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A study on limit theorems related to stochastic models in pupulation genetics

幼虫遗传学随机模型相关极限定理的研究

基本信息

批准号：
10640132
负责人：
IIZUKA Masaru
金额：
$ 0.77万
依托单位：
Kyushu Dental College
依托单位国家：
日本
项目类别：
Grant-in-Aid for Scientific Research (C)
财政年份：
1998
资助国家：
日本
起止时间：
1998 至 1999
项目状态：
已结题

项目摘要

Let N(t) = N(t,ωィイD21ィエD2) be the size of population in generation t. We assume that {N(t,ωィイD21ィエD2)}t=0,1,2,…, in a Markov chain of {NィイD21ィエD2, NィイD22ィエD2}, with the transition probability/P(N(t+1,ωィイD21ィエD2)≠ N(t, ωィイD21ィエD2)|N(t, ωィイD21ィエD2) =NィイD2jィエD2) = qィイD2jィエD2,P(N(t+1,ωィイD21ィエD2) = N(t, ωィイD21ィエD2)|N(t, ωィイD21ィエD2) =NィイD2jィエD2) = 1 - qィイD2jィエD2,(1 < NィイD21ィエD2 < NィイD22ィエD2). We consider a locus with two alleles AィイD21ィエD2 and AィイD22ィエD2. Let Z(t,ωィイD21ィエD2,ωィイD22ィエD2) be the number of AィイD21ィエD2 allele in generation t. For given {N(t,ωィイD21ィエD2)}t=0,1,2,…, a Markov chain {Z(t,ωィイD21ィエD2)}t=0,1,2,…, whose probability law is given by P(Z(t+1,ωィイD21ィエD2,ωィイD22ィエD2) = j|Z(t,ωィイD21ィエD2,ωィイD22ィエD2) = I) =(ィイD72N(t+1,ωィイD21ィエD2)(/)jィエD7)j) (ィイD7i(/)2N(t,ωィイD21ィエD2ィエD7)ィイD1jィエD1 (1-ィイD7i(/)2N(t,ωィイD21ィエD2)ィエD7)ィイD12N(t+1+ω1)-JィエD1, 0【less than or equal】I【less than or equal】"N(t,ωィイD21ィエD2), 0【less than or equal】j【less than or equal】2N(t+1,ωィイD21ィエD2), is referred to as a Wright-Fis … More her model with fluctuation of population size.In population genetics theory, a concept of effective size of population has played and important role. For the above model, we can define two effective sizes. The one is the variance effective size NィイD3(v)(/)eィエD3 and the other is the inbreeding effective size NィイD3(I)(/)eィエD3. In this study, the following properties are shown for the Wright-Fisher model with fluctuation of population size.1. The variance effective size is the same as the inbreeding effective size NィイD2eィエD2 = NィイD3(v)(/)eィエD3 = NィイD3(I)(/)eィエD3).2. The explicit form of the effective size NィイD2eィエD2 is determined.3. The effective size NィイD2eィエD2 is larger (resp. smaller) than the harmonic mean NィイD2HィエD2 of NィイD21ィエD2 and NィイD22ィエD2 if the fluctuation of population size is positively (resp. negatively) autocorrelated. The effective size is the same as the harmonic mean (NィイD2eィエD2 = NィイD2HィエD2) if and only if the fluctuation is uncorrelated.4. The dependence of qィイD21ィエD2 and qィイD22ィエD2 on the effective size NィイD2eィエD2 is determined.5. Asymptotic expression for the effective size NィイD2eィエD2 is determined for cases of qィイD21ィエD2, qィイD22ィエD2 → 0 or NィイD21ィエD2, NィイD22ィエD2 → ∞. Less

设N(T)=N(t，ωィイD21ィエD2)是第t代的种群大小，我们假设{N(t，ωィイD21ィエD2)}t=0，1，2，…，在{NィイD21ィエD2，NィイD22ィエD2}的马尔可夫链中，转移概率为/P(N(t+1，ωィイD21ィエD2)≠N(t，ωィイD21ィエD2)|N(t，ωィイD21ィエD2)=NィイD2jィエD2)=QィイD2jィエD2，P(N(t+1，ωィイD21ィエD2)=N(t，ωィイD21ィエD2)|N(t，ωィイD21ィエD2)=NィイD2jィエD2)=1-QィイD2jィエD2，(1&lt；NィイD21ィエD2&lt；NィイD22ィエD2)。我们考虑一个具有两个等位基因AィイD21ィエD2和AィイD22ィエD2的基因座。设Z(t，ωィイD21ィエD2，ωィイD22ィエD2)为第t代AィイD21ィエD2等位基因个数。对于给定的{N(t，ωィイD21ィエD2)}t=0，1，2，…，一个马尔可夫链{Z(t，ωィイD21ィエD2)}t=0，1，2，…其概率律由P(Z(t+1，ωィイD21ィエD2，ωィイD22ィエD2)=j|Z(t，ωィイD21ィエD2，ωィイD22ィエD2)=i)=i)=(ィイD72N(t+1，ωィイD21ィエD2)(/)jィエD7)j)(ィイD7i(/)2N(t，ωィイD21ィエD2ィエD7)ィイD1jィエd1(1-ィイD7i(/)2N(t，ωィイD21ィエD2)ィエD7)ィイD12N(t+1+ω1)-Jィエd1，0[小于等于]i[小于等于]N(t，ωィイD21ィエD2)，0[小于等于]j[小于等于]2N(t+1，ωィイD21ィエD2)，称为莱特菲斯…在种群遗传学理论中，有效种群数量的概念起到了重要的作用。对于上述模型，我们可以定义两个有效尺寸。一个是方差有效大小NィイD3(V)(/)eィエD3，另一个是近交有效大小NィイD3(I)(/)eィエD3。在这项研究中，具有种群规模涨落的Wright-Fisher模型表现出以下性质。方差有效大小与近交有效大小NィイD2eィエD2=NィイD3(V)(/)eィエD3=NィイD3(I)(/)eィエD3)相同。确定了有效尺寸NィイD2eィエD2的显式形式。有效尺寸NィイD2eィエD2较大(分别为小于NィイD21ィエD2和NィイD22ィエD2的调和平均值Nィイd2hD2和NィイD22ィエD2。负)自相关。当且仅当涨落不相关时，有效尺寸等于调和平均值(NィイD2eィエD2=Nィイd2hD2ィエD2)。确定了QィイD21ィエD2和QィイD22ィエD2与有效尺寸NィイD2eィエD2的关系。对于QィイD21ィエD2，QィイD22 D2→0或NィイD21ィエD2，NィイD22ィエD2→∞的情况，确定了有效尺寸NィイD2eィエD2的渐近表达式。较少