Linkage disequilibrium

Linkage disequilibrium, often abbreviated to LD, is a term in population genetics referring to the association of genes, usually linked genes, in a population. It has become an important tool in medical genetics and other fields.

In defining LD, it is important first to distinguish two different concepts: linkage disequilibrium and linkage (genetic linkage). Linkage disequilibrium refers to the association of genes in a population. Linkage, on the other hand, tells us whether genes are on the same chromosome in an individual. There is no necessary relationship between the two. Genes that are closely linked may or may not be associated in populations. Looking at parents and offspring, if genes at closely linked loci are present together in the parent, then they will usually be found together in the offspring. But looking at individuals in a population with no known common ancestry, it is much more difficult to see any relationships.

To give a concrete, although imaginary, example in terms of frequencies of characters, consider a case where the "gene for red hair" is closely linked to the "gene for blue eyes". What does that tell us about the expected population frequency of individuals with red hair and blue eyes? Are all redheads expected to have blue eyes, just because the genes controlling these characters are closely linked?

Formal definition

Consider an allele A at the A locus with frequency p_A in a particular population. At a linked B locus, the frequency of the allele B is p_B. The question is, what is the expected frequency p_AB of the allele pair, or haplotype, AB? (See note below about genetic nomenclature.)

If the A and B alleles are independent in a population, then, by definition, p_AB is simply the product p_Ap_B. The difference between these two is given by D, the coefficient of linkage disequilbrium:

D = p_AB - p_Ap_B

Departure of D from zero indicates LD.

Note on genetic nomenclature

The descriptors "allele A at the A locus" and "allele B at the B locus" seem unnecessarily complicated. Why not just the "A gene" and the "B gene"? The problem is that the term "gene" has been used since the foundation of genetics without a clear understanding of what a gene actually is. So, despite its widespread popular usage, its use is now avoided in genetics journals (see for a discussion about the changing definition of the gene). This is unfortunate for discussions of population frequencies where the nature of the gene is not important.

Use of the term "allele" rather than "gene" sidesteps this problem, but in a way that is not entirely satisfactory. Allele was originally defined and still understood as meaning "alternative", and allele A and allele B are not alleles (of each other). The easiest way of talking about these linked "things" would be to use the term "gene".

Historical

The expectation, dating back to 1918, is that LD is NOT to be expected, even for loci that are closely linked. Robbins following the studies of Lewontin and Hubby in Drosophila and Harris in humans. Using protein electrophoresis, these authors showed that around one third of loci must be 'polymorphic', having some genetic differences between individuals in the population. Given the large number of loci in the genome and the limited amount of recombination, it followed that there must be many very closely linked polymorphic loci.

Subsequent DNA sequencing, e.g. the International HapMap Project has shown that protein studies considerably underestimate the amount of polymorphism. There will usually be thousands of genetic differences, titled Single Nucleotide Polymorphism or SNPs, within short regions of the genome. Cases of zero or very low recombination must be common.

A second important finding pertaining to LD was the realisation that LD can arise simply because of population structure. Studies such as those of Robbins

LD as a covariance or correlation of frequencies

Haplotype frequencies can be set out in the form of a table with x and y columns. Allele A is given the value '1' and allele a the value '0' in the x column. Similarly for B in the y column. Gamete frequencies are of the form g_i, summing to 1.

{| class="wikitable"

|+

!Haplotype

!x value

!y value

!frequency (f)

|-

|AB

|1

|1

|g₁

|-

|Ab

|1

|0

|g₂

|-

|aB

|0

|1

|g₃

|-

|ab

|0

|0

|g₄

|}

Then summing over the four classes:

Σfxy = 1.g₁ + 0.g₂ + 0.g₃ + 0.g₄ = g₁

Σfx = g₁ + g₂ = p_A

Σfy = g₁ + g₃ = p_B

The covariance between x and y values is

Σfxy - Σfx Σfy = g₁ - p_A p_B

which is equivalent to the LD coefficient, D, as defined above.

It is usually convenient to calculate the correlation rather than the covariance, normalising by the variances:

V(x) = Σfx² - (Σfx)² = p_A - p_A² = p_A ( 1 - p_A )

V(y) = Σfy² - (Σfy)² = p_B - p_B² = p_B ( 1 - p_B )

Substituting gives the correlation, which can be given the designation r_AB, as:

_{<math>r_{AB} = \frac{D}{\sqrt{p_A ( 1 - p_A ) p_B ( 1 - p_B )} }</math>}

or

<math>r_{AB}^2 = \frac{D^2}{p_A ( 1 - p_A ) p_B ( 1 - p_B ) }</math>

This LD measure was introduced by Sewall Wright and its use popularised by Hill and Robertson. Gao et al show that the diploid covariance is the same as "Burrows' composite LD measure". The table below shows x and y values for diploid genotypes. It also shows the expected frequencies on the assumption of random mating.

{| class="wikitable"

|+

!Genotype

!x value

!y value

!frequency (f)

|-

|AABB

|1

|1

|g₁²

|-

|AABb

|1

|1/2

|2g₁g₂

|-

|AAbb

|1

|0

|g₂²

|-

|AaBB

|1/2

|1

|2g₁g₃

|-

|AaBb

|1/2

|1/2

|2g₁g₄+2g₂g₃

|-

|Aabb

|1/2

|0

|2g₂g₄

|-

|aaBB

|0

|1

|g₃²

|-

|aaBb

|0

|1/2

|2g₃g₄

|-

|aabb

|0

|0

|g₄²

|}

Covariance and correlation calculations for these frequencies are as follows:

Σfxy = g₁² + g₁g₂ + g₁g₃ + g₁g₄/2 + g₂g₃/2

Noting the alternative definition of D = g₁g₄ - g₂g₃, this simplifies to

Σfxy = g₁ - D/2.

Σfx = g₁² + 2g₁g₂ + g₂² + g₁g₃ + g₁g₄ + g₂g₃ + g₃g₄

which simplifies, as in the haploid calculation, to

Σfx = g₁ + g₂ = p_A

Similary, Σfy = g₁ + g₃ = p_B

The covariance between x and y values is

Σfxy - Σfx Σfy = g₁ - D/2 - p_A p_B

which is simply D/2.

V(x) = Σfx² - (Σfx)² which can be shown to be p_A ( 1 - p_A )/2

V(y) = Σfy² - (Σfy)² = p_B ( 1 - p_B )/2

Normalising by the variances, the factor 2 cancels out. The diploid correlation which can be designated as R_AB, has expectation:

<math> E(R_{AB}) = \frac{D}{\sqrt{p_A ( 1 - p_A ) p_B ( 1 - p_B )} }</math>

Surprisingly, this is identical to the haploid LD correlation r_AB. The result is, as mentioned above, an expectation based on the assumption of random mating. But this assumption can be relaxed.

If the deviation from random mating is expressed in terms of the inbreeding coefficient F, the expected frequency of AABB homozygotes is equal to (1-F)g₁² + Fg₁, the expected frequency of non-homozygotes such as AABb is equal to (1-F)g₁g₂ etc.  Using these frequencies, the covariance and variance statistics simplify to:

Cov(x,y) = (1+F)D/2

V(x) = (1+F)p_A(1-p_A)/2  [equivalent to (p_A(1-p_A) + D_A)/2, where D_A is the A locus 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. suggested calculating the normalized linkage disequilibrium (also referred to as relative linkage disequilibrium) <math>D'</math> by dividing <math>D</math> by the theoretical maximum difference between the observed and expected allele frequencies as follows:

: <math>D' = \frac D {D_\max}</math>

where

: <math>D_\max= \begin{cases}

\min\{p_A p_B,\,(1-p_A)(1-p_B)\} & \text{when } D < 0\\

\min\{p_A (1-p_B),\,p_B(1-p_A)\} & \text{when } D > 0

\end{cases} </math>

The value of <math>D'</math> will be within the range <math>-1\leq D'\leq 1</math>. When <math>D' = 0</math>, the loci are independent. When <math>-1 \leq D' < 0</math>, the alleles are found less often than expected. When <math>0 < D' \leq 1</math>, the alleles are found more often than expected.

Note that <math>|D'|</math> may be used in place of <math>D'</math> when measuring how close two alleles are to linkage equilibrium.

r² Method

An alternative to <math>D'</math> is the correlation coefficient between pairs of loci, usually expressed as its square, <math>r^2</math>,

: <math>r^2=\frac{D^2}{p_A(1-p_A)p_B (1-p_B)}</math>

The value of <math>r^2</math> will be within the range <math>0 \leq r^2 \leq 1</math>. When <math>r^2 = 0</math>, there is no correlation between the pair. When <math>r^2 = 1</math>, the correlation is either perfect positive or perfect negative according to the sign of <math>r</math>.

d Method

Another alternative normalizes <math>D</math> by the product of two of the four allele frequencies when the two frequencies represent alleles from the same locus. This allows comparison of asymmetry between a pair of loci. This is often used in case-control studies where <math>B</math> is the locus containing a disease allele.

<math>d =\frac{D}{p_B (1-p_B)}</math>

ρ Method

Similar to the d method, this alternative normalizes <math>D</math> by the product of two of the four allele frequencies when the two frequencies represent alleles from different loci. While <math>D'</math> can always take a maximum value of 1, its minimum value for two loci is equal to <math>|r|</math> 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:

{| class=wikitable

|-

|Haplotype

|Frequency

|-

|<math>A_1B_1</math>||<math>x_{11}</math>

|-

|<math>A_1B_2</math>||<math>x_{12}</math>

|-

|<math>A_2B_1</math>||<math>x_{21}</math>

|-

|<math>A_2B_2</math>||<math>x_{22}</math>

|}

Note that these are relative frequencies. One can use the above frequencies to determine the frequency of each of the alleles:

{| class=wikitable

|-

|Allele||Frequency

|-

|<math>A_1</math>||<math>p_1=x_{11}+x_{12}</math>

|-

|<math>A_2</math>||<math>p_2=x_{21}+x_{22}</math>

|-

|<math>B_1</math>||<math>q_1=x_{11}+x_{21}</math>

|-

|<math>B_2</math>||<math>q_2=x_{12}+x_{22}</math>

|}

If the two loci and the alleles are independent from each other, then we would expect the frequency of each haplotype to be equal to the product of the frequencies of its corresponding alleles (e.g. <math>x_{11} = p_1 q_1</math>).

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:

: <math>D = x_{11} - p_1q_1</math>

Thus, if the loci were inherited independently, then <math>x_{11} = p_1 q_1</math>, so <math>D = 0</math>, and there is linkage equilibrium. However, if the observed frequency of haplotype <math>A_1B_1</math> were higher than what would be expected based on the individual frequencies of <math>A_1</math> and <math>B_1</math> then <math>x_{11} > p_1 q_1</math>, so <math>D > 0</math>, and there is positive linkage disequilibrium. Conversely, if the observed frequency were lower, then <math>x_{11} < p_1 q_1</math>, <math>D < 0</math>, and there is negative linkage disequilibrium.

The following table illustrates the relationship between the haplotype frequencies and allele frequencies and D.

{| class=wikitable

|-

| ||<math>A_1</math> ||<math>A_2</math> ||Total

|-

|<math>B_1</math>||<math>x_{11}=p_1q_1+D</math>&nbsp;&nbsp;&nbsp;&nbsp;||<math>x_{21}=p_2q_1-D</math>&nbsp;&nbsp;&nbsp;||<math>q_1</math>

|-

|<math>B_2</math>||<math>x_{12}=p_1q_2-D</math> ||<math>x_{22}=p_2q_2+D</math>||<math>q_2</math>

|-

|Total&nbsp;&nbsp;&nbsp;||<math>p_1</math>||<math>p_2</math>||<math>1</math>

|}

Additionally, we can normalize our data based on what we are trying to accomplish. For example, if we aim to create an association map in a case-control study, then we may use the d method due to its asymmetry. If we are trying to find the probability that a given haplotype will descend in a population without being recombined by other haplotypes, then it may be better to use the ρ method. But for most scenarios, <math>r^2</math> tends to be the most popular method due to the usefulness of the correlation coefficient in statistics. A couple examples of where <math>r^2</math> may be very useful would include measuring the recombination rate in an evolving population, or detecting disease associations.

Another visualization option is forests of hierarchical latent class models (FHLCM). All loci are plotted along the top layer of the graph, and below this top layer, boxes representing latent variables are added with links to the top level. Lines connect the loci at the top level to the latent variables below, and the lower the level of the box that the loci are connected to, the greater the linkage disequilibrium and the smaller the distance between the loci. While this method does not have the same advantages of the textile plot, it does allow for the visualization of loci that are far apart without requiring the sequence to be rearranged, as is the case with the textile plot.

This is not an exhaustive list of visualization methods, and multiple methods may be used to display a data set in order to give a better picture of the data based on the information that the researcher aims to highlight.

Resources

A comparison of different measures of LD is provided by Devlin & Risch

The International HapMap Project enables the study of LD in human populations online. The Ensembl project integrates HapMap data with other genetic information from dbSNP.

Analysis software

• PLINK – whole genome association analysis toolset, which can calculate LD among other things
• LDHat
• Haploview
• LdCompare&mdash; open-source software for calculating LD.
• 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
• rAggr – finds proxy markers (SNPs and indels) that are in linkage disequilibrium with a set of queried markers, using the 1000 Genomes Project 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 &mdash; a C library for population genetic simulation (GPL)

See also

• Haploview
• Hardy–Weinberg principle
• Genetic hitchhiking
• Genetic linkage
• Co-adaptation
• Genealogical DNA test
• Tag SNP
• Association mapping
• Family based QTL mapping

References

Further reading

• Bibliography: Linkage Disequilibrium Analysis : a bibliography of more than one thousand articles on Linkage disequilibrium published since 1918.