Properties of Two-Locus Genealogies and Linkage Disequilibrium in Temporally Structured Samples

Archaeogenetics has been revolutionary, revealing insights into demographic history and recent positive selection in many organisms. However, most studies to date have ignored the non-random association of genetic variants at different loci (i.e., linkage disequilibrium, LD). This may be in part because basic properties of LD in samples from different times are still not well understood. Here, we derive several results for summary statistics of haplotypic variation under a model with time-stratified sampling: 1) The correlation between the number of pairwise differences observed between time-staggered samples ({pi}{Delta}t) in models with and without strict population continuity; 2) The product of the LD coefficient, D, between ancient and modern samples, which is a measure of haplotypic similarity between modern and ancient samples; and 3) The expected switch rate in the Li and Stephens haplotype copying model. The latter has implications for genotype imputation and phasing in ancient samples with modern reference panels. Overall, these results provide a characterization of how haplotype patterns are affected by sample age, recombination rates, and population sizes. We expect these results will help guide the interpretation and analysis of haplotype data from ancient and modern samples.