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In population genetics, linkage disequilibrium (LD) is a measure of non-random association between segments of DNA (alleles) at different positions on the chromosome (loci) in a given population based on a comparison between the frequency at which two alleles are detected together at the same loci and the frequencies at which each allele is detected at that loci overall, whether it occurs with or without the other allele of interest. Loci are said to be in linkage disequilibrium when the frequency of being detected together (the frequency of association of their different alleles) is higher or lower than expected if the loci were independent and associated randomly.^[1]

While the pattern of linkage disequilibrium in a genome is a powerful signal of the population genetic processes that are structuring it, it does not indicate why the pattern emerges by itself. 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.

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

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

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

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

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

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

The linkage disequilibrium ${\displaystyle D}$ 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. Normalization of linkage disequilibrium allows these alleles to be compared more easily.

D' Method

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

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

where

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

The value of ${\displaystyle D'}$ will be within the range ${\displaystyle -1\leq D'\leq 1}$. When ${\displaystyle D'=0}$, the loci are independent. When ${\displaystyle -1\leq D'<0}$Zdroj:https://en.wikipedia.org?pojem=Linkage_disequilibrium
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