The Sampling Distribution of Linkage Disequilibrium under an Infinite Allele Model without Selection

The sampling distributions of several statistics that measure the association of alleles on gametes (linkage disequilibrium) are estimated under a two-locus neutral infinite allele model using an efficient Monte Carlo method. An often used approximation for the mean squared linkage disequilibrium is shown to be inaccurate unless the proper statistical conditioning is used. The joint distribution of linkage disequilibrium and the allele frequencies in the sample is studied. This estimated joint distribution is sufficient for obtaining an approximate maximum likelihood estimate of C = 4Nc, where N is the population size and c is the recombination rate. It has been suggested that observations of high linkage disequilibrium might be a good basis for rejecting a neutral model in favor of a model in which natural selection maintains genetic variation. It is found that a single sample of chromosomes, examined at two loci cannot provide sufficient information for such a test if C less than 10, because with C this small, very high levels of linkage disequilibrium are not unexpected under the neutral model. In samples of size 50, it is found that, even when C is as large as 50, the distribution of linkage disequilibrium conditional on the allele frequencies is substantially different from the distribution when there is no linkage between the loci. When conditioned on the number of alleles at each locus in the sample, all of the sample statistics examined are nearly independent of theta = 4N mu, where mu is the neutral mutation rate.     

Evidence that Egfr contributes to cryptic genetic variation for photoreceptor determination in natural populations of Drosophila melanogaster

One objective of quantitative genetics is to identify the nucleotide variants within genes that contribute to phenotypic variation and susceptibility [1]. In an evolutionary context, this means characterizing the molecular polymorphisms that modify the penetrance and expressivity of perturbed traits. A survey of association between 267 SNPs in almost 11 kb of the D. melanogaster Egfr and the degree of eye roughening due to a gain-of-function Egfr(E1) allele crossed into 210 isogenic wild-type lines provides evidence that a handful of synonymous substitutions supply cryptic variation for photoreceptor determination. Ten sites exceed Bonferroni threshold for association in two sets of crosses to different Egfr(E1) backgrounds including a particularly significant cluster of sites in tight linkage disequilibrium toward the 3' end of the coding region. Epistatic interaction of this cluster with one other site enhances the expressivity of this haplotype. Replication of the strongest associations with an independent sample of 302 phenotypically extreme individuals derived from 1000 crosses of Egfr(E1) to freshly trapped males was achieved using modified case-control and transmission-disequilibrium tests. A tendency for the rarer alleles to have more disrupted eye development suggests that mutation-selection balance is a possible mechanism contributing to maintaining cryptic variation for Egfr.     

Linkage disequilibrium under genetic hitchhiking in finite populations

The model of genetic hitchhiking predicts a reduction in sequence diversity at a neutral locus closely linked to a beneficial allele. In addition, it has been shown that the same process results in a specific pattern of correlations (linkage disequilibrium) between neutral polymorphisms along the chromosome at the time of fixation of the beneficial allele. During the hitchhiking event, linkage disequilibrium on either side of the beneficial allele is built up whereas it is destroyed across the selected site. We derive explicit formulas for the expectation of the covariance measure D and standardized linkage disequilibrium sigma 2D between a pair of polymorphic sites. For our analysis we use the approximation of a star-like genealogy at the selected site. The resulting expressions are approximately correct in the limit of large selection coefficients. Using simulations we show that the resulting pattern of linkage disequilibrium is quickly-i.e., in <0.1N generations-destroyed after the fixation of the beneficial allele for moderately distant neutral loci, where N is the diploid population size.     

Models and tests of linkage and association studies of quantitative trait locus for multi-allele marker Loci

In this paper, we explore models and tests for association and linkage studies of a quantitative trait locus (QTL) linked to a multi-allele marker locus. Based on the difference between an offspring's conditional trait means of receiving and not receiving an allele from a parent at marker locus, we propose three statistics T(m), T(m,row) and T(m,col) to test association or linkage disequilibrium between the marker locus and the QTL. These tests are composite tests, and use the offspring marginal sample means including offspring data of both homozygous and heterozygous parents. For the linkage study, we calculate the offspring's conditional trait mean given the allele transmission status of a heterozygous parent at the marker locus. Based on the difference between the conditional means of a transmitted and a nontransmitted allele from a heterozygous parent, we propose statistics T(parsi), T(satur), T(gen) and T(m,het) to perform composite tests of linkage between the marker locus and the quantitative trait locus in the presence of association. These tests only use the offspring data that are related to the heterozygous parents at the marker locus. T(parsi) is a parsimonious or allele-wise statistic, T(satur) and T(gen )are satured or genotype-wise statistics, and T(m,het) compares the row and column sample means for offspring data of heterozygous parents. After comparing the powers and the sample sizes, we conclude that T(parsi) has higher power than those of the bi-allele tests, T(satur), T(gen), and T(m,het). If there is tight linkage between the marker and the trait locus, T(parsi) is powerful in detecting linkage between the marker and the trait locus in the presence of association. By investigating the goodness-of-fit of T(parsi), we find that T(satur) does not gain much power compared to that of T(parsi). Moreover, T(parsi) takes into account the pattern of the data that is consistent with linkage and linkage disequilibrium. As the number of alleles at the marker locus increases, T(parsi) is very conservative, and can be useful even for sparse data. To illustrate the usefulness and the power of the methods proposed in this paper, we analyze the chromosome 6 data of the Oxford asthma data, Genetic Analysis Workshop 12.     

The brain-derived neurotrophic factor gene confers susceptibility to bipolar disorder: evidence from a family-based association study

Bipolar disorder (BP) is a severe psychiatric disease, with a strong genetic component, that affects 1% of the population worldwide and is characterized by recurrent episodes of mania and depression. Brain-derived neurotrophic factor (BDNF) has been implicated in the pathogenesis of mood disorders, and the aim of the present study was to test for the presence of linkage disequilibrium between two polymorphisms in the BDNF gene and BP in 283 nuclear families. Family-based association test (FBAT) results for the dinucleotide repeat (GT)(N) polymorphism at position -1040 bp showed that allele A3 was preferentially transmitted to the affected individuals (Z=2.035 and P=.042). FBAT results for the val66met SNP showed a significant association for allele G (Z=3.415 and P=.00064). Transmission/disequilibrium test (TDT) haplotype analysis showed a significant result for the 3-G allele combination (P=.000394), suggesting that a DNA variant in the vicinity of the BDNF locus confers susceptibility to BP. Given that there is no direct evidence that either of the polymorphisms we examined alters function, it is unlikely that the actual risk-conferring allele is from these two sites. Rather, the causative site is likely nearby and in linkage disequilibrium with the 3-G haplotype that we have identified.     

Joint linkage and linkage disequilibrium mapping of quantitative trait loci in natural populations

Linkage analysis and allelic association (also referred to as linkage disequilibrium) studies are two major approaches for mapping genes that control simple or complex traits in plants, animals, and humans. But these two approaches have limited utility when used alone, because they use only part of the information that is available for a mapping population. More recently, a new mapping strategy has been designed to integrate the advantages of linkage analysis and linkage disequilibrium analysis for genome mapping in outcrossing populations. The new strategy makes use of a random sample from a panmictic population and the open-pollinated progeny of the sample. In this article, we extend the new strategy to map quantitative trait loci (QTL), using molecular markers within the EM-implemented maximum-likelihood framework. The most significant advantage of this extension is that both linkage and linkage disequilibrium between a marker and QTL can be estimated simultaneously, thus increasing the efficiency and effectiveness of genome mapping for recalcitrant outcrossing species. Simulation studies are performed to test the statistical properties of the MLEs of genetic and genomic parameters including QTL allele frequency, QTL effects, QTL position, and the linkage disequilibrium of the QTL and a marker. The potential utility of our mapping strategy is discussed.     

Transmission-disequilibrium tests for quantitative traits.

The transmission-disequilibrium test (TDT) of Spielman et al. is a family-based linkage-disequilibrium test that offers a powerful way to test for linkage between alleles and phenotypes that is either causal (i.e., the marker locus is the disease/trait allele) or due to linkage disequilibrium. The TDT is equivalent to a randomized experiment and, therefore, is resistant to confounding. When the marker is extremely close to the disease locus or is the disease locus itself, tests such as the TDT can be far more powerful than conventional linkage tests. To date, the TDT and most other family-based association tests have been applied only to dichotomous traits. This paper develops five TDT-type tests for use with quantitative traits. These tests accommodate either unselected sampling or sampling based on selection of phenotypically extreme offspring. Power calculations are provided and show that, when a candidate gene is available (1) these TDT-type tests are at least an order of magnitude more efficient than two common sib-pair tests of linkage; (2) extreme sampling results in substantial increases in power; and (3) if the most extreme 20% of the phenotypic distribution is selectively sampled, across a wide variety of plausible genetic models, quantitative-trait loci explaining as little as 5% of the phenotypic variation can be detected at the .0001 alpha level with <300 observations.     

Polymorphic DNA haplotypes at the human phenylalanine hydroxylase locus and their relationship with phenylketonuria

Summary Eight polymorphic restriction enzyme sites at the phenylalanine hydroxylase (PAH) locus were analyzed from the parental chromosomes in 33 Danish nuclear families with at least one phenylketonuric (PKU) child. Determination of haplotypes of 66 normal chromosomes and 66 chromosomes bearing mutant allele (S) demonstrated that there are at least two haplotypes which occur predominantly on PKU chromosomes and rarely otherwise. Overall, the relative frequencies of the various haplotypes are significantly different on PKU-and normal-allele bearing chromosomes, even though there is no predominantly occurring unique haplotype which can characterize the PKU chromosomes. In addition, no significant association (linkage disequilibrium) between any single polymorphic site and the mutant allele (s) was observed. The results suggest that either the phenylketonuric mutation was very ancient so that the polymorphic sites and the mutation have reached linkage equilibrium or the mutant allele (s) are the results of multiple mutations in the phenylalanine hydroxylase gene in man. Furthermore, a crude relationship between standardized linkage disequilibria and physical map distances of the polymorphic sites indicates that there is no apparent recombination hot-spot in the human phenylalanine hydroxylase gene, since the recombination rate within the locus apears to be uniform and likely to be occurring at a rate similar to that within the HLA gene cluster. The limitations of this later analysis are discussed in view of the sampling errors of disequilibrium measure used, and the potential untility of the PAH haplotypes for prenatal diagnosis and detection of PKU carriers is established.     

X-APL: an improved family-based test of association in the presence of linkage for the X chromosome

Family-based association methods have been developed primarily for autosomal markers. The X-linked sibling transmission/disequilibrium test (XS-TDT) and the reconstruction-combined TDT for X-chromosome markers (XRC-TDT) are the first association-based methods for testing markers on the X chromosome in family data sets. These are valid tests of association in family triads or discordant sib pairs but are not theoretically valid in multiplex families when linkage is present. Recently, XPDT and XMCPDT, modified versions of the pedigree disequilibrium test (PDT), were proposed. Like the PDT, XPDT compares genotype transmissions from parents to affected offspring or genotypes of discordant siblings; however, the XPDT can have low power if there are many missing parental genotypes. XMCPDT uses a Monte Carlo sampling approach to infer missing parental genotypes on the basis of true or estimated population allele frequencies. Although the XMCPDT was shown to be more powerful than the XPDT, variability in the statistic due to the use of an estimate of allele frequency is not properly accounted for. Here, we present a novel family-based test of association, X-APL, a modification of the test for association in the presence of linkage (APL) test. Like the APL, X-APL can use singleton or multiplex families and properly infers missing parental genotypes in linkage regions by considering identity-by-descent parameters for affected siblings. Sampling variability of parameter estimates is accounted for through a bootstrap procedure. X-APL can test individual marker loci or X-chromosome haplotypes. To allow for different penetrances in males and females, separate sex-specific tests are provided. Using simulated data, we demonstrated validity and showed that the X-APL is more powerful than alternative tests. To show its utility and to discuss interpretation in real-data analysis, we also applied the X-APL to candidate-gene data in a sample of families with Parkinson disease.     

Population genetics of coagulant factor IX: frequencies of two DNA polymorphisms in five ethnic groups.

Two frequently used restriction-enzyme polymorphisms (RFLPs) of coagulant F.IX, TaqI and XmnI, have been examined in five ethnic groups: white Americans, black Americans, East Indians, Chinese, and Malays. There is a distinct "cline" in the frequencies of both polymorphisms, from white Americans to Malays. The rarer type 2 alleles of both polymorphisms, in which middle recognition sites are present--and which in our sample reach their highest frequencies in white Americans--are marginally higher in four groups of Europeans previously reported by others. The frequencies of the rarer alleles are significantly higher in Europeans than in black Americans and East Indians, and these alleles are essentially absent in Chinese and Malays. The frequency of heterozygosity diminishes in the same order, being zero in Malays for both polymorphisms. The polymorphisms are in strong linkage disequilibrium, and in all groups the type 1 allele for TaqI is disproportionately accompanied by the type 1 allele for XmnI. The paucity of type 2 alleles and the low rate of heterozygosity in four non-European groups suggest that the polymorphisms will be of little diagnostic value south of Gibraltar and east of Suez. This prediction is confirmed by the observed haplotype frequencies in the black American and the Oriental groups.     

Nucleotide variability at G6pd and the signature of malarial selection in humans

Glucose-6-phosphate dehydrogenase (G6PD) deficiency is the most common enzymopathy in humans. Deficiency alleles for this X-linked disorder are geographically correlated with historical patterns of malaria, and the most common deficiency allele in Africa (G6PD A-) has been shown to confer some resistance to malaria in both hemizygous males and heterozygous females. We studied DNA sequence variation in 5.1 kb of G6pd from 47 individuals representing a worldwide sample to examine the impact of selection on patterns of human nucleotide diversity and to infer the evolutionary history of the G6PD A- allele. We also sequenced 3.7 kb of a neighboring locus, L1cam, from the same set of individuals to study the effect of selection on patterns of linkage disequilibrium. Despite strong clinical evidence for malarial selection maintaining G6PD deficiency alleles in human populations, the overall level of nucleotide heterozygosity at G6pd is typical of other genes on the X chromosome. However, the signature of selection is evident in the absence of genetic variation among A- alleles from different parts of Africa and in the unusually high levels of linkage disequilibrium over a considerable distance of the X chromosome. In spite of a long-term association between Plasmodium falciparum and the ancestors of modern humans, patterns of nucleotide variability and linkage disequilibrium suggest that the A- allele arose in Africa only within the last 10,000 years and spread due to selection.     

Genetic variation at the matrix metalloproteinase-9 locus on chromosome 20q12.2–13.1

Allelic association methods are better suited than linkage analysis for mapping of susceptibility genes that confer modest increases in risk in complex diseases. In both family- and population-based association studies, it is very useful to have prior knowledge of all sequence variants and the degree of linkage disequilibrium in a candidate gene region. In this study, we scanned sequence variants in a 2.2-kb promoter sequence and all 13 exons (totalling 3.3 kb) of the matrix metalloproteinase-9 gene, which is associated with coronary heart disease and a candidate for other diseases involving connective tissue remodelling, such as cancer metastasis. The sequences had a total of ten variable sites, four in the promoter, five in the coding region (three of which alter the amino acid encoded) and one in the 3' untranslated sequence. Sequence inspection suggests that some of the variants will have a functional impact on either level of expression or enzymatic activity. Tight linkage disequilibrium was detected between variants across the entire length of the gene (approximately 9 kb), and frequencies of different haplotypes were determined. The data provide an essential tool for studies of the possible contribution of genetic variation at the matrix metalloproteinase-9 locus to genetically determined susceptibility to a number of important diseases. The results also provide experimental data on the extent of linkage disequilibrium in the general population, which is yet to be resolved.     

A test of transmission/disequilibrium for quantitative traits in pedigree data, by multiple regression.

The transmission/disequilibrium (TD) test (TDT), proposed, by Spielman et al., for binary traits is a powerful method for detection of linkage between a marker locus and a disease locus, in the presence of allelic association. As a test for linkage disequilibrium, the TDT makes the assumption that any allelic association present is due to linkage. Allison proposed a series of TD-type tests for quantitative traits and calculated their power, assuming that the marker locus is the disease locus. All these tests assume that the observations are independent, and therefore they are applicable, as a test for linkage, only for nuclear-family data. In this report, we propose a regression-based TD-type test for linkage between a marker locus and a quantitative trait locus, using information on the parent-to-offspring transmission status of the associated allele at the marker locus. This method does not require independence of observations, thus allowing for analysis of pedigree data as well, and allows adjustment for covariates. We investigate the statistical power and validity of the test by simulating markers at various recombination fractions from the disease locus.     

Co-selection of the H63D mutation and the HLA-A29 allele: a new paradigm of linkage disequilibrium?

The major histocompatibility complex (MHC) shows a remarkable conservation of particular HLA antigens and haplotypes in linkage disequilibrium in most human populations, suggesting the existence of a convergent evolution. A recent example of such conservation is the association of particular HLA haplotypes with the HFE mutations. With the objective of exploring the significance of that association, the present paper offers an analysis of the linkage disequilibrium between HLA alleles or haplotypes and the HFE mutations in a Portuguese population. Allele and haplotype associations between HLA and HFE mutations were first reviewed in a population of 43 hemochromatosis families. The results confirmed the linkage disequilibrium of the HLA haplotype HLA-A3-B7 and the HLA-A29 allele, respectively, with the HFE mutations C282Y and H63D. In order to extend the study of the linkage disequilibrium between H63D and the HLA-A29-containing haplotypes in a normal, random population, an additional sample of 398 haplotypes was analyzed. The results reveal significant linkage disequilibrium between the H63D mutation and all HLA-A29-containing haplotypes, favoring the hypothesis of a co-selection of H63D and the HLA-A29 allele itself. An insight into the biological significance of this association is given by the finding of significantly higher CD8(+) T-lymphocyte counts in subjects simultaneously carrying the H63D mutation and the HLA-A29 allele.     

A locus on chromosome 11p with multiple restriction site polymorphisms.

We have discovered and characterized a new polymorphic locus on chromosome 11p, D11S12, defined by an arbitrary genomic DNA segment cloned in the plasmid pADJ762. Four different polymorphic restriction sites with minor allele frequencies greater than 5% are revealed by Southern hybridization of this probe and its derivatives to digests of human DNAs. These include two MspI sites, a TaqI site, and a BclI site. The frequencies of the common haplotypes at this locus have been determined in a Utah population. Significant linkage disequilibrium has been demonstrated to exist between some pairs of polymorphic sites. A molecular map of this region has been determined, and the polymorphic sites have been localized. Comparison of physical separation with degree of linkage disequilibrium reveals an interesting case where an MspI site and a TaqI site that are separated by 6.8 kilobases (kb) show a greater degree of disequilibrium with each other than they do with two polymorphic sites located between them. One of the two interior sites is a BclI site that is approximately 0.2 kb away from the TaqI site but shows the same degree of disequilibrium with the TaqI site as with the MspI site 6.7 kb away. Although there is significant linkage disequilibrium at this locus, there are four major haplotypes with frequencies of 5% or greater, and the polymorphic information content (PIC) of this locus is .64.     

A log-linear approach to case-parent-triad data: assessing effects of disease genes that act either directly or through maternal effects and that may be subject to parental imprinting.

We describe a log-linear method for analysis of case-parent-triad data, based on maximum likelihood with stratification on parental mating type. The method leads to estimates of association parameters, such as relative risks, for a single allele, and also to likelihood ratio chi2 tests (LRTs) of linkage disequilibrium. Hardy-Weinberg equilibrium need not be assumed. Our simulations suggest that the LRT has power similar to that of the chi2 "score" test proposed by Schaid and Sommer and that both can outperform the transmission/disequilibrium test (TDT), although the TDT can perform better under an additive model of inheritance. Because a restricted version of the LRT is asymptotically equivalent to the TDT, the proposed test can be regarded as a generalization of the TDT. The method that we describe generalizes easily to accommodate maternal effects on risk and, in fact, produces powerful and orthogonal tests of the contribution of fetal versus maternal genetic factors. We further generalize the model to allow for effects of parental imprinting. Imprinting effects can be fitted by a simple, iterative procedure that relies on the expectation-maximization algorithm and that uses standard statistical software for the maximization steps. Simulations reveal that LRT tests for detection of imprinting have very good operating characteristics. When a single allele is under study, the proposed method can yield powerful tests for detection of linkage disequilibrium and is applicable to a broader array of causal scenarios than is the TDT.     

A Study of Linkage Disequilibrium in Populations of DROSOPHILA MELANOGASTER

This paper presents the results of a study of linkage disequilibrium between five polymorphic enzyme genes located on chromosome 3 of D. melanogaster. Three sets of chromosomes were examined: two represented samples from successive years of the same natural population, and one came from a large laboratory population. Out of the thirty possible tests for linkage disequilibrium between pairs of loci, two were significant at the 5% level and two at the 1% level. This result cannot reasonably be ascribed to chance alone. The pairs of loci that had a significant correlation in one sample had higher than average correlations in the other samples (though not necessarily in the same direction); this effect was highly significant statistically. There was no tendency for the high correlations to be associated with tightness of linkage between the loci concerned. All five loci were involved in at least one significant effect. It was concluded that these results are difficult to explain on the neutral allele theory of protein polymorphism, but are consistent with the concept of selective control of allele frequencies.     

A global perspective on genetic variation at the ADH genes reveals unusual patterns of linkage disequilibrium and diversity

Variants of different Class I alcohol dehydrogenase (ADH) genes have been shown to be associated with an effect that is protective against alcoholism. Previous work from our laboratory has shown that the two sites showing the association are in linkage disequilibrium and has identified the ADH1B Arg47His site as causative, with the ADH1C Ile349Val site showing association only because of the disequilibrium. Here, we describe an initial study of the nature of linkage disequilibrium and genetic variation, in population samples from different regions of the world, in a larger segment of the ADH cluster (including the three Class I ADH genes and ADH7). Linkage disequilibrium across approximately 40 kb of the Class I ADH cluster is moderate to strong in all population samples that we studied. We observed nominally significant pairwise linkage disequilibrium, in some populations, between the ADH7 site and some Class I ADH sites, at moderate values and at a molecular distance as great as 100 kb. Our data indicate (1) that most ADH-alcoholism association studies have failed to consider many sites in the ADH cluster that may harbor etiologically significant alleles and (2) that the relevance of the various ADH sites will be population dependent. Some individual sites in the Class I ADH cluster show Fst values that are among the highest seen among several dozen unlinked sites that were studied in the same subset of populations. The high Fst values can be attributed to the discrepant frequencies of specific alleles in eastern Asia relative to those in other regions of the world. These alleles are part of a single haplotype that exists at high (>65%) frequency only in the eastern-Asian samples. It seems unlikely that this haplotype, which is rare or unobserved in other populations, reached such high frequency because of random genetic drift alone.     

Transmission/Disequilibrium Tests Incorporating Unaffected Offspring

We propose a new method for family-based tests of association and linkage called transmission/disequilibrium tests incorporating unaffected offspring (TDTU). This new approach, constructed based on transmission/disequilibrium tests for quantitative traits (QTDT), provides a natural extension of the transmission/disequilibrium test (TDT) to utilize transmission information from heterozygous parents to their unaffected offspring as well as the affected offspring from ascertained nuclear families. TDTU can be used in various study designs and can accommodate all types of independent nuclear families with at least one affected offspring. When the study sample contains only case-parent trios, the TDTU is equivalent to TDT. Informative-transmission disequilibrium test (i-TDT) and generalized disequilibrium test(GDT) are another two methods that can use information of both unaffected offspring and affected offspring. In contract to i-TDT and GDT, the test statistic of TDTU is simpler and more explicit, and can be implemented more easily. Through computer simulations, we demonstrate that power of the TDTU is slightly higher compared to i-TDT and GDT. All the three methods are more powerful than method that uses affected offspring only, suggesting that unaffected siblings also provide information about linkage and association.     

Inferring linkage disequilibrium between a polymorphic marker locus and a trait locus in natural populations

Three approaches are proposed in this study for detecting or estimating linkage disequilibrium between a polymorphic marker locus and a locus affecting quantitative genetic variation using the sample from random mating populations. It is shown that the disequilibrium over a wide range of circumstances may be detected with a power of 80% by using phenotypic records and marker genotypes of a few hundred individuals. Comparison of ANOVA and regression methods in this article to the transmission disequilibrium test (TDT) shows that, given the genetic variance explained by the trait locus, the power of TDT depends on the trait allele frequency, whereas the power of ANOVA and regression analyses is relatively independent from the allelic frequency. The TDT method is more powerful when the trait allele frequency is low, but much less powerful when it is high. The likelihood analysis provides reliable estimation of the model parameters when the QTL variance is at least 10% of the phenotypic variance and the sample size of a few hundred is used. Potential use of these estimates in mapping the trait locus is also discussed.