Linkage disequilibrium

In population genetics, linkage disequilibrium (LD) is a measure of non-random association between segments of DNA (alleles) at different positions on the chromosome (loci) in a given population based on a comparison between the frequency at which two alleles are detected together at the same loci and the frequencies at which each allele is detected at that loci overall, whether it occurs with or without the other allele of interest. Loci are said to be in linkage disequilibrium when the frequency of being detected together (the frequency of association of their different alleles) is higher or lower than expected if the loci were independent and associated randomly.

While the pattern of linkage disequilibrium in a genome is a powerful signal of the population genetic processes that are structuring it, it does not indicate why the pattern emerges by itself. Linkage disequilibrium is influenced by many factors, including selection, the rate of genetic recombination, mutation rate, genetic drift, the system of mating, population structure, and genetic linkage.

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 $$ p_A $$ 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 $$ p_{AB} $$ be the frequency with which both A and B occur together in the same gamete (i.e. $$ p_{AB} $$ 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—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 $$ p_{A} p_{B} $$ of the probabilities. There is said to be a linkage disequilibrium between the two alleles whenever $$ p_{AB} $$ 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 $$ D_{AB}$$, which is defined as


 * $$ D_{AB} = p_{AB} - p_A p_B,$$

Linkage disequilibrium corresponds to $$ D_{AB} \neq 0 $$. In the case $$D_{AB}=0$$ we have $$ p_{AB} = p_A p_B $$ and the alleles A and B are said to be in linkage equilibrium. The subscript "AB" on $$ D_{AB} $$ emphasizes that linkage disequilibrium is a property of the pair $$\{A, B\}$$ 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, $$D_{AB} = -D_{Ab} = -D_{aB} = D_{ab}$$. Their relationships can be characterized as follows.

$$D = P_{AB} -P_{A}P_{B}$$

$$-D = P_{Ab} -P_{A}P_{b}$$

$$-D = P_{aB} -P_{a}P_{B}$$

$$D = P_{ab} -P_{a}P_{b}$$

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.

Normalization
The linkage disequilibrium $$D$$ reflects both changes in the intensity of the linkage correlation and changes in gene frequency. This poses an issue when comparing linkage disequilibrium between alleles with differing frequencies. Normalization of linkage disequilibrium allows these alleles to be compared more easily.

D' Method
Lewontin suggested calculating the normalized linkage disequilibrium (also referred to as relative linkage disequilibrium) $$D'$$ by dividing $$D$$ by the theoretical maximum difference between the observed and expected allele frequencies as follows:


 * $$D' = \frac D {D_\max}$$

where


 * $$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} $$

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

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

r² Method
An alternative to $$D'$$ is the correlation coefficient between pairs of loci, usually expressed as its square, $$r^2$$.


 * $$r^2=\frac{D^2}{p_A(1-p_A)p_B (1-p_B)}$$

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

d Method
Another alternative normalizes $$D$$ 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 $$B$$ is the locus containing a disease allele.

$$d =\frac{D}{p_B (1-p_B)}$$

ρ Method
Similar to the d method, this alternative normalizes $$D$$ by the product of two of the four allele frequencies when the two frequencies represent alleles from different loci.

$$\rho =\frac{D}{(1-p_A) p_B}$$

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 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 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. $$x_{11} = p_1 q_1$$).

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:


 * $$D = x_{11} - p_1q_1$$

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

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

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, $$r^2$$ tends to be the most popular method due to the usefulness of the correlation coefficient in statistics. A couple examples of where $$r^2$$ may be very useful would include measuring the recombination rate in an evolving population, or detecting disease associations.

Role of recombination
In the absence of evolutionary forces other than random mating, Mendelian segregation, random chromosomal assortment, and chromosomal crossover (i.e. in the absence of natural selection, inbreeding, and genetic drift), 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, $$D= x_{11}-p_1 q_1$$, we can demonstrate this convergence to zero as follows. In the next generation, $$x_{11}'$$, the frequency of the haplotype $$A_1 B_1$$, becomes


 * $$x_{11}' = (1-c)\,x_{11} + c\,p_1 q_1$$

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 $$x_{11}$$ 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


 * $$x_{11}' - p_1 q_1 = (1-c)\,(x_{11} - p_1 q_1)$$

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.

Visualization
Once linkage disequilibrium has been calculated for a dataset, a visualization method is often chosen to display the linkage disequilibrium to make it more easily understandable.

The most common method is to use a heatmap, where colors are used to indicate the loci with positive linkage disequilibrium, and linkage equilibrium. This example displays the full heatmap, but because the heatmap is symmetrical across the diagonal (that is, the linkage disequilibrium between loci A and B is the same as between B and A), a triangular heatmap that shows the pairs only once is also commonly employed. This method has the advantage of being easy to interpret, but it also cannot display information about other variables that may be of interest.

More robust visualization options are also available, like the textile plot. In a textile plot, combinations of alleles at a certain loci can be linked with combinations of alleles at a different loci. Each genotype (combination of alleles) is represented by a circle which has an area proportional to the frequency of that genotype, with a column for each loci. Lines are drawn from each circle to the circles in the other column(s), and the thickness of the connecting line is proportional to the frequency that the two genotypes occur together. Linkage disequilibrium is seen through the number of line crossings in the diagram, where a greater number of line crossings indicates a low linkage disequilibrium and fewer crossings indicate a high linkage disequilibrium. The advantage of this method is that it shows the individual genotype frequencies and includes a visual difference between absolute (where the alleles at the two loci always appear together) and complete (where alleles at the two loci show a strong connection but with the possibility of recombination) linkage disequilibrium by the shape of the graph.

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)