by the QTL is estimated by residual maximum likelihood using the correlation structure defined by the IBD probabilities. Unlinked background genes were accounted for by fitting a polygenic variance component. The method was used to fine map a QTL for twinning rate in cattle, previously mapped on chromosome 5 by linkage analysis. The data consisted of large half-sib families, but the method could also handle more complex pedigrees. The likelihood of the putative QTL was very small along most of the chromosome, except for a sharp likelihood peak in the ninth marker bracket, which positioned the QTL within a region <1 cM in the middle part of bovine chromosome 5. The method was expected to be robust against multiple genes affecting the trait, multiple mutations at the QTL, and relatively low marker density.     

Detection of quantitative trait Loci influencing recombination using recombinant inbred lines

The genetic basis of variation in recombination in higher plants is polygenic and poorly understood, despite its theoretical and practical importance. Here a method of detecting quantitative trait loci (QTL) influencing recombination in recombinant inbred lines (RILs) is proposed that relies upon the fact that genotype data within RILs carry the signature of past recombination. Behavior of the segregational genetic variance in numbers of chromosomal crossovers (recombination) over generations is described for self-, full-sib-, and half-sib-generated RILs with no dominance in true crossovers. This genetic variance, which as a fraction of the total phenotypic variance contributes to the statistical power of the method, was asymptotically greatest with half sibbing, less with sibbing, and least with selfing. The statistical power to detect a recombination QTL declined with diminishing QTL effect, genome target size, and marker density. For reasonably tight marker linkage power was greater with less intense inbreeding for later generations and vice versa for early generations. Generational optima for segregation variance and statistical power were found, whose onset and narrowness varied with marker density and mating design, being more pronounced for looser marker linkage. Application of this method to a maize RIL population derived from inbred lines Mo17 and B73 and developed by selfing suggested two putative QTL (LOD > 2.4) affecting certain chromosomes, and using a canonical transformation another putative QTL was detected. However, permutation tests failed to support their presence (experimentwise alpha = 0.05). Other populations with more statistical power and chosen specifically for recombination QTL segregation would be more effective.     

人类混血群体连锁不平衡的遗传学模型与基因定位

人类混血群体可以说是混合群体的一种特例．在无选择、无突变、无限随机交配群体的假定前提下,研究了亲本群体的基因频率对混血群体及其衍生后代群体连锁不平衡结构的影响,导出了各群体连锁不平衡值的表达式,建立了一个估计基因间重组率的简便方法;同时, 采用估算分子标记与QTL之间连锁不平衡系数的统计分析方法,分析了人类混血群体及其衍生后代群体QTL检测与估计的关系,建立了该关系的系列理论公式．研究结果表明,本方法不仅适用于人类疾病(包括复杂遗传疾病)基因定位,而且适合于人类正常基因的定位,同时也适用于人类普通多基因性状的QTL分析．     

基于选择基因型对数量性状进行关联分析

数量性状的遗传分析可以通过＂选择基因型＂的方式完成。本文提出了一个利用极端样本来对数量性状位点（QTL）进行关联分析的统计量T。统计量T比较上极端群体样本中具有纯合子标记的性状值差异。通过计算机模拟考察了无关联情形时T的分布和Ⅰ型错误率,结果表明,在各种样本选择策略下,T的分布近似于χ^2-分布,Ⅰ型错误率接近设定的显著性水平。同时,考察了各种遗传模型下不同遗传率,不同样本大小,及不同样本选择阈值对T的统计功效的影响,结果表明,T的功效随着标记和QTL间连锁不平衡程度的增强及遗传率和样本大小的增大而增大,当样本选择阈值更严格时,功效也越大。     

Combined linkage and association mapping of quantitative trait loci by multiple markers

Using multiple diallelic markers, variance component models are proposed for high-resolution combined linkage and association mapping of quantitative trait loci (QTL) based on nuclear families. The objective is to build a model that may fully use marker information for fine association mapping of QTL in the presence of prior linkage. The measures of linkage disequilibrium and the genetic effects are incorporated in the mean coefficients and are decomposed into orthogonal additive and dominance effects. The linkage information is modeled in variance-covariance matrices. Hence, the proposed methods model both association and linkage in a unified model. On the basis of marker information, a multipoint interval mapping method is provided to estimate the proportion of allele sharing identical by descent (IBD) and the probability of sharing two alleles IBD at a putative QTL for a sib-pair. To test the association between the trait locus and the markers, both likelihood-ratio tests and F-tests can be constructed on the basis of the proposed models. In addition, analytical formulas of noncentrality parameter approximations of the F-test statistics are provided. Type I error rates of the proposed test statistics are calculated to show their robustness. After comparing with the association between-family and association within-family (AbAw) approach by Abecasis and Fulker et al., it is found that the method proposed in this article is more powerful and advantageous based on simulation study and power calculation. By power and sample size comparison, it is shown that models that use more markers may have higher power than models that use fewer markers. The multiple-marker analysis can be more advantageous and has higher power in fine mapping QTL. As an application, the Genetic Analysis Workshop 12 German asthma data are analyzed using the proposed methods.     

Selective genotyping for determination of linkage between a marker locus and a quantitative trait locus

Summary Selective genotyping is the term used when the determination of linkage between marker loci and quantitative trait loci (QTL) affecting some particular trait is carried out by genotyping only individuals from the high and low phenotypic tails of the entire sample population. Selective genotyping can markedly decrease the number of individuals genotyped for a given power at the expense of an increase in the number of individuals phenotyped. The optimum proportion of individuals genotyped from the point of view of minimizing costs for a given experimental power depends strongly on the cost of completely genotyping an individual for all of the markers included in the experiment (including the costs of obtaining a DNA sample) relative to the cost of rearing and trait evaluation of an individual. However, in single trait studies, it will almost never be useful to genotype more than the upper and lower 25% of a population. It is shown that the observed difference in quantitative trait values associated with alternative marker genotypes in the selected population can be much greater than the actual gene effect at the quantitative trait locus when the entire population is considered. An expression and a figure is provided for converting observed differences under selective genotyping to actual gene effects.     

Efficiency of triple test cross for detecting epistasis with marker information

The triple test cross (TTC) is an experimental design for detecting epistasis and estimating the components of genetic variance for quantitative traits. In this paper, we extend the analysis to include molecular information. The statistical power of the mating design was assessed under a model assuming that a finite number of loci affect the trait in question. Formulae are developed for the analysis with or without marker information relating to the recombination fraction between loci, the genetical properties of quantitative trait controlled by the quantitative trait loci (QTL), the linkage phases of the parents and population size. Application of these formulae showed that the recombination fraction between genes and the magnitude and the types of epistasis have important interactions in their effects on power. The results demonstrate that the TTC may have increased power to detect epistasis when marker information is present. However, the simulation experiments show that the standard deviation of the estimated expected mean square was higher with one marker than that with two, whereas the corresponding value without marker information was the lowest. In addition, we demonstrate that the relative position of QTL and markers and the number of markers can both affect the power of epistatic detection.     

Linkage between quantitative trait loci and marker loci: resolution power of three statistical approaches in single marker analysis

This paper presents a comparison of three methods of parameter estimation in analysis of linkage between a quantitative trait locus (QTL) and a marker locus: maximum likelihood, mean square for trait cumulative distribution function, and method of moments, employing simulated backcross data. The sensitivity of estimates to violation of assumptions of normality and equal variances were also studied. Some measures of discrepancy between the trait distributions in the QTL groups are considered to evaluate the potential dependence of the resolution capacity of the QTL substitution effect with respect to trait mean value and variance.     

Localization of a quantitative trait locus via a Bayesian approach

A Bayesian approach to the direct mapping of a quantitative trait locus (QTL), fully utilizing information from multiple linked gene markers, is presented in this paper. The joint posterior distribution (a mixture distribution modeling the linkage between a biallelic QTL and N gene markers) is computationally challenging and invites exploration via Markov chain Monte Carlo methods. The parameter's complete marginal posterior densities are obtained, allowing a diverse range of inferences. Parameters estimated include the QTL genotype probabilities for the sires and the offspring, the allele frequencies for the QTL, and the position and additive and dominance effects of the QTL. The methodology is applied through simulation to a half-sib design to form an outbred pedigree structure where there is an entire class of missing information. The capacity of the technique to accurately estimate parameters is examined for a range of scenarios.     

An improved formulation of marker heterozygosity in recurrent selection and backcross schemes

This report presents a theoretical formulation for predicting heterozygosity of a putative marker locus linked to two quantitative trait loci (QTL) in a recurrent selection and backcross (RSB) scheme. Since the heterozygosity at any given marker locus maintained in such a breeding programme reflects its map location relative to QTL, the present study develops the theoretical analysis of the QTL mapping method that recently appeared in the literature. The formulae take into account selection, recombination and finite population size during the multiple-generation breeding scheme. The single-marker and two-QTL model was compared numerically with the model involving two linked marker loci and two QTL. Without recombination interference, the two models predict the same expected heterozygosity at the linked marker loci, indicating that the model is valid for predicting marker heterozygosity maintained at any loci in an RSB breeding scheme. The formulation is demonstrated numerically for several RSB schemes and its implications in developing a likelihood-based statistical framework for modeling the RSB experiments are discussed.     

Detection of closely linked multiple quantitative trait loci using a genetic algorithm

The existence of a quantitative trait locus (QTL) is usually tested using the likelihood of the quantitative trait on the basis of phenotypic character data plus the recombination fraction between QTL and flanking markers. When doing this, the likelihood is calculated for all possible locations on the linkage map. When multiple QTL are suspected close by, it is impractical to calculate the likelihood for all possible combinations of numbers and locations of QTL. Here, we propose a genetic algorithm (GA) for the heuristic solution of this problem. GA can globally search the optimum by improving the "genotype" with alterations called "recombination" and "mutation." The "genotype" of our GA is the number and location of QTL. The "fitness" is a function based on the likelihood plus Akaike's information criterion (AIC), which helps avoid false-positive QTL. A simulation study comparing the new method with existing QTL mapping packages shows the advantage of the new GA. The GA reliably distinguishes multiple QTL located in a single marker interval.