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In
population genetics Population genetics is a subfield of genetics that deals with genetic differences within and between populations, and is a part of evolutionary biology. Studies in this branch of biology examine such phenomena as adaptation, speciation, and po ...
, linkage disequilibrium (LD) is the non-random association of
allele An allele (, ; ; modern formation from Greek ἄλλος ''állos'', "other") is a variation of the same sequence of nucleotides at the same place on a long DNA molecule, as described in leading textbooks on genetics and evolution. ::"The chrom ...
s 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 what would be expected if the loci were independent and associated randomly. Linkage disequilibrium is influenced by many factors, including selection, the rate of
genetic recombination Genetic recombination (also known as genetic reshuffling) is the exchange of genetic material between different organisms which leads to production of offspring with combinations of traits that differ from those found in either parent. In eukaryo ...
,
mutation rate In genetics, the mutation rate is the frequency of new mutations in a single gene or organism over time. Mutation rates are not constant and are not limited to a single type of mutation; there are many different types of mutations. Mutation rates ...
,
genetic drift Genetic drift, also known as allelic drift or the Wright effect, is the change in the frequency of an existing gene variant (allele) in a population due to random chance. Genetic drift may cause gene variants to disappear completely and there ...
, 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 sep ...
. 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 phase disequilibrium; however, the concept also applies to asexual organisms and therefore does not depend on the presence of
gametes A gamete (; , ultimately ) is a haploid cell that fuses with another haploid cell during fertilization in organisms that reproduce sexually. Gametes are an organism's reproductive cells, also referred to as sex cells. In species that produce ...
.


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 genotype) is a group of alleles in an organism that are inherited together from a single parent. Many organisms contain genetic material ( DNA) which is inherited from two parents. Normally these organisms have their DNA o ...
). The association between the alleles ''A'' and ''B'' can be regarded as completely random—which is known in
statistics Statistics (from German: '' Statistik'', "description of a state, a country") is the discipline that concerns the collection, organization, analysis, interpretation, and presentation of data. In applying statistics to a scientific, indust ...
as ''
independence Independence is a condition of a person, nation, country, or state in which residents and population, or some portion thereof, exercise self-government, and usually sovereignty, over its territory. The opposite of independence is the stat ...
''—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 Product may refer to: Business * Product (business), an item that serves as a solution to a specific consumer problem. * Product (project management), a deliverable or set of deliverables that contribute to a business solution Mathematics * Produ ...
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' = \frac D where : D_\max= \begin \max\ & \text D < 0\\ \min\ & \text D > 0 \end An alternative to D' is the
correlation coefficient A correlation coefficient is a numerical measure of some type of correlation, meaning a statistical relationship between two variables. The variables may be two columns of a given data set of observations, often called a sample, or two components ...
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' 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' can always take a maximum value of 1, its 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 that are inherited together from a single parent. Many organisms contain genetic material ( DNA) which is inherited from two parents. Normally these organisms have their DNA org ...
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, the frequency (or absolute frequency) of an event i is the number n_i of times the observation has occurred/recorded in an experiment or study. These frequencies are often depicted graphically or in tabular form. Types The cumul ...
. 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 * Independe ...
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 forB_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, 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' non-sister chromatids that results in recombinant chromosomes. It is one of the final phases of ge ...
(i.e. in the absence of
natural selection Natural selection is the differential survival and reproduction of individuals due to differences in phenotype. It is a key mechanism of evolution, the change in the heritable traits characteristic of a population over generations. Cha ...
,
inbreeding Inbreeding is the production of offspring from the mating or breeding of individuals or organisms that are closely related genetically. By analogy, the term is used in human reproduction, but more commonly refers to the genetic disorders a ...
, and
genetic drift Genetic drift, also known as allelic drift or the Wright effect, is the change in the frequency of an existing gene variant (allele) in a population due to random chance. Genetic drift may cause gene variants to disappear completely and there ...
), 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_', the frequency of the haplotype A_1 B_1, becomes : x_' = (1-c)\,x_ + 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_ 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_' - p_1 q_1 = (1-c)\,(x_ - 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.


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 Epistasis is a phenomenon in genetics in which the effect of a gene mutation is dependent on the presence or absence of mutations in one or more other genes, respectively termed modifier genes. In other words, the effect of the mutation is dep ...
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 f_\sqrt and the estimated frequency of haplotype ''xy'' is :e f_\sqrt. Then LD measure \Delta_ is expressed as :\Delta_=o f_e f_\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=(1-c)^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 of an outcome in an exposed group to the probability of an outcome in an unexposed group. Together with risk difference and odds ratio, relative risk measures the association be ...
of an HLA allele for a disease is approximated by the
odds ratio An odds ratio (OR) is a statistic that quantifies the strength of the association between two events, A and B. The odds ratio is defined as the ratio of the odds of A in the presence of B and the odds of A in the absence of B, or equivalently (due ...
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 xof this allele is approximated by : x=\frac=\frac\;(=39.7,\text). Woolf's method is applied to see if there is statistical significance. Let :y=\ln (x)\;(=3.68) and :\frac=\frac+\frac+\frac+\frac\;(=0.0703). Then :\chi^2=wy^2\;\left 193>\chi^2(p=0.001,\; df=1)=10.8 \right /math> follows the chi-square distribution with df=1. In the data of Table 3, a significant association exists at the 0.1% level. Haldane's modification applies to the case when either of a,\; b,\;c,\textd is zero, where x and 1/w are replaced with :x=\frac and :\frac=\frac+\frac+\frac+\frac, respectively. In Table 4, some examples of association between HLA alleles and diseases are presented. (1a) Allele frequency excess among patients over controls Even high relative risks between HLA alleles and the diseases were observed, only the magnitude of relative risk would not be able to determine the strength of association.\delta value is expressed by :\delta=\frac,\;\;0\le \delta \le 1, where FAD and FAP are HLA allele frequencies among patients and healthy populations, respectively. In Table 4, \delta column was added in this quotation. Putting aside 2 diseases with high relative risks both of which are also with high \delta values, among other diseases, juvenile diabetes mellitus (type 1) has a strong association with DR4 even with a low relative risk=6. (2) Discrepancies from expected values from marginal frequencies in 2×2 association table of HLA alleles and disease This can be confirmed by \chi^2 test calculating :\chi^2=\frac\;(=336,\textP<0.001). where df=1. For data with small sample size, such as no marginal total is greater than 15 (and consequently N \le 30), one should utilize
Yates's correction for continuity In statistics, Yates's correction for continuity (or Yates's chi-squared test) is used in certain situations when testing for independence in a contingency table. It aims at correcting the error introduced by assuming that the discrete probabilities ...
or
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 sizes are small, it is valid for all sample sizes. It is named after its inventor, Ronald Fisher, a ...
.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, disease ...
enables the study of LD in human population
online
The
Ensembl Ensembl genome database project is a scientific project at the European Bioinformatics Institute, which provides a centralized resource for geneticists, molecular biologists and other researchers studying the genomes of our own species and other v ...
project integrates HapMap data with other genetic information from
dbSNP The Single Nucleotide Polymorphism Database (dbSNP) is a free public archive for genetic variation within and across different species developed and hosted by the National Center for Biotechnology Information (NCBI) in collaboration with the Natio ...
.


Analysis software


PLINK
– whole genome association analysis toolset, which can calculate LD among other things
LDHat
*
Haploview Haploview is a commonly used bioinformatics software which is designed to analyze and visualize patterns of linkage disequilibrium (LD) in genetic data. Haploview can also perform association studies, choosing tagSNPs{{cite journal, author1=de Bakk ...

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 human genetic variation. Scientists planned to sequence the genomes of at least one th ...
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


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


See also

*
Haploview Haploview is a commonly used bioinformatics software which is designed to analyze and visualize patterns of linkage disequilibrium (LD) in genetic data. Haploview can also perform association studies, choosing tagSNPs{{cite journal, author1=de Bakk ...
*
Hardy–Weinberg principle In population genetics, the Hardy–Weinberg principle, also known as the Hardy–Weinberg equilibrium, model, theorem, or law, states that allele and genotype frequencies in a population will remain constant from generation to generation in t ...
*
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 sep ...
* Co-adaptation *
Genealogical DNA test A genealogical DNA test is a DNA-based test used in genetic genealogy that looks at specific locations of a person's genome in order to find or verify ancestral genealogical relationships, or (with lower reliability) to estimate the ethnic mixt ...
* Tag SNP * Association mapping *
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 ...


References


Further reading

*
Bibliography: Linkage Disequilibrium Analysis
: a bibliography of more than one thousand articles on Linkage disequilibrium published since 1918. {{DEFAULTSORT:Linkage Disequilibrium Population genetics