Quantitative Trait Loci Genotypes Research Articles

Iteratively reweighted least squares mapping of quantitative trait loci.

Mapping quantitative trait loci (QTL) is a typical problem of regression with uncertain independent variables because the genotype of a putative QTL is not observed. Rather, the genotype is inferred from marker information. The method of maximum likelihood (ML) methods is considered to be the optimal solution for this problem because the distribution of the unobserved QTL genotype is fully taken into account. The simple linear regression method (REG) is a first-order approximation to ML and usually performs very well. In this study, an iteratively reweighted least squares method (IRWLS) is proposed. The new method is a second-order approximation to ML because both the expectation and the variance of the unobserved QTL genotype are taken into consideration. The IRWLS is developed in the context of a single large outbred family. The properties of IRWLS are demonstrated and compared with REG and ML via replicated Monte Carlo simulations. The conclusions are: (1) when marker information content is high, the three methods perform equally well, but ML and IRWLS outperform REG when marker information content is low and the variance explained by the QTL is high; (2) when the residual distribution is not normal, ML can fail or have low power to detect small QTLs, but REG and IRWLS are robust to non-normality; and (3) when the residual distribution is normal, the performance of IRWLS is almost identical to ML, but the computational speed of IRWLS is many times faster than that of ML.

Mapping quantitative trait loci for binary traits using a heterogeneous residual variance model: an application to Marek's disease susceptibility in chickens.

A typical problem in mapping quantitative trait loci (QTLs) comes from missing QTL genotype. A routine method for parameter estimation involving missing data is the mixture model maximum likelihood method. We developed an alternative QTL mapping method that describes a mixture of several distributions by a single model with a heterogeneous residual variance. The two methods produce similar results, but the heterogeneous residual variance method is computationally much faster than the mixture model approach. In addition, the new method can automatically generate sampling variances of the estimated parameters. We derive the new method in the context of QTL mapping for binary traits in a F2 population. Using the heterogeneous residual variance model, we identified a QTL on chromosome IV that controls Marek's disease susceptibility in chickens. The QTL alone explains 7.2% of the total disease variation.

General Formulas for Obtaining the MLEs and the Asymptotic Variance- Covariance Matrix in Mapping Quantitative Trait Loci When Using the EM Algorithm

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

Estimation of effects of quantitative trait loci in large complex pedigrees.

A method was derived to estimate effects of quantitative trait loci (QTL) using incomplete genotype information in large outbreeding populations with complex pedigrees. The method accounts for background genes by estimating polygenic effects. The basic equations used are very similar to the usual linear mixed model equations for polygenic models, and segregation analysis was used to estimate the probabilities of the QTL genotypes for each animal. Method R was used to estimate the polygenic heritability simultaneously with the QTL effects. Also, initial allele frequencies were estimated. The method was tested in a simulated data set of 10,000 animals evenly distributed over 10 generations, where 0, 400 or 10,000 animals were genotyped for a candidate gene. In the absence of selection, the bias of the QTL estimates was < 2%. Selection biased the estimate of the Aa genotype slightly, when zero animals were genotyped. Estimates of the polygenic heritability were 0.251 and 0.257, in absence and presence of selection, respectively, while the simulated value was 0.25. Although not tested in this study, marker information could be accommodated by adjusting the transmission probabilities of the genotypes from parent to offspring according to the marker information. This renders a QTL mapping study in large multi-generation pedigrees possible.

Utilisation of marker assisted selection in a commercial dairy cow population

Abstract Genetic and economic benefits of marker assisted selection (MAS) to a commercial dairy cow population were evaluated by calculating selection advance resulting from four pathways of selection. In addition to background polygenic effects, a single additive quantitative trait loci (QTL) was segregating. Three sizes of QTL were assessed; 0.5, 1.0 and 2.0 genetic standard deviations ( σ G ) (difference between homozygotes), at each of four starting QTL frequencies; 0.01, 0.10, 0.35 and 0.75 over a thirty year time horizon. No recombination existed between QTL and a marker. Two MAS strategies were evaluated by comparison to a breeding scheme that had no genotypic knowledge of the QTL, over a thirty year time horizon. Economic benefits were calculated from the returns of extra milk produced (accounting for increased feed costs) less identification costs of the QTL and subsequent genotyping costs. In both strategies increase in QTL frequency was not immediate and thus returns were received in the later years of the analysis. The MAS strategy that utilised knowledge of QTL genotype for bull dams and bull sires had superior genetic gain at all QTL sizes and frequencies. However, it was only profitable for a 1.0 σ G QTL at 0.1 and 0.35 frequencies, and a 2.0 σ G QTL at all except the highest frequency (0.75). Long-term genetic loss was observed. The second MAS strategy of progeny testing only homozygous and heterozygous QTL bulls required a QTL of size 1.0 σ G at frequencies of 0.1 or 0.35 or a 2.0 σ G QTL at frequencies of 0.01, 0.1 and 0.35 to be more profitable than the current breeding scheme. The choice of MAS strategy should depend on QTL size and frequency.

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 alone 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.

Models for mapping quantitative trait loci (QTL) in progeny of non-inbred parents and their behaviour in presence of distorted segregation ratios

SummaryIn plants, models for mapping quantitative trait loci (QTL) based on flanking markers have been mainly developed for progenies of inbred lines. We propose twoflanking marker models for QTL mapping in F1progenies of non-inbred parents. The first is based on the segregation of four different scorable alleles at a marker locus (the four-allele model) and the second (the commonallele model) on one scorable allele per marker locus segregating in both parents. These models are suitable for the majority of the allelic configurations which may occur in crosses between heterozygous parents. For both cases, when four scorable or one common-allele per marker locus segregate, additional algorithms were developed to estimate the recombination frequency between two marker loci. Tests carried out with simulated populations of various sizes indicate that the models provide a good estimate of QTL genotypic means and of recombination frequencies between flanking markers and between the marker loci and the QTL.The estimates of QTL genotypic means have a higher precision than the estimates of recombination frequencies. The four-allele model shows a higher ability to detect QTLs than the common-allele model. If segregation ratios are distorted, the power of both models and the precision of the estimates of recombination frequencies are reduced, whereas the accuracy of estimates of QTL genotype means is not affected by distorted segregation ratios. The power of the common-allele model is substantially reduced if QTL genotypic means depend on additive allelic interactions, whereas the four-allele model is less affected by the non-additive behaviour of QTL alleles.

Genetic mapping of QTLs controlling vegetative propagation in Eucalyptus grandis and E. urophylla using a pseudo-testcross strategy and RAPD markers

We have extended the combined use of the "pseudo-testcross" mapping strategy and RAPD markers to map quantitative trait loci (QTLs) controlling traits related to vegetative propagation in Eucalyptus. QTL analyses were performed using two different interval mapping approaches, MAPMAKER-QTL (maximum likelihood) and QTL-STAT (non-linear least squares). A total of ten QTLs were detected for micropropagation response (measured as fresh weight of shoots, FWS), six for stump sprouting ability (measured as # stump sprout cuttings, #Cutt) and four for rooting ability (measured as % rooting of cuttings, %Root). With the exception of three QTLs, both interval-mapping methods yielded similar results in terms of QTL detection. Discrepancies in the most likely QTL location were observed between the two methods. In 75% of the cases the most likely position was in the same, or in an adjacent, interval. Standardized gene substitution effects for the QTLs detected were typically between 0.46 and 2.1 phenotypic standard deviations (σp), while differences between the family mean and the favorable QTL genotype were between 0.25 and 1.07 (σp). Multipoint estimates of the total genetic variation explained by the QTLs (89.0% for FWS, 67.1 % for#Cutt, 62.7% for %Root) indicate that a large proportion of the variation in these traits is controlled by a relatively small number of major-effect QTLs. In this cross, E. grandis is responsible for most of the inherited variation in the ability to form shoots, while E. urophylla contributes most of the ability in rooting. QTL mapping in the pseudo-testcross configuration relies on withinfamily linkage disequilibrium to establish marker/trait associations. With this approach QTL analysis is possible in any available full-sib family generated from undomesticated and highly heterozygous organisms such as forest trees. QTL mapping on two-generation pedigrees opens the possibility of using already existing families in retrospective QTL analyses to gather the quantitative data necessary for marker-assisted tree breeding.

Derivation of single-locus relationship coefficients conditional on marker information

The coefficient of relationship is defined as the correlation between the additive genetic values of two individuals. This coefficient can be defined specifically for a single quantitative trait locus (QTL) and may deviate considerably from the overall expectation if it is taken conditional on information from linked marker loci. Conditional halfsib correlations are derived under a simple genetic model with a biallelic QTL linked to a biallelic marker locus. The conditional relationship coefficients are shown to depend on the recombination rate between the marker and the QTL and the population frequency of the marker alleles, but not on parameters of the QTL, i.e. number and frequency of QTL alleles, degree of dominance etc., nor on the (usually unknown) QTL genotype of the sire. Extensions to less simplified cases (multiple alleles at the marker locus and the QTL, two marker loci flanking the QTL) are given. For arbitrary pedigrees, conditional relationship coefficients can also be derived from the conditional gametic covariance matrix suggested by Fernando and Grossman (1989). The connection of these two approaches is discussed. The conditional relationship coefficient can be used for marker-assisted genetic evaluation as well as for the detection of QTL and the estimation of their effects.

Statistical analysis of a linkage experiment in barley involving quantitative trait loci for height and ear-emergence time and two genetic markers on chromosome 4

The quantitative traits height and ear-emergence date were analyzed in the F2 progeny of a cross between a tall winter barley cultivar (Gerbel) and a short spring barley cultivar (Heriot). The trait distributions were found to be related to the genotypes at two biochemical loci, β-amylase (Bmy1) and water-soluble protein (Wsp3), which are known to lie on the long arm of chromosome 4. Linkages between each trait and the markers were investigated using normal mixture models. The two parental phenotypes and the heterozygote phenotype of Bmy1 were distinguishable so the model could be used directly to estimate linkage between Bmy1 and a quantitative trait locus (QTL) for height (Height). The Gerbel homozygote and heterozygote phenotype of Wsp3 could not be distinguished and the model was adapted accordingly. The proportion of plants requiring vernalization was consistent with control by two independent genes acting epistatically, and a normal mixture model based on a two-gene hypothesis was fitted to the distribution of ear-emergence date to estimate linkage between the marker loci and a QTL for ear-emergence date (Vrn1). The parameters of each model were the recombination fraction between the marker locus and the QTL and the means and standard deviations associated with each QTL genotype; these were estimated by maximum likelihood. The fitted distributions correspond well to those observed and the order of the loci along the chromosome is inferred to be Height - Vrn1 - Bmy1 - Wsp3, with Wsp3 being the most distal.

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

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.

Using molecular markers to estimate quantitative trait locus parameters: power and genetic variances for unreplicated and replicated progeny.

Many of the progeny types used to estimate quantitative trait locus (QTL) parameters can be replicated, e.g., recombinant inbred, doubled haploid, and F3 lines. These parameters are estimated using molecular markers or QTL genotypes estimated from molecular markers as independent variables. Experiment designs for replicated progeny are functions of the number of replications per line (r) and the number of replications per QTL genotype (n). The value of n is determined by the size of the progeny population (N), the progeny type, and the number of simultaneously estimated QTL parameters (q - 1). Power for testing hypotheses about means of QTL genotypes is increased by increasing r and n, but the effects of these factors have not been quantified. In this paper, we describe how power is affected by r, n, and other factors. The genetic variance between lines nested in QTL genotypes (sigma 2n:q) is the fraction of the genetic variance between lines (sigma 2n) which is not explained by simultaneously estimated intralocus and interlocus QTL parameters (phi 2Q); thus, sigma 2n:q = sigma 2n - phi 2Q. If sigma 2n:q not equal to 0, then power is not efficiently increased by increasing r and is maximized by maximizing n and using r = 1; however, if sigma 2n:q = 0, then r and n affect power equally and power is efficiently increased by increasing r and is maximized by maximizing N.r. Increasing n efficiently increases power for a wide range of values of sigma 2n:q.sigma 2n:q = 0 when the genetic variance between lines is fully explained by QTL parameters (sigma 2n = phi 2Q).(ABSTRACT TRUNCATED AT 250 WORDS)

Mapping quantitative trait loci using molecular marker linkage maps

High-density restriction fragment length polymorphism (RFLP) and allozyme linkage maps have been developed in several plant species. These maps make it technically feasible to map quantitative trait loci (QTL) using methods based on flanking marker genetic models. In this paper, we describe flanking marker models for doubled haploid (DH), recombinant inbred (RI), backcross (BC), F1 testcross (F1TC), DH testcross (DHTC), recombinant inbred testcross (RITC), F2, and F3 progeny. These models are functions of the means of quantitative trait locus genotypes and recombination frequencies between marker and quantitative trait loci. In addition to the genetic models, we describe maximum likelihood methods for estimating these parameters using linear, nonlinear, and univariate or multivariate normal distribution mixture models. We defined recombination frequency estimators for backcross and F2 progeny group genetic models using the parameters of linear models. In addition, we found a genetically unbiased estimator of the QTL heterozygote mean using a linear function of marker means. In nonlinear models, recombination frequencies are estimated less efficiently than the means of quantitative trait locus genotypes. Recombination frequency estimation efficiency decreases as the distance between markers decreases, because the number of progeny in recombinant marker classes decreases. Mean estimation efficiency is nearly equal for these methods.

A Series of FORTRAN Computer Programs for Estimating Plant Mating Systems

loid, backcross, recombinant inbred, various testcross, F2, and F3 progeny in which the phenotypes of flanking molecular marker loci are used as independent variable values (Knapp et al. 1990). The statistical models arising from these genetic models are nonlinear. The parameters of these models are the means of quantitative trait locus genotypes and recombination frequencies between marker and quantitative trait loci. We are distributing SAS (1987) PROC NLIN programs for estimating the parameters of these models using maximum-likelihood methods. (PROC NLIN is the SAS [1987] procedure for estimating the parameters of nonlinear models.) The Gauss-Newton algorithm (Gallant 1987) and ordinary or generalized least squares methods are used in our programs. Maximum-likelihood estimates of the parameters are output. In addition, we are distributing PROC IML, PROC REG, and PROC NLIN programs (SAS 1985, 1987), which test hypotheses about gene effects and recombination frequencies using Wald statistics or log-likelihood ratios (Gallant 1987). The model and derivative source code from our programs, which are specific to the genetic and statistical models, can be used in C or FORTRAN implementations of Marquardt or GaussNewton nonlinear least-squares algorithms (Press et al. 1986, 1988).