Sainudiin R, Thatte B, Véber A
Biomathematics Research Centre and School of Mathematics and Statistics, University of Canterbury, Private Bag 4800, Christchurch, 8041, New Zealand.
Departamento de Matemática, Universidade Federal de Minas Gerais, Av. Antônio Carlos, 6627, Belo Horizonte, CEP: 31270-901, Brazil.
J Math Biol. 2016 Jan;72(1-2):363-408. doi: 10.1007/s00285-015-0886-z. Epub 2015 Apr 30.
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).
我们推导了在具有n个二倍体个体的双亲子代群体的离散赖特-费希尔模型下，当前世代随机样本的祖先谱系数量的精确一步转移概率。我们的模型允许每代的重组概率为r。当r = 1时，我们的模型等同于Chang（《应用概率进展》31:1002 - 1038，1999）的真核生物谱系模型。当r = 0时，我们的模型等同于Kingman（《随机过程及其应用》13:235 - 248，1982）的离散合并模型，用于细胞质树或不含位点内重组的DNA位点的亚核树。当0 < r < 1时，我们的模型可以被认为是追踪一个包含来自常染色体的DNA序列的亚核祖先图，该序列具有位点内重组概率r。因此，我们以r ∈ [0, 1]为索引的模型族随着r从0到1的变化，以连续的方式将Kingman的离散合并与Chang的谱系联系起来。对于大群体，我们还研究了与给定的r ∈ (0, 1)相对应的祖先过程的三个性质：群体到最近共同祖先（MRCA）的时间Tn、所有个体要么是所有当今个体的共同祖先要么不是任何当今个体祖先的时间Un，以及在时间Un时是共同祖先的个体比例。这些结果推广了Chang（《应用概率进展》31:1002 - 1038，1999）的三个主要结果。当我们通过群体大小对时间和重组概率进行适当的重新缩放时，我们的模型导致了被称为Hudson（《理论种群生物学》23:183 - 201，1983）和Griffiths（《双位点祖先图》，数理统计研究所100 - 117，1991）的祖先重组图的连续时间马尔可夫链。
