Analytic computation of the expectation of the linkage disequilibrium coefficient r²

The squared correlation coefficient r² (sometimes denoted Δ²) is a measure of linkage disequilibrium that is widely used, but computing its expectation E [r²] in the population has remained an intriguing open problem. The expectation E [r²] is often approximated by the standard linkage deviation σ_d², which is a ratio of two expectations amenable to analytic computation. In this paper, a method of computing the population-wide E [r²] is introduced for a model with recurrent mutation, genetic drift and recombination. The approach is algebraic and is based on the diffusion process approximation. In the limit as the population-scaled recombination rate ρ approaches ∞, it is shown rigorously that the asymptotic behavior of E [r²] is given by 1 / ρ + O (ρ^{- 2}), which, incidentally, is the same as that of σ_d². A computer software that computes E [r²] numerically is available upon request.