An improved formulation of marker heterozygosity in recurrent selection and backcross schemes

This report presents a theoretical formulation for predicting heterozygosity of a putative marker locus linked to two quantitative trait loci (QTL) in a recurrent selection and backcross (RSB) scheme. Since the heterozygosity at any given marker locus maintained in such a breeding programme reflects its map location relative to QTL, the present study develops the theoretical analysis of the QTL mapping method that recently appeared in the literature. The formulae take into account selection, recombination and finite population size during the multiple-generation breeding scheme. The single-marker and two-QTL model was compared numerically with the model involving two linked marker loci and two QTL. Without recombination interference, the two models predict the same expected heterozygosity at the linked marker loci, indicating that the model is valid for predicting marker heterozygosity maintained at any loci in an RSB breeding scheme. The formulation is demonstrated numerically for several RSB schemes and its implications in developing a likelihood-based statistical framework for modeling the RSB experiments are discussed.     

Detection and parameter estimation for quantitative trait loci using regression models and multiple markers

A strategy of multi-step minimal conditional regression analysis has been developed to determine the existence of statistical testing and parameter estimation for a quantitative trait locus (QTL) that are unaffected by linked QTLs. The estimation of marker-QTL recombination frequency needs to consider only three cases: 1) the chromosome has only one QTL, 2) one side of the target QTL has one or more QTLs, and 3) either side of the target QTL has one or more QTLs. Analytical formula was derived to estimate marker-QTL recombination frequency for each of the three cases. The formula involves two flanking markers for case 1), two flanking markers plus a conditional marker for case 2), and two flanking markers plus two conditional markers for case 3). Each QTL variance and effect, and the total QTL variance were also estimated using analytical formulae. Simulation data show that the formulae for estimating marker-QTL recombination frequency could be a useful statistical tool for fine QTL mapping. With 1 000 observations, a QTL could be mapped to a narrow chromosome region of 1.5 cM if no linked QTL is present, and to a 2.8 cM chromosome region if either side of the target QTL has at least one linked QTL.     

Fine-Scale Mapping of Quantitative Trait Loci Using Historical Recombinations

With increasing popularity of QTL mapping in economically important animals and experimental species, the need for statistical methodology for fine-scale QTL mapping becomes increasingly urgent. The ability to disentangle several linked QTL depends on the number of recombination events. An obvious approach to increase the recombination events is to increase sample size, but this approach is often constrained by resources. Moreover, increasing the sample size beyond a certain point will not further reduce the length of confidence interval for QTL map locations. The alternative approach is to use historical recombinations. We use analytical methods to examine the properties of fine QTL mapping using historical recombinations that are accumulated through repeated intercrossing from an F(2) population. We demonstrate that, using the historical recombinations, both simple and multiple regression models can reduce significantly the lengths of support intervals for estimated QTL map locations and the variances of estimated QTL map locations. We also demonstrate that, while the simple regression model using historical recombinations does not reduce the variances of the estimated additive and dominant effects, the multiple regression model does. We further determine the power and threshold values for both the simple and multiple regression models. In addition, we calculate the Kullback-Leibler distance and Fisher information for the simple regression model, in the hope to further understand the advantages and disadvantages of using historical recombinations relative to F(2) data.     

Disequilibrium likelihoods for fine-scale mapping of a rare allele.

Genetic linkage studies based on pedigree data have limited resolution, because of the relatively small number of segregations. Disequilibrium mapping, which uses population associations to infer the location of a disease mutation, provides one possible strategy for narrowing the candidate region. The coalescent process provides a model for the ancestry of a sample of disease alleles, and recombination events between disease locus and marker may be placed on this ancestral phylogeny. These events define the recombinant classes, the sets of sampled disease copies descending from the meiosis at which a given recombination occurred. We show how Monte Carlo generation of the recombinant classes leads to a linkage likelihood for fine-scale mapping from disease haplotypes. We compare single-marker disequilibrium mapping with interval-disequilibrium mapping and discuss how the approach may be extended to multipoint-disequilibrium mapping. The method and its properties are illustrated with an example of simulated data, constructed to be typical of fine-scale mapping of a rare disease in the Japanese population. The method can take into account known features of population history, such as changing patterns of population growth.     

Detection of closely linked multiple quantitative trait loci using a genetic algorithm

The existence of a quantitative trait locus (QTL) is usually tested using the likelihood of the quantitative trait on the basis of phenotypic character data plus the recombination fraction between QTL and flanking markers. When doing this, the likelihood is calculated for all possible locations on the linkage map. When multiple QTL are suspected close by, it is impractical to calculate the likelihood for all possible combinations of numbers and locations of QTL. Here, we propose a genetic algorithm (GA) for the heuristic solution of this problem. GA can globally search the optimum by improving the "genotype" with alterations called "recombination" and "mutation." The "genotype" of our GA is the number and location of QTL. The "fitness" is a function based on the likelihood plus Akaike's information criterion (AIC), which helps avoid false-positive QTL. A simulation study comparing the new method with existing QTL mapping packages shows the advantage of the new GA. The GA reliably distinguishes multiple QTL located in a single marker interval.     

A quantitative trait locus mixture model that avoids spurious LOD score peaks

In standard interval mapping of quantitative trait loci (QTL), the QTL effect is described by a normal mixture model. At any given location in the genome, the evidence of a putative QTL is measured by the likelihood ratio of the mixture model compared to a single normal distribution (the LOD score). This approach can occasionally produce spurious LOD score peaks in regions of low genotype information (e.g., widely spaced markers), especially if the phenotype distribution deviates markedly from a normal distribution. Such peaks are not indicative of a QTL effect; rather, they are caused by the fact that a mixture of normals always produces a better fit than a single normal distribution. In this study, a mixture model for QTL mapping that avoids the problems of such spurious LOD score peaks is presented.     

Simultaneous detection and fine mapping of quantitative trait loci in mice using heterogeneous stocks

We describe a method to simultaneously detect and fine map quantitative trait loci (QTL) that is especially suited to the mapping of modifier loci in mouse mutant models. The method exploits the high level of historical recombination present in a heterogeneous stock (HS), an outbred population of mice derived from known founder strains. The experimental design is an F(2) cross between the HS and a genetically distinct line, such as one carrying a knockout or transgene. QTL detection is performed by a standard genome scan with approximately 100 markers and fine mapping by typing the same animals using densely spaced markers over those candidate regions detected by the scan. The analysis uses an extension of the dynamic-programming technique employed previously to fine map QTL in HS mice. We show by simulation that a QTL accounting for 5% of the total variance can be detected and fine mapped with >50% probability to within 3 cM by genotyping approximately 1500 animals.     

Advanced Intercross Lines, an Experimental Population for Fine Genetic Mapping

An advanced intercrossed line (AIL) is an experimental population that can provide more accurate estimates of quantitative trait loci (QTL) map location than conventional mapping populations. An AIL is produced by randomly and sequentially intercrossing a population that initially originated from a cross between two inbred lines or some variant thereof. This provides increasing probability of recombination between any two loci. Consequently, the genetic length of the entire genome is stretched, providing increased mapping resolution. In this way, for example, with the same population size and QTL effect, a 95% confidence interval of QTL map location of 20 cM in the F(2) is reduced fivefold after eight additional random mating generations (F(10)). Simulation results showed that to obtain the anticipated reduction in the confidence interval, breeding population size of the AIL in all generations should comprise an effective number of >/=100 individuals. It is proposed that AILs derived from crosses between known inbred lines may be a useful resource for fine genetic mapping.     

基于分离亚群体QTL定位的模拟研究

在数量性状QTL的精细定位中, 通过数量性状目标QTL的近等基因系可构建分离群体。在目标QTL效应较大的情况下, 数量性状表型值可反映目标QTL的基因型。若目标QTL附近的标记密度大时, 大样本才能定位该QTL。但是, 这增加了试验费用。为节约试验经费, 若只利用QTL纯合隐性基因型植株的分子标记信息, 也可比较准确地定位该QTL。利用极大似然法, 分别推导出F2、BC、DH以及RIL群体中重组率及其标准误的估计公式。Monte Carlo模拟研究表明, 基于定位群体中全部数据或隐性纯合基因型数据所获得的重组率估计值是一致的, 且在相同样本容量条件下, 二者精度相当。     

Fine mapping of quantitative trait loci using linkage disequilibria with closely linked marker loci

A multimarker linkage disequilibrium mapping method was developed for the fine mapping of quantitative trait loci (QTL) using a dense marker map. The method compares the expected covariances between haplotype effects given a postulated QTL position to the covariances that are found in the data. The expected covariances between the haplotype effects are proportional to the probability that the QTL position is identical by descent (IBD) given the marker haplotype information, which is calculated using the genedropping method. Simulation results showed that a QTL was correctly positioned within a region of 3, 1.5, or 0.75 cM in 70, 62, and 68%, respectively, of the replicates using markers spaced at intervals of 1, 0.5, and 0.25 cM, respectively. These results were rather insensitive to the number of generations since the QTL occurred and to the effective population size, except that 10 generations yielded rather poor estimates of the QTL position. The position estimates of this multimarker disequilibrium mapping method were more accurate than those from a single marker transmission disequilibrium test. A general approach for identifying QTL is suggested, where several stages of disequilibrium mapping are used with increasingly dense marker spacing.     

An algorithm for molecular dissection of tumor progression

The volumetric growth of tumor cells as a function of time is most often likely to be a complex trait, controlled by the combined influences of multiple genes and environmental influences. Genetic mapping has proven to be a powerful tool for detecting and identifying specific genes affecting complex traits, i.e., quantitative trait loci (QTL), based on polymorphic markers. In this article, we present a novel statistical model for genetic mapping of QTL governing tumor growth trajectories in humans. In principle, this model is a combination of functional mapping proposed to map function-valued traits and linkage disequilibrium mapping designed to provide high resolution mapping of QTL by making use of recombination events created at a historic time. We implement an EM-simplex hybrid algorithm for parameter estimation, in which a closed-form solution for the EM algorithm is derived to estimate the population genetic parameters of QTL including the allele frequencies and the coefficient of linkage disequilibrium, and the simplex algorithm incorporated to estimate the curve parameters describing the dynamic changes of cancer cells for different QTL genotypes. Extensive simulations are performed to investigate the statistical properties of our model. Through a number of hypothesis tests, our model allows for cutting-edge studies aimed to decipher the genetic mechanisms underlying cancer growth, development and differentiation. The implications of our model in gene therapy for cancer research are discussed.     

High-resolution patterns of meiotic recombination across the human major histocompatibility complex

Definitive characteristics of meiotic recombination events over large (i.e., >1 Mb) segments of the human genome remain obscure, yet they are essential for establishing the haplotypic structure of the genome and for efficient mapping of complex traits. We present a high-resolution map of recombination at the kilobase level across a 3.3-Mb interval encompassing the major histocompatibility complex (MHC). Genotyping of 20,031 single sperm from 12 individuals resulted in the identification and fine mapping of 325 recombinant chromosomes within genomic intervals as small as 7 kb. Several principal characteristics of recombination in this region were observed: (1) rates of recombination can differ significantly between individuals; (2) intense hot spots of recombination occur at least every 0.8 Mb but are not necessarily evenly spaced; (3) distribution in the location of recombination events can differ significantly among individuals; (4) between hot spots, low levels of recombination occur fairly evenly across 100-kb segments, suggesting the presence of warm spots of recombination; and (5) specific sequence motifs associate significantly with recombination distribution. These data provide a plausible model for recombination patterns of the human genome overall.     

Statistical Properties of the Number of Recombination Events in the History of a Sample of DNA Sequences

Some statistical properties of samples of DNA sequences are studied under an infinite-site neutral model with recombination. The two quantities of interest are R, the number of recombination events in the history of a sample of sequences, and RM, the number of recombination events that can be parsimoniously inferred from a sample of sequences. Formulas are derived for the mean and variance of R. In contrast to R, RM can be determined from the sample. Since no formulas are known for the mean and variance of RM, they are estimated with Monte Carlo simulations. It is found that RM is often much less than R, therefore, the number of recombination events may be greatly under-estimated in a parsimonious reconstruction of the history of a sample. The statistic RM can be used to estimate the product of the recombination rate and the population size or, if the recombination rate is known, to estimate the population size. To illustrate this, DNA sequences from the Adh region of Drosophila melanogaster are used to estimate the effective population size of this species.     

Interval Mapping of Multiple Quantitative Trait Loci

The interval mapping method is widely used for the mapping of quantitative trait loci (QTLs) in segregating generations derived from crosses between inbred lines. The efficiency of detecting and the accuracy of mapping multiple QTLs by using genetic markers are much increased by employing multiple QTL models instead of the single QTL models (and no QTL models) used in interval mapping. However, the computational work involved with multiple QTL models is considerable when the number of QTLs is large. In this paper it is proposed to combine multiple linear regression methods with conventional interval mapping. This is achieved by fitting one QTL at a time in a given interval and simultaneously using (part of) the markers as cofactors to eliminate the effects of additional QTLs. It is shown that the proposed method combines the easy computation of the single QTL interval mapping method with much of the efficiency and accuracy of multiple QTL models.     

Mapping quantitative trait loci by genotyping haploid tissues.

Mapping strategies based on a half- or full-sib family design have been developed to map quantitative trait loci (QTL) for outcrossing species. However, these strategies are dependent on controlled crosses where marker-allelic frequency and linkage disequilibrium between the marker and QTL may limit their application. In this article, a maximum-likelihood method is developed to map QTL segregating in an open-pollinated progeny population using dominant markers derived from haploid tissues from single meiotic events. Results from the haploid-based mapping strategy are not influenced by the allelic frequencies of markers and their linkage disequilibria with QTL, because the probabilities of QTL genotypes conditional on marker genotypes of haploid tissues are independent of these population parameters. Parameter estimation and hypothesis testing are implemented via expectation/conditional maximization algorithm. Parameters estimated include the additive effect, the dominant effect, the population mean, the chromosomal location of the QTL in the interval, and the residual variance within the QTL genotypes, plus two population parameters, outcrossing rate and QTL-allelic frequency. Simulation experiments show that the accuracy and power of parameter estimates are affected by the magnitude of QTL effects, heritability levels of a trait, and sample sizes used. The application and limitation of the method are discussed.     

Potential of a tomato MAGIC population to decipher the genetic control of quantitative traits and detect causal variants in the resequencing era

Identification of the polymorphisms controlling quantitative traits remains a challenge for plant geneticists. Multiparent advanced generation intercross (MAGIC) populations offer an alternative to traditional linkage or association mapping populations by increasing the precision of quantitative trait loci (QTL) mapping. Here, we present the first tomato MAGIC population and highlight its potential for the valorization of intraspecific variation, QTL mapping and causal polymorphism identification. The population was developed by crossing eight founder lines, selected to include a wide range of genetic diversity, whose genomes have been previously resequenced. We selected 1536 SNPs among the 4 million available to enhance haplotype prediction and recombination detection in the population. The linkage map obtained showed an 87% increase in recombination frequencies compared to biparental populations. The prediction of the haplotype origin was possible for 89% of the MAGIC line genomes, allowing QTL detection at the haplotype level. We grew the population in two greenhouse trials and detected QTLs for fruit weight. We mapped three stable QTLs and six specific of a location. Finally, we showed the potential of the MAGIC population when coupled with whole genome sequencing of founder lines to detect candidate SNPs underlying the QTLs. For a previously cloned QTL on chromosome 3, we used the predicted allelic effect of each founder and their genome sequences to select putative causal polymorphisms in the supporting interval. The number of candidate polymorphisms was reduced from 12 284 (in 800 genes) to 96 (in 54 genes), including the actual causal polymorphism. This population represents a new permanent resource for the tomato genetics community.     

Genomewide markers as cofactors for precision mapping of quantitative trait loci

In composite interval mapping of quantitative trait loci (QTL), subsets of background markers are used to account for the effects of QTL outside the marker interval being tested. Here, I propose a QTL mapping approach (called G model) that utilizes genomewide markers as cofactors. The G model involves backward elimination on a given chromosome after correcting for genomewide marker effects, calculated under a random effects model, at all the other chromosomes. I simulated a trait controlled by 15 or 30 QTL, mapping populations of N = 96, 192, and 384 recombinant inbreds, and N _M = 192 and 384 evenly spaced markers. In the C model, which utilized subsets of background markers, the number of QTL detected and the number of false positives depended on the number of cofactors used, with five cofactors being too few with N = 384 and 20–40 cofactors being too many with N = 96. A window size of 0 cM for excluding cofactors maintained the number of true QTL detected while decreasing the number of false positives. The number of true QTL detected was generally higher with the G model than with the C model, and the G model led to good control of the type I error rate in simulations where the null hypothesis of no marker–QTL linkage was true. Overall, the results indicated that the G model is useful in QTL mapping because it is less subjective and has equal, if not better, performance when compared with the traditional approach of using subsets of markers to account for background QTL.     

Detecting Marker-Qtl Linkage and Estimating Qtl Gene Effect and Map Location Using a Saturated Genetic Map

A simulation study was carried out on a backcross population in order to determine the effect of marker spacing, gene effect and population size on the power of marker-quantitative trait loci (QTL) linkage experiments and on the standard error of maximum likelihood estimates (MLE) of QTL gene effect and map location. Power of detecting a QTL was virtually the same for a marker spacing of 10 cM as for an infinite number of markers and was only slightly decreased for marker spacing of 20 or even 50 cM. The advantage of using interval mapping as compared to single-marker analysis was slight. ``Resolving power' of a marker-QTL linkage experiment was defined as the 95% confidence interval for the QTL map location that would be obtained when scoring an infinite number of markers. It was found that reducing marker spacing below the resolving power did not add appreciably to narrowing the confidence interval. Thus, the 95% confidence interval with infinite markers sets the useful marker spacing for estimating QTL map location for a given population size and estimated gene effect.     

Look before you leap: a new approach to mapping QTL

In this paper, we present an innovative and powerful approach for mapping quantitative trait loci (QTL) in experimental populations. This deviates from the traditional approach of (composite) interval mapping which uses a QTL profile to simultaneously determine the number and location of QTL. Instead, we look before we leap by employing separate detection and localization stages. In the detection stage, we use an iterative variable selection process coupled with permutation to identify the number and synteny of QTL. In the localization stage, we position the detected QTL through a series of one-dimensional interval mapping scans. Results from a detailed simulation study and real analysis of wheat data are presented. We achieve impressive increases in the power of QTL detection compared to composite interval mapping. We also accurately estimate the size and position of QTL. An R library, DLMap, implements the methods described here and is freely available from CRAN ().