Exact distribution of linkage disequilibrium in the presence of mutation, selection, or minor allele frequency filtering

Linkage disequilibrium (LD), often expressed in terms of the squared correlation (r²) between allelic values at two loci, is an important concept in many branches of genetics and genomics. Genetic drift and recombination have opposite effects on LD, and thus r² will keep changing until the effects of these two forces are counterbalanced. Several approximations have been used to determine the expected value of r² at equilibrium in the presence or absence of mutation. In this paper, we propose a probability-based approach to compute the exact distribution of allele frequencies at two loci in a finite population at any generation t conditional on the distribution at generation t-1. As r² is a function of this distribution of allele frequencies, this approach can be used to examine the distribution of r² over generations as it approaches equilibrium. The exact distribution of LD from our method is used to describe, quantify, and compare LD at different equilibria, including equilibrium in the absence or presence of mutation, selection, and filtering by minor allele frequency. We also propose a deterministic formula for expected LD in the presence of mutation at equilibrium based on the exact distribution of LD.