Linkage disequilibrium 681230 222923159 2008-07-01T19:28:54Z DOI bot 6652755 Citation maintenance. Formatted: title. You can [[WP:DOI|use this bot]] yourself! Please [[User:DOI_bot/bugs|report any bugs]]. In [[population genetics]], '''linkage disequilibrium''' is the non-random association of [[allele]]s at two or more [[locus (genetics)|loci]], not necessarily on the same [[chromosome]]. It is not the same as [[Genetic linkage|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 [[haplotype]]s from [[allele]]s based on their frequencies. Non-random associations between polymorphisms at different loci are measured by the degree of linkage disequilibrium (LD). Linkage disequilibrium is generally caused by genetic linkage and the rate of [[recombination]]; rate of [[mutation]]; [[random drift]] or 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. ==Linkage disequilibrium measure, ''&delta;''== Formally, if we define pairwise LD, we consider [[indicator variable]]s on [[allele]]s at two loci, say <math>I_1, I_2</math>. We define the LD parameter <math>\delta</math> (delta) as: : <math>\delta := \operatorname{cov}(I_1, I_2) = h_{12} - p_1 p_2= h_{12}h_{21} - h_{11}h_{22} </math> Here <math>p_1, p_2 </math> denote the marginal [[allele frequency|allele frequencies]] at the two loci and <math>h_{12}</math> 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 <math>\delta \ne 0</math>. Contrariwise, linkage equilibrium, denotes the case <math>\delta = 0</math>. ==Linkage disequilibrium measure, ''D''== If inspecting the 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 |- |<math>A_1B_1</math> |<math>x_{11}</math> |- |<math>A_1B_2</math> |<math>x_{12}</math> |- |<math>A_2B_1</math> |<math>x_{21}</math> |- |<math>A_2B_2</math> |<math>x_{22}</math> |} Note that these are [[Frequency (statistics)|relative frequencies]]. One can use the above frequencies to determine the frequency of each of the alleles: {| |Allele||Frequency |- |<math>A_1</math>||<math>p_{1}=x_{11}+x_{12}</math> |- |<math>A_2</math>||<math>p_{2}=x_{21}+x_{22}</math> |- |<math>B_1</math>||<math>q_{1}=x_{11}+x_{21}</math> |- |<math>B_2</math>||<math>q_{2}=x_{12}+x_{22}</math> |} if the two loci and the alleles are independent from each other, then one can express the observation A1B1 as "A1 must be found and B1 must be found". The table above lists the frequencies for <math>A_1, p_1</math>, and <math>B_1, q_1</math>, hence the frequency of <math>A_1B_1</math>, <math>x_{11}</math>, equals according to the rules of elementary statistics <math>x_{11} = p_{1} * q_{1}</math>. A deviation of the observed frequencies from the expected is referred to as the linkage disequilibrium parameter<ref>{{cite journal|author=Robbins, R.B.|title=Some applications of mathematics to breeding problems III|journal=Genetics|year=1918|volume=3|pages=375–389|url=http://www.genetics.org/cgi/reprint/3/4/375}}</ref>, and is commonly denoted by a capital D <ref>{{cite journal|author=R.C. Lewontin and K. Kojima| year=1960| title=The evolutionary dynamics of complex polymorphisms| journal=Evolution| volume=14| pages=458–472| url=http://links.jstor.org/sici?sici=0014-3820%28196012%2914%3A4%3C458%3ATEDOCP%3E2.0.CO%3B2-4|doi=10.2307/2405995| issue=4}}</ref> as defined by: {| | |<math>D = x_{11} - p_1q_1</math> |} The following table illustrates the relationship between the haplotype and allele frequencies and D. {| | ||<math>A_1</math> ||<math>A_2</math> ||Total |- |<math>B_1</math>||<math>x_{11}=p_1q_1+D</math>&nbsp;&nbsp;&nbsp;&nbsp;||<math>x_{21}=p_2q_1-D</math>&nbsp;&nbsp;&nbsp;||<math>q_1</math> |- |<math>B_2</math>||<math>x_{12}=p_1q_2-D</math> ||<math>x_{22}=p_2q_2+D</math>||<math>q_2</math> |- |Total&nbsp;&nbsp;&nbsp;||<math>p_1</math>||<math>p_2</math>||<math>1</math> |} When extending these formula for [[diploid]] cells rather than investigating the [[gamete]]s/[[haplotype]]s directly, the laid out principle prevails, the [[Genetic_recombination|recombination rate]] between the two loci <math>A</math> and <math>B</math> must be taken into account, though, which is commonly denoted by the letter <math>c</math>. <math>D</math> is nice to calculate with but has the disadvantage of depending on the frequency of the alleles inspected. This is evident since frequencies are between 0 and 1. There can be no <math>D</math> 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 <math>D'=\frac{D}{D_\max}</math> when <math>D \ge 0</math> When <math>D < 0 </math>, <math>D'=\frac{D}{D_\min}</math>. <math>D_\max</math> is given by the smaller of <math>p_1 q_2</math> and <math>p_2 q_1</math>. <math>D_\min</math> is given by the larger of <math>-p_1 q_1</math> and <math>-p_2 q_2</math> Another measure of LD commonly reported with D' is the [[correlation coefficient]] between pairs of loci, denoted as <math>r^2=\frac{D^2}{p_1p_2q_1q_2}</math>. This however is not adjusted to the loci having different allele frequencies. If it was, <math>r</math>, the square root of <math>r^2</math> if given the sign of <math>D</math> would be equivalent to <math>D'</math> <ref>{{cite journal|author=P.W. Hedrick and S. Kumar| year=2001| title=Mutation and linkage disequilibrium in human mtDNA| journal=Eur. J. Hum. Genet.| volume=9| pages=969–972| doi=10.1038/sj.ejhg.5200735}}</ref> ==Resources== A comparison of different measures of LD is provided by Devlin & Risch <ref>{{cite journal|author=Devlin B., Risch N. |title=A Comparison of Linkage Disequilibrium Measures for Fine-Scale Mapping|journal=Genomics|year=1995|volume=29|pages=311–322|url=http://www.sciencedirect.com/science?_ob=MImg&_imagekey=B6WG1-45S9156-30-1&_cdi=6809&_user=128590&_orig=browse&_coverDate=09%2F30%2F1995&_sk=999709997&view=c&wchp=dGLbVtb-zSkzk&md5=71c2158ad4c51ae80b12a68c68814f78&ie=/sdarticle.pdf |doi=10.1006/geno.1995.9003}}</ref> The [[International HapMap Project]] enables the study of LD in human populations [http://www.hapmap.org/cgi-perl/gbrowse/hapmap/ online]. The [[Ensembl]] project integrates HapMap data and such from [[dbSNP]] in general with other genetic information. ==Analysis software== * [[Haploview]] * [http://www.affymetrix.com/support/developer/tools/devnettools.affx LdCompare]<ref>{{cite journal|author=Hao K., Di X., Cawley S.|title=LdCompare: rapid computation of single- and multiple-marker r2 and genetic coverage|journal=Bioinformatics|year=2007|volume=23|pages=252–254|url=http://bioinformatics.oxfordjournals.org/cgi/reprint/23/2/252|doi=10.1093/bioinformatics/btl574|pmid=17148510}}</ref> &mdash; open-source software for calculating LD. * [http://www.pypop.org/ PyPop] * [http://www.goldenhelix.com/SNP_Variation/HelixTree/linkage_disequilibrium_analysis.html HelixTree] - commercial software with interactive LD plot. == References == <references/> ==See also== *[[Haploview]] *[[Hardy-Weinberg principle]] *[[Genetic linkage]] *[[Co-adaptation]] *[[Genealogical DNA test]] *[[Tag SNP]] ==Further reading== *{{cite book| first=Philip W.| last=Hedrick| title=Genetics of Populations| year=2005 | edition=3rd| id=ISBN 0763747726| location=Sudbury, Boston, Toronto, London, Singapore | publisher=Jones and Bartlett Publishers}} * [http://www.nslij-genetics.org/ld/ Bibliography: Linkage Disequilibrium Analysis] : a bibliography of more than one thousand articles on Linkage disequilibrium published since 1918. {{popgen}} [[Category:Population genetics]] [[it:Linkage disequilibrium]] [[he:תאחיזה לא שוויונית]] [[ja:連鎖不平衡]] [[es:Desequilibrio de ligamiento]]