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 (biology), adaptation, ...
, 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 chro ...
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
Selection may refer to:
Science
* Selection (biology), also called natural selection, selection in evolution
** Sex selection, in genetics
** Mate selection, in mating
** Sexual selection in humans, in human sexuality
** Human mating strateg ...
, the rate of
genetic recombination,
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. 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 In 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 influence ...
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
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).
The association between the alleles ''A'' and ''B'' can be regarded as completely random—which is known in
statistics 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 statu ...
''—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 ...
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
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,
:
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
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. 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
* Independ ...
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
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 genetic nor behavioural, upon the population and that therefore all ...
,
Mendelian segregation, random
chromosomal assortment, and
chromosomal crossover (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 and o ...
, 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
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 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 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 of this allele is approximated by
:
Woolf's method] is applied to see if there is statistical significance. Let
:
and
:
Then
: