Ancestries of a recombining diploid population.
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 (Adv Appl Probab 31:1002-1038, 1999) model for the karyotic pedigree. When r = 0, our model is equivalent to Kingman's (Stoch Process Appl 13:235-248, 1982) discrete coalescent model 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 ∈ [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 ancestral process corresponding to a given r ∈ (0, 1): the time Tn to a most recent common ancestor (MRCA) of the population, the time Un at which all individuals are either common ancestors of 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 of Chang's (Adv Appl Probab 31:1002-1038, 1999). 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 (Theor Popul Biol 23:183-201, 1983) and Griffiths (The two-locus ancestral graph, Institute of Mathematical Statistics 100-117, 1991).
Biomathematics Research Centre and School of Mathematics and Statistics, University of Canterbury, Private Bag 4800, Christchurch, 8041, New Zealand. r.sainudiin@math.canterbury.ac.nz.
R
R
Sainudiin
Departamento de Matemática, Universidade Federal de Minas Gerais, Av. Antônio Carlos, 6627, Belo Horizonte, CEP: 31270-901, Brazil. bhalchandra@mat.ufmg.br.
B
B
Thatte
Centre de Mathématiques Appliquées, École Polytechnique, Route de Saclay, 91128, Palaiseau Cedex, France. amandine.veber@cmap.polytechnique.fr.
A
A
Véber
D000818:Animals; D004171:Diploidy; D005260:Female; D005828:Genetics, Population; D006801:Humans; D008297:Male; D008390:Markov Chains; D055641:Mathematical Concepts; D008957:Models, Genetic; D010375:Pedigree; D011336:Probability; D011995:Recombination, Genetic; D012641:Selection, Genetic
Chang’s pedigree; Griffiths’ and Hudson’s ancestral recombination graphs; Kingman’s coalescent; Zygotic/karyotic/cytoplasmic/sub-karyotic ancestral graphs