Centre for Computational Systems Biology, Fudan University, Shanghai 20433, People's Republic of China.
Genetics. 2013 Aug;194(4):973-85. doi: 10.1534/genetics.113.152017. Epub 2013 Jun 7.
A numerical method is presented to solve the diffusion equation for the random genetic drift that occurs at a single unlinked locus with two alleles. The method was designed to conserve probability, and the resulting numerical solution represents a probability distribution whose total probability is unity. We describe solutions of the diffusion equation whose total probability is unity as complete. Thus the numerical method introduced in this work produces complete solutions, and such solutions have the property that whenever fixation and loss can occur, they are automatically included within the solution. This feature demonstrates that the diffusion approximation can describe not only internal allele frequencies, but also the boundary frequencies zero and one. The numerical approach presented here constitutes a single inclusive framework from which to perform calculations for random genetic drift. It has a straightforward implementation, allowing it to be applied to a wide variety of problems, including those with time-dependent parameters, such as changing population sizes. As tests and illustrations of the numerical method, it is used to determine: (i) the probability density and time-dependent probability of fixation for a neutral locus in a population of constant size; (ii) the probability of fixation in the presence of selection; and (iii) the probability of fixation in the presence of selection and demographic change, the latter in the form of a changing population size.
本文提出了一种数值方法，用于求解单一位点、两个等位基因随机遗传漂变的扩散方程。该方法旨在保持概率守恒，得到的数值解代表一个总概率为 1 的概率分布。我们将总概率为 1 的扩散方程解称为完全解。因此，本文中引入的数值方法可以生成完全解，并且这些解具有这样的性质，即只要固定和丢失都可能发生，它们就会自动包含在解中。这一特性表明，扩散近似不仅可以描述内部等位基因频率，还可以描述边界频率 0 和 1。本文提出的数值方法构成了一个单一的综合框架，可用于进行随机遗传漂变的计算。它的实现简单直接，允许它应用于各种问题，包括那些具有时变参数的问题，如种群大小的变化。作为对数值方法的测试和说明，它用于确定：(i)在大小恒定的种群中中性基因座的固定概率密度和时变概率；(ii)在存在选择的情况下的固定概率；(iii)在存在选择和人口变化的情况下的固定概率，后者以种群大小变化的形式出现。
