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Linkage disequilibrium (LD) is a term used in the study of population genetics for 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. LD 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.

Linkage disequilibrium is caused by fitness interactions between genes or by such non-adaptive processes as population structure, inbreeding, and stochastic effects. In population genetics, linkage disequilibrium is said to characterize the haplotype distribution at two or more loci. Formally, if we define pairwise LD, we consider indicator variables on alleles at two loci, say I_1, I_2. We define the LD parameter \delta as:

\delta := \operatorname{cov}(I_1, I_2) = p_1 p_2 - h_{12}.

Here p_1, p_2 denote the marginal allele frequencies at the two loci and h_{12} denotes the haplotype frequency in the joint distribution of both alleles. Various derivatives of this parameter have been developed. In the genetic literature the wording "two alleles are in LD" usually means to imply \delta \ne 0. Contrariwise, linkage equilibrium, denotes the case \delta = 0.

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.

Linkage disequilibrium measure, DEdit

If inspecting the two loci A and B with two alleles each, a two-locus, two-allele model, the following table shall denote the frequencies of each combination:

Haplotype Frequency
A_1B_1 x_{11}
A_1B_2 x_{12}
A_2B_1 x_{21}
A_2B_2 x_{22}

From there one can 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 otherjhjkk, then one can express the oservation A1B1 as "A1 must be found and B1 must be found". The table above lists the frequencies for A_1, p_1, and B_1, q_1, hence the frequency of A_1B_1, x_{11}, equals according to the rules of elementary statistics x_{11} = p_{1} * q_{1}.

A deviation of the observed frequencies from the expected is referred to as the linkage disequilibrium parameter of Lewontin and Kojima (1960)[1] and commonly denoted by a capital D as defined by D = x_{11} - p_1q_1. It is vividly presented in the following table.

A_1 A_2 Total
B_1 x_{11}=p_1q_1+D x_{21}=p_2q_1-D q_1
B_2 x_{12}=p_1q_2-D x_{22}=p_2q_2+D q_2
Total p_1 p_2 1

When extending these formula for diploid cells rather than investigating the gametes/haplotypes directly, the laid out principle prevails, the recombination rate between the two loci A and B must be taken into account, though, which is commonly denoted by the letter c.

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

Another value is the correlation coefficient as also laid out in the initial paragraphs of this page, denoted as r^2=\frac{D^2}{p_1p_2q_1q_2}. This however is not adjusted to the loci having different allele frequencies. If it was, r, the square root of r^2 if given the sign of D would be equivalent to D' [2]

See alsoEdit

References Edit

  1. R.C. Lewontin and K. Kojima (1960). The evolutionary dynamics of complex polymorphisms.. Evolution 14: 458-472.
  2. P.W. Hendrick and S. Kumar (2001). Mutation and linkage disequilibrium in human mtDNA. Eur. J. Hum. Genet. 9: 969-972.

Further readingEdit

  • Hedrick, Philip W. (2005). Genetics of Populations, 3rd, Sudbury, Boston, Toronto, London, Singapure: Jones and Bartlett Publishers. ISBN 0-7637-4882-6.
Topics in population genetics(edit)
Key concepts: Hardy-Weinberg law | linkage disequilibrium | Fisher's fundamental theorem | neutral theory
Selection: natural | sexual | artificial | ecological
Genetic drift: small population size | population bottleneck | founder effect | coalescence
Founders: Ronald Fisher | J.B.S. Haldane | Sewall Wright
Related topics: evolution | microevolution | evolutionary game theory | fitness landscape
List of evolutionary biology topics
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