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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 expected if the loci were independent and associated randomly.^[1]

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).^[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[edit]

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 $p_{AB}$ be the frequency with which both A and B occur together in the same gamete (i.e. $p_{AB}$ 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 $p_{A}p_{B}$ of the probabilities. There is said to be a linkage disequilibrium between the two alleles whenever $p_{AB}$ 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 $D_{AB}$ , which is defined as

D_{AB}=p_{AB}-p_{A}p_{B},

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

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, $D_{AB}=-D_{Ab}=-D_{aB}=D_{ab}$ . Their relationships can be characterized as follows.^[3]

$D=P_{AB}-P_{A}P_{B}$

$-D=P_{Ab}-P_{A}P_{b}$

$-D=P_{aB}-P_{a}P_{B}$

$D=P_{ab}-P_{a}P_{b}$

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.^[4] 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.^[1]

Normalization[edit]

The linkage disequilibrium $D$ 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.

D' Method[edit]

Lewontin^[5] suggested calculating the normalized linkage disequilibrium (also referred to as relative linkage disequilibrium) $D'$ by dividing $D$ by the theoretical maximum difference between the observed and expected allele frequencies as follows:

D'={\frac {D}{D_{\max }}}

where

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

Note that $|D'|$ may be used in place of $D'$ when measuring how close two alleles are to linkage equilibrium.

r² Method[edit]

An alternative to $D'$ is the correlation coefficient between pairs of loci, usually expressed as its square, $r^{2}$ .^[6]

r^{2}={\frac {D^{2}}{p_{A}(1-p_{A})p_{B}(1-p_{B})}}

d Method[edit]

Another alternative normalizes $D$ 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 $B$ is the locus containing a disease allele.^[7]

$d={\frac {D}{p_{B}(1-p_{B})}}$

ρ Method[edit]

Similar to the d method, this alternative normalizes $D$ by the product of two of the four allele frequencies when the two frequencies represent alleles from different loci.^[7]

$\rho ={\frac {D}{(1-p_{A})p_{B}}}$

Limits for the ranges of linkage disequilibrium measures[edit]

The measures $r^{2}$ and $D'$ have limits to their ranges and do not range over all values of zero to one for all pairs of loci. The maximum of $r^{2}$ 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, $P_{A}=P_{B}$ where $D>0$ , or when the allele frequencies have the relationship $P_{A}=1-P_{B}$ when $D<0$ .^[8] While $D'$ can always take a maximum value of 1, its minimum value for two loci is equal to $|r|$ for those loci.^[9]

Example: Two-loci and two-alleles[edit]

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
$A_{1}B_{1}$	$x_{11}$
$A_{1}B_{2}$	$x_{12}$
$A_{2}B_{1}$	$x_{21}$
$A_{2}B_{2}$	$x_{22}$

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

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

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^[10] called the linkage disequilibrium^[11] and is commonly denoted by a capital D:

D=x_{11}-p_{1}q_{1}

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

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

Role of recombination[edit]

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

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 $x_{11}$ 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

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 $D_{n}$ . Thus we have

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

If $n\to \infty$ , then $(1-c)^{n}\to 0$ 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.

Resources[edit]

A comparison of different measures of LD is provided by Devlin & Risch^[12]

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[edit]

PLINK – whole genome association analysis toolset, which can calculate LD among other things
LDHat Archived 2016-05-13 at the Wayback Machine
Haploview
LdCompare^[13]— 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.
LDlink – A suite of web-based applications to easily and efficiently explore linkage disequilibrium in population subgroups. All population genotype data originates from Phase 3 of the 1000 Genomes Project and variant RS numbers are indexed based on dbSNP build 151.
Bcftools – utilities for variant calling and manipulating VCFs and BCFs.

Simulation software[edit]

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

References[edit]

^ ^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. PMC 5124487. PMID 18427557.
^ Falconer, DS; Mackay, TFC (1996). Introduction to Quantitative Genetics (4th ed.). Harlow, Essex, UK: Addison Wesley Longman. ISBN 978-0-582-24302-6.
^ 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. ISSN 1471-0056. PMC 5124487. PMID 18427557.
^ Calabrese, Barbara (2019-01-01), "Linkage Disequilibrium", in Ranganathan, Shoba; Gribskov, Michael; Nakai, Kenta; Schönbach, Christian (eds.), Encyclopedia of Bioinformatics and Computational Biology, Oxford: Academic Press, pp. 763–765, doi:10.1016/b978-0-12-809633-8.20234-3, ISBN 978-0-12-811432-2, S2CID 226248080, retrieved 2020-10-21
^ Lewontin, R. C. (1964). "The interaction of selection and linkage. I. General considerations; heterotic models". Genetics. 49 (1): 49–67. doi:10.1093/genetics/49.1.49. PMC 1210557. PMID 17248194.
^ Hill, W.G. & Robertson, A. (1968). "Linkage disequilibrium in finite populations". Theoretical and Applied Genetics. 38 (6): 226–231. doi:10.1007/BF01245622. PMID 24442307. S2CID 11801197.{{cite journal}}: CS1 maint: multiple names: authors list (link)
^ ^a ^b Kang, Jonathan T.L.; Rosenberg, Noah A. (2019). "Mathematical Properties of Linkage Disequilibrium Statistics Defined by Normalization of the Coefficient D = pAB – pApB". Human Heredity. 84 (3): 127–143. doi:10.1159/000504171. ISSN 0001-5652. PMC 7199518. PMID 32045910.
^ VanLiere, J.M. & Rosenberg, N.A. (2008). "Mathematical properties of the $r^{2}$ measure of linkage disequilibrium". Theoretical Population Biology. 74 (1): 130–137. doi:10.1016/j.tpb.2008.05.006. PMC 2580747. PMID 18572214.{{cite journal}}: CS1 maint: multiple names: authors list (link)
^ Smith, R.D. (2020). "The nonlinear structure of linkage disequilibrium". Theoretical Population Biology. 134: 160–170. doi:10.1016/j.tpb.2020.02.005. PMID 32222435. S2CID 214716456.
^ Robbins, R.B. (1 July 1918). "Some applications of mathematics to breeding problems III". Genetics. 3 (4): 375–389. doi:10.1093/genetics/3.4.375. PMC 1200443. PMID 17245911.
^ 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.
^ Devlin B.; Risch N. (1995). "A Comparison of Linkage Disequilibrium Measures for Fine-Scale Mapping" (PDF). Genomics. 29 (2): 311–322. CiteSeerX 10.1.1.319.9349. doi:10.1006/geno.1995.9003. PMID 8666377.
^ 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.

Population genetics
Key concepts	Hardy–Weinberg principle Genetic linkage Identity by descent Linkage disequilibrium Fisher's fundamental theorem Neutral theory Shifting balance theory Price equation Coefficient of inbreeding Coefficient of relationship Selection coefficient Fitness Heritability Population structure Constructive neutral evolution
Selection	Natural Artificial Sexual Ecological
Effects of selection on genomic variation	Genetic hitchhiking Background selection
Genetic drift	Small population size Population bottleneck Founder effect Coalescence Balding–Nichols model
Founders	R. A. Fisher J. B. S. Haldane Sewall Wright
Related topics	Biogeography Evolution Evolutionary game theory Fitness landscape Genetic genealogy Landscape genetics and genomics Microevolution Population genomics Phylogeography Quantitative genetics
Index of evolutionary biology articles