Linkage disequilibrium

In population genetics, linkage disequilibrium (LD) is the non-random association of alleles at different loci in a given population. Loci are said to be in linkage disequilibrium when the frequency of association of their different alleles is higher or lower than what would be 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.
Formal definition.
Suppose that among the gametes that are formed in a sexually reproducing population, allele "A" occurs with frequency formula_1 at one locus (i.e. formula_1 is the proportion of gametes with "A" at that locus), while at a different locus allele "B" occurs with frequency formula_3. Similarly, let formula_4 be the frequency with which both "A" and "B" occur together in the same gamete (i.e. formula_4 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 formula_6 of the probabilities. There is said to be a linkage disequilibrium between the two alleles whenever formula_4 differs from formula_8 for any reason.
The level of linkage disequilibrium between "A" and "B" can be quantified by the "coefficient of linkage disequilibrium" formula_9, which is defined as
provided that both formula_11 and formula_12 are greater than zero.
Linkage disequilibrium corresponds to formula_13. In the case formula_14 we have formula_15 and the alleles "A" and "B" are said to be in "linkage equilibrium". The subscript "AB" on formula_16 emphasizes that linkage disequilibrium is a property of the pair {"A", "B"} 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, formula_17. Their relationships can be characterized as follows.
formula_18
formula_19
formula_20
formula_21
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.
Measures derived from formula_22.
The coefficient of linkage disequilibrium formula_23 is not always a convenient measure of linkage disequilibrium because its range of possible values depends on the frequencies of the alleles it refers to. This makes it difficult to compare the level of linkage disequilibrium between different pairs of alleles.
Lewontin suggested normalising "D" by dividing it by the theoretical maximum difference between the observed and expected haplotype frequencies as follows:
where
An alternative to formula_26 is the correlation coefficient between pairs of loci, expressed as
Example: Two-loci and two-alleles.
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:
Note that these are relative frequencies. One can use the above frequencies to determine the frequency of each of the alleles:
If the two loci and the alleles are independent from each other, then one can express the observation formula_28 as "formula_29 is found and formula_30 is found". The table above lists the frequencies for formula_29, formula_32, and forformula_30, formula_34, hence the frequency of formula_28 is formula_36, and according to the rules of elementary statistics formula_37.
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":
The following table illustrates the relationship between the haplotype frequencies and allele frequencies and D.
Role of recombination.
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 formula_23 converges to zero along the time axis at a rate
depending on the magnitude of the recombination rate formula_40 between the two loci.
Using the notation above, formula_41, we can demonstrate this convergence to zero
as follows. In the next generation, formula_42, the frequency of the haplotype formula_43, becomes
This follows because a fraction formula_45 of the haplotypes in the offspring have not
recombined, and are thus copies of a random haplotype in their parents. A fraction formula_36 of those are formula_43. A fraction formula_40
have recombined these two loci. If the parents result from random mating, the probability of the
copy at locus formula_49 having allele formula_29 is formula_32 and the probability
of the copy at locus formula_52 having allele formula_30 is formula_34, 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 formula_23 at the formula_58-th generation is designated as formula_59. Thus we have
If formula_61, then formula_62 so that formula_59 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 formula_23 to zero.
Example: Human leukocyte antigen (HLA) alleles.
HLA constitutes a group of cell surface antigens also known as the MHC of humans. Because HLA genes are located at adjacent loci on the particular region of a chromosome and presumed to exhibit epistasis with each other or with other genes, a sizable fraction of alleles are in linkage disequilibrium.
An example of such linkage disequilibrium is between HLA-A1 and B8 alleles in unrelated Danes referred to by Vogel and Motulsky (1997).
Because HLA is codominant and HLA expression is only tested locus by locus in surveys, LD measure is to be estimated from such a 2×2 table to the right.
expression (formula_65) frequency of antigen formula_66 :
expression (formula_65) frequency of antigen formula_69 :
frequency of gene formula_66, given that individuals with '+/−', '+/+', and '−/+' genotypes are all positive for antigen formula_66:
and
Denoting the '―' alleles at antigen "i" to be "x", and at antigen "j" to be "y", the observed frequency of haplotype "xy" is
and the estimated frequency of haplotype "xy" is
Then LD measure formula_77 is expressed as
Standard errors formula_79 are obtained as follows:
Then, if
exceeds 2 in its absolute value, the magnitude of formula_77 is statistically significantly large. For data in Table 1 it is 20.9, thus existence of statistically significant LD between A1 and B8 in the population is admitted.
Table 2 shows some of the combinations of HLA-A and B alleles where significant LD was observed among pan-Europeans.
Vogel and Motulsky (1997) argued how long would it take that linkage disequilibrium between loci of HLA-A and B disappeared. Recombination between loci of HLA-A and B was considered to be of the order of magnitude 0.008. We will argue similarly to Vogel and Motulsky below. In case LD measure was observed to be 0.003 in pan-Europeans in the list of Mittal it is mostly non-significant. If formula_85 had reduced from 0.07 to 0.003 under recombination effect as shown by formula_86, then formula_87. Suppose a generation took 25 years, this means 10,000 years. The time span seems rather short in the history of humans. Thus observed linkage disequilibrium between HLA-A and B loci might indicate some sort of interactive selection.
The presence of linkage disequilibrium between an HLA locus and a presumed major gene of disease susceptibility corresponds to any of the following phenomena:
(1) Relative risk
Relative risk of an HLA allele for a disease is approximated by the odds ratio in the 2×2 association table of the allele with the disease. Table 3 shows association of HLA-B27 with ankylosing spondylitis among a Dutch population. Relative risk formula_89of this allele is approximated by
Woolf's method is applied to see if there is statistical significance. Let
and
Then
follows the chi-square distribution with formula_94. In the data of Table 3, a significant association exists at the 0.1% level. Haldane's modification applies to the case when either of formula_95 is zero, where formula_89 and formula_97 are replaced with
and
respectively.
In Table 4, some examples of association between HLA alleles and diseases are presented.
(1a) Allele frequency excess among patients over controls
Even high relative risks between HLA alleles and the diseases were observed, only the magnitude of relative risk would not be able to determine the strength of association.formula_88 value is expressed by
where formula_102 and formula_103 are HLA allele frequencies among patients and healthy populations, respectively. In Table 4, formula_88 column was added in this quotation. Putting aside 2 diseases with high relative risks both of which are also with high formula_88 values, among other diseases, juvenile diabetes mellitus (type 1) has a strong association with DR4 even with a low relative riskformula_106.
(2) Discrepancies from expected values from marginal frequencies in 2×2 association table of HLA alleles and disease
This can be confirmed by formula_107 test calculating
where formula_94. For data with small sample size, such as no marginal total is greater than 15 (and consequently formula_110), one should utilize Yates's correction for continuity or Fisher's exact test.
Resources.
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.