In
population genetics, linkage disequilibrium (LD) is the non-random association of
alleles 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,
genetic drift, 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 separ ...
. 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.
Formal definition
Suppose that among the gametes that are formed in a sexually reproducing population, allele ''A'' occurs with frequency
at one locus (i.e.
is the proportion of gametes with ''A'' at that locus), while at a different locus allele ''B'' occurs with frequency
. Similarly, let
be the frequency with which both ''A'' and ''B'' occur together in the same gamete (i.e.
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 or ...
).
The association between the alleles ''A'' and ''B'' can be regarded as completely random—which is known in
statistics
Statistics (from German language, German: ''wikt:Statistik#German, Statistik'', "description of a State (polity), state, a country") is the discipline that concerns the collection, organization, analysis, interpretation, and presentation of ...
as ''
independence''—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 of the probabilities. There is said to be a linkage disequilibrium between the two alleles whenever
differs from
for any reason.
The level of linkage disequilibrium between ''A'' and ''B'' can be quantified by the ''coefficient of linkage disequilibrium''
, which is defined as
:
provided that both
and
are greater than zero.
Linkage disequilibrium corresponds to
. In the case
we have
and the alleles ''A'' and ''B'' are said to be in ''linkage equilibrium''. The subscript "AB" on
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,
. Their relationships can be characterized as follows.
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
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:
:
where
:
An alternative to
is the
correlation coefficient between pairs of loci, usually expressed as its square,
:
Limits for the ranges of linkage disequilibrium measures
The measures
and
have limits to their ranges and do not range over all values of zero to one for all pairs of loci. The maximum of
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,
where
, or when the allele frequencies have the relationship
when
. While
can always take a maximum value of 1, its minimum value for two loci is equal to
for those loci.
Example: Two-loci and two-alleles
Consider the
haplotypes 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 (probability theory), 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 tabu ...
. 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'':
:
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 geneti ...
(i.e. in the absence of
natural selection,
inbreeding, and
genetic drift),
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 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
:
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 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 Danes
[Svejgaard 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 :
:
expression () frequency of antigen :
:
frequency of gene , given that individuals with '+/−', '+/+', and '−/+' genotypes are all positive for antigen :
:
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 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 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.
*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 bet ...
of an HLA allele for a disease is approximated by the odds ratio 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 of this allele is approximated by
:
Woolf's method] is applied to see if there is statistical significance. Let
:
and
:
Then
: