突变与Shannon信息熵
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摘要
本文以Shannon信息量为工具，对群体遗传学中的突变问题进行了以下四方面分析：
1．两对中性基因平衡群体的Shannon信息熵性质，得到如下结果：
（1）两对中性等位基因平衡群体的基因型信息熵最大，配子信息熵最大，即最大信息熵分布就是平衡群体分布。
（2）两对中性等位基因平衡群体中，各位点基因型间相互独立，两性配子间相互独立，即它们的互信息为0。
（3）两对中性等位基因平衡群体中，杂合度和相对信息熵都可以作为群体多样度的测度，相对信息熵反映了群体中所有基因型的信息，而杂合度反映了杂合体的信息，因而作为基因多样度的测度，优于。计算机模拟结果表明，两者呈正相关，且相关系数为0.9747.
（4）两对中性等位基因的非平衡群体，经过一代随机交配，各位点的基因型信息熵达到最大；随着随机交配代数的增加，配子信息熵逐代增大，直到平衡时达到最大。
（5）两对中性等位基因平衡群体中，基因型信息熵等于2倍的配子信息熵，配子信息熵等于各位点的基因库信息熵之和。
2．中性突变基因在世代传递中的Shannon信息熵性质，得到如下结果：
（1）在频发突变中，设基因和的频率为0.5的时间为，则时间小于时，基因库信息熵单调上升；时间大于时，基因库信息熵单调下降；时间等于时，基因库信息熵达到最大。
（2）在频发突变中，最终基因在群体固定。基因发生替换时，群体所付出的代价为。
（3）在往复突变Aa（u＞v）中，基因频率随世代变化是一个稳定平衡过程，平衡频率为平衡时的基因库信息熵为，进化结果为稳定多态性。
（4）在往复突变Aa（u＞v）中，当基因的初始频率小于0.5时，基因库信息熵随世代先单调上升到，然后单调下降到；当基因的初始频率大于或等于0.5而小于平衡频率时，基因库信息熵随世代单调下降到；当基因的初始频率大于平衡频率时，基因库信息熵随世代单调上升到；当基因的初始频率等于平衡频率，基因库信息熵不随世代变化。
（5）在往复突变Aa（u＞v）中，群体由初始到平衡的变化过程中，群体所付出的代价为，其中为平衡时的群体信息熵。
3．中性突变基因在有限群体中的Shannon信息熵性质，得到如下结果：
（1）在群体大小为的随机交配的中性突变基因群体中，初始群体的基因库为随机漂变的最终结果是群体分布为。
（2）在群体大小为的随机交配的中性突变基因群体中，群体的信息熵随世代减小。
（3）在群体大小为的随机交配的中性突变基因群体中，群体由于随机遗传漂
变而达到固定，群体所付出的代价为。
4．自然选择与稳定多态现象的Shannon信息熵性质，得到如下结果：
（1）初始群体的基因库为基因型的选择系数分别为，则当时，基因的平衡频率为其中，群体为稳定平衡；当时，基因的平衡频率仍为群体为不稳定平衡。
（2）在的情况下，若，平衡时的基因库信息熵为，当基因的初始频率小于0.5时，基因库信息熵随世代先单调上升到，然后单调下降到；当基因的初始频率大于或等于0.5而小于平衡频率时，基因库信息熵随世代单调下降到；当基因的初始频率大于平衡频率时，基因库信息熵随世代单调上升到；当基因的初始频率等于平衡频率，基因库信息熵不随世代变化。
（3）在的情况下，若，当基因的初始频率小于0.5时，基因库信息熵随世代单调下降到0，基因替换；当基因的初始频率大于0.5而小于平衡频率时，基因库信息熵随世代单调上升到，然后下降到0，基因替换；当基因的初始频率大于平衡频率时，基因库信息熵随世代单调下降到0，基因替换；当基因的初始频率等于平衡频率，基因库信息熵不随世代变化。
（4）在针对基因的完全及部分选择中，群体的最终结果为基因替换，群体发生基因替换所付出的代价为。
（5）在超显性选择、突变与选择共同作用下，群体由初始状态到平衡，所付出的代价为，其中为平衡时的群体信息熵。
Based on the Shannon information entropy, this paper discusses the mutation in population genetics from four aspects:
1 From the analysis on the entropy character of two pairs alleles equilibrium population, we get the following results:
(1) The genotype information entropy and gametal information entropy of two pairs alleles equilibrium are the largest, that means, the largest information entropy distribution is the distribution of equilibrium population.
(2) Among the equilibrium population, each position genotype is independent , the same is true for bisextral gametal. That means their interinformation is zero.
(3) Among the equilibrium population, the degree of heterozygosity H and relative information entropy can be used for the measurement of population diversity. The relative information entropy reflects the information of all genotypes in the population and the degree of heterozygosity H reflects the information of heterogeneous species. So as the measurement of gene diversity, is superior to H. The simulated result by computer shows that the two are positively relative and their correlative modulus is 0.9747.
(4) The non-equilibrium population, after random copulation for one generation, the information entropy of each position genotype reaches the largest; with the increase of random copulation generation, gametal information entropy increases generation after generation. It reaches the largest at the balanced state.
(5) Among the equilibrium population, the information entropy of genotype is twice
that of gamete. Gametal information entropy is equal to the sum of information entropy of the two loci’s gene pool.
2 From the analysis on the entropy character of Shannon information entropy of neutral mutation gene, we get the following results:
(1) In the frequent mutation , suppose the time for gene “A” and “a”’s frequency to reach 0.5 is . Before the time reach , the gene pool information entropy is continuously increasing. After that point, the gene pool information entropy is continuously decreasing. When the time is ,the gene pool information entropy reaches the maximum.
(2) ) In the frequent mutation , the end gene “a” is fixed in population. When gene replacement happens, the information entropy cost offered by the population is .
(3) In the reciprocating mutation, gene frequency’s change with different time is a steady and balancing process. The balanced frequency is . The information entropy of gene pool at the balanced point is . The evolution result is stable diversity.
(4) In the reciprocating mutation, when the initial frequency of gene “a” is below 0.5, the gene pool information entropy first continuously rises to and then decreases to ;When the former is equal to or bigger than 0.5 but below the balanced frequency , the gene pool information entropy continuously decreases to ; when the former is bigger than the latter, the gene pool information entropy continuously increases to ; when the former is equal to the latter, the gene pool information entropy dosen’t change.
(5) In the reciprocating mutation, during the changes from the initial point to the balanced point, the cost spent by the population is , among which is the population information entropy at the balanced point.
3 From the analysis on the entropy character of Shannon information entropy of neutral mutation gene in the limited population, we get the following results:
(1) In the random copulation population whose number is N, supposed the initial gene
pool distribution is , the eventual result of gene random drift is that the population’s genotype distributions is .
(2) In the above population, the information entropy of the population decreases with time change.
(3) In the above population, the population becomes steady with random drift. The cost of it is .
4 From the study of the character of natural selection and steady diversity, we get the following results:
(1) Suppose the gene pool of the initial population is , the selection modulus of the genotyp
1．两对中性基因平衡群体的Shannon信息熵性质，得到如下结果：
（1）两对中性等位基因平衡群体的基因型信息熵最大，配子信息熵最大，即最大信息熵分布就是平衡群体分布。
（2）两对中性等位基因平衡群体中，各位点基因型间相互独立，两性配子间相互独立，即它们的互信息为0。
（3）两对中性等位基因平衡群体中，杂合度和相对信息熵都可以作为群体多样度的测度，相对信息熵反映了群体中所有基因型的信息，而杂合度反映了杂合体的信息，因而作为基因多样度的测度，优于。计算机模拟结果表明，两者呈正相关，且相关系数为0.9747.
（4）两对中性等位基因的非平衡群体，经过一代随机交配，各位点的基因型信息熵达到最大；随着随机交配代数的增加，配子信息熵逐代增大，直到平衡时达到最大。
（5）两对中性等位基因平衡群体中，基因型信息熵等于2倍的配子信息熵，配子信息熵等于各位点的基因库信息熵之和。
2．中性突变基因在世代传递中的Shannon信息熵性质，得到如下结果：
（1）在频发突变中，设基因和的频率为0.5的时间为，则时间小于时，基因库信息熵单调上升；时间大于时，基因库信息熵单调下降；时间等于时，基因库信息熵达到最大。
（2）在频发突变中，最终基因在群体固定。基因发生替换时，群体所付出的代价为。
（3）在往复突变Aa（u＞v）中，基因频率随世代变化是一个稳定平衡过程，平衡频率为平衡时的基因库信息熵为，进化结果为稳定多态性。
（4）在往复突变Aa（u＞v）中，当基因的初始频率小于0.5时，基因库信息熵随世代先单调上升到，然后单调下降到；当基因的初始频率大于或等于0.5而小于平衡频率时，基因库信息熵随世代单调下降到；当基因的初始频率大于平衡频率时，基因库信息熵随世代单调上升到；当基因的初始频率等于平衡频率，基因库信息熵不随世代变化。
（5）在往复突变Aa（u＞v）中，群体由初始到平衡的变化过程中，群体所付出的代价为，其中为平衡时的群体信息熵。
3．中性突变基因在有限群体中的Shannon信息熵性质，得到如下结果：
（1）在群体大小为的随机交配的中性突变基因群体中，初始群体的基因库为随机漂变的最终结果是群体分布为。
（2）在群体大小为的随机交配的中性突变基因群体中，群体的信息熵随世代减小。
（3）在群体大小为的随机交配的中性突变基因群体中，群体由于随机遗传漂
变而达到固定，群体所付出的代价为。
4．自然选择与稳定多态现象的Shannon信息熵性质，得到如下结果：
（1）初始群体的基因库为基因型的选择系数分别为，则当时，基因的平衡频率为其中，群体为稳定平衡；当时，基因的平衡频率仍为群体为不稳定平衡。
（2）在的情况下，若，平衡时的基因库信息熵为，当基因的初始频率小于0.5时，基因库信息熵随世代先单调上升到，然后单调下降到；当基因的初始频率大于或等于0.5而小于平衡频率时，基因库信息熵随世代单调下降到；当基因的初始频率大于平衡频率时，基因库信息熵随世代单调上升到；当基因的初始频率等于平衡频率，基因库信息熵不随世代变化。
（3）在的情况下，若，当基因的初始频率小于0.5时，基因库信息熵随世代单调下降到0，基因替换；当基因的初始频率大于0.5而小于平衡频率时，基因库信息熵随世代单调上升到，然后下降到0，基因替换；当基因的初始频率大于平衡频率时，基因库信息熵随世代单调下降到0，基因替换；当基因的初始频率等于平衡频率，基因库信息熵不随世代变化。
（4）在针对基因的完全及部分选择中，群体的最终结果为基因替换，群体发生基因替换所付出的代价为。
（5）在超显性选择、突变与选择共同作用下，群体由初始状态到平衡，所付出的代价为，其中为平衡时的群体信息熵。
Based on the Shannon information entropy, this paper discusses the mutation in population genetics from four aspects:
1 From the analysis on the entropy character of two pairs alleles equilibrium population, we get the following results:
(1) The genotype information entropy and gametal information entropy of two pairs alleles equilibrium are the largest, that means, the largest information entropy distribution is the distribution of equilibrium population.
(2) Among the equilibrium population, each position genotype is independent , the same is true for bisextral gametal. That means their interinformation is zero.
(3) Among the equilibrium population, the degree of heterozygosity H and relative information entropy can be used for the measurement of population diversity. The relative information entropy reflects the information of all genotypes in the population and the degree of heterozygosity H reflects the information of heterogeneous species. So as the measurement of gene diversity, is superior to H. The simulated result by computer shows that the two are positively relative and their correlative modulus is 0.9747.
(4) The non-equilibrium population, after random copulation for one generation, the information entropy of each position genotype reaches the largest; with the increase of random copulation generation, gametal information entropy increases generation after generation. It reaches the largest at the balanced state.
(5) Among the equilibrium population, the information entropy of genotype is twice
that of gamete. Gametal information entropy is equal to the sum of information entropy of the two loci’s gene pool.
2 From the analysis on the entropy character of Shannon information entropy of neutral mutation gene, we get the following results:
(1) In the frequent mutation , suppose the time for gene “A” and “a”’s frequency to reach 0.5 is . Before the time reach , the gene pool information entropy is continuously increasing. After that point, the gene pool information entropy is continuously decreasing. When the time is ,the gene pool information entropy reaches the maximum.
(2) ) In the frequent mutation , the end gene “a” is fixed in population. When gene replacement happens, the information entropy cost offered by the population is .
(3) In the reciprocating mutation, gene frequency’s change with different time is a steady and balancing process. The balanced frequency is . The information entropy of gene pool at the balanced point is . The evolution result is stable diversity.
(4) In the reciprocating mutation, when the initial frequency of gene “a” is below 0.5, the gene pool information entropy first continuously rises to and then decreases to ;When the former is equal to or bigger than 0.5 but below the balanced frequency , the gene pool information entropy continuously decreases to ; when the former is bigger than the latter, the gene pool information entropy continuously increases to ; when the former is equal to the latter, the gene pool information entropy dosen’t change.
(5) In the reciprocating mutation, during the changes from the initial point to the balanced point, the cost spent by the population is , among which is the population information entropy at the balanced point.
3 From the analysis on the entropy character of Shannon information entropy of neutral mutation gene in the limited population, we get the following results:
(1) In the random copulation population whose number is N, supposed the initial gene
pool distribution is , the eventual result of gene random drift is that the population’s genotype distributions is .
(2) In the above population, the information entropy of the population decreases with time change.
(3) In the above population, the population becomes steady with random drift. The cost of it is .
4 From the study of the character of natural selection and steady diversity, we get the following results:
(1) Suppose the gene pool of the initial population is , the selection modulus of the genotyp
引文
达尔文, 周建人，叶笃庄，方宗熙译.物种起源[M].北京：商务印书馆，1991.
D. T. 铃木，A. J. F. 格里菲斯，R. C. 雷文廷，兰斌等译.遗传分析导论[M].西安：陕西人民教育出版社，1990.
季道藩.遗传学[M].北京：农业出版社，1986.
罗辽复.生物进化的物理观[M].上海：上海科学技术出版社，2000
张昀.生物进化[M].北京：北京大学出版社，1998.
C. C. Li著,吴忠贤译. 群体遗传学[M]. 农业出版社, 1981.
高平仲. 群体遗传学导论[M]. 农业出版社, 1993.
根井正利，王家玉译.. 分子群体遗传学与进化论[M].农业出版社,1983.
孟庆生.信息论[M].西安交通大学出版社，1986.
朱雪龙.应用信息论基础[M].清华大学出版社，2001.
[英] D.S.Falconer,[美] T.F.C.Mackay著，储明星译.数量遗传学导论.中国农业出版社,2000.
王身立..生物物理遗传学[M].湖南科学技术出版社，1992.
中国科学院生物多样性委员会.生物多样性译丛（三）[M].北京：科学出版社，1997.
季维智，宿兵.遗传多样性研究的原理与方法[M].杭州：浙江科学出版社，1999.
胡志昂，张亚平.中国动植物的遗传多样性[M].杭州：浙江科学出版社，1997.
T. K Attwood, D J Parry-Smith著，罗静初等译.生物信息学概论[M].北京：北京大学出版社，2002.
Minoru Kanehisa等著，孙之荣等译.后基因组信息学[M].北京：清华大学出版社，2002.
李衍达，孙之荣.生物信息学——基因和蛋白质分析的实用指南[M].北京：清华大学出版社，2000.
R. Durbin S. Eddy, A. Krogh, G. Mitchison..生物序列分析-蛋白质和核酸的概率论模型[M].北京：清华大学出版社.2002.
张成岗，贺福初.生物信息学方法与实践[M].北京：科学出版社.2002.
赵国屏，陈军，陈凯先等.生物信息学[M].北京：科学出版社.2002.
[巴西]J.塞图宝，J.梅丹尼斯著,朱浩等译.计算分子生物学[M].北京：科学出版社.2003.
将彦，王小行等著.基础生物信息学及其应用[M].北京：清华大学出版社.2003.
李越中，闫章才，高培基.基因组研究与生物信息学[M].济南：山东大学出版
社.2001.
杨群.分子古生物学原理与方法[M].北京：科学出版社.2003.
张劳, 李玉奎编著. 群体遗传学概论[M]. 中国农业出版社 1999
袁志发，宋世德，周静芋等.非平衡群体基因变异测量[J].西北农业大学学报.1998，26（4）：30-34.
郭满才，宋世德，周静芋等.非平衡群体基因变异测量的Shannon信息量方法[J].生物数学学报.2001，16（3）：341-347.
刘建军，郭满才，解小莉等.性连锁群体平衡的信息模型研究[J].西北农林科技大学学报（自然科学版）.2002，30（4）：119-122.
汪小龙，袁志发，郭满才.最大信息熵原理与群体遗传平衡[J].遗传学报.2002，29（6）：562-564.
张宏礼，郭满才，解小莉等.近亲繁殖下一对等基因群体的熵性质[J].西北农林科技大学学报（自然科学版）.2003.31（1）：148-150.
解小莉，郭满才，张宏礼等.两对等位基因群体熵的性质[J].生物数学学报.2003,18（4）:482-486..
郭满才.群体遗传变异的信息论模型研究.西北农林科技大学博士学位论文[D].2002.
袁志发，周静芋.多元统计分析[M].科学出版社.2002.
Mayr E. Populations, species, and Evolution[M]. Cambridge, Massachusetts: Harvard University Press, 1977.
Dobzhansky T. Genetics and the Origin of Speies[M]. New York: Columbia Univ Press, 1937
Fisher R. A. The General Theory of Natural Selection[M]. Oxford: Clarendon Press, 1930
Haldane J. B. S. The Causes of Evolution[M]. New York: Harper and Row,1932.
Dobzhansky T., Ayala F. J., Stebbins G. L., Valentine J. W. Evolution[M]. Freeman H.and company, san Francisco, 1977.
Li W-H, Graur D.Fundamentals of Molecular Evolution[M]. Sunderland (Massachusetts): Sinauer Associates,1991.
Crow, J. F. and Kimura, M. An Introduction to Population Genetics Theory[M]. Burgess, Minnesota, USA,1970.
Cynthia Gibas & Per Jambeck. Developing Bioinformatics Computer Skill[M]. O’Reilly & Associates, 2001.
Darnell. J. E.. H. F. Lodish and D. Baltimore. Molecular Cell Biology[M]. 2nd Ed. Scientific American Books, New York,1990.
David W. Mount. Bioinformatics: Sequence and Genome Analysis[M]. Cold Spring Harbor Lab(CSHL) Press. 2001.
Ewens, W. J. Population Genetics[M]. USA,1969.
Falconer, D. S. Introduction to Quantitative Genetics[M], 2nd ed. Longman, London,1981.
Fisher, H. D. The Genetical Theroy of Natural Selection[M]. oxford: Clarendon Press,1930.
Gale, J. S. Population Genetic[M]s. John Wiley, New York,1980.
Guo Man-cai, Yuan Zhi-fa, Chen Hong, et al. Research of the Genetic Diversity Index Sysaem. Animal Biotechnology Bulletin[A].Vol. 8(1):359-364.
Guo Man-cai, Yuan Zhi-fa, Chen Hong, et al. Research of the Homological DNA Similarity Index[A]. Animal Biotechnology Bulletin. 8(1): 370-372.
Haldane, J. B. S. The cost of natual selection. J. Genet. 1957,55, 511-524.
Haldane, J. B. S. More Precise expressions for the cost of natural Selection. J. Genet. 1960,57, 351-360
Hartl, D. L. Principies of Population Genetics[M]. Sinauer, Massachusetts, USA,1980.
KIMURA, M. . Some problems of stochastic processes in genetics[J]. Ann. Math. Statist.1957,28, 882-901
KIMURA, M. 1962. On the probability of fixation of mutant genes in a population[J]. Genetics .1962,47, 713-719
KIMURA, M. Diffusion models in population genetics[J]. Appl. Probab. 1964,1, 177-232
KIMURA, M. Attainment of quasi linkage equilibrium when gene frequencies are changing by natural selection[J]. Genetics. 1965,52, 875-890
KIMURA, M. Evolutionary rate at the molecular level[J]. Nature .1968a,217, 624-626
KIMURA, M. Genetic variability maintained in a finite population due to mutational production of neutral and nearly neutral isoalleles[J]. Genet. Res. 1968b,11, 247-269
KIMURA, M. The number of heterozygous nucleotide sites maintained in a finitc population due to steady flux of mutation[J]. Genctics. 1969a,61, 893-903
KIMURA, M. THE RATE OF MOLECULAR EVOLUTION CONSIDERED FROM THE STANDPOINT OF POPULATION GENETICS[J]. pROC. Natl. Acad. Sci. U. S. 1969b,63, 1181-1188
KIMURA, M., Theoretical foundation of population genetics at the molecular level[J]. Theoret. Popul. Biol.1971, 2, 174-208.
KIMURA, M. Gene pool of higher organisms as a product of evolution[J]. Cold Spring
Harbor Symp. Quant. Biol. 1974,38, 515-524
Kimura, M. and Ohta, T. Theoretical Aspects of Population Genetics[M]. Princeton Univ. Press, USA,1971.
KIMURA, M. and T. OHTA . On some principles governing molecular evolution[J]. Proc. Natl. Acad. Sci. U. S. 1974,1, 2848-2852
Kojima, K. Mathematical Topics in Population Genetics[M]. New York,.1970.
Kolmogorov, A Deviations from Hardy's formula in partial isolation Compt rend[J]. Acad. d-sc. de I'U. R. S. S. 1935,3(7), 129132
Lewin. B. Genes Ⅳ[M]. Oxford University Press. New York,1990.
Li, C. C. First Course in Population Genetics[M]. California, USA,1976.
Merrell, D. J. Ecological Genetics[M]. London,1981.
Mettler, L. E. and Gregg G. Population Gentics and Evolution[M]. New Jerkey, USA,1939.
Nei, M. Moleculer Population Genetics and Evolution[M]. New York,1975.
Nei, M. Molecular Evolutinary Genetics[M]. Columbia Univ. Press, New York,1987.
Pirchner, F. Population Genetics in Animal Breeding[M]. 2nd ed. New York,1983.
Roughgarden, J. , Theory of Population Genetics and Evolutionary Ecology[M]. New York,1979.
Wright, S. The differential equation of the distribution of gene frequencies. Proc,1945. 39-59
Solbrig, O. T. and Solbrig, D. J., Introduction to Population Biology and Evolution. Massachusetts[M], USA,1979.
Spiess, E. B. , Genes in Populations[M]. New York,1977.
Stryer. L. Biochemistry[M]. 3rd Ed. Freman. New York,1988.
Yuan Zhi-fa, Zhou Jing-yu,, Guo Man-cai, et al. Gene Diversity and Shannon Information Entropy. Animal[A] Biotechnology Bulletin.8(1):353-358.
Wallace, B. , Basic Population Genetics[M]. New York,1981.
Guo Man-cai, Yuan Zhi-fa, Chen Hong, et al. Research of the Genetic Diversity Index Sysaem. Animal Biotechnology Bulletin[A].Vol. 8(1):359-364.
Guo Man-cai, Yuan Zhi-fa, Chen Hong, et al. Research of the Homological DNA Similarity Index[A]. Animal Biotechnology Bulletin. 8(1): 370-372.
Wilson, E. O. and Bossert, W. H. ,A Primer of Population Biology[M]. Massachusetts, USA,1977.
Wright, S. Evolution in Mondelian populations. Genetics, 1931,16: 97-159
Schrqdinger,E. What’slife.Macmillanco,NewYork,1946,CamdriggeUniv,Press,1951.
D. T. 铃木，A. J. F. 格里菲斯，R. C. 雷文廷，兰斌等译.遗传分析导论[M].西安：陕西人民教育出版社，1990.
季道藩.遗传学[M].北京：农业出版社，1986.
罗辽复.生物进化的物理观[M].上海：上海科学技术出版社，2000
张昀.生物进化[M].北京：北京大学出版社，1998.
C. C. Li著,吴忠贤译. 群体遗传学[M]. 农业出版社, 1981.
高平仲. 群体遗传学导论[M]. 农业出版社, 1993.
根井正利，王家玉译.. 分子群体遗传学与进化论[M].农业出版社,1983.
孟庆生.信息论[M].西安交通大学出版社，1986.
朱雪龙.应用信息论基础[M].清华大学出版社，2001.
[英] D.S.Falconer,[美] T.F.C.Mackay著，储明星译.数量遗传学导论.中国农业出版社,2000.
王身立..生物物理遗传学[M].湖南科学技术出版社，1992.
中国科学院生物多样性委员会.生物多样性译丛（三）[M].北京：科学出版社，1997.
季维智，宿兵.遗传多样性研究的原理与方法[M].杭州：浙江科学出版社，1999.
胡志昂，张亚平.中国动植物的遗传多样性[M].杭州：浙江科学出版社，1997.
T. K Attwood, D J Parry-Smith著，罗静初等译.生物信息学概论[M].北京：北京大学出版社，2002.
Minoru Kanehisa等著，孙之荣等译.后基因组信息学[M].北京：清华大学出版社，2002.
李衍达，孙之荣.生物信息学——基因和蛋白质分析的实用指南[M].北京：清华大学出版社，2000.
R. Durbin S. Eddy, A. Krogh, G. Mitchison..生物序列分析-蛋白质和核酸的概率论模型[M].北京：清华大学出版社.2002.
张成岗，贺福初.生物信息学方法与实践[M].北京：科学出版社.2002.
赵国屏，陈军，陈凯先等.生物信息学[M].北京：科学出版社.2002.
[巴西]J.塞图宝，J.梅丹尼斯著,朱浩等译.计算分子生物学[M].北京：科学出版社.2003.
将彦，王小行等著.基础生物信息学及其应用[M].北京：清华大学出版社.2003.
李越中，闫章才，高培基.基因组研究与生物信息学[M].济南：山东大学出版
社.2001.
杨群.分子古生物学原理与方法[M].北京：科学出版社.2003.
张劳, 李玉奎编著. 群体遗传学概论[M]. 中国农业出版社 1999
袁志发，宋世德，周静芋等.非平衡群体基因变异测量[J].西北农业大学学报.1998，26（4）：30-34.
郭满才，宋世德，周静芋等.非平衡群体基因变异测量的Shannon信息量方法[J].生物数学学报.2001，16（3）：341-347.
刘建军，郭满才，解小莉等.性连锁群体平衡的信息模型研究[J].西北农林科技大学学报（自然科学版）.2002，30（4）：119-122.
汪小龙，袁志发，郭满才.最大信息熵原理与群体遗传平衡[J].遗传学报.2002，29（6）：562-564.
张宏礼，郭满才，解小莉等.近亲繁殖下一对等基因群体的熵性质[J].西北农林科技大学学报（自然科学版）.2003.31（1）：148-150.
解小莉，郭满才，张宏礼等.两对等位基因群体熵的性质[J].生物数学学报.2003,18（4）:482-486..
郭满才.群体遗传变异的信息论模型研究.西北农林科技大学博士学位论文[D].2002.
袁志发，周静芋.多元统计分析[M].科学出版社.2002.
Mayr E. Populations, species, and Evolution[M]. Cambridge, Massachusetts: Harvard University Press, 1977.
Dobzhansky T. Genetics and the Origin of Speies[M]. New York: Columbia Univ Press, 1937
Fisher R. A. The General Theory of Natural Selection[M]. Oxford: Clarendon Press, 1930
Haldane J. B. S. The Causes of Evolution[M]. New York: Harper and Row,1932.
Dobzhansky T., Ayala F. J., Stebbins G. L., Valentine J. W. Evolution[M]. Freeman H.and company, san Francisco, 1977.
Li W-H, Graur D.Fundamentals of Molecular Evolution[M]. Sunderland (Massachusetts): Sinauer Associates,1991.
Crow, J. F. and Kimura, M. An Introduction to Population Genetics Theory[M]. Burgess, Minnesota, USA,1970.
Cynthia Gibas & Per Jambeck. Developing Bioinformatics Computer Skill[M]. O’Reilly & Associates, 2001.
Darnell. J. E.. H. F. Lodish and D. Baltimore. Molecular Cell Biology[M]. 2nd Ed. Scientific American Books, New York,1990.
David W. Mount. Bioinformatics: Sequence and Genome Analysis[M]. Cold Spring Harbor Lab(CSHL) Press. 2001.
Ewens, W. J. Population Genetics[M]. USA,1969.
Falconer, D. S. Introduction to Quantitative Genetics[M], 2nd ed. Longman, London,1981.
Fisher, H. D. The Genetical Theroy of Natural Selection[M]. oxford: Clarendon Press,1930.
Gale, J. S. Population Genetic[M]s. John Wiley, New York,1980.
Guo Man-cai, Yuan Zhi-fa, Chen Hong, et al. Research of the Genetic Diversity Index Sysaem. Animal Biotechnology Bulletin[A].Vol. 8(1):359-364.
Guo Man-cai, Yuan Zhi-fa, Chen Hong, et al. Research of the Homological DNA Similarity Index[A]. Animal Biotechnology Bulletin. 8(1): 370-372.
Haldane, J. B. S. The cost of natual selection. J. Genet. 1957,55, 511-524.
Haldane, J. B. S. More Precise expressions for the cost of natural Selection. J. Genet. 1960,57, 351-360
Hartl, D. L. Principies of Population Genetics[M]. Sinauer, Massachusetts, USA,1980.
KIMURA, M. . Some problems of stochastic processes in genetics[J]. Ann. Math. Statist.1957,28, 882-901
KIMURA, M. 1962. On the probability of fixation of mutant genes in a population[J]. Genetics .1962,47, 713-719
KIMURA, M. Diffusion models in population genetics[J]. Appl. Probab. 1964,1, 177-232
KIMURA, M. Attainment of quasi linkage equilibrium when gene frequencies are changing by natural selection[J]. Genetics. 1965,52, 875-890
KIMURA, M. Evolutionary rate at the molecular level[J]. Nature .1968a,217, 624-626
KIMURA, M. Genetic variability maintained in a finite population due to mutational production of neutral and nearly neutral isoalleles[J]. Genet. Res. 1968b,11, 247-269
KIMURA, M. The number of heterozygous nucleotide sites maintained in a finitc population due to steady flux of mutation[J]. Genctics. 1969a,61, 893-903
KIMURA, M. THE RATE OF MOLECULAR EVOLUTION CONSIDERED FROM THE STANDPOINT OF POPULATION GENETICS[J]. pROC. Natl. Acad. Sci. U. S. 1969b,63, 1181-1188
KIMURA, M., Theoretical foundation of population genetics at the molecular level[J]. Theoret. Popul. Biol.1971, 2, 174-208.
KIMURA, M. Gene pool of higher organisms as a product of evolution[J]. Cold Spring
Harbor Symp. Quant. Biol. 1974,38, 515-524
Kimura, M. and Ohta, T. Theoretical Aspects of Population Genetics[M]. Princeton Univ. Press, USA,1971.
KIMURA, M. and T. OHTA . On some principles governing molecular evolution[J]. Proc. Natl. Acad. Sci. U. S. 1974,1, 2848-2852
Kojima, K. Mathematical Topics in Population Genetics[M]. New York,.1970.
Kolmogorov, A Deviations from Hardy's formula in partial isolation Compt rend[J]. Acad. d-sc. de I'U. R. S. S. 1935,3(7), 129132
Lewin. B. Genes Ⅳ[M]. Oxford University Press. New York,1990.
Li, C. C. First Course in Population Genetics[M]. California, USA,1976.
Merrell, D. J. Ecological Genetics[M]. London,1981.
Mettler, L. E. and Gregg G. Population Gentics and Evolution[M]. New Jerkey, USA,1939.
Nei, M. Moleculer Population Genetics and Evolution[M]. New York,1975.
Nei, M. Molecular Evolutinary Genetics[M]. Columbia Univ. Press, New York,1987.
Pirchner, F. Population Genetics in Animal Breeding[M]. 2nd ed. New York,1983.
Roughgarden, J. , Theory of Population Genetics and Evolutionary Ecology[M]. New York,1979.
Wright, S. The differential equation of the distribution of gene frequencies. Proc,1945. 39-59
Solbrig, O. T. and Solbrig, D. J., Introduction to Population Biology and Evolution. Massachusetts[M], USA,1979.
Spiess, E. B. , Genes in Populations[M]. New York,1977.
Stryer. L. Biochemistry[M]. 3rd Ed. Freman. New York,1988.
Yuan Zhi-fa, Zhou Jing-yu,, Guo Man-cai, et al. Gene Diversity and Shannon Information Entropy. Animal[A] Biotechnology Bulletin.8(1):353-358.
Wallace, B. , Basic Population Genetics[M]. New York,1981.
Guo Man-cai, Yuan Zhi-fa, Chen Hong, et al. Research of the Genetic Diversity Index Sysaem. Animal Biotechnology Bulletin[A].Vol. 8(1):359-364.
Guo Man-cai, Yuan Zhi-fa, Chen Hong, et al. Research of the Homological DNA Similarity Index[A]. Animal Biotechnology Bulletin. 8(1): 370-372.
Wilson, E. O. and Bossert, W. H. ,A Primer of Population Biology[M]. Massachusetts, USA,1977.
Wright, S. Evolution in Mondelian populations. Genetics, 1931,16: 97-159
Schrqdinger,E. What’slife.Macmillanco,NewYork,1946,CamdriggeUniv,Press,1951.
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