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.     

Maximum likelihood techniques for the mapping and analysis of quantitative trait loci with the aid of genetic markers

A method is presented to estimate the biometric parameters of a quantitative trait locus linked to a genetic marker when both loci are segregating in the F-2 generation of a cross between two inbred lines. The method, which assumes underlying normal distributions, is a combination of maximum likelihood and moments methods and uses the statistics of the genetic marker genotype samples for the quantitative trait to estimate the recombination frequency between the two loci and the means and variances of the genotypes of the quantitative trait locus. With this method, the genetic parameters of a locus affecting plant height linked to an electrophoretic marker for esterase were accurately estimated from a sample of 1596 F-2 progeny of a cross between two species of Lycopersicon (tomato). Linkage distance between the two loci was 38 map units and the effect of the quantitative trait locus was 1.6 phenotypic standard deviation units. Accurate estimates of the genetic parameters and linkage distance for populations of 2000 individuals simulated with a segregating codominant locus with an effect of 1.63 standard deviations linked to a genetic marker with .2 recombination were also derived by this method. The method is not effective in distinguishing between complete and partial linkage in samples of only 500 individuals or for quantitative loci with effects less than a phenotypic standard deviation. The method is more effective for codominant than for dominant loci.     

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.     

High-resolution association mapping of quantitative trait loci: a population-based approach

In this article, population-based regression models are proposed for high-resolution linkage disequilibrium mapping of quantitative trait loci (QTL). Two regression models, the "genotype effect model" and the "additive effect model," are proposed to model the association between the markers and the trait locus. The marker can be either diallelic or multiallelic. If only one marker is used, the method is similar to a classical setting by Nielsen and Weir, and the additive effect model is equivalent to the haplotype trend regression (HTR) method by Zaykin et al. If two/multiple marker data with phase ambiguity are used in the analysis, the proposed models can be used to analyze the data directly. By analytical formulas, we show that the genotype effect model can be used to model the additive and dominance effects simultaneously; the additive effect model takes care of the additive effect only. On the basis of the two models, F-test statistics are proposed to test association between the QTL and markers. By a simulation study, we show that the two models have reasonable type I error rates for a data set of moderate sample size. The noncentrality parameter approximations of F-test statistics are derived to make power calculation and comparison. By a simulation study, it is found that the noncentrality parameter approximations of F-test statistics work very well. Using the noncentrality parameter approximations, we compare the power of the two models with that of the HTR. In addition, a simulation study is performed to make a comparison on the basis of the haplotype frequencies of 10 SNPs of angiotensin-1 converting enzyme (ACE) genes.     

The Contribution of Quantitative Trait Loci and Neutral Marker Loci to the Genetic Variances and Covariances among Quantitative Traits in Random Mating Populations

Using Cockerham's approach of orthogonal scales, we develop genetic models for the effect of an arbitrary number of multiallelic quantitative trait loci (QTLs) or neutral marker loci (NMLs) upon any number of quantitative traits. These models allow the unbiased estimation of the contributions of a set of marker loci to the additive and dominance variances and covariances among traits in a random mating population. The method has been applied to an analysis of allozyme and quantitative data from the European oyster. The contribution of a set of marker loci may either be real, when the markers are actually QTLs, or apparent, when they are NMLs that are in linkage disequilibrium with hidden QTLs. Our results show that the additive and dominance variances contributed by a set of NMLs are always minimum estimates of the corresponding variances contributed by the associated QTLs. In contrast, the apparent contribution of the NMLs to the additive and dominance covariances between two traits may be larger than, equal to or lower than the actual contributions of the QTLs. We also derive an expression for the expected variance explained by the correlation between a quantitative trait and multilocus heterozygosity. This correlation explains only a part of the genetic variance contributed by the markers, i.e., in general, a combination of additive and dominance variances and, thus, provides only very limited information relative to the method supplied here.     

QTL mapping under truncation selection in homozygous lines derived from biparental crosses

In plant breeding, a large number of progenies that will be discarded later in the breeding process must be phenotyped and marker genotyped for conducting QTL analysis. In many cases, phenotypic preselection of lines could be useful. However, in QTL analyses even moderate preselection can have a significant effect on the power of QTL detection and estimation of effects of the target traits. In this study, we provide exact formulas for quantifying the change of allele frequencies within marker classes, expectations of marker contrasts and the variance of the marker contrasts under truncation selection, for the general case of two QTL affecting the target trait and a correlated trait. We focused on homozygous lines derived at random from biparental crosses. The effects of linkage between the marker and the QTL under selection as well as the effect of selection on a correlated trait can be quantified with the given formulas. Theoretical results clearly show that depending on the magnitude of QTL effects, high selection intensities can lead to a dramatic reduction in power of QTL detection and that approximations based on the infinitesimal model deviate substantially from exact solutions. The presented formulas are valuable for choosing appropriate selection intensity when performing QTL mapping experiments on the data on phenotypically preselected traits and enable the calculation and bias correction of the effects of QTL under selection. Application of our theory to experimental data revealed that selection-induced bias of QTL effects can be successfully corrected.     

雄性不交换条件下F2群体区间标记——QTL定位的相关方法（英文）

提出了雄性不交换条件下F2群体区间标记定位QTL的相关方法,并且对其适用的条件进行了讨论,通过对分子区间标记进行赋值,计算在无交叉干涉条件下分子标记与表型值的简单相关系数,并在此基础上进行连锁检验,在特定条件下可以估计数量性状座位（QTL）与分子标记座位间的连锁值。     

Natural and Sexual Selection on Many Loci

A method is developed that describes the effects on an arbitrary number of autosomal loci of selection on haploid and diploid stages, of nonrandom mating between haploid individuals, and of recombination. We provide exact recursions for the dynamics of allele frequencies and linkage disequilibria (nonrandom associations of alleles across loci). When selection is weak relative to recombination, our recursions provide simple approximations for the linkage disequilibria among arbitrary combinations of loci. We show how previous models of sex-independent natural selection on diploids, assortative mating between haploids, and sexual selection on haploids can be analyzed in this framework. Using our weak-selection approximations, we derive new results concerning the coevolution of male traits and female preferences under natural and sexual selection. In particular, we provide general expressions for the intensity of linkage-disequilibrium induced selection experienced by loci that contribute to female preferences for specific male traits. Our general results support the previous observation that these indirect selection forces are so weak that they are unlikely to dominate the evolution of preference-producing loci.     

Simulation of marker-assisted selection utilizing linkage disequilibrium: the effects of several additional factors

Use was made of our published model and methods to investigate the effects of several additional factors on marker-assisted selection (MAS) utilizing linkage disequilibrium. The additional factors were: size of the sample used to estimate the marker quantitative trait locus (MQTL) association effects, the method used to estimate the MQTL effects, use of the average of the top MQTL estimates in selection rather than individual estimates, size of the selection population, and the crossing of duplicate selection lines to generate further linkage disequilibrium and further selection response. The average map distance between the quantitative trait loci (QTLs) and their nearest marker was 0.15 Morgans. Use of estimates of MQTL effects derived by least squares yielded smaller selection responses than estimates derived by mixed-model methods. Selection responses were also reduced by using a smaller sample for estimating the associations because MQTL effects were less well estimated. This applied to selection on the MQTL effects themselves and to selection combining the MQTL with phenotypic information. Thus, poorly estimated MQTL effects added noise to the system and reduced selection response in combined selection. Using the average of the top MQTL estimates, rather than individual estimates, also reduced selection response. New linkage disequilibrium, generated by crossing two lines selected from the same population, did not lead to additional selection response in the cross line. These results show limitations to MAS using linkage disequilibrium until close linkages of markers and QTLs are available.     

猪的基因图谱及数量性状位点定位

在人类基因组计划的带动下,猪的遗传连锁图谱和细胞遗传学图谱有了较大的进步,利用目前猪基因组图谱的研究成果,通过基因组扫描法和候选基因法,可以对猪重要经济性状的主效基因位点进行区域定位,进而图位克隆,找到主效基因,为现代遗传育种奠定理论基础。     

基于三点测交的双标记 -QTL基因定位的相关方法

提出在测交群体中,对区间标记座位赋值后求其与待定位的数量性状表型值间的简单相关系数Ｒ,以此进行连锁测验,并且在一定条件下用Ｒ值求出该数量性状座位（ＱＴＬ）与各标记座位（ＭＬ）间的重组值。     

A multilocus analysis of intraspecific competition and stabilizing selection on a quantitative trait

The equilibrium structure of an additive, diallelic multilocus model of a quantitative trait under frequency- and density-dependent selection is derived. The trait is under stabilizing selection and mediates intraspecific competition as induced, for instance, by differential resource utilization. It is assumed that stabilizing selection is weak, but the strength of competition may be arbitrary relative to it. Density dependence is caused by population regulation, which may be of a very general kind. The number and effects of loci are arbitrary, and stabilizing selection is not necessarily symmetric with respect to the range of phenotypic values. All previously studied models of intraspecific competition for a continuum of resources known to the author reduce to a special case of the present model if overall selection is weak. Therefore, in this case our results are applicable as approximations to all these models. Our central result is the (nearly) complete characterization of the equilibrium and stability structure in terms of all parameters. It is derived under the sole assumption that selection is weak enough relative to recombination to ignore linkage disequilibrium. In particular, necessary and sufficient conditions on the strength of competition relative to stabilizing selection are found that ensure the maintenance of multilocus polymorphism and the occurrence of disruptive selection. In this case, explicit formulas for the number of polymorphic loci at equilibrium, the allele frequencies, the genetic variance, and the strength of disruptive selection are obtained. For two loci, the effects of linkage are investigated analytically; for several loci, they are studied numerically.     

具交叉干涉的三点测交区间标记定位（QTL）的相关方法

本文提出在有交叉干涉条件下,在测交群体中对区间标记座位赋值后求其与待定位的数量性状表型值间的简单相关系数R,以此进行连锁测验,并且在一定条件下用R值求出该数量性状座位（QTL）与各标记座位（ML）间的重组值。     

甘蓝型油菜花瓣缺失基因的图谱定位

在无花瓣品系APT02和正常有花瓣品种中双4号构建的的F2分离群体中,运用AFLP和SRAP两种标记技术对甘蓝型油菜花瓣缺失基因进行分子标记和图谱定位。在两亲本间筛选20对AFLP引物和170对SRAP 引物,进一步通过BSA法筛选,获得了与甘蓝型油菜花瓣缺失基因WHB连锁的1个SRAP标记e8m3_4（600bp）和1个AFLP标记E3247_15（150bp）,标记与基因WHB之间的遗传距离分别为5 cM和13.5cM;构建了一个甘蓝型油菜(Brassica napus.L )的分子标记遗传连锁图谱,该图谱共包含213个AFLP标记、56个SRAP标记和1个形态标记,分布于17个主要连锁群、两个三联体和4个连锁对中,遗传图距总长2487.1cM,标记间平均距离为10.09 cM。通过图谱定位,控制花瓣缺失性状的基因WHB被定位到第4连锁群(LG4)上。     

The extent and distribution of linkage disequilibrium in a multi-hierarchic outbred canine pedigree

A canine integrated linkage-radiation map has been recently constructed by using microsatellite markers. This map, with a good coverage of the canine genome, allows for a genome-wide search for the extent and distribution of linkage disequilibrium derived from linkage and evolutionary forces. In this study, we genotyped an outbred pedigree between Labrador retriever and Greyhound breeds with a set of microsatellite markers (240) from the canine linkage map. Linkage disequilibrium was measured between all syntenic and nonsyntenic marker pairs. Analysis of syntenic pairs revealed a significant correlation (–0.229, P < 0.001) between linkage disequilibrium and genetic distance (log transformed). Significant linkage disequilibria were observed more frequently between syntenic pairs spaced <40 cM than those paced >40 cM. There is a clear trend for linkage disequilibrium to decline with marker distance. From our results, a genome-wide screen with markers at low to moderate density (1–2 per 10 cM) should take full advantage of linkage disequilibrium for quantitative trait locus mapping in dogs. This study supports the appropriateness of linkage disequilibrium analysis to detect and map quantitative trait loci underlying complex traits in dogs.     

A Random Model Approach to Interval Mapping of Quantitative Trait Loci

Mapping quantitative trait loci in outbred populations is important because many populations of organisms are noninbred. Unfortunately, information about the genetic architecture of the trait may not be available in outbred populations. Thus, the allelic effects of genes can not be estimated with ease. In addition, under linkage equilibrium, marker genotypes provide no information about the genotype of a QTL (our terminology for a single quantitative trait locus is QTL while multiple loci are referred to as QTLs). To circumvent this problem, an interval mapping procedure based on a random model approach is described. Under a random model, instead of estimating the effects, segregating variances of QTLs are estimated by a maximum likelihood method. Estimation of the variance component of a QTL depends on the proportion of genes identical-by-descent (IBD) shared by relatives at the locus, which is predicted by the IBD of two markers flanking the QTL. The marker IBD shared by two relatives are inferred from the observed marker genotypes. The procedure offers an advantage over the regression interval mapping in terms of high power and small estimation errors and provides flexibility for large sibships, irregular pedigree relationships and incorporation of common environmental and fixed effects.     

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.     

An SSR genetic map of Sorghum bicolor (L.) Moench and its comparison to a published genetic map.

Sorghum bicolor (L.) Moench is an important grain and forage crop grown worldwide. We developed a simple sequence repeat (SSR) linkage map for sorghum using 352 publicly available SSR primer pairs and a population of 277 F2 individuals derived from a cross between the Westland A line and PI 550610. A total of 132 SSR loci appeared polymorphic in the mapping population, and 118 SSRs were mapped to 16 linkage groups. These mapped SSR loci were distributed throughout 10 chromosomes of sorghum, and spanned a distance of 997.5 cM. More important, 38 new SSR loci were added to the sorghum genetic map in this study. The mapping result also showed that chromosomes SBI-01, SBI-02, SBI-05, and SBI-06 each had 1 linkage group; the other 6 chromosomes were composed of 2 linkage groups each. Except for 5 closely linked marker flips and 1 locus (Sb6_34), the marker order of this map was collinear to a published sorghum map, and the genetic distances of common marker intervals were similar, with a difference ratio 相似文献   

An apricot (Prunus armeniaca L.) F2 progeny linkage map based on SSR and AFLP markers,mapping plum pox virus resistance and self-incompatibility traits

A genetic linkage map of apricot ( Prunus armeniaca L.) was constructed using AFLP and SSR markers. The map is based on an F(2) population (76 individuals) derived from self-pollination of an F(1) individual ('Lito') originated from a cross between 'Stark Early Orange' and 'Tyrinthos'. This family, designated as 'Lito' x 'Lito', segregated for two important agronomical traits: plum pox virus resistance (PPV) and self-incompatibility. A total of 211 markers (180 AFLPs, 29 SSRs and two agronomic traits) were assigned to 11 linkage groups covering 602 cM of the apricot genome. The average distance (cM/marker) between adjacent markers is 3.84 cM. The PPV resistance trait was mapped on linkage group G1 and the self-incompatibility trait was mapped on linkage group G6. Twenty two loci held in common with other Prunus maps allowed us to compare and establish homologies among the respective linkage groups.     

A Bayesian Approach to Detect Quantitative Trait Loci Using Markov Chain Monte Carlo

Markov chain Monte Carlo (MCMC) techniques are applied to simultaneously identify multiple quantitative trait loci (QTL) and the magnitude of their effects. Using a Bayesian approach a multi-locus model is fit to quantitative trait and molecular marker data, instead of fitting one locus at a time. The phenotypic trait is modeled as a linear function of the additive and dominance effects of the unknown QTL genotypes. Inference summaries for the locations of the QTL and their effects are derived from the corresponding marginal posterior densities obtained by integrating the likelihood, rather than by optimizing the joint likelihood surface. This is done using MCMC by treating the unknown QTL genotypes, and any missing marker genotypes, as augmented data and then by including these unknowns in the Markov chain cycle along with the unknown parameters. Parameter estimates are obtained as means of the corresponding marginal posterior densities. High posterior density regions of the marginal densities are obtained as confidence regions. We examine flowering time data from double haploid progeny of Brassica napus to illustrate the proposed method.