Linkage between loci of quantitative traits and marker loci: multi-trait analysis with a single marker

Novel Resampling Improves Statistical Power for Multiple-Trait QTL Mapping

Multiple-trait analysis typically employs models that associate a quantitative trait locus (QTL) with all of the traits. As a result, statistical power for QTL detection may not be optimal if the QTL contributes to the phenotypic variation in only a small proportion of the traits. Excluding QTL effects that contribute little to the test statistic can improve statistical power. In this article, we show that an optimal power can be achieved when the number of QTL effects is best estimated, and that a stringent criterion for QTL effect selection may improve power when the number of QTL effects is small but can reduce power otherwise. We investigate strategies for excluding trivial QTL effects, and propose a method that improves statistical power when the number of QTL effects is relatively small, and fairly maintains the power when the number of QTL effects is large. The proposed method first uses resampling techniques to determine the number of nontrivial QTL effects, and then selects QTL effects by the backward elimination procedure for significance test. We also propose a method for testing QTL-trait associations that are desired for biological interpretation in applications. We validate our methods using simulations and Arabidopsis thaliana transcript data.

Selecting informative traits for multivariate quantitative trait locus mapping helps to gain optimal power

A major consideration in multitrait analysis is which traits should be jointly analyzed. As a common strategy, multitrait analysis is performed either on pairs of traits or on all of traits. To fully exploit the power of multitrait analysis, we propose variable selection to choose a subset of informative traits for multitrait quantitative trait locus (QTL) mapping. The proposed method is very useful for achieving optimal statistical power for QTL identification and for disclosing the most relevant traits. It is also a practical strategy to effectively take advantage of multitrait analysis when the number of traits under consideration is too large, making the usual multivariate analysis of all traits challenging. We study the impact of selection bias and the usage of permutation tests in the context of variable selection and develop a powerful implementation procedure of variable selection for genome scanning. We demonstrate the proposed method and selection procedure in a backcross population, using both simulated and real data. The extension to other experimental mapping populations is straightforward.

Merino sheep: a further look at quantitative trait loci for wool production

A quantitative trait loci (QTL) analysis of wool traits from experimental half-sib data of Merino sheep is presented. A total of 617 animals distributed in 10 families were genotyped for 36 microsatellite markers on four ovine chromosomes OAR1, OAR3, OAR4 and OAR11. The markers covering OAR3 and OAR11 were densely spaced, at an average distance of 2.8 and 1.2 cM, respectively. Body weight and wool traits were measured at first and second shearing. Analyses were conducted under three hypotheses: (i) a single QTL controlling a single trait (for multimarker regression models); (ii) two linked QTLs controlling a single trait (using maximum likelihood techniques) and (iii) a single QTL controlling more than one trait (also using maximum likelihood techniques). One QTL was identified for several wool traits on OAR1 (average curvature of fibre at first and second shearing, and clean wool yield measured at second shearing) and on OAR11 (weight and staple strength at first shearing, and coefficient of variation of fibre diameter at second shearing). In addition, one QTL was detected on OAR4 affecting weight measured at second shearing. The results of the single trait method and the two-QTL hypotheses showed an additional QTL segregating on OAR11 (for greasy fleece weight at first shearing and clean wool yield trait at second shearing). Pleiotropic QTLs (controlling more than one trait) were found on OAR1 (clean wool yield, average curvature of fibre, clean and greasy fleece weightand staple length, all measured at second shearing).

Bayesian analysis for genetic architecture of dynamic traits

The dissection of the genetic architecture of quantitative traits, including the number and locations of quantitative trait loci (QTL) and their main and epistatic effects, has been an important topic in current QTL mapping. We extend the Bayesian model selection framework for mapping multiple epistatic QTL affecting continuous traits to dynamic traits in experimental crosses. The extension inherits the efficiency of Bayesian model selection and the flexibility of the Legendre polynomial model fitting to the change in genetic and environmental effects with time. We illustrate the proposed method by simultaneously detecting the main and epistatic QTLs for the growth of leaf age in a doubled-haploid population of rice. The behavior and performance of the method are also shown by computer simulation experiments. The results show that our method can more quickly identify interacting QTLs for dynamic traits in the models with many numbers of genetic effects, enhancing our understanding of genetic architecture for dynamic traits. Our proposed method can be treated as a general form of mapping QTL for continuous quantitative traits, being easier to extend to multiple traits and to a single trait with repeat records.

Quantification of variation in expression networks

Gene expression microarrays allow rapid and easy quantification of transcript accumulation for almost transcripts present in a genome. This technology has been utilized for diverse investigations from studying gene regulation in response to genetic or environmental fluctuation to global expression QTL (eQTL) analyses of natural variation. Typical analysis techniques focus on responses of individual genes in isolation of other genes. However, emerging evidence indicates that genes are organized into regulons, i.e., they respond as groups due to individual transcription factors binding multiple promoters, creating what is commonly called a network. We have developed a set of statistical approaches that allow researchers to test specific network hypothesis using a priori-defined gene networks. When applied to Arabidopsis thaliana this approach has been able to identify natural genetic variation that controls networks. In this chapter we describe approaches to develop and test specific network hypothesis utilizing natural genetic variation. This approach can be expanded to facilitate direct tests of the relationship between phenotypic trait and transcript genetic architecture. Finally, the use of a priori network definitions can be applied to any microarray experiment to directly conduct hypothesis testing at a genomics level.

Estimating variance effect of QTL: an important prospect to increase the resolution power of interval mapping

Equal variances within quantitative trait locus (QTL) groups in the segregating population are a usual simplifying assumption in QTL mapping. The objective of this paper is to demonstrate the advantages of taking into account potential variance effect of QTLs within the framework of standard interval mapping approach. Using backcross case as an example, we show that the resolution power of the analysis may be increased in the presence of variance effect, if the latter is allowed for in the model. For a putative QTL (say,A/a) one can compare two situations, (i)and (ii). It was found that, if the variance effect ofA/ais large enough, then in spite of the necessity to evaluate an increased number of parameters, the more correctly specified model provides an increase in the resolution power, as compared to the situation (i). This is not unexpected, if eitherin (ii) is lower thanfrom (i). But our conclusion holds even if. These advantages are illustrated on sweet corn data data (F3families of F2genotypes). In particular, the log-likelihood test statistics and the parameter estimates obtained for a QT locus in the distal region of chromosome 2 show that the allele enhancing the trait is recessive over the opposite allele simultaneously for the mean value and variance.

Strategies to Assess Structural Variation in the Chicken Genome and its Associations with Biodiversity and Biological Performance

A primary goal in the assessment of structural variation in the avian genome is to understand the relationship of this variation with biodiversity and with biological performance. To develop such knowledge, certain essential tools are needed. One set of tools includes the laboratory techniques used to assess molecular genetic variation. The current time is a transitional one for this field, in that the recently sequenced chicken genome will add significantly to the portfolio of existing methods used to identify molecular markers. To most efficiently discover marker-trait associations, the experimental mapping populations must be appropriately designed and the relevant statistical analyses applied. This paper reviews methods for assessment of molecular markers in poultry and their use in the characterization of avian biodiversity and in studies to identify marker associations with biological traits, including important considerations of population structure and statistical analysis.

Bayesian reanalysis of a quantitative trait locus accounting for multiple environments by scaling in broilers1

A Bayesian method was developed to handle QTL analyses of multiple experimental data of outbred populations with heterogeneity of variance between sexes for all random effects. The method employed a scaled reduced animal model with random polygenic and QTL allelic effects. A parsimonious model specification was applied by choosing assumptions regarding the covariance structure to limit the number of parameters to estimate. Markov chain Monte Carlo algorithms were applied to obtain marginal posterior densities. Simulation demonstrated that joint analysis of multiple environments is more powerful than separate single trait analyses of each environment. Measurements on broiler BW obtained from 2 experiments concerning growth efficiency and carcass traits were used to illustrate the method. The population consisted of 10 full-sib families from a cross between 2 broiler lines. Microsatellite genotypes were determined on generations 1 and 2, and phenotypes were collected on groups of generation 3 animals. The model included a polygenic correlation, which had a posterior mean of 0.70 in the analyses. The reanalysis agreed on the presence of a QTL in marker bracket MCW0058-LEI0071 accounting for 34% of the genetic variation in males and 24% in females in the growth efficiency experiment. In the carcass experiment, this QTL accounted for 19% of the genetic variation in males and 6% in females.

Mapping quantitative trait loci for longitudinal traits in line crosses

Quantitative traits whose phenotypic values change over time are called longitudinal traits. Genetic analyses of longitudinal traits can be conducted using any of the following approaches: (1) treating the phenotypic values at different time points as repeated measurements of the same trait and analyzing the trait under the repeated measurements framework, (2) treating the phenotypes measured from different time points as different traits and analyzing the traits jointly on the basis of the theory of multivariate analysis, and (3) fitting a growth curve to the phenotypic values across time points and analyzing the fitted parameters of the growth trajectory under the theory of multivariate analysis. The third approach has been used in QTL mapping for longitudinal traits by fitting the data to a logistic growth trajectory. This approach applies only to the particular S-shaped growth process. In practice, a longitudinal trait may show a trajectory of any shape. We demonstrate that one can describe a longitudinal trait with orthogonal polynomials, which are sufficiently general for fitting any shaped curve. We develop a mixed-model methodology for QTL mapping of longitudinal traits and a maximum-likelihood method for parameter estimation and statistical tests. The expectation-maximization (EM) algorithm is applied to search for the maximum-likelihood estimates of parameters. The method is verified with simulated data and demonstrated with experimental data from a pseudobackcross family of Populus (poplar) trees.

Regression-based quantitative trait loci mapping: robust, efficient and effective

Regression has always been an important tool for quantitative geneticists. The use of maximum likelihood (ML) has been advocated for the detection of quantitative trait loci (QTL) through linkage with molecular markers, and this approach can be very effective. However, linear regression models have also been proposed which perform similarly to ML, while retaining the many beneficial features of regression and, hence, can be more tractable and versatile than ML in some circumstances. Here, the use of linear regression to detect QTL in structured outbred populations is reviewed and its perceived shortfalls are revisited. It is argued that the approach is valuable now and will remain so in the future.

Genetic mapping of variation in spatial learning in the mouse

Inbred strains of mice are known to differ in their performance in the Morris water maze task, a test of spatial discrimination and place navigation in rodents, but the genetic basis of individual variation in spatial learning is unknown. We have mapped genetic effects that contribute to the difference between two strains, DBA/2 and C57BL6/J, using an F2 intercross and methods to detect quantitative trait loci (QTL). We found two QTL, one on chromosome 4 and one on chromosome 12, that influence behavior in the probe trial of the water maze (genome-wide significance p = 0.017 and 0.015, respectively). By including tests of avoidance conditioning and behavior in a novel environment, we show that the QTL on chromosomes 4 and 12 specifically influence variation in spatial learning. QTL that influence differences in fearful behavior (on chromosomes 1, 3, 7, 15, and 19) operate while mice are trained in the water maze apparatus.

Functional mapping of quantitative trait loci underlying the character process: a theoretical framework

Unlike a character measured at a finite set of landmark points, function-valued traits are those that change as a function of some independent and continuous variable. These traits, also called infinite-dimensional characters, can be described as the character process and include a number of biologically, economically, or biomedically important features, such as growth trajectories, allometric scalings, and norms of reaction. Here we present a new statistical infrastructure for mapping quantitative trait loci (QTL) underlying the character process. This strategy, termed functional mapping, integrates mathematical relationships of different traits or variables within the genetic mapping framework. Logistic mapping proposed in this article can be viewed as an example of functional mapping. Logistic mapping is based on a universal biological law that for each and every living organism growth over time follows an exponential growth curve (e.g., logistic or S-shaped). A maximum-likelihood approach based on a logistic-mixture model, implemented with the EM algorithm, is developed to provide the estimates of QTL positions, QTL effects, and other model parameters responsible for growth trajectories. Logistic mapping displays a tremendous potential to increase the power of QTL detection, the precision of parameter estimation, and the resolution of QTL localization due to the small number of parameters to be estimated, the pleiotropic effect of a QTL on growth, and/or residual correlations of growth at different ages. More importantly, logistic mapping allows for testing numerous biologically important hypotheses concerning the genetic basis of quantitative variation, thus gaining an insight into the critical role of development in shaping plant and animal evolution and domestication. The power of logistic mapping is demonstrated by an example of a forest tree, in which one QTL affecting stem growth processes is detected on a linkage group using our method, whereas it cannot be detected using current methods. The advantages of functional mapping are also discussed.

Mapping and analysis of quantitative trait loci in experimental populations

Simple statistical methods for the study of quantitative trait loci (QTL), such as analysis of variance, have given way to methods that involve several markers and high-resolution genetic maps. As a result, the mapping community has been provided with statistical and computational tools that have much greater power than ever before for studying and locating multiple and interacting QTL. Apart from their immediate practical applications, the lessons learnt from this evolution of QTL methodology might also be generally relevant to other types of functional genomics approach that are aimed at the dissection of complex phenotypes, such as microarray assessment of gene expression.

Enhanced efficiency of quantitative trait loci mapping analysis based on multivariate complexes of quantitative traits

An approach to increase the efficiency of mapping quantitative trait loci (QTL) was proposed earlier by the authors on the basis of bivariate analysis of correlated traits. The power of QTL detection using the log-likelihood ratio (LOD scores) grows proportionally to the broad sense heritability. We found that this relationship holds also for correlated traits, so that an increased bivariate heritability implicates a higher LOD score, higher detection power, and better mapping resolution. However, the increased number of parameters to be estimated complicates the application of this approach when a large number of traits are considered simultaneously. Here we present a multivariate generalization of our previous two-trait QTL analysis. The proposed multivariate analogue of QTL contribution to the broad-sense heritability based on interval-specific calculation of eigenvalues and eigenvectors of the residual covariance matrix allows prediction of the expected QTL detection power and mapping resolution for any subset of the initial multivariate trait complex. Permutation technique allows chromosome-wise testing of significance for the whole trait complex and the significance of the contribution of individual traits owing to: (a) their correlation with other traits, (b) dependence on the chromosome in question, and (c) both a and b. An example of application of the proposed method on a real data set of 11 traits from an experiment performed on an F(2)/F(3) mapping population of tetraploid wheat (Triticum durum x T. dicoccoides) is provided.

Multitrait least squares for quantitative trait loci detection

A multiple-trait QTL mapping method using least squares is described. It is presented as an extension of a single-trait method for use with three-generation, outbred pedigrees. The multiple-trait framework allows formal testing of whether the same QTL affects more than one trait (i.e., a pleiotropic QTL) or whether more than one linked QTL are segregating. Several approaches to the testing procedure are presented and their suitability discussed. The performance of the method is investigated by simulation. As previously found, multitrait analyses increase the power to detect a pleiotropic QTL and the precision of its location estimate. With enough information, discrimination between alternative genetic models is possible.

Joint multipoint linkage analysis of multivariate qualitative and quantitative traits. I. Likelihood formulation and simulation results

We describe a variance-components method for multipoint linkage analysis that allows joint consideration of a discrete trait and a correlated continuous biological marker (e.g., a disease precursor or associated risk factor) in pedigrees of arbitrary size and complexity. The continuous trait is assumed to be multivariate normally distributed within pedigrees, and the discrete trait is modeled by a threshold process acting on an underlying multivariate normal liability distribution. The liability is allowed to be correlated with the quantitative trait, and the liability and quantitative phenotype may each include covariate effects. Bivariate discrete-continuous observations will be common, but the method easily accommodates qualitative and quantitative phenotypes that are themselves multivariate. Formal likelihood-based tests are described for coincident linkage (i.e., linkage of the traits to distinct quantitative-trait loci [QTLs] that happen to be linked) and pleiotropy (i.e., the same QTL influences both discrete-trait status and the correlated continuous phenotype). The properties of the method are demonstrated by use of simulated data from Genetic Analysis Workshop 10. In a companion paper, the method is applied to data from the Collaborative Study on the Genetics of Alcoholism, in a bivariate linkage analysis of alcoholism diagnoses and P300 amplitude of event-related brain potentials.

Single- and multiple-trait mapping analysis of linked quantitative trait loci. Some asymptotic analytical approximations

Estimating the resolution power of mapping analysis of linked quantitative trait loci (QTL) remains a difficult problem, which has been previously addressed mainly by Monte Carlo simulations. The analytical method of evaluation of the expected LOD developed in this article spreads the "deterministic sampling" approach for the case of two linked QTL for single- and two-trait analysis. Several complicated questions are addressed through this evaluation: the dependence of QTL detection power on the QTL effects, residual correlation between the traits, and the effect of epistatic interaction between the QTL for one or both traits on expected LOD (ELOD), etc. Although this method gives only an asymptotic estimation of ELOD, it allows one to get an approximate assessment of a broad spectrum of mapping situations. A good correspondence was found between the ELODs predicted by the model and LOD values averaged over Monte Carlo simulations.

A nonparametric bootstrap method for testing close linkage vs. pleiotropy of coincident quantitative trait loci

A novel method using the nonparametric bootstrap is proposed for testing whether a quantitative trait locus (QTL) at one chromosomal position could explain effects on two separate traits. If the single-QTL hypothesis is accepted, pleiotropy could explain the effect on two traits. If it is rejected, then the effects on two traits are due to linked QTLs. The method can be used in conjunction with several QTL mapping methods as long as they provide a straightforward estimate of the number of QTLs detectable from the data set. A selection step was introduced in the bootstrap procedure to reduce the conservativeness of the test of close linkage vs. pleiotropy, so that the erroneous rejection of the null hypothesis of pleiotropy only happens at a frequency equal to the nominal type I error risk specified by the user. The approach was assessed using computer simulations and proved to be relatively unbiased and robust over the range of genetic situations tested. An example of its application on a real data set from a saline stress experiment performed on a recombinant population of wheat (Triticum aestivum L. ) doubled haploid lines is also provided.

Approximate analysis of QTL-environment interaction with no limits on the number of environments

An approach is presented here for quantitative trait loci (QTL) mapping analysis that allows for QTL x environment (E) interaction across multiple environments, without necessarily increasing the number of parameters. The main distinction of the proposed model is in the chosen way of approximation of the dependence of putative QTL effects on environmental states. We hypothesize that environmental dependence of a putative QTL effect can be represented as a function of environmental mean value of the trait. Such a description can be applied to take into account the effects of any cosegregating QTLs from other genomic regions that also may vary across environments. The conducted Monte-Carlo simulations and the example of barley multiple environments experiment demonstrate a high potential of the proposed approach for analyzing QTL x E interaction, although the results are only approximated by definition. However, this drawback is compensated by the possibility to utilize information from a potentially unlimited number of environments with a remarkable reduction in the number of parameters, as compared to previously proposed mapping models with QTL x E interactions.

QTL analysis in plants; where are we now?

We have briefly reviewed the methods currently available for QTL analysis in segregating populations and summarized some of the conclusions arising from such analyses in plant populations. We show that the analytical methods locate QTL with poor precision (10-30 cM), unless the heritability of an individual QTL is high. Also the estimates of the QTL effects, particularly the dominance effects tend to be inflated because only large estimates are significant. Estimates of numbers of QTL per trait are generally low (< 8) for individual trials. This may suggest that there are few QTL but probably reflects the power of the methods. There is no large correlation between the numbers of QTL found and the amount of the variation explained. Of those cases where dominance is measurable, dominance ratios are often > 1, but seldom significantly greater. These latter cases need further analysis. Many QTL map close to candidate genes, and there is growing evidence from synteny studies of corresponding chromosome regions carrying similar QTL in different species. However, unreliability of QTL location may suggest false candidates.