Linkage disequilibrium

Formal definition

Suppose that among the gametes that are formed in a sexually reproducing population, allele A occurs with frequency p A at one locus (i.e. p A is the proportion of gametes with A at that locus), while at a different locus allele B occurs with frequency p B . Similarly, let p A B be the frequency with which both A and B occur together in the same gamete (i.e. p A B 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 p A p B of the probabilities. There is said to be a linkage disequilibrium between the two alleles whenever p A B differs from p A p B for any reason.

The level of linkage disequilibrium between A and B can be quantified by the coefficient of linkage disequilibrium D A B , which is defined as

provided that both p A and p B are greater than zero. Linkage disequilibrium corresponds to D A B ≠ 0 . In the case D A B = 0 we have p A B = p A p B and the alleles A and B are said to be in linkage equilibrium. The subscript "AB" on D A B 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.

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 D {displaystyle D}

The coefficient of linkage disequilibrium D 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 allele frequencies as follows:

where

An alternative to D ′ is the correlation coefficient between pairs of loci, expressed as

r = D p A ( 1 − p A ) p B ( 1 − p B ) .

Example: Two-loci and two-alleles

Consider the haplotypes for two loci A and B with two alleles each—a two-locus, 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 A 1 B 1 as " A 1 is found and B 1 is found". The table above lists the frequencies for A 1 , p 1 , and for B 1 , q 1 , hence the frequency of A 1 B 1 is x 11 , and according to the rules of elementary statistics x 11 = p 1 q 1 .

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 D converges to zero along the time axis at a rate depending on the magnitude of the recombination rate c between the two loci.

Using the notation above, D = x 11 − p 1 q 1 , we can demonstrate this convergence to zero as follows. In the next generation, x 11 ′ , the frequency of the haplotype A 1 B 1 , becomes

This follows because a fraction ( 1 − c ) of the haplotypes in the offspring have not recombined, and are thus copies of a random haplotype in their parents. A fraction x 11 of those are A 1 B 1 . A fraction c have recombined these two loci. If the parents result from random mating, the probability of the copy at locus A having allele A 1 is p 1 and the probability of the copy at locus B having allele B 1 is q 1 , 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 D at the n -th generation is designated as D n . Thus we have

If n → ∞ , then ( 1 − c ) n → 0 so that D n 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 D to zero.

Example: Human leukocyte antigen (HLA) alleles

HLA constitutes a group of cell surface antigens as 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 2x2 table to the right.

expression ( + ) frequency of antigen i  :

expression ( + ) frequency of antigen j  :

frequency of gene i  :

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 Δ i j is expressed as

Standard errors S E s are obtained as follows:

Then, if

exceeds 2 in its absolute value, the magnitude of Δ i j 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 Δ 0 had reduced from 0.07 to 0.003 under recombination effect as shown by Δ n = ( 1 − c ) n Δ 0 , then n ≈ 400 . 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 2x2 association table of the allele with the disease. Table 3 shows association of HLA-B27 with ankylosing spondylitis among a Dutch population. Relative risk x of 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 d f = 1 . In the data of Table 3, the significant association exists at the 0.1% level. Haldane's modification applies to the case when either of a , b , c ,  and  d is zero, where replace x and 1 / w 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. δ value is expressed by

where F A D and F A P are HLA allele frequencies among patients and healthy populations, respectively. In Table 4, δ column was added in this quotation. Putting aside 2 diseases with high relative risks both of which are also with high δ values, among other diseases, juvenile diabetes mellitus (type 1) has a strong association with DR4 even with a low relative risk = 6 .

(2) Discrepancies from expected values from marginal frequencies in 2x2 association table of HLA alleles and disease

This can be confirmed by χ 2 test calculating

where d f = 1 . For data with small sample size, such as no marginal total is greater than 15 (and consequently N ≤ 30 ), one should utilize Yates's correction for continuity or Fisher's exact test.

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