In population genetics, linkage disequilibrium (LD) is a measure of non-random association between segments of DNA (alleles) at different positions on the chromosome (loci) in a given population based on a comparison between the frequency at which two alleles are detected together at the same loci and the frequencies at which each allele is detected at that loci overall, whether it occurs with or without the other allele of interest.
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Loci are said to be in linkage disequilibrium when the frequency of being detected together (the frequency of association of their different alleles) is higher or lower than expected if the loci were independent and associated randomly.
Linkage disequilibrium is influenced by many factors, including selection, the rate of genetic recombination, mutation rate, genetic drift, the system of mating, population structure, and genetic linkage. As a result, the pattern of linkage disequilibrium in a genome is a powerful signal of the population genetic processes that are structuring it.
In spite of its name, linkage disequilibrium may exist between alleles at different loci without any genetic linkage between them and independently of whether or not allele frequencies are in equilibrium (not changing with time). Furthermore, linkage disequilibrium is sometimes referred to as gametic phase disequilibrium; however, the concept also applies to asexual organisms and therefore does not depend on the presence of gametes.
Suppose that among the gametes that are formed in a sexually reproducing population, allele A occurs with frequency at one locus (i.e. is the proportion of gametes with A at that locus), while at a different locus allele B occurs with frequency . Similarly, let be the frequency with which both A and B occur together in the same gamete (i.e. is the frequency of the AB haplotype).
The association between the alleles A and B can be regarded as completely random—which is known in statistics as independence—when the occurrence of one does not affect the occurrence of the other, in which case the probability that both A and B occur together is given by the product of the probabilities. There is said to be a linkage disequilibrium between the two alleles whenever differs from for any reason.
The level of linkage disequilibrium between A and B can be quantified by the coefficient of linkage disequilibrium , which is defined as
provided that both and are greater than zero. Linkage disequilibrium corresponds to . In the case we have and the alleles A and B are said to be in linkage equilibrium. The subscript "AB" on emphasizes that linkage disequilibrium is a property of the pair of alleles and not of their respective loci. Other pairs of alleles at those same two loci may have different coefficients of linkage disequilibrium.
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.
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.
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
Note that may be used in place of when measuring how close two alleles are to linkage equilibrium.
An alternative to is the correlation coefficient between pairs of loci, usually expressed as its square, .
Another alternative normalizes by the product of two of the four allele frequencies when the two frequencies represent alleles from the same locus. This allows comparison of asymmetry between a pair of loci. This is often used in case-control studies where is the locus containing a disease allele.
Similar to the d method, this alternative normalizes by the product of two of the four allele frequencies when the two frequencies represent alleles from different loci.
The measures and have limits to their ranges and do not range over all values of zero to one for all pairs of loci. The maximum of depends on the allele frequencies at the two loci being compared and can only range fully from zero to one where either the allele frequencies at both loci are equal, where , or when the allele frequencies have the relationship when . While can always take a maximum value of 1, its minimum value for two loci is equal to for those loci.
Consider the haplotypes for two loci A and B with two alleles each—a two-loci, two-allele model. Then the following table defines the frequencies of each combination:
Haplotype | Frequency |
Note that these are relative frequencies. One can use the above frequencies to determine the frequency of each of the alleles:
Allele | Frequency |
If the two loci and the alleles are independent from each other, then we would expect the frequency of each haplotype to be equal to the product of the frequencies of its corresponding alleles (e.g. ).
The deviation of the observed frequency of a haplotype from the expected is a quantity called the linkage disequilibrium and is commonly denoted by a capital D:
Thus, if the loci were inherited independently, then , so , and there is linkage equilibrium. However, if the observed frequency of haplotype were higher than what would be expected based on the individual frequencies of and then , so , and there is positive linkage disequilibrium. Conversely, if the observed frequency were lower, then , , and there is negative linkage disequilibrium.
The following table illustrates the relationship between the haplotype frequencies and allele frequencies and D.
Total | |||
Total |
In the absence of evolutionary forces other than random mating, Mendelian segregation, random chromosomal assortment, and chromosomal crossover (i.e. in the absence of natural selection, inbreeding, and genetic drift), the linkage disequilibrium measure converges to zero along the time axis at a rate depending on the magnitude of the recombination rate between the two loci.
Using the notation above, , we can demonstrate this convergence to zero as follows. In the next generation, , the frequency of the haplotype , becomes
This follows because a fraction of the haplotypes in the offspring have not recombined, and are thus copies of a random haplotype in their parents. A fraction of those are . A fraction have recombined these two loci. If the parents result from random mating, the probability of the copy at locus having allele is and the probability of the copy at locus having allele is , and as these copies are initially in the two different gametes that formed the diploid genotype, these are independent events so that the probabilities can be multiplied.
This formula can be rewritten as
so that
where at the -th generation is designated as . Thus we have
If , then so that converges to zero.
If at some time we observe linkage disequilibrium, it will disappear in the future due to recombination. However, the smaller the distance between the two loci, the smaller will be the rate of convergence of to zero.
A comparison of different measures of LD is provided by Devlin & Risch
The International HapMap Project enables the study of LD in human populations online. The Ensembl project integrates HapMap data with other genetic information from dbSNP.
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