Shafiey Hassan, Waxman David
Centre for Computational Systems Biology, Fudan University, Shanghai 200433, PR China.
Centre for Computational Systems Biology, Fudan University, Shanghai 200433, PR China.
J Theor Biol. 2017 Oct 7;430:64-77. doi: 10.1016/j.jtbi.2017.06.026. Epub 2017 Jun 22.
In this work we consider fixation of an allele in a population. Fixation is key to understanding the way long-term evolutionary change occurs at the gene and molecular levels. Two basic aspects of fixation are: (i) the chance it occurs and (ii) the way the gene frequency progresses to fixation. We present exact results for both aspects of fixation for the Wright-Fisher model. We give the exact fixation probability for some different schemes of frequency-dependent selection. We also give the corresponding exact stochastic difference equation that generates frequency trajectories which ultimately fix. Exactness of the results means selection need not be weak. There are possible applications of this work to data analysis, modelling, and tests of approximations. The methodology employed illustrates that knowledge of the fixation probability, for all initial frequencies, fully characterises the dynamics of the Wright-Fisher model. The stochastic equations for fixing trajectories allow insight into the way fixation occurs. They provide the alternative picture that fixation is driven by the injection of one carrier of the fixing allele into the population each generation. The stochastic equations allow explicit calculation of some properties of fixing trajectories and their efficient simulation. The results are illustrated and tested with simulations.
在这项工作中，我们考虑群体中等位基因的固定。固定是理解基因和分子水平上长期进化变化发生方式的关键。固定的两个基本方面是：（i）其发生的概率，以及（ii）基因频率达到固定的方式。我们给出了赖特 - 费希尔模型固定这两个方面的精确结果。我们给出了一些频率依赖选择不同方案的精确固定概率。我们还给出了相应的精确随机差分方程，该方程生成最终达到固定的频率轨迹。结果的精确性意味着选择不一定是微弱的。这项工作在数据分析、建模和近似检验方面可能有应用。所采用的方法表明，对于所有初始频率的固定概率知识，完全刻画了赖特 - 费希尔模型的动态。用于固定轨迹的随机方程有助于洞察固定发生的方式。它们提供了另一种观点，即固定是由每代向群体中注入一个固定等位基因的携带者驱动的。随机方程允许明确计算固定轨迹的一些性质及其有效模拟。通过模拟对结果进行了说明和检验。
