小尺度突变率极限下的双等位基因突变-漂移扩散
Biallelic Mutation-Drift Diffusion in the Limit of Small Scaled Mutation Rates
摘要：
在扩散模型中研究了受突变和漂移力影响的双等位基因位点的等位基因比例$x$的演变，假设小规模的突变率。总体缩放突变率用$θ=(μ_1+μ_0)N$参数化，突变率的比率用$α=μ_1/(μ_1+μ_0)=1-β$参数化。该过程的平衡密度为β，参数为$αθ$和$βθ$。远离平衡态时，跃迁密度可以展开成一系列修正的雅可比多项式。如果缩放的突变率很小，即，$θ\ll 1$，可以假设多态性来源于边界处的突变。一个模型，其中的内部动力学符合纯漂移扩散模型和突变是从边界进入。在平衡状态下，多态性区域$[1/N,1-1/N]$内的多态性等位基因的比例密度\ie \ $x$为$αβθ(\tfrac1x+\tfrac1{1-x})=\tfrac{αβθ}{x(1-x)}$，而突变偏好$α$影响0和1处的单态等位基因的比例。类似于用修正的Jacobi多项式展开，导出了跃迁密度的级数展开式，它与Kimura用Gegenbauer多项式对纯漂移模型的著名解相联系。两个时间和两个空间区域被分离。代表多态区域内的空间分量的特征向量既不依赖于缩放的突变率$θ$也不依赖于突变偏置$α$。因此，参数变化，例如，增长或收缩的群体或突变偏差的变化可以相对容易地建模，而无需改变雅可比多项式级数展开所需的本征函数。
英文摘要：
The evolution of the allelic proportion $x$ of a biallelic locus subject to the forces of mutation and drift is investigated in a diffusion model, assuming small scaled mutation rates. The overall scaled mutation rate is parametrized with $θ=(μ_1+μ_0)N$ and the ratio of mutation rates with $α=μ_1/(μ_1+μ_0)=1-β$. The equilibrium density of this process is beta with parameters $αθ$ and $βθ$. Away from equilibrium, the transition density can be expanded into a series of modified Jacobi polynomials. If the scaled mutation rates are small, i.e., $θ\ll 1$, it may be assumed that polymorphism derives from mutations at the boundaries. A model, where the interior dynamics conform to the pure drift diffusion model and the mutations are entering from the boundaries is derived. In equilibrium, the density of the proportion of polymorphic alleles, \ie\ $x$ within the polymorphic region $[1/N,1-1/N]$, is $αβθ(\tfrac1x+\tfrac1{1-x})=\tfrac{αβθ}{x(1-x)}$, while the mutation bias $α$ influences the proportion of monomorphic alleles at 0 and 1. Analogous to the expansion with modified Jacobi polynomials, a series expansion of the transition density is derived, which is connected to Kimura's well known solution of the pure drift model using Gegenbauer polynomials. Two temporal and two spatial regions are separated. The eigenvectors representing the spatial component within the polymorphic region depend neither on the on the scaled mutation rate $θ$ nor on the mutation bias $α$. Therefore parameter changes, e.g., growing or shrinking populations or changes in the mutation bias, can be modeled relatively easily, without the change of the eigenfunctions necessary for the series expansion with Jacobi polynomials.
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