封闭近交群体内等位墓因纯化的数学模型

目的：试建立线性代数模型，以拟合封闭近交群体内等位基因纯化的过程。方法：通过考察与性染色体Ｘ连锁的等位基因频率的变化规律，获得邻近两代（第ｎ－１代和第ｎ代）自交对基因型概率之间的关系为Ｘ~（ｎ）＝ＭＸ~（ｎ－１）。经矩阵相似变换，求得系数矩阵Ｍ的特征值、特征向量和可逆矩阵Ｐ。结果：第ｎ代自交对各种基因型的概率与起始状态基因型的概率之间的关系为Ｘ~（ｎ）＝Ｐ·Ｄ~ｎ·Ｐ~(－１)·Ｘ~（０），当ｎ→∞时，杂合自交对的基因型概率全为零，而只有纯合自交对的基因型概率非零，即自交对的等位基因都趋于纯化，只留下（Ａ，ＡＡ）和（ａ， ａ ａ）型。结论：线性代数模型定量地模拟了封闭近交群体内等位基因逐步纯化的过程，同时能客观分析不同自交代之间的生物数学特征及内在联系，根据模型还能由初始条件预报纯化基因型的频率。

A mathematical model imitating the allele homologizing process in a closed inbreeding population

Objective: To develop a linear algebra model to imitate the process of the allele homologizing within a closed inbreeding population. Methods: By examining the change of the frequency of X-linked alleles, the formula,X~(n) = MX~(n-1), expressing the relationship of the frequencies of alleies between 2 continued generations in a closed inbreeding population was obtained. Through matrix transformation of similitude, the characteristic values and vectors of coefficient matrix M, and invertible matrix P could be cacutated. Results:The relationship between the probabilities of each kind of genotypes in inbreeding pairs of generation n and initial generation is X~(n)=P' D~n. P~(-1)' X~(o). If generation number n approaches to the infinite, the frequencies of any kind of heterozygotes should approach to zero, however only the frequency of homozygotes, (A,AA) or (a, a a),would not be to zero. Conclusion:The gradual process of allele homologizing in a closed inbreeding population is imitated quantitatively by using linear algebra model. By the way, the biomathematic feature and the internal relationship among different selfing generations ware revealed and analysed objectively. At last, the frequencies of homologizing genotypes could be predicated based on the initial conditions.