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。
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。
Population genetics which study mainly on genetic structure of biological population and
its 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 of
population genetics relates to the theory of biological evolution closely. Population genetic
variation is affected by many factors. It is difficult to solve many important problems only by
the 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 studying
system. Beforehand mathematical model in population genetics is basically statistic model.
But in fact, hereditary process itself is the process of informational pass. So information
model should be also applied to the genetic science as another mathematical method. Since
1998, Prof. Yuan Zhi-fa and Prof. Guo Man-cai et al. argue that Shannon entropy can be used
as the metrical indexof genetic diversity, providing a new method for the development of
population genetics. The study will apply information model to further discuss some problems
on inbreeding population.
When information method is applied in the other subjects, although basic statistic
property 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 of
reciprocal crosses is the same, when Shannon entropy and interentropy related to distribution
of frequency are involved.
2.All information concepts, formula and limited condition in population genetics have
nothing to do with the value of frequency.
According to studying on inbreeding populationwith the information model, the main
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 genotype
frequency or inbreeding coefficient in inbreeding system. From the beginning of equilibrium
population of random mating system, in inbreeding system, its genotype entropy decreases by
generation number, meeting: S(G1) = S(G0 ) 2 ≤ S(G) ≤ S(G0); after enough generations, the
genotype entropy decreases by generation becomes slower and slower, finally tends to fixed
value, population reaches equilibrium of inbreeding; with the decrease of genotype entropy,
the genetic diversity degree of population also decreases; the process that genotype entropy
tends to fixed value also is the process that population tends to inbreeding equilibrium.
2.Interentropy among gametes is monotonically increasing concave function by
genotype frequency or inbreeding coefficient. From the beginning of equilibrium of random
matting system, in inbreeding system, interentropy among gametes increases by generation
number, 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 by
generation number. Inter-entropy between parents is not always monotonically function about
generation numbers, but meeting:0 ≤ I(GX (t),GY (t)) ≤ S(G0).
4.The associated entropy between a female genotypes and its descendant genotypes is
monotonic- ally decreasing function about generation number. From the beginning of
equilibrium of random mating system, inbreeding is given in mating proportion to w, in
alternate generation, the associated entropybetween a female genotypes and its descendant
genotypes will decrease by generation, and
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