Linkage disequilibrium

Per Unneberg

A first look at haplotype structure

Figure 1: Shared haplotypes in the Drosophila ADH gene (Kreitman, 1983)

Observations

• Particular combinations of alleles at different SNPs occur together
• Non-random assortment of alleles is called linkage disequilibrium (LD)
  • Note: detecting LD does not ensure linkage or lack of equilibrium
• Unfortunate term: barrier to understanding (Slatkin, 2008)

Going forward: discuss how linkage generates LD and recombination breaks it down

Linkage generates haplotype structure

Figure 2: A simple genealogy with mutations mapped onto branches

Figure 3: Individual haplotypes and their mutations.

• Mutations 1, 2 on same branch \Rightarrow appear together
• Mutations 1, 5 on derived branches \Rightarrow usually appear together

Key assumption in genealogies: infinite sites model; a mutation happens once and once only at a given site

Linkage disequilibrium in the absence of recombination

Figure 4: In the absence of recombination we can observe 2 or 3 possible haplotypes out of 4. Here AB is the ancestral haplotype.

• we assume no homoplasy (back mutations)
• haplotypes consistent with single tree (without recombination) form perfect phylogeny

Note the implicit assumption of the infinite sites model, that is, we are only allowed to place each mutation once on the tree.

Recombination breaks association between loci

Miller (2020), Fig. 5.12.3

One (at least) crossover in meiosis I per chromosome! But: rates vary between loci (hotspots), sex chromosomes vs autosomes, and in some species, recombination only occurs in one sex (e.g., D.melanogaster).

Genetic distance and recombination rate

d - physical distance in bp

Definiton: genetic distance between two points is x centiMorgans(cM) if the average number of crossovers between points x/100 per meiosis

For short distances, genetic distance in cM \approx \mathrm{Pr}(\text{crossover})

Definition: recombination rate r relates genetic distance to base pair distance; commonly measured in cM/Mb

Example: in human average is 1.2 cM per Mb - the probability of crossover is a 1.2%

The mechanism that breaks up association between loci is recombination. Cartoon to right shows the possible (haploid) outcomes of meiosis. r (the recombination rate) denotes the probability that there is a recombination event between the two loci.

The farther apart two loci on the same chromosome, the larger the probability that a recombination event occurs between them. Note that this can be applied to loci on different chromosomes as well; they segregate independently.

Recombination generates new combinations of alleles

Figure 5: Recombination mixes haplotypes

Rules of thumb (human)

• For close SNPs (less than ~0.01-0.1 cM, or ~10-100Kb) linkage is stronger force than recombination
• At larger (>0.1cM) recombination is stronger

Measuring LD

Figure 6: Allele and haplotype frequencies at two SNPs. p denotes a frequency.

Given only information about SNP allele frequencies p_A and p_B, what would guess be for p_{AB}?

If independent, then p_{AB} = p_Ap_B, else p_{AB} \neq p_Ap_B. We measure the deviation D

D = p_{AB} - p_Ap_B

and say that there is linkage equilibrium if D=0!

Alternative measures

Unfortunate property of D: its magnitude depends on allele frequencies!

\begin{align*} D^\prime = \frac{D}{D_\mathrm{max}} & = \frac{D}{\min(p_Ap_b,p_ap_B)}, \quad\mathrm{for\ }D>0\\ & = \frac{D}{\min(p_Ap_B,p_ap_b)}, \quad\mathrm{for\ }D<0 \end{align*}

|D^\prime| < 1 implies there must have been recombination

r^2 = \frac{D^2}{p_Ap_ap_Bp_b}

• r^2=1 perfect LD
• r^2 natural parameter for measuring contribution of LD to genetic associations

Alternatively put:

\max{|D|} = \min(p(1-q), (1-p)q)

Better use R^2 or D^\prime

Square of correlation coefficient between alleles at two loci:

R^2 = \frac{D^2}{pq(1-p)(1-q)}

Toy example to show how mixing subpopulations can introduce linkage: assume two subpopulations fixed for AB and ab. Then for either population D=p_{AB} - p_Ap_B=1-1=0. Mixing the populations leads to p_{AB}=p_{ab}=p_A=p_A=0.5, so D=0.5-0.5^2=0.25, D_{max}=\min(p_Ap_B, p_ap_b)=0.25 and D^\prime=1.

LD decay

Figure 7: Decay of LD over time due to recombination, starting from D_0.

Observation

Even for free recombination (r=0.5) LD decay takes time. Comparison: HWE which takes one generation.

Figure 8: LD between pairs of autosomal SNPs for human and mouse. From (Laurie et al., 2007, Figure 2)

LD importance and applications

Importance

• information about past events
• constrains potential response to natural and artificial selection
• genome-wide LD is reflection of
  • population history
  • breeding system
  • pattern of geographic subdivision
• regional LD reflection of
  • natural selection
  • gene conversion
  • mutation and more

Due to lack of data LD didn’t become important until the end of the 70’s

Applications

Mutation and gene mapping

Most loci are in close linkage with a variable site which can be used as a marker to study the inheritance of a trait of interest.

Linked selection

Reduced variation close to a site under selection due to linkage

Estimating allele age

Strong LD in large region indication of young allele.

Determining window size for genome scans

Ideal window size ~ distance at which LD between markers approaches background levels

(Slatkin, 2008)

Bibliography

Kreitman, M. (1983). Nucleotide polymorphism at the alcohol dehydrogenase locus of Drosophila melanogaster. Nature, 304(5925), 412. https://doi.org/10.1038/304412a0

Laurie, C. C., Nickerson, D. A., Anderson, A. D., Weir, B. S., Livingston, R. J., Dean, M. D., Smith, K. L., Schadt, E. E., & Nachman, M. W. (2007). Linkage Disequilibrium in Wild Mice. PLOS Genetics, 3(8), e144. https://doi.org/10.1371/journal.pgen.0030144

Miller, C. (2020). Human Biology. Thompson Rivers University.

Pritchard, J. K. (n.d.). An Owner’s Guide to the Human Genome. Retrieved August 18, 2025, from https://web.stanford.edu/group/pritchardlab/HGbook.html

Slatkin, M. (2008). Linkage disequilibrium — understanding the evolutionary past and mapping the medical future. Nature Reviews Genetics, 9(6), 477–485. https://doi.org/10.1038/nrg2361