Linkage equilibrium between rare alleles

Recombination is ubiquitous among bacteria, but the extent to which it shapes genetic diversity within bacterial populations remains elusive. Classical approaches for measuring recombination focus on the correlations between alleles ("linkage disequilibrium"), and how they decay with the distance between loci. However, the overall levels of linkage disequilibrium are influenced by other evolutionary forces like natural selection and genetic drift, which makes it difficult to tease out the effects of recombination. Here, we introduce an alternative metric ("linkage equilibrium") that vanishes in the absence of recombination and approaches one for unlinked loci. We derive analytical expressions that predict how this metric scales with the rate of recombination, the strength of selection, and the present-day frequencies of the two alleles. We find that our linkage equilibrium metric strongly depends on this frequency scale, which reflects the underlying timescales over which these mutations occurred. We show how this scaling could be used to quantify the relative strengths of selection and recombination, and discuss their implications for recombination patterns in bacteria.