QTL analysis in arbitrary pedigrees with incomplete marker information

Mapping quantitative trait loci (QTL) in arbitrary outbred pedigrees is complicated by the combinatorial possibilities of allele flow relationships and of the founder allelic configurations. Exact methods are only available for rather short and simple pedigrees. Stochastic simulation using Markov chain Monte Carlo (MCMC) integration offers more flexibility. MCMC methods are less natural in a frequentist than in a Bayesian context, which we therefore adopt. Among the MCMC algorithms for updating marker locus genotypes, we implement the descent-graph algorithm. It can be used to update marker locus allele flow relationships and can handle arbitrarily complex pedigrees and missing marker information. Compared with updating marker genotypic information, updating QTL parameters, such as position, effects, and the allele flow relationships is relatively easy with MCMC. We treat the effect of each diploid combination of founder alleles as a random variable and only estimate the variance of these effects, ie, we model diploid genotypic effects instead of the usual partition in additive and dominance effects. This is a variant of the random model approach. The number of QTL alleles is generally unknown. In the Bayesian context, the number of QTL present on a linkage group can be treated as variable. Computer simulations suggest that the algorithm can indeed handle complex pedigrees and detect two QTL on a linkage group, but that the number of individuals in a single extended family is limited to about 50 to 100 individuals.     

Irreducibility and efficiency of ESIP to sample marker genotypes in large pedigrees with loops

Markov chain Monte Carlo (MCMC) methods have been proposed to overcome computational problems in linkage and segregation analyses. This approach involves sampling genotypes at the marker and trait loci. Among MCMC methods, scalar-Gibbs is the easiest to implement, and it is used in genetics. However, the Markov chain that corresponds to scalar-Gibbs may not be irreducible when the marker locus has more than two alleles, and even when the chain is irreducible, mixing has been observed to be slow. Joint sampling of genotypes has been proposed as a strategy to overcome these problems. An algorithm that combines the Elston-Stewart algorithm and iterative peeling (ESIP sampler) to sample genotypes jointly from the entire pedigree is used in this study. Here, it is shown that the ESIP sampler yields an irreducible Markov chain, regardless of the number of alleles at a locus. Further, results obtained by ESIP sampler are compared with other methods in the literature. Of the methods that are guaranteed to be irreducible, ESIP was the most efficient.     

Inference of genetic architecture from chromosome partitioning analyses is sensitive to genome variation,sample size,heritability and effect size distribution

Genomewide association studies have contributed immensely to our understanding of the genetic basis of complex traits. One major conclusion arising from these studies is that most traits are controlled by many loci of small effect, confirming the infinitesimal model of quantitative genetics. A popular approach to test for polygenic architecture involves so‐called “chromosome partitioning” where phenotypic variance explained by each chromosome is regressed on the size of the chromosome. First developed for humans, this has now been repeatedly used in other species, but there has been no evaluation of the suitability of this method in species that can differ in their genome characteristics such as number and size of chromosomes. Nor has the influence of sample size, heritability of the trait, effect size distribution of loci controlling the trait or the physical distribution of the causal loci in the genome been examined. Using simulated data, we show that these characteristics have major influence on the inferences of the genetic architecture of traits we can infer using chromosome partitioning analyses. In particular, small variation in chromosome size, small sample size, low heritability, a skewed effect size distribution and clustering of loci can lead to a loss of power and consequently altered inference from chromosome partitioning analyses. Future studies employing this approach need to consider and derive an appropriate null model for their study system, taking these parameters into consideration. Our simulation results can provide some guidelines on these matters, but further studies examining a broader parameter space are needed.     

Genealogical inference from microsatellite data.

Ease and accuracy of typing, together with high levels of polymorphism and widespread distribution in the genome, make microsatellite (or short tandem repeat) loci an attractive potential source of information about both population histories and evolutionary processes. However, microsatellite data are difficult to interpret, in particular because of the frequency of back-mutations. Stochastic models for the underlying genetic processes can be specified, but in the past they have been too complicated for direct analysis. Recent developments in stochastic simulation methodology now allow direct inference about both historical events, such as genealogical coalescence times, and evolutionary parameters, such as mutation rates. A feature of the Markov chain Monte Carlo (MCMC) algorithm that we propose here is that the likelihood computations are simplified by treating the (unknown) ancestral allelic states as auxiliary parameters. We illustrate the algorithm by analyzing microsatellite samples simulated under the model. Our results suggest that a single microsatellite usually does not provide enough information for useful inferences, but that several completely linked microsatellites can be informative about some aspects of genealogical history and evolutionary processes. We also reanalyze data from a previously published human Y chromosome microsatellite study, finding evidence for an effective population size for human Y chromosomes in the low thousands and a recent time since their most recent common ancestor: the 95% interval runs from approximately 15, 000 to 130,000 years, with most likely values around 30,000 years.     

Two new recursive likelihood calculation methods for genetic analysis

Recursive likelihood calculations for genetic analysis with ungenotyped pedigree data employ variations of the Elston-Stewart (ES) or the Lander-Green (LG) algorithms. With the ES algorithm, the number of loci may be limited but not the pedigree size. With the LG algorithm, the reverse is the case. We introduce two new algorithms for the computation of regressive likelihoods for pedigrees with multivariate traits. The first is an alternative formulation of our existing model, which leads to a simpler form in the binary trait, polygenic and mixed model cases. The second is an approximation model, which is computationally efficient. These methods apply to both continuous and binary traits, in the oligogenic and polygenic cases. Both methods coincide in the binary case. We considered these methods for cases in which all the traits are controlled by a single locus, with each trait controlled by one locus independent to the others. Simulation studies and analysis of a real data are presented for segregation analysis as illustrations. These methods can also be used in other model-based analyses. These methods are implemented in G.E.M.S., the genetic epidemiology models software.     

A mixture model approach to the mapping of QTL controlling endosperm traits with bulked samples

Endosperm traits are of triploid inheritance and have become a focus of breeding effort for their close relations with the grain quality. Current methods for mapping quantitative trait loci (QTL) underlying endosperm traits are restricted to the use of the phenotypes of single grain samples as input data set, which are often not available in practice due to the small size of the cereal seeds. This paper proposed a statistical model for one specially tailored mapping strategy, where the marker genotypes are obtained from the maternal plants in the segregation population and the phenotypic responses are replaced by the trait means of composite endosperm samples pooled from each plant. It should therefore be more practical and have wide applicability in mapping endosperm traits. The method was implemented by fitting the phenotypic means of endosperms into a Gaussian mixture model. Both the exact and approximate Expectation-Maximization algorithms were proposed to estimate the model parameters. The presence of the QTL was determined by likelihood ratio test statistics. Statistical power and other properties of the new method were investigated and compared to the current single-seed method under a variety of scenarios through simulation studies. The simulations suggest a reasonable sample size should be used to ensure reliable results. The proposed method was also applied to a simulated genome data for further evaluation. As an illustration, a real data of maize was analyzed to find the loci responsible for the popping expansion volume.     

Bayesian mapping of quantitative trait loci under complicated mating designs

Quantitative trait loci (QTL) are easily studied in a biallelic system. Such a system requires the cross of two inbred lines presumably fixed for alternative alleles of the QTL. However, development of inbred lines can be time consuming and cost ineffective for species with long generation intervals and severe inbreeding depression. In addition, restriction of the investigation to a biallelic system can sometimes be misleading because many potentially important allelic interactions do not have a chance to express and thus fail to be detected. A complicated mating design involving multiple alleles mimics the actual breeding system. However, it is difficult to develop the statistical model and algorithm using the classical maximum-likelihood method. In this study, we investigate the application of a Bayesian method implemented via the Markov chain Monte Carlo (MCMC) algorithm to QTL mapping under arbitrarily complicated mating designs. We develop the method under a mixed-model framework where the genetic values of founder alleles are treated as random and the nongenetic effects are treated as fixed. With the MCMC algorithm, we first draw the gene flows from the founders to the descendants for each QTL and then draw samples of the genetic parameters. Finally, we are able to simultaneously infer the posterior distribution of the number, the additive and dominance variances, and the chromosomal locations of all identified QTL.     

Estimating linkage disequilibrium between a polymorphic marker locus and a trait locus in natural populations.

Positional cloning of gene(s) underlying a complex trait requires a high-resolution linkage map between the trait locus and genetic marker loci. Recent research has shown that this may be achieved through appropriately modeling and screening linkage disequilibrium between the candidate marker locus and the major trait locus. A quantitative genetics model was developed in the present study to estimate the coefficient of linkage disequilibrium between a polymorphic genetic marker locus and a locus underlying a quantitative trait as well as the relevant genetic parameters using the sample from randomly mating populations. Asymptotic covariances of the maximum-likelihood estimates of the parameters were formulated. Convergence of the EM-based statistical algorithm for calculating the maximum-likelihood estimates was confirmed and its utility to analyze practical data was exploited by use of extensive Monte-Carlo simulations. Appropriateness of calculating the asymptotic covariance matrix in the present model was investigated for three different approaches. Numerical analyses based on simulation data indicated that accurate estimation of the genetic parameters may be achieved if a sample size of 500 is used and if segregation at the trait locus explains not less than a quarter of phenotypic variation of the trait, but the study reveals difficulties in predicting the asymptotic variances of these maximum-likelihood estimates. A comparison was made between the statistical powers of the maximum-likelihood analysis and the previously proposed regression analysis for detecting the disequilibrium.     

Bayesian LASSO for quantitative trait loci mapping

The mapping of quantitative trait loci (QTL) is to identify molecular markers or genomic loci that influence the variation of complex traits. The problem is complicated by the facts that QTL data usually contain a large number of markers across the entire genome and most of them have little or no effect on the phenotype. In this article, we propose several Bayesian hierarchical models for mapping multiple QTL that simultaneously fit and estimate all possible genetic effects associated with all markers. The proposed models use prior distributions for the genetic effects that are scale mixtures of normal distributions with mean zero and variances distributed to give each effect a high probability of being near zero. We consider two types of priors for the variances, exponential and scaled inverse-chi(2) distributions, which result in a Bayesian version of the popular least absolute shrinkage and selection operator (LASSO) model and the well-known Student's t model, respectively. Unlike most applications where fixed values are preset for hyperparameters in the priors, we treat all hyperparameters as unknowns and estimate them along with other parameters. Markov chain Monte Carlo (MCMC) algorithms are developed to simulate the parameters from the posteriors. The methods are illustrated using well-known barley data.     

Multivariate segregation analysis for quantitative traits in line crosses

Segregation analysis is a method of detecting major genes for quantitative traits without using marker information. It serves as an important tool in helping investigators to plan further studies such as quantitative trait loci mapping or more sophisticated genomic analyses. However, current methods of segregation analysis for a single trait typically have low statistical power. We propose a multivariate segregation analysis (MSA) that takes advantage of the correlation structure of multiple quantitative traits to detect major genes. This method not only increases the statistical power, but allows dissection of the genetic architecture underlying the trait complex. In MSA the observed phenotypes of multiple correlated traits are fitted to a multivariate Gaussian mixture model. Model parameters are estimated under the maximum likelihood framework via the expectation-maximization algorithm. The presence of major genes is tested using likelihood ratio test statistics. Pleiotropy is distinguished from close linkage by comparing three possible models using the Bayesian information criterion. Two simulation experiments were performed based on the F(2) mating design. In the first, the statistical properties of MSA under varying heritabilities and sample sizes were investigated and the results compared with those obtained from single-trait analysis. In the second simulation the efficacy of MSA in separating pleiotropy from close linkage was demonstrated. Finally, the new method was applied to real data and detected a major gene responsible for both plant height and tiller number in rice.     

Detection of multiple QTL with epistatic effects under a mixed inheritance model in an outbred population

A quantitative trait depends on multiple quantitative trait loci (QTL) and on the interaction between two or more QTL, named epistasis. Several methods to detect multiple QTL in various types of design have been proposed, but most of these are based on the assumption that each QTL works independently and epistasis has not been explored sufficiently. The objective of the study was to propose an integrated method to detect multiple QTL with epistases using Bayesian inference via a Markov chain Monte Carlo (MCMC) algorithm. Since the mixed inheritance model is assumed and the deterministic algorithm to calculate the probabilities of QTL genotypes is incorporated in the method, this can be applied to an outbred population such as livestock. Additionally, we treated a pair of QTL as one variable in the Reversible jump Markov chain Monte Carlo (RJMCMC) algorithm so that two QTL were able to be simultaneously added into or deleted from a model. As a result, both of the QTL can be detected, not only in cases where either of the two QTL has main effects and they have epistatic effects between each other, but also in cases where neither of the two QTL has main effects but they have epistatic effects. The method will help ascertain the complicated structure of quantitative traits.     

The nature of quantittative genetic variation revisited: Lessons from Drosophila bristles

Most characters that distinguish one individual from another, like height or weight, vary continuously in populations. Continuous variation of these ‘quantitative’ traits is due to the simultaneous segregation of multiple quantitative trait loci (QTLs) as well as environmental influences. A major challenge in human medicine, animal and plant breeding and evolutionary genetics is to identify QTLs and determine their genetic properties. Studies of the classic quantitative traits, abdominal and sternopleural bristle numbers of Drosophila, have shown that: (1) many loci have small effects on bristle number, but a few have large effects and cause most of the genetic variation; (2) ‘candidate’ loci involved in bristle development often have large quantitative effects on bristle number; and (3) alleles at QTLs affecting bristle number have variable degrees of dominance, interact with each other, and affect other quantitative traits, including fitness. Lessons learned from this model system will be applicable to studies of the genetic basis of quantitative variation in other species.     

A study on the minimum number of loci required for genetic evaluation using a finite locus model

For a finite locus model, Markov chain Monte Carlo (MCMC) methods can be used to estimate the conditional mean of genotypic values given phenotypes, which is also known as the best predictor (BP). When computationally feasible, this type of genetic prediction provides an elegant solution to the problem of genetic evaluation under non-additive inheritance, especially for crossbred data. Successful application of MCMC methods for genetic evaluation using finite locus models depends, among other factors, on the number of loci assumed in the model. The effect of the assumed number of loci on evaluations obtained by BP was investigated using data simulated with about 100 loci. For several small pedigrees, genetic evaluations obtained by best linear prediction (BLP) were compared to genetic evaluations obtained by BP. For BLP evaluation, used here as the standard of comparison, only the first and second moments of the joint distribution of the genotypic and phenotypic values must be known. These moments were calculated from the gene frequencies and genotypic effects used in the simulation model. BP evaluation requires the complete distribution to be known. For each model used for BP evaluation, the gene frequencies and genotypic effects, which completely specify the required distribution, were derived such that the genotypic mean, the additive variance, and the dominance variance were the same as in the simulation model. For lowly heritable traits, evaluations obtained by BP under models with up to three loci closely matched the evaluations obtained by BLP for both purebred and crossbred data. For highly heritable traits, models with up to six loci were needed to match the evaluations obtained by BLP.     

Bayesian mapping of quantitative trait loci for multiple complex traits with the use of variance components

Complex traits important for humans are often correlated phenotypically and genetically. Joint mapping of quantitative-trait loci (QTLs) for multiple correlated traits plays an important role in unraveling the genetic architecture of complex traits. Compared with single-trait analysis, joint mapping addresses more questions and has advantages for power of QTL detection and precision of parameter estimation. Some statistical methods have been developed to map QTLs underlying multiple traits, most of which are based on maximum-likelihood methods. We develop here a multivariate version of the Bayes methodology for joint mapping of QTLs, using the Markov chain-Monte Carlo (MCMC) algorithm. We adopt a variance-components method to model complex traits in outbred populations (e.g., humans). The method is robust, can deal with an arbitrary number of alleles with arbitrary patterns of gene actions (such as additive and dominant), and allows for multiple phenotype data of various types in the joint analysis (e.g., multiple continuous traits and mixtures of continuous traits and discrete traits). Under a Bayesian framework, parameters--including the number of QTLs--are estimated on the basis of their marginal posterior samples, which are generated through two samplers, the Gibbs sampler and the reversible-jump MCMC. In addition, we calculate the Bayes factor related to each identified QTL, to test coincident linkage versus pleiotropy. The performance of our method is evaluated in simulations with full-sib families. The results show that our proposed Bayesian joint-mapping method performs well for mapping multiple QTLs in situations of either bivariate continuous traits or mixed data types. Compared with the analysis for each trait separately, Bayesian joint mapping improves statistical power, provides stronger evidence of QTL detection, and increases precision in estimation of parameter and QTL position. We also applied the proposed method to a set of real data and detected a coincident linkage responsible for determining bone mineral density and areal bone size of wrist in humans.     

Bayesian quantitative trait loci mapping for multiple traits

Most quantitative trait loci (QTL) mapping experiments typically collect phenotypic data on multiple correlated complex traits. However, there is a lack of a comprehensive genomewide mapping strategy for correlated traits in the literature. We develop Bayesian multiple-QTL mapping methods for correlated continuous traits using two multivariate models: one that assumes the same genetic model for all traits, the traditional multivariate model, and the other known as the seemingly unrelated regression (SUR) model that allows different genetic models for different traits. We develop computationally efficient Markov chain Monte Carlo (MCMC) algorithms for performing joint analysis. We conduct extensive simulation studies to assess the performance of the proposed methods and to compare with the conventional single-trait model. Our methods have been implemented in the freely available package R/qtlbim (http://www.qtlbim.org), which greatly facilitates the general usage of the Bayesian methodology for unraveling the genetic architecture of complex traits.     

Study on mapping Quantitative Trait Loci for animal complex binary traits using Bayesian-Markov chain Monte Carlo approach

It is a challenging issue to map Quantitative Trait Loci (QTL) underlying complex discrete traits,which usually show discontinuous distribution and less information,using conventional statisti-cal methods. Bayesian-Markov chain Monte Carlo (Bayesian-MCMC) approach is the key procedure in mapping QTL for complex binary traits,which provides a complete posterior distribution for QTL parameters using all prior information. As a consequence,Bayesian estimates of all interested vari-ables can be obtained straightforwardly basing on their posterior samples simulated by the MCMC algorithm. In our study,utilities of Bayesian-MCMC are demonstrated using simulated several ani-mal outbred full-sib families with different family structures for a complex binary trait underlied by both a QTL and polygene. Under the Identity-by-Descent-Based variance component random model,three samplers basing on MCMC,including Gibbs sampling,Metropolis algorithm and reversible jump MCMC,were implemented to generate the joint posterior distribution of all unknowns so that the QTL parameters were obtained by Bayesian statistical inferring. The results showed that Bayesian-MCMC approach could work well and robust under different family structures and QTL effects. As family size increases and the number of family decreases,the accuracy of the parameter estimates will be im-proved. When the true QTL has a small effect,using outbred population experiment design with large family size is the optimal mapping strategy.     

Pedigree analysis of vitiligo: Further support for multilocus involvement

Vitiligo is a dermatological disorder in man that shows familial aggregation. We performed segregation analysis on data pertaining to vitiligo on members of 147 pedigrees each ascertained through a single proband, and tested various non-genetic, and one-locus and two-locus genetic models. Non-genetic and one-locus genetic models were rejected in favour of a two-locus model postulating epistatic interaction of recessive alleles in the aetiology of vitiligo. The present results show that vitiligo is not a single-locus disorder and substantiate our earlier inference, drawn on the basis of nuclear-family data, of multilocus involvement in the pathogenesis of vitiligo.     

Study on mapping Quantitative Trait Loci for animal complex binary traits using Bayesian-Markov chain Monte Carlo approach

It is a challenging issue to map Quantitative Trait Loci (QTL) underlying complex discrete traits, which usually show discontinuous distribution and less information, using conventional statistical methods. Bayesian-Markov chain Monte Carlo (Bayesian-MCMC) approach is the key procedure in mapping QTL for complex binary traits, which provides a complete posterior distribution for QTL parameters using all prior information. As a consequence, Bayesian estimates of all interested variables can be obtained straightforwardly basing on their posterior samples simulated by the MCMC algorithm. In our study, utilities of Bayesian-MCMC are demonstrated using simulated several animal outbred full-sib families with different family structures for a complex binary trait underlied by both a QTL and polygene. Under the Identity-by-Descent-Based variance component random model, three samplers basing on MCMC, including Gibbs sampling, Metropolis algorithm and reversible jump MCMC, were implemented to generate the joint posterior distribution of all unknowns so that the QTL parameters were obtained by Bayesian statistical inferring. The results showed that Bayesian-MCMC approach could work well and robust under different family structures and QTL effects. As family size increases and the number of family decreases, the accuracy of the parameter estimates will be improved. When the true QTL has a small effect, using outbred population experiment design with large family size is the optimal mapping strategy.     

Modeling population genetic data in autotetraploid species

Allozyme and PCR-based molecular markers have been widely used to investigate genetic diversity and population genetic structure in autotetraploid species. However, an empirical but inaccurate approach was often used to infer marker genotype from the pattern and intensity of gel bands. Obviously, this introduces serious errors in prediction of the marker genotypes and severely biases the data analysis. This article developed a theoretical model to characterize genetic segregation of alleles at genetic marker loci in autotetraploid populations and a novel likelihood-based method to estimate the model parameters. The model properly accounts for segregation complexities due to multiple alleles and double reduction at autotetrasomic loci in natural populations, and the method takes appropriate account of incomplete marker phenotype information with respect to genotype due to multiple-dosage allele segregation at marker loci in tetraploids. The theoretical analyses were validated by making use of a computer simulation study and their utility is demonstrated by analyzing microsatellite marker data collected from two populations of sycamore maple (Acer pseudoplatanus L.), an economically important autotetraploid tree species. Numerical analyses based on simulation data indicate that the model parameters can be adequately estimated and double reduction is detected with good power using reasonable sample size.     

Model choice in gene mapping: what and why

The choice of an appropriate genetic model describing the genetic architecture underlying a character of interest is an inherent part of the gene mapping studies of human and other living organisms. The genetic model specifies the statistical parameters for the number of genes, their positions, and the types and magnitudes of their contributions to the phenotype. There are many considerations involved in model formulation (choice) ranging from the assumptions concerning the data, the role of environment, and the number of oligogenes (or quantitative trait loci) influencing the trait behavior. There are several model selection procedures and criteria under specific sampling designs in the genetic literature. These approaches often have their origin in computer science or in general statistical theory. Our aim here is to give an overview of the most popular statistical criteria and to present principles behind them. Bayesian model averaging is suggested as a robust alternative for such methods.