Sequential sampling in determining linkage between marker loci and quantitative trait loci

Summary As compared to classical, fixed sample size techniques, simulation studies showed that a proposed sequential sampling procedure can provide a substantial decrease (up to 50%, in some cases) in the mean sample size required for the detection of linkage between marker loci and quantitative trait loci. Sequential sampling with truncation set at the required sample size for the non-sequential test, produced a modest further decrease in mean sample size, accompanied by a modest increase in error probabilities. Sequential sampling with observations taken in groups produced a noticeable increase in mean sample size, with a considerable decrease in error probabilities, as compared to straightforward sequential sampling. It is concluded that sequential sampling has a particularly useful application to experiments aimed at investigating the genetics of differences between lines or strains that differ in some single outstanding trait.     

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.     

Bayesian analysis of linkage between genetic markers and quantitative trait loci. I. Prior knowledge

Summary Prior information on gene effects at individual quantitative trait loci (QTL) and on recombination rates between marker loci and QTL is derived. The prior distribution of QTL gene effects is assumed to be exponential with major effects less likely than minor ones. The prior probability of linkage between a marker and another single locus is a function of the number and length of chromosomes, and of the map function relating recombination rate to genetic distance among loci. The prior probability of linkage between a marker locus and a quantitative trait depends additionally on the number of detectable QTL, which may be determined from total additive genetic variance and minimum detectable QTL effect. The use of this prior information should improve linkage tests and estimates of QTL effects.     

Mapping quantitative trait loci with dominant and missing markers in various crosses from two inbred lines

Dominant phenotype of a genetic marker provides incomplete information about the marker genotype of an individual. A consequence of using this incomplete information for mapping quantitative trait loci (QTL) is that the inference of the genotype of a putative QTL flanked by a marker with dominant phenotype will depend on the genotype or phenotype of the next marker. This dependence can be extended further until a marker genotype is fully observed. A general algorithm is derived to calculate the probability distribution of the genotype of a putative QTL at a given genomic position, conditional on all observed marker phenotypes in the region with dominant and missing marker information for an individual. The algorithm is implemented for various populations stemming from two inbred lines in the context of mapping QTL. Simulation results show that if only a proportion of markers contain missing or dominant phenotypes, QTL mapping can be almost as efficient as if there were no missing information in the data. The efficiency of the analysis, however, may decrease substantially when a very large proportion of markers contain missing or dominant phenotypes and a genetic map has to be reconstructed first on the same data as well. So it is important to combine dominant markers with codominant markers in a QTL mapping study. This revised version was published online in July 2006 with corrections to the Cover Date.     

Detection of linkage between marker loci and loci affecting quantitative traits in crosses between segregating populations

Summary By making use of pedigree information and information on marker-genotypes of the parent and F-1 individuals crossed to form an F-2 population, it is possible to carry out a linkage analysis between marker loci and loci affecting quantitative traits in a cross between segregating parent populations that are at fixation for alternative alleles at the QTL, but share the same alleles at the marker loci. For two-allele systems, depending on marker allele frequencies in the parent populations, 2–4 times as many F-2 offspring will have to be raised and scored for markers and quantitative traits in order to provide power equivalent to that obtained in a cross between fully inbred lines. Major savings in number of F-2 offspring raised can be achieved by scoring each parent pair for a large number of markers in each chromosomal region and scoring F-1 and F-2 offspring only for those markers for which the parents were homozygous for alternative alleles. For multiple allele systems, particularly when dealing with hypervariable loci, only 10%–20% additional F-2 offspring will have to be raised and scored to provide power equivalent to that obtained in a cross between inbred lines. When a resource population contains novel favorable alleles at quantitative trait loci that are not present (or rare) in a commercial population, analyses of this sort will enable the loci of interest to be identified, mapped and manipulated effectively in breeding programs.Contribution no. 2124-E, 1987 series from The Agricultural Research Organization, The Volcani Center, Bet Dagan, Israel     

Selective genotyping for determination of linkage between a marker locus and a quantitative trait locus

Summary Selective genotyping is the term used when the determination of linkage between marker loci and quantitative trait loci (QTL) affecting some particular trait is carried out by genotyping only individuals from the high and low phenotypic tails of the entire sample population. Selective genotyping can markedly decrease the number of individuals genotyped for a given power at the expense of an increase in the number of individuals phenotyped. The optimum proportion of individuals genotyped from the point of view of minimizing costs for a given experimental power depends strongly on the cost of completely genotyping an individual for all of the markers included in the experiment (including the costs of obtaining a DNA sample) relative to the cost of rearing and trait evaluation of an individual. However, in single trait studies, it will almost never be useful to genotype more than the upper and lower 25% of a population. It is shown that the observed difference in quantitative trait values associated with alternative marker genotypes in the selected population can be much greater than the actual gene effect at the quantitative trait locus when the entire population is considered. An expression and a figure is provided for converting observed differences under selective genotyping to actual gene effects.     

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     

Bayesian analysis of linkage between genetic markers and quantitative trait loci. II. Combining prior knowledge with experimental evidence

Summary A Bayesian method was developed for identifying genetic markers linked to quantitative trait loci (QTL) by analyzing data from daughter or granddaughter designs and single markers or marker pairs. Traditional methods may yield unrealistic results because linkage tests depend on number of markers and QTL gene effects associated with selected markers are overestimated. The Bayesian or posterior probability of linkage combines information from a daughter or granddaughter design with the prior probability of linkage between a marker locus and a QTL. If the posterior probability exceeds a certain quantity, linkage is declared. Upon linkage acceptance, Bayesian estimates of marker-QTL recombination rate and QTL gene effects and frequencies are obtained. The Bayesian estimates of QTL gene effects account for different amounts of information by shrinking information from data toward the mean or mode of a prior exponential distribution of gene effects. Computation of the Bayesian analysis is feasible. Exact results are given for biallelic QTL, and extensions to multiallelic QTL are suggested.     

On the power of experimental designs for the detection of linkage between marker loci and quantitative loci in crosses between inbred lines

Summary The power of experiments aimed at detecting linkage between a quantitative locus and a marker locus, both segregating in the backross or F₂ generation of a cross between two inbred lines, is examined. Given that the two lines are close to fixation for alternative alleles of both marker locus and quantitative locus, it is concluded that experiments involving a few thousand offspring should be able to detect close linkages involving quantitative loci (or groups of loci) having rather modest effects (i.e., that contribute, say, 1% of the total phenotypic variance in the F₂).     

Estimation of the contribution of quantitative trait loci (QTL) to the variance of a quantitative trait by means of genetic markers

The estimation of the contribution of an individual quantitative trait locus (QTL) to the variance of a quantitative trait is considered in the framework of an analysis of variance (ANOVA). ANOVA mean squares expectations which are appropriate to the specific case of QTL mapping experiments are derived. These expectations allow the specificities associated with the limited number of genotypes at a given locus to be taken into account. Discrepancies with classical expectations are particularly important for two-class experiments (backcross, recombinant inbred lines, doubled haploid populations) and F₂ populations. The result allows us firstly to reconsider the power of experiments (i.e. the probability of detecting a QTL with a given contribution to the variance of the trait). It illustrates that the use of classical formulae for mean squares expectations leads to a strong underestimation of the power of the experiments. Secondly, from the observed mean squares it is possible to estimate directly the variance associated with a locus and the fraction of the total variance associated to this locus (r _l ² ). When compared to other methods, the values estimated using this method are unbiased. Considering unbiased estimators increases in importance when (1) the experimental size is limited; (2) the number of genotypes at the locus of interest is large; and (3) the fraction of the variation associated with this locus is small. Finally, specific mean squares expectations allows us to propose a simple analytical method by which to estimate the confidence interval of r _l ² . This point is particularly important since results indicate that 95% confidence intervals for r _l ² can be rather wide:2–23% for a 10% estimate and 8–34% for a 20% estimate if 100 individuals are considered.     

Detection of linkage between quantitative trait loci and restriction fragment length polymorphisms using inbred lines

Summary In segregating populations, large numbers of individuals are needed to detect linkage between markers, such as restriction fragment length polymorphisms (RFLPs), and quantitative trait loci (QTL), limiting the potential use of such markers for detecting linkage. Fewer individuals from inbred lines are needed to detect linkage. Simulation data were used to test the utility of two methods to detect linkage: maximum likelihood and comparison of marker genotype means. When there is tight linkage, the two methods have similar power, but when there is loose linkage, maximum likelihood is much more powerful. Once inbred lines have been established, they can be screened rapidly to detect QTL for several traits simultaneously. If there is sufficient coverage of the genome with RFLPs, several QTL for each trait may be detected.     

Mapping quantitative trait loci with dominant markers in four-way crosses

 A common problem in mapping quantitative trait loci (QTLs) is that marker data are often incomplete. This includes missing data, dominant markers, and partially informative markers, arising in outbred populations. Here we briefly present an iteratively re-weighted least square method (IRWLS) to incorporate dominant and missing markers for mapping QTLs in four-way crosses under a heterogeneous variance model. The algorithm uses information from all markers in a linkage group to infer the QTL genotype. Monte Carlo simulations indicate that with half dominant markers, QTL detection is almost as efficient as with all co-dominant markers. However, the precision of the estimated QTL parameters generally decreases as more markers become missing or dominant. Notable differences are observed on the standard deviation of the estimated QTL position for varying levels of marker information content. The method is relatively simple so that more complex models including multiple QTLs or fixed effects can be fitted. Finally, the method can be readily extended to QTL mapping in full-sib families. Received: 16 June 1998 / Accepted: 29 September 1998     

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.     

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.     

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.     

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.     

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.     

Methods for multiple-marker mapping of quantitative trait loci in half-sib populations

In this paper we consider the detection of individual loci controlling quantitative traits of interest (quantitative trait loci or QTLs) in the large half-sib family structure found in some species. Two simple approaches using multiple markers are proposed, one using least squares and the other maximum likelihood. These methods are intended to provide a relatively fast screening of the entire genome to pinpoint regions of interest for further investigation. They are compared with a more traditional single-marker least-squares approach. The use of multiple markers is shown to increase power and has the advantage of providing an estimate for the location of the QTL. The maximum-likelihood and the least-squares approaches using multiple markers give similar power and estimates for the QTL location, although the likelihood approach also provides estimates of the QTL effect and sire heterozygote frequency. A number of assumptions have been made in order to make the likelihood calculations feasible, however, and computationally it is still more demanding than the least-squares approach. The least-squares approach using multiple markers provides a fast method that can easily be extended to include additional effects.     

Identification of quantitative trait loci controlling developmental characteristics of Brassica oleracea L

A segregating population of F₁-derived doubled haploid (DH) lines of Brassica oleracea was used to detect and locate QTLs controlling 27 morphological and developmental traits, including leaf, flowering, axillary bud and stem characters. The population resulted from a cross between two very different B. oleracea crop types, an annual cauliflower and a biennial Brussels sprout. A principal component analysis (PCA), based on line means, allowed all the traits to be grouped into distinct categories according to the first five Principal Components. These were: leaf traits (PC1), flowering traits (PC2), axillary bud traits (PC3 and 5) and stem traits (PC4). Between zero and four putative QTL were located per trait, which individually explained between 6% and 43% of the additive genetic variation, using the multiple-marker regression approach to QTL mapping. For lamina width, bare petiole length and stem length two QTL with opposite effects were detected on the same linkage groups. Intra- and inter-specific comparative mapping using RFLP markers identified a QTL on linkage group O8 accounting for variation in vernalisation, which is probably synonymous with a QTL detected on linkage group N19 of Brassica napus. In addition, a QTL for petiole length detected on O3 of this study appeared to be homologous to a QTL detected on another B. oleracea genetic map (Camargo et al. 1995). Received: 28 March 2001 / Accepted: 25 June 2001