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].