A general mixture model for mapping quantitative trait loci by using molecular markers

Summary In a segregating population a quantitative trait may be considered to follow a mixture of (normal) distributions, the mixing proportions being based on Mendelian segregation rules. A general and flexible mixture model is proposed for mapping quantitative trait loci (QTLs) by using molecular markers. A method is discribed to fit the model to data. The model makes it possible to (1) analyse non-normally distributed traits such as lifetimes, counts or percentages in addition to normally distributed traits, (2) reduce environmental variation by taking into account the effects of experimental design factors and interaction between genotype and environment, (3) reduce genotypic variation by taking into account the effects of two or more QTLs simultaneously, (4) carry out a (combined) analysis of different population types, (5) estimate recombination frequencies between markers or use known marker distances, (6) cope with missing marker observations, (7) use markers as covariables in detection and mapping of QTLs, and finally to (8) implement the mapping in standard statistical packages.     

AFLP mapping of quantitative trait loci for yield-determining physiological characters in spring barley

 An amplified fragment length polymorphism (AFLP) map covering 965 cM was constructed using 94 recombinant inbred lines of a cross between the spring barley varieties Prisma and Apex. This map was employed to identify quantitative trait loci (QTLs) controlling plant height, yield and yield-determining physiological characters using an approximate multiple-QTL model, the MQM method. The seven physiological traits were parameters used in a process-based crop-growth model that predicts barley biomass production as affected by daily temperature and radiation. The traits were measured in experiments conducted over 2 years. Except for the relative growth rate of leaf area, all traits examined had at least one QTL in each year. QTLs and their effects were found to vary with developmental stages for one trait, the fraction of shoot biomass partitioned to leaves, that was measured at several stages. Most of the traits were associated, though to different extents, with the denso dwarfing gene (the height-reducing allele in Prisma) located on the long arm of chromosome 3. Some of the QTLs were mapped to similar positions in both years. The results in relation to effects of the dwarfing gene, the physiological basis for QTL×environment interaction, and the relative importance of the parameter traits with respect to yield, are discussed. Received: 17 September 1998 / Accepted: 28 December 1998     

Lineage-specific mapping of quantitative trait loci

We present an approach for quantitative trait locus (QTL) mapping, termed as ‘lineage-specific QTL mapping'', for inferring allelic changes of QTL evolution along with branches in a phylogeny. We describe and analyze the simplest case: by adding a third taxon into the normal procedure of QTL mapping between pairs of taxa, such inferences can be made along lineages to a presumed common ancestor. Although comparisons of QTL maps among species can identify homology of QTLs by apparent co-location, lineage-specific mapping of QTL can classify homology into (1) orthology (shared origin of QTL) versus (2) paralogy (independent origin of QTL within resolution of map distance). In this light, we present a graphical method that identifies six modes of QTL evolution in a three taxon comparison. We then apply our model to map lineage-specific QTLs for inbreeding among three taxa of yellow monkey-flower: Mimulus guttatus and two inbreeders M. platycalyx and M. micranthus, but critically assuming outcrossing was the ancestral state. The two most common modes of homology across traits were orthologous (shared ancestry of mutation for QTL alleles). The outbreeder M. guttatus had the fewest lineage-specific QTL, in accordance with the presumed ancestry of outbreeding. Extensions of lineage-specific QTL mapping to other types of data and crosses, and to inference of ancestral QTL state, are discussed.     

Selection bias in quantitative trait loci mapping

A simulation study was performed to see whether selection affected quantitative trait loci (QTL) mapping. Populations under random selection, under selection among full-sib families, and under selection within a full-sib family were simulated each with heritability of 0.3, 0.5, and 0.7. They were analyzed with the marker spacing of 10 cM and 20 cM. The accuracy for QTL detection decreased for the populations under selection within full-sib family. Estimates of QTL effects and positions differed (P < .05) from their input values. The problems could be ignored when mapping a QTL for the populations under selection among full-sib families. A large heritability helped reduction of such problems. When the animals were selected within a full-sib family, the QTL was detected for the populations with heritability of 0.5 or larger using the marker spacing of 10 cM, and with heritability of 0.7 using the marker spacing of 20 cM. This study implied that when selection was introduced, the accuracy for QTL detection decreased and the estimates of QTL effects were biased. A caution was warranted on the decision of data (including selected animals to be genotyped) for QTL mapping.     

Nonparametric functional mapping of quantitative trait loci

Summary .  Functional mapping is a useful tool for mapping quantitative trait loci (QTL) that control dynamic traits. It incorporates mathematical aspects of biological processes into the mixture model-based likelihood setting for QTL mapping, thus increasing the power of QTL detection and the precision of parameter estimation. However, in many situations there is no obvious functional form and, in such cases, this strategy will not be optimal. Here we propose to use nonparametric function estimation, typically implemented with B-splines, to estimate the underlying functional form of phenotypic trajectories, and then construct a nonparametric test to find evidence of existing QTL. Using the representation of a nonparametric regression as a mixed model, the final test statistic is a likelihood ratio test. We consider two types of genetic maps: dense maps and general maps, and the power of nonparametric functional mapping is investigated through simulation studies and demonstrated by examples.     

A mixed-model approach to mapping quantitative trait loci in barley on the basis of multiple environment data

In this article, I propose a mixed-model method to detect QTL with significant mean effect across environments and to characterize the stability of effects across multiple environments. I demonstrate the method using the barley dataset by the North American Barley Genome Mapping Project. The analysis raises the need for mixed modeling in two different ways. First, it is reasonable to regard environments as a random sample from a population of target environments. Thus, environmental main effects and QTL-by-environment interaction effects are regarded as random. Second, I expect a genetic correlation among pairs of environments caused by undetected QTL. I show how random QTL-by-environment effects as well as genetic correlations are straightforwardly handled in a mixed-model framework. The main advantage of this method is the ability to assess the stability of QTL effects. Moreover, the method allows valid statistical inferences regarding average QTL effects.     

Mapping of quantitative trait loci affecting Drechslera teres resistance in barley with molecular markers

 Resistance loci for seedling-stage resistance to net blotch disease (Drechslera teres) in barley were mapped with molecular markers in an F₂ population derived from a cross between the susceptible barley cultivar ‘Arena’ and the resistant Ethiopian landrace ‘Hor 9088’. Disease reactions were scored with first and second leaves of 2-week-old plants 7 and 9 days after inoculation with a single spore-derived isolate. For linkage analysis, 22 RFLP markers and 284 AFLP markers were used. The seven linkage groups covered 1153.3 cM with an average marker interval of 3.76 cM. The resistance was determined to be inherited in a quantitative manner. Altogether, 12 QTLs were mapped with positions depending on the leaf used for testing and the time period after infection. Heritability in the broad sense ranged between 0.21 and 0.37. Received: 26 May 1998 / Accepted: 9 June 1998     

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.     

Light-response quantitative trait loci identified with composite interval and eXtreme array mapping in Arabidopsis thaliana

Genetic analysis of natural variation in ecotypes of Arabidopsis thaliana can facilitate the discovery of new genes or of allelic variants of previously identified genes controlling physiological processes in plants. We mapped quantitative trait loci (QTL) for light response in recombinant inbred lines (RILs) derived from the Columbia and Kashmir accessions via two methods: composite interval mapping and eXtreme array mapping (XAM). After measuring seedling hypocotyl lengths in blue, red, far-red, and white light, and in darkness, eight QTL were identified by composite interval mapping and five localized near photoreceptor loci. Two QTL in blue light were associated with CRY1 and CRY2, two in red light were near PHYB and PHYC, and one in far-red light localized near PHYA. The RED2 and RED5 QTL were verified in segregating lines. XAM was tested for the identification of QTL in red light with pools of RILs selected for extreme phenotypes. Thousands of single feature polymorphisms detected by differential DNA hybridized to high-density oligo-nucleotide arrays were used to estimate allele frequency differences between the pools. The RED2 QTL was identified clearly; differences exceeded a threshold of significance determined by simulations. The sensitivities of XAM to population type and size and genetic models were also determined by simulation analysis.     

High-resolution mapping of quantitative trait loci by selective recombinant genotyping

Selective recombinant genotyping (SRG) is a three-stage procedure for high-resolution mapping of a QTL that has previously been mapped to a known confidence interval (target C.I.). In stage 1, a large mapping population is accessed and phenotyped, and a proportion, P, of the high and low tails is selected. In stage 2, the selected individuals are genotyped for a pair of markers flanking the target C.I., and a group of R individuals carrying recombinant chromosomes in the target interval are identified. In stage 3, the recombinant individuals are genotyped for a set of M markers spanning the target C.I. Extensive simulations showed that: (1) Standard error of QTL location (SEQTL) decreased when QTL effect (d) or population size (N) increased, but was constant for given "power factor" (PF = d(2)N); (2) increasing the proportion selected in the tails beyond 0.25 had only a negligible effect on SEQTL; and (3) marker spacing in the target interval had a remarkably powerful effect on SEQTL, yielding a reduction of up to 10-fold in going from highest (24 cM) to lowest (0.29 cM) spacing at given population size and QTL effect. At the densest marker spacing, SEQTL of 1.0-0.06 cM were obtained at PF = 500-16,000. Two new genotyping procedures, the half-section algorithm and the golden section/half-section algorithm, allow the equivalent of complete haplotyping of the target C.I. in the recombinant individuals to be achieved with many fewer data points than would be required by complete individual genotyping.     

Fine mapping and syntenic integration of the semi-dwarfing gene sdw3 of barley

The barley mutant allele sdw3 confers a gibberellin-insensitive, semi-dwarf phenotype with potential for breeding of new semi-dwarfed barley cultivars. Towards map-based cloning, sdw3 was delimited by high-resolution genetic mapping to a 0.04 cM interval in a “cold spot” of recombination of the proximal region of the short arm of barley chromosome 2H. Extensive synteny between the barley Sdw3 locus (Hvu_sdw3) and the orthologous regions (Osa_sdw3, Sbi_sdw3, Bsy_sdw3) of three other grass species (Oryza sativa, Sorghum bicolor, Brachypodium sylvaticum) allowed for efficient synteny-based marker saturation in the target interval. Comparative sequence analysis revealed colinearity for 23 out of the 38, 35, and 29 genes identified in Brachypodium, rice, and Sorghum, respectively. Markers co-segregating with Hvu_sdw3 were generated from two of these genes. Initial attempts at chromosome walking in barley were performed with seven orthologous gene probes which were delimiting physical distances of 223, 123, and 127 kb in Brachypodium, rice, and Sorghum, respectively. Six non-overlapping small bacterial artificial chromosome (BAC) clone contigs (cumulative length of 670 kb) were obtained, which indicated a considerably larger physical size of Hvu_sdw3. Low-pass sequencing of selected BAC clones from these barley contigs exhibited a substantially lower gene frequency per physical distance and the presence of additional non-colinear genes. Four candidate genes for sdw3 were identified within barley BAC sequences that either co-segregated with the gene sdw3 or were located adjacent to these co-segregating genes. Identification of genic sequences in the sdw3 context provides tools for marker-assisted selection. Eventual identification of the actual gene will contribute new information for a basic understanding of the mechanisms underlying growth regulation in barley.     

A method to optimize selection on multiple identified quantitative trait loci

A mathematical approach was developed to model and optimize selection on multiple known quantitative trait loci (QTL) and polygenic estimated breeding values in order to maximize a weighted sum of responses to selection over multiple generations. The model allows for linkage between QTL with multiple alleles and arbitrary genetic effects, including dominance, epistasis, and gametic imprinting. Gametic phase disequilibrium between the QTL and between the QTL and polygenes is modeled but polygenic variance is assumed constant. Breeding programs with discrete generations, differential selection of males and females and random mating of selected parents are modeled. Polygenic EBV obtained from best linear unbiased prediction models can be accommodated. The problem was formulated as a multiple-stage optimal control problem and an iterative approach was developed for its solution. The method can be used to develop and evaluate optimal strategies for selection on multiple QTL for a wide range of situations and genetic models.     

In silico mapping of quantitative trait loci in maize

Quantitative trait loci (QTL) are most often detected through designed mapping experiments. An alternative approach is in silico mapping, whereby genes are detected using existing phenotypic and genomic databases. We explored the usefulness of in silico mapping via a mixed-model approach in maize (Zea mays L.). Specifically, our objective was to determine if the procedure gave results that were repeatable across populations. Multilocation data were obtained from the 1995–2002 hybrid testing program of Limagrain Genetics in Europe. Nine heterotic patterns comprised 22,774 single crosses. These single crosses were made from 1,266 inbreds that had data for 96 simple sequence repeat (SSR) markers. By a mixed-model approach, we estimated the general combining ability effects associated with marker alleles in each heterotic pattern. The numbers of marker loci with significant effects—37 for plant height, 24 for smut [Ustilago maydis (DC.) Cda.] resistance, and 44 for grain moisture—were consistent with previous results from designed mapping experiments. Each trait had many loci with small effects and few loci with large effects. For smut resistance, a marker in bin 8.05 on chromosome 8 had a significant effect in seven (out of a maximum of 18) instances. For this major QTL, the maximum effect of an allele substitution ranged from 5.4% to 41.9%, with an average of 22.0%. We conclude that in silico mapping via a mixed-model approach can detect associations that are repeatable across different populations. We speculate that in silico mapping will be more useful for gene discovery than for selection in plant breeding programs.     

Improved confidence intervals in quantitative trait loci mapping by permutation bootstrapping

The nonparametric bootstrap approach is known to be suitable for calculating central confidence intervals for the locations of quantitative trait loci (QTL). However, the distribution of the bootstrap QTL position estimates along the chromosome is peaked at the positions of the markers and is not tailed equally. This results in conservativeness and large width of the confidence intervals. In this study three modified methods are proposed to calculate nonparametric bootstrap confidence intervals for QTL locations, which compute noncentral confidence intervals (uncorrected method I), correct for the impact of the markers (weighted method I), or both (weighted method II). Noncentral confidence intervals were computed with an analog of the highest posterior density method. The correction for the markers is based on the distribution of QTL estimates along the chromosome when the QTL is not linked with any marker, and it can be obtained with a permutation approach. In a simulation study the three methods were compared with the original bootstrap method. The results showed that it is useful, first, to compute noncentral confidence intervals and, second, to correct the bootstrap distribution of the QTL estimates for the impact of the markers. The weighted method II, combining these two properties, produced the shortest and less biased confidence intervals in a large number of simulated configurations.     

Accuracy of mapping quantitative trait loci in autogamous species

Summary The development of linkage maps with large numbers of molecular markers has stimulated the search for methods to map genes involved in quantitative traits (QTLs). A promising method, proposed by Lander and Botstein (1989), employs pairs of neighbouring markers to obtain maximum linkage information about the presence of a QTL within the enclosed chromosomal segment. In this paper the accuracy of this method was investigated by computer simulation. The results show that there is a reasonable probability of detecting QTLs that explain at least 5% of the total variance. For this purpose a minimum population of 200 backcross or F₂ individuals is necessary. Both the number of individuals and the relative size of the genotypic effect of the QTL are important factors determining the mapping precision. On the average, a QTL with 5% or 10% explained variance is mapped on an interval of 40 or 20 centiMorgans, respectively. Of course, QTLs with a larger genotypic effect will be located more precisely. It must be noted, however, that the interval length is rather variable.     

A new approach for mapping quantitative trait loci using complete genetic marker linkage maps

A new approach based on nonlinear regression for the mapping of quantitative trait loci (QTLs) using complete genetic marker linkage maps is advanced in this paper. We call the approach joint mapping as it makes comprehensive use of the information from every marker locus on a chromosome. With this approach, both the detection of the existence of QTLs and the estimation of their positions, with corresponding confidence intervals, and effects can be realized simultaneously. This approach is widely applicable because only moments are used. It is simple and can save considerable computer time. It is especially useful when there are multiple QTLs and/or interactions between them on a chromosome.     

Association mapping of seed oil content in Brassica napus and comparison with quantitative trait loci identified from linkage mapping

Association mapping has been used increasingly in natural populations with rich genetic diversity to detect DNA-based markers that are associated with important agronomic traits. Brassica napus is an important oil crop with limited genetic diversity. "New-type" B. napus that is introgressed with subgenomic components from related species has been developed to broaden the genetic basis of "traditional" B. napus. In this study, new-type B. napus lines and a collection of traditional B. napus varieties from different countries were used as two different populations to evaluate seed oil content and to determine the efficacy of association mapping by comparison with previous study of linkage mapping. Relatively rich genetic diversity, but a higher level of linkage disequilibrium was observed in the new-type B. napus as compared with the traditional B. napus. Similarly, a larger variation in oil content and a greater number of associated markers were detected in the population of new-type B. napus. Meanwhile, more than half of the genetic loci, to which the associated markers corresponded, were located within the quantitative trait loci intervals identified previously in linkage mapping experiments, which demonstrated the power of association mapping in B. napus.     

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.