Hössjer Ola, Tyvand Peder A, Miloh Touvia
Department of Mathematics, Div. of Mathematical Statistics, Stockholm University, Stockholm SE 106 91, Sweden.
Department of Mathematical Sciences and Technology, Norwegian University of Life Sciences, Ås 1432, Norway.
Math Biosci. 2016 Feb;272:100-12. doi: 10.1016/j.mbs.2015.12.006. Epub 2015 Dec 24.
The classical Kimura solution of the diffusion equation is investigated for a haploid random mating (Wright-Fisher) model, with one-way mutations and initial-value specified by the founder population. The validity of the transient diffusion solution is checked by exact Markov chain computations, using a Jordan decomposition of the transition matrix. The conclusion is that the one-way diffusion model mostly works well, although the rate of convergence depends on the initial allele frequency and the mutation rate. The diffusion approximation is poor for mutation rates so low that the non-fixation boundary is regular. When this happens we perturb the diffusion solution around the non-fixation boundary and obtain a more accurate approximation that takes quasi-fixation of the mutant allele into account. The main application is to quantify how fast a specific genetic variant of the infinite alleles model is lost. We also discuss extensions of the quasi-fixation approach to other models with small mutation rates.
针对单倍体随机交配（赖特 - 费希尔）模型，研究了扩散方程的经典木村解，该模型具有单向突变且初始值由奠基群体指定。通过对转移矩阵进行约当分解的精确马尔可夫链计算，检验了瞬态扩散解的有效性。结论是，尽管收敛速度取决于初始等位基因频率和突变率，但单向扩散模型大多运行良好。对于突变率极低以至于非固定边界规则的情况，扩散近似效果不佳。当这种情况发生时，我们在非固定边界附近扰动扩散解，并得到一个更准确的近似值，该近似值考虑了突变等位基因的准固定。主要应用是量化无限等位基因模型的特定遗传变异丢失的速度。我们还讨论了将准固定方法扩展到其他具有小突变率的模型。
