Linkage analysis of "necessary" disease loci versus "susceptibility" loci.

The association of some diseases with specific alleles of certain genetic markers has been difficult to explain. Several explanations have been proposed for the phenomenon of association, e.g. the existence of multiple, interacting genes (epistasis) or a disease locus in linkage disequilibrium with the marker locus. One might suppose that when marker data from families with associated diseases are analyzed for linkage, the existence of the association would assure that linkage will be found, and found at a tight recombination fraction. In fact, however, linkage analyses of some diseases associated with HLA, as well as diseases associated with alleles at other loci located throughout the genome, show significant evidence against linkage, and others show loose linkage, to the puzzlement of many researchers. In part, the puzzlement arises because linkage analysis is ideal for looking for loci that are necessary, even if not sufficient, for disease expression but may be much less useful for finding loci that are neither necessary nor sufficient for disease expression (so-called susceptibility loci). This work explores what happens when one looks for linkage to susceptibility loci. A susceptibility locus in this case means that the allele increases risk but is neither necessary nor sufficient for disease expression. It might be either an allele at the marker locus itself that is increasing susceptibility or an allele at a locus in linkage disequilibrium with the marker. This work uses computer simulation to examine how linkage analyses behave when confronted with data from such a model.(ABSTRACT TRUNCATED AT 250 WORDS)     

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.     

Effect of genetic heterogeneity and assortative mating on linkage analysis: a simulation study.

Linkage studies of complex genetic traits raise questions about the effects of genetic heterogeneity and assortative mating on linkage analysis. To further understand these problems, I have simulated and analyzed family data for a complex genetic disease in which disease phenotype is determined by two unlinked disease loci. Two models were studied, a two-locus threshold model and a two-locus heterogeneity model. Information was generated for a marker locus linked to one of the disease-defining loci. Random-mating and assortative-mating samples were generated. Linkage analysis was then carried out by use of standard methods, under the assumptions of a single-locus disease trait and a random-mating population. Results were compared with those from analysis of a single-locus homogeneous trait in samples with the same levels of assortative mating as those considered for the two-locus traits. The results show that (1) introduction of assortative mating does not, in itself, markedly affect the estimate of the recombination fraction; (2) the power of the analysis, reflected in the LOD scores, is somewhat lower with assortative rather than random mating. Loss of power is greater with increasing levels of assortative mating; and (3) for a heterogeneous genetic disease, regardless of mating type, heterogeneity analysis permits more accurate estimate of the recombination fraction but may be of limited use in distinguishing which families belong to each homogeneous subset. These simulations also confirmed earlier observations that linkage to a disease "locus" can be detected even if the disease is incorrectly defined as a single-locus (homogeneous) trait, although the estimated recombination fraction will be significantly greater than the true recombination fraction between the linked disease-defining locus and the marker locus.     

Haplotypes at ATM identify coding-sequence variation and indicate a region of extensive linkage disequilibrium

Genetic variation in the human population may lead to functional variants of genes that contribute to risk for common chronic diseases such as cancer. In an effort to detect such possible predisposing variants, we constructed haplotypes for a candidate gene and tested their efficacy in association studies. We developed haplotypes consisting of 14 biallelic neutral-sequence variants that span 142 kb of the ATM locus. ATM is the gene responsible for the autosomal recessive disease ataxia-telangiectasia (AT). These ATM noncoding single-nucleotide polymorphisms (SNPs) were genotyped in nine CEPH families (89 individuals) and in 260 DNA samples from four different ethnic origins. Analysis of these data with an expectation-maximization algorithm revealed 22 haplotypes at this locus, with three major haplotypes having frequencies > or = .10. Tests for recombination and linkage disequilibrium (LD) show reduced recombination and extensive LD at the ATM locus, in all four ethnic groups studied. The most striking example was found in the study population of European ancestry, in which no evidence for recombination could be discerned. The potential of ATM haplotypes for detection of genetic variants through association studies was tested by analysis of 84 individuals carrying one of three ATM coding SNPs. Each coding SNP was detected by association with an ATM haplotype. We demonstrate that association studies with haplotypes for candidate genes have significant potential for the detection of genetic backgrounds that contribute to disease.     

Joint modeling of linkage and association: identifying SNPs responsible for a linkage signal

Once genetic linkage has been identified for a complex disease, the next step is often association analysis, in which single-nucleotide polymorphisms (SNPs) within the linkage region are genotyped and tested for association with the disease. If a SNP shows evidence of association, it is useful to know whether the linkage result can be explained, in part or in full, by the candidate SNP. We propose a novel approach that quantifies the degree of linkage disequilibrium (LD) between the candidate SNP and the putative disease locus through joint modeling of linkage and association. We describe a simple likelihood of the marker data conditional on the trait data for a sample of affected sib pairs, with disease penetrances and disease-SNP haplotype frequencies as parameters. We estimate model parameters by maximum likelihood and propose two likelihood-ratio tests to characterize the relationship of the candidate SNP and the disease locus. The first test assesses whether the candidate SNP and the disease locus are in linkage equilibrium so that the SNP plays no causal role in the linkage signal. The second test assesses whether the candidate SNP and the disease locus are in complete LD so that the SNP or a marker in complete LD with it may account fully for the linkage signal. Our method also yields a genetic model that includes parameter estimates for disease-SNP haplotype frequencies and the degree of disease-SNP LD. Our method provides a new tool for detecting linkage and association and can be extended to study designs that include unaffected family members.     

Association studies under general disease models

There is great expectation that the levels of association found between genetic markers and disease status will play a role in the location of disease genes. This expectation follows from regarding association as being proportional to linkage disequilibrium and therefore inversely related to recombination value. For disease genes with more than two alleles, the association measure is instead a weighted average of linkage disequilibria, with the weights depending on allele frequencies and genotype susceptibilities at the disease loci. There is no longer a simple relationship, even in expectation, with recombination. We adopt a general framework to examine association mapping methods which helps to clarify the nature of case-control and transmission/disequilibrium-type tests and reveals the relationship between measures of association and coefficients of linkage disequilibrium. In particular, we can show the consequences of additive and nonadditive effects at the trait locus on the behavior of these tests. These concepts have a natural extension to marker haplotypes. The association of two-locus marker haplotypes with disease phenotype depends on a weighted average of three-locus disequilibria (two markers with each disease locus). It is likely that these two-marker analyses will provide additional information in association mapping studies.     

A semi-symmetric two-locus model

The two-locus symmetric viability model characterized by its invariance with respect to the exchange of alleles at each locus, is a well-studied model of classical two-locus theory. The symmetric model introduced by Lewontin and Kojima is among the few multi-locus models with epistatic interactions between loci for which a polymorphism with linkage equilibrium can be stable and this happens when recombination is sufficiently large. We show that an analogous property holds true for a different model, in which symmetry need exist at only one locus. The properties of this new semi-symmetric model are compared with those of the classical symmetric model. For tight linkage, two classes of polymorphisms are possible, depending on the magnitude of additive epistasis. The recombination rate above which linkage equilibrium becomes stable is derived analytically. As in the symmetric model, intervals of recombination in which no polymorphism is stable are possible, and stable polymorphisms can coexist with stable fixations.     

The posterior probability of linkage allowing for linkage disequilibrium and a new estimate of disequilibrium between a trait and a marker

The posterior probability of linkage (PPL) statistic has been developed as a method for the rigorous accumulation of evidence for or against linkage allowing for both intra- and inter-sample heterogeneity. To date, the method has assumed linkage equilibrium between alleles at the trait locus and the marker locus. We now generalize the PPL to allow for linkage disequilibrium (LD), by incorporating variable phase probabilities into the underlying linkage likelihood. This enables us to recover the marginal posterior density of the recombination fraction, integrating out nuisance parameters of the trait model, including the locus heterogeneity (admixture) parameter, as well as a vector of LD parameters. The marginal posterior density can then be updated across data subsets or new data as they become available, while allowing parameters of the trait model to vary between data sets. The method applies immediately to general pedigree structures and to markers with multiple alleles. In the case of SNPs, the likelihood is parameterized in terms of the standard single LD parameter D'; and it therefore affords a mechanism for estimation of D' between the marker and the trait, again, without fixing the parameters of the trait model and allowing for updating across data sets. It is even possible to allow for a different associated allele in different populations, while accumulating information regarding the strength of LD. While a computationally efficient implementation for multi-allelic markers is still in progress, we have implemented a version of this new LD-PPL for SNPs and evaluated its performance in nuclear families. Our simulations show that LD-PPLs tend to be larger than PPLs (stronger evidence in favor of linkage/LD) with increased LD level, under a variety of generating models; while in the absence of linkage and LD, LD-PPLs tend to be smaller than PPLs (stronger evidence against linkage). The estimate of D' also behaves well even in relatively small, heterogeneous samples.     

Linkage disequilibrium under recurrent bottlenecks

To model deviations from selectively neutral genetic variation caused by different forms of selection, it is necessary to first understand patterns of neutral variation. Best understood is neutral genetic variation at a single locus. But, as is well known, additional insights can be gained by investigating multiple loci. The resulting patterns reflect the degree of association (linkage) between loci and provide information about the underlying multilocus gene genealogies. The statistical properties of two-locus gene genealogies have been intensively studied for populations of constant size, as well as for simple demographic histories such as exponential population growth and single bottlenecks. By contrast, the combined effect of recombination and sustained demographic fluctuations is poorly understood. Addressing this issue, we study a two-locus Wright-Fisher model of a population subject to recurrent bottlenecks. We derive coalescent approximations for the covariance of the times to the most recent common ancestor at two loci in samples of two chromosomes. This covariance reflects the degree of association and thus linkage disequilibrium between these loci. We find, first, that an effective population-size approximation describes the numerically observed association between two loci provided that recombination occurs either much faster or much more slowly than the population-size fluctuations. Second, when recombination occurs frequently between but rarely within bottlenecks, we observe that the association of gene histories becomes independent of physical distance over a certain range of distances. Third, we show that in this case, a commonly used measure of linkage disequilibrium, σ(2)(d) (closely related to r(2)), fails to capture the long-range association between two loci. The reason is that constituent terms, each reflecting the long-range association, cancel. Fourth, we analyze a limiting case in which the long-range association can be described in terms of a Xi coalescent allowing for simultaneous multiple mergers of ancestral lines.     

渐近混合群体的混合连锁不平衡及其遗传连锁推断

在渐近混合模型中,混合现象发生在每一世代,通过对其混合连锁不平衡的理论分析,发现混合连锁不平衡与两个子群体间的基因频率差成正比。基于这一点,构造了一个对重组率严格单调的函数（△ker=△／（p1-p2）,其中△代表连锁不平衡）,进而据此推断标记基因座与疾病基因座的遗传连锁。应用人类基因组上不连锁的标记基因提供的连锁不平衡信息,基于病人组数据构造了一个准似然比统计量。模拟结果显示,此检验可用于精确的基因定位。文章亦讨论了参数对检验的影响。     

Disequilibrium likelihoods for fine-scale mapping of a rare allele.

Genetic linkage studies based on pedigree data have limited resolution, because of the relatively small number of segregations. Disequilibrium mapping, which uses population associations to infer the location of a disease mutation, provides one possible strategy for narrowing the candidate region. The coalescent process provides a model for the ancestry of a sample of disease alleles, and recombination events between disease locus and marker may be placed on this ancestral phylogeny. These events define the recombinant classes, the sets of sampled disease copies descending from the meiosis at which a given recombination occurred. We show how Monte Carlo generation of the recombinant classes leads to a linkage likelihood for fine-scale mapping from disease haplotypes. We compare single-marker disequilibrium mapping with interval-disequilibrium mapping and discuss how the approach may be extended to multipoint-disequilibrium mapping. The method and its properties are illustrated with an example of simulated data, constructed to be typical of fine-scale mapping of a rare disease in the Japanese population. The method can take into account known features of population history, such as changing patterns of population growth.     

A method for estimating the mutation,gene conversion and recombination parameters in small multigene families

A simple two-locus gene conversion model is considered to investigate the amounts of DNA variation and linkage disequilibrium in small multigene families. The exact solutions for the expectations and variances of the amounts of variation within and between two loci are obtained. It is shown that gene conversion increases the amount of variation within each locus and decreases the amount of variation between two loci. The expectation and variance of the amount of linkage disequilibrium are also obtained. Gene conversion generates positive linkage disequilibrium and the degree of linkage disequilibrium decreases as the recombination rate is increased. Using the theoretical results, a method for estimating the mutation, gene conversion, and recombination parameters is developed and applied to the data of the Amy multigene family in Drosophila melanogaster. The gene conversion rate is estimated to be approximately 60-165 times higher than the mutation rate for synonymous sites.     

Detecting linkage for genetically heterogeneous diseases and detecting heterogeneity with linkage data.

Interest in searching for genetic linkage between diseases and marker loci has been greatly increased by the recent introduction of DNA polymorphisms. However, even for the most well-behaved Mendelian disorders, those with clear-cut mode of inheritance, complete penetrance, and no phenocopies, genetic heterogeneity may exist; that is, in the population there may be more than one locus that can determine the disease, and these loci may not be linked. In such cases, two questions arise: (1) What sample size is necessary to detect linkage for a genetically heterogeneous disease? (2) What sample size is necessary to detect heterogeneity given linkage between a disease and a marker locus? We have answered these questions for the most important types of matings under specified conditions: linkage phase known or unknown, number of alleles involved in the cross at the marker locus, and different numbers of affected and unaffected children. In general, the presence of heterogeneity increases the recombination value at which lod scores peak, by an amount that increases with the degree of heterogeneity. There is a corresponding increase in the number of families necessary to establish linkage. For the specific case of backcrosses between disease and marker loci with two alleles, linkage can be detected at recombination fractions up to 20% with reasonable numbers of families, even if only half the families carry the disease locus linked to the marker. The task is easier if more than two informative children are available or if phase is known. For recessive diseases, highly polymorphic markers with four different alleles in the parents greatly reduce the number of families required.     

A combined segregation and linkage analysis of insulin-dependent diabetes mellitus.

In an effort to clarify the mode of inheritance of insulin-dependent diabetes mellitus (IDDM), a total of 230 nuclear families with pointers were analyzed using the computer program COMBIN. Each family was ascertained without deliberate selection for multiplex families, and most families were completely typed for HLA-B, HLA-DR, and properdin factor B (Bf). There were 186 families with normal parents, 44 families with one affected parent, and no families with two affected parents. The computer program COMBIN evaluates evidence for a major locus of disease susceptibility, linkage of the major locus to a known genetic marker locus, linkage disequilibrium between the marker haplotypes and disease susceptibility, pleiotropic effects, and presence of an unlinked modifier. The parameters of COMBIN are T, Q, and D, representing the displacement, gene frequency of the IDDM allele, and dominance, respectively, of the major locus--and TM, QM, and DM being the analogous parameters of the modifier. In addition, the recombination fraction, theta, between the IDDM locus and HLA as well as the coupling frequencies are estimated. Finally, COMBIN simultaneously performs segregation and linkage analysis, with the optimal model being adjusted by the fit to the haplotype sharing distribution of IDDM. The results of these analyses indicated that the best-fitting genetic model of diabetic susceptibility appears to be a single major locus with near recessivity on a scale of standardized genetic liability, with gene frequency of the IDDM susceptibility allele of approximately 14%. In addition, the recombination fraction between the major locus and HLA is zero in all models; that is, for the B-BF-DR haplotype, the IDDM locus is tightly linked, probably (according to data from previous studies) to HLA-DR. Information determined by magnitude of coupling frequencies indicated that there is significant positive linkage disequilibrium with the haplotypes B8-BfS-DR4 and B15-BfS-DR4, significant negative linkage disequilibrium with B7-BfS-DR2, and intermediate disequilibrium for B8-BfS-DR3, B18-BfF1-DR3, and B40-BfS-DR4. Significant evidence in favor of an unlinked (to HLA) modifier (either single major locus or polygenes) could not be demonstrated. In conclusion, genetic susceptibility to IDDM appears to be most consistent with a single major locus with near recessivity that is tightly linked to HLA.     

Effects of misspecifying genetic parameters in lod score analysis

The lod score method is widely used to test linkage and to estimate the recombination fraction between a disease locus and a marker locus. The parameters (gene frequency, penetrance, and degree of dominance) are assumed to be known at each locus. This condition may not be fulfilled at the disease locus. In this paper, we evaluate the errors due to the use of wrong parameters. The power of the linkage test is sensitive to the degree of dominance, and slightly to the penetrance, but not to the gene frequency. In contrast, the estimation of the recombination fraction may be strongly affected by an error on any genetic parameter.     

Randomly distributed crossovers may generate block-like patterns of linkage disequilibrium: an act of genetic drift

There is considerable interest in identifying and characterizing block-like patterns of linkage disequilibrium (LD; haplotype blocks) in the human genome as these may facilitate the identification of complex disease genes via genome-wide association studies. Although recombination hot-spots have been suggested as the primary mechanism to explain the block-like pattern of LD, other forces, such as genetic drift, may also be important. To this end, we have studied the effect of various recombination models on patterns of LD by using extensive simulations. As expected, haplotype blocks were observed under a model allowing recombination hot-spots. However, we also observed similar block-like patterns in the models where recombination crossovers are randomly and uniformly distributed, and we demonstrate that these blocks are generated by genetic drift. We caution that genetic drift may be an alternative mechanism (in addition to recombination hot-spots) that can lead to block-like patterns of LD. Our findings highlight the necessity of characterizing haplotype blocks in world-wide populations.     

Analytical results concerning linkage disequilibrium in models with genetic transformation and conjugation

Expressions are obtained for the expected levels of linkage disequilibrium under three different equilibrium neutral models that make different assumptions about how recombination takes place. A transformation model is considered in which exchange events involve only one locus at a time. Two conjugation models are considered one with a linear genome and one with a circular genome. In the conjugation models large blocks of genes can be transferred with each conjugation. Consistent with published simulation results, it is found that if the transformation rate per locus is more than twenty times the mutation rate per locus, then the levels of linkage disequilibrium are quite low. If the number of loci being sampled is greater than 10, conjugation with a circular genome can be considerably more effective than transformation in reducing linkage disequilibrium. When recombination rates are high, expected linkage disequilibrium is shown to be proportional to the inverse of the transformation rate (or conjugation rate.)