The covariance between relatives conditional on genetic markers

The development of molecular genotyping techniques makes it possible to analyze quantitative traits on the basis of individual loci. With marker information, the classical theory of estimating the genetic covariance between relatives can be reformulated to improve the accuracy of estimation. In this study, an algorithm was derived for computing the conditional covariance between relatives given genetic markers. Procedures for calculating the conditional relationship coefficients for additive, dominance, additive by additive, additive by dominance, dominance by additive and dominance by dominance effects were developed. The relationship coefficients were computed based on conditional QTL allelic transmission probabilities, which were inferred from the marker allelic transmission probabilities. An example data set with pedigree and linked markers was used to demonstrate the methods developed. Although this study dealt with two QTLs coupled with linked markers, the same principle can be readily extended to the situation of multiple QTL. The treatment of missing marker information and unknown linkage phase between markers for calculating the covariance between relatives was discussed.     

Relationship between heterosis and heterozygosity at marker loci: a theoretical computation

Summary In this paper we have studied the linear correlation between a genetic distance index between two parent lines (based on marker loci information) and the heterosis observed in the F₁ hybrid from the two lines, for a quantitative character (determined by several loci, or QTL). Theoretical computations of the correlation coefficient () between the distance index and the heterosis were made, assuming the biallelic model (defined by Fisher). When the alleles at both marker loci and QTL are equally distributed among the whole population of considered lines, the coefficient is a function of the squares of linkage disequilibria between alleles at marker loci and alleles at QTL. The QTL that are not marked by marker loci and marker loci that do not mark any QTL play symmetrical roles and can decrease greatly. We conclude that the prediction of F₁ hybrid heterosis based on marker loci would be more efficient if these markers were selected for their relationship to the alleles implicated in the heterotic traits considered.     

Identification of Quantitative Trait Loci Influencing Wood Specific Gravity in an Outbred Pedigree of Loblolly Pine

We report the identification of quantitative trait loci (QTL) influencing wood specific gravity (WSG) in an outbred pedigree of loblolly pine (Pinus taeda L.). QTL mapping in an outcrossing species is complicated by the presence of multiple alleles (>2) at QTL and marker loci. Multiple alleles at QTL allow the examination of interaction among alleles at QTL (deviation from additive gene action). Restriction fragment length polymorphism (RFLP) marker genotypes and wood specific gravity phenotypes were determined for 177 progeny. Two RFLP linkage maps were constructed, representing maternal and paternal parent gamete segregations as inferred from diploid progeny RFLP genotypes. RFLP loci segregating for multiple alleles were vital for aligning the two maps. Each RFLP locus was assayed for cosegregation with WSG QTL using analysis of variance (ANOVA). Five regions of the genome contained one or more RFLP loci showing differences in mean WSG at or below the P = 0.05 level for progeny as grouped by RFLP genotype. One region contained a marker locus (S6a) whose QTL-associated effects were highly significant (P > 0.0002). Marker S6a segregated for multiple alleles, a prerequisite for determining the number of alleles segregating at the linked QTL and analyzing the interactions among QTL alleles. The QTL associated with marker S6a appeared to be segregating for multiple alleles which interacted with each other and with environments. No evidence for digenic epistasis was found among the five QTL.     

Theory for identification of marker locus-QTL associations in population by line crosses

The objective of this paper is to present genetic theory demonstrating the conditions under which it should be possible to identify molecular marker-quantitative trait locus (QTL) associations in crosses of random-mating populations to inbreds. Using as an example the cross of a corn (Zea mays L.) population to an inbred, the expected disequilibrium for testcross and per se performance of F₂, F₃, BC₁ (to the inbred) and recombinant inbred generations was derived for cases where a marker allele is linked to an unfavorable QTL allele in the inbred and where the marker allele is linked to a favorable QTL allele in the inbred. Disequilibrium in segregating generations was shown to be a function of disequilibrium in the parent population, the frequency of marker and QTL alleles in the parent population, and the recombination distance between the marker and the QTL. To maximize the opportunity to identify a favorable QTL the following procedures are suggested:

Detection and parameter estimation for quantitative trait loci using regression models and multiple markers

A strategy of multi-step minimal conditional regression analysis has been developed to determine the existence of statistical testing and parameter estimation for a quantitative trait locus (QTL) that are unaffected by linked QTLs. The estimation of marker-QTL recombination frequency needs to consider only three cases: 1) the chromosome has only one QTL, 2) one side of the target QTL has one or more QTLs, and 3) either side of the target QTL has one or more QTLs. Analytical formula was derived to estimate marker-QTL recombination frequency for each of the three cases. The formula involves two flanking markers for case 1), two flanking markers plus a conditional marker for case 2), and two flanking markers plus two conditional markers for case 3). Each QTL variance and effect, and the total QTL variance were also estimated using analytical formulae. Simulation data show that the formulae for estimating marker-QTL recombination frequency could be a useful statistical tool for fine QTL mapping. With 1 000 observations, a QTL could be mapped to a narrow chromosome region of 1.5 cM if no linked QTL is present, and to a 2.8 cM chromosome region if either side of the target QTL has at least one linked QTL.     

Identifying quantitative trait locus by genetic background interactions in association studies

Association studies are designed to identify main effects of alleles across a potentially wide range of genetic backgrounds. To control for spurious associations, effects of the genetic background itself are often incorporated into the linear model, either in the form of subpopulation effects in the case of structure or in the form of genetic relationship matrices in the case of complex pedigrees. In this context epistatic interactions between loci can be captured as an interaction effect between the associated locus and the genetic background. In this study I developed genetic and statistical models to tie the locus by genetic background interaction idea back to more standard concepts of epistasis when genetic background is modeled using an additive relationship matrix. I also simulated epistatic interactions in four-generation randomly mating pedigrees and evaluated the ability of the statistical models to identify when a biallelic associated locus was epistatic to other loci. Under additive-by-additive epistasis, when interaction effects of the associated locus were quite large (explaining 20% of the phenotypic variance), epistasis was detected in 79% of pedigrees containing 320 individuals. The epistatic model also predicted the genotypic value of progeny better than a standard additive model in 78% of simulations. When interaction effects were smaller (although still fairly large, explaining 5% of the phenotypic variance), epistasis was detected in only 9% of pedigrees containing 320 individuals and the epistatic and additive models were equally effective at predicting the genotypic values of progeny. Epistasis was detected with the same power whether the overall epistatic effect was the result of a single pairwise interaction or the sum of nine pairwise interactions, each generating one ninth of the epistatic variance. The power to detect epistasis was highest (94%) at low QTL minor allele frequency, fell to a minimum (60%) at minor allele frequency of about 0.2, and then plateaued at about 80% as alleles reached intermediate frequencies. The power to detect epistasis declined when the linkage disequilibrium between the DNA marker and the functional polymorphism was not complete.     

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.     

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.     

QTL fine mapping by measuring and testing for Hardy-Weinberg and linkage disequilibrium at a series of linked marker loci in extreme samples of populations

It has recently been demonstrated that fine-scale mapping of a susceptibility locus for a complex disease can be accomplished on the basis of deviations from Hardy-Weinberg (HW) equilibrium at closely linked marker loci among affected individuals. We extend this theory to fine-scale localization of a quantitative-trait locus (QTL) from extreme individuals in populations, by means of HW and linkage-disequilibrium (LD) analyses. QTL mapping and/or linkage analyses can establish a large genomic region ( approximately 30 cM) that contains a QTL. The QTL can be fine mapped by examination of the degree of deviation from HW and LD at a series of closely linked marker loci. The tests can be performed for samples of individuals belonging to either high or low percentiles of the phenotype distribution or for combined samples of these extreme individuals. The statistical properties (the power and the size) of the tests of this fine-mapping approach are investigated and are compared extensively, under various genetic models and parameters for the QTL and marker loci. On the basis of the results, a two-stage procedure that uses extreme samples and different tests (for HW and LD) is suggested for QTL fine mapping. This two-step procedure is economic and powerful and can accurately narrow a genomic region containing a QTL from approximately 30-1 cM, a range that renders physical mapping feasible for identification of the QTL. In addition, the relationship between parameterizations of complex diseases, by means of penetrance, and those of complex quantitative traits, by means of genotypic values, is outlined. This means that many statistical genetic methods developed for searching for susceptibility loci of complex diseases can be directly adopted and/or extended to QTL mapping for quantitative traits.     

Quantitative trait locus mapping of pyrethroid resistance in Colorado potato beetle, Leptinotarsa decemlineata (Say) (Coleoptera: Chrysomelidae)

Quantitative trait locus (QTL) mapping is a valuable new tool for locating genomic regions that underlie variation in important traits such as insecticide resistance. Because QTL mapping complements a candidate gene strategy for understanding the genetic architecture of important traits, it may also facilitate the identification of genes causing important variation. After mapping the QTL locations, markers closely linked to QTL can be used for genetic analysis of population structure and to measure the spread and increase of resistance-causing QTL alleles. In this study, QTL influencing resistance to the pyrethroid insecticide esfenvalerate were mapped in the Colorado potato beetle Leptinotarsa decemlineata (Say) (CPB). Three QTL contributing to esfenvalerate resistance were identified from a mapping population of 79 individuals analyzed at 90 marker loci. One QTL had a large effect and two QTL had smaller effects. The major QTL occurs on the X chromosome, overlapping the position of a candidate gene (Leptinotarsa decemlineata Voltage sensitive sodium channel [LdVssc1]) previously implicated in pyrethroid resistance. Resistance-increasing alleles at the two minor-effect QTL originated with the susceptible parent, suggesting that alleles of small effect may be segregating in susceptible populations. Comparison of the New York population from which the susceptible parent originated with a more-susceptible population from North Carolina suggests that the minor-effect loci identified here may explain some of the variation in tolerance observed among susceptible populations. DNA sequencing of a portion of LdVssc1 shows that the resistance-conferring allele from the resistant parent does not contain the kdr mutation previously found in CPB and typically observed in other insects that are resistant to pyrethroid insecticides because of changes in this gene.     

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.     

人类复杂遗传疾病QTL分析方法的理论探讨

利用连锁不平衡理论,人类遗传学家已能把影响人类疾病的质量基因定位在小至1cM区域内,有些基因已被克隆出来。罗泽伟等进一步发展统计分析方法检测及估算分子标记与QTL之间的连锁不平衡系数,从而提出了人类复杂遗传病高解析度基因定位的理论策略。以此为基础,进一步探讨了供试群体在双亲基因频率存在差异时检测QTL和检测QTL互作的方法,给出了有关的理论结果。     

Mapping and Analysis of Dairy Cattle Quantitative Trait Loci by Maximum Likelihood Methodology Using Milk Protein Genes as Genetic Markers

Maximum likelihood methodology was used to estimate effects of both a marker gene and a linked quantitative trait locus (QTL) on quantitative traits in a segregating population. Two alleles were assumed for the QTL. In addition to the effects of genotypes at both loci on the mean of the quantitative trait, recombination frequency between the loci, frequency of the QTL alleles and the residual standard deviation were also estimated. Thus six parameters were estimated in addition to the marker genotype means. The statistical model was tested on simulated data, and used to estimate direct and linked effects of the milk protein genes, β-lactoglobulin, κcasein, and β-casein, on milk, fat, and protein production and fat and protein percent in the Dutch dairy cattle population. β-Lactoglobulin had significant direct effects on milk yield and fat percent. κ-Casein had significant direct effects on milk yield, protein percent and fat yield. β-Casein had significant direct effects on milk yield, fat and protein percent and fat and protein yield. Linked QTL with significant effects on fat percent were found for κ-casein and β-casein. Since the β-casein and κ-casein genes are closely linked, it is likely that the same QTL was detected for those two markers. Further, a QTL with a significant effect on fat yield was found to be linked to κ-casein and a QTL with a significant effect on protein yield was linked to β-lactoglobulin.     

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.     

Efficient computation of the inverse of gametic relationship matrix for a marked QTL

Best linear unbiased prediction of genetic merits for a marked quantitative trait locus (QTL) using mixed model methodology includes the inverse of conditional gametic relationship matrix (G^-1) for a marked QTL. When accounting for inbreeding, the conditional gametic relationships between two parents of individuals for a marked QTL are necessary to build G^-1 directly. Up to now, the tabular method and its adaptations have been used to compute these relationships. In the present paper, an indirect method was implemented at the gametic level to compute these few relationships. Simulation results showed that the indirect method can perform faster with significantly less storage requirements than adaptation of the tabular method. The efficiency of the indirect method was mainly due to the use of the sparseness of G^-1. The indirect method can also be applied to construct an approximate G^-1 for populations with incomplete marker data, providing approximate probabilities of descent for QTL alleles for individuals with incomplete marker data.     

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.     

Using molecular markers to map multiple quantitative trait loci: models for backcross,recombinant inbred,and doubled haploid progeny

Summary To maximize parameter estimation efficiency and statistical power and to estimate epistasis, the parameters of multiple quantitative trait loci (QTLs) must be simultaneously estimated. If multiple QTL affect a trait, then estimates of means of QTL genotypes from individual locus models are statistically biased. In this paper, I describe methods for estimating means of QTL genotypes and recombination frequencies between marker and quantitative trait loci using multilocus backcross, doubled haploid, recombinant inbred, and testcross progeny models. Expected values of marker genotype means were defined using no double or multiple crossover frequencies and flanking markers for linked and unlinked quantitative trait loci. The expected values for a particular model comprise a system of nonlinear equations that can be solved using an interative algorithm, e.g., the Gauss-Newton algorithm. The solutions are maximum likelihood estimates when the errors are normally distributed. A linear model for estimating the parameters of unlinked quantitative trait loci was found by transforming the nonlinear model. Recombination frequency estimators were defined using this linear model. Certain means of linked QTLs are less efficiently estimated than means of unlinked QTLs.     

Selection mapping of loci for quantitative disease resistance in a diverse maize population

The selection response of a complex maize population improved primarily for quantitative disease resistance to northern leaf blight (NLB) and secondarily for common rust resistance and agronomic phenotypes was investigated at the molecular genetic level. A tiered marker analysis with 151 simple sequence repeat (SSR) markers in 90 individuals of the population indicated that on average six alleles per locus were available for selection. An improved test statistic for selection mapping was developed, in which quantitative trait loci (QTL) are identified through the analysis of allele-frequency shifts at mapped multiallelic loci over generations of selection. After correcting for the multiple tests performed, 25 SSR loci showed evidence of selection. Many of the putatively selected loci were unlinked and dispersed across the genome, which was consistent with the diffuse distribution of previously published QTL for NLB resistance. Compelling evidence for selection was found on maize chromosome 8, where several putatively selected loci colocalized with published NLB QTL and a race-specific resistance gene. Analysis of F(2) populations derived from the selection mapping population suggested that multiple linked loci in this chromosomal segment were, in part, responsible for the selection response for quantitative resistance to NLB.     

The use of intraallelic variability for testing neutrality and estimating population growth rate

To better understand the forces affecting individual alleles, we introduce a method for finding the joint distribution of the frequency of a neutral allele and the extent of variability at closely linked marker loci (the intraallelic variability). We model three types of intraallelic variability: (a) the number of nonrecombinants at a linked biallelic marker locus, (b) the length of a conserved haplotype, and (c) the number of mutations at a linked marker locus. If the population growth rate is known, the joint distribution provides the basis for a test of neutrality by testing whether the observed level of intraallelic variability is consistent with the observed allele frequency. If the population growth rate is unknown but neutrality can be assumed, the joint distribution provides the likelihood of the growth rate and leads to a maximum-likelihood estimate. We apply the method to data from published data sets for four loci in humans. We conclude that the Delta32 allele at CCR5 and a disease-associated allele at MLH1 arose recently and have been subject to strong selection. Alleles at PAH appear to be neutral and we estimate the recent growth rate of the European population to be approximately 0.027 per generation with a support interval of (0.017-0.037). Four of the relatively common alleles at CFTR also appear to be neutral but DeltaF508 appears to be significantly advantageous to heterozygous carriers.     

Matrix representation of the Haseman-Elston method.

The Hasemann-Elston method of linkage detection is based on the probabilities of a sib pair having 0, 1, or 2 alleles identical by descent (IBD) at a marker and a trait locus. These probabilities form a 3x3 matrix. Here, the characteristic values and characteristic vectors of this matrix were used to clarify the structure of the equations and to simplify calculations. As examples, the regression coefficients were derived for three genetic systems: a trait and a marker, two epistatic traits and two markers, and one trait locus and two markers. The last model was studied under the assumption of no crossover interference, the expression for allele IBD sharing at a trait locus was derived as a function of allele IBD sharing at two marker loci, and the regression is shown to be non-linear.