Blythe R A
School of Physics, University of Edinburgh, Mayfield Road, Edinburgh EH9 3JZ, UK.
Theor Popul Biol. 2007 Jun;71(4):454-72. doi: 10.1016/j.tpb.2007.01.006. Epub 2007 Jan 30.
We study fixation probabilities and times as a consequence of neutral genetic drift in subdivided populations, motivated by a model of the cultural evolutionary process of language change that is described by the same mathematics as the biological process. We focus on the growth of fixation times with the number of subpopulations, and variation of fixation probabilities and times with initial distributions of mutants. A general formula for the fixation probability for arbitrary initial condition is derived by extending a duality relation between forwards- and backwards-time properties of the model from a panmictic to a subdivided population. From this we obtain new formulae(formally exact in the limit of extremely weak migration) for the mean fixation time from an arbitrary initial condition for Wright's island model, presenting two cases as examples. For more general models of population subdivision, formulae are introduced for an arbitrary number of mutants that are randomly located, and a single mutant whose position is known. These formulae contain parameters that typically have to be obtained numerically, a procedure we follow for two contrasting clustered models. These data suggest that variation of fixation time with the initial condition is slight, but depends strongly on the nature of subdivision. In particular, we demonstrate conditions under which the fixation time remains finite even in the limit of an infinite number of demes. In many cases-except this last where fixation in a finite time is seen--the time to fixation is shown to be in precise agreement with predictions from formulae for the asymptotic effective population size.
我们研究了在细分种群中由中性遗传漂变导致的固定概率和时间，其动机来自于一个语言变化的文化进化过程模型，该模型由与生物过程相同的数学描述。我们关注固定时间随亚种群数量的增长，以及固定概率和时间随突变体初始分布的变化。通过将模型的正向和反向时间属性之间的对偶关系从随机交配种群扩展到细分种群，推导出了任意初始条件下固定概率的一般公式。由此，我们得到了赖特岛模型从任意初始条件出发的平均固定时间的新公式（在极弱迁移极限下形式上精确），并给出两个案例作为示例。对于更一般的种群细分模型，引入了随机分布的任意数量突变体以及位置已知的单个突变体的公式。这些公式包含通常必须通过数值方法获得的参数，我们针对两个对比的聚类模型采用了该方法。这些数据表明，固定时间随初始条件的变化很小，但强烈依赖于细分的性质。特别是，我们证明了即使在无限多个deme的极限情况下固定时间仍保持有限的条件。在许多情况下——除了最后一种能在有限时间内固定的情况——固定时间与渐近有效种群大小公式的预测精确一致。
