Linkage Disequilibrium in Humans: Models and Data

Abstract

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.

Introduction

The extent and distribution of linkage disequilibrium (LD) in humans is a topic of great current interest. LD plays a fundamental role in gene mapping, both as a tool for fine mapping of complex disease genes (e.g., see Horikawa et al. ²⁰⁰⁰) and in proposed genomewide association studies (Risch and Merikangas 1996). LD is also of interest for what it can reveal about human history and human origins (e.g., see Tishkoff et al. ¹⁹⁹⁶), because the distribution of LD is determined, in part, by population history. Finally, studies of LD may enable us to learn more about the biology of recombination in humans. It is difficult to use pedigrees to estimate rates of homologous gene conversion, or variation in recombination rates over very short distances, because the events of interest occur at very low rates. However, studies of LD may offer insight (e.g., see Chakravarti et al. ¹⁹⁸⁴; Awadalla et al. ¹⁹⁹⁹; Przeworski and Wall ²⁰⁰¹).

In the present review article, we describe some predictions about the extent of LD, using simple models of population genetics. After summarizing the main empirical findings in humans, we turn to applications in association mapping and discuss the implications of the data collected thus far. We have not aimed to cover the literature exhaustively but, instead, have highlighted some issues that we think are of particular interest.

Models and Measures of LD

LD refers to the nonindependence of alleles at different sites. For example, suppose that allele A at locus 1 and allele B at locus 2 are at frequencies π_A and π_B, respectively, in the population. If the two loci are independent, then would we expect to see the AB haplotype at frequency π_Aπ_B. If the population frequency of the AB haplotype is either higher or lower than this—implying that particular alleles tend to be observed together—then the two loci are said to be in LD.

A wide variety of statistics have been proposed to measure the amount of LD, and these have different strengths, depending on the context. The measurement of LD is a large and complex topic and will not be reviewed in detail here; but see the work of Devlin and Risch (1995); Jorde (2000) and Hudson (2001). Most of the measures of LD that are in wide use quantify the degree of association between pairs of markers. In part, they differ according to the way in which they depend on the marginal allele frequencies. In the present article, we use one popular measure of LD between pairs of biallelic markers, commonly denoted by r² (elsewhere, r² is also denoted by Δ²). We also discuss a multilocus approach, based on an underlying population genetic model, that we feel has some advantages as a summary of the overall amount of LD in a region.

Consider two biallelic loci on the same chromosome, with alleles A and a at the first locus and with alleles B and b at the second locus, where the labeling is arbitrary. The allele frequencies will be written as π_A, π_a, π_B, and π_b, and the four haplotype frequencies will be written as π_AB, π_Ab, π_aB, and π_ab. Then,

In some of the figures, we plot , because this can make it easier to see the data points. For brevity, we will refer to as “r.”

In practice, one typically has a sample of m chromosomes from the population. Then, estimates of r² () are usually obtained by plugging the sample frequencies , , , etc., into equation (1).

Besides estimating the amount of disequilibrium between pairs of markers, it is also natural to test the null hypothesis of independence between marker pairs (i.e., linkage equilibrium). This can be done by a χ² test, and it turns out that, for biallelic markers, is the standard χ²-test statistic divided by the number of chromosomes in the sample (Weir 1996, p. 113). As shown later in the present article, r² also arises naturally in the context of association mapping.

The actual value of the disequilibrium coefficient r² (or any other measure of LD) between two loci is drawn from a probability distribution that results from the evolutionary process. This process can be described in terms of a population genetics tool called the “coalescent” (for reviews, see, e.g., Hudson ¹⁹⁹³; Nordborg ²⁰⁰¹). When we draw a sample of chromosomes from a population, all the chromosomes are related by some unknown ancestral genealogy, known as a “coalescent tree.” Genetic markers that are very close together on a chromosome have either the same or similar genealogies, and this induces dependence between the alleles at different markers. Markers that are farther apart may have different ancestral genealogies, because of recombination. For this reason, the strength of LD between pairs of markers decreases as a function of the genetic distance between markers.

The expected value of r² is a function of the parameter ρ≡4N_ec, where c is the recombination rate between the two markers and where N_e is the effective population size. For large ρ, E(r²)≈1/ρ (reviewed by Hudson ²⁰⁰¹). Below, we will show simulations of the distribution of r under various models.

Despite their convenience, the use of r² and other standard summaries of LD has some shortcomings. Each pair of loci produces an estimate, and it is not clear how to combine these in a sensible way, in part because they are not independent. More seriously, it is not straightforward to compare different regions. For instance, we might want to know whether a difference in the values of r² in two different regions is biologically meaningful. (There are two kinds of statistical significance in this context: first, we might want to know whether the amount of LD, appropriately defined, in one region is significantly more than that in another [e.g., for repeated sampling of loci from the same region]; second, we might want to know whether there is evidence that the underlying biological parameters—in particular, the recombination rate—differ either between regions or from predicted values. We use the latter sense in this article.) As we will argue below, for these and other reasons, it seems useful to supplement the standard pairwise summaries of LD by a model-based LD measure from population genetics.

As mentioned above, E(r²) is a function of ρ=4N_ec. It turns out to be a general result that ρ (also called “C” in the literature) is a key determinant of the extent of LD (Long and Langley 1999), with the strength of LD decreasing as ρ increases. Essentially, ρ is a scaled recombination rate, where the scaling constant 4N_e arises naturally from consideration of the genealogical process (e.g., see Nordborg ²⁰⁰¹; also see the Nordborg web site). For a given demographic model of population history, the extent and distribution of LD depends primarily on ρ, and we can readily simulate the distribution of any summary of LD (we also need to specify a mutation model). (Note a matter of notation: when we consider ρ=4N_ec for a region containing a series of markers, c is normally taken to refer to the total recombination rate across the entire region.)

The central role of ρ as a determinant of LD suggests that it is of interest to estimate this parameter from data. To do this, we need to assume an explicit model describing the population history and the processes of mutation and recombination. Then, for a given set of haplotype data, we can (in principle) compute the likelihood of the data as a function of ρ, and the mutation rate (Stephens 2001). Unfortunately, computing this likelihood is both technically challenging and computationally intensive, even for relatively small data sets, and methods for doing so are in their infancy. Existing approaches estimate the full likelihood by using either importance sampling (Griffiths and Marjoram 1996; also see the Paul Fearnhead web site) or Markov chain Monte Carlo (Kuhner et al. 2000; Nielsen ²⁰⁰⁰). To lessen the computational load, other methods simplify the structure of the data to approximate the shape of the likelihood function (Wall 2000; also see the Hudson Lab Home Page); confidence intervals can be obtained by simulation.

Despite the computational difficulties, it is now becoming possible to estimate ρ in cases of interest, and we report some results here. There are several potential advantages to using the model-based estimate as a summary statistic: (1) We can obtain a single number that summarizes the amount of LD in a region. (2) We can compare the amount of LD observed in studies with different marker spacing or from microsatellite and single-nucleotide polymorphism (SNP) studies. (3) The quantity is an estimate of the recombination rate per generation, c. The latter is of particular interest for studying variation in recombination rate at small scales; it also allows us to test the models used to predict patterns of LD.

Model Predictions

Population history can have a large effect on the distribution of LD. To illustrate this, we plot the decay of for a sample of 400 chromosomes, under four simple demographic models (see fig. 1). Each plot shows the results for one data set simulated from a neutral coalescent model with recombination (Hudson 1993). Points above the horizontal line are in significant LD at the 5% level. The first column in figure 1 shows realizations of the standard null model of a randomly mating population of constant size N_e=10⁴. (The estimate of N_e is obtained from observed diversity levels, where the mutation rate is assumed to be ≈2×10^-8/site/generation [Li and Sadler ¹⁹⁹¹; Przeworski et al. ²⁰⁰⁰].) As can be seen by comparison of the five replicates, considerable variability is expected across runs with the same underlying parameters, especially between tightly linked sites.

Figure 1.

Simulated decay of , as a function of genetic distance, for five independent replicates for each of four demographic models. Each plot shows the results for one sample of 400 chromosomes simulated under a particular demographic model of population history. On the X-axis is the genetic distance, in centimorgans, separating pairs of markers; at the genome average recombination rate of ≈1 cM/Mb, this is equivalent to distances in megabases. Each plot shows all pairwise comparisons for 40 biallelic markers, chosen randomly from the available markers whose minor-allele frequencies were ⩾.2. Points above the horizontal lines are in significant LD at the .05 level; for a sample size of 400, this corresponds to a value of r=.098 (see the “Models and Measures of LD” section). The first column shows results for a panmictic population of constant size N_e=10⁴; the second column shows results for the model of population growth considered by Kruglyak (1999) and described in the “Model Predictions” section; and the third and fourth columns show results for a simple model of population structure. In the third column, all individuals are drawn from the same subpopulation; in the last column they are drawn equally from both subpopulations. We used the symmetric two-island migration model (Wright 1951), with N_e=5,000 for each deme, and migration rates of one individual per deme per generation.

In the second column of figure 1, we plot the decay of for the model used by Kruglyak (1999). In this scenario, the effective population size increases exponentially, from 10⁴ to 5×10⁹, starting 5,000 generations ago. As pointed out by Kruglyak (1999), very little LD is expected under this model, and, in fact, few significant values of are observed beyond 10 kb. It should be noted, however, that, even in the absence of growth, LD is expected to be smaller in larger populations. In the Kruglyak model, it is assumed that N_e has always been ⩾10⁴, the value for the model of constant population size. If we want to know the effect of population growth per se, rather than that of a large population size, then we need to match the effective population sizes.

In figure 2, we plot the expected levels of for different growth models with comparable levels of diversity. Under a neutral model, diversity levels are determined by the mutation rate and the effective population size. Thus, matching the diversity levels is one way to match effective population sizes. Here, we fix the current effective population size (taken to be 10⁵) and the time at which population growth started; we then pick the growth rate that yields the same average number of segregating sites (for 100 chromosomes) as does the model of constant population size with N_e=10⁴. Once population sizes are matched in this way, population growth still leads to a reduction in LD, but the effect is smaller than that in the Kruglyak model.

Figure 2.

Decay of expected value of , as a function of genetic distance, for different growth models (sample size of 100 chromosomes). From top to bottom, the four models are as follows: constant population size of N_e=10⁴ (unbroken line); population-growth onset 500 generations ago—that is, 10⁴ years ago, under the assumption that there are 20 years per generation (dotted line); population-growth onset 5,000 generations ago (short-dashed line); and the model used by Kruglyak (1999) (see the “Model Predictions” section) (long-dashed line). In two growth models (dotted line and short-dashed line), the current population was fixed at 10⁵. Except in the Kruglyak model, the parameters were chosen to match the amount of genetic diversity seen in humans (see text). In each of 10⁴ simulations, 10 SNPs with a minor-allele frequency ⩾.2 were chosen.

It should be noted that, in addition to diversity levels and levels of LD, other aspects of the data (e.g., the frequency spectrum of segregating sites) can be used to gauge the plausibility of particular demographic models. Population growth leads to an excess of low-frequency variants relative to a model of constant population size (Tajima 1989). Under the model used by Kruglyak, >80% of all minor alleles would be found only once in a sample of 400 chromosomes (results not shown). Such a pronounced skew in the frequency spectrum is not seen in actual data. Indeed, the frequency spectrum at synonymous sites (as detected by variant-detection arrays; Cargill et al. ¹⁹⁹⁹) indicates little skew from the predictions of a model of constant population size. Although the size of the human population clearly has increased—at least over recent time—there are not yet enough data to allow us to know which growth model will be appropriate (Wall and Przeworski 2000).

In contrast to population growth, population structure tends to increase levels of LD. It can even lead to significant associations between unlinked markers; however, we still expect the strongest LD between tightly linked markers. An example of this can be seen in the third and fourth columns of figure 1, where we present the results for a model with two subpopulations, corresponding to a level of population differentiation of F_ST≈.2 (Wright 1951; Hartl and Clark ¹⁹⁹⁷).

In the third column of figure 1, all individuals are sampled from one subpopulation, whereas, in the fourth column of figure 1, they are drawn equally from both. Both situations lead to increased LD, and, particularly in the latter case, there is strong LD across the entire distance plotted.

Empirical Patterns

Long-Distance LD

There are now more than a dozen studies that characterize the extent and range of background LD over large distances (e.g., see Peterson et al. ¹⁹⁹⁵; Laan and Pääbo ¹⁹⁹⁷; Huttley et al. ¹⁹⁹⁹; Dunning et al. ²⁰⁰⁰; Eaves et al. ²⁰⁰⁰; Gordon et al. ²⁰⁰⁰; Taillon-Miller et al. ²⁰⁰⁰; Wilson and Goldstein ²⁰⁰⁰; Zavattari et al. ²⁰⁰⁰; Abecasis et al. ²⁰⁰¹; Service et al. ²⁰⁰¹). Although a few recent studies consider SNPs (Dunning et al. 2000; Taillon-Miller et al. ²⁰⁰⁰; Abecasis et al. ²⁰⁰¹), most data sets consist of microsatellites. The populations sampled are usually either European or of European descent (e.g., the Amish or Afrikaners). Many are thought to be either relatively isolated or to have experienced rapid growth from a small number of founders (e.g., Finns and Sardinians). Sample sizes vary from 50 to several hundred haplotypes.

Several of these studies find that LD extends over very long distances. For example, Peterson et al. (1995), Laan and Pääbo (1997), Huttley et al. (1999), Gordon et al. (2000), Wilson and Goldstein (2000), and Service et al. (2001) all observe significant associations between pairs of microsatellites separated by ⩾1 cM. Significant LD extends out to 2 cM in Ashkenazim and Bantus, in an X-chromosome survey conducted by Wilson and Goldstein (2000). In a sample of Afrikaners, Gordon et al. (2000) find two pairs of markers in significant LD (after a Bonferroni correction for multiple tests), despite the markers being separated by >3 cM.

Huttley et al. (1999) describe results from a genomewide search for LD in the microsatellite data of the Centre d'Étude du Polymorphisme Humain (CEPH). Those authors used the LD level observed in the major histocompatibility (MHC) region as a benchmark for “high” levels of LD. (The MHC region shows extreme levels of LD, which are thought to reflect the action of natural selection.) By this criterion, eight regions, each spanning several centimorgans, show “excess LD” in a sample of CEPH individuals.

One of the SNP studies (Taillon-Miller et al. ²⁰⁰⁰) also finds long-range LD in regions Xq25 and Xq28, with highly significant associations between several pairs of markers separated by >500 kb. However, SNP data from four autosomal regions (from Dunning et al. [2000] and Abecasis et al. [2001]) seem to show a different pattern, with a more rapid decay of LD with distance (fig. 3).

Figure 3.

Plots of , as a function of physical distance (in kb), for SNP data from five regions (Dunning et al. 2000; Taillon-Miller et al. ²⁰⁰⁰; Abecasis et al. ²⁰⁰¹). On each plot, points above the unbroken line are in significant LD at the .05 level, and points above the dotted line correspond to what Kruglyak (1999) has called “useful LD”; these lines are set at r=.316, the equivalent of r²=.1.

A visual inspection of figure 1 suggests that, under the standard models, the finding that LD extends over multiple centimorgans is surprising, at least in the absence of substantial population structure. Ideally, one would like to compare the patterns of LD found by different studies more formally and, also, to test the predictions of alternative models. However, this is not straightforward. First, the power to detect LD may be higher for markers with many alleles (e.g., microsatellites) than for biallelic markers (Slatkin 1994). Second, the results from different studies are reported in terms of a variety of measures of LD. Most common among these are D′ (Lewontin 1964), r², and P values from pairwise significance tests of LD. As illustrated in figure 4, D′ and r² behave very differently, and high values of D′ may not be inconsistent with low values of r². In particular, there seems to be much more random variation in values of D′ at a given recombination distance.

Figure 4.

Simulated decay of (+) and (♦), as a function of genetic distance. These data correspond to one realization from figure 1 (first column, second row), simulated under a model of constant population size (i.e., N_e=10⁴) and random mating.

The studies also differ in the way in which haplotypes are obtained. In studies of X-linked regions, the haplotypes can be determined in males. For autosomal data, the haplotypes are sometimes reconstructed on the basis of pedigrees. Otherwise, they are usually estimated by statistical methods, with unknown effects on the accuracy of inferences about LD (especially if the populations show pronounced departures from Hardy-Weinberg equilibrium, as found by Dunning et al. [2000]). Significance tests of LD can also be performed on the genotype data directly, at the cost of some loss in power.

Another complication in comparing the data from different studies is that some report only physical distances (as plotted in fig. 4). These are not necessarily comparable, since it is known that recombination rates vary widely across the genome (Payseur and Nachman 2000; Yu et al. ²⁰⁰¹). Moreover, in comparisons of LD on the X chromosome and the autosomes, it should be noted that higher levels of LD are expected on the X chromosome. In fact, we expect ρ to be halved, since (1) N_e is 3/4 of the value for autosomes (when a sex ratio of 1:1 and no sexual selection are assumed) and (2) recombination between X chromosomes occurs only in females.

As discussed above, one approach to quantifying the extent of LD in a region is to estimate ρ=4N_ec; this estimate, , can be thought of as providing a summary of the amount of LD. We can usually obtain external estimates of N_e (on the basis of diversity data) and of c (on the basis of genetic maps), and this allows us to predict what ρ “should” be. To conclude that there is more LD than what would be predicted under the model, we need to show that is less than the predicted value of ρ.

Estimates of ρ depend on the demographic model. The choice of model influences both the average rate of decay of LD (fig. 2) and the amount of chance variation in the distribution of LD across independent realizations of the evolutionary process (e.g., compare the second and fourth columns of fig. 1). As noted above, there is considerable uncertainty in the choice of an appropriate model for human populations. Here, we report estimates of ρ for a model of constant population size. Since we suspect that there may be an excess of LD, this choice is conservative: more LD is expected with constant population size than is expected in the presence of population growth (fig. 2; Slatkin ¹⁹⁹⁴).

Estimates of ρ are available for three SNP regions (see the first, second, and fifth plots in fig. 3). For the data reported by Taillon-Miller et al. (2000), the maximum-likelihood estimate of ρ is five times lower than would be expected if c=1 cM/Mb and N_e=7,500 (for the X chromosome), lending support to the qualitative view that there is “excess” LD in this region (see the Hudson Lab Home Page). Alternative physical maps yield divergent estimates of the recombination rate for this region (see the data reported by Payseur and Nachman ²⁰⁰⁰; also see the Human Recombination Rates web site); the rate may be as high as 3 cM/Mb, which would make the discrepancy 15-fold.

In contrast, for the two regions with SNP data reported by Abecasis et al. (2001), the plot of decay of (see fig. 3) appears more like the plots for constant N_e (see the first column of fig. 1). Estimates of ρ were obtained by a pair of similar methods (see the Hudson Lab Home Page and the Gil McVean web site). Because of computational limitations, the analysis was performed on subsets of 50 of the available haplotypes. For N_e=10⁴, the maximum likelihood estimate of c is 1.8 cM/Mb for the region on chromosome 2, and 2.5 cM/Mb for the region on chromosome 14; estimates of c, from comparison of a genetic map and a physical map, are ≈1 and ≈2.5 cM/Mb, respectively (as estimated by Payseur and Nachman [2000]). There are considerable uncertainties in all of these estimates; nonetheless, a model of constant population size roughly predicts the decay of pairwise LD found in the data of Abecasis et al. (2001). We do not have estimates of ρ for the data reported by Dunning et al. (2000), but the plots of decay of (see fig. 3) also appear to be consistent with either a model of constant population size or a model with some population growth.

The parameter ρ has also been estimated for data from a microsatellite study (Laan and Pääbo 1997) that, in the Saami, found strong pairwise LD extending over several centimorgans. The maximum-likelihood estimate of ρ, obtained under a stepwise mutation model, is ∼100 times smaller than that expected on the basis of the genetic distance between markers if N_e=10⁴ (see the Paul Fearnhead web site); use of a geometric mutation model yields similar results (see Pritchard et al. ¹⁹⁹⁹). (Again, because of computational limitations, estimates of ρ were obtained from a series of subsets of the markers.) One explanation for the high LD is simply that the Saami have a tiny effective population size (i.e., N_e≈10²). However, the levels of genetic diversity at the same microsatellite loci are similar in the Saami and in neighboring populations that have much lower LD, arguing against the hypothesis of very small N_e. An alternative hypothesis is that recent admixture or gene flow into the Saami is responsible for these observations.

There seems to be mounting evidence that, for some regions, there is more LD than would be expected under simple demographic models. Not all data sets, however, show such departures from the model predictions. For some of the reports of excess LD, there is independent knowledge of demographic departures from model assumptions—for example, of recent admixture between diverged populations of the Lemba (Wilson and Goldstein 2000) or of admixture and inbreeding in a Costa Rican population (Service et al. 2001); for other examples, there is no obvious explanation (e.g., see Laan and Pääbo ¹⁹⁹⁷; Taillon-Miller et al. ²⁰⁰⁰).

Short-Scale LD

A poor fit of certain regions and/or populations is perhaps not surprising, given that the models are simplifications of the history of human populations. In that light, it is interesting to note that, at short physical distances, there does not appear to be an excess of LD. In fact, polymorphism data from short (i.e., <10 kb) scales appear to show less LD than would be expected if N_e=10⁴ and c=1 cM/Mb. For example, for data on Lpl (Clark et al. 1998), β-globin (Harding et al. 1997), and Dmd44 (Nachman and Crowell 2000), estimates of ρ are all an order of magnitude higher than would be expected under either a model of constant population size or a simple model with population growth (Przeworski and Wall 2001). Recent analysis of SNPs in a series of sequence-tagged-site regions reached a similar conclusion (K. Ardlie and L. Kruglyak, personal communication).

In summary, we have several examples in which large regions exhibit more LD than would be expected under either a model of constant population size or a model with rapid population growth. Yet, at the same time, studies of polymorphism at a small scale reveal less LD than would be expected. These observations at different scales are hard to accommodate in a single explanation, since factors that increase long-distance LD will tend to have an even larger effect on closely linked sites (e.g., see the third and fourth columns of fig. 1). In what follows, we will discuss potentially salient biological features not included in the simple models described thus far.

Demographic Departures from Model Assumptions

LD can be inflated by demographic factors, including inbreeding, population structure, and bottlenecks. The potential importance of inbreeding is highlighted by a recent study that examined genotypes of eight reference families from the CEPH (Broman and Weber 1999). That study found homozygous chromosomal segments of >10 cM in several families, a much larger distance than would be expected with random mating (Clark 1999). As pointed out by the study's authors, these long identical tracts could have resulted from mating of related individuals.

The effect of inbreeding among relatives is to lower diversity levels and to increase LD (i.e., decrease ρ). In this context, it is known that very high rates of inbreeding can have a greater impact on LD than on diversity (as seen in highly selfing plants [Nordborg ²⁰⁰⁰]). However, this effect is likely to be minor in human populations, in which, by comparison, the rates of inbreeding are modest.

The recent admixture of populations with different allele frequencies is a known factor in the generation of LD. At the time of admixture, there can be LD even between unlinked sites (e.g., see Pritchard and Rosenberg ¹⁹⁹⁹). However, this decays very rapidly: with random mating, LD breaks down at a rate of (1-c) per generation (Hartl and Clark 1997). The net result is that in recently admixed populations, there can be substantial LD over centimorgan distances (e.g., see Wilson and Goldstein ²⁰⁰⁰; Service et al. ²⁰⁰¹).

Note that extensive LD is also seen in populations that appear relatively homogeneous. Low rates of gene flow from more divergent populations could introduce haplotypes that increase LD (as shown in the third column of fig. 1). The importance of this effect is unknown.

The extent of LD may also be increased by temporary reductions in population size (“bottlenecks”). Under some models of changing population size, N_e is given by the harmonic mean of the population size over time (Hartl and Clark 1997). Since the harmonic mean is very sensitive to the smallest terms, severe or long-term bottlenecks can lead to sharp reductions in N_e. Several authors have appealed to a population bottleneck in the history of non-African populations as an explanation for what appear to be higher LD levels outside Africa (e.g., see Tishkoff et al. ¹⁹⁹⁶; Kidd et al. ¹⁹⁹⁸). However, there is still considerable uncertainty about the appropriate model for the evolution of modern humans. In particular, it remains to be tested whether a model with a bottleneck accompanying an emigration from Africa ∼100,000 years ago is consistent with the data on LD, as well as with levels of genetic diversity and with the frequency spectrum.

When recent demographic effects (e.g., the founding events in the Finns or Sardinians) are being considered, it is worth keeping in mind that most surveys of SNPs focus on markers where both alleles are common. These high-frequency variants tend to be older than random polymorphisms and therefore reflect events that occurred farther in the past. Thus, unless they are very severe, recent demographic events may have little impact on the extent and distribution of LD among high-frequency SNPs (see Kruglyak ¹⁹⁹⁹). This might explain why levels of LD appear roughly similar across European populations that probably share most of their evolutionary history (Dunning et al. 2000; Eaves et al. ²⁰⁰⁰; Taillon-Miller et al. ²⁰⁰⁰).

Natural Selection

Significant allelic associations over large genetic distances may also result from the action of natural selection (as noted by Peterson et al. ¹⁹⁹⁵; Huttley et al. ¹⁹⁹⁹; Abecasis et al. ²⁰⁰¹)—for example, from balancing selection acting in the MHC region. This said, studies of polymorphism in Drosophila and humans suggest that long-lived balancing selection is rare (Hey 1999; Przeworski et al. ²⁰⁰⁰). A second mode of selection known to generate LD is epistasis (nonadditive interactions between sites [Hartl and Clark ¹⁹⁹⁷; Kelly and Wade ²⁰⁰⁰]). However, epistatic selection would have to be very strong to maintain allelic associations at the scale of megabases, in the face of substantial recombination. The same is true for “selective sweeps,” in which a rare, favorable mutation is quickly swept to fixation. It is unknown to what extent this mode of selection increases pairwise LD between high-frequency alleles. However, selective sweeps affect sites over a genetic distance on the order of the selection coefficient (Kaplan et al. 1989), so, for a single sweep to affect ⩾1 Mb, the selective advantage of the variant would have to be large (i.e., at least ≈.01).

The Relationship between Physical Distance and Genetic Distance

Estimates of the recombination rate at a centimorgan scale can be obtained by comparison of the genetic and physical maps. Although these are not yet very precise, they indicate that recombination rates vary by an order of magnitude across the human genome (Payseur and Nachman 2000; Yu et al. ²⁰⁰¹). Regions of several megabases have rates <0.3 cM/Mb, whereas others have rates >3 cM/Mb. As would be expected, these “jungles and deserts” of recombination correspond to blocks of low and high linkage disequilibrium, respectively (Yu et al. 2001). Thus, comparison of LD levels in different regions may not be meaningful unless the local recombination rates are taken into account.

Recombination rates may also vary at a scale that is not detected by markers separated by megabases (see Yu et al. ²⁰⁰¹). Dramatic changes in recombination rate have been reported over distances as short as a few kilobases (e.g., see Chakravarti et al. ¹⁹⁸⁴). The molecular basis of hotspots for recombination remains unknown, but contributing factors might include high GC content (Eisenbarth et al. 2000; Yu et al. ²⁰⁰¹) or whether the region is transcribed (Nicolas 1998). Thus, genetic maps may provide limited information about rates at short scales; more-refined estimates can be obtained by means of single-sperm typing (e.g., see Lien et al. ²⁰⁰⁰).

One known aspect of recombination not taken into account by most models is homologous gene conversion (here, gene conversion and crossing-over are thought of as alternative outcomes of a common recombination mechanism). For markers a megabase or so apart, the contribution of gene conversion to the overall level of genetic exchange is negligible (Andolfatto and Nordborg 1998). As a result, genetic map–based estimates of the recombination rate are essentially estimates of the crossing-over rates alone. The latter should accurately predict the extent of pairwise LD between polymorphisms far apart (given an adequate demographic model). For closely linked markers, however, LD may also be broken up by gene conversion. Indeed, in Drosophila and yeast, it appears that the rate of initiation of recombination events resolved as gene conversions and as crossovers is similar. Thus, pedigree-based estimates may substantially underestimate the total rate of recombination at small scales.

Currently, very little is known about gene conversion in humans. However, as noted above, there appears to be less LD at small scales than would be expected from estimates of crossing-over rates and observed levels of diversity. An intriguing explanation for this pattern is that gene conversion is quite frequent in humans. In support of this, Przeworski and Wall (2001) show that the data that they have analyzed are more likely under a model in which two-thirds of recombination events are gene-conversion events than under a model of crossing-over alone. An analysis of 10 anonymous intergenic regions also finds evidence for extensive gene conversion (A. Di Rienzo, L. Frisse, and R. R. Hudson, personal communication).

The relationship between LD and distance might also be shaped by the segregation of inversions. Inversion polymorphisms are thought to suppress recombination in heterozygotes throughout much of the length of the inverted segment (Roberts 1976; Martin ¹⁹⁹⁹), although the precise details are unknown. Thus, the presence of inversion polymorphisms in a given region will reduce the rate of recombination. For instance, if 50% of the individuals used to construct a genetic map are heterozygous for an inversion, and if there is no recombination in heterozygotes, within the inverted region, then the average recombination rate in the region will be halved.

Moreover, inversion polymorphism can potentially have a second, much stronger effect on the extent of LD. Because recombination between the standard and inverted types is rare or absent, strong LD can develop between the two kinds of chromosomes. In the extreme case where there is complete suppression of recombination, a mutation within the inverted region arises on one type of chromosome and cannot move to the other via recombination. As mutations accumulate on both genetic backgrounds, the two arrangements diverge, potentially leading to a buildup of substantial LD. The extent of the effect will depend on the history and frequency of the inversion—including what kind of natural selection, if any, is acting on the inversion (see Andolfatto et al. ²⁰⁰¹).

Little is known about the number, size, and frequency of inversions in the human genome. In particular, inversions shorter than a few megabases are currently difficult to detect by cytological methods but, if they were to reach intermediate frequencies, could have a substantial impact on the extent of LD. There are now a number of findings of common inversion polymorphisms: several studies of disease-associated inversions have reported inversion-heterozygote frequencies of 21%–33% in controls (Small et al. 1997; Saunier et al. ²⁰⁰⁰; Giglio et al. ²⁰⁰¹). The length of the inverted segments varies from ∼50 kb (Small et al. 1997) to 3 Mb (J. Weber, personal communication), long enough for these rearrangements to potentially contribute to islands of extensive LD.

It appears that many of these chromosome rearrangements are mediated by nonhomologous meiotic exchange between inverted repeats (Small et al. 1997; Saunier et al. ²⁰⁰⁰; Giglio et al. ²⁰⁰¹; Tilford et al. ²⁰⁰¹). Since much of the human genome consists of repetitive DNA, it is possible that chromosomal rearrangements resulting in inversion polymorphisms are fairly common.

Implications for LD Mapping

The recent interest in LD in humans is due in large part to the prospect of large-scale association studies to locate complex disease genes. Risch and Merikangas (1996) showed that, under ideal circumstances, the power to detect disease mutations of small effect is much greater with association mapping than it is with linkage analysis. On the basis of this result, they argued that the future of complex-disease genetics lies in the use of genomewide screens of association.

However, the practical aspects of genomewide association mapping are currently daunting. It is clear that the number of markers that will be needed to scan the genome for association is very large. Recent progress has expanded the set of available SNPs across the genome (International SNP Map Working Group 2001), but the costs of genotyping a large sample of cases and controls at sufficient marker density would still be extremely high (although dropping). Clearly, we need good estimates of the marker density that will be required to achieve acceptable power in these studies.

The required density will depend on which statistical tests are used to detect association. Currently, it is most common to test for association at each marker in turn (or, sometimes, by combining pairs of nearby markers). In what follows, we consider the case in which only one marker is used, while noting that there seems to be a great need for the development of multilocus tests of association that make use of haplotype information, since these might prove to be much more efficient (see the Nordborg web site).

Density of Markers

Suppose that we test for association at a marker locus that is near a disease-susceptibility mutation. It can be shown that, in order to achieve roughly the same power at the marker locus as we would have if we could test the disease mutation itself, we need to increase the sample size by a factor of 1/r², where r² is the coefficient of LD between the marker and the disease mutation (see the sidebar; also see Kruglyak ¹⁹⁹⁹). Hence, for small values of r², there is little power to detect association at the marker locus.

r² and the Power of Association Studies.

Several recent articles have referred to the connection between various measures of LD and the power of association studies (Kruglyak 1999; Dunning et al. ²⁰⁰⁰; Abecasis et al. ²⁰⁰¹). Here we clarify this relationship.

Suppose that we genotype a case-control sample of N₁ individuals at locus 1, a (true) disease-susceptibility locus, and that we genotype N₂ individuals at locus 2, a nearby marker locus. We want to compare the power of association tests at these two loci.

Assume that both loci are biallelic, with alleles A and a at locus 1 and with alleles B and b at locus 2. Let π_DA and π_CA be the frequencies of allele A in individuals with the disease and in controls, respectively, and let π_DB and π_CB be the analogous frequencies at locus 2. Let q_AB (respectively, q_aB) be the probability that a chromosome with allele A (respectively, allele a) at locus 1 has the B allele at locus 2. Then,

The standard χ² test statistic of association at locus 1 (call this “X²₁”) can be written as

where , , and are the sample frequencies of A in affected individuals, in controls, and in the overall sample, respectively, and where φ and 1-φ are the fractions of the sample that are cases and controls, respectively. The test statistic for association at locus 2 (X²₂) is similar—but with B in place of A and with N₂ in place of N₁. The distributions of X²₁ and X²₂ are approximately the squares of normal random variables, with means

and

where

and similarly for , and where the variances are ≈1 if the difference, in frequency of A, between cases and controls is small. Then, since

it follows that the distributions of X²₁ and X²₂ are approximately the same if N₂=N₁/r². In other words, to achieve (approximately) the same power at the marker locus as is achieved at the susceptibility locus, the sample size must be increased by a factor of 1/r².

In a highly influential paper, Kruglyak ran simulations of the coalescent with recombination to predict the rate of decay of LD; he predicted that “a useful level of LD is unlikely to extend beyond an average distance of roughly 3 kb in the general population” (Kruglyak 1999, p. 139). None of his models predicted “useful” LD over >30 kb. His criterion for useful LD was that the sample size necessary to detect association at the marker should not be increased more than 10-fold (in our terms, this corresponds to r²⩾.1). (The formal criterion used by Kruglyak (1999) is slightly different from ours: d²⩾.1.) The predictions of more-realistic models of growth are less drastic (see fig. 2).

As data on LD among SNPs become available, we can start to get an empirical sense of the rate of decay of r². Figure 3 shows plots of for SNPs in five regions. If we make the assumption that the distribution of r² between two random SNPs is the same as that between a SNP and a disease mutation, then we can use plots such as those in figure 3 to study the decay of r² directly (see Dunning et al. ²⁰⁰⁰; Taillon-Miller et al. ²⁰⁰⁰; Abecasis et al. ²⁰⁰¹). In practice, it seems likely that disease mutations at polymorphic frequencies will usually be deleterious, which would have the helpful effect of increasing the average LD with nearby sites (Pritchard 2001 [in this issue]).

In figure 3, points above the dotted lines meet Kruglyak’s criterion of useful LD. Even at very short distances, many pairs of markers are in low LD (except, possibly, in the data reported by Taillon-Miller et al. (2000)). Thus, even if a disease mutation is very close to the nearest marker, there may be a substantial probability of failing to detect association at that marker.

In a genome-scan for association, there would be a series of markers spaced along the chromosome. For a given location (and frequency) of the disease mutation, there is some probability that each of the nearby markers is in useful LD with the disease mutation; this probability drops as a function of the distance between marker and mutation. In effect, the goal is to choose the marker density in such a way that there is a high probability that the mutation will be in strong LD with at least one of the markers. This density might differ substantially from what would be suggested by consideration of average r² values.

At this time, it seems premature to provide a formal estimate of the required marker density. However, suppose that the first four plots in figure 3 are typical of LD across the genome. Then the genome scans will need to have numerous (perhaps 5–10 or more) markers within, say, 50 kb of each disease mutation, to ensure a high probability that at least one marker is in strong LD with the mutation.

In the discussion so far, we have implicitly assumed that the value of r² between the disease mutation and each nearby marker is independent. This is not true: each recombination event in the ancestral genealogy can affect multiple markers. This correlation effect will further increase the required density of markers, although the size of this effect is unknown.

Ultimately, the marker map will also need to reflect variation in the extent of LD across regions, placing more markers in some regions than in others. As discussed above, some of the reasons for variation may be predictable—for example, variation in recombination rate and the predicted difference between the X chromosome and the autosomes. Some may be due to factors that we do not yet understand, whereas some variation will result from the inherent randomness of the coalescent process.

In closing, we should mention one situation in which LD is expected to extend over quite long distances—namely, in admixed populations such as African Americans (Pfaff et al. 2001). It has been proposed that, for diseases that differ in frequency between the parental groups, the resulting “admixture disequilibrium” could enable LD mapping using relatively small numbers of markers, spaced centimorgans apart (Chakraborty and Weiss 1988; McKeigue ¹⁹⁹⁷).

Fine-Scale Mapping

So far, we have discussed the issue of testing for association, but LD also plays a central role in the problem of gene localization. These two goals are rather different: in the first, the aim is either to identify chromosomal regions that contain disease-susceptibility loci (as in the genome-scan model) or to confirm regions with weak evidence from pedigree studies; in fine-scale mapping, one already has strong evidence that a region of interest contains a disease locus, and the goal is to use a series of markers to estimate its location (e.g., see McPeek and Strahs ¹⁹⁹⁹; Horikawa et al. ²⁰⁰⁰; Morris et al. ²⁰⁰⁰).

The extent and distribution of LD affect these two kinds of goals in different ways. When one is testing for association, it is helpful if LD extends over long distances around the disease mutation, because then not so many markers are needed to scan for association; but at the later stage, when the goal is to infer location, long-ranging LD is potentially problematic. It means that strong associations might be observed far from the causative site(s), and these could lead to effort being spent on the wrong location.

The possibility of rampant gene conversion is worrying for the use of LD in fine-scale mapping. The presence of gene conversion increases the recombination rate at very-short scales (Andolfatto and Nordborg 1998), which is helpful. However, it may also make the relationship between LD and distance less predictable, since gene conversion affects only short regions, leaving flanking haplotypes unchanged. Currently, little is known about the extent to which the spatial distribution of LD matches model predictions, such as those used in fine-mapping (e.g., see McPeek and Strahs ¹⁹⁹⁹). For now at least, studies at the fine-mapping stage should take great care to map the entire extent of the region of association, in case the region contains local peaks of association far from the causative site(s).

Future Directions

We have aimed to place the new empirical data on LD in the context of various population genetics models. One point that emerges clearly is that, to understand LD in humans, we need to have a much better understanding of human demography. This includes the history of changes in population size, as well as population structure and other forms of nonrandom mating. We also need to have more data on recombination rates across the genome. It is known that recombination rates vary greatly, and accurate data are needed to allow comparisons of LD from different regions. When the time comes to construct marker sets for association mapping, we will need accurate and fine-scale recombination maps, so that the density of markers in each region reflects local rates. As discussed above, little is known about either the rate of gene conversion or the frequency of submicroscopic inversion polymorphisms, but both have the potential to play an important role in determining the distribution of LD.

At this time, our conclusions about LD between SNPs are somewhat limited by the amount of data available. Ideally, one would want a large number of SNPs from many regions, to enable studies of differences among regions. The markers should be placed so that there are data on LD both over short distances (i.e., <5 kb) and over long distances (i.e., ⩾1 Mb). In terms of experimental design, there is probably little to be gained from the genotyping of huge numbers of individuals at each SNP; instead it makes more sense to genotype many SNPs.

Our review has concentrated primarily on the extent of LD, as measured by pairwise measures (mainly r²). This emphasis reflects that of the majority of the literature. However, we feel strongly that, for many problems, it will be important to think about the spatial structure of LD in more detail. For example, how much local variation in LD is there, as one moves through a small region? In LD mapping, how often will there be isolated peaks of association far from the causative site? What is the probability that a disease mutation sits in a local trough of low LD, making it virtually undetectable for any plausible marker density? In other contexts, the use of complete haplotype data may also offer major advantages over pairwise measures—for instance, in the estimation of ρ; in making more efficient use of marker data for detecting associations, particularly if there is allelic heterogeneity (Pritchard 2001 [in this issue]); and in fine-structure mapping (e.g., see McPeek and Strahs ¹⁹⁹⁹).

Note added in proof.—

A recent article by Reich et al. (2001) reports extensive LD among SNPs at 19 loci. They report more LD than would be expected under a model of stepwise population growth (and, also, more than would be expected with a constant population size of 10,000 [D. Reich, personal communication]). As a possible explanation, Reich et al. suggest that there may have been a recent bottleneck in non-African populations.