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.     

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.     

Optimum spacing of genetic markers for determining linkage between marker loci and quantitative trait loci

The cost of experiments aimed at determining linkage between marker loci and quantitative trait loci (QTL) was investigated as a function of marker spacing and number of individuals scored. It was found that for a variety of experimental designs, fairly wide marker spacings (ca. 50 cM) are optimum or close to optimum for initial studies of marker-QTL linkage, in the sense of minimizing overall cost of the experiment. Thus, even when large numbers of more or less evenly spaced markers are available, it will not always be cost effective to make full utilization of this capacity. This is particularly true when costs of rearing and trait evaluation per individual scored are low, as when marker data are obtained on individuals raised and evaluated for quantitative traits as part of existing programs. When costs of rearing and trait evaluation per individual scored are high, however, as in human family data collection carried out primarily for subsequent marker — QTL analyses, or when plants or animals are raised specifically for purposes of marker — QTL linkage experiments, optimum spacing may be rather narrow. It is noteworthy that when marginal costs of additional markers or individuals are constant, total resources allocated to a given experiment will determine total number of individuals sampled, but not the optimal marker spacing.     

Model fitting and model testing in the method of joint mapping of quantitative trait loci

A previous paper proposed a new method of QTL mapping called joint mapping (JM). Some problems have been found in model fitting and model testing due to the neglect of the correlations among different observations of the dependent variable in this model. The present paper reports a method of solving the problems. The coefficient of correlation between two observations of the dependent variable is derived. A generalized least square (GLS) approach is developed for model fitting and a strategy and procedure of model testing based on a chi-square test is suggested. A simulated example is given. The example shows that the JM method is quite efficient in mapping multiple linked QTLs.     

Correcting for bias in estimation of quantitative trait loci effects

Estimates of quantitative trait loci (QTL) effects derived from complete genome scans are biased, if no assumptions are made about the distribution of QTL effects. Bias should be reduced if estimates are derived by maximum likelihood, with the QTL effects sampled from a known distribution. The parameters of the distributions of QTL effects for nine economic traits in dairy cattle were estimated from a daughter design analysis of the Israeli Holstein population including 490 marker-by-sire contrasts. A separate gamma distribution was derived for each trait. Estimates for both the α and β parameters and their SE decreased as a function of heritability. The maximum likelihood estimates derived for the individual QTL effects using the gamma distributions for each trait were regressed relative to the least squares estimates, but the regression factor decreased as a function of the least squares estimate. On simulated data, the mean of least squares estimates for effects with nominal 1% significance was more than twice the simulated values, while the mean of the maximum likelihood estimates was slightly lower than the mean of the simulated values. The coefficient of determination for the maximum likelihood estimates was five-fold the corresponding value for the least squares estimates.     

Mapping quantitative trait loci using molecular marker linkage maps

Summary High-density restriction fragment length polymorphism (RFLP) and allozyme linkage maps have been developed in several plant species. These maps make it technically feasible to map quantitative trait loci (QTL) using methods based on flanking marker genetic models. In this paper, we describe flanking marker models for doubled haploid (DH), recombinant inbred (RI), backcross (BC), F₁ testcross (F₁TC), DH testcross (DHTC), recombinant inbred testcross (RITC), F₂, and F₃ progeny. These models are functions of the means of quantitative trait locus genotypes and recombination frequencies between marker and quantitative trait loci. In addition to the genetic models, we describe maximum likelihood methods for estimating these parameters using linear, nonlinear, and univariate or multivariate normal distribution mixture models. We defined recombination frequency estimators for backcross and F₂ progeny group genetic models using the parameters of linear models. In addition, we found a genetically unbiased estimator of the QTL heterozygote mean using a linear function of marker means. In nonlinear models, recombination frequencies are estimated less efficiently than the means of quantitative trait locus genotypes. Recombination frequency estimation efficiency decreases as the distance between markers decreases, because the number of progeny in recombinant marker classes decreases. Mean estimation efficiency is nearly equal for these methods.     

Bayesian point estimation of quantitative trait loci

In this article, we consider the problem of the estimation of quantitative trait loci (QTL), those chromosomal regions at which genetic information affecting some quantitative trait is encoded. Generally the number of such encoding sites is unknown, and associations between neutral molecular marker genotypes and observed trait phenotypes are sought to locate them. We consider a Bayesian model for simple experimental designs, and discuss the existing approaches to inference for this problem. In particular, we focus on locating positions of the best candidate markers segregating for the trait, a situation which is of primary interest in comparative mapping. We introduce a loss function for estimating both the number of QTL and their location, and we illustrate its application via simulated and real data.     

Joint mapping of quantitative trait loci using F2 populations

In this paper, the theory of joint mapping of quantitative trait loci is extended to F₂ populations. Two independent regression equations, related to the additive and dominance effects respectively, are derived. Therefore, there are three alternative strategies for mapping QTLs, called additive-based mapping (ABM), dominance-based mapping (DBM) and additive-dominance-based mapping (ADBM). Simulation results have shown that ADBM is the most appropriate in most situations.     

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.     

Shared quantitative trait loci underlying the genetic correlation between continuous traits

We review genetic correlations among quantitative traits in light of their underlying quantitative trait loci (QTL). We derive an expectation of genetic correlation from the effects of underlying loci and test whether published genetic correlations can be explained by the QTL underlying the traits. While genetically correlated traits shared more QTL (33%) on average than uncorrelated traits (11%), the actual number of shared QTL shared was small. QTL usually predicted the sign of the correlation with good accuracy, but the quantitative prediction was poor. Approximately 25% of trait pairs in the data set had at least one QTL with antagonistic effects. Yet a significant minority (20%) of such trait pairs have net positive genetic correlations due to such antagonistic QTL 'hidden' within positive genetic correlations. We review the evidence on whether shared QTL represent single pleiotropic loci or closely linked monotropic genes, and argue that strict pleiotropy can be viewed as one end of a continuum of recombination rates where r=0. QTL studies of genetic correlation will likely be insufficient to predict evolutionary trajectories over long time spans in large panmictic populations, but will provide important insights into the trade-offs involved in population and species divergence.     

Power analysis for quantitative trait locus mapping in populations derived by multiple backcrosses

 Populations derived by multiple backcrosses are potentially useful for quantitative trait locus (QTL) mapping studies. Comparisons of relative power to detect QTL using populations derived by multiple back-crosses are needed to make decisions when mapping projects are initiated. The objective of this study was to theoretically compare the power to detect QTL in populations derived by multiple backcrosses relative to mapping in a recombinant inbred population of equal size. Backcrossing results in a reduction in genetic variance with each generation and also results in an increasing frequency of the recurrent parent marker genotype. The relevant outcome for QTL mapping is a reduction in genetic variance to partition between marker genotype classes and increasing unbalance of the number of individuals contributing to the mean of the marker genotypes. Both of these factors lead to a decrease in the power to detect a QTL as the number of backcross generations increases. Experimental error was held constant with the populations compared. From a theoretical standpoint, backcross-derived populations offer few advantages for QTL detection. If, however, a backcrossing approach is the most efficient method to achieve a desired breeding objective and if QTL detection is an objective of equal or less importance, backcross-derived populations are a reasonable approach to QTL detection. Received: 4 August 1996 / Accepted: 4 April 1997     

Genetic mapping of quantitative trait loci for resistance to Haemonchus contortus in sheep

This paper presents results from a mapping experiment to detect quantitative trait loci (QTL) for resistance to Haemonchus contortus infestation in merino sheep. The primary trait analysed was faecal worm egg count in response to artificial challenge at 6 months of age. In the first stage of the experiment, whole genome linkage analysis was used for broad-scale mapping. The animal resource used was a designed flock comprising 571 individuals from four half-sib families. The average marker spacing was about 20 cM. For the primary trait, 11 QTL (as chromosomal/family combinations) were significant at the 5% chromosome-wide level, with allelic substitution effects of between 0.19 and 0.38 phenotypic standard deviation units. In general, these QTL did not have a significant effect on faecal worm egg count recorded at 13 months of age. In the second stage of the experiment, three promising regions (located on chromosomes 1, 3 and 4) were fine-mapped. This involved typing more closely spaced markers on individuals from the designed flock as well as an additional 495 individuals selected from a related population with a deeper pedigree. Analysis was performed using a linkage disequilibrium–linkage approach, under additive, dominant and multiple QTL models. Of these, the multiple QTL model resulted in the most refined QTL positions, with resolutions of <10 cM achieved for two regions. Because of the moderate size of effect of the QTL, and the apparent age and/or immune status specificity of the QTL, it is suggested that a panel of QTL will be required for significant genetic gains to be achieved within industry via marker-assisted selection.     

Linkage disequilibrium fine mapping of quantitative trait loci: A simulation study

Recently, the use of linkage disequilibrium (LD) to locate genes which affect quantitative traits (QTL) has received an increasing interest, but the plausibility of fine mapping using linkage disequilibrium techniques for QTL has not been well studied. The main objectives of this work were to (1) measure the extent and pattern of LD between a putative QTL and nearby markers in finite populations and (2) investigate the usefulness of LD in fine mapping QTL in simulated populations using a dense map of multiallelic or biallelic marker loci. The test of association between a marker and QTL and the power of the test were calculated based on single-marker regression analysis. The results show the presence of substantial linkage disequilibrium with closely linked marker loci after 100 to 200 generations of random mating. Although the power to test the association with a frequent QTL of large effect was satisfactory, the power was low for the QTL with a small effect and/or low frequency. More powerful, multi-locus methods may be required to map low frequent QTL with small genetic effects, as well as combining both linkage and linkage disequilibrium information. The results also showed that multiallelic markers are more useful than biallelic markers to detect linkage disequilibrium and association at an equal distance.     

Age-dependent quantitative trait loci affecting growth traits in Scottish Blackface sheep

To dissect age-dependent quantitative trait loci (QTL) associated with growth and to examine changes in QTL effects over time, the Gompertz growth model was fitted to longitudinal live weight data on 788 Scottish Blackface lambs from nine half-sib families. QTL were mapped for model parameters and weekly live weights and growth rates using microsatellite markers on chromosomes 1, 2, 3, 5, 14, 18, 20 and 21. QTL significance (using α = 0.05 chromosome-wide significance thresholds, unless otherwise stated) varied with age, and those for growth rate occurred earlier than equivalent QTL for live weight. A chromosome 20 QTL for growth rate was significant from 4 to 9 weeks (maximum significance at 6 weeks) and for maximum growth rate. For live weight, this QTL was significant from 8 to 16 weeks (maximum significance at 12 weeks). A nominally significant chromosome 14 QTL was detected for growth rates from birth to week 2 in the same families and location as an 8-week weight QTL. In addition, at the same position on chromosome 14, a QTL was significant for growth rate for 17–28 weeks (maximum significance at 24 weeks). A chromosome 3 QTL was significant for weights at early ages (birth to week 4) and a growth rate QTL on chromosome 18 was significant from 8 to 12 weeks. Fitting growth curves allowed the combination of information from multiple measurements into a few biologically meaningful variables, and the detection of growth QTL that were not observed from analyses of raw weight data. These QTL describe distinct parts of an animal's growth curve trajectory, possibly enabling manipulation of this trajectory.     

Theobroma cacao L.: a genetic linkage map and quantitative trait loci analysis

A genetic linkage map of Theobroma cacao (cocoa) has been constructed from 131 backcross trees derived from a cross between a single tree of the variety Catongo and an F1 tree from the cross of Catongo by Pound 12. The map comprises 138 markers: 104 RAPD loci, 32 RFLP loci and two morphologic loci. Ten linkage groups were found which cover 1068 centimorgans (cM). Only six (4%) molecular-marker loci show a significant deviation from the expected 11 segregation ratio.The average distance between two adjacent markers is 8.3 cM. The final genome-size estimates based on two-point linkage data ranged from 1078 to 1112 cM for the cocoa genome. This backcross progeny segregates for two apparently single gene loci controlling (1) anthocyanidin synthesis (Anth) in seeds, leaves and flowers and (2) self-compatibility (Autoc). The Anth locus was found to be 25 cM from Autoc and two molecular markers co-segregate with Anth. The genetic linkage map was used to localize QTLs for early flowering, trunk diameter, jorquette height and ovule number in the BC₁ generation using both single-point ANOVA and interval mapping. A minimum number of 2–4 QTLs (P<0.01) involved in the genetic expression of the traits studied was detected. Coincident map locations of a QTL for jorquette height and trunk diameter suggests the possibility of pleiotropic effects in cocoa for these traits. The combined estimated effects of the different mapped QTLs explained between 11.2% and 25.8% of the phenotypic variance observed in the BC₁ population.     

Marker-based mapping of quantitative trait loci using replicated progenies

Summary When heritability of the trait under investigation is low, replicated progenies can bring about a major reduction in the number of individuals that need to be scored for marker genotype in determining linkage between marker loci and quantitative trait loci (QTL). Savings are greatest when heritability of the trait is low, but are much reduced when heritability of the quantitative trait is moderate to high. Required numbers for recombinant inbred lines will be greater than those required for a simple F₂ population when heritabilities are moderate to high and the proportion of recombination between marker locus and quantitative trait locus is substantial.Contribution No. 2613-E of the Agricultural Research Organization, 1989 series     

The effects of selective genotyping on estimates of proportion of recombination between linked quantitative trait loci

Selective genotyping is the marker assay of only the more extreme phenotypes for a quantitative trait and is intended to increase the efficiency of quantitative trait loci (QTL) mapping. We show that selective genotyping can bias estimates of the recombination frequency between linked QTLs — upwardly when QTLs are in repulsion phase, and downwardly when QTLs are in coupling phase. We examined these biases under simple models involving two QTLs segregating in a backcross or F₂ population, using both analytical models and computer simulations. We found that bias is a function of the proportion selected, the magnitude of QTL effects, distance between QTLs and the dominance of QTLs. Selective genotyping thus may decrease the power of mapping multiple linked QTLs and bias the construction of a marker map. We suggest a large proportion than previously suggested (50%) or the entire population be genotyped if linked QTLs of large effects (explain > 10% phenotypic variance) are evident. New models need to be developed to explicitly incorporate selection into QTL map construction.