Joint linkage and linkage disequilibrium mapping of quantitative trait loci in natural populations

Linkage analysis and allelic association (also referred to as linkage disequilibrium) studies are two major approaches for mapping genes that control simple or complex traits in plants, animals, and humans. But these two approaches have limited utility when used alone, because they use only part of the information that is available for a mapping population. More recently, a new mapping strategy has been designed to integrate the advantages of linkage analysis and linkage disequilibrium analysis for genome mapping in outcrossing populations. The new strategy makes use of a random sample from a panmictic population and the open-pollinated progeny of the sample. In this article, we extend the new strategy to map quantitative trait loci (QTL), using molecular markers within the EM-implemented maximum-likelihood framework. The most significant advantage of this extension is that both linkage and linkage disequilibrium between a marker and QTL can be estimated simultaneously, thus increasing the efficiency and effectiveness of genome mapping for recalcitrant outcrossing species. Simulation studies are performed to test the statistical properties of the MLEs of genetic and genomic parameters including QTL allele frequency, QTL effects, QTL position, and the linkage disequilibrium of the QTL and a marker. The potential utility of our mapping strategy is discussed.     

Multitrait fine mapping of quantitative trait loci using combined linkage disequilibria and linkage analysis

A novel multitrait fine-mapping method is presented. The method is implemented by a model that treats QTL effects as random variables. The covariance matrix of allelic effects is proportional to the IBD matrix, where each element is the probability that a pair of alleles is identical by descent, given marker information and QTL position. These probabilities are calculated on the basis of similarities of marker haplotypes of individuals of the first generation of genotyped individuals, using "gene dropping" (linkage disequilibrium) and transmission of markers from genotyped parents to genotyped offspring (linkage). A small simulation study based on a granddaughter design was carried out to illustrate that the method provides accurate estimates of QTL position. Results from the simulation also indicate that it is possible to distinguish between a model postulating one pleiotropic QTL affecting two traits vs. one postulating two closely linked loci, each affecting one of the traits.     

Comparing linkage disequilibrium-based methods for fine mapping quantitative trait loci

Recently, a method for fine mapping quantitative trait loci (QTL) using linkage disequilibrium was proposed to map QTL by modeling covariance between individuals, due to identical-by-descent (IBD) QTL alleles, on the basis of the similarity of their marker haplotypes under an assumed population history. In the work presented here, the advantage of using marker haplotype information for fine mapping QTL was studied by comparing the IBD-based method with 10 markers to regression on a single marker, a pair of markers, or a two-locus haplotype under alternative population histories. When 10 markers were genotyped, the IBD-based method estimated the position of the QTL more accurately than did single-marker regression in all populations. When 20 markers were genotyped for regression, as single-marker methods do not require knowledge of haplotypes, the mapping accuracy of regression in all populations was similar to or greater than that of the IBD-based method using 10 markers. Thus for populations similar to those simulated here, the IBD-based method is comparable to single-marker regression analysis for fine mapping QTL.     

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.     

High-resolution joint linkage disequilibrium and linkage mapping of quantitative trait loci based on sibship data

This paper proposes variance component models for high resolution joint linkage disequilibrium (LD) and linkage mapping of quantitative trait loci (QTL) based on sibship data; this can include population data if independent individuals are treated as single sibships. One application of these models is late onset complex disease gene mapping, when parental data are not available. The models simultaneously incorporate both LD and linkage information. The LD information is contained in mean coefficients of sibship data. The linkage information is contained in the variance-covariance matrices of trait values for sibships with at least two siblings. We derive formulas for calculating the probability of sharing two trait alleles identical by descent (IBD) for sibpairs in interval mapping of QTL; this is the coefficient of dominant variance of the trait covariance of sibpairs on major QTL. To investigate the performance of the formulas, we calculate the numerical values via the formulas and get satisfactory approximations. We compare the power and sample sizes for both LD and linkage mapping. By simulation and theoretical analysis, we compare the results with those of Fulker and Abecasis "AbAw" approach. It is well known that the resolution of linkage analysis can be low for complex disease gene mapping. LD mapping, on the other hand, can increase mapping precision and is useful in high resolution mapping. Linkage analysis is less sensitive to population subdivisions and admixtures. The level of LD is sensitive to population stratification which may easily lead to spurious association. Performing a joint analysis of LD and linkage mapping can help to overcome the limits of both approaches. Moreover, the advantages of the two complementary strategies can be utilized maximally. In practice, linkage analysis may be performed using pedigree data to identify suggestive linkage between markers and trait loci based on a sparse marker map. In the presence of linkage, joint LD and linkage mapping can be carried out to do fine gene mapping based on a dense genetic map using both pedigree and population data. Population and pedigree data of any type can be combined to perform a joint analysis of high resolution LD and linkage mapping of QTL by generalizing the method.     

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.     

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.     

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.     

Fine mapping of quantitative trait loci using linkage disequilibria with closely linked marker loci

A multimarker linkage disequilibrium mapping method was developed for the fine mapping of quantitative trait loci (QTL) using a dense marker map. The method compares the expected covariances between haplotype effects given a postulated QTL position to the covariances that are found in the data. The expected covariances between the haplotype effects are proportional to the probability that the QTL position is identical by descent (IBD) given the marker haplotype information, which is calculated using the genedropping method. Simulation results showed that a QTL was correctly positioned within a region of 3, 1.5, or 0.75 cM in 70, 62, and 68%, respectively, of the replicates using markers spaced at intervals of 1, 0.5, and 0.25 cM, respectively. These results were rather insensitive to the number of generations since the QTL occurred and to the effective population size, except that 10 generations yielded rather poor estimates of the QTL position. The position estimates of this multimarker disequilibrium mapping method were more accurate than those from a single marker transmission disequilibrium test. A general approach for identifying QTL is suggested, where several stages of disequilibrium mapping are used with increasingly dense marker spacing.     

The efficiency of designs for fine-mapping of quantitative trait loci using combined linkage disequilibrium and linkage

In a simulation study, different designs were compared for efficiency of fine-mapping of QTL. The variance component method for fine-mapping of QTL was used to estimate QTL position and variance components. The design of many families with small size gave a higher mapping resolution than a design with few families of large size. However, the difference is small in half sib designs. The proportion of replicates with the QTL positioned within 3 cM of the true position is 0.71 in the best design, and 0.68 in the worst design applied to 128 animals with a phenotypic record and a QTL explaining 25% of the phenotypic variance. The design of two half sib families each of size 64 was further investigated for a hypothetical population with effective size of 1000 simulated for 6000 generations with a marker density of 0.25 cM and with marker mutation rate 4 × 10^-4 per generation. In mapping using bi-allelic markers, 42~55% of replicated simulations could position QTL within 0.75 cM of the true position whereas this was higher for multi allelic markers (48~76%). The accuracy was lowest (48%) when mutation age was 100 generations and increased to 68% and 76% for mutation ages of 200 and 500 generations, respectively, after which it was about 70% for mutation ages of 1000 generations and older. When effective size was linearly decreasing in the last 50 generations, the accuracy was decreased (56 to 70%). We show that half sib designs that have often been used for linkage mapping can have sufficient information for fine-mapping of QTL. It is suggested that the same design with the same animals for linkage mapping should be used for fine-mapping so gene mapping can be cost effective in livestock populations.     

Optimal selection on two quantitative trait loci with linkage

A mathematical approach to optimize selection on multiple quantitative trait loci (QTL) and an estimate of residual polygenic effects was applied to selection on two linked or unlinked additive QTL. Strategies to maximize total or cumulative discounted response over ten generations were compared to standard QTL selection on the sum of breeding values for the QTL and an estimated breeding value for polygenes, and to phenotypic selection. Optimal selection resulted in greater response to selection than standard QTL or phenotypic selection. Tight linkage between the QTL (recombination rate 0.05) resulted in a slightly lower response for standard QTL and phenotypic selection but in a greater response for optimal selection. Optimal selection capitalized on linkage by emphasizing selection on favorable haplotypes. When the objective was to maximize total response after ten generations and QTL were unlinked, optimal selection increased QTL frequencies to fixation in a near linear manner. When starting frequencies were equal for the two QTL, equal emphasis was given to each QTL, regardless of the difference in effects of the QTL and regardless of the linkage, but the emphasis given to each of the two QTL was not additive. These results demonstrate the ability of optimal selection to capitalize on information on the complex genetic basis of quantitative traits that is forthcoming.     

A correlation method for detecting and estimating linkage between a marker locus and a quantitative trait locus using inbred lines

The advent of molecular genetic markers has stimulated interest in detecting linkage between a marker locus and a quantitative trait locus (QTL) because the marker locus, even without direct effect on the quantitative trait, could be useful in increasing the response to selection. A correlation method for detecting and estimating linkage between a marker locus and a QTL is described using selfing and sib-mating populations. Computer simulations were performed to estimate the power of the method, the sample size (N) needed to detect linkage, and the recombination value (r). The power of this method was a function of the expected recombination value E(r), the standardized difference (d) between the QTL genotypic means, and N. The power was highest at complete linkage, decreased with an increase in E(r), and then increased at E(r)=0.5. A larger d and N led to a higher power. The sample size needed to detect linkage was dependent upon E(r) and d. The sample size had a minimum value at E(r)=0, increased with an increase in E(r) and a decrease in d. In general, the r was overestimated. With an increase in d, the r was closer to its expectation. Detection of linkage by the proposed method under incomplete linkage was more efficient than estimation of recombination values. The correlation method and the method of comparison of marker-genotype means have a similar power when there is linkage, but the former has a slightly higher power than the latter when there is no linkage.     

Optimum spacing of genetic markers for determining linkage between marker loci and quantitative trait loci

The cost of experiments aimed at determining linkage between marker loci and quantitative trait loci (QTL) was investigated as a function of marker spacing and number of individuals scored. It was found that for a variety of experimental designs, fairly wide marker spacings (ca. 50 cM) are optimum or close to optimum for initial studies of marker-QTL linkage, in the sense of minimizing overall cost of the experiment. Thus, even when large numbers of more or less evenly spaced markers are available, it will not always be cost effective to make full utilization of this capacity. This is particularly true when costs of rearing and trait evaluation per individual scored are low, as when marker data are obtained on individuals raised and evaluated for quantitative traits as part of existing programs. When costs of rearing and trait evaluation per individual scored are high, however, as in human family data collection carried out primarily for subsequent marker — QTL analyses, or when plants or animals are raised specifically for purposes of marker — QTL linkage experiments, optimum spacing may be rather narrow. It is noteworthy that when marginal costs of additional markers or individuals are constant, total resources allocated to a given experiment will determine total number of individuals sampled, but not the optimal marker spacing.     

A haplotype-based algorithm for multilocus linkage disequilibrium mapping of quantitative trait loci with epistasis

For tightly linked loci, cosegregation may lead to nonrandom associations between alleles in a population. Because of its evolutionary relationship with linkage, this phenomenon is called linkage disequilibrium. Today, linkage disequilibrium-based mapping has become a major focus of recent genome research into mapping complex traits. In this article, we present a new statistical method for mapping quantitative trait loci (QTL) of additive, dominant, and epistatic effects in equilibrium natural populations. Our method is based on haplotype analysis of multilocus linkage disequilibrium and exhibits two significant advantages over current disequilibrium mapping methods. First, we have derived closed-form solutions for estimating the marker-QTL haplotype frequencies within the maximum-likelihood framework implemented by the EM algorithm. The allele frequencies of putative QTL and their linkage disequilibria with the markers are estimated by solving a system of regular equations. This procedure has significantly improved the computational efficiency and the precision of parameter estimation. Second, our method can detect marker-QTL disequilibria of different orders and QTL epistatic interactions of various kinds on the basis of a multilocus analysis. This can not only enhance the precision of parameter estimation, but also make it possible to perform whole-genome association studies. We carried out extensive simulation studies to examine the robustness and statistical performance of our method. The application of the new method was validated using a case study from humans, in which we successfully detected significant QTL affecting human body heights. Finally, we discuss the implications of our method for genome projects and its extension to a broader circumstance. The computer program for the method proposed in this article is available at the webpage http://www.ifasstat.ufl.edu/genome/~LD.     

Interval mapping of quantitative trait loci with selective DNA pooling data

Selective DNA pooling is an efficient method to identify chromosomal regions that harbor quantitative trait loci (QTL) by comparing marker allele frequencies in pooled DNA from phenotypically extreme individuals. Currently used single marker analysis methods can detect linkage of markers to a QTL but do not provide separate estimates of QTL position and effect, nor do they utilize the joint information from multiple markers. In this study, two interval mapping methods for analysis of selective DNA pooling data were developed and evaluated. One was based on least squares regression (LS-pool) and the other on approximate maximum likelihood (ML-pool). Both methods simultaneously utilize information from multiple markers and multiple families and can be applied to different family structures (half-sib, F2 cross and backcross). The results from these two interval mapping methods were compared with results from single marker analysis by simulation. The results indicate that both LS-pool and ML-pool provided greater power to detect the QTL than single marker analysis. They also provide separate estimates of QTL location and effect. With large family sizes, both LS-pool and ML-pool provided similar power and estimates of QTL location and effect as selective genotyping. With small family sizes, however, the LS-pool method resulted in severely biased estimates of QTL location for distal QTL but this bias was reduced with the ML-pool.     

Optimal haplotype structure for linkage disequilibrium-based fine mapping of quantitative trait loci using identity by descent

A linkage disequilibrium-based method for fine mapping quantitative trait loci (QTL) has been described that uses similarity between individuals' marker haplotypes to determine if QTL alleles are identical by descent (IBD) to model covariances among individuals' QTL alleles for a mixed linear model. Mapping accuracy with this method was found to be sensitive to the number of linked markers that was included in the haplotype when fitting the model at a putative position of the QTL. The objective of this study was to determine the optimal haplotype structure for this IBD-based method for fine mapping a QTL in a previously identified QTL region. Haplotypes consisting of 1, 2, 4, 6, or all 10 available markers were fit as a "sliding window" across the QTL region under ideal and nonideal simulated population conditions. It was found that using haplotypes of 4 or 6 markers as a sliding "window" resulted in the greatest mapping accuracy under nearly all conditions, although the true IBD state at a putative QTL position was most accurately predicted by IBD probabilities obtained using all markers. Using 4 or 6 markers resulted in greater discrimination of IBD probabilities between positions while maintaining sufficient accuracy of IBD probabilities to detect the QTL. Fitting IBD probabilities on the basis of a single marker resulted in the worst mapping accuracy under all conditions because it resulted in poor accuracy of IBD probabilities. In conclusion, for fine mapping using IBD methods, marker information must be used in a manner that results in sensitivity of IBD probabilities to the putative position of the QTL while maintaining sufficient accuracy of IBD probabilities to detect the QTL. Contrary to expectation, use of haplotypes of 4-6 markers to derive IBD probabilities, rather than all available markers, best fits these criteria. Thus for populations similar to those simulated here, optimal mapping accuracy for this IBD-based fine-mapping method is obtained with a haplotype structure including a subset of all available markers.     

Simultaneous fine mapping of multiple closely linked quantitative trait Loci using combined linkage disequilibrium and linkage with a general pedigree

Within a small region (e.g., <10 cM), there can be multiple quantitative trait loci (QTL) underlying phenotypes of a trait. Simultaneous fine mapping of closely linked QTL needs an efficient tool to remove confounded shade effects among QTL within such a small region. We propose a variance component method using combined linkage disequilibrium (LD) and linkage information and a reversible jump Markov chain Monte Carlo (MCMC) sampling for model selection. QTL identity-by-descent (IBD) coefficients between individuals are estimated by a hybrid MCMC combining the random walk and the meiosis Gibbs sampler. These coefficients are used in a mixed linear model and an empirical Bayesian procedure combines residual maximum likelihood (REML) to estimate QTL effects and a reversible jump MCMC that samples the number of QTL and the posterior QTL intensities across the tested region. Note that two MCMC processes are used, i.e., an (internal) MCMC for IBD estimation and an (external) MCMC for model selection. In a simulation study, the use of the multiple-QTL model clearly removes the shade effects between three closely linked QTL located at 1.125, 3.875, and 7.875 cM across the region of 10 cM, using 40 markers at 0.25-cM intervals. It is shown that the use of combined LD and linkage information gives much more useful information compared to using linkage information alone for both single- and multiple-QTL analyses. When using a lower marker density (11 markers at 1-cM intervals), the signal of the second QTL can disappear. Extreme values of past effective size (resulting in extreme levels of LD) decrease the mapping accuracy.     

On the detection of imprinted quantitative trait loci in line crosses: effect of linkage disequilibrium

Imprinted quantitative trait loci (QTL) are commonly reported in studies using line-cross designs, especially in livestock species. It was previously shown that such parent-of-origin effects might result from the nonfixation of QTL alleles in one or both parental lines, rather than from genuine molecular parental imprinting. We herein demonstrate that if linkage disequilibrium exists between marker loci and nonfixed QTL, spurious detection of pseudo-imprinting is increased by an additional 40–80% in scenarios mimicking typical livestock situations. This is due to the fact that imprinting can be tested only in F₂ offspring whose sire and dam have distinct marker genotypes. In the case of linkage disequilibrium between markers and QTL, such parents have a higher chance to have distinct QTL genotypes as well, thus resulting in distinct padumnal and madumnal allele substitution effects, i.e., QTL pseudo-imprinting.     

Linkage disequilibrium fine mapping of quantitative trait loci: A simulation study

Recently, the use of linkage disequilibrium (LD) to locate genes which affect quantitative traits (QTL) has received an increasing interest, but the plausibility of fine mapping using linkage disequilibrium techniques for QTL has not been well studied. The main objectives of this work were to (1) measure the extent and pattern of LD between a putative QTL and nearby markers in finite populations and (2) investigate the usefulness of LD in fine mapping QTL in simulated populations using a dense map of multiallelic or biallelic marker loci. The test of association between a marker and QTL and the power of the test were calculated based on single-marker regression analysis. The results show the presence of substantial linkage disequilibrium with closely linked marker loci after 100 to 200 generations of random mating. Although the power to test the association with a frequent QTL of large effect was satisfactory, the power was low for the QTL with a small effect and/or low frequency. More powerful, multi-locus methods may be required to map low frequent QTL with small genetic effects, as well as combining both linkage and linkage disequilibrium information. The results also showed that multiallelic markers are more useful than biallelic markers to detect linkage disequilibrium and association at an equal distance.     

Detection of linkage between quantitative trait loci and restriction fragment length polymorphisms using inbred lines

Summary In segregating populations, large numbers of individuals are needed to detect linkage between markers, such as restriction fragment length polymorphisms (RFLPs), and quantitative trait loci (QTL), limiting the potential use of such markers for detecting linkage. Fewer individuals from inbred lines are needed to detect linkage. Simulation data were used to test the utility of two methods to detect linkage: maximum likelihood and comparison of marker genotype means. When there is tight linkage, the two methods have similar power, but when there is loose linkage, maximum likelihood is much more powerful. Once inbred lines have been established, they can be screened rapidly to detect QTL for several traits simultaneously. If there is sufficient coverage of the genome with RFLPs, several QTL for each trait may be detected.