Linkage disequilibrium
Encyclopedia
In population genetics
, linkage disequilibrium is the non-random association of allele
s at two or more loci
, not necessarily on the same chromosome
. It is also referred to as to as gametic phase disequilibrium , or simply gametic disequilibrium. In other words, linkage disequilibrium is the occurrence of some combinations of alleles or genetic markers in a population more often or less often than would be expected from a random formation of haplotype
s from alleles based on their frequencies. It is not the same as linkage
, which is the association of two or more loci on a chromosome with limited recombination
between them. The amount of linkage disequilibrium depends on the difference between observed and expected (assuming random distributions) allelic frequencies.
The level of linkage disequilibrium is influenced by a number of factors, including genetic linkage, selection
, the rate of recombination, the rate of mutation
, genetic drift
, non-random mating
, and population structure
. A limiting example of the effect of rate of recombination may be seen in some organisms (such as bacteria
) that reproduce asexually
and hence exhibit no recombination to break down the linkage disequilibrium. An example of the effect of population structure is the phenomenon of Finnish disease heritage, which is attributed to a population bottleneck
.
Note that these are relative frequencies
. One can use the above frequencies to determine the frequency of each of the alleles:
If the two loci and the alleles are independent from each other, then one can express the observation
as "
is found and 
is found". The table above lists the frequencies for 
, 
, and for
, 
, hence the frequency of 
is 
, and according to the rules of elementary statistics 
.
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:
In the genetic literature the phrase "two alleles are in LD" usually means that D ≠ 0. Contrariwise, "linkage equilibrium" means D = 0.
The following table illustrates the relationship between the haplotype frequencies and allele frequencies and D.
is easy to calculate with, but has the disadvantage of depending on the frequencies of the alleles. This is evident since frequencies are between 0 and 1. If any locus has an allele frequency 0 or 1 no disequilibrium 
can be observed. When the allelic frequencies are 0.5, the disequilibrium 
is maximal. Lewontin suggested normalising D by dividing it by the theoretical maximum for the observed allele frequencies.
Thus
= 
where
when
, and
when
.
Another measure of LD which is an alternative to
is the correlation coefficient between pairs of loci, expressed as 
. This is also adjusted to the loci having different allele frequencies.
In summary, linkage disequilibrium reflects the difference between the expected haplotype frequencies under the assumption of independence, and observed haplotype frequencies. A value of 0 for
indicates that the examined loci are in fact independent of one another, while a value of 1 demonstrates complete dependency.
the linkage disequilibrium measure
converges to zero along the time axis at a rate
depending on the magnitude of the recombination rate
between the two loci.
Using the notation above,
, we can demonstrate this convergence to zero
as follows. In the next generation,
, the frequency of the haplotype 
, becomes
This follows because a fraction
of the haplotypes in the offspring have not
recombined, and are thus copies of a random haplotype in their parents. A fraction
of those are 
. A fraction 
have recombined these two loci. If the parents result from random mating, the probability of the
copy at locus
having allele 
is 
and the probability
of the copy at locus
having allele 
is 
, and as these copies are initially on different loci, these are independent events so that the probabilities can be multiplied.
This formula can be rewritten as
so that
where
at the 
-th generation is designated as 
. Thus we have
If
, then 
so that 
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
to zero.
of humans. Because HLA genes are located at adjacent loci on the particular region of a chromosome and presumed to exhibit epistasis
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 Danes referred to by Vogel and Motulsky (1997).
Because HLA is codominant and HLA expression is only tested locus by locus in surveys, LD measure is to be estimated from such a 2x2 table to the right.
expression (
) frequency of antigen 
:
;
expression (
) frequency of antigen 
:
;
frequency of gene
:
,
and
.
Denoting the '―' alleles at antigen i to be 'x,' and at antigen j to be 'y,' the observed frequency of haplotype xy is
and the estimated frequency of haplotype xy is
.
Then LD measure
is expressed as
.
Standard errors
are obtained as follows:
,
.
Then, if
exceeds 2 in its absolute value, the magnitude of
is large statistically significantly. 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 Caucasians.
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 Caucasians in the list of Mittal it is mostly non-significant. If
had reduced from 0.07 to 0.003 under recombination effect as shown by 
, then 
. 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:
(1) Relative risk
Relative risk
of an HLA allele for a disease is approximated by the odds ratio
in the 2x2 association table of the allele with the disease. Table 3 shows association of HLA-B27 with ankylosing spondylitis among a Dutch population. Relative risk
of this allele is approximated by
.
Woolf's method is applied to see if there is statistical significance. Let
and
.
Then
follows the chi-square distribution with
. In the data of Table 3, the significant association exists at the 0.1% level. Haldane's modification applies to the case when either of
is zero, where replace 
and 
with
and
,
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.
value is expressed by
,
where
and 
are HLA allele frequencies among patients and healthy populations, respectively. In Table 4, 
column was added in this quotation. Putting aside 2 diseases with high relative risks both of which are also with high 
values, among other diseases, juvenile diabetes mellitus (type 1) has a strong association with DR4 even with a low relative risk
.
(2) Discrepancies from expected values from marginal frequencies in 2x2 association table of HLA alleles and disease
This can be confirmed by
test calculating
.
where
. For data with small sample size, such as no marginal total is greater than 15 (and consequently 
), one should utilize Yates's correction for continuity or Fisher's exact test
.
The International HapMap Project
enables the study of LD in human populationsonline. The Ensembl
project integrates HapMap data and such from dbSNP
in general with other genetic information.
Population genetics
Population genetics is the study of allele frequency distribution and change under the influence of the four main evolutionary processes: natural selection, genetic drift, mutation and gene flow. It also takes into account the factors of recombination, population subdivision and population...
, linkage disequilibrium is the non-random association of allele
Allele
An allele is one of two or more forms of a gene or a genetic locus . "Allel" is an abbreviation of allelomorph. Sometimes, different alleles can result in different observable phenotypic traits, such as different pigmentation...
s at two or more loci
Locus (genetics)
In the fields of genetics and genetic computation, a locus is the specific location of a gene or DNA sequence on a chromosome. A variant of the DNA sequence at a given locus is called an allele. The ordered list of loci known for a particular genome is called a genetic map...
, not necessarily on the same chromosome
Chromosome
A chromosome is an organized structure of DNA and protein found in cells. It is a single piece of coiled DNA containing many genes, regulatory elements and other nucleotide sequences. Chromosomes also contain DNA-bound proteins, which serve to package the DNA and control its functions.Chromosomes...
. It is also referred to as to as gametic phase disequilibrium , or simply gametic disequilibrium. In other words, linkage disequilibrium is the occurrence of some combinations of alleles or genetic markers in a population more often or less often than would be expected from a random formation of haplotype
Haplotype
A haplotype in genetics is a combination of alleles at adjacent locations on the chromosome that are transmitted together...
s from alleles based on their frequencies. It is not the same as linkage
Genetic linkage
Genetic linkage is the tendency of certain loci or alleles to be inherited together. Genetic loci that are physically close to one another on the same chromosome tend to stay together during meiosis, and are thus genetically linked.-Background:...
, which is the association of two or more loci on a chromosome with limited recombination
Homologous recombination
Homologous recombination is a type of genetic recombination in which nucleotide sequences are exchanged between two similar or identical molecules of DNA. It is most widely used by cells to accurately repair harmful breaks that occur on both strands of DNA, known as double-strand breaks...
between them. The amount of linkage disequilibrium depends on the difference between observed and expected (assuming random distributions) allelic frequencies.
The level of linkage disequilibrium is influenced by a number of factors, including genetic linkage, selection
Selection
In the context of evolution, certain traits or alleles of genes segregating within a population may be subject to selection. Under selection, individuals with advantageous or "adaptive" traits tend to be more successful than their peers reproductively—meaning they contribute more offspring to the...
, the rate of recombination, the rate of mutation
Mutation
In molecular biology and genetics, mutations are changes in a genomic sequence: the DNA sequence of a cell's genome or the DNA or RNA sequence of a virus. They can be defined as sudden and spontaneous changes in the cell. Mutations are caused by radiation, viruses, transposons and mutagenic...
, genetic drift
Genetic drift
Genetic drift or allelic drift is the change in the frequency of a gene variant in a population due to random sampling.The alleles in the offspring are a sample of those in the parents, and chance has a role in determining whether a given individual survives and reproduces...
, non-random mating
Assortative mating
Assortative mating , and the related concept Disassortative mating, is the phenomenon where a sexually reproducing organism chooses to mate with individuals that are similar or dissimilar to itself in some specific manner...
, and population structure
Population stratification
Population stratification is the presence of a systematic difference in allele frequencies between subpopulations in a population possibly due to different ancestry, especially in the context of association studies...
. A limiting example of the effect of rate of recombination may be seen in some organisms (such as bacteria
Bacteria
Bacteria are a large domain of prokaryotic microorganisms. Typically a few micrometres in length, bacteria have a wide range of shapes, ranging from spheres to rods and spirals...
) that reproduce asexually
Asexual reproduction
Asexual reproduction is a mode of reproduction by which offspring arise from a single parent, and inherit the genes of that parent only, it is reproduction which does not involve meiosis, ploidy reduction, or fertilization. A more stringent definition is agamogenesis which is reproduction without...
and hence exhibit no recombination to break down the linkage disequilibrium. An example of the effect of population structure is the phenomenon of Finnish disease heritage, which is attributed to a population bottleneck
Population bottleneck
A population bottleneck is an evolutionary event in which a significant percentage of a population or species is killed or otherwise prevented from reproducing....
.
Definition
Consider the haplotypes for two loci A and B with two alleles each—a two-locus, two-allele model. Then the following table defines the frequencies of each combination:| Haplotype | Frequency |
 |
 |
 |
 |
 |
 |
 |
 |
Note that these are relative frequencies
Frequency (statistics)
In statistics the frequency of an event i is the number ni of times the event occurred in the experiment or the study. These frequencies are often graphically represented in histograms....
. One can use the above frequencies to determine the frequency of each of the alleles:
| Allele | Frequency |
 |
 |
 |
 |
 |
 |
 |
 |
If the two loci and the alleles are independent from each other, then one can express the observation
as " is found and is found". The table above lists the frequencies for , , and for, , hence the frequency of is , and according to the rules of elementary statistics .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:
 |
In the genetic literature the phrase "two alleles are in LD" usually means that D ≠ 0. Contrariwise, "linkage equilibrium" means D = 0.
The following table illustrates the relationship between the haplotype frequencies and allele frequencies and D.
 |
 |
Total | |
 |
 |
 |
 |
 |
 |
 |
 |
| Total |  |
 |
 |
is easy to calculate with, but has the disadvantage of depending on the frequencies of the alleles. This is evident since frequencies are between 0 and 1. If any locus has an allele frequency 0 or 1 no disequilibrium can be observed. When the allelic frequencies are 0.5, the disequilibrium is maximal. Lewontin suggested normalising D by dividing it by the theoretical maximum for the observed allele frequencies.Thus
= where
when, and when.Another measure of LD which is an alternative to
is the correlation coefficient between pairs of loci, expressed as . This is also adjusted to the loci having different allele frequencies.In summary, linkage disequilibrium reflects the difference between the expected haplotype frequencies under the assumption of independence, and observed haplotype frequencies. A value of 0 for
indicates that the examined loci are in fact independent of one another, while a value of 1 demonstrates complete dependency.Role of recombination
In the absence of evolutionary forces other than random mating and Mendelian segregation,the linkage disequilibrium measure
converges to zero along the time axis at a ratedepending on the magnitude of the recombination rate
between the two loci.Using the notation above,
, we can demonstrate this convergence to zeroas follows. In the next generation,
, the frequency of the haplotype , becomes
 |
This follows because a fraction
of the haplotypes in the offspring have notrecombined, and are thus copies of a random haplotype in their parents. A fraction
of those are . A fraction have recombined these two loci. If the parents result from random mating, the probability of the
copy at locus
having allele is and the probabilityof the copy at locus
having allele is , and as these copies are initially on different loci, these are independent events so that the probabilities can be multiplied.This formula can be rewritten as
 |
so that
 |
where
at the -th generation is designated as . Thus we have
 . |
If
, then so that 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
to zero.Example: Human Leukocyte Antigen (HLA) alleles
HLA constitutes a group of cell surface antigens as MHCMHC
-Biology:*Myosin heavy chain - part of the motor protein myosin's quaternary protein structure*Major histocompatibility complex - a highly polymorphic region on chromosome 6 with genes particularly involved in immune functions-Colleges:...
of humans. Because HLA genes are located at adjacent loci on the particular region of a chromosome and presumed to exhibit epistasis
Epistasis
In genetics, epistasis is the phenomenon where the effects of one gene are modified by one or several other genes, which are sometimes called modifier genes. The gene whose phenotype is expressed is called epistatic, while the phenotype altered or suppressed is called hypostatic...
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 Danes referred to by Vogel and Motulsky (1997).
| Antigen j | Total | ||||
|---|---|---|---|---|---|
 |
 |
||||
 |
 |
||||
| Antigen i |  |
 |
 |
 |
 |
 |
 |
 |
 |
 |
|
| Total |  |
 |
 |
||
| No. of individuals | |||||
Because HLA is codominant and HLA expression is only tested locus by locus in surveys, LD measure is to be estimated from such a 2x2 table to the right.
expression (
) frequency of antigen : ;expression (
) frequency of antigen : ;frequency of gene
: ,and
.Denoting the '―' alleles at antigen i to be 'x,' and at antigen j to be 'y,' the observed frequency of haplotype xy is
and the estimated frequency of haplotype xy is
.Then LD measure
is expressed as.Standard errors
are obtained as follows:,.Then, if
exceeds 2 in its absolute value, the magnitude of
is large statistically significantly. For data in Table 1 it is 20.9, thus existence of statistically significant LD between A1 and B8 in the population is admitted.| HLA-A alleles i | HLA-B alleles j |  |
 |
|---|---|---|---|
| A1 | B8 | 0.065 | 16.0 |
| A3 | B7 | 0.039 | 10.3 |
| A2 | Bw40 | 0.013 | 4.4 |
| A2 | Bw15 | 0.01 | 3.4 |
| A1 | Bw17 | 0.014 | 5.4 |
| A2 | B18 | 0.006 | 2.2 |
| A2 | Bw35 | -0.009 | -2.3 |
| A29 | B12 | 0.013 | 6.0 |
| A10 | Bw16 | 0.013 | 5.9 |
Table 2 shows some of the combinations of HLA-A and B alleles where significant LD was observed among Caucasians.
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 Caucasians in the list of Mittal it is mostly non-significant. If
had reduced from 0.07 to 0.003 under recombination effect as shown by , then . 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
value to exceed 0.
Table 3. Association of ankylosing spondylitis with HLA-B27 allele Ankylosing spondylitis Total Patients Healthy controls HLA alleles 
Total 
- 2x2 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
Relative risk
In statistics and mathematical epidemiology, relative risk is the risk of an event relative to exposure. Relative risk is a ratio of the probability of the event occurring in the exposed group versus a non-exposed group....
of an HLA allele for a disease is approximated by the odds ratio
Odds ratio
The odds ratio is a measure of effect size, describing the strength of association or non-independence between two binary data values. It is used as a descriptive statistic, and plays an important role in logistic regression...
in the 2x2 association table of the allele with the disease. Table 3 shows association of HLA-B27 with ankylosing spondylitis among a Dutch population. Relative risk
of this allele is approximated by.Woolf's method is applied to see if there is statistical significance. Let
and
.Then
follows the chi-square distribution with
. In the data of Table 3, the significant association exists at the 0.1% level. Haldane's modification applies to the case when either of is zero, where replace and withand
,respectively.
| Disease | HLA allele | Relative risk (%) | FAD (%) | FAP (%) |  |
|---|---|---|---|---|---|
| Ankylosing spondylitis Ankylosing spondylitis Ankylosing spondylitis , previously known as Bekhterev's disease, Bekhterev syndrome, and Marie-Strümpell disease is a chronic inflammatory disease of the axial skeleton with variable involvement of peripheral joints and nonarticular structures... |
B27 | 90 | 90 | 8 | 0.89 |
| Reiter's syndrome | B27 | 40 | 70 | 8 | 0.67 |
| Spondylitis Spondylitis Spondylitis is an inflammation of the vertebra. It is a form of spondylopathy. In many cases, spondylitis involves one or more vertebral joint as well, which itself is called spondylarthritis.-Examples:... in inflammatory bowel disease |
B27 | 10 | 50 | 8 | 0.46 |
| Rheumatoid arthritis Rheumatoid arthritis Rheumatoid arthritis is a chronic, systemic inflammatory disorder that may affect many tissues and organs, but principally attacks synovial joints. The process produces an inflammatory response of the synovium secondary to hyperplasia of synovial cells, excess synovial fluid, and the development... |
DR4 | 6 | 70 | 30 | 0.57 |
| Systemic lupus erythematosus Systemic lupus erythematosus Systemic lupus erythematosus , often abbreviated to SLE or lupus, is a systemic autoimmune disease that can affect any part of the body. As occurs in other autoimmune diseases, the immune system attacks the body's cells and tissue, resulting in inflammation and tissue damage... |
DR3 | 3 | 45 | 20 | 0.31 |
| Multiple sclerosis Multiple sclerosis Multiple sclerosis is an inflammatory disease in which the fatty myelin sheaths around the axons of the brain and spinal cord are damaged, leading to demyelination and scarring as well as a broad spectrum of signs and symptoms... |
DR2 | 4 | 60 | 20 | 0.5 |
| Diabetes mellitus type 1 Diabetes mellitus type 1 Diabetes mellitus type 1 is a form of diabetes mellitus that results from autoimmune destruction of insulin-producing beta cells of the pancreas. The subsequent lack of insulin leads to increased blood and urine glucose... |
DR4 | 6 | 75 | 30 | 0.64 |
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.
value is expressed by,where
and are HLA allele frequencies among patients and healthy populations, respectively. In Table 4, column was added in this quotation. Putting aside 2 diseases with high relative risks both of which are also with high values, among other diseases, juvenile diabetes mellitus (type 1) has a strong association with DR4 even with a low relative risk.(2) Discrepancies from expected values from marginal frequencies in 2x2 association table of HLA alleles and disease
This can be confirmed by
test calculating.where
. For data with small sample size, such as no marginal total is greater than 15 (and consequently ), one should utilize Yates's correction for continuity or Fisher's exact testFisher's exact test
Fisher's exact test is a statistical significance test used in the analysis of contingency tables where sample sizes are small. It is named after its inventor, R. A...
.
Resources
A comparison of different measures of LD is provided by Devlin & RischThe International HapMap Project
International HapMap Project
The International HapMap Project is an organization that aims to develop a haplotype map of the human genome, which will describe the common patterns of human genetic variation. HapMap is a key resource for researchers to find genetic variants affecting health, disease and responses to drugs and...
enables the study of LD in human populationsonline. The Ensembl
Ensembl
Ensembl is a joint scientific project between the European Bioinformatics Institute and the Wellcome Trust Sanger Institute, which was launched in 1999 in response to the imminent completion of the Human Genome Project...
project integrates HapMap data and such from dbSNP
DbSNP
The Single Nucleotide Polymorphism Database is a free public archive for genetic variation within and across different species developed and hosted by the National Center for Biotechnology Information in collaboration with the National Human Genome Research Institute...
in general with other genetic information.
Analysis software
- LDHat
- HaploviewHaploviewHaploview is a commonly used bioinformatics software which is designed to analyze and visualize patterns of linkage disequilibrium in genetic data. Haploview can also perform association studies, choosing tagSNPs and estimating haplotype frequencies. Haploview is developed and maintained by Dr...
- LdCompare— open-source software for calculating LD.
- PyPop
- SNP and Variation Suite- commercial software with interactive LD plot.
- GOLD - Graphical Overview of Linkage Disequilibrium
- TASSEL -software to evaluate linkage disequilibrium, traits associations, and evolutionary patterns
Simulation software
- Haploid — a CC (programming language)C is a general-purpose computer programming language developed between 1969 and 1973 by Dennis Ritchie at the Bell Telephone Laboratories for use with the Unix operating system....
library for population genetic simulation (GPL)
See also
- HaploviewHaploviewHaploview is a commonly used bioinformatics software which is designed to analyze and visualize patterns of linkage disequilibrium in genetic data. Haploview can also perform association studies, choosing tagSNPs and estimating haplotype frequencies. Haploview is developed and maintained by Dr...
- Hardy-Weinberg principleHardy-Weinberg principleThe Hardy–Weinberg principle states that both allele and genotype frequencies in a population remain constant—that is, they are in equilibrium—from generation to generation unless specific disturbing influences are introduced...
- Genetic linkageGenetic linkageGenetic linkage is the tendency of certain loci or alleles to be inherited together. Genetic loci that are physically close to one another on the same chromosome tend to stay together during meiosis, and are thus genetically linked.-Background:...
- Co-adaptationCo-adaptationIn biology, co-adaptation, or coadaptation refers to the mutual adaptation of:* Species: see mutualism, symbiosis* organs: see the evolution of the eye.* Genes or gene complexes: see Linkage disequilibrium, epistasis...
- Genealogical DNA testGenealogical DNA testA genealogical DNA test examines the nucleotides at specific locations on a person's DNA for genetic genealogy purposes. The test results are not meant to have any informative medical value and do not determine specific genetic diseases or disorders ; they are intended only to give genealogical...
- Tag SNPTag SNPA tag SNP is a representative single nucleotide polymorphism in a region of the genome with high linkage disequilibrium . It is possible to identify genetic variation without genotyping every SNP in a chromosomal region...
Further reading
- Bibliography: Linkage Disequilibrium Analysis : a bibliography of more than one thousand articles on Linkage disequilibrium published since 1918.