Linkage disequilibrium, often abbreviated to LD, is a term in population genetics referring to the association of genes, usually linked genes, in a population. It has become an important tool in medical genetics and other fields.
In defining LD, it is important first to distinguish two different concepts: linkage disequilibrium and linkage (genetic linkage). Linkage disequilibrium refers to the association of genes in a population. Linkage, on the other hand, tells us whether genes are on the same chromosome in an individual. There is no necessary relationship between the two. Genes that are closely linked may or may not be associated in populations. Looking at parents and offspring, if genes at closely linked loci are present together in the parent, then they will usually be found together in the offspring. But looking at individuals in a population with no known common ancestry, it is much more difficult to see any relationships.
To give a concrete, although imaginary, example in terms of frequencies of characters, consider a case where the "gene for red hair" is closely linked to the "gene for blue eyes". What does that tell us about the expected population frequency of individuals with red hair and blue eyes? Are all redheads expected to have blue eyes, just because the genes controlling these characters are closely linked?
Formal definition
Consider an allele A at the A locus with frequency pA in a particular population. At a linked B locus, the frequency of the allele B is pB. The question is, what is the expected frequency pAB of the allele pair, or haplotype, AB? (See note below about genetic nomenclature.)
If the A and B alleles are independent in a population, then, by definition, pAB is simply the product pApB. The difference between these two is given by D, the coefficient of linkage disequilbrium:
D = pAB - pApB
Departure of D from zero indicates LD.
Note on genetic nomenclature The descriptors "allele A at the A locus" and "allele B at the B locus" seem unnecessarily complicated. Why not just the "A gene" and the "B gene"? The problem is that the term "gene" has been used since the foundation of genetics without a clear understanding of what a gene actually is. So, despite its widespread popular usage, its use is now avoided in genetics journals (see for a discussion about the changing definition of the gene). This is unfortunate for discussions of population frequencies where the nature of the gene is not important.
Use of the term "allele" rather than "gene" sidesteps this problem, but in a way that is not entirely satisfactory. Allele was originally defined and still understood as meaning "alternative", and allele A and allele B are not alleles (of each other). The easiest way of talking about these linked "things" would be to use the term "gene".
Historical The expectation, dating back to 1918, is that LD is NOT to be expected, even for loci that are closely linked. Robbins showed that recombination is expected to decrease the value of D in each generation by a factor (1 - c), where c is the frequency of recombination.
If D between alleles at two loci at generation 0 is given the designation D0, then in the following generation :
D1 = D0 (1 - c)
and in generation t :
Dt = D0 (1 - c)t
If there is some recombination, c will be greater than zero, Dt approaches zero as t becomes large.
For the example given previously, no association is expected between hair colour and eye colour. The frequency of individuals who are both red-headed and blue-eyed is expected to become simply the product of the frequency of red-headed individuals multiplied by the frequency of blue-eyed individuals, despite the fact that the two characters are controlled by closely linked loci.
Selection provides one possible way in which LD might be expected, in spite of the above argument. If particular gene combinations are favoured, selection can cause LD to arise in the population, maintaining the frequency of the favoured gene combinations. Such "equilibrium models" require reasonably high levels of selection, specifically "selective interaction", which is only possible for a minority of gene pairs.
The term LD remains as a legacy from this period. It was introduced for cases where there is known recombination but where the population has not come to equilibrium for the gene pair in question. But the most prominent uses of LD now involve very closely linked DNA bases (see below). Independence cannot be expected in such cases. The 'disequilibrium' description seems inappropriate, with its implication that this situation is temporary and/or unexpected.
The molecular era The molecular era for population genetics can be said to date from 1966 following the studies of Lewontin and Hubby in Drosophila and Harris in humans. Using protein electrophoresis, these authors showed that around one third of loci must be 'polymorphic', having some genetic differences between individuals in the population. Given the large number of loci in the genome and the limited amount of recombination, it followed that there must be many very closely linked polymorphic loci.
Subsequent DNA sequencing, e.g. the International HapMap Project has shown that protein studies considerably underestimate the amount of polymorphism. There will usually be thousands of genetic differences, titled Single Nucleotide Polymorphism or SNPs, within short regions of the genome. Cases of zero or very low recombination must be common.
A second important finding pertaining to LD was the realisation that LD can arise simply because of population structure. Studies such as those of Robbins referred to above essentially assume an infinite population size. Small population size, in particular, can lead to LD quite independently of any selection. It became clear that LD, rather than being rare and of secondary importance, must be widespread.
This has had enormous importance in diverse fields of human genetics and animal breeding. It means that any gene of importance is likely to be surrounded by DNA SNPs in high LD with the gene of interest. The position of the gene may be unknown, but the position of all DNA SNPs is known exactly. This has allowed the mapping of causal genes in human genetics, using Genome-wide association studies (GWAS). It has allowed DNA 'breeding values' to be used as predictors, leading to advances in animal and plant breeding.
LD as a covariance or correlation of frequencies Haplotype frequencies can be set out in the form of a table with x and y columns. Allele A is given the value '1' and allele a the value '0' in the x column. Similarly for B in the y column. Gamete frequencies are of the form gi, summing to 1. +Haplotypex valuey valuefrequency (f)AB11g1Ab10g2aB01g3ab00g4 Then summing over the four classes:
Σfxy = 1.g1 + 0.g2 + 0.g3 + 0.g4 = g1
Σfx = g1 + g2 = pA
Σfy = g1 + g3 = pB
The covariance between x and y values is
Σfxy - Σfx Σfy = g1 - pA pB
which is equivalent to the LD coefficient, D, as defined above.
It is usually convenient to calculate the correlation rather than the covariance, normalising by the variances:
V(x) = Σfx2 - (Σfx)2 = pA - pA2 = pA ( 1 - pA )
V(y) = Σfy2 - (Σfy)2 = pB - pB2 = pB ( 1 - pB )
Substituting gives the correlation, which can be given the designation rAB, as:
or
This LD measure was introduced by Sewall Wright and its use popularised by Hill and Robertson.
LD for diploid frequencies The above LD theory is based on haploid frequencies. In practice, direct observation of such frequencies is rarely possible, since in most species of interest only diploid genotypes can be observed. Assumptions need to be made to infer haploid frequencies.
A different approach to estimating LD from diploid frequencies is to calculate the covariance and correlation of frequencies, just as for haploid frequencies. Gao et al show that the diploid covariance is the same as "Burrows' composite LD measure". The table below shows x and y values for diploid genotypes. It also shows the expected frequencies on the assumption of random mating. +Genotypex valuey valuefrequency (f)AABB11g12AABb11/22g1g2AAbb10g22AaBB1/212g1g3AaBb1/21/22g1g4+2g2g3Aabb1/202g2g4aaBB01g32aaBb01/22g3g4aabb00g42 Covariance and correlation calculations for these frequencies are as follows:
Σfxy = g12 + g1g2 + g1g3 + g1g4/2 + g2g3/2
Noting the alternative definition of D = g1g4 - g2g3, this simplifies to
Σfxy = g1 - D/2.
Σfx = g12 + 2g1g2 + g22 + g1g3 + g1g4 + g2g3 + g3g4
which simplifies, as in the haploid calculation, to
Σfx = g1 + g2 = pA
Similarly, Σfy = g1 + g3 = pB
The covariance between x and y values is
Σfxy - Σfx Σfy = g1 - D/2 - pA pB
which is simply D/2.
V(x) = Σfx2 - (Σfx)2 which can be shown to be pA ( 1 - pA )/2
V(y) = Σfy2 - (Σfy)2 = pB ( 1 - pB )/2
Normalising by the variances, the factor 2 cancels out. The diploid correlation which can be designated as RAB, has expectation:
Surprisingly, this is identical to the haploid LD correlation rAB. The result is, as mentioned above, an expectation based on the assumption of random mating. But this assumption can be relaxed.
If the deviation from random mating is expressed in terms of the inbreeding coefficient F, the expected frequency of AABB homozygotes is equal to (1-F)g12 + Fg1, the expected frequency of non-homozygotes such as AABb is equal to (1-F)g1g2 etc. Using these frequencies, the covariance and variance statistics simplify to:
Cov(x,y) = (1+F)D/2
V(x) = (1+F)pA(1-pA)/2 [equivalent to (pA(1-pA) + DA)/2, where DA is the A locus disequilibrium]
V(y) = (1+F)pB(1-pB)/2
Terms in (1+F) cancel, so that the diploid correlation still estimates the haploid correlation:
E(RAB) = rAB
More calculations involving D For two biallelic loci, where a and b are the other alleles at these two loci, the restrictions are so strong that only one value of D is sufficient to represent all linkage disequilibrium relationships between these alleles. In this case, . Their relationships can be characterized as follows.
The sign of D in this case is chosen arbitrarily. The magnitude of D is more important than the sign of D because the magnitude of D is representative of the degree of linkage disequilibrium. However, positive D value means that the gamete is more frequent than expected while negative means that the combination of these two alleles are less frequent than expected.
Linkage disequilibrium in asexual populations can be defined in a similar way in terms of population allele frequencies. Furthermore, it is also possible to define linkage disequilibrium among three or more alleles, however these higher-order associations are not commonly used in practice.
Normalization
The linkage disequilibrium reflects both changes in the intensity of the linkage correlation and changes in gene frequency. This poses an issue when comparing linkage disequilibrium between alleles with differing frequencies. Normalization of linkage disequilibrium allows these alleles to be compared more easily.
D' Method Lewontin suggested calculating the normalized linkage disequilibrium (also referred to as relative linkage disequilibrium) by dividing by the theoretical maximum difference between the observed and expected allele frequencies as follows:
where
The value of will be within the range . When , the loci are independent. When , the alleles are found less often than expected. When , the alleles are found more often than expected.
Note that may be used in place of when measuring how close two alleles are to linkage equilibrium.
r2 Method An alternative to is the correlation coefficient between pairs of loci, usually expressed as its square, ,
The value of will be within the range . When , there is no correlation between the pair. When , the correlation is either perfect positive or perfect negative according to the sign of .
d Method