Linkage disequilibrium

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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 versus the frequencies at which each allele is simply detected (alone or with the second allele) at that same loci. 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. [1]

Contents

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). [1] Furthermore, linkage disequilibrium is sometimes referred to as gametic phase disequilibrium; [2] 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).

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 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

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. [3]

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. [4] 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. [1]

Normalization

The linkage disequilibrium 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 [5] suggested calculating the normalized linkage disequilibrium (also referred to as relative linkage disequilibrium) by dividing by the theoretical maximum difference between the observed and expected allele frequencies as follows:

where

The value of will be within the range . When , the loci are independent. When , the alleles are found less often than expected. When , the alleles are found more often than expected.

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

r² Method

An alternative to is the correlation coefficient between pairs of loci, usually expressed as its square, . [6]

The value of will be within the range . When , there is no correlation between the pair. When , the correlation is either perfect positive or perfect negative according to the sign of .

d Method

Another alternative normalizes 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 is the locus containing a disease allele. [7]

ρ Method

Similar to the d method, this alternative normalizes by the product of two of the four allele frequencies when the two frequencies represent alleles from different loci. [7]

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 . [8] While can always take a maximum value of 1, its minimum value for two loci is equal to for those loci. [9]

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:

HaplotypeFrequency

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

AlleleFrequency

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. ).

The deviation of the observed frequency of a haplotype from the expected is a quantity [10] called the linkage disequilibrium [11] and is commonly denoted by a capital D:

Thus, if the loci were inherited independently, then , so , and there is linkage equilibrium. However, if the observed frequency of haplotype were higher than what would be expected based on the individual frequencies of and then , so , and there is positive linkage disequilibrium. Conversely, if the observed frequency were lower, then , , and there is negative linkage disequilibrium.

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

Total
       
Total   

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, tends to be the most popular method due to the usefulness of the correlation coefficient in statistics. A couple examples of where may be very useful would include measuring the recombination rate in an evolving population, or detecting disease associations. [7]

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 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.

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.

A heatmap showing the linkage disequilibrium between genetic loci, detected using the GAM method. Linkage Disequilibrium Heatmap.png
A heatmap showing the linkage disequilibrium between genetic loci, detected using the GAM method.

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. [12]

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. [13]

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 [14]

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

Simulation software

See also

Related Research Articles

An allele, or allelomorph, is a variant of the sequence of nucleotides at a particular location, or locus, on a DNA molecule.

Population genetics is a subfield of genetics that deals with genetic differences within and among populations, and is a part of evolutionary biology. Studies in this branch of biology examine such phenomena as adaptation, speciation, and population structure.

<span class="mw-page-title-main">Hardy–Weinberg principle</span> Principle in genetics

In population genetics, the Hardy–Weinberg principle, also known as the Hardy–Weinberg equilibrium, model, theorem, or law, states that allele and genotype frequencies in a population will remain constant from generation to generation in the absence of other evolutionary influences. These influences include genetic drift, mate choice, assortative mating, natural selection, sexual selection, mutation, gene flow, meiotic drive, genetic hitchhiking, population bottleneck, founder effect,inbreeding and outbreeding depression.

Allele frequency, or gene frequency, is the relative frequency of an allele at a particular locus in a population, expressed as a fraction or percentage. Specifically, it is the fraction of all chromosomes in the population that carry that allele over the total population or sample size. Microevolution is the change in allele frequencies that occurs over time within a population.

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 separated onto different chromatids during chromosomal crossover, and are therefore said to be more linked than markers that are far apart. In other words, the nearer two genes are on a chromosome, the lower the chance of recombination between them, and the more likely they are to be inherited together. Markers on different chromosomes are perfectly unlinked, although the penetrance of potentially deleterious alleles may be influenced by the presence of other alleles, and these other alleles may be located on other chromosomes than that on which a particular potentially deleterious allele is located.

<span class="mw-page-title-main">Haplotype</span> Group of genes from one parent

A haplotype is a group of alleles in an organism that are inherited together from a single parent.

Genetic association is when one or more genotypes within a population co-occur with a phenotypic trait more often than would be expected by chance occurrence.

The transmission disequilibrium test (TDT) was proposed by Spielman, McGinnis and Ewens (1993) as a family-based association test for the presence of genetic linkage between a genetic marker and a trait. It is an application of McNemar's test.

Genetic hitchhiking, also called genetic draft or the hitchhiking effect, is when an allele changes frequency not because it itself is under natural selection, but because it is near another gene that is undergoing a selective sweep and that is on the same DNA chain. When one gene goes through a selective sweep, any other nearby polymorphisms that are in linkage disequilibrium will tend to change their allele frequencies too. Selective sweeps happen when newly appeared mutations are advantageous and increase in frequency. Neutral or even slightly deleterious alleles that happen to be close by on the chromosome 'hitchhike' along with the sweep. In contrast, effects on a neutral locus due to linkage disequilibrium with newly appeared deleterious mutations are called background selection. Both genetic hitchhiking and background selection are stochastic (random) evolutionary forces, like genetic drift.

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Haploview is a commonly used bioinformatics software which is designed to analyze and visualize patterns of linkage disequilibrium (LD) in genetic data. Haploview can also perform association studies, choosing tagSNPs and estimating haplotype frequencies. Haploview is developed and maintained by Dr. Mark Daly's lab at the MIT/Harvard Broad Institute.

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HLA A30-Cw5-B18-DR3-DQ2 (A30::DQ2) is a multigene haplotype that extends across a majority of the major histocompatibility complex on human chromosome 6. A multigene haplotype is a set of inherited alleles covering several genes, or gene-alleles. Long haplotypes, like A30::DQ2, are generally the result of descent by common ancestry. As haplotypes increase in size, Chromosomal recombination fragments them in a generation dependent process.

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In genetics, haplotype estimation refers to the process of statistical estimation of haplotypes from genotype data. The most common situation arises when genotypes are collected at a set of polymorphic sites from a group of individuals. For example in human genetics, genome-wide association studies collect genotypes in thousands of individuals at between 200,000-5,000,000 SNPs using microarrays. Haplotype estimation methods are used in the analysis of these datasets and allow genotype imputation of alleles from reference databases such as the HapMap Project and the 1000 Genomes Project.

In genetics, when multiple copies of a beneficial mutation become established and fix together it is called soft sweep. Depending on the origin of these copies, linked variants might then be retained and emerge as haplotype structures in the population. There are two major forms of soft sweeps:

  1. A beneficial mutation previously separated in the population neutrally and therefore existed as multiple haplotypes at the time of the selective shift in which the mutation became beneficial. In this way, a single beneficial mutation may carry multiple haplotypes to an intermediate frequency, while itself becomes fixed.
  2. Another model happening when multiple beneficial mutations independently occur in short succession of one another — consequently, a second copy occur through mutation before the selective fixation of the first copy.

References

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Further reading