Advances in Statistical Methods to Map Quantitative Trait Loci in Outbred Populations

Statistical methods to map quantitative trait loci (QTL) in outbred populations are reviewed, extensions and applications to human and plant genetic data are indicated, and areas for further research are identified. Simple and computationally inexpensive methods include (multiple) linear regression of phenotype on marker genotypes and regression of squared phenotypic differences among relative pairs on estimated proportions of identity-by-descent at a locus. These methods are less suited for genetic parameter estimation in outbred populations but allow the determination of test statistic distributions via simulation or data permutation; however, further inferences including confidence intervals of QTL location require the use of Monte Carlo or bootstrap sampling techniques. A method which is intermediate in computational requirements is residual maximum likelihood (REML) with a covariance matrix of random QTL effects conditional on information from multiple linked markers. Testing for the number of QTLs on a chromosome is difficult in a classical framework. The computationally most demanding methods are maximum likelihood and Bayesian analysis, which take account of the distribution of multilocus marker-QTL genotypes on a pedigree and permit investigators to fit different models of variation at the QTL. The Bayesian analysis includes the number of QTLs on a chromosome as an unknown.     

A centromere-based linkage group on the long arm of human chromosome 17

A genetic linkage group on the long arm of chromosome 17 is reported. A maximum likelihood of theta = 0.20 between the centromere-based locus D17Z1 and COL1A1 has been found, as well as a theta = 0.10 between COL1A1 and GH1. The most likely order of the three loci is D17Z1-COL1A1-GH1.     

A study of genetic linkage heterogeneity in 35 adult-onset polycystic kidney disease families

A genetic heterogeneity analysis of 35 kindreds with adult-onset polycystic kidney disease (ADPKD) was carried out using the D16S85, D16S84, D16S125 and D16S94 loci that are closely linked to the PKD1 locus on chromosome 16. The results show that the likelihood of two ADPKD loci is 2,514.9 times greater than for a single locus (P < 0.0001). The maximum likelihood lod score is 27.38 under heterogeneity with PKD1 lying 4.9 cM proximal to D16S85 (in males). At least 3% of kindreds are unlinked to PKD1, since the 95% confidence limits of alpha, the proportion of families linked to PKD1, are 0.54–0.97. Only 2 out of 35 kindreds (5.7%) show statistically significant evidence of non-linkage to PKD1, with conditional probabilities of 0.987 and 0.993 that the disease locus is unlinked. This confirms the existence of a small subgroup of ADPKD kindreds that are unlinked to PKD1 and provides a firm basis for genetic counselling of this population on the basis of DNA probes.     

Evidence of genetic heterogeneity of Leber's congenital amaurosis (LCA) and mapping of LCA1 to chromosome 17p13

Leber’s congenital amaurosis (LCA) is an autosomal recessive disease responsible for congenital blindness. It is the earliest and most severe inherited retinal dystrophy in human and its genetic heterogeneity has long been recognised. We have recently reported on the first localisation of a disease gene (LCA1) to the short arm of chromosome 17 by homozygosity mapping in five families of North African origin. Here, we refine the genetic mapping of LCA1 to chromosome 17p13 between loci D17S938 and D17S1353 and provide strong support for the genetic heterogeneity of this condition (maximum likelihood for heterogeneity, 17.20 in lnL; heterogeneity versus homogeneity, P = 0.0002, heterogeneity versus no linkage, P < 0.0001) Received: 23 October 1995 / Revised: 11 January 1996     

The Distribution of Mutation Effects on Viability in Drosophila Melanogaster

Parameters of continuous distributions of effects and rates of spontaneous mutation for relative viability in Drosophila are estimated by maximum likelihood from data of two published experiments on accumulation of mutations on protected second chromosomes. A model of equal mutant effects gives a poor fit to the data of the two experiments; higher likelihoods are obtained with leptokurtic distributions or for models in which there is more than one class of mutation effect. Minimum estimates of mutation rates (events per generation) at polygenes affecting viability on chromosome 2 are 0.14 and 0.068, but estimates are strongly confounded with other parameters in the model. Separate information on rates of molecular divergence between Drosophila species and from rates of movement of transposable elements is used to infer the overall genomic mutation rate in Drosophila, and the viability data are analyzed with mutation rate as a known parameter. If, for example, a mutation rate for chromosome 2 of 0.4 is assumed, maximum likelihood estimates of mean mutant effect on relative viability are 0.4% and 1%, but the majority of mutations have very much smaller effects than these values as distributions are highly leptokurtic. The methodology is applied to estimate viability effects of single P element insertional mutations. The mean effect per insertion is found to be higher, and their distribution is found to be less leptokurtic than for spontaneous mutations. The equilibrium genetic variance of viability predicted by a mutation-selection balance model with parameters estimated from the mutation accumulation experiments is similar to laboratory estimates of genetic variance of viability from natural populations of Drosophila.     

A general program for estimation of haplotype frequencies from population diploid data.

The program which is written in FORTRAN estimates haplotype frequencies in two-locus and three-locus genetic systems from population diploid data. It is based on the gene counting method which leads to maximum likelihood estimates, and can be used whenever the possible antigens (one or more) on each chromosome can be specified for each person and for each locus, i.e., ABO-like systems and inclusions are permitted. The number of alleles per locus may be rather large, and both grouped and ungrouped data can be used. Log likelihoods are calculated on the basis of various assumptions, so that likelihood ratio tests can be carried out.     

A Maximum Likelihood Method for Estimating Genome Length Using Genetic Linkage Data

The genetic length of a genome, in units of Morgans or centimorgans, is a fundamental characteristic of an organism. We propose a maximum likelihood method for estimating this quantity from counts of recombinants and nonrecombinants between marker locus pairs studied from a backcross linkage experiment, assuming no interference and equal chromosome lengths. This method allows the calculation of the standard deviation of the estimate and a confidence interval containing the estimate. Computer simulations have been performed to evaluate and compare the accuracy of the maximum likelihood method and a previously suggested method-of-moments estimator. Specifically, we have investigated the effects of the number of meioses, the number of marker loci, and variation in the genetic lengths of individual chromosomes on the estimate. The effect of missing data, obtained when the results of two separate linkage studies with a fraction of marker loci in common are pooled, is also investigated. The maximum likelihood estimator, in contrast to the method-of-moments estimator, is relatively insensitive to violation of the assumptions made during analysis and is the method of choice. The various methods are compared by application to partial linkage data from Xiphophorus.     

Consistency of estimators of the population-scaled recombination rate

We consider (approximate) likelihood methods for estimating the population-scaled recombination rate from population genetic data. We show that the dependence between the data from two regions of a chromosome decays inversely with the amount of recombination between the two regions. We use this result to show that the maximum likelihood estimator (mle) for the recombination rate, based on the composite likelihood of Fearnhead and Donnelly, is consistent. We also consider inference based on the pairwise likelihood of Hudson. We consider two approximations to this likelihood, and prove that the mle based on one of these approximations is consistent, while the mle based on the other approximation (which is used by McVean, Awadalla and Fearnhead) is not.     

Tightly linked markers for the neurofibromatosis type 1 gene

Relationships among genetic markers in the region of the neurofibromatosis type 1 (NF1) gene on chromosome 17 were investigated by linkage studies in a large sample set of affected families and in a panel of 58 normal families. A new marker, pHHH202 (D17S33), was included along with two markers known to be closely linked to NF. The maximum likelihood estimate of the recombination rate between the pHHH202 and NF1 loci was found to be O. Multilocus analysis suggested the following marker order: pA10-41-(p3-6, pHHH202); the NF1 gene fell with equal likelihood between either pA10-41-p3-6 or p3-6-pHHH202. The odds against NF1 being outside this cluster of tightly linked markers were greater than 15:1.     

A genetic linkage map of 96 loci on the short arm of human chromosome 3.

We constructed a genetic map of 96 loci on the short arm of human chromosome 3 (3p) in 59 families provided by the Centre d'Etude du Polymorphisme Humaine (CEPH). Twenty-nine continuously linked loci were placed on the map with likelihood support of at least 1000:1; one locus, D3S213, was placed on the map with likelihood support of 871:1; D3Z1, an alpha satellite centromeric repeat probe, was placed on the map with likelihood support of 159:1; 65 loci were assigned regional locations. The average heterozygosity of the uniquely ordered markers was 49%. The map extends from 3p26, the terminal band of 3p, to the centromere (from D3S211 to D3Z1). Multipoint linkage analysis indicated that the male, female, and sex-averaged maps extend for 102, 147, and 116 cM, respectively. The mean genetic distance between uniquely ordered loci on the sex-averaged map was 4.0 cM. Probe density was greatest for the region of 3p between D3F15S2e and the telomere. The sex-averaged map contained two intervals greater than 10 cM. Seventeen probes were localized by fluorescence in situ hybridization. The loci described in this report will be useful in building an integrated genetic and physical map of this chromosome.     

A Radiation Hybrid Map of 95 STSs Spanning Human Chromosome 13q

We have constructed a high-resolution physical map of the long arm of human chromosome 13 using a panel of 94 radiation hybrids. A comprehensive map of 95 chromosome 13-specific sequence tagged sites (STSs) spanning 13q from the presumed centromere at D13Z1 to the known telomere was obtained by multipoint maximum likelihood statistical methods. The 95 markers have an average retention frequency of 10%, with markers closer to the centromere having much greater retention frequencies (22-49%) than distal 13q markers (2-12%) The most likely radiation hybrid map localized the 95 STSs into 54 unique map positions, 34 with odds of 1000:1 or greater; the comprehensive map localized all but 17 STSs with odds exceeding 10:1. The total map length of 13q was 1302 cR₉₀₀₀ (range 6.4-94.4 cR₉₀₀₀) and a physical distance of 98 Mb, so that 1% breakage in the RH panel corresponds to 75 kb. A comparison of the comprehensive RH map to genetic maps of chromosome 13q shows identical locus orders for the common markers, with two exceptions over 1-cM distances. We discuss the possible relationships between the genetic and the radiation hybrid maps.     

Mapping of five human chromosome 19 DNA markers to owl monkey chromosomes.

Hybridization of DNA from three panels of karyotypically distinct owl monkey x rodent somatic cell hybrids with human DNA probes resulted in the syntenic assignments of INSR-LDLR-TGFB1-APOE-D19S8 to owl monkey chromosome 25 of karyotype VI (2n = 49/50), INSR-LDLR-TGFB1-D19S8 to chromosome 2 of karyotype II (2n = 54), and INSR-APOE to chromosome 2 of karyotype V (2n = 46). The APOE and D19S8 loci are on adjacent regions proximal to the centromere of chromosomes 25q (K-VI) and 2p (K-II), as determined by in situ chromosomal hybridization analysis. These findings support our previous proposals on (1) the homology of these chromosomes of three owl monkey karyotypes, (2) the evolutionary derivation of chromosome 2 of karyotypes II and V as the result of two separate centric fusion events, and (3) the likelihood that owl monkey chromosome 25 (K-VI) (and its homologs) is a conserved genetic homoeolog of human chromosome 19.     

Two statistical tests for meiotic breakpoint analysis.

Meiotic breakpoint analysis (BPA), a statistical method for ordering genetic markers, is increasing in importance as a method for building genetic maps of human chromosomes. Although BPA does not provide estimates of genetic distances between markers, it efficiently locates new markers on already defined dense maps, when likelihood analysis becomes cumbersome or the sample size is small. However, until now no assessments of statistical significance have been available for evaluating the possibility that the results of a BPA were produced by chance. In this paper, we propose two statistical tests to determine whether the size of a sample and its genetic information content are sufficient to distinguish between "no linkage" and "linkage" of a marker mapped by BPA to a certain region. Both tests are exact and should be conducted after a BPA has assigned the marker to an interval on the map. Applications of the new tests are demonstrated by three examples: (1) a synthetic data set, (2) a data set of five markers on human chromosome 8p, and (3) a data set of four markers on human chromosome 17q.     

MIXED MODEL APPROACHES FOR ESTIMATING GENETIC VARIANCES AND COVARIANCES

The limitations of methods for analysis of variance(ANOVA)in estimating genetic variances are discussed. Among the three methods(maximum likelihood ML, restricted maximum likelihood REML, and minimum norm quadratic unbiased estimation MINQUE)for mixed linear models, MINQUE method is presented with formulae for estimating variance components and covariances components and for predicting genetic effects. Several genetic models, which cannot be appropriately analyzed by ANOVA methods, are introduced in forms of mixed linear models. Genetic models with independent random effects can be analyzed by MINQUE(1)method whieh is a MINQUE method with all prior values setting 1. MINQUE(1)method can give unbiased estimation for variance components and covariance components, and linear unbiased prediction (LUP) for genetic effects. There are more complicate genetic models for plant seeds which involve correlated random effects. MINQUE(0/1)method, which is a MINQUE method with all prior covariances setting 0 and all prior variances setting 1, is suitable for estimating variance and covariance components in these models. Mixed model approaches have advantage over ANOVA methods for the capacity of analyzing unbalanced data and complicated models. Some problems about estimation and hypothesis test by MINQUE method are discussed.     

Polylink: to support two-point linkage analysis in autotetraploids

SUMMARY: Polylink runs under Microsoft Windows (95 or later). It performs various calculations that are useful for investigating two-point linkage analysis for autopolyploids, based on the random chromosome pairing model. These include calculation of offspring phenotypic probabilities as functions of the recombination fraction, calculation of theoretical standard errors for the maximum likelihood estimator of and numerical computation of maximum likelihood estimates. It also includes simulation facilities. AVAILABILITY: Polylink is free and available from Xiangming Xu via email     

A chromosome-based model for estimating the number of conserved segments between pairs of species from comparative genetic maps

Comparative genetic maps of two species allow insights into the rearrangements of their genomes since divergence from a common ancestor. When the map details the positions of genes (or any set of orthologous DNA sequences) on chromosomes, syntenic blocks of one or more genes may be identified and used, with appropriate models, to estimate the number of chromosomal segments with conserved content conserved between species. We propose a model for the distribution of the lengths of unobserved segments on each chromosome that allows for widely differing chromosome lengths. The model uses as data either the counts of genes in a syntenic block or the distance between extreme members of a block, or both. The parameters of the proposed segment length distribution, estimated by maximum likelihood, give predictions of the number of conserved segments per chromosome. The model is applied to data from two comparative maps for the chicken, one with human and one with mouse.     

Assignment of a novel locus for autosomal recessive congenital ichthyosis to chromosome 19p13.1-p13.2

Autosomal recessive congenital ichthyosis (ARCI) is a rare, clinically and genetically heterogeneous genodermatosis. One gene (transglutaminase 1, on 14q11) and one additional locus (on 2q33-35, with an unidentified gene) have been shown to be associated with a lamellar, nonerythrodermic type of ARCI. We performed a genomewide scan, with 370 highly polymorphic microsatellite markers, on five affected individuals from one large Finnish family with nonerythrodermic, nonlamellar ARCI. The only evidence for linkage emerged from markers in a 6.0-cM region on chromosome 19p13.1-2. The maximum two-point LOD score of 7.33 was obtained with the locus D19S252, and multipoint likelihood calculations gave a maximum location score of 5.2. The affected individuals share two common core haplotypes, which makes compound heterozygosity possible. The novel disease locus is the third locus linked to ARCI, supporting previous evidence for genetic heterogeneity of ARCI. This is also the first locus for a nonlamellar, nonerythrodermic phenotype of ARCI.     

Interval Mapping of Viability Loci Causing Heterosis in Arabidopsis

The genetic basis of heterosis has implications for many problems in genetics and evolution. Heterosis and inbreeding depression affect human genetic diseases, maintenance of genetic variation, evolution of breeding systems, agricultural productivity, and conservation biology. Despite decades of theoretical and empirical studies, the genetic basis of heterosis has remained unclear. I mapped viability loci contributing to heterosis in Arabidopsis. An overdominant factor with large effects on viability mapped to a short interval on chromosome I. Homozygotes had 50% lower viability than heterozygotes in this chromosomal region. Statistical analysis of viability data in this cross indicates that observed viability heterosis is better explained by functional overdominance than by pseudo-overdominance. Overdominance sometimes may be an important cause of hybrid vigor, especially in habitually inbreeding species. Finally, I developed a maximum likelihood interval mapping procedure that can be used to examine chromosomal regions showing segregation distortion or viability selection.     

Map error reduction: using genetic and sequence-based physical maps to order closely linked markers

The Marshfield comprehensive genetic maps are frequently used for linkage and association studies, however, for some regions of these maps the marker order has low level of likelihood ratio support. In order to investigate the level of statistical support and the accuracy of the genetic maps compared to sequence-based physical maps, two approximately 30 cM autosomal regions were selected. The first region was selected from chromosome 3 and consisted predominately of draft sequence. The second region was selected from chromosome 21 and consisted of finished sequence data. The physical order of these markers was based upon their position on Celera (CEL) and Human Genome Project-Santa Cruz (HGP-sc) sequence-based physical maps. The chromosome 3 and 21 regions contained 100 and 61 markers, respectively, on the Marshfield genetic map. The genetic and physical map order was consistent for 88.9 and 89.2% of the markers in the region on chromosome 3 and 21, respectively. Using a novel scoring criterion to assess inconsistent marker order between genetic and physical maps, it was determined that the physical order was likely the correct order for 3.3 and 7.1% of the markers in the chromosome 3 and 21 regions, respectively. To increase the accuracy of the order of markers selected for fine mapping a method is presented which combines information from genetic and sequence-based physical maps.     

Statistical aspects of genetic mapping in autopolyploids.

Many plant species of agriculture importance are polyploid, having more than two copies of each chromosome per cell. In this paper, we describe statistical methods for genetic map construction in autopolyploid species with particular reference to the use of molecular markers. The first step is to determine the dosage of each DNA fragment (electrophoretic band) from its segregation ratio. Fragments present in a single dose can be used to construct framework maps for individual chromosomes. Fragments present in multiple doses can often be used to link the single chromosome maps into homologous groups and provide additional ordering information. Marker phenotype probabilities were calculated for pairs of markers arranged in different configurations among the homologous chromosomes. These probabilities were used to compute a maximum likelihood estimator of the recombination fraction between pairs of markers. A likelihood ratio test for linkage of multidose markers was derived. The information provided by each configuration and power and sample size considerations are also discussed. A set of 294 RFLP markers scored on 90 plants of the species Saccharum spontaneum L. was used to illustrate the construction of an autopolyploid map. Previous studies conducted on the same data revealed that this species of sugar cane is an autooctaploid with 64 chromosomes arranged into eight homologous groups. The methodology described permitted consolidation of 54 linkage groups into ten homologous groups.