TheInfoList

In
population genetics Population genetics is a subfield of that deals with genetic differences within and between s, and is a part of . Studies in this branch of examine such phenomena as , , and . Population genetics was a vital ingredient in the of the . Its pri ...
, linkage disequilibrium (LD) is the non-random association of
allele An allele (, ; ; modern formation from Greek ἄλλος ''állos'', "other") is one of two, or more, forms of a given gene In biology, a gene (from ''genos'' "...Wilhelm Johannsen coined the word gene to describe the Mendelian_inheritance ...
s at different
loci Locus (plural loci) is Latin for "place". It may refer to: Entertainment * Locus (comics), a Marvel Comics mutant villainess, a member of the Mutant Liberation Front * Locus (magazine), ''Locus'' (magazine), science fiction and fantasy magazine ...
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 what would be expected if the loci were independent and associated randomly. Linkage disequilibrium is influenced by many factors, including
selection Selection may refer to: In science: * Selection (biology) Natural selection is the differential survival and reproduction of individuals due to differences in phenotype right , Here the relation between genotype and phenotype is ill ...
, the rate of
genetic recombination Genetic recombination (also known as genetic reshuffling) is the exchange of genetic material between different organism In biology Biology is the natural science that studies life and living organisms, including their anatomy, ph ...
,
mutation rate In genetics Genetics is a branch of biology concerned with the study of genes, genetic variation, and heredity in organisms.Hartl D, Jones E (2005) Though heredity had been observed for millennia, Gregor Mendel, Moravia, Moravian scientist an ...
,
genetic drift Genetic drift (allelic drift or the Sewall Wright effect) is the change in the frequency of an existing gene In biology, a gene (from ''genos'' "...Wilhelm Johannsen coined the word gene to describe the Mendelian_inheritance#History, M ...

, the system of mating, population structure, and
genetic linkage Genetic linkage is the tendency of DNA sequences that are close together on a chromosome to be inherited together during the meiosis phase of sexual reproduction. Two genetic markers that are physically near to each other are unlikely to be separa ...
. 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). Furthermore, linkage disequilibrium is sometimes referred to as
gametic phaseIn genetics, a gametic phase represents the original allelic combinations that a diploid individual inherits from both parents. It is therefore a particular association of alleles at different loci on the same chromosome. Gametic phase is influenced ...
disequilibrium; however, the concept also applies to asexual organisms and therefore does not depend on the presence of
gametes A gamete ( /ˈɡæmiːt/; from Ancient Greek Ancient Greek includes the forms of the Greek language used in ancient Greece and the classical antiquity, ancient world from around 1500 BC to 300 BC. It is often roughly divided into the foll ...
.

# Formal definition

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 $p_$ be the frequency with which both ''A'' and ''B'' occur together in the same gamete (i.e. $p_$ is the frequency of the ''AB''
haplotype A haplotype (haploid Ploidy () is the number of complete sets of chromosome A chromosome is a long DNA molecule with part or all of the genetic material of an organism. Most eukaryotic chromosomes include packaging proteins called hi ...
). The association between the alleles ''A'' and ''B'' can be regarded as completely random—which is known in
statistics Statistics is the discipline that concerns the collection, organization, analysis, interpretation, and presentation of data Data (; ) are individual facts, statistics, or items of information, often numeric. In a more technical sens ...

as ''
independence upright=1.0, Pedro surrounded by a crowd in Brazil's independence on September 7, 1822.">Independence of Brazil">Brazil's independence on September 7, 1822. Independence is a condition of a person, nation, country, or state State may ref ...
''—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 $p_ p_$ of the probabilities. There is said to be a linkage disequilibrium between the two alleles whenever $p_$ 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'' $D_$, which is defined as : $D_ = p_ - p_A p_B,$ provided that both $p_A$ and $p_B$ are greater than zero. Linkage disequilibrium corresponds to $D_ \neq 0$. In the case $D_=0$ we have $p_ = p_A p_B$ and the alleles ''A'' and ''B'' are said to be in ''linkage equilibrium''. The subscript "AB" on $D_$ emphasizes that linkage disequilibrium is a property of the pair 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, $D_ = -D_ = -D_ = D_$. Their relationships can be characterized as follows. $D = P_ -P_*P_$ $-D = P_ -P_*P_$ $-D = P_ -P_*P_$ $D = P_ -P_*P_$ 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. 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.

# Measures derived from ''D''

The coefficient of linkage disequilibrium $D$ 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 suggested normalising ''D'' by dividing it by the theoretical maximum difference between the observed and expected haplotype frequencies as follows: : $D\text{'} = \frac D$ where : $D_\max= \begin \max\ & \text D < 0\\ \min\ & \text D > 0 \end$ An alternative to $D\text{'}$ is the
correlation coefficient A correlation coefficient is a numerical measure of some type of correlation and dependence, correlation, meaning a statistical relationship between two variable (mathematics), variables. The variables may be two column (database), columns of a give ...
between pairs of loci, usually expressed as its square, $r^2$ : $r^2=\frac.$

## Limits for the ranges of linkage disequilibrium measures

The measures $r^2$ and $D\text{'}$ 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 $D<0$. While $D\text{'}$ can always take a maximum value of 1, it's minimum value for two loci is equal to $, r,$ for those loci.

# Example: Two-loci and two-alleles

Consider the
haplotypes A haplotype ( haploid genotype) is a group of alleles in an organism In biology, an organism (from Ancient Greek, Greek: ὀργανισμός, ''organismos'') is any individual contiguous system that embodies the Life#Biology, propertie ...
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: Note that these are
relative frequencies In statistics Statistics is the discipline that concerns the collection, organization, analysis, interpretation, and presentation of data. In applying statistics to a scientific, industrial, or social problem, it is conventional to begin wit ...
. One can use the above frequencies to determine the frequency of each of the alleles: If the two loci and the alleles are
independent Independent or Independents may refer to: Arts, entertainment, and media Artist groups * Independents (artist group), a group of modernist painters based in the New Hope, Pennsylvania, area of the United States during the early 1930s * Independent ...

from each other, then one can express the observation $A_1B_1$ 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 $A_1B_1$ is $x_$, and according to the rules of elementary statistics $x_ = p_1 q_1$. The deviation of the observed frequency of a haplotype from the expected is a quantity called the linkage disequilibrium and is commonly denoted by a capital ''D'': : $D = x_ - p_1q_1$ The following table illustrates the relationship between the haplotype frequencies and allele frequencies and D.

# Role of recombination

In the absence of evolutionary forces other than
random mating Panmixia (or panmixis) means random mating. A panmictic population is one where all individuals are potential partners. This assumes that there are no mating restrictions, neither Genetics, genetic nor behavioural, upon the population and that there ...
, Mendelian segregation, random chromosomal assortment, and
chromosomal crossover Chromosomal crossover, or crossing over, is the exchange of genetic material during sexual reproduction between two homologous chromosomes' sister chromatids, non-sister chromatids that results in recombinant chromosomes. It is one of the final ...
(i.e. in the absence of
natural selection Natural selection is the differential survival and reproduction of individuals due to differences in phenotype right , Here the relation between genotype and phenotype is illustrated, using a Punnett square, for the character of peta ...
,
inbreeding Inbreeding is the production of offspring In biology, offspring are the young creation of living organisms, produced either by a Asexual reproduction, single organism or, in the case of sexual reproduction, two organisms. Collective offspri ...
, and
genetic drift Genetic drift (allelic drift or the Sewall Wright effect) is the change in the frequency of an existing gene In biology, a gene (from ''genos'' "...Wilhelm Johannsen coined the word gene to describe the Mendelian_inheritance#History, M ...

), 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, $D= x_-p_1 q_1$, we can demonstrate this convergence to zero as follows. In the next generation, $x_\text{'}$, the frequency of the haplotype $A_1 B_1$, becomes : $x_\text{'} = \left(1-c\right)\,x_ + c\,p_1 q_1$ This follows because a fraction $\left(1-c\right)$ of the haplotypes in the offspring have not recombined, and are thus copies of a random haplotype in their parents. A fraction $x_$ 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 : $x_\text{'} - p_1 q_1 = \left(1-c\right)\,\left(x_ - p_1 q_1\right)$ so that : $D_1 = \left(1-c\right)\;D_0$ where $D$ at the $n$-th generation is designated as $D_n$. Thus we have : $D_n = \left(1-c\right)^n\; D_0.$ If $n \to \infty$, then $\left(1-c\right)^n \to 0$ 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.

# Example: Human leukocyte antigen (HLA) alleles

HLA constitutes a group of cell surface antigens also known as the MHC of humans. Because HLA genes are located at adjacent loci on the particular region of a chromosome and presumed to exhibit
epistasis File:Epistasis.png, Example of epistasis in coat colour genetics: If no pigments can be produced the other coat colour genes have no effect on the phenotype, no matter if they are dominant or if the individual is homozygous. Here the genotype "c ...

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 DanesSvejgaard 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, (pbk). referred to by Vogel and Motulsky (1997).Vogel F, Motulsky AG (1997). ''Human Genetics : Problems and Approaches, 3rd ed.''Berlin; London: Springer, . Because HLA is codominant and HLA expression is only tested locus by locus in surveys, LD measure is to be estimated from such a 2×2 table to the right.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, . expression ($+$) frequency of antigen $i$ : :$pf_i = \frac C N = 0.311;$ expression ($+$) frequency of antigen $j$ : :$pf_j = \frac A N = 0.237;$ frequency of gene $i$, given that individuals with '+/−', '+/+', and '−/+' genotypes are all positive for antigen $i$: :$gf_i = 1 - \sqrt = 0.170,$ and :$hf_ = \text ij = gf_i \; gf_j = 0.0215.$ Denoting the '―' alleles at antigen ''i'' to be ''x'', and at antigen ''j'' to be ''y'', the observed frequency of haplotype ''xy'' is : $o$\sqrt and the estimated frequency of haplotype ''xy'' is :$e$\sqrt. Then LD measure $\Delta_$ is expressed as :$\Delta_=o$e\frac=0.0769. Standard errors $SEs$ are obtained as follows: :$SE\text gf_i = \frac = 0.00628.$ :$SE\text hf_ = \sqrt=0.00514$ :$SE\text\Delta_=\frac\sqrt=0.00367.$ Then, if :$t = \frac$ exceeds 2 in its absolute value, the magnitude of $\Delta_$ 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 shows some of the combinations of HLA-A and B alleles where significant LD was observed among pan-Europeans. Vogel and Motulsky (1997) 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 it is mostly non-significant. If $\Delta_0$ had reduced from 0.07 to 0.003 under recombination effect as shown by $\Delta_n=\left(1-c\right)^n \Delta_0$, then $n\approx 400$. 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. 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. * The HLA antigen frequency among patients exceeds more than that among a healthy population. This is evaluated by $\delta$ value to exceed 0. *2×2 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 The relative risk (RR) or risk ratio is the ratio of the probability Probability is the branch of mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and relat ...
of an HLA allele for a disease is approximated by the
odds ratio An odds ratio (OR) is a statistic A statistic (singular) or sample statistic is any quantity computed from values in a sample which is considered for a statistical purpose. Statistical purposes include estimating a population Population ty ...
in the 2×2 association table of the allele with the disease. Table 3 shows association of HLA-B27 with ankylosing spondylitis among a Dutch population. Relative risk $x$of this allele is approximated by : $x=\frac=\frac\;\left(=39.7,\text\right).$ Woolf's method is applied to see if there is statistical significance. Let :$y=\ln \left(x\right)\;\left(=3.68\right)$ and :$\frac=\frac+\frac+\frac+\frac\;\left(=0.0703\right).$ Then :
Fisher's exact test Fisher's exact test is a statistical significance test used in the analysis of contingency tables. Although in practice it is employed when sample (statistics), sample sizes are small, it is valid for all sample sizes. It is named after its inventor ...
.Sokal RR, Rohlf FJ (1981). ''Biometry: The Principles and Practice of Statistics in Biological Research.'' Oxford: W.H. Freeman, .

# Resources

A comparison of different measures of LD is provided by Devlin & Risch The
International HapMap Project The International HapMap Project was an organization that aimed to develop a haplotype map (HapMap) of the human genome, to describe the common patterns of human genetic variation. HapMap is used to find genetic variants affecting health, diseas ...
enables the study of LD in human population
online
The
EnsemblEnsembl genome database project is a scientific project at the European Bioinformatics Institute, which was launched in 1999 in response to the imminent completion of the Human Genome Project. Ensembl aims to provide a centralized resource for geneti ...
project integrates HapMap data with other genetic information from
dbSNP The Single Nucleotide Polymorphism Database (dbSNP) is a free public archive for genetic variation thumb File:Genetic Variation and Inheritance.svg, Parents have similar gene coding in this specific situation where they reproduce and variatio ...
.

# Analysis software

– whole genome association analysis toolset, which can calculate LD among other things
LDHat
*
Haploview Haploview is a commonly used bioinformatics softwareThe list of bioinformatics software tools can be split up according to the license used: *List of proprietary bioinformatics software *List of open-source bioinformatics software Alternatively, he ...

LdCompare
ref>
— open-source software for calculating LD.

– commercial software with interactive LD plot.

– 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 The 1000 Genomes Project (abbreviated as 1KGP), launched in January 2008, was an international research effort to establish by far the most detailed catalogue of . Scientists planned to the s of at least one thousand anonymous participants from a ...

and
HapMap The International HapMap Project was an organization that aimed to develop a haplotype A haplotype (haploid Ploidy () is the number of complete sets of chromosome A chromosome is a long DNA molecule with part or all of the genetic mat ...
genotype databases.
SNeP
– Fast computation of LD and Ne for large genotype datasets in PLINK format.
– 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

Haploid
— a C library for population genetic simulation (
GPL The GNU General Public License (GNU GPL or simply GPL) is a series of widely used free software license A free-software license is a notice that grants the recipient of a piece of software extensive rights to modify and redistribute that ...
)

*
Haploview Haploview is a commonly used bioinformatics softwareThe list of bioinformatics software tools can be split up according to the license used: *List of proprietary bioinformatics software *List of open-source bioinformatics software Alternatively, he ...
*
Hardy–Weinberg principle In population genetics, the Hardy–Weinberg principle, also known as the Hardy–Weinberg equilibrium, model, theorem, or law, states that allele An allele (, ; ; modern formation from Greek ἄλλος ''állos'', "other") is one of two, o ...
*
Genetic linkage Genetic linkage is the tendency of DNA sequences A nucleic acid sequence is a succession of bases signified by a series of a set of five different letters that indicate the order of nucleotides Nucleotides are organic molecules , CH4; is ...
*
Co-adaptationIn biology, co-adaptation is the process by which two or more species, genes or phenotypic traits undergo adaptation as a pair or group. This occurs when two or more interacting characteristics undergo natural selection together in response to the sa ...
*
Genealogical DNA test A genealogical DNA test is a DNA The structure of part of a DNA double helix Deoxyribonucleic acid (; DNA) is a molecule File:Pentacene on Ni(111) STM.jpg, A scanning tunneling microscopy image of pentacene molecules, which consist o ...
*
Tag SNP A tag SNP is a representative Single-nucleotide polymorphism, single nucleotide polymorphism (SNP) in a region of the genome with high linkage disequilibrium that represents a group of SNPs called a haplotype. It is possible to identify genetic vari ...
*
Association mappingIn genetics, association mapping, also known as " linkage disequilibrium mapping", is a method of mapping quantitative trait loci (QTLs) that takes advantage of historic linkage disequilibrium to link phenotype right , Here the relation between ...
*
Family based QTL mapping Quantitative trait loci mapping or QTL mapping is the process of identifying genomic regions that potentially contain genes responsible for important economic, health or environmental characters. Mapping QTLs is an important activity that plant bre ...