A Two-Locus Neutrality Test: Applications to Humans, E. COLI and Lodgepole Pine

The expected disequilibrium between two loci with k alleles at one locus and l alleles at the other is given for a sample of size n drawn from a population under neutrality equilibrium. Three different measures of disequilibrium with 95% intervals are tabulated for combinations of n, k, l and 4Nc, where N is the effective population size and c is the amount of recombination between the loci. The extent and pattern of disequilibrium are strongly dependent upon 4Nc and are somewhat dependent on n, k and l. The 95% intervals are large, particularly for low numbers of alleles and low values of 4Nc. As examples, observed disequilibrium from histocompatibility loci in humans (HLA) and electrophoretic data in E. coli and lodgepole pine were compared to these theoretical values. Using information about recombination rates, the HLA data showed more disequilibrium than neutrality expectations, whereas electrophoretic data from E. coli and lodgepole pine had somewhat less disequilibrium than neutrality expectations.     

Modeling extent and distribution of zygotic disequilibrium: implications for a multigenerational canine pedigree

Unlike gametic linkage disequilibrium defined for a random-mating population, zygotic disequilibrium describes the nonrandom association between different loci in a nonequilibrium population that deviates from Hardy-Weinberg equilibrium. Zygotic disequilibrium specifies five different types of disequilibria simultaneously that are (1) Hardy-Weinberg disequilibria at each locus, (2) gametic disequilibrium (including two alleles in the same gamete, each from a different locus), (3) nongametic disequilibrium (including two alleles in different gametes, each from a different locus), (4) trigenic disequilibrium (including a zygote at one locus and an allele at the other), and (5) quadrigenic disequilibrium (including two zygotes each from a different locus). However, because of the uncertainty on the phase of the double heterozygote, gametic and nongametic disequilibria need to be combined into a composite digenic disequilibrium and further define a composite quadrigenic disequilibrium together with the quadrigenic disequilibrium. To investigate the extent and distribution of zygotic disequilibrium across the canine genome, a total of 148 dogs were genotyped at 247 microsatellite markers located on 39 pairs of chromosomes for an outbred multigenerational pedigree, initiated with a limited number of unrelated founders. A major portion of zygotic disequilibrium was contributed by the composite digenic and quadrigenic disequilibrium whose values and numbers of significant marker pairs are both greater than those of trigenic disequilibrium. All types of disequilibrium are extensive in the canine genome, although their values tend to decrease with extended map distances, but with a greater slope for trigenic disequilibrium than for the other types of disequilibrium. Considerable variation in the pattern of disequilibrium reduction was observed among different chromosomes. The results from this study provide scientific guidance about the determination of the number of markers used for whole-genome association studies.     

Measuring the extent of linkage disequilibrium in commercial pig populations

To evaluate the extent of linkage disequilibrium in domestic pigs, we genotyped 33 and 44 unrelated individuals from two commercial populations for 29 and five microsatellite markers located on chromosomes 15 and 2 respectively. A high proportion of marker pairs up to 40 cM apart exhibited significant linkage disequilibrium in both populations. Pair-wise r(2) values averaged between 0.15 and 0.50 (depending on chromosome and population) for markers <1 cM apart and declined to values of 0.05 for more distant syntenic markers. Our results suggest that both populations underwent a bottleneck approximately 20 generations ago, which reduced the effective population size from thousands to <200 animals.     

The extent of linkage disequilibrium in a large cattle population of western Africa and its consequences for association studies

Several previous studies concluded that linkage disequilibrium (LD) in livestock populations from developed countries originated from the impact of strong selection. Here, we assessed the extent of LD in a cattle population from western Africa that was bred in an extensive farming system. The analyses were performed on 363 individuals in a Bos indicus x Bos taurus population using 42 microsatellite markers on BTA04, BTA07 and BTA13. A high level of expected heterozygosity (0.71), a high mean number of alleles per locus (9.7) and a mild shift in Hardy-Weinberg equilibrium were found. Linkage disequilibrium extended over shorter distances than what has been observed in cattle from developed countries. Effective population size was assessed using two methods; both methods produced large values: 1388 when considering heterozygosity (assuming a mutation rate of 10(-3)) and 2344 when considering LD on whole linkage groups (assuming a constant population size over generations). However, analysing the decay of LD as a function of marker spacing indicated a decreasing trend in effective population size over generations. This decrease could be explained by increasing selective pressure and/or by an admixture process. Finally, LD extended over small distances, which suggested that whole-genome scans will require a large number of markers. However, association studies using such populations will be effective.     

Estimation of the extent of linkage disequilibrium in seven regions of the porcine genome

Linkage disequilibrium (LD) refers to the correlation among neighboring alleles, reflecting non-random patterns of association between alleles at (nearby) loci. A better understanding of LD in the porcine genome is of direct relevance for identification of genes and mutations with a certain effect on the traits of interest. Here, 215 SNPs in seven genomic regions were genotyped in individuals of three breeds. Pairwise linkage disequilibrium was calculated for all marker pairs. To estimate the extent of LD, all pairwise LD values were plotted against the distance between the markers. Based on SNP markers in four genomic regions analyzed in three panels from populations of Large White, Dutch Landrace, and Meishan origin, useful LD is estimated to extend for approximately 40 to 60 kb in the porcine genome.     

Linkage disequilibrium fine mapping of quantitative trait loci: A simulation study

Recently, the use of linkage disequilibrium (LD) to locate genes which affect quantitative traits (QTL) has received an increasing interest, but the plausibility of fine mapping using linkage disequilibrium techniques for QTL has not been well studied. The main objectives of this work were to (1) measure the extent and pattern of LD between a putative QTL and nearby markers in finite populations and (2) investigate the usefulness of LD in fine mapping QTL in simulated populations using a dense map of multiallelic or biallelic marker loci. The test of association between a marker and QTL and the power of the test were calculated based on single-marker regression analysis. The results show the presence of substantial linkage disequilibrium with closely linked marker loci after 100 to 200 generations of random mating. Although the power to test the association with a frequent QTL of large effect was satisfactory, the power was low for the QTL with a small effect and/or low frequency. More powerful, multi-locus methods may be required to map low frequent QTL with small genetic effects, as well as combining both linkage and linkage disequilibrium information. The results also showed that multiallelic markers are more useful than biallelic markers to detect linkage disequilibrium and association at an equal distance.     

Haplotypes and linkage disequilibrium at the phenylalanine hydroxylase locus, PAH, in a global representation of populations

Because defects in the phenylalanine hydroxylase gene (PAH) cause phenylketonuria (PKU), PAH was studied for normal polymorphisms and linkage disequilibrium soon after the gene was cloned. Studies in the 1980s concentrated on European populations in which PKU was common and showed that haplotype-frequency variation exists between some regions of the world. In European populations, linkage disequilibrium generally was found not to exist between RFLPs at opposite ends of the gene but was found to exist among the RFLPs clustered at each end. We have now undertaken the first global survey of normal variation and disequilibrium across the PAH gene. Four well-mapped single-nucleotide polymorphisms (SNPs) spanning approximately 75 kb, two near each end of the gene, were selected to allow linkage disequilibrium across most of the gene to be examined. These SNPs were studied as PCR-RFLP markers in samples of, on average, 50 individuals for each of 29 populations, including, for the first time, multiple populations from Africa and from the Americas. All four sites are polymorphic in all 29 populations. Although all but 5 of the 16 possible haplotypes reach frequencies >5% somewhere in the world, no haplotype was seen in all populations. Overall linkage disequilibrium is highly significant in all populations, but disequilibrium between the opposite ends is significant only in Native American populations and in one African population. This study demonstrates that the physical extent of linkage disequilibrium can differ substantially among populations from different regions of the world, because of both ancient genetic drift in the ancestor common to a large regional group of modern populations and recent genetic drift affecting individual populations.     

Two loci with two alleles: Linkage equilibrium and linkage disequilibrium can be simultaneously stable

A class of two-locus two-allele viability matrices is exhibited which has the property that, for a large range of recombination values, both linkage equilibrium and linkage disequilibrium are stable. The model is specified by five viabilities; the classical schemes previously analyzed involve at most four selection parameters.     

Likelihood analysis of disequilibrium mapping, and related problems.

In this paper a theory is developed that provides the sampling distribution of alleles at a diallelic marker locus closely linked to a low-frequency allele that arose as a single mutant. The sampling distribution provides a basis for maximum-likelihood estimation of either the recombination rate, the mutation rate, or the age of the allele, provided that the two other parameters are known. This theory is applied to (1) the data of Hästbacka et al., to estimate the recombination rate between a locus associated with diastrophic dysplasia and a linked RFLP marker; (2) the data of Risch et al., to estimate the age of a presumptive allele causing idiopathic distortion dystonia in Ashkenazi jews; and (3) the data of Tishkoff et al., to estimate the date at which, at the CD4 locus, non-African lineages diverged from African lineages. We conclude that the extent of linkage disequilibrium can lead to relatively accurate estimates of recombination and mutation rates and that those estimates are not very sensitive to parameters, such as the population age, whose values are not known with certainty. In contrast, we also conclude that, in many cases, linkage disequilibrium may not lead to useful estimates of allele age, because of the relatively large degree of uncertainly in those estimates.     

Three-Locus Systems Impose Additional Constraints on Pairwise Disequilibria

Combinations of allele frequencies and pairwise linkage disequilibrium terms, each of which is permissible at the two-locus level, may not always be permissible at the three-locus level. These additional constraints on the possible maximum and minimum values for the pairwise disequilibrium terms are formally determined and numerically analyzed. In some cases, the three-locus constraints on a pairwise disequilibrium (D) may be equivalent to the usual two-locus constraints, while in others, the positive or negative range may be restricted. This can result in situations where the allowable values of D are limited to only positive or only negative values up to the extreme case where there is only a single admissible value. No additional restrictions are placed on pairwise disequilibrium values when four loci are considered, other than those imposed by the three-way combinations containing the two loci of interest. A new measure of normalized pairwise linkage disequilibrium, allowing for the three-locus constraints, is defined and illustrated by an application to data from the human histocompatibility antigen (HLA) system. An analogous normalized three-way disequilibrium measure is also formulated.     

Gametic Disequilibrium Measures: Proceed with Caution

Five different measures of gametic disequilibrium in current use and a new one based on R. C. Lewontin's D', are examined and compared. All of them, except the measure based on Lewontin's D', are highly dependent upon allelic frequencies, including four measures that are normalized in some manner. In addition, the measures suggested by A. H. D. Brown, M. F. Feldman and E. Nevo, and T. Ohta can have negative values when there is maximum disequilibrium and have rates of decay in infinite populations that are a function of the initial gametic array. The variances were large for all the measures in samples taken from populations at equilibrium under neutrality, with the measure based on D' having the lowest variance. In these samples, three of the measures were highly correlated, D2, D (equal to the correlation coefficient when there are two alleles at each locus) and the measure X(2) of Brown et al. Using frequency-dependent measures may result in mistaken conclusions, a fact illustrated by discussion of studies inferring recombinational hot spots and the effects of population bottlenecks from disequilibrium values.     

Selection, Hitchhiking and Disequilibrium Analysis at Three Linked Loci with Application to Hla Data

The HLA system has been extensively studied from an evolutionary perspective. Although it is clear that selection has acted on the genes in the HLA complex, the nature of this selection has yet to be fully clarified. A study of constrained disequilibrium values is presented that is applicable to HLA and other less polymorphic systems with three or more linked loci, with the purpose of identifying selection events. The method uses the fact that three locus systems impose additional constraints on the range of possible disequilibrium values for any pair of loci. We have thus examined the behavior of the normalized pairwise disequilibrium measures using two locus (D'), and also three locus (D"), constraints on pairwise disequilibria in a three locus system when one of the three loci is under positive selection. The difference between these measures, delta = magnitude of D' - magnitude of D", has a distribution for the two unselected loci differing from that for the selected locus with either of the unselected loci (the hallmark is a high positive value of delta for the two unselected loci). An examination of genetic drift indicates that positive delta values are unlikely to be found in human populations in the absence of selection when recombination is greater than about 0.1%. This measure can thus provide insight into which allele of several linked loci might have been subject to selection. Application of this method to HLA haplotypes from a large French population study (Provinces Francaise) identifies selected alleles on particular haplotypes. Application of a complementary method, disequilibrium pattern analysis also confirms the action of selection on these haplotypes.     

On the detection of nonrandom associations between DNA polymorphisms in natural populations of Drosophila

The capacity to detect nonrandom associations between restriction-map variants was examined in eight gene regions of Drosophila melanogaster (yellow-achaetescute, white, Zw, Adh, Est6, and rosy) and D. pseudoobscura (Adh and Xdh), on the basis of published population data. The statistical power from individual pairwise tests was both heterogeneous and generally low across gene regions. Sample sizes larger than those currently being used are needed to ensure any power to detect disequilibrium by individual tests. It is found that the heterogeneity in power is mostly explained by large differences in the intensity of sample disequilibrium among regions. The yellow-achaete- scute, Zw, and Adh loci of D. melanogaster displayed both the highest mean power (approximately 0.4) and a very great disequilibrium (mean absolute values of D' were 0.8-1). By contrast, all the other gene regions exhibited lower mean power (approximately 0.2) and moderate levels of disequilibrium (0.4-0.6). Although the proportion of significant pairwise associations, especially for white, Est6, and rosy in D. melanogaster and for Adh and Xdh in D. pseudoobscura, is more or less close to the type I error, simultaneous-inference significance tests show that gametic disequilibrium is occurring at the eight DNA regions examined.      

Association studies in consanguineous populations.

To study the genetic determinism of multifactorial diseases in large panmictic populations, a strategy consists in looking for an association with markers closely linked to candidate genes. A distribution of marker genotypes different in patients and controls may indicate that the candidate gene is involved in the disease. In panmictic populations, the power to detect the role of a candidate gene depends on the gametic disequilibrium with the marker locus. In consanguineous populations, we show that it depends on the inbreeding coefficient F as well. Inbreeding increases the power to detect the role of a recessive or quasi-recessive disease-susceptibility factor. The gain in power turns out to be greater for small values of the gametic disequilibrium. Moreover, even in the absence of gametic disequilibrium, the presence of inbreeding may allow to detect the role of a recessive factor. Ignoring inbreeding when it exists may lead to reject falsely a recessive model if the mode of inheritance is inferred on the distribution of genotypes among patients.     

The importance of identity-by-state information for the accuracy of genomic selection

Background

It is commonly assumed that prediction of genome-wide breeding values in genomic selection is achieved by capitalizing on linkage disequilibrium between markers and QTL but also on genetic relationships. Here, we investigated the reliability of predicting genome-wide breeding values based on population-wide linkage disequilibrium information, based on identity-by-descent relationships within the known pedigree, and to what extent linkage disequilibrium information improves predictions based on identity-by-descent genomic relationship information.

Methods

The study was performed on milk, fat, and protein yield, using genotype data on 35 706 SNP and deregressed proofs of 1086 Italian Brown Swiss bulls. Genome-wide breeding values were predicted using a genomic identity-by-state relationship matrix and a genomic identity-by-descent relationship matrix (averaged over all marker loci). The identity-by-descent matrix was calculated by linkage analysis using one to five generations of pedigree data.

Results

We showed that genome-wide breeding values prediction based only on identity-by-descent genomic relationships within the known pedigree was as or more reliable than that based on identity-by-state, which implicitly also accounts for genomic relationships that occurred before the known pedigree. Furthermore, combining the two matrices did not improve the prediction compared to using identity-by-descent alone. Including different numbers of generations in the pedigree showed that most of the information in genome-wide breeding values prediction comes from animals with known common ancestors less than four generations back in the pedigree.

Conclusions

Our results show that, in pedigreed breeding populations, the accuracy of genome-wide breeding values obtained by identity-by-descent relationships was not improved by identity-by-state information. Although, in principle, genomic selection based on identity-by-state does not require pedigree data, it does use the available pedigree structure. Our findings may explain why the prediction equations derived for one breed may not predict accurate genome-wide breeding values when applied to other breeds, since family structures differ among breeds.     

Linkage disequilibrium under skewed offspring distribution among individuals in a population

Correlations in coalescence times between two loci are derived under selectively neutral population models in which the offspring of an individual can number on the order of the population size. The correlations depend on the rates of recombination and random drift and are shown to be functions of the parameters controlling the size and frequency of these large reproduction events. Since a prediction of linkage disequilibrium can be written in terms of correlations in coalescence times, it follows that the prediction of linkage disequilibrium is a function not only of the rate of recombination but also of the reproduction parameters. Low linkage disequilibrium is predicted if the offspring of a single individual frequently replace almost the entire population. However, high linkage disequilibrium can be predicted if the offspring of a single individual replace an intermediate fraction of the population. In some cases the model reproduces the standard Wright-Fisher predictions. Contrary to common intuition, high linkage disequilibrium can be predicted despite frequent recombination, and low linkage disequilibrium under infrequent recombination. Simulations support the analytical results but show that the variance of linkage disequilibrium is very large.     

Size and Power of Test Statistics for Gene Correlation in 2 × 2 Contingency Tables

Monte Carlo simulation of size and power of two proposed tests for linkage disequilibrium between two genes each with two alleles were investigated. Results were compared with two commonly used statistics, the correlation coefficient r and the log-odds ratio tests. Depending on the sign of the linkage disequilibrium, the new tests were found to be more powerful than either of the correlation or log-odds ratio tests. However, on average (positive and negative linkage disequilibrium) the Chi-square test using the correlation coefficient was to a small extent more powerful than the other tests.     

SimPed: a simulation program to generate haplotype and genotype data for pedigree structures

With the widespread availability of SNP genotype data, there is great interest in analyzing pedigree haplotype data. Intermarker linkage disequilibrium for microsatellite markers is usually low due to their physical distance; however, for dense maps of SNP markers, there can be strong linkage disequilibrium between marker loci. Linkage analysis (parametric and nonparametric) and family-based association studies are currently being carried out using dense maps of SNP marker loci. Monte Carlo methods are often used for both linkage and association studies; however, to date there are no programs available which can generate haplotype and/or genotype data consisting of a large number of loci for pedigree structures. SimPed is a program that quickly generates haplotype and/or genotype data for pedigrees of virtually any size and complexity. Marker data either in linkage disequilibrium or equilibrium can be generated for greater than 20,000 diallelic or multiallelic marker loci. Haplotypes and/or genotypes are generated for pedigree structures using specified genetic map distances and haplotype and/or allele frequencies. The simulated data generated by SimPed is useful for a variety of purposes, including evaluating methods that estimate haplotype frequencies for pedigree data, evaluating type I error due to intermarker linkage disequilibrium and estimating empirical p values for linkage and family-based association studies.     

A fast method for computing high-significance disease association in large population-based studies

Because of rapid progress in genotyping techniques, many large-scale, genomewide disease-association studies are now under way. Typically, the disorders examined are multifactorial, and, therefore, researchers seeking association must consider interactions among loci and between loci and other factors. One of the challenges of large disease-association studies is obtaining accurate estimates of the significance of discovered associations. The linkage disequilibrium between SNPs makes the tests highly dependent, and dependency worsens when interactions are tested. The standard way of assigning significance (P value) is by a permutation test. Unfortunately, in large studies, it is prohibitively slow to compute low P values by this method. We present here a faster algorithm for accurately calculating low P values in case-control association studies. Unlike with several previous methods, we do not assume a specific distribution of the traits, given the genotypes. Our method is based on importance sampling and on accounting for the decay in linkage disequilibrium along the chromosome. The algorithm is dramatically faster than the standard permutation test. On data sets mimicking medium-to-large association studies, it speeds up computation by a factor of 5,000-100,000, sometimes reducing running times from years to minutes. Thus, our method significantly increases the problem-size range for which accurate, meaningful association results are attainable.     

The D' measure of overall gametic disequilibrium between pairs of multiallelic loci

Abstract The D ' coefficient is one of the most commonly used measures of the extent of gametic disequilibrium between multiallelic loci. It has been suggested that the range of the D ' measure of overall disequilibrium between pairs of multiallelic loci depends on allele frequencies, except under some very restricted conditions. Nevertheless, the problem of dependence of the range of D ' has not been characterized under a wide set of possible polymorphisms. Evaluation of the utility of D ' as a measure of the strength of overall disequilibrium between all possible pairs of alleles at two multiallelic loci requires better knowledge of its range than is currently available. In this work, the conditions of polymorphism under which the range of D ' is frequency independent are given. It is found that the range of D ' is more often independent of allelic frequencies than is commonly thought. Furthermore, the range of D ' undergoes only small fluctuations as a function of the polymorphisms at the loci. Numerical cases and microsatellite data from humans are used for illustration. These observations indicate that the D ' coefficient is a useful tool for the estimation and comparison of the extent of overall disequilibrium across pairs of multiallelic loci.