In population genetics, linkage disequilibrium is the non-random association of alleles at different loci. 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.[1]

Linkage disequilibrium is influenced by many factors, including selection, the rate of recombination, the rate of mutation, 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).[1] Furthermore, linkage disequilibrium is sometimes referred to as gametic phase disequilibrium;[2] 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 ${\displaystyle p_{A}}$ at one locus (i.e. ${\displaystyle p_{A}}$ is the proportion of gametes with A at that locus), while at a different locus allele B occurs with frequency ${\displaystyle p_{B}}$. Similarly, let ${\displaystyle p_{AB}}$ be the frequency with which both A and B occur together in the same gamete (i.e. ${\displaystyle p_{AB}}$ 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 ${\displaystyle p_{A}p_{B}}$ of the probabilities. There is said to be a linkage disequilibrium between the two alleles whenever ${\displaystyle p_{AB}}$ differs from ${\displaystyle p_{A}p_{B}}$ for any reason.

The level of linkage disequilibrium between A and B can be quantified by the coefficient of linkage disequilibrium ${\displaystyle D_{AB}}$, which is defined as

 ${\displaystyle D_{AB}=p_{AB}-p_{A}p_{B}}$,

provided that both ${\displaystyle p_{A}}$ and ${\displaystyle p_{B}}$ are greater than zero. Linkage disequilibrium corresponds to ${\displaystyle D_{AB}\neq 0}$. In the case ${\displaystyle D_{AB}=0}$ we have ${\displaystyle p_{AB}=p_{A}p_{B}}$ and the alleles A and B are said to be in linkage equilibrium. The subscript "AB" on ${\displaystyle D_{AB}}$ 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.[1]

## Measures derived from ${\displaystyle D}$

The coefficient of linkage disequilibrium ${\displaystyle D}$ 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[3] suggested normalising D by dividing it by the theoretical maximum difference between the observed and expected allele frequencies as follows:

 ${\displaystyle D'=D/D_{\min }}$

where

 ${\displaystyle D_{\min }={\begin{cases}\max\{-p_{A}p_{B},\,-(1-p_{A})(1-p_{B})\}&{\text{when }}D<0\\\min\{p_{A}(1-p_{B}),\,(1-p_{A})p_{B}\}&{\text{when }}D>0\end{cases}}}$

An alternative to ${\displaystyle D'}$ is the correlation coefficient between pairs of loci, expressed as

${\displaystyle r={\frac {D}{\sqrt {p_{A}(1-p_{A})p_{B}(1-p_{B})}}}}$.

## 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:

 Haplotype Frequency ${\displaystyle A_{1}B_{1}}$ ${\displaystyle x_{11}}$ ${\displaystyle A_{1}B_{2}}$ ${\displaystyle x_{12}}$ ${\displaystyle A_{2}B_{1}}$ ${\displaystyle x_{21}}$ ${\displaystyle A_{2}B_{2}}$ ${\displaystyle x_{22}}$

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

 Allele Frequency ${\displaystyle A_{1}}$ ${\displaystyle p_{1}=x_{11}+x_{12}}$ ${\displaystyle A_{2}}$ ${\displaystyle p_{2}=x_{21}+x_{22}}$ ${\displaystyle B_{1}}$ ${\displaystyle q_{1}=x_{11}+x_{21}}$ ${\displaystyle B_{2}}$ ${\displaystyle q_{2}=x_{12}+x_{22}}$

If the two loci and the alleles are independent from each other, then one can express the observation ${\displaystyle A_{1}B_{1}}$ as "${\displaystyle A_{1}}$ is found and ${\displaystyle B_{1}}$ is found". The table above lists the frequencies for ${\displaystyle A_{1}}$, ${\displaystyle p_{1}}$, and for${\displaystyle B_{1}}$, ${\displaystyle q_{1}}$, hence the frequency of ${\displaystyle A_{1}B_{1}}$ is ${\displaystyle x_{11}}$, and according to the rules of elementary statistics ${\displaystyle x_{11}=p_{1}q_{1}}$.

The deviation of the observed frequency of a haplotype from the expected is a quantity[4] called the linkage disequilibrium[5] and is commonly denoted by a capital D:

 ${\displaystyle D=x_{11}-p_{1}q_{1}}$

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

 ${\displaystyle A_{1}}$ ${\displaystyle A_{2}}$ Total ${\displaystyle B_{1}}$ ${\displaystyle x_{11}=p_{1}q_{1}+D}$ ${\displaystyle x_{21}=p_{2}q_{1}-D}$ ${\displaystyle q_{1}}$ ${\displaystyle B_{2}}$ ${\displaystyle x_{12}=p_{1}q_{2}-D}$ ${\displaystyle x_{22}=p_{2}q_{2}+D}$ ${\displaystyle q_{2}}$ Total ${\displaystyle p_{1}}$ ${\displaystyle p_{2}}$ ${\displaystyle 1}$

## 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 ${\displaystyle D}$ converges to zero along the time axis at a rate depending on the magnitude of the recombination rate ${\displaystyle c}$ between the two loci.

Using the notation above, ${\displaystyle D=x_{11}-p_{1}q_{1}}$, we can demonstrate this convergence to zero as follows. In the next generation, ${\displaystyle x_{11}'}$, the frequency of the haplotype ${\displaystyle A_{1}B_{1}}$, becomes

 ${\displaystyle x_{11}'=(1-c)\,x_{11}+c\,p_{1}q_{1}}$

This follows because a fraction ${\displaystyle (1-c)}$ of the haplotypes in the offspring have not recombined, and are thus copies of a random haplotype in their parents. A fraction ${\displaystyle x_{11}}$ of those are ${\displaystyle A_{1}B_{1}}$. A fraction ${\displaystyle c}$ have recombined these two loci. If the parents result from random mating, the probability of the copy at locus ${\displaystyle A}$ having allele ${\displaystyle A_{1}}$ is ${\displaystyle p_{1}}$ and the probability of the copy at locus ${\displaystyle B}$ having allele ${\displaystyle B_{1}}$ is ${\displaystyle q_{1}}$, 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

 ${\displaystyle x_{11}'-p_{1}q_{1}=(1-c)\,(x_{11}-p_{1}q_{1})}$

so that

 ${\displaystyle D_{1}=(1-c)\;D_{0}}$

where ${\displaystyle D}$ at the ${\displaystyle n}$-th generation is designated as ${\displaystyle D_{n}}$. Thus we have

 ${\displaystyle D_{n}=(1-c)^{n}\;D_{0}}$.

If ${\displaystyle n\to \infty }$, then ${\displaystyle (1-c)^{n}\to 0}$ so that ${\displaystyle D_{n}}$ 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 ${\displaystyle D}$ 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[6] referred to by Vogel and Motulsky (1997).[7]

Table 1. Association of HLA-A1 and B8 in unrelated Danes[6]
Antigen j Total
${\displaystyle +}$ ${\displaystyle -}$
${\displaystyle B8^{+}}$ ${\displaystyle B8^{-}}$
Antigen i ${\displaystyle +}$ ${\displaystyle A1^{+}}$ ${\displaystyle a=376}$ ${\displaystyle b=237}$ ${\displaystyle C}$
${\displaystyle -}$ ${\displaystyle A1^{-}}$ ${\displaystyle c=91}$ ${\displaystyle d=1265}$ ${\displaystyle D}$
Total ${\displaystyle A}$ ${\displaystyle B}$ ${\displaystyle N}$
No. of individuals

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.[7][8][9][10]

expression (${\displaystyle +}$) frequency of antigen ${\displaystyle i}$ :

${\displaystyle pf_{i}=C/N=0.311\!}$ ;

expression (${\displaystyle +}$) frequency of antigen ${\displaystyle j}$ :

${\displaystyle pf_{j}=A/N=0.237\!}$ ;

frequency of gene ${\displaystyle i}$ :

${\displaystyle gf_{i}=1-{\sqrt {1-pf_{i}}}=0.170\!}$ ,

and

${\displaystyle hf_{ij}={\text{estimated frequency of haplotype }}ij=gf_{i}\;gf_{j}=0.0215\!}$ .

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

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

and the estimated frequency of haplotype xy is

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

Then LD measure ${\displaystyle \Delta _{ij}}$ is expressed as

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

Standard errors ${\displaystyle SEs}$ are obtained as follows:

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

Then, if

${\displaystyle t=\Delta _{ij}/(SE{\text{ of }}\Delta _{ij})}$

exceeds 2 in its absolute value, the magnitude of ${\displaystyle \Delta _{ij}}$ 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. Linkage disequilibrium among HLA alleles in Pan-europeans[10]
HLA-A alleles i HLA-B alleles j ${\displaystyle \Delta _{ij}}$ ${\displaystyle t}$
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 pan-europeans.[10]

Vogel and Motulsky (1997)[7] 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[10] it is mostly non-significant. If ${\displaystyle \Delta _{0}}$ had reduced from 0.07 to 0.003 under recombination effect as shown by ${\displaystyle \Delta _{n}=(1-c)^{n}\Delta _{0}}$, then ${\displaystyle n\approx 400}$. 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.[7]

Further information: HLA A1-B8 haplotype

The 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 greater than 1.[11]
• The HLA antigen frequency among patients exceeds more than that among a healthy population. This is evaluated by ${\displaystyle \delta }$ value[12] to exceed 0.
Ankylosing spondylitis Total Patients ${\displaystyle a=96}$ ${\displaystyle b=77}$ ${\displaystyle C}$ ${\displaystyle c=22}$ ${\displaystyle d=701}$ ${\displaystyle D}$ ${\displaystyle A}$ ${\displaystyle B}$ ${\displaystyle N}$
• 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.[13] Relative risk ${\displaystyle x}$of this allele is approximated by

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

Woolf's method[14] is applied to see if there is statistical significance. Let

${\displaystyle y=\ln(x)\;(=3.68)}$

and

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

Then

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

follows the chi-square distribution with ${\displaystyle df=1}$. In the data of Table 3, the significant association exists at the 0.1% level. Haldane's[15] modification applies to the case when either of${\displaystyle a,\;b,\;c,{\text{ and }}d}$ is zero, where replace ${\displaystyle x}$ and ${\displaystyle 1/w}$with

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

and

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

respectively.

Table 4. Association of HLA alleles with rheumatic and autoimmune diseases among white populations[11]
Disease HLA allele Relative risk (%) FAD (%) FAP (%) ${\displaystyle \delta }$
Ankylosing spondylitis B27 90 90 8 0.89
Reactive arthritis 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.[11]

(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.[12]${\displaystyle \delta }$ value is expressed by

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

where ${\displaystyle FAD}$ and ${\displaystyle FAP}$ are HLA allele frequencies among patients and healthy populations, respectively.[12] In Table 4, ${\displaystyle \delta }$ column was added in this quotation. Putting aside 2 diseases with high relative risks both of which are also with high ${\displaystyle \delta }$ values, among other diseases, juvenile diabetes mellitus (type 1) has a strong association with DR4 even with a low relative risk${\displaystyle =6}$.

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

This can be confirmed by ${\displaystyle \chi ^{2}}$ test calculating

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

where ${\displaystyle df=1}$. For data with small sample size, such as no marginal total is greater than 15 (and consequently ${\displaystyle N\leq 30}$), one should utilize Yates's correction for continuity or Fisher's exact test.[16]

## Resources

A comparison of different measures of LD is provided by Devlin & Risch[17]

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.

## Analysis software

• PLINK - whole genome association analysis toolset, which can calculate LD among other things
• LDHat
• Haploview
• LdCompare[18]— open-source software for calculating LD.
• SNP and Variation Suite- commercial software with interactive LD plot.
• GOLD - Graphical Overview of Linkage Disequilibrium
• TASSEL -software to evaluate linkage disequilibrium, traits associations, and evolutionary patterns
• rAggr - finds proxy markers (SNPs and indels) that are in linkage disequilibrium with a set of queried markers, using the 1000 Genomes Project and HapMap genotype databases.
• SNeP - Fast computation of LD and Ne for large genotype datasets in PLINK format.

## Simulation software

• Haploid — a C library for population genetic simulation (GPL)

## References

1. ^ a b c Slatkin, Montgomery (June 2008). "Linkage disequilibrium — understanding the evolutionary past and mapping the medical future". Nature Reviews Genetics. 9 (6): 477–485. doi:10.1038/nrg2361.
2. ^ Falconer, DS; Mackay, TFC (1996). Introduction to Quantitative Genetics (4th ed.). Harlow, Essex, UK: Addison Wesley Longman. ISBN 0-582-24302-5.
3. ^ Lewontin, R. C. (1964). "The interaction of selection and linkage. I. General considerations; heterotic models". Genetics. 49 (1): 49–67. PMC . PMID 17248194.
4. ^ Robbins, R.B. (1 July 1918). "Some applications of mathematics to breeding problems III". Genetics. 3 (4): 375–389. PMC . PMID 17245911.
5. ^ R.C. Lewontin & K. Kojima (1960). "The evolutionary dynamics of complex polymorphisms". Evolution. 14 (4): 458–472. doi:10.2307/2405995. ISSN 0014-3820. JSTOR 2405995.
6. ^ 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).
7. ^ a b c d Vogel F, Motulsky AG (1997). Human Genetics : Problems and Approaches, 3rd ed.Berlin; London: Springer, ISBN 3-540-60290-9.
8. ^ 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 87-16-01101-5.
9. ^ 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.
10. ^ a b c d Mittal KK (1976). "The HLA polymorphism and susceptibility to disease." Vox Sang 31: 161-173, PMID 969389.
11. ^ a b c Gregersen PK (2009). "Genetics of rheumatic diseases," InFirestein 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 978-1-4160-3285-4.
12. ^ a b c Bengtsson BO, Thomson G (1981). "Measuring the strength of associations between HLA antigens and diseases." Tissue Antigens18(5): 356-363, PMID 7344182.
13. ^ a b Nijenhuis LE (1977). "Genetic considerations on association between HLA and disease." Hum Genet38(2): 175-182, PMID 908564.
14. ^ Woolf B (1955). "On estimating the relation between blood group and disease." Ann Hum Genet 19(4): 251-253, PMID 14388528.
15. ^ Haldane JB (1956). "The estimation and significance of the logarithm of a ratio of frequencies." Ann Hum Genet20(4): 309-311, PMID 13314400.
16. ^ Sokal RR, Rohlf FJ (1981). Biometry: The Principles and Practice of Statistics in Biological Research. Oxford: W.H. Freeman, ISBN 0-7167-1254-7.
17. ^ Devlin B.; Risch N. (1995). "A Comparison of Linkage Disequilibrium Measures for Fine-Scale Mapping" (PDF). Genomics. 29 (2): 311–322. doi:10.1006/geno.1995.9003. PMID 8666377.
18. ^ 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.