Linkage disequilibrium(LD)

LD Definition

In meiosis, homologous chromosomes are recombined. Recombination rates at different DNA regions are not equal. The fragments can be detected after tens of generations, causing Linkage disequilibrium, which refers to the non-random association of alleles of different loci.

Factors affecting LD

Recombination rates
Mutation
Natural selection
Genetic drift
Population subdivision and bottlenecks
Inbreeding

LD Estimation

Suppose we have two SNPs whose alleles are \(A/a\) and \(B/b\).

The haplotype frequencies are:

Haplotype	Frequency
AB	\(p_{AB}\)
Ab	\(p_{Ab}\)
aB	\(p_{aB}\)
ab	\(p_{ab}\)

The allele frequencies are:

Allele	Frequency
A	\(p_A=p_{AB}+p_{Ab}\)
a	\(p_A=p_{aB}+p_{ab}\)
B	\(p_A=p_{AB}+p_{aB}\)
b	\(p_A=p_{Ab}+p_{ab}\)

D : the level of LD between A and B can be estimated using coefficient of linkage disequilibrium (D), which is defined as:

\[D_{AB} = p_{AB} - p_Ap_B\]

If A and B are in linkage equilibrium, we can get

\[D_{AB} = p_{AB} - p_Ap_B = 0\]

which means the coefficient of linkage disequilibrium is 0 in this case.

D can be calculated for each pair of alleles and their relationships can be expressed as:

\[D_{AB} = -D_{Ab} = -D_{aB} = D_{ab} \]

So we can simply denote \(D = D_{AB}\), and the relationship between haplotype frequencies and allele frequencies can be summarized in the following table.

Allele	A	a	Total
B	\(p_{AB}=p_Ap_B+D\)	\(p_{aB}=p_ap_B-D\)	\(p_B\)
b	\(p_{AB}=p_Ap_b-D\)	\(p_{AB}=p_ap_b+D\)	\(p_b\)
Total	\(p_A\)	\(p_a\)	1

The range of possible values of D depends on the allele frequencies, which is not suitable for comparison between different pairs of alleles.

Lewontin suggested a method for the normalization of D :

\[D_{normalized} = {{D}\over{D_{max}}}\]

where

\[ D_{max} = \begin{cases} max\{-p_Ap_B, -(1-p_A)(1-p_B)\} & \text{when } D \lt 0 \\ min\{ p_A(1-p_B), p_B(1-p_A) \} & \text{when } D \gt 0 \\ \end{cases} \]

It measures how much proportion of the haplotypes had undergone recombination.

In practice, the most commonly used alternative metric to \(D_{normalized}\) is \(r^2\), the correlation coefficient, which can be obtained by:

\[ r^2 = {{D^2}\over{p_A(1-p_A)p_B(1-p_B)}} \]

Reference: Slatkin, M. (2008). Linkage disequilibrium—understanding the evolutionary past and mapping the medical future. Nature Reviews Genetics, 9(6), 477-485.

LD Calculation using software

LDstore2

LDstore2: http://www.christianbenner.com/#

Reference: Benner, C. et al. Prospects of fine-papping trait-associated genomic regions by using summary statistics from genome-wide association studies. Am. J. Hum. Genet. (2017).

PLINK LD

Please check Calculate LD using PLINK.

LD Lookup using LDlink

LDlink

LDlink is a suite of web-based applications designed to easily and efficiently interrogate linkage disequilibrium in population groups. Each included application is specialized for querying and displaying unique aspects of linkage disequilibrium.

https://ldlink.nci.nih.gov/?tab=home

Reference: Machiela, M. J., & Chanock, S. J. (2015). LDlink: a web-based application for exploring population-specific haplotype structure and linking correlated alleles of possible functional variants. Bioinformatics, 31(21), 3555-3557.

LDlink is a very useful tool for quick lookups of any information related to LD.

LDlink-LDpair

LDpair

LDlink-LDproxy

LDproxy for rs671

Query in batch using LDlink API

LDlink provides API for queries using command line.

You need to register and get a token first.

https://ldlink.nci.nih.gov/?tab=apiaccess

Query LD proxies for variants using LDproxy API

curl -k -X GET 'https://ldlink.nci.nih.gov/LDlinkRest/ldproxy?var=rs3&pop=MXL&r2_d=r2&window=500000&    genome_build=grch37&token=faketoken123'

LDlinkR

There is also a related R package for LDlink.

Query LD proxies for variants using LDlinkR

install.packages("LDlinkR")

library(LDlinkR)

my_proxies <- LDproxy(snp = "rs671", 
                      pop = "EAS", 
                      r2d = "r2", 
                      token = "YourTokenHere123",
                      genome_build = "grch38"
                     )

Reference: Myers, T. A., Chanock, S. J., & Machiela, M. J. (2020). LDlinkR: an R package for rapidly calculating linkage disequilibrium statistics in diverse populations. Frontiers in genetics, 11, 157.

LD-pruning

Please check LD-pruning

LD-clumping

Please check LD-clumping

LD score

Definition: https://cloufield.github.io/GWASTutorial/08_LDSC/#ld-score

LDSC

LD score can be estimated with LDSC using PLINK format genotype data as the reference panel.

plinkPrefix=chr22

python ldsc.py \
    --bfile ${plinkPrefix}
    --l2 \
    --ld-wind-cm 1\
    --out ${plinkPrefix}

Check here for details.

GCTA

GCTA also provides a function to estimate LD scores using PLINK format genotype data.

plinkPrefix=chr22

gcta64 \
    --bfile  ${plinkPrefix} \
    --ld-score \
    --ld-wind 1000 \
    --ld-rsq-cutoff 0.01 \
    --out  ${plinkPrefix}

Check here for details.

LD score regression

Please check LD score regression

Reference

LD review : Slatkin, M. (2008). Linkage disequilibrium—understanding the evolutionary past and mapping the medical future. Nature Reviews Genetics, 9(6), 477-485.
LD normalization : Lewontin, R. C. (1964). The interaction of selection and linkage. I. General considerations; heterotic models. Genetics, 49(1), 49.
plink: Purcell, S., Neale, B., Todd-Brown, K., Thomas, L., Ferreira, M. A., Bender, D., ... & Sham, P. C. (2007). PLINK: a tool set for whole-genome association and population-based linkage analyses. The American journal of human genetics, 81(3), 559-575.
gcta: Yang, J., Lee, S. H., Goddard, M. E., & Visscher, P. M. (2011). GCTA: a tool for genome-wide complex trait analysis. The American Journal of Human Genetics, 88(1), 76-82.
ldstore: Benner, C. et al. Prospects of fine-papping trait-associated genomic regions by using summary statistics from genome-wide association studies. Am. J. Hum. Genet. (2017).
ldlink: Machiela, M. J., & Chanock, S. J. (2015). LDlink: a web-based application for exploring population-specific haplotype structure and linking correlated alleles of possible functional variants. Bioinformatics, 31(21), 3555-3557.
ldlinkR: Myers, T. A., Chanock, S. J., & Machiela, M. J. (2020). LDlinkR: an R package for rapidly calculating linkage disequilibrium statistics in diverse populations. Frontiers in genetics, 11, 157.