Ancestries of a Recombining Diploid Population,
R Sainudiin, B. Thatte and A. Veber, UCDMS Research Report 2014/3, 42 pages, 2014

We derive the exact one-step transition probabilities of the number of lineages
that are ancestral to a random sample from the current generation of a bi-parental
population that is evolving under the discrete Wright-Fisher model with n diploid
individuals. Our model allows for a per-generation recombination probability of
r. When r = 1, our model is equivalent to Chang’s model [4] for the karyotic
pedigree. When r = 0, our model is equivalent to Kingman’s discrete coalescent
model [16] for the cytoplasmic tree or sub-karyotic tree containing a DNA locus that
is free of intra-locus recombination. When 0 < r < 1 our model can be thought to
track a sub-karyotic ancestral graph containing a DNA sequence from an autosomal
chromosome that has an intra-locus recombination probability r. Thus, our family
of models indexed by r 2 [0; 1] connects Kingman's discrete coalescent to Chang's
pedigree in a continuous way as r goes from 0 to 1. For large populations, we
also study three properties of the r-specific ancestral process: the time Tn to a
most recent common ancestor (MRCA) of the population, the time Un at which all
individuals are either common ancestors to all present day individuals or ancestral
to none of them, and the fraction of individuals that are common ancestors at time
Un. These results generalize the three main results in [4]. When we appropriately
rescale time and recombination probability by the population size, our model leads
to the continuous time Markov chain called the ancestral recombination graph of
Hudson [12] and Griffiths [9].