Linkage disequilibrium with multiple alleles and loci

Question

Linkage disequilibrium $\left(D\right)$ for two bi-allelic loci is defined as:

$$D=X_{11}X_{22} - X_{12}X_{21}$$

where $X_{11},\ X_{12},\ X_{21},\ X_{22}$ are the frequencies of the haplotypes $A_1B_1$, $A_1B_2$, $A_2B_1$, $A_2B_2$ respectively (the subscript number correspond to the two different alleles and $A$ and $B$ correspond to the two different loci)

My questions are

• How is $D$ defined for more loci and a greater number of alleles?

• How is $D$ defined when the number of alleles differ from locus to locus?

Answer 1

LD describes the associations among alleles at different loci. For anything more than two bi-allelic loci, you need more than just one number to fully describe the associations. For instance, if we have three bi-allelic loci 1-3, we need the three pairwise LD coefficients, plus one extra three-way coefficient. In general, if we have $K$ loci, and the $k^\text{th}$ locus has $n_k$ alleles, there are $\prod_k n_k$ possible haplotypes, so to fully describe the state of the population we need $\prod_k n_k-1$ haplotype frequencies. If we instead want to use allele frequencies and associations, we'll therefore need $$\prod_k n_k-1-\sum_k(n_k-1)=\prod_k n_k-\sum_k n_k+k-1$$ LD coefficients. (Note that each locus is described by $n_k-1$ allele frequencies.) This means that things get complicated pretty quickly with many loci! If you want to dive into the details, check out Barton and Turelli (1991) and Kirkpatrick, Johnson, and Barton (2002).

Answer 2

Being a typical molecular biologist, I am a little uncomfortable with classical genetics terms. I might redefine some symbols (perhaps to mean the same) [It is like talking to oneself while thinking].

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There are four DNA-blocks : A₁, B₁, A₂ and B₂. A_k and B_k are adjacent blocks. [Perhaps this is same as what you defined the symbols as]. A and B are contiguous DNA regions whereas 1 and 2 are different chromosomes (homologous regions). See the figure below.

                                    

The formula:

D=X₁₁X₂₂ - X₁₂X₂₁

basically means that the difference between the probability that the loci are in configuration I (11 & 22) or configuration II (swapped) (12 & 21). Lets say there is a third region (3) which is homologous to the two alleles which has DNA-blocks A₃ and B₃.

As LD is defined it is a difference in probabilities. Depending on how you define, it will take a negative or a positive value. When you have 3 recombinationally feasible alleles then there can be six cases:

1. C₁: [A₁B₁] [A₂B₂] [A₃B₃]
2. C₂: [A₁B₁] [A₂B₃] [A₃B₂]
3. C₃: [A₁B₃] [A₂B₂] [A₃B₁]
4. C₄: [A₁B₂] [A₂B₁] [A₃B₃]
5. C₅: [A₁B₂] [A₂B₃] [A₃B₁]
6. C₆: [A₁B₃] [A₂B₁] [A₃B₂]

In this case there is no single measure for difference. You can also do something like this:

LD1' = P(C1) + P(C2) - (P(C3) + P(C4) + P(C5) + P(C6))

Where LD_1' is LD of region 1 and P(C_m) is the frequency of m^th configuration.

Now, if there are 3 or more adjacent DNA blocks, you can only define LD, pairwise. It however, makes sense because linkage reduces with distance and you can calculate LD_AC = LD_AB x LD_BC
(A,B and C are contiguous in alphabetical order. If A and B are not linked then A and C cannot be linked.)