Lecture 5 of the Molecular Genomics series. The measure that makes marker-based genomics possible.
Recombination breaks up associations between loci, and it does so slowly when the loci are close together. Linkage disequilibrium is the name for what is left: the association that has not yet been broken. It is the single most important concept in this course, because it is what allows a marker that is not causal to carry information about a variant that is.
Linkage disequilibrium (LD) is the non-random association of alleles at two loci within a population. It measures the degree to which knowing an animal's allele at one locus tells you something about its allele at the other.
Two points about the name are worth making immediately, because both mislead.
First, LD is a property of a population, not of an individual or of a chromosome. It is a statement about allele frequencies across many gametes.
Second, LD is not the same thing as linkage. Linkage is physical proximity on a chromosome, and it is a property of the genome that does not change. LD is a statistical association, and it can exist between unlinked loci and be absent between tightly linked ones. Physical proximity makes LD persist, because recombination erodes it slowly, but proximity does not create it. LD is created by population history: mutation, drift, migration, admixture, selection and bottlenecks. The size of the blocks of associated alleles reflects both recombination and those historical events.
The reference state is what Mendel's second law describes. Under independent assortment, alleles at two loci are sorted into gametes independently of one another, so every combination occurs at the frequency expected from the separate allele frequencies.
Take two loci, the first with alleles \(A\) and \(a\) at frequencies \(p_A\) and \(p_a\), the second with alleles \(B\) and \(b\) at frequencies \(p_B\) and \(p_b\). There are four possible haplotypes. Under independence,
A population in which these equalities hold is in linkage equilibrium. Any departure from them is linkage disequilibrium. All the standard measures are ways of quantifying the size of that departure.
The most direct measure is simply the difference between the observed haplotype frequency and the frequency expected under independence.
Because the four haplotype frequencies must sum to one and the marginal allele frequencies are fixed, the same \(D\) appears, with alternating sign, in all four cells:
An equivalent and often more convenient form follows by substitution:
which is the determinant of the two-by-two table of haplotype frequencies. It is zero exactly when the rows are proportional, that is, when the loci are independent.
In a sample of gametes the four haplotypes occur with frequencies
The allele frequencies are \(p_A = 0.45+0.15 = 0.60\) and \(p_B = 0.45+0.15 = 0.60\).
Under independence we would expect \(p_Ap_B = 0.36\), but we observe 0.45, so
Checking with the determinant form: \(0.45\times 0.25 - 0.15\times 0.15 = 0.1125-0.0225=0.09\). The \(AB\) and \(ab\) haplotypes are in excess, and \(Ab\) and \(aB\) are correspondingly rare.
Both problems are about scale, which is what the next two measures address.
The first repair is to divide \(D\) by the largest value it could have taken, given the allele frequencies.
This forces \(D'\) into the range \(-1\) to \(+1\), and \(|D'| = 1\) has a clean interpretation: at least one of the four haplotypes is absent from the population.
The last point is serious. In livestock datasets with modest sample sizes and many low-frequency markers, \(D'\) close to one is common and often means nothing.
There is a more useful way to think about the whole problem. Define, for a randomly drawn gamete, an indicator variable at each locus: \(X = 1\) if the gamete carries \(A\) and 0 otherwise, and \(Y = 1\) if it carries \(B\) and 0 otherwise. Then LD is just the statistical association between two random variables, and the natural measure is their correlation.
The measure used in practice is the square of this correlation.
The last property is the one that makes \(r^{2}\) indispensable, and it has a concrete consequence for study design. If a causal variant is not genotyped but a marker in LD with it is, then the sample size required to detect the association through the marker is inflated by a factor of \(1/r^{2}\) relative to testing the causal variant directly. An \(r^{2}\) of 0.5 doubles the number of animals needed. This is the argument that sets marker density.
Using the haplotype frequencies from before, \(D=0.09\), \(p_A=p_B=0.60\) and \(p_a=p_b=0.40\).
Now compare with \(D'\). Since \(D>0\), \(D_{\max}=\min(p_Ap_b,\,p_ap_B)=\min(0.24,0.24)=0.24\), so \(D'=0.09/0.24=0.375\).
The two measures give quite different impressions of the same data: \(D'=0.375\) sounds moderate, \(r^{2}=0.141\) says that only 14% of the variation at one locus is captured by the other. The second is the number that matters for whether the marker is a usable proxy.
Because recombination separates loci at a rate that increases with the distance between them, \(r^{2}\) declines as physical distance grows. Plotting average \(r^{2}\) against distance gives the LD decay curve, which is the standard way of summarising the LD structure of a population.
The level of the curve, not only its shape, is informative. A population with a small effective population size has high LD extending over long distances, because drift generates associations faster than recombination removes them. A large, outbred population has low LD that decays quickly. This is the basis for estimating effective population size from marker data, and it is also the practical reason that the marker density needed to cover a genome differs between commercial breeds and indigenous populations.
What counts as "useful" LD is a matter of convention rather than biology. A threshold such as \(r^{2}>0.2\) or \(r^{2}>0.3\) is commonly adopted, and the distance at which the average curve crosses it is then quoted as the distance over which markers are informative. That distance translates directly into the number of markers needed.
Almost every remaining lecture in this course rests on LD.
| Application | How LD enters |
|---|---|
| Association studies | LD is what allows an ungenotyped causal variant to be detected through a genotyped marker, and what determines how wide the implicated region is |
| Genomic selection | Prediction works only if every part of the genome is in LD with at least one marker; this sets the required density |
| Effective population size | The level of LD at a given genetic distance reflects \(N_e\) some number of generations ago, so the decay curve is a record of population history |
| Selection signatures | A sweep drags a whole haplotype to high frequency, producing a local excess of LD, which is one of the signals used to detect it |
| Evolutionary inference | Admixture, bottlenecks and migration each leave a characteristic imprint on the LD structure |
Computing all three measures. In a population the haplotype frequencies at two SNPs are \(p_{AB}=0.30\), \(p_{Ab}=0.10\), \(p_{aB}=0.20\), \(p_{ab}=0.40\). Compute \(D\), \(D'\) and \(r^{2}\), and say which you would report.
Allele frequencies: \(p_A = 0.30+0.10 = 0.40\), so \(p_a=0.60\); \(p_B = 0.30+0.20 = 0.50\), so \(p_b=0.50\).
D. \(D = p_{AB}-p_Ap_B = 0.30 - 0.40\times0.50 = 0.30-0.20 = 0.10\). Check: \(0.30\times0.40 - 0.10\times0.20 = 0.12-0.02=0.10\).
D'. Since \(D>0\), \(D_{\max}=\min(p_Ap_b,\,p_ap_B)=\min(0.20,\,0.30)=0.20\), so \(D' = 0.10/0.20 = 0.50\).
r-squared.
Report \(r^{2}\). It is the measure with a direct interpretation, namely that one marker explains about 17% of the variation at the other, and it is the quantity that determines the sample size penalty for using this marker as a proxy. \(D'=0.50\) would give a misleadingly favourable impression.
Why D-prime misleads. Construct a numerical example with two loci where \(D' = 1\) but \(r^{2}\) is small, and explain what feature of the data produces the discrepancy.
Take \(p_{AB}=0.05\), \(p_{Ab}=0\), \(p_{aB}=0.45\), \(p_{ab}=0.50\). Then \(p_A=0.05\), \(p_a=0.95\), \(p_B=0.50\), \(p_b=0.50\).
\(D = 0.05 - 0.05\times0.50 = 0.025\). Since \(D>0\), \(D_{\max}=\min(p_Ap_b,\,p_ap_B)=\min(0.025,\,0.475)=0.025\), so \(D'=1\) exactly.
But
\(D'=1\) because the haplotype \(Ab\) is absent, which is all \(D'\) requires. The feature producing the discrepancy is the rare allele: \(A\) has frequency 0.05, so the \(Ab\) haplotype would be rare even under independence, and its absence is weak evidence.
Substantively, the two markers are nearly useless as proxies for one another. Knowing an animal's genotype at the second locus says almost nothing about the first, because 95% of gametes carry \(a\) regardless. This is exactly the situation in which \(D'\) is inflated, and it is common in real data.
Sample size and marker density. A GWAS would need 800 animals to detect a causal variant if that variant were genotyped directly. It is not genotyped, but a marker with \(r^{2}=0.25\) to it is. How many animals are needed now? What density of markers would you need so that a typical causal variant has \(r^{2}\ge 0.8\) with some marker, given an LD decay curve that falls below 0.8 at 20 kb?
The required sample size scales as \(1/r^{2}\), so \(800/0.25 = 3{,}200\) animals. Using a proxy with \(r^{2}=0.25\) quadruples the cost of the study, which is why LD structure, not statistics alone, drives study design.
For the density: if \(r^{2}\) falls below 0.8 beyond 20 kb, then a causal variant is adequately tagged only if a marker lies within about 20 kb of it. Markers spaced 40 kb apart place every position within 20 kb of a marker, so on a 3,000 Mb genome that is \(3{,}000{,}000/40 = 75{,}000\) markers, and spacing at 20 kb to be safe gives 150,000.
Two caveats. This treats markers as evenly spaced and LD as uniform, and neither is true: real chips have gaps and LD is patchy, so the effective coverage is worse than the arithmetic suggests. And the calculation is population-specific, since the 20 kb figure comes from one decay curve. A population with faster decay needs proportionally more markers, which is precisely the difficulty with applying commercial chip densities to indigenous African populations.
Interpreting a decay curve. Two cattle populations are compared. Population 1 has mean \(r^{2}=0.30\) at 100 kb, population 2 has \(r^{2}=0.10\) at the same distance. What does this suggest about their histories, and which will give higher genomic prediction accuracy at a fixed marker density and reference size?
LD is generated by drift and eroded by recombination, so at a fixed genetic distance the equilibrium level of \(r^{2}\) is inversely related to effective population size. Population 1's higher LD indicates a smaller \(N_e\), consistent with a recent bottleneck, intense selection through few sires, or a founder event. Population 2's lower LD indicates a larger, more outbred population, which is typical of indigenous populations that have not been through a commercial breeding programme.
At fixed marker density and reference size, population 1 will give higher prediction accuracy. Prediction relies on markers being in LD with causal variants, and in population 1 each marker tags a longer stretch of chromosome. The same chip therefore covers the genome more effectively.
This is worth stating plainly because it is counterintuitive and it has a policy implication: genomic selection transfers less readily to diverse indigenous populations than to commercial breeds, not because the animals are less well studied but because low LD is a direct consequence of the genetic diversity that makes those populations valuable. Higher marker density or sequence data is the technical answer, and it costs more precisely where budgets are smallest.
