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Proc Natl Acad Sci U S A. Aug 16, 2005; 102(33): 11835–11839.
Published online Aug 9, 2005. doi:  10.1073/pnas.0505262102
PMCID: PMC1188000
Medical Sciences

A map of the human genome in linkage disequilibrium units

Abstract

Two genetic maps with additive distances contribute information about recombination patterns, recombinogenic sequences, and discovery of genes affecting a particular phenotype. Recombination is measured in morgans (w) over a single generation in a linkage map but may cover thousands of generations in a linkage disequilibrium (LD) map measured in LD units (LDU). We used a subset of single nucleotide polymorphisms from the HapMap Project to create a genome-wide map in LDU. Recombination accounts for 96.8% of the LDU variance in chromosome arms and 92.4% in their deciles. However, deeper analysis shows that LDU/w, an estimate of the effective bottleneck time (t), is significantly variable among chromosome arms because (i) the linkage map is approximated from the Haldane function, then adjusted toward the Kosambi function that is more accurate but still exaggerates w for all chromosomes, especially shorter ones; (ii) the nonpseudoautosomal region of the X chromosome is subject to hemizygous selection; and (iii) at resolution less than ≈40,000 markers per w, there are indeterminacies (holes) in the LD map reflecting intervals of very high recombination. Selection and stochastic variation in small regions must have effects, which remain to be investigated by comparisons among populations. These considerations suggest an optimal strategy to eliminate holes quickly, greatly enhance the resolution of sex-specific linkage maps, and maximize the gain in association mapping by using LD maps.

Keywords: effective bottleneck time, HapMap, recombination, interference, selection

Alinkage map with 14,759 polymorphic markers has recently been published, far exceeding the density and coverage of earlier maps (1). Although the human genome sequence facilitated construction of this genetic map by determining the physical order of its markers, 56% of its intervals have recombination rates of zero. This observation demonstrates the relatively low resolution of the linkage map, for which a costly solution would be the analysis of many more pedigrees and markers. Alternatively, sperm typing offers recombination data of the highest resolution currently available (2) but is feasible only for very small regions in male chromosomes and does not reflect historical recombination events. Using the HapMap data (3) to construct linkage disequilibrium (LD) maps measured in LD units (LDU) (4) and interpolating them into the sex-specific linkage map enhances the resolution of the linkage map as a by-product of our main objective, which is to construct genome-wide LD maps with additive LDU distances. Such maps are applicable to association mapping (5, 6), population comparisons (79), and identification of genomic regions that are influenced by selection (10). LD mapping has begun the task of explaining and exploiting complexities (8) that do not affect application of the Malecot model (4) to create an LD map but determine how the map should be used to increase resolution of the linkage map.

Materials and Methods

Genotypic Data for LD Map Construction. The HapMap data (www.hapmap.org) were obtained on 60 parental DNA samples from Utah Mormons of northwestern European ancestry collected by the Centre d'Etude du Polymorphisme Humain. These samples are a subset of 270 in the database that includes 3 other populations. A total of 665,335 single nucleotide polymorphism (SNP) genotypes were downloaded from the September 2004 public release of the HapMap data. A little more than one-quarter of the downloaded SNPs (25.8%, 171,927) were removed by a screening procedure that rejected 6,795 SNPs with χ21 > 10 for the Hardy–Weinberg test (11) or a minor allele frequency <5% (165,132), leaving 493,408 SNPs for LD map construction. These SNPs are a subset of the ≈1 million SNPs included in the HapMap release after our analysis was completed. Both samples take their nucleotide position from the July 2003 Golden Path database (http://genome.ucsc.edu). The number of SNPs is expected to reach 3 million next year, with a subsequent increase likely in an updated physical map.

LD Map Construction. LD maps were constructed by the methods in the next paragraph for 23 chromosomes, 1–22 and X, covering 98% (2,790 Mb) of the euchromatin. Each SNP has an LD location, and distance between adjacent SNPs in the LD map was constrained to a maximum of 3 LDU. Such intervals are called holes (12), constituting 0.6% of the total map intervals, 17.8% of the total LDU length, and 2.2% of the total physical length. Ignoring stochastic variation and selection in the LD map and errors in estimating the linkage map in morgans (w), the Malecot model predicts that the ratio of corresponding distances in LD and linkage maps estimates t, the number of generations over which recombination has accumulated after one or more population bottlenecks (8). On these assumptions, t would be constant between chromosomal arms and their deciles. To test this hypothesis, the initial unit of analysis, neglecting acrocentric short arms, centromeres, and pseudoautosomal regions, was the length of a chromosome arm between the first and last physical locations shared by the two genetic maps (linkage and LD). For the X chromosome, the two pseudoautosomal regions were removed and the residual linkage map length in females was multiplied by 2/3 to allow for the absence of crossing over in males. ldmap was used for autosomes and the female X chromosome to calculate pairwise association probabilities ρ and information Kρ under the null hypothesis that ρ = 0, so that ρ2Kρ = χ21 (1315). Haplotype frequencies for pairs of loci were directly observed for male X chromosomes, and the association probabilities and information were determined from these counts. Association probabilities were then obtained as the weighted mean of sex-specific estimates ρm and ρf, and the information was taken as the sum of the weights Kρm and Kρf. Default parameters were used to construct LD maps by estimating the length of an interval between adjacent SNPs (maximal window of adjacent intervals = 100, maximum distance between any SNP pair = 500 kb, segment size = 500 markers with a 25-marker overlap in adjacent segments, and overlap distance is averaged). These defaults give rapid construction of a good LD map, with the flanking intervals contributing efficiently to each interval estimate.

The predicted value of association ρ between two SNPs at a distance Σdi kb is ρ = (1 - L)Me-Σεidi + L, where di is the kb length of the ith included interval between adjacent SNPs and Σεidi is the corresponding LDU distance (4). On certain assumptions, Σεidi has expectation wt (8). However, w and t are not used when constructing the LD map. The model is fitted from the composite likelihood equation M1, where equation M2 is an estimate with prediction ρ and the summation is over n pairs of SNPs used for LD analysis within a given window containing r SNPs. The asymptote L was predicted for each segment as the weighted mean deviation for a normal distribution (4).

Statistical Analysis of LD Maps. LD maps were analyzed by chromosome arms between the first and last physical location shared by the linkage and LD map, omitting heterochromatic, centromeric, and pseudoautosomal regions. This procedure excludes 3,709 SNPs distal to the linkage map, comprising 48.1 Mb of the LD map span and leaving 489,699 SNPs covering 2,790 Mb to be analyzed. Correlations were examined to suggest models for stepwise linear regression. Inferences from chromosome arms were confirmed by partitioning each physical map into deciles. Assuming that the variance of LDU is proportional to w, we calculated effective bottleneck time in generations (t) and its variance (Vt). Weighting each of s arms or deciles by length on the linkage map in w, with length LDU on the LD map, these estimates are

equation M3
[1]

equation M4
[2]

For both arms and deciles, the smallest values of Vt were obtained when using the most recent linkage map (1), compared with earlier linkage maps constructed from deCODE data by using a multipoint approach (16) and a composite likelihood method (17). We confirmed this choice by verifying that it gave the smallest reduction in error variance when independent parameters were introduced with LDU/w as a dependent variable that is weighted by w. The independent variables for all maps were h/w, Mb/w, m/w, and their squares, where h and m are counts of holes and markers, respectively. We verified that forward and backward stepwise regression gave the same model. When linear and quadratic terms were both significant for a particular variable, we accepted an exponential function if it gave a better fit.

Results

Recombination Dominates Patterns of LD. Regressing LDU on w through the origin and weighting by w gives the same slope t = ΣLDU/Σw = 1,435 for the 41 chromosome arms and their 410 deciles. This result is the ratio of their capabilities to resolve causal from predictive markers in association mapping. On simple assumptions, it is also the number of generations over which recombination has accumulated in the LD map during a succession of bottlenecks in population size. Deviations from this regression account for only 3.2% of the variance of LDU/w for arms and 7.6% for deciles, including random errors in both LDU and w and significant effects of other variables (Fig. 1). Assuming a maximum of 25 years per generation, the effective bottleneck time is no more than 25t = 35,875 years for this population, diminished if generation time were reduced. This finding is less than half the time since the out-of-Africa bottleneck (≈100,000 years), reflecting subsequent population bottlenecks and justifying the term effective bottleneck time as an analogue of effective population size. Recovery of diversity after a bottleneck can be rapid if determined by migration into a local population but is extremely slow if dominated by mutation within a species (18).

Fig. 1.
The relationship in the human genome between LDU and linkage in w in chromosome arms (red) and deciles (blue).

Unexpected Variation in Estimates of Effective Bottleneck Time. Although recombination accounts for agreement with the Malecot prediction of uniform t, deviations may be significant. We therefore examine a range of variables, including LDU, w, markers (m), megabases (Mb), and holes (h), for deciles and performed stepwise regression with LDU/w as the dependent variable weighted by w. Chromosome arm size (Mb/w), marker density (m/w), and hole density (h/w) are all significant predictors and were next studied separately.

Chiasma Interference. The first analysis examined the relationship between t as LDU/w and arm length as Mb/w. Computer programs in common use for constructing linkage maps assume no chiasma interference for the computation of multilocus likelihoods and may then convert the resulting map to approximate conformity with the Kosambi function, which is not multipoint feasible. Studies of chiasma interference in humans have shown that it exists at greater levels than the Kosambi function provides and appears to vary among chromosomes (1921). The estimate of t is significantly lower for smaller autosomes (Fig. 2), supporting evidence that chiasma interference becomes more intense with decreasing chromosome size, as in the mouse (21).

Fig. 2.
A graph showing variation among chromosome arms for the ratio of the LD map in LDU/w to the physical map in Mb/w.

Natural Selection. The X chromosome is exceptional in having unusually low LDU/w (high LD), despite correcting for the absence of recombination in males and is therefore excluded from the weighted exponential fit. This finding is consistent with more rapid selection against deleterious mutations when the X is monosomic in males, and to a lesser extent, under random inactivation (Lyonization) in females (2223). The smaller effective population size Ne for the X chromosome than for autosomes is not a factor, because Ne affects the M parameter of the Malecot equation but not the LDU length Σεidi (15).

Incomplete Data. Some values of εidi are indeterminate and are assigned a maximum value of 3 LDU, which may become as little as 2.5 in subsequent iterations. These holes reflect segments with elevated recombination and/or insufficient SNP density in steps where holes occur. This uncertainty will not be resolved until the density of SNPs within holes increases, which would also increase the power of association mapping. Here we take εidi > 2.5 as defining a hole. There is a significant relationship between density of holes (h/w) and markers (m/w). Fig. 3 suggests that the number of holes will decline with progress of the HapMap Project and would decline more rapidly if SNP selection focused on the 3,144 holes in the LD map or if holes were defined on a cosmopolitan map that averages data for two or more populations (7). However, the factors that determine holes are complex, dominated by recombination but also including SNP distribution, kb width of holes, the criteria to declare a hole, and errors in estimation of εi. The number of holes in future databases cannot be estimated reliably but is likely to decrease to a nonzero limit as LD maps evolve.

Fig. 3.
The declining density of holes with marker density among chromosome arms.

High Resolution Sex-Specific Linkage Maps. We see that both linkage and LDU maps are more complex than their simple models, despite nearly a century of development for the former and nearly three years for the latter. Nevertheless, the low resolution of the sex-specific linkage maps can be greatly increased by interpolation from the LD map. To motivate this increase, we examined LDU/Mb and cM/Mb (cM, centimorgans) in 2-Mb sliding windows for chromosome 19 (Fig. 4) comparing the high-resolution LD map with the low-resolution linkage map. At this resolution, blocks and steps in the LD map are not visible, but major peaks and troughs in recombination rate (cM/Mb) and strength of association (LDU/Mb) are apparent and show high concordance between LD and linkage maps. This agreement remains but to a lesser extent as window size is reduced. Such graphs illustrate variation within chromosomes and the degree of sharing between maps at arbitrary resolution but do not provide either an LD map or linkage map. The q arm was previously examined in a similar way by a coalescent method, with less detail and conspicuous distortion near the centromere (24).

Fig. 4.
A graph of chromosome 19 for LDU/Mb (blue) and centimorgans (cM)/Mb (red) against the physical scale in Mb.

The sex of ancestors in whom recombination took place is unknown for LD maps, but this fact does not imply that LD maps cannot be used to create sex-specific linkage maps at high resolution. All that is required is to conserve framework loci from the appropriate linkage map, interpolating locations from the LD map between adjacent framework loci. Because the same physical map was used for linkage and LD files, all linkage markers not assigned to the same base pair in the LD file were interpolated from the physical map and used as framework loci if the distance to the preceding framework locus on the LD map was zero (i.e., in the same block) or if the distance in the linkage map was nonzero (i.e., recombination had been detected and, therefore, not in the same block). In this way, the sex-specific linkage framework was maintained during interpolation from the high-resolution LD map. Fig. 5 illustrates the profound sex differences in recombination, with males accounting for most crossovers near the telomeres and females responsible for most recombination near the centromere. The ratio cM/Mb is proportional to the Malecot parameter ε = Σεididi, the rate of change of the LDU map with respect to the physical map. The LDU length Σεidi is much smoother and increases monotonically (Fig. 6), but it identifies the same peaks. Coalescent methods conceal these differences by interpolating between the pair of most distant markers shared between the sex-averaged linkage map and the coalescent construct, which is neither a linkage map nor an LD map (24). Applied to LD for a particular chromosome in different populations, coalescence adjusts all of them to the sex-averaged linkage map and therefore to the same length determined by misrepresentation of interference, leaving only selected sequences as a memento of LD differences.

Fig. 5.
A graph of chromosome 19 for centimorgans (cM)/Mb in males (blue) and females (red) against the physical scale in Mb.
Fig. 6.
A graph of chromosome 19 for centimorgans (cM) in males (blue) and females (red) against the physical map in Mb.

Discussion

Progress in human genomics has been so rapid that terminology has not kept up. Cloning is no longer necessary to localize a gene, and so the term positional cloning is currently being replaced by association mapping. PubMed lists many papers on linkage disequilibrium mapping, meaning that some aspect of LD is being used. We argue that an LD map should not refer to the annotation of genomic sequence with LD blocks and other features, but most usefully defines a map with additive distances that describes the pattern of LD. It is not sufficient to construct a high-resolution recombination map, because association mapping requires characterization of LD patterns that reflect duration together with the effects of drift, selection, and mutation. An LD map provides a tool for localizing genetic effects that takes the same role for association mapping at high resolution that the linkage map provides for low-resolution mapping. However, an LD map is not merely a scaled linkage map, but it is a logically different entity with its own story to tell of recombination, selection, population history, and gene expression.

In contrast, the linkage map is uniquely able to identify and use sex-specific recombination patterns. Although the properties of linkage and LD maps are different, their relation is, at present, one-sided. Because of its much greater resolution, the LD map can be usefully interpolated into nonzero intervals of sex-specific linkage maps. On the contrary, current linkage maps have nothing to contribute to LD maps, which have no chiasma interference because multiple recombination within small regions takes place in different generations, giving a composite likelihood that provides a benchmark for association mapping (5, 6). The best sex-averaged linkage map should be most nearly proportional to an LD map, and conversely, without trying to impose one entity on the other. Evidence on selection and population history is fully retained, with no incentive to incorporate information from a linkage map at a much lower resolution. Linkage and LD are separate but complementary, with their different evidence as unconfounded, as is consistent with the goal of enhancing resolution of the linkage map without degrading the LD map. It remains to be determined whether one linkage map will suffice for different populations and whether a single composite LD map will be efficient for association mapping in samples with different Malecot parameters (7).

Mathematical geneticists frequently warn about problems absent from mutation models but central to coalescent theory of recombination. “The full likelihood model will be very difficult to specify without unrealistically stringent assumptions about population history” (25). “The lineages we follow never recombine with each other (the probability of such an event is vanishingly small). They always recombine with the (infinitely many) nonancestral chromosomes” to generate a tree that bifurcates with no junctions (26). We may agree that such a model is only “slightly nonintuitive” if applied to speciation over many millions of years, leaving few surviving haplotypes. However, the assumption cannot be so easily swallowed that each lineage always traces in a much shorter time to a single ancestral haplotype within a restricted population of a single species that lacks an alternative population to provide “infinitely many nonancestral chromosomes.” The assumption that each lineage always traces in estimable time to a single ancestral haplotype (“most recent common ancestor”) that can be reliably inferred has other flaws: the rarest allele for each polymorphism in a haploset has a finite probability of being the oldest one, and so the probability that m polymorphisms all trace to a single ancestral haplotype tends to zero as m increases, and the most recent common ancestor and its associated time vary greatly along a chromosome (27).

Inevitably these theoretical arguments have practical corollaries. In the Malecot model, the effective size is the harmonic mean over generations and affects the M parameter but not LDU, coalescent time is irrelevant, bottleneck time is estimable, a founder haplotype is not inferred, the LD map estimate is direct, degrees of freedom are specified, neighboring intervals are not smoothed for either linkage or LD, and the population is not assumed to be at equilibrium. In contrast, a coalescent model assumes constant effective size N with proportional time, does not recognize bottlenecks, assumes a unique founder genotype, cannot directly estimate LD map length, has unspecified degrees of freedom, arbitrarily smoothes adjacent intervals, and assumes equilibrium. Despite their logical differences, the two coalescent maps currently available can be scaled to fairly good agreement with LD maps of the same small regions (24, 28). A definitive comparison cannot be made until the coalescent model is applied to the whole genome. Association mapping poses a greater problem, because coalescence assigns the most recent common ancestor to a common haplotype composed of markers with high minor allele frequencies (MAF), excluding markers with a smaller MAF that may be predictive or causal for association in an LD map. Experience with the two approaches will determine the most powerful (8), but coalescent models are heavily handicapped.

Competition between LDU maps and arbitrarily scaled substitutes is part of a larger development of analytic genomics that will enrich the colored diagrams that conventionally represent genomes (29) by location databases (http://cedar.genetics.soton.ac.uk/public_html) in which each point on the physical map is associated with a vector of locations on other maps including, but not restricted to, LDU in multiple populations, sex-specific linkage maps, chromosome bands, and isochores. Only such location databases can provide the composite likelihoods on which efficient tests of significance and support intervals are based for association mapping by linkage and LD and for identification of sequences that are recombinogenic or subject to regional selection.

Acknowledgments

We thank the Utah participants in the HapMap Project and all of the many research groups cited by the HapMap Consortium that were involved in producing the data and making them publicly available as a community resource project. This work was supported by research grants from the U.K. Biotechnology and Biological Sciences Research Council and the National Institutes of Health. The data used here are a subset of one sample from release 13 of the International HapMap Consortium (3).

Notes

Author contributions: W.T., A.C., and N.E.M. designed research; W.T. performed research; W.T., J.G., N.M., and S.E. analyzed data; and W.T., A.C., and N.E.M. wrote the paper.

Abbreviations: LD, linkage disequilibrium; LDU, linkage disequilibrium unit; w, morgans; cM, centimorgans.

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