Linkage analysis of quantitative trait loci in multiple line crosses

Simple line crosses, for example, backcross and F₂, are commonly used in mapping quantitative trait loci (QTL). However, these simple crosses are rarely used alone in commercial plant breeding; rather, crosses involving multiple inbred lines or several simple crosses but connected by shared inbred lines may be common in plant breeding. Mapping QTL using crosses of multiple lines is more relevant to plant breeding. Unfortunately, current statistical methods and computer programs of QTL mapping are all designed for simple line crosses or multiple line crosses but under a regular mating system. It is not straightforward to extend the existing methods to handle multiple line crosses under irregular and complicated mating designs. The major hurdle comes from irregular inbreeding, multiple generations, and multiple alleles. In this study, we develop a Bayesian method implemented via the Markov chain Monte Carlo (MCMC) algorithm for mapping QTL using complicated multiple line crosses. With the MCMC algorithm, we are able to draw a complete path of the gene flow from founder alleles to their descendents via a recursive process. This has greatly simplified the problem caused by irregular mating and inbreeding in the mapping population. Adopting the reversible jump MCMC algorithm, we are able to simultaneously search for multiple QTL along the genome. We can even infer the posterior distribution of the number of QTL, one of the most important parameters in QTL study. Application of the new MCMC based QTL mapping procedure is demonstrated using two different mating designs. Design I involves two inbred lines and their derived F₁, F₂, and BC populations. Design II is a half-diallel cross involving three inbred lines. The two designs appear different, but can be handled with the same robust computer program.     

Mapping quantitative trait loci in noninbred mosquito crosses

The identification of genes that affect quantitative traits has been of great interest to geneticists for many decades, and many statistical methods have been developed to map quantitative trait loci (QTL). Most QTL mapping studies in experimental organisms use purely inbred lines, where the two homologous chromosomes in each individual are identical. As a result, many existing QTL mapping methods developed for experimental organisms are applicable only to genetic crosses between inbred lines. However, it may be difficult to obtain inbred lines for certain organisms, e.g., mosquitoes. Although statistical methods for QTL mapping in outbred populations, e.g., humans, can be applied for such crosses, these methods may not fully take advantage of the uniqueness of these crosses. For example, we can generally assume that the two grandparental lines are homozygous at the QTL of interest, but such information is not be utilized through methods developed for outbred populations. In addition, mating types and phases can be relatively easy to establish through the analysis of adjacent markers due to the large number of offspring that can be collected, substantially simplifying the computational need. In this article, motivated by a mosquito intercross experiment involving two selected lines that are not genetically homozygous across the genome, we develop statistical methods for QTL mapping for genetic crosses involving noninbred lines. In our procedure, we first infer parental mating types and use likelihood-based methods to infer phases in each parent on the basis of genotypes of offspring and one parent. A hidden Markov model is then employed to estimate the number of high-risk alleles at marker positions and putative QTL positions between markers in each offspring, and QTL mapping is finally conducted through the inferred QTL configuration across all offspring in all crosses. The performance of the proposed methods is assessed through simulation studies, and the usefulness of this method is demonstrated through its application to a mosquito data set.     

Bayesian model selection for genome-wide epistatic quantitative trait loci analysis

The problem of identifying complex epistatic quantitative trait loci (QTL) across the entire genome continues to be a formidable challenge for geneticists. The complexity of genome-wide epistatic analysis results mainly from the number of QTL being unknown and the number of possible epistatic effects being huge. In this article, we use a composite model space approach to develop a Bayesian model selection framework for identifying epistatic QTL for complex traits in experimental crosses from two inbred lines. By placing a liberal constraint on the upper bound of the number of detectable QTL we restrict attention to models of fixed dimension, greatly simplifying calculations. Indicators specify which main and epistatic effects of putative QTL are included. We detail how to use prior knowledge to bound the number of detectable QTL and to specify prior distributions for indicators of genetic effects. We develop a computationally efficient Markov chain Monte Carlo (MCMC) algorithm using the Gibbs sampler and Metropolis-Hastings algorithm to explore the posterior distribution. We illustrate the proposed method by detecting new epistatic QTL for obesity in a backcross of CAST/Ei mice onto M16i.     

Detection of putative quantitative trait loci in line crosses under infinitesimal genetic models

Quantitative trait locus (QTL) mapping studies often employ segregating generations derived from a cross between genetically divergent inbred lines. In the analysis of such data it is customary to fit a single QTL and use a null hypothesis which assumes that the genomic region under study contributes no genetic variance. To explore the situation in which multiple linked genes contribute to the genetic variance, we simulated an F₂-mapping experiment in which the genetic difference between the two original inbred strains was caused by a large number of loci, each having equal effect on the quantitative trait. QTLs were either in coupling, dispersion or repulsion phase in the base population of inbred lines, with the expected F₂ genetic variance explained by the QTLs being equivalent in the three models. Where QTLs were in coupling phase, one inbred line was fixed for all plus alleles, and the other line was fixed for minus alleles. Where QTLs were in dispersion phase, they were assumed to be randomly fixed for one or other allele (as if the inbred lines had evolved from a common ancestor by random drift). Where QTLs were in repulsion phase alleles within an inbred line were alternating plus and minus at adjacent loci, and alternative alleles were fixed in the two inbred lines. In all these genetic models a standard interval mapping test statistic used to determine whether there is a QTL of large effect segregating in the population was inflated on average. Furthermore, the use of a threshold for QTL detection derived under the assumption that no QTLs were segregating would often lead to spurious conclusions regards the presence of genes of large effects (i.e. type I errors). The employment of an alternative model for the analysis, including linked markers as cofactors in the analysis of a single interval, reduced the problem of type I error rate, although test statistics were still inflated relative to the case of no QTLs. It is argued that in practice one should take into account the difference between the strains or the genetic variance in the F₂ population when setting significance thresholds. In addition, tests designed to probe the adequacy of a single-QTL model or of an alternative infinitesimal coupling model are described. Such tests should be applied in QTL mapping studies to help dissect the true nature of genetic variation.     

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.     

Estimating allelic number and identity in state of QTLs in interconnected families

When multiple related families derived from inbred lines are jointly analysed to detect quantitative trait loci (QTLs), the analysis should estimate allelic effects as accurately as possible and estimate the probability that different parents carry alleles that are identical in state. Analyses exist that assume that all parents carry unique alleles or that all parents but one carry the same allele. In practice, many configurations are possible that group different parents according to their identity-in-state condition at a putative QTL allele. Here, we propose a variable model Bayesian analysis that selects among possible identity-in-state configurations and jointly estimates the allelic effects of identical-in-state parents. We contrast this analysis with a fixed model analysis that estimates unique allelic effects for all parents. We analyse two simulated mating designs: an experimental design in which three inbred parents were crossed to generate two families of 150 doubled haploid lines; and a breeding design in which 20 inbred parents were crossed to generate 60 families of 20 doubled haploid lines, with each parent contributing to six families. In all cases where some parents were simulated to carry alleles of identical effect (that is, they were identical in state), the variable analysis estimated allelic effects with lower mean-squared error than the fixed analysis. The variable analysis showed that, unless each family contains many individuals (more than 100), there is insufficient information in DNA-marker and phenotypic data to determine with high probability the QTL allelic number.     

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.     

Recent approaches into the genetic basis of inbreeding depression in plants

Predictions for the evolution of mating systems and genetic load vary, depending on the genetic basis of inbreeding depression (dominance versus overdominance, epistasis and the relative frequencies of genes of large and small effect). A distinction between the dominance and overdominance hypotheses is that deleterious recessive mutations should be purged in inbreeding populations. Comparative studies of populations differing in their level of inbreeding and experimental approaches that allow selection among inbred lines support this prediction. More direct biometric approaches provide strong support for the importance of partly recessive deleterious alleles. Investigators using molecular markers to study quantitative trait loci (QTL) often find support for overdominance, though pseudo-overdominance (deleterious alleles linked in repulsion) may bias this perception. QTL and biometric studies of inbred lines often find evidence for epistasis, which may also contribute to the perception of overdominance, though this may be because of the divergent lines initially crossed in QTL studies. Studies of marker segregation distortion commonly uncover genes of major effect on viability, but these have only minor contributions to inbreeding depression. Although considerable progress has been made in understanding the genetic basis of inbreeding depression, we feel that all three aspects merit more study in natural plant populations.     

Quantitative trait loci mapping in F(2) crosses between outbred lines

We develop a mixed-model approach for QTL analysis in crosses between outbred lines that allows for QTL segregation within lines as well as for differences in mean QTL effects between lines. We also propose a method called "segment mapping" that is based in partitioning the genome in a series of segments. The expected change in mean according to percentage of breed origin, together with the genetic variance associated with each segment, is estimated using maximum likelihood. The method also allows the estimation of differences in additive variances between the parental lines. Completely fixed random and mixed models together with segment mapping are compared via simulation. The segment mapping and mixed-model behaviors are similar to those of classical methods, either the fixed or random models, under simple genetic models (a single QTL with alternative alleles fixed in each line), whereas they provide less biased estimates and have higher power than fixed or random models in more complex situations, i.e., when the QTL are segregating within the parental lines. The segment mapping approach is particularly useful to determining which chromosome regions are likely to contain QTL when these are linked.     

Mapping quantitative trait loci using naturally occurring genetic variance among commercial inbred lines of maize (Zea mays L.)

Many commercial inbred lines are available in crops. A large amount of genetic variation is preserved among these lines. The genealogical history of the inbred lines is usually well documented. However, quantitative trait loci (QTL) responsible for the genetic variances among the lines are largely unexplored due to lack of statistical methods. In this study, we show that the pedigree information of the lines along with the trait values and marker information can be used to map QTL without the need of further crossing experiments. We develop a Monte Carlo method to estimate locus-specific identity-by-descent (IBD) matrices. These IBD matrices are further incorporated into a mixed-model equation for variance component analysis. QTL variance is estimated and tested at every putative position of the genome. The actual QTL are detected by scanning the entire genome. Applying this new method to a well-documented pedigree of maize (Zea mays L.) that consists of 404 inbred lines, we mapped eight QTL for the maize male flowering trait, growing degree day heat units to pollen shedding (GDUSHD). These detected QTL contributed >80% of the variance observed among the inbred lines. The QTL were then used to evaluate all the inbred lines using the best linear unbiased prediction (BLUP) technique. Superior lines were selected according to the estimated QTL allelic values, a technique called marker-assisted selection (MAS). The MAS procedure implemented via BLUP may be routinely used by breeders to select superior lines and line combinations for development of new cultivars.     

Joint estimates of quantitative trait locus effect and frequency using synthetic recombinant populations of Drosophila melanogaster

We develop and implement a strategy to map QTL in two synthetic populations of Drosophila melanogaster each initiated with eight inbred founder strains. These recombinant populations allow simultaneous estimates of QTL location, effect, and frequency. Five X-linked QTL influencing bristle number were resolved to intervals of approximately 1.3 cM. We confirm previous observations of bristle number QTL distal to 4A at the tip of the chromosome and identify two novel QTL in 7F-8C, an interval that does not include any classic bristle number candidate genes. If QTL at the tip of the X are biallelic they appear to be intermediate in frequency, although there is evidence that these QTL may reside in multiallelic haplotypes. Conversely, the two QTL mapping to the middle of the X chromosome are likely rare: in each case the minor allele is observed in only 1 of the 16 founders. Assuming additivity and biallelism we estimate that identified QTL contribute 1.0 and 8.7%, respectively, to total phenotypic variation in male abdominal and sternopleural bristle number in nature. Models that seek to explain the maintenance of genetic variation make different predictions about the population frequency of QTL alleles. Thus, mapping QTL in eight-way recombinant populations can distinguish between these models.     

Quantitative trait loci (QTL) detection in multicross inbred designs: recovering QTL identical-by-descent status information from marker data

Mapping quantitative trait loci in plants is usually conducted using a population derived from a cross between two inbred lines. The power of such QTL detection and the parameter estimates depend largely on the choice of the two parental lines. Thus, the QTL detected in such populations represent only a small part of the genetic architecture of the trait. In addition, the effects of only two alleles are characterized, which is of limited interest to the breeder, while common pedigree breeding material remains unexploited for QTL mapping. In this study, we extend QTL mapping methodology to a generalized framework, based on a two-step IBD variance component approach, applicable to any type of breeding population obtained from inbred parents. We then investigate with simulated data mimicking conventional breeding programs the influence of different estimates of the IBD values on the power of QTL detection. The proposed method would provide an alternative to the development of specifically designed recombinant populations, by utilizing the genetic variation actually managed by plant breeders. The use of these detected QTL in assisting breeding would thus be facilitated.     

Quantitative trait locus analysis of male mating success and sperm competition in Drosophila melanogaster

Much of sexual selection theory depends on assumptions about the genetic basis of variation in male mating success and sperm competitive ability. Despite intense interest in this topic, few genes have been identified that contribute to variation in these traits. Here we report the results of quantitative trait locus (QTL) analyses of mating success of male Drosophila melanogaster when exposed to virgin females, remating success of males with previously mated females, and both defense and offense components of sperm competition. We found two to four significant QTLs for remating success, but no QTLs for mating success, even though mating success was more genetically variable than remating success in the recombinant inbred lines used in this study. By combining these results with data from previous gene-expression experiments, we were able to identify three X-linked candidate genes for variation in remating ability. For two of these genes, QTL and expression data were completely concordant with respect to directionality of effects: high mating success was associated with high levels of gene expression and with beneficial QTL effects on the trait. We found equivocal evidence for genetic variation in sperm offense and defense in the recombinant inbred lines, and we did not find any significant QTLs for either sperm competition trait.     

Strategies for genetic mapping of categorical traits

The search for efficient and powerful statistical methods and optimal mapping strategies for categorical traits under various experimental designs continues to be one of the main tasks in genetic mapping studies. Methodologies for genetic mapping of categorical traits can generally be classified into two groups, linear and non-linear models. We develop a method based on a threshold model, termed mixture threshold model to handle ordinal (or binary) data from multiple families. Monte Carlo simulations are done to compare its statistical efficiencies and properties of the proposed non-linear model with a linear model for genetic mapping of categorical traits using multiple families. The mixture threshold model has notably higher statistical power than linear models. There may be an optimal sampling strategy (family size vs number of families) in which genetic mapping reaches its maximal power and minimal estimation errors. A single large-sibship family does not necessarily produce the maximal power for detection of quantitative trait loci (QTL) due to genetic sampling of QTL alleles. The QTL allelic model has a marked impact on efficiency of genetic mapping of categorical traits in terms of statistical power and QTL parameter estimation. Compared with a fixed number of QTL alleles (two or four), the model with an infinite number of QTL alleles and normally distributed allelic effects results in loss of statistical power. The results imply that inbred designs (e.g. F₂ or four-way crosses) with a few QTL alleles segregating or reducing number of QTL alleles (e.g. by selection) in outbred populations are desirable in genetic mapping of categorical traits using data from multiple families. This revised version was published online in July 2006 with corrections to the Cover Date.     

A unified Markov chain Monte Carlo framework for mapping multiple quantitative trait loci

In this article, a unified Markov chain Monte Carlo (MCMC) framework is proposed to identify multiple quantitative trait loci (QTL) for complex traits in experimental designs, based on a composite space representation of the problem that has fixed dimension. The proposed unified approach includes the existing Bayesian QTL mapping methods using reversible jump MCMC algorithm as special cases. We also show that a variety of Bayesian variable selection methods using Gibbs sampling can be applied to the composite model space for mapping multiple QTL. The unified framework not only results in some new algorithms, but also gives useful insight into some of the important factors governing the performance of Gibbs sampling and reversible jump for mapping multiple QTL. Finally, we develop strategies to improve the performance of MCMC algorithms.     

A general mixture model approach for mapping quantitative trait loci from diverse cross designs involving multiple inbred lines

Most current statistical methods developed for mapping quantitative trait loci (QTL) based on inbred line designs apply to crosses from two inbred lines. Analysis of QTL in these crosses is restricted by the parental genetic differences between lines. Crosses from multiple inbred lines or multiple families are common in plant and animal breeding programmes, and can be used to increase the efficiency of a QTL mapping study. A general statistical method using mixture model procedures and the EM algorithm is developed for mapping QTL from various cross designs of multiple inbred lines. The general procedure features three cross design matrices, W, that define the contribution of parental lines to a particular cross and a genetic design matrix, D, that specifies the genetic model used in multiple line crosses. By appropriately specifying W matrices, the statistical method can be applied to various cross designs, such as diallel, factorial, cyclic, parallel or arbitrary-pattern cross designs with two or multiple parental lines. Also, with appropriate specification for the D matrix, the method can be used to analyse different kinds of cross populations, such as F2 backcross, four-way cross and mixed crosses (e.g. combining backcross and F2). Simulation studies were conducted to explore the properties of the method, and confirmed its applicability to diverse experimental designs.     

Mapping-Linked Quantitative Trait Loci Using Bayesian Analysis and Markov Chain Monte Carlo Algorithms

A Bayesian method for mapping linked quantitative trait loci (QTL) using multiple linked genetic markers is presented. Parameter estimation and hypothesis testing was implemented via Markov chain Monte Carlo (MCMC) algorithms. Parameters included were allele frequencies and substitution effects for two biallelic QTL, map positions of the QTL and markers, allele frequencies of the markers, and polygenic and residual variances. Missing data were polygenic effects and multi-locus marker-QTL genotypes. Three different MCMC schemes for testing the presence of a single or two linked QTL on the chromosome were compared. The first approach includes a model indicator variable representing two unlinked QTL affecting the trait, one linked and one unlinked QTL, or both QTL linked with the markers. The second approach incorporates an indicator variable for each QTL into the model for phenotype, allowing or not allowing for a substitution effect of a QTL on phenotype, and the third approach is based on model determination by reversible jump MCMC. Methods were evaluated empirically by analyzing simulated granddaughter designs. All methods identified correctly a second, linked QTL and did not reject the one-QTL model when there was only a single QTL and no additional or an unlinked QTL.     

Derivation of single-locus relationship coefficients conditional on marker information

The coefficient of relationship is defined as the correlation between the additive genetic values of two individuals. This coefficient can be defined specifically for a single quantitative trait locus (QTL) and may deviate considerably from the overall expectation if it is taken conditional on information from linked marker loci. Conditional halfsib correlations are derived under a simple genetic model with a biallelic QTL linked to a biallelic marker locus. The conditional relationship coefficients are shown to depend on the recombination rate between the marker and the QTL and the population frequency of the marker alleles, but not on parameters of the QTL, i.e. number and frequency of QTL alleles, degree of dominance etc., nor on the (usually unknown) QTL genotype of the sire. Extensions to less simplified cases (multiple alleles at the marker locus and the QTL, two marker loci flanking the QTL) are given. For arbitrary pedigrees, conditional relationship coefficients can also be derived from the conditional gametic covariance matrix suggested by Fernando and Grossman (1989). The connection of these two approaches is discussed. The conditional relationship coefficient can be used for marker-assisted genetic evaluation as well as for the detection of QTL and the estimation of their effects.     

A method for computing identity by descent probabilities and quantitative trait loci mapping with dominant (AFLP) markers

Amplified fragment length polymorphisms (AFLPs) are a widely used marker system: the technique is very cost-effective, easy and rapid, and reproducibly generates hundreds of markers. Unfortunately, AFLP alleles are typically scored as the presence or absence of a band and, thus, heterozygous and dominant homozygous genotypes cannot be distinguished. This results in a significant loss of information, especially as regards mapping of quantitative trait loci (QTLs). We present a Monte Carlo Markov Chain method that allows us to compute the identity by descent probabilities (IBD) in a general pedigree whose individuals have been typed for dominant markers. The method allows us to include the information provided by the fluorescent band intensities of the markers, the rationale being that homozygous individuals have on average higher band intensities than heterozygous individuals, as well as information from linked markers in each individual and its relatives. Once IBD probabilities are obtained, they can be combined into the QTL mapping strategy of choice. We illustrate the method with two simulated populations: an outbred population consisting of full sib families, and an F2 cross between inbred lines. Two marker spacings were considered, 5 or 20 cM, in the outbred population. There was almost no difference, for the practical purpose of QTL estimation, between AFLPs and biallelic codominant markers when the band density is taken into account, especially at the 5 cM spacing. The performance of AFLPs every 5 cM was also comparable to that of highly polymorphic markers (microsatellites) spaced every 20 cM. In economic terms, QTL mapping with a dense map of AFLPs is clearly better than microsatellite QTL mapping and little is lost in terms of accuracy of position. Nevertheless, at low marker densities, AFLPs or other biallelic markers result in very inaccurate estimates of QTL position.     

Quantitative Trait Loci Differentiating the Outbreeding Mimulus Guttatus from the Inbreeding M. Platycalyx

Theoretical predictions about the evolution of selfing depend on the genetic architecture of loci controlling selfing (monogenic vs. polygenic determination, large vs. small effect of alleles, dominance vs. recessiveness), and studies of such architecture are lacking. We inferred the genetic basis of mating system differences between the outbreeding Mimulus guttatus and the inbreeding M. platycalyx by quantitative trait locus (QTL) mapping using random amplified polymorphic DNA and isozyme markers. One to three QTL were detected for each of five mating system characters, and each QTL explained 7.6-28.6% of the phenotypic variance. Taken together, QTL accounted for up to 38% of the variation in mating system characters, and a large proportion of variation was unaccounted for. Inferred QTL often affected more than one trait, contributing to the genetic correlation between those traits. These results are consistent with the hypothesis that quantitative variation in plant mating system characters is primarily controlled by loci with small effect.