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In population genetics, linkage disequilibrium is the non-random association of alleles at two or more loci, not necessarily on the same chromosome. It is not the same as linkage, which describes the association of two or more loci on a chromosome with limited recombination between them. Linkage disequilibrium describes a situation in which some combinations of alleles or genetic markers occur more or less frequently in a population than would be expected from a random formation of haplotypes from alleles based on their frequencies. Non-random associations between polymorphisms at different loci are measured by the degree of linkage disequilibrium (LD).

An example is the prevalence of two rare diseases in Finland: there, compared to elsewhere in Europe, cystic fibrosis is less prevalent but congenital chloride diarrhea is more prevalent (see Finnish disease heritage). Both diseases are due to mutations on chromosome 7, in adjacent genes.[1]

The level of linkage disequilibrium is influenced by a number of factors including genetic linkage, selection, the rate of recombination, the rate of mutation, genetic drift, non-random mating, and population structure. For example, some organisms (such as bacteria) may show linkage disequilibrium because they reproduce asexually and there is no recombination to break down the linkage disequilibrium.

If we look at haplotypes for two loci A and B with two alleles each—a two-locus, two-allele model—the following table denotes the frequencies of each combination:

 Haplotype Frequency A1B1 x11 A1B2 x12 A2B1 x21 A2B2 x22

Note that these are relative frequencies. One can use the above frequencies to determine the frequency of each of the alleles:

 Allele Frequency A1 p1 = x11 + x12 A2 p2 = x21 + x22 B1 q1 = x11 + x21 B2 q2 = x12 + x22

If the two loci and the alleles are independent from each other, then one can express the observation A1B1 as "A1 is found and B1 is found". The table above lists the frequencies for A1, p1, and for B1, q1, hence the frequency of A1B1 is x11, and according to the rules of elementary statistics x11 = p1q1.

The deviation of the observed frequency of a haplotype from the expected is a quantity[2] called the linkage disequilibrium[3] and is commonly denoted by a capital D:

 D = x11 − p1q1

In the genetic literature the phrase "two alleles are in LD" usually means that $D \ne 0$. Contrariwise, "linkage equilibrium" denotes the case D = 0.

The following table illustrates the relationship between the haplotype frequencies and allele frequencies and D.

 A1 A2 Total B1 x11 = p1q1 + D x21 = p2q1 − D q1 B2 x12 = p1q2 − D x22 = p2q2 + D q2 Total p1 p2 1

D is easy to calculate with, but has the disadvantage of depending on the frequency of the alleles. This is evident since frequencies are between 0 and 1. There can be no D observed if any locus has an allele frequency 0 or 1 and is maximal when frequencies are at 0.5. Lewontin (1964) suggested normalising D by dividing it with the theoretical maximum for the observed allele frequencies. Thus $D'=\frac{D}{D_\max}$ when $D \ge 0$ When D < 0, $D'=\frac{D}{D_\min}$.

Dmax is given by the smaller of p1q2 and p2q1. Dmin is given by the larger of p1q1 and p2q2

Another measure of LD which is an alternative to D' is the correlation coefficient between pairs of loci, denoted as $r=\frac{D}{\sqrt{p_1p_2q_1q_2}}$. This is also adjusted to the loci having different allele frequencies. There is some relationship between r and D'. [4]

In summary, linkage disequilibrium reflects the difference between the expected haplotype frequencies under the assumption of independence, and observed haplotype frequencies. A value of 0 for D' indicates that the examined loci are in fact independent of one another, while a value of 1 demonstrates complete dependency.

## Role of recombination

In the absence of evolutionary forces other than random mating and Mendelian segregation, 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 = x11p1q1, we can demonstrate this convergence to zero as follows. In the next generation, x11', the frequency of the haplotype A1B1, becomes

 $x_{11}' = (1-c)\,x_{11} + c\,p_1 q_1$

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 x11 of those are A1B1. 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 A1 is p1 and the probability of the copy at locus B having allele B1 is q1, and as these copies are initially on different haplotypes, these are independent events so that the probabilities can be multiplied.

This formula can be rewritten as

 $x_{11}' - p_1 q_1 = (1-c)\,(x_{11} - p_1 q_1)$

so that

 $D_1 = (1-c)\;D_0$

where D at the n-th generation is designated as Dn. Thus we have

 $D_n = (1-c)^n\; D_0$.

If $n \to \infty$, then $(1-c)^n \to 0$ so that Dn converges to zero.

If at some time we observe linkage disequilibrium, it will disappear in 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.

## Linkage disequilibrium appears frequently in genetic systems

### Human leucocyte antigen (HLA)

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[5] referred to by Vogel and Motulsky (1997).[6]

No. of individuals Antigen j Total a = 376 b = 237 C c = 91 d = 1265 D A B N

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.[6][7][8][9]

pfi = frequency of antigen i = C / N = 0.311,
pfj = 0.237,
$gf_i=\text{frequency of gene }i=1-\sqrt{1-pf_i}=0.170$,

and

$hf_{ij}=\text{estimated frequency of haplotype }ij=gf_i\; gf_j=0.0215$.

Denoting the '―' alleles at antigen i to be 'x,' and at antigen j to be 'y,' the observed frequency of haplotype xy is

$o[hf_{xy}]=\sqrt{d/N}$

and the estimated frequency of haplotype xy is

$e[hf_{xy}]=\sqrt{(D/N)(B/N)}$.

Then LD measure Δij is expressed as

$\Delta_{ij}=o[hf_{xy}]-e[hf_{xy}]=\frac{\sqrt{Nd}-\sqrt{BD}}{N}=0.0769$.

Standard errors SEs are obtained as follows:

$SE\text{ of }gf_i=\sqrt{C}/(2N)=0.00628$,
$SE\text{ of }hf_{ij}=\sqrt{\frac{(1-\sqrt{d/B})(1-\sqrt{d/D})-hf_{ij}-hf_{ij}^2/2}{2N}}=0.00514$
$SE\text{ of }\Delta_{ij}=\frac{1}{2N}\sqrt{a-4N\Delta_{ij}\left (\frac{B+D}{2\sqrt{BD}}-\frac{\sqrt{BD}}{N}\right )}=0.00367$.

Then, if

t = Δij / (SE of Δij)

exceeds 2 in its absolute value, the magnitude of Δij is large statistically significantly. 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. Linkage disequilibrium among HLA alleles in Caucasians[9]
HLA-A alleles i HLA-B alleles j Δij t
A1 B8 0.065 16.0
A3 B7 0.039 10.3
A2 Bw40 0.013 4.4
A2 Bw15 0.01 3.4
A1 Bw17 0.014 5.4
A2 B18 0.006 2.2
A2 Bw35 -0.009 -2.3
A29 B12 0.013 6.0
A10 Bw16 0.013 5.9

Table 2 shows some of the combinations of HLA-A and B alleles where significant LD was observed among Caucasians.[9]

Vogel and Motulsky (1997)[6] 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 Caucasians in the list of Mittal[9] it is mostly non-significant. If Δ0 had reduced from 0.07 to 0.003 under recombination effect as shown by Δn = (1 − c)nΔ0, then $n\approx 400$. 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.[6]

### Between an HLA locus and a presumed major gene locus having disease susceptibility

Presence of linkage disequilibrium between an HLA locus and a presumed major gene of disease susceptibility corresponds to any of the following phenomena:

• Relative risk for the person having a specific HLA allele to become suffered from a particular disease is larger than one.[10]
• The HLA antigen frequency among patients exceeds more than that among a healthy population. This is evaluated by δ value[11] to exceed 0.
Ankylosing spondylitis Total Patients a = 96 b = 77 C c = 22 d = 701 D A B N
• 2x2 association table of patients and healthy controls with HLA alleles shows a significant deviation from the equilibrium state deduced from the marginal frequencies.

(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.[12] Relative risk x of this allele is approximated by

$x=\frac{a/b}{c/d}=\frac{ad}{bc}\;(=39.7,\text{ in Table 3 })$.

Woolf's method[13] is applied to see if there is statistical significance. Let

$y=\ln (x)\;(=3.68)$

and

$\frac{1}{w}=\frac{1}{a}+\frac{1}{b}+\frac{1}{c}+\frac{1}{d}\;(=0.0703)$.

Then

$\chi^2=wy^2\;\left [=193>\chi^2(p=0.001,\; df=1)=10.8 \right ]$

follows the chi-square distribution with df = 1. In the data of Table 3, the significant association exists at the 0.1% level. Haldane's[14] modification applies to the case when either of $a,\; b,\;c,\text{ and }d$ is zero, where replace x and 1 / w with

$x=\frac{(a+1/2)(d+1/2)}{(b+1/2)(c+1/2)}$

and

$\frac{1}{w}=\frac{1}{a+1}+\frac{1}{b+1}+\frac{1}{c+1}+\frac{1}{d+1}$,

respectively.

Table 4. Association of HLA alleles with rheumatic and autoimmune diseases among white populations[10]
Disease HLA allele Relative risk (%) FAD (%) FAP (%) δ
Ankylosing spondylitis B27 90 90 8 0.89
Reiter's syndrome B27 40 70 8 0.67
Spondylitis in inflammatory bowel disease B27 10 50 8 0.46
Rheumatoid arthritis DR4 6 70 30 0.57
Systemic lupus erythematosus DR3 3 45 20 0.31
Multiple sclerosis DR2 4 60 20 0.5
Diabetes mellitus type 1 DR4 6 75 30 0.64

In Table 4, some examples of association between HLA alleles and diseases are presented.[10]

(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.[11] δ value is expressed by

$\delta=\frac{FAD-FAP}{1-FAP},\;\;0\le \delta \le 1$,

where FAD and FAP are HLA allele frequencies among patients and healthy populations, respectively.[11] 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 χ2 test calculating

$\chi^2=\frac{(ad-bc)^2 N}{ABCD}\;(=336,\text{ for data in Table 3; }P<0.001)$.

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

## Resources

A comparison of different measures of LD is provided by Devlin & Risch [16]

The International HapMap Project enables the study of LD in human populations online. The Ensembl project integrates HapMap data and such from dbSNP in general with other genetic information.

## References

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2. ^ Robbins, R.B. (1 July 1918). "Some applications of mathematics to breeding problems III". Genetics 3 (4): 375–389. PMID 17245911. PMC 1200443.
3. ^ R.C. Lewontin and K. Kojima (1960). "The evolutionary dynamics of complex polymorphisms". Evolution 14 (4): 458–472. doi:10.2307/2405995.
4. ^ P.W. Hedrick and S. Kumar (2001). "Mutation and linkage disequilibrium in human mtDNA". Eur. J. Hum. Genet. 9 (12): 969–972. doi:10.1038/sj.ejhg.5200735. PMID 11840186.
5. ^ a b Svejgaard A, Hauge M, Jersild C, Plaz P, Ryder LP, Staub Nielsen L, Thomsen M (1979). The HLA System: An Introductory Survey, 2nd ed. Basel; London; Chichester: Karger; Distributed by Wiley, ISBN 3805530498(pbk).
6. ^ a b c d Vogel F, Motulsky AG (1997). Human Genetics : Problems and Approaches, 3rd ed. Berlin; London: Springer, ISBN 3540602909.
7. ^ Mittal KK, Hasegawa T, Ting A, Mickey MR, Terasaki PI (1973). "Genetic variation in the HL-A system between Ainus, Japanese, and Caucasians," In Dausset J, Colombani J, eds. Histocompatibility Testing, 1972, pp. 187-195, Copenhagen: Munksgaard, ISBN 8716011015.
8. ^ Yasuda N, Tsuji K (1975). "A counting method of maximum likelihood for estimating haplotype frequency in the HL-A system." Jinrui Idengaku Zasshi 20(1): 1-15, PMID 1237691.
9. ^ a b c d Mittal KK (1976). "The HLA polymorphism and susceptibility to disease." Vox Sang 31: 161-173, PMID 969389.
10. ^ a b c Gregersen PK (2009). "Genetics of rheumatic diseases," In Firestein GS, Budd RC, Harris ED Jr, McInnes IB, Ruddy S, Sergent JS, eds. (2009). Kelley's Textbook of Rheumatology, pp. 305-321, Philadelphia, PA: Saunders/Elsevier, ISBN 9781416032854.
11. ^ a b c Bengtsson BO, Thomson G (1981). "Measuring the strength of associations between HLA antigens and diseases." Tissue Antigens 18(5): 356-363, PMID 7344182.
12. ^ a b Nijenhuis LE (1977). "Genetic considerations on association between HLA and disease." Hum Genet 38(2): 175-182, PMID 908564.
13. ^ Woolf B (1955). "On estimating the relation between blood group and disease." Ann Hum Genet 19(4): 251-253, PMID 14388528.
14. ^ Haldane JB (1956). "The estimation and significance of the logarithm of a ratio of frequencies." Ann Hum Genet 20(4): 309-311, PMID 13314400.
15. ^ Sokal RR, Rohlf FJ (1981). Biometry: The Principles and Practice of Statistics in Biological Research. Oxford: W.H. Freeman, ISBN 0716712547.
16. ^
17. ^ Hao K., Di X., Cawley S. (2007). "LdCompare: rapid computation of single- and multiple-marker r2 and genetic coverage". Bioinformatics 23 (2): 252–254. doi:10.1093/bioinformatics/btl574. PMID 17148510.