Bayesian versus frequentist analysis of multiple quantitative trait loci with an application to an outbred apple cross

Two methods, following different statistical paradigms for mapping multiple quantitative trait loci (QTLs), were compared: the first is a frequentist, the second a Bayesian approach. Both methods were applied to previously published experimental data from an outbred progeny of a single cross between two apple cultivars (Malus pumila Mill.). These approaches were compared with respect to (1) the models used, (2) the number of putative QTLs, (3) their estimated map positions and accuracies thereof and (4) the choice of cofactor markers. In general, the strongest evidence for QTLs, provided by both methods, was for the same linkage groups and for similar map positions. However, some differences were found with respect to evidence for QTLs on other linkage groups. The effect of using cofactor markers which were selected differently was also somewhat different. Received: 17 July 2000 / Accepted: 13 January 2001     

The statistics of bulk segregant analysis using next generation sequencing

We describe a statistical framework for QTL mapping using bulk segregant analysis (BSA) based on high throughput, short-read sequencing. Our proposed approach is based on a smoothed version of the standard G statistic, and takes into account variation in allele frequency estimates due to sampling of segregants to form bulks as well as variation introduced during the sequencing of bulks. Using simulation, we explore the impact of key experimental variables such as bulk size and sequencing coverage on the ability to detect QTLs. Counterintuitively, we find that relatively large bulks maximize the power to detect QTLs even though this implies weaker selection and less extreme allele frequency differences. Our simulation studies suggest that with large bulks and sufficient sequencing depth, the methods we propose can be used to detect even weak effect QTLs and we demonstrate the utility of this framework by application to a BSA experiment in the budding yeast Saccharomyces cerevisiae.     

基于F3种子的胚乳性状QTL区间定位

文章提出了包括胚乳效应和母体效应的胚乳性状QTL定位的统计方法,该方法的实验设计是分子标记基因型信息来自F2母体植株和F3种子胚(或植株),胚乳性状表型值来自F3单粒种子胚乳,称之为两步等级设计。同时,用计算机全面模拟以验证该模型的可行性,模拟结果表明,只要群体足够大,该模型能较有效地进行胚乳性状QTL定位并精确地估计出胚乳QTL的各种遗传效应和母体效应。     

R/qtl: QTL mapping in experimental crosses

SUMMARY: R/qtl is an extensible, interactive environment for mapping quantitative trait loci (QTLs) in experimental populations derived from inbred lines. It is implemented as an add-on package for the freely-available statistical software, R, and includes functions for estimating genetic maps, identifying genotyping errors, and performing single-QTL and two-dimensional, two-QTL genome scans by multiple methods, with the possible inclusion of covariates. AVAILABILITY: The package is freely available at http://www.biostat.jhsph.edu/~kbroman/qtl.     

Mapping quantitative trait loci with epistatic effects

Epistatic variance can be an important source of variation for complex traits. However, detecting epistatic effects is difficult primarily due to insufficient sample sizes and lack of robust statistical methods. In this paper, we develop a Bayesian method to map multiple quantitative trait loci (QTLs) with epistatic effects. The method can map QTLs in complicated mating designs derived from the cross of two inbred lines. In addition to mapping QTLs for quantitative traits, the proposed method can even map genes underlying binary traits such as disease susceptibility using the threshold model. The parameters of interest are various QTL effects, including additive, dominance and epistatic effects of QTLs, the locations of identified QTLs and even the number of QTLs. When the number of QTLs is treated as an unknown parameter, the dimension of the model becomes a variable. This requires the reversible jump Markov chain Monte Carlo algorithm. The utility of the proposed method is demonstrated through analysis of simulation data.     

基于BC_1F_（1：2）种子的胚乳QTL区间定位

提出了基于分子标记基因型信息来自BC_1F_1母体植株,胚乳性状表型值来自BC_1F_（1：2）单粒种子胚乳的试验设计的胚乳QTL定位的区间作图方法.同时,用计算机全面模拟以验证该模型的可行性,模拟结果表明,只要群体足够大,该模型能有效地进行胚乳性状QTL定位并能估计出胚乳QTL的各种遗传效应和母体效应.     

Semiparametric methods for mapping quantitative trait loci with censored data

Statistical methods for the detection of genes influencing quantitative traits with the aid of genetic markers are well developed for normally distributed, fully observed phenotypes. Many experiments are concerned with failure-time phenotypes, which have skewed distributions and which are usually subject to censoring because of random loss to follow-up, failures from competing causes, or limited duration of the experiment. In this article, we develop semiparametric statistical methods for mapping quantitative trait loci (QTLs) based on censored failure-time phenotypes. We formulate the effects of the QTL genotype on the failure time through the Cox (1972, Journal of the Royal Statistical Society, Series B 34, 187-220) proportional hazards model and derive efficient likelihood-based inference procedures. In addition, we show how to assess statistical significance when searching several regions or the entire genome for QTLs. Extensive simulation studies demonstrate that the proposed methods perform well in practical situations. Applications to two animal studies are provided.     

Mapping of Quantitative Trait Loci Underlying Cold Tolerance in Rice Seedlings via High-Throughput Sequencing of Pooled Extremes

Low temperature is a major limiting factor in rice growth and development. Mapping of quantitative trait loci (QTLs) controlling cold tolerance is important for rice breeding. Recent studies have suggested that bulked segregant analysis (BSA) combined with next-generation sequencing (NGS) can be an efficient and cost-effective way for QTL mapping. In this study, we employed NGS-assisted BSA to map QTLs conferring cold tolerance at the seedling stage in rice. By deep sequencing of a pair of large DNA pools acquired from a very large F₃ population (10,800 individuals), we obtained ∼450,000 single nucleotide polymorphisms (SNPs) after strict screening. We employed two statistical methods for QTL analysis based on these SNPs, which yielded consistent results. Six QTLs were mapped on chromosomes 1, 2, 5, 8 and 10. The three most significant QTLs on chromosomes 1, 2 and 8 were validated by comparison with previous studies. Two QTLs on chromosomes 2 and 5 were also identified previously, but at the booting stage rather than the seedling stage, suggesting that some QTLs may function at different developmental stages, which would be useful for cold tolerance breeding in rice. Compared with previously reported QTL mapping studies for cold tolerance in rice based on the traditional approaches, the results of this study demonstrated the advantages of NGS-assisted BSA in both efficiency and statistical power.     

Bayesian interval mapping of count trait loci based on zero-inflated generalized Poisson regression model

Count phenotypes with excessive zeros are often observed in the biological world. Researchers have studied many statistical methods for mapping the quantitative trait loci (QTLs) of zero-inflated count phenotypes. However, most of the existing methods consist of finding the approximate positions of the QTLs on the chromosome by genome-wide scanning. Additionally, most of the existing methods use the EM algorithm for parameter estimation. In this paper, we propose a Bayesian interval mapping scheme of QTLs for zero-inflated count data. The method takes advantage of a zero-inflated generalized Poisson (ZIGP) regression model to study the influence of QTLs on the zero-inflated count phenotype. The MCMC algorithm is used to estimate the effects and position parameters of QTLs. We use the Haldane map function to realize the conversion between recombination rate and map distance. Monte Carlo simulations are conducted to test the applicability and advantage of the proposed method. The effects of QTLs on the formation of mouse cholesterol gallstones were demonstrated by analyzing an mouse data set.     

Genetic Relationships Among Panicle Characteristics of Rice (<Emphasis Type="Italic">Oryza sativa</Emphasis> L.) Using Unconditional and Conditional QTL Analyses

Using mixed-model-based composite interval mapping and conditional statistical methods, we studied quantitative trait loci (QTLs) with epistatic effects and QTLs by environment interaction effects for rice seed set percent (SSP), filled grain number per plant (FGP), and panicle length (PL). A population of 241 recombinant inbred lines was used which was derived from a cross between “Zhenshan 97” and “Minghui 63.” Its linkage map included 221 molecular markers. Our QTL analysis detected 28, 25, and 32 QTLs for SSP, FGP, and PL, respectively. Each QTL explained 1.37%∼13.19% of the mean phenotypic variation. A comparison of conventional and conditional mapping provided information about the genetic control system involved in the synthetic process of SSP, FGP, and PL at the level of individual QTLs. Conditional QTLs with reduced (or increased) effects were identified for SSP, which were significantly influenced by FGP or PL. Some QTLs could express independently for the given traits, thereby providing possibilities for simultaneous improvement of SSR and PL, and SSR and FGP. Epistasis was more sensitive to environmental conditions than were additive effects.     

Epistat : a computer program for identifying and testing interactions between pairs of quantitative trait loci

 We describe a computer program, Epistat, which combines statistical methods and color-graphic displays to facilitate the analysis of interactions between pairs of quantitative trait loci (QTLs). Epistat organizes genetic-mapping data and quantitative-trait values into graphic displays which illustrate the individual effects of single loci as well as the interactions between any two loci. Keyboard commands allow the user to search the data set for individual QTLs and to test for interactions between QTLs. For a given trait, the program displays the effects of the alleles at each of two loci on the quantitative-trait value, as well as the effects of the interactions between these alleles. Loglikelihood ratios are used to compare the likelihood of explaining the effects by null, additive, or epistatic models. Examples of interactions in soybean are presented for near-infrared transmittance (NIT), seed number, and reproductive period. Epistat has been used to find numerous interactions between QTLs in soybean in which trait variation at one locus is conditional upon a specific allele at another. Received: 16 January 1996 / Accepted: 27 September 1996     

A Multi-Marker Model for Detecting Chromosomal Segments Displaying Qtl Activity

A statistical method is presented for detecting quantitative trait loci (QTLs), based on the linear model. Unlike methods able to detect a few well separated QTLs and to estimate their effects and positions, this method considers the genome as a whole and enables the detection of chromosomal segments involved in the differences between two homozygous lines, and their backcross, doubled haploid, or F(2) progenies, for a quantitative trait. Genetic markers must be codominant, but missing markers are accepted, provided they are missing independently from the experiment. Asymptotic properties, which are of practical use, are developed. This method does not rely on strong genetic hypotheses, and thus does not permit any precise genetic analysis of the trait under study, but it does assess which regions of the genome are involved, whatever the complexity of the genetic determinism (number, effects and interactions among QTLs). Simultaneous use of several methods, including this one, should lead to better efficiency in QTL detection.     

Analysis of quantitative trait loci (QTLs) and quantitative trait alleles (QTAs) for potato tuber yield and starch content

 Using RFLP markers, QTLs for tuber starch-content and tuber yield were mapped in two F₁ populations derived from crossing non-inbred di-haploid potato breeding lines. QTLs were identified and mapped, based on both single-marker tests and interval analyses. A model specifically developed for interval QTL analysis in non-inbred plant species was successfully applied for the first time to experimental data. Results of both methods of QTL analysis were similar but not identical. QTLs for tuber starch-content and tuber yield were analysed in segregating populations K31 and LH in five and two environments, respectively. Population K31 was fully genotyped whereas population LH was selectively genotyped according to high and low tuber-starch content. Eighteen putative QTLs for tuber starch-content were identified on all 12 potato linkage groups and eight putative QTLs for tuber yield were identified on eight linkage groups. Twenty of twenty six putative QTLs were reproducibly detected in at least two environments and/or mapping populations. Few major QTLs for tuber starch-content were highly stable across environments but were detected in only one of the two mapping populations analysed. Most QTLs for tuber yield were linked with QTLs for tuber starch-content suggesting that the effects on both traits are controlled by the same genetic factors. The results are discussed with respect to marker-assisted selection in potato. Received: 9 March 1998 / Accepted: 29 April 1998     

Experimental designs and statistical methods for mapping quantitative trait loci underlying triploid endosperm traits without maternal genetic variation

Many endosperm traits are related to grain quality in cereal crops. Endosperm traits are mainly controlled by the endosperm genome but may be affected by the maternal genome. Studies have shown that maternal genotypic variation could greatly influence the estimation of the direct effects of quantitative trait loci (QTLs) underlying endosperm traits. In this paper, we propose methods of interval mapping of endosperm QTLs using seeds of F2 or BC1 (an equal mixture of F1 x P1 and F1 x P2 with F1 as the female parent) derived from a cross between 2 pure lines (P1 x P2). The most significant advantage of our experimental designs is that the maternal effects do not contribute to the genetic variation of endosperm traits and therefore the direct effects of endosperm QTLs can be estimated without the influence of maternal effects. In addition, the experimental designs can greatly reduce environmental variation because a few F1 plants grown in a small block of field will produce sufficient F2 or BC1 seeds for endosperm QTL analysis. Simulation studies show that the methods can efficiently detect endosperm QTLs and unbiasedly estimate their positions and effects. The BC1 design is better than the F2 design.     

Mapping Mendelian Factors Underlying Quantitative Traits Using RFLP Linkage Maps

The advent of complete genetic linkage maps consisting of codominant DNA markers [typically restriction fragment length polymorphisms (RFLPs)] has stimulated interest in the systematic genetic dissection of discrete Mendelian factors underlying quantitative traits in experimental organisms. We describe here a set of analytical methods that modify and extend the classical theory for mapping such quantitative trait loci (QTLs). These include: (i) a method of identifying promising crosses for QTL mapping by exploiting a classical formula of SEWALL WRIGHT; (ii) a method (interval mapping) for exploiting the full power of RFLP linkage maps by adapting the approach of LOD score analysis used in human genetics, to obtain accurate estimates of the genetic location and phenotypic effect of QTLs; and (iii) a method (selective genotyping) that allows a substantial reduction in the number of progeny that need to be scored with the DNA markers. In addition to the exposition of the methods, explicit graphs are provided that allow experimental geneticists to estimate, in any particular case, the number of progeny required to map QTLs underlying a quantitative trait.     

A statistical model for estimating maternal-zygotic interactions and parent-of-origin effects of QTLs for seed development

Proper development of a seed requires coordinated exchanges of signals among the three components that develop side by side in the seed. One of these is the maternal integument that encloses the other two zygotic components, i.e., the diploid embryo and its nurturing annex, the triploid endosperm. Although the formation of the embryo and endosperm contains the contributions of both maternal and paternal parents, maternally and paternally derived alleles may be expressed differently, leading to a so-called parent-of-origin or imprinting effect. Currently, the nature of how genes from the maternal and zygotic genomes interact to affect seed development remains largely unknown. Here, we present a novel statistical model for estimating the main and interaction effects of quantitative trait loci (QTLs) that are derived from different genomes and further testing the imprinting effects of these QTLs on seed development. The experimental design used is based on reciprocal backcrosses toward both parents, so that the inheritance of parent-specific alleles could be traced. The computing model and algorithm were implemented with the maximum likelihood approach. The new strategy presented was applied to study the mode of inheritance for QTLs that control endoreduplication traits in maize endosperm. Monte Carlo simulation studies were performed to investigate the statistical properties of the new model with the data simulated under different imprinting degrees. The false positive rate of imprinting QTL discovery by the model was examined by analyzing the simulated data that contain no imprinting QTL. The reciprocal design and a series of analytical and testing strategies proposed provide a standard procedure for genomic mapping of QTLs involved in the genetic control of complex seed development traits in flowering plants.     

Mapping quantitative trait loci using binned genotypes

Precise mapping of quantitative trait loci(QTLs)is critical for assessing genetic effects and identifying candidate genes for quantitative traits.Interval and composite interval mappings have been the methods of choice for several decades,which have provided tools for identifying genomic regions harboring causal genes for quantitative traits.Historically,the concept was developed on the basis of sparse marker maps where genotypes of loci within intervals could not be observed.Currently,genomes of many organisms have been saturated with markers due to the new sequencing technologies.Genotyping by sequencing usually generates hundreds of thousands of single nucleotide polymorphisms(SNPs),which often include the causal polymorphisms.The concept of interval no longer exists,prompting the necessity of a norm change in QTL mapping technology to make use of the high-volume genomic data.Here we developed a statistical method and a software package to map QTLs by binning markers into haplotype blocks,called bins.The new method detects associations of bins with quantitative traits.It borrows the mixed model methodology with a polygenic control from genome-wide association studies(GWAS)and can handle all kinds of experimental populations under the linear mixed model(LMM)framework.We tested the method using both simulated data and data from populations of rice.The results showed that this method has higher power than the current methods.An R package named binQTL is available from GitHub.     

Assessing the importance of genotype × environment interaction for root traits in rice using a mapping population II: conventional QTL analysis

Modifying plant root systems is considered a means of crop improvement targeted to low-resource environments, particularly low nutrient and drought-prone agriculture. The identification of quantitative trait loci (QTLs) for root traits has stimulated marker-assisted breeding to this end, but different QTLs have been detected in different populations of the same species, and importantly, in the same population when grown in different experimental environments. The presence of QTL × environment interaction is implicated, and this must be characterised if the utility of the target QTLs is to be realised. Previous attempts to do this suffer from a lack of control over replicate environments and inadequate statistical rigour. The Bala × Azucena mapping population was grown in two replicate experiments of four treatment environments, a control, a low light, a low soil nitrogen and a low soil water treatment. After a 4 weeks growth, maximum root length, maximum root thickness, root mass below 50 cm, total plant dry mass, % root mass and shoot length were measured. A summary of the overall results is presented in an accompanying paper. Here, QTL analysis by composite interval mapping is presented. A total of 145 QTLs were detected, mapping to 37 discrete loci on all chromosomes. Superficial evidence of QTL × E (great difference in LOD score) was tested by single-marker analysis which confirmed QTL × E for five loci representing only five individual trait-loci interactions. Some loci appeared to be stable across environments. Some QTLs were clearly more or less active under low light, low nitrogen or drought. A few notable loci on chromosomes 1, 2, 3, 5, 7 and 9 are briefly discussed. Also discussed are some remaining statistical shortcomings that will be addressed in another companion paper.     

Composite Interval Mapping Based on Lattice Design for Error Control May Increase Power of Quantitative Trait Locus Detection

Experimental error control is very important in quantitative trait locus (QTL) mapping. Although numerous statistical methods have been developed for QTL mapping, a QTL detection model based on an appropriate experimental design that emphasizes error control has not been developed. Lattice design is very suitable for experiments with large sample sizes, which is usually required for accurate mapping of quantitative traits. However, the lack of a QTL mapping method based on lattice design dictates that the arithmetic mean or adjusted mean of each line of observations in the lattice design had to be used as a response variable, resulting in low QTL detection power. As an improvement, we developed a QTL mapping method termed composite interval mapping based on lattice design (CIMLD). In the lattice design, experimental errors are decomposed into random errors and block-within-replication errors. Four levels of block-within-replication errors were simulated to show the power of QTL detection under different error controls. The simulation results showed that the arithmetic mean method, which is equivalent to a method under random complete block design (RCBD), was very sensitive to the size of the block variance and with the increase of block variance, the power of QTL detection decreased from 51.3% to 9.4%. In contrast to the RCBD method, the power of CIMLD and the adjusted mean method did not change for different block variances. The CIMLD method showed 1.2- to 7.6-fold higher power of QTL detection than the arithmetic or adjusted mean methods. Our proposed method was applied to real soybean (Glycine max) data as an example and 10 QTLs for biomass were identified that explained 65.87% of the phenotypic variation, while only three and two QTLs were identified by arithmetic and adjusted mean methods, respectively.     

Meta - and combined - QTL analysis of different experiments on immune traits in chickens

Meta and/or combined QTL analysis from multiple studies can improve quantitative trait loci (QTL) position estimates compared to the individual experiments. Hereby we present results of a meta-analysis of QTL on chicken chromosome 9, 14 and 18 using data from three separate experiments and joint QTL analysis for chromosome 14 and 18. Meta QTL analysis uses information from multiple QTLs studies. Joint QTL analysis is based on combining raw data from different QTL experimental populations. QTLs under the study were related to specific antibody response to keyhole lymphet hemocyanin (KLH), and natural antibodies to environmental antigens, lipopolisaccharide (LPS) and lipoteichoic acid (LTA). Meta QTL analysis resulted in narrowing down the confidence interval for two QTLs on GGA14. The first one for natural antibodies against LTA and the second one for specific antibody response toward KLH. Also, a confidence interval of a QTL for natural antibodies against LPS located on GGA18 was narrowed down. Combined QTL analysis was successful for two QTLs: for specific antibody response toward KLH on GGA14, and for natural antibodies against LPS on GGA18. The greatest statistical power for QTL detection in joint analysis was achieved when raw data from segregating half–sib families from different populations under the study was used.