F between archaic and present-day samples.

The increasing abundance of DNA sequences obtained from fossils calls for new population genetics theory that takes account of both the temporal and spatial separation of samples. Here, we exploit the relationship between Wright's F and average coalescence times to develop an analytic theory describing how F depends on both the distance and time separating pairs of sampled genomes. We apply this theory to several simple models of population history. If there is a time series of samples, partial population replacement creates a discontinuity in pairwise F values. The magnitude of the discontinuity depends on the extent of replacement. In stepping-stone models, pairwise F values between archaic and present-day samples reflect both the spatial and temporal separation. At long distances, an isolation by distance pattern dominates. At short distances, the time separation dominates. Analytic predictions fit patterns generated by simulations. We illustrate our results with applications to archaic samples from European human populations. We compare present-day samples with a pair of archaic samples taken before and after a replacement event.