Lappo Egor, Rosenberg Noah A
Department of Biology, Stanford University, Stanford, CA 94305, USA.
bioRxiv. 2025 Jun 24:2025.06.18.660412. doi: 10.1101/2025.06.18.660412.
The directionality index was introduced by Peter & Slatkin ( 67: 3274-3289, 2013) to infer the direction of range expansions from single-nucleotide polymorphism variation. Computed from the joint site frequency spectrum for two populations, uses shared genetic variants to measure the difference in the amount of genetic drift experienced by the populations, associating excess drift with greater distance from the origin of the range expansion. Although has been successfully applied in natural populations, its statistical properties have not been well understood. In this paper, we define as a random variable originating from a coalescent process in a two-population demography. For samples consisting of a pair of diploid genomes, one from each of two populations, we derive expressions for moments for standard parameterizations of bottlenecks during a founder event. For the expectation , we identify parameter combinations that represent distinct demographic scenarios yet yield the same value of . We also show that the variance increases with the time since the bottleneck, bottleneck severity, and size of the ancestral population. For a sample containing many SNPs, the central limit theorem allows us to approximate the distribution of the observed value of the index conditional on the demographic history. Finally, we analyze the values of computed from existing demographic models of and compare them with empirically computed . Our work builds the foundation for theoretical treatments of the index and can help in evaluating its behavior in empirical applications.
方向性指数由彼得和斯莱特金（《遗传学》67卷：3274 - 3289页，2013年）引入，用于从单核苷酸多态性变异推断范围扩张的方向。它根据两个种群的联合位点频率谱计算得出，利用共享遗传变异来衡量种群经历的遗传漂变量的差异，将过多的漂变与离范围扩张起源地的更大距离联系起来。尽管它已在自然种群中成功应用，但其统计特性尚未得到很好的理解。在本文中，我们将其定义为源自双种群人口统计学中合并过程的随机变量。对于由一对二倍体基因组组成的样本，每个样本分别来自两个种群之一，我们推导了在奠基者事件期间瓶颈标准参数化下矩的表达式。对于期望 ，我们确定了代表不同人口统计场景但产生相同 值的参数组合。我们还表明，方差 随着自瓶颈以来的时间、瓶颈严重程度和祖先种群大小的增加而增加。对于包含许多单核苷酸多态性的样本，中心极限定理使我们能够在人口统计历史条件下近似方向性指数观测值的分布。最后，我们分析了从现有的 人口统计模型计算出的 值，并将它们与经验计算的 值进行比较。我们的工作为方向性指数的理论处理奠定了基础，并有助于评估其在实证应用中的行为。
