The Sampling Distribution of Linkage Disequilibrium

The probabilities of obtaining particular samples of gametes with two completely linked loci are derived. It is assumed that the population consists of N diploid, randomly mating individuals, that each of the two loci mutate according to the infinite allele model at a rate µ and that the population is at equilibrium. When 4Nµ is small, the most probable samples of gametes are those that segregate only two alleles at either locus. The probabilities of various samples of gametes are discussed. The results show that most samples with completely linked loci have either a very small or a very large association between the alleles of each locus. This causes the distribution of linkage disequilibrium to be skewed and the distribution of the correlation coefficient to be bimodal. The correlation coefficient is commonly used as a test statistic with a chi square distribution and yet has a bimodal distribution when the loci are completely linked. Thus, such a test is not likely to be accurate unless the rate of recombination between the loci and/or the effective population size are sufficiently large enough so that the loci can be treated as unlinked.     

Linkage Disequilibrium in Subdivided Populations

The linkage disequilibrium in a subdivided populaton is shown to be equal to the sum of the average linkage disequilibrium for all subpopulations and the covariance between gene frequencies of the loci concerned. Thus, in a subdivided population the linkage disequilibrium may not be 0 even if the linkage disequilibrium in each subpopulation is 0. If a population is divided into two subpopulations between which migration occurs, the asymptotic rate of approach to linkage equilibrium is equal to either r or 2(m(1) + m(2)) - (m(1) + m(2))(2), whichever is smaller, where r is the recombination value and m(1) and m(2) are the proportions of immigrants in subpopulations 1 and 2, respectively. Thus, if migration rate is high compared with recombination value, the change of linkage disequilibrium in subdivided populations is similar to that of a single random mating population. On the other hand, if migration rate is low, the approach to lnkage equilibrium may be retarded in subdivided populations. If isolated populations begin to exchange genes by migration, linkage disequilibrium may increase temporarily even for neutral loci. If overdominant selection operates and the equilibrium gene frequencies are different in the two subpopulations, a permanent linkage disequilibrium may be produced without epistasis in each subpopulation.     

The Detection of Linkage Disequilibrium in Molecular Sequence Data

Studies of genetic variation in natural populations at the sequence level usually show that most polymorphic sites are very asymmetrical in allele frequencies, with the rarer allele at a site near fixation. When the rarer allele at a site is present only a few times in the sample, say below five representatives, it becomes very difficult to detect linkage disequilibrium between sites from tests of association. This is a consequence of the numerical properties of even the most powerful test of association, Fisher's exact test. Sites with fewer than five representatives in the sample should be excluded from association tests, but this generally leaves few site pairs eligible for testing. A test for overall linkage disequilibrium, based on the sign of the observed linkage disequilibria, is derived which can use all the data. It is shown that more power can be achieved by increasing the length of sequence determined than by increasing the number of genomes sampled for the same total work.     

CIS-TRANS Effects Induced by Linkage Disequilibrium

This note concerns theoretical and experimental studies of multifactorial traits, especially fitness and its components, in which (1) the loci studied are only a subset of those relevant to the character of interest and (2) the genotypes at the loci studied are in nonrandom association (linkage disequilibrium) with genotypes at the loci ignored. In these cases, phenotypic differences between cis and trans double heterozygotes can occur even though no linkage phase effects are inherent in the genetic determination of the trait. Examples are drawn from both theoretical and experimental work, and implications in both areas are discussed.     

The Admixture Linkage Disequilibrium and Genetic Linkage Inference on the Gradual Admixture Population

Through the theoretical analysis of the admixture linkage disequilibrium (ALD) in the gradual admixture (GA) model, in which admixture occurs in every generation, the ALD is found to be proportional to the difference in marker allele frequencies, p₁-p₂, between two subpopulations. Based on this property, we can employ a strict monotonic function (Δ_ker=Δ/(p₁-p₂), where Δ denotes the linkage disequilibrium (LD)) of the recombination fraction between the marker locus and the disease locus to infer the true genetic linkage. We construct a quasi likelihood ratio test (LRT) for the case-only data utilizing the information of unlinked markers in the human genome. The simulation results show that our tests can be used to fine map a disease locus. The effects of parameter values in the ALD mapping are also discussed.     

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。     

How Population Growth Affects Linkage Disequilibrium

The “LD curve” relates the linkage disequilibrium (LD) between pairs of nucleotide sites to the distance that separates them along the chromosome. The shape of this curve reflects natural selection, admixture between populations, and the history of population size. This article derives new results about the last of these effects. When a population expands in size, the LD curve grows steeper, and this effect is especially pronounced following a bottleneck in population size. When a population shrinks, the LD curve rises but remains relatively flat. As LD converges toward a new equilibrium, its time path may not be monotonic. Following an episode of growth, for example, it declines to a low value before rising toward the new equilibrium. These changes happen at different rates for different LD statistics. They are especially slow for estimates of σd2, which therefore allow inferences about ancient population history. For the human population of Europe, these results suggest a history of population growth.     

Linkage Disequilibrium in Growing and Stable Populations

Nonrandom associations between alleles at different loci can be tested for using Fisher's exact test. Extensive simulations show that there is a substantial probability of obtaining significant nonrandom associations between closely or completely linked polymorphic neutral loci in a population of constant size at equilibrium under mutation and genetic drift. In a rapidly growing population, however, there will be little chance of finding significant nonrandom associations even between completely linked loci if the growth has been sufficiently rapid. This result is illustrated by the analysis of mitochondrial DNA sequence data from humans. In comparing all pairs of informative sites, fewer than 5% of the pairs show significant disequilibrium in Sardinians, which have apparently undergone rapid population growth, while 20% to 30% in !Kung and Pygmies, which apparently have not undergone rapid growth, show significance. The extent of linkage disequilibrium in a population is closely related to the gene genealogies of the loci examined, with ``starlike' genealogies making significant linkage disequilibrium unlikely.     

Linkage Disequilibrium in Natural Populations of DROSOPHILA MELANOGASTER

Two large, stable populations (Texas and Japan) of Drosophila melanogaster were surveyed at 21 allozyme loci on the second and third chromosomes and for chromosomal gene arrangements on those two chromosomes. Over 220 independent gametes were sampled from each population. The types and frequencies of the surveyed genetic variation are similar to those observed previously and suggest only slight differentiation among geographically distant populations. Linkage disequilibrium among linked allozymes loci is only slightly, if at all, detectable with these sample sizes. Linkage disequilibrium between linked inversions and allozymes loci is common especially when located in the same arm. These disequilibria appear to be in the same direction for most comparisons in the two population samples. This result is interpreted as evidence of similar selective environments (ecological and genetic) in the two populations. It is also noted that the direction of these linkage disequilibria appears to be oriented with respect to the gene frequencies at the component loci.     

Linkage Disequilibrium and Genetic Variances under Mutation-Selection Balance

I determine the contribution of linkage disequilibrium to genetic variances using results for two loci and for induced or marginal systems. The analysis allows epistasis and dominance, but assumes that mutation is weak relative to selection. The linkage disequilibrium component of genetic variance is shown to be unimportant for unlinked loci if the gametic mutation rate divided by the harmonic mean of the pairwise recombination rates is much less than one. For tightly linked loci, linkage disequilibrium is unimportant if the gametic mutation rate divided by the (induced) per locus selection is much less than one.     

Divergence between Human Populations Estimated from Linkage Disequilibrium

Observed linkage disequilibrium (LD) between genetic markers in different populations descended independently from a common ancestral population can be used to estimate their absolute time of divergence, because the correlation of LD between populations will be reduced each generation by an amount that, approximately, depends only on the recombination rate between markers. Although drift leads to divergence in allele frequencies, it has less effect on divergence in LD values. We derived the relationship between LD and time of divergence and verified it with coalescent simulations. We then used HapMap Phase II data to estimate time of divergence between human populations. Summed over large numbers of pairs of loci, we find a positive correlation of LD between African and non-African populations at levels of up to ~0.3 cM. We estimate that the observed correlation of LD is consistent with an effective separation time of approximately 1,000 generations or ~25,000 years before present. The most likely explanation for such relatively low separation times is the existence of substantial levels of migration between populations after the initial separation. Theory and results from coalescent simulations confirm that low levels of migration can lead to a downward bias in the estimate of separation time.     

Long Range Linkage Disequilibrium across the Human Genome

Long-range linkage disequilibria (LRLD) between sites that are widely separated on chromosomes may suggest that population admixture, epistatic selection, or other evolutionary forces are at work. We quantified patterns of LRLD on a chromosome-wide level in the YRI population of the HapMap dataset of single nucleotide polymorphisms (SNPs). We calculated the disequilibrium between all pairs of SNPs on each chromosome (a total of >2×10¹¹ values) and evaluated significance of overall disequilibrium using randomization. The results show an excess of associations between pairs of distant sites (separated by >0.25 cM) on all of the 22 autosomes. We discuss possible explanations for this observation.     

Linkage Disequilibrium in Humans: Models and Data

In this review, we describe recent empirical and theoretical work on the extent of linkage disequilibrium (LD) in the human genome, comparing the predictions of simple population-genetic models to available data. Several studies report significant LD over distances longer than those predicted by standard models, whereas some data from short, intergenic regions show less LD than would be expected. The apparent discrepancies between theory and data present a challenge-both to modelers and to human geneticists-to identify which important features are missing from our understanding of the biological processes that give rise to LD. Salient features may include demographic complications such as recent admixture, as well as genetic factors such as local variation in recombination rates, gene conversion, and the potential segregation of inversions. We also outline some implications that the emerging patterns of LD have for association-mapping strategies. In particular, we discuss what marker densities might be necessary for genomewide association scans.     

The Effect of Marker Heterozygosity on the Power to Detect Linkage Disequilibrium

The relationship between marker heterozygosity and the power to detect linkage disequilibrium is examined through the analysis of an example and through a simulation study. The analysis suggests that, despite the penalties for multiple testing incurred with multiple alleles, greater heterozygosity results in greater power. The results of the simulation study are in accord with those of the analysis.     

Testing Hypotheses about Linkage Disequilibrium with Multiple Alleles

For loci with multiple alleles, hypotheses about linkage disequilibrium may be tested on the complete set of gametic data, or on various collapsed sets of data. Collapsing data into a few alleles at each locus can change the power of the tests, as implied in a recent paper by Zouros, Golding and Mackay (1977). We show that the nature of such changes can be found from properties of the noncentral chi-square distribution, and that the magnitude and direction of these changes depend on the levels of linkage disequilibria, allelic frequencies and degrees of freedom.     

Linkage Disequilibrium Analysis of Chromosomal Inversion Polymorphisms of Drosophila

The linkage disequilibrium pattern analysis is a method that allows one to detect present and past events of selection. We applied it to our data on the inversions of Drosophila mediopunctata and to published data of Drosophila subobscura, which are systems probably under complex modes of selection. The method gave meaningful results showing that it might be an excellent exploratory tool to identify problems worthy of further study. It is also suggested that it can be used as a criterion to determine the roots of inversion phylogenies.     

Modeling Linkage Disequilibrium Increases Accuracy of Polygenic Risk Scores

Polygenic risk scores have shown great promise in predicting complex disease risk and will become more accurate as training sample sizes increase. The standard approach for calculating risk scores involves linkage disequilibrium (LD)-based marker pruning and applying a p value threshold to association statistics, but this discards information and can reduce predictive accuracy. We introduce LDpred, a method that infers the posterior mean effect size of each marker by using a prior on effect sizes and LD information from an external reference panel. Theory and simulations show that LDpred outperforms the approach of pruning followed by thresholding, particularly at large sample sizes. Accordingly, predicted R² increased from 20.1% to 25.3% in a large schizophrenia dataset and from 9.8% to 12.0% in a large multiple sclerosis dataset. A similar relative improvement in accuracy was observed for three additional large disease datasets and for non-European schizophrenia samples. The advantage of LDpred over existing methods will grow as sample sizes increase.     

Effect of Mating Structure on Variation in Linkage Disequilibrium

Measurement of linkage disequilibrium involves two sampling processes. First, there is the sampling of gametes in the population to form successive generations, and this generates disequilibrium dependent on the effective population size (N_e) and the mating structure. Second, there is sampling of a finite number (n) of individuals to estimate the population disequilibrium.——Two-locus descent measures are used to describe the mating system and are transformed to disequilibrium moments at the final sampling. Approximate eigenvectors for the transition matrix of descent measures are used to obtain formulae for the variance of the observed disequilibria as a function of N_e, mating structure, n, and linkage or recombination parameter.——The variance of disequilibrium is the same for monoecious populations with or without random selfing and for dioecious populations with random pairing for each progeny. With monogamy, the variance is slightly higher, the proportional difference being greater for unlinked loci.