Tree measures and the number of segregating sites in time-structured population samples

Time-structured genetic samples are a valuable source of information in population genetics because they provide several correlated observations of the underlying evolutionary processes. In this paper we study basic properties of the genetic variation in time-structured samples as reflected in the genealogies relating individuals and the number of segregating sites observed. Our emphasis is on "measurably evolving populations" i.e. populations from which it is possible to obtain time-structured samples that span a significant interval of evolutionary time.

We use results from the coalescent process to derive properties of time-structured samples. In the first section we extend existing results to attain measures on coalescent trees relating time-structured samples. These include the expected time to a most recent common ancestor, the expected total branch length and the expected length of branches subtending only ancient individuals. The effect of different sampling schemes on the latter measure is studied. In the second section we study the special case where the full sample consists of a group of contemporary extant samples and a group of contemporary ancient samples. As regards this case, we present results and applications concerning the probability distribution of the number of segregating sites where a mutation is unique to the ancient individuals and the number of segregating sites where a mutation is shared between ancient and extant individuals.

The methodology and results presented here is of use to the design and interpretation of ancient DNA experiments. Furthermore, the results may be useful in further development of statistical tests of e.g. population dynamics and selection, which include temporal information.