【作者】 张宏礼；

【导师】 袁志发； 周静芋； 郭满才；

【作者基本信息】 西北农林科技大学 ， 应用数学， 2004， 硕士

【摘要】 群体遗传学是研究生物群体的遗传结构及其变化规律的遗传学分支学科。生物进化从基因水平上看就是群体遗传结构的变化，群体遗传学的发展和生物进化理论密切相关。群体的遗传变异受多种因子的影响，进化中许多重要问题是无法通过对自然群体或实验群体的观察或实验来解决的，要研究群体遗传变异的程度，进行的速度，限制的条件，进行数学处理是必不可少的。 数学模型是对有关系统的特性的高度概括，以往群体遗传学中的数学模型基本上是统计学模型。群体遗传学所研究的世代传递过程本身也是一个信息传递的过程，因此信息论模型也应该是研究该门学科的一种数学模型。1998 年以来，袁志发、郭满才等提出用 Shannon信息熵作为度量群体遗传多样性的数量指标，为群体遗传学的发展提供了新思路。本研究正是在他们工作的基础上，应用信息论模型进一步研究群体遗传学中近亲繁殖群体的有关问题。 当信息论方法用于具体学科时，尽管信息的基本统计学性质本质是一样的，但在具体学科中会有其特殊的具体内容。在本研究中中，始终坚持下面的两个原则：　 １．当涉及到与频率分布有关的 Ｓｈａｎｎｏｎ 信息熵、互信息等概念及推理时，总是把正反交分开，并且假定正反交频率相等。　 ２．一切应用到群体遗传学中的信息概念、公式以及约束条件等均不依赖取值，与取值无关。 通过对近亲繁殖群体的信息论模型研究，得出以下主要结论： 1．基因型信息熵是关于基因型频率改变量或近交系数的单调递减的上凸函数。从随机交配下的平衡群体开始，在近亲繁殖制度下，其基因型信息熵 S(G)逐代减少，且满足S(G1) = S(G0 ) 2 ≤ S(G) ≤ S(G0)；经过足够多的世代后，基因型信息熵S(G)逐代减少的趋势越来越慢，最终趋于固定值，群体达到近亲繁殖下的平衡；随着世代交替中基因型信息熵S(G)的逐代减少，群体的遗传多样性程度也逐渐减小；基因型信息熵S(G)趋于固定值的过程，也是群体趋向于近亲繁殖平衡的过程。<WP=7>II 近亲繁殖群体的信息论模型研究 2．配子间互信息是关于基因型频率改变量或近交系数的单调递增的上凹函数。从随机交配下的平衡群体开始，在近亲繁殖制度下，配子间互信息 I(X,Y)逐代增加，且满足0≤ I(X,Y) ≤ S(X ) = S(Y) = S(A) = S(G1) = S(G0) 2；经过足够多的世代后，配子间互信息 I(X,Y)逐代增加的趋势越来越慢，最终趋于固定值，群体达到近亲繁殖下的平衡。 3．从随机交配下的平衡群体开始，在近亲繁殖制度下，配偶间的基因型联合信息熵逐代减少。配偶间互信息I(GX (t),GY (t))并不一定是关于世代数t的单调函数，但满足0 ≤ I(GX (t), GY(t)) ≤ S(G0)。 4．母子间基因型联合信息熵 S(G(t)G(t +1)) 是关于世代数 t的减函数。从随机交配下的平衡群体开始施以自交比例为w的近亲繁殖，则在世代交替中，母子间的基因型联合信息熵逐代减少；母子间的基因型联合信息熵满足m ≤ S(G(t)G(t +1)) ≤ M ，其中m = lim S(G(t)G(t +1)) = S(G(∞)G(∞))， M = S(G( G( )，m 的意义是：近亲繁殖的平 0) 1) t→∞衡状态下母子间的基因型联合信息熵；随世代交替，母子间基因型联合信息熵的变化率逐渐趋向于零，直到群体平衡。 5．定义并讨论了强相对基因型信息熵、弱相对基因型信息熵、强近交关联信息系数、弱近交关联信息系数等指标，用以进一步刻画群体的遗传、变异。更多还原

【Abstract】 Population genetics which study mainly on genetic structure of biological population andits disciplinarian is the branch subject of genetics. From the aspect of gene standard,biological evolution is the change of genetic structure of population. The development ofpopulation genetics relates to the theory of biological evolution closely. Population geneticvariation is affected by many factors. It is difficult to solve many important problems only bythe observation or experiment to natural population or experimental population in evolution.So mathematical method is absolutely necessary to research on the degree of genetic variation,the evolutionary speed, and the limited condition. Mathematical model is the high summary about some property of a certain studyingsystem. Beforehand mathematical model in population genetics is basically statistic model.But in fact, hereditary process itself is the process of informational pass. So informationmodel should be also applied to the genetic science as another mathematical method. Since1998, Prof. Yuan Zhi-fa and Prof. Guo Man-cai et al. argue that Shannon entropy can be usedas the metrical indexof genetic diversity, providing a new method for the development ofpopulation genetics. The study will apply information model to further discuss some problemson inbreeding population. When information method is applied in the other subjects, although basic statisticproperty of information is the same, it holds special content in different subjects. In this study,the two principles following are observed from beginning to end: 1．We always suppose that reciprocal crosses are divided, and that the frequency ofreciprocal crosses is the same, when Shannon entropy and interentropy related to distributionof frequency are involved. 2．All information concepts, formula and limited condition in population genetics havenothing to do with the value of frequency. According to studying on inbreeding populationwith the information model, the main<WP=9>IV 近亲繁殖群体的信息论模型研究results following are given: 1．Genotype entropy is monotonically decreasing and convex function by the genotypefrequency or inbreeding coefficient in inbreeding system. From the beginning of equilibriumpopulation of random mating system, in inbreeding system, its genotype entropy decreases bygeneration number, meeting: S(G1) = S(G0 ) 2 ≤ S(G) ≤ S(G0); after enough generations, thegenotype entropy decreases by generation becomes slower and slower, finally tends to fixedvalue, population reaches equilibrium of inbreeding; with the decrease of genotype entropy,the genetic diversity degree of population also decreases; the process that genotype entropytends to fixed value also is the process that population tends to inbreeding equilibrium. 2．Interentropy among gametes is monotonically increasing concave function bygenotype frequency or inbreeding coefficient. From the beginning of equilibrium of randommatting system, in inbreeding system, interentropy among gametes increases by generationnumber, meeting:0 ≤ I(X,Y) ≤ S(X ) = S(Y) = S(A) = S(G1) = S(G0) 2 ; after enough genera-tion, the interentropy among gametes increases by generation becomes slower and slower,finally tends to the fixed value, population reaches inbreeding equilibrium. 3．Associated genotype entropy between parents is monotonically decreasing function bygeneration number. Inter-entropy between parents is not always monotonically function aboutgeneration numbers, but meeting:0 ≤ I(GX (t),GY (t)) ≤ S(G0). 4．The associated entropy between a female genotypes and its descendant genotypes ismonotonic- ally decreasing function about generation number. From the beginning ofequilibrium of random mating system, inbreeding is given in mating proportion to w, inalternate generation, the associated entropybetween a female genotypes and its descendantgenotypes will decrease by generation, and更多还原