Asymptotic Distributions of Coalescence Times and Ancestral Lineage Numbers for Populations with Temporally Varying Size
Asymptotic Distributions of Coalescence Times and Ancestral Lineage Numbers for Populations with Temporally Varying Size
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DOI：
10.1534/genetics.113.151522
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发表时间：
2013-07-01
期刊：
影响因子：
3.3
通讯作者：
Chen, Kun
中科院分区：
文献类型：
--
作者：
Chen, Hua;Chen, Kun
合并时间和祖先谱系数量的分布在合并建模和祖先推断中起着至关重要的作用。合并时间和祖先谱系数的精确分布都表示为交替序列的总和，并且序列中的项对于大样本来说在数值上变得难以处理。在计算上更具吸引力的是它们的渐近分布，该分布是在 Griffiths (1984) 中针对具有恒定大小的群体得出的。在本文中，我们推导了随时间变化的种群的合并时间和祖先谱系数的渐近分布。对于大小为 n 的样本，用 T-m 表示 m + 1 个谱系合并为 m 个谱系时的第 m 个合并时间，A(n)(t) 表示从当前一代返回的时间 t 时祖先谱系的数量。与 Griffiths (1984) 的结果类似，祖先谱系的数量 A(n)(t) 和合并时间 T-m 呈渐近正态分布，这些分布的均值和方差取决于种群规模函数 N(t)。在合并的早期阶段，当 t -> 0 时，合并谱系数 n - A(n)(t) 遵循泊松分布，当 m -> n 时，n(n - 1)T-m/2N(0) 遵循伽马分布。我们通过与精确分布和合并模拟进行比较来证明渐近近似的准确性。还展示了理论结果的几个应用：导出与基因谱系特性相关的统计数据，例如谱系的最近共同祖先的时间（TMRCA）和总分支长度（TBL），以及导出大型谱系的等位基因频谱。随着大样本基因组水平测序数据的出现，渐近分布有望在群体遗传推断的理论和方法发展中得到广泛应用。
The distributions of coalescence times and ancestral lineage numbers play an essential role in coalescent modeling and ancestral inference. Both exact distributions of coalescence times and ancestral lineage numbers are expressed as the sum of alternating series, and the terms in the series become numerically intractable for large samples. More computationally attractive are their asymptotic distributions, which were derived in Griffiths (1984) for populations with constant size. In this article, we derive the asymptotic distributions of coalescence times and ancestral lineage numbers for populations with temporally varying size. For a sample of size n, denote by T-m the mth coalescent time, when m + 1 lineages coalesce into m lineages, and A(n)(t) the number of ancestral lineages at time t back from the current generation. Similar to the results in Griffiths (1984), the number of ancestral lineages, A(n)(t), and the coalescence times, T-m, are asymptotically normal, with the mean and variance of these distributions depending on the population size function, N(t). At the very early stage of the coalescent, when t -> 0, the number of coalesced lineages n - A(n)(t) follows a Poisson distribution, and as m -> n, n(n - 1)T-m/2N(0) follows a gamma distribution. We demonstrate the accuracy of the asymptotic approximations by comparing to both exact distributions and coalescent simulations. Several applications of the theoretical results are also shown: deriving statistics related to the properties of gene genealogies, such as the time to the most recent common ancestor (TMRCA) and the total branch length (TBL) of the genealogy, and deriving the allele frequency spectrum for large genealogies. With the advent of genomic-level sequencing data for large samples, the asymptotic distributions are expected to have wide applications in theoretical and methodological development for population genetic inference.