【作者】 刘建军；

【导师】 袁志发； 周静芋；

【作者基本信息】 西北农林科技大学 ， 应用数学， 2009， 硕士

【摘要】 本文以Shannon信息论为工具,对群体遗传学中的连锁遗传进行了以下三方面的研究:1有连锁的两对等位基因群体的Shannon信息熵研究,得出以下结果:(1)有连锁的两对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)有连锁的两对等位基因非平衡群体,经过一代随机交配,各位点基因型信息熵达到最大,随着交配代数的增加,群体信息熵,配子信息熵逐渐增大,直到平衡达到最大;(3)有连锁的两对等位基因非平衡群体,信息熵增大的速度随着连锁强度的减弱,即重组率c的增加而加快;(4)有连锁的两对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;(5)有连锁的两对等位基因平衡群体中,基因型信息熵等于配子信息熵的2倍,配子信息熵等于各位点基因库信息熵之和。2性连锁平衡群体信息模型研究,得出以下结果:(1)当雌性群体与雄性群体信息熵之和达到最大且等于基因信源信息熵3倍,也就是[S[G(♀)]+ S[G(♂)]]max=3 S[A(P)]时,,性连锁群体达到平衡.即p1 1 = p13=pq, P1 4 = q2, p 15 =p, p 16 =q(2)经过一代随机交配,性连锁群体的雄性基因信源为上代雌性基因信源,因而有S[A1(♂)]=S[A0(♀)].这个性质说明,平衡过程是下代雄性基因信源及其信息量变为上代雌性基因信源及其信息量的交替振荡过程,即S[Ai(♂)]=S[Ai-1(♀)], i=1,2,…,(3)平衡过程是用上代雌、雄基因信源信息量表达下代雌性群体信息量的过程,即S[Gi(♀)]=S[Ai-1(♂)]+S[Ai-1(♀)]. i=1,2,…,当S[A(♂)]=S[A(♀)]=S[A0(P)]时,再经过一代随机交配群体就达到平衡.(4)群体中基因平均信源信息量不变,即S[A1(P)]=S[A0(P)].这说明,在随机交配下,群体基因平均信息量逐代不变,即S[Ai(P)]=S[A0(P)]. i=1,2,…(5)经过一代随机交配雌性基因信源A, a的频率为上代雌、雄基因信源同种基因频率的平均值,即1p (A♀(6)性连锁不平衡群体,逐代随机交配下去,终将平衡,使S[G(♂)]+S[G(♀)]达到最大值3S[A0(P)]。由性连锁群体的平衡及其信息熵的分析知,平衡时信息熵最大,这在伴性遗传基因定位上很有意义。另外,性连锁不平衡群体,在随机交配下,其平衡过程就是信息熵增大的过程,即种内进化是熵增大过程,也是生物保持其遗传多样性的过程。3群体连锁在QTL分析中的信息模型研究,得出以下结论:(1)回交群体信息熵S (c)随着连锁强度的减弱而逐渐增大,由完全连锁时的ln 2增加到无连锁时的2 ln2.并且关于重组率c是增加的凸函数。(2)在回交群体对重组率c的信息论估计中,不论是利用亲祖型频率与重组型频率的观察值估计c.还是利用亲祖型频率与重组型频率的观察值的合并值估计c。重组率c的估计值c?就是使回交群体观察值信息熵S (p)与期望值信息熵S (c)之差的平方f ( c)= [S(p)?S(c)]2取得最小值时的取值,这时最小值为零。即[ f (c?)]min =0(3)设θ=(1 ?c)2,F2代群体表现型期望值信息熵为S (θ),当14≤θ≤1时, S (θ)是单调递减的凸函数,且(4)在用F2代群体对重组率c的信息论估计中,使函数f (θ)= [S(p)?S(θ)]2取得最小值时θ的取值所对应的c为重组率。重组率c的估计值为: c? =1?θ?.函数f (θ)最小值为零,即[ f (θ?)]min =0更多还原

【Abstract】 Based on the Shannon information entropy, this paper discusses the linkage heredity in population genetics from three aspects:1 From the analysis on the Shannon information entropy character of two pairs linkage alleles population, we get the following results:(1) The genotype information entropy and gametal information entropy of two pairs linkage alleles equilibrium population are the largest, that means the largest information entropy distribution is the distribution of equilibrium population.(2) The non-equilibrium population with two pair linkage alleles genotype, after random copulation for one generation, the information entropy of each position genotype reaches the largest; with the increase of random copulation generation, the population information entropy and gametal information entropy increase generation after generation. It reaches the largest at the balanced state.(3) Among the non-equilibrium population with two pairs linkage alleles genotype, the increases speed of information entropy will gradually get larger as decreases in intensity of linkage, that means as the increases speed of recombination frequencyc .(4) Among the equilibrium population with two pair linkage alleles genotype, each position genotype is independent, the same is true for bisextral gametal.(5)Among the equilibrium population with two pair linkage allele genotype, the genotype information entropy is twice of that of gamete, gametal information entropy is equal to the sum of the two loci′s gene pool.2 From the analysis on the information model of sex-linkage alleles equilibrium population, we get the following results:(1) When the sum information entropy in female population and male population reaches the largest, and is three times of the entropy in population gene information resources, [S[G(♀)]+ [S[G(♂)]max=3[S[A(P)], the sex-linkage population reaches equilibrium state. that is p1 1 = p13=pq, p14=q2,p15=p,p16=q(2) After random copulation for one generation, the male gene information resources in the sex-linkage population comes from the female gene information resources in former generation, that is S[A 1(♂)]+ S[A0(♀)].it explains: equilibrium process is a alternant vibration process with the increase of the gene information resources between the female gene information resources in former generation and the male gene information resources in the in later generation, that is S[A i(♂)]+ S[Ai-1(♀)], i=1,2,…,(3) Equilibrium process is a population gene expression process that the female gene information resources in the later generation comes from the female and the male gene information resources in the former generation, that is S[Gi(♀)]+ S[Ai-1(♂)]+ S[A i-1(♀)], i=1,2,…, whenS[A (♂)]= S[A (♀)]= S[A0(p)], after random copulation for one generation, equilibrium population is reached.(4) Among the population the information entropy value of gene average information resources is constant. it explains: after random copulation for one generation to it’s offspring, population gene average information entropy value is constant with generations. that is S[A i(P)]= S[A0(p)], i=1,2,…,(5) After random copulation for one generation, A, a, the frequency of the female gene information resources is equal to the average value of the same gene frequency from the female and the male gene information resources in former generation, that is(6) Among the non-equilibrium population with sex-linkage, in random matting generation after generation, the non-equilibrium population will reach the largest at the balanced state at last, that is 3[S[A(P)].From the analysis on the Shannon information entropy character of sex-linkage equilibrium population, the largest information entropy; the largest information entropy is at the balanced state. It is very important to locate gene with sex-linkage. Besides, among the non-equilibrium population with sex-linkage, in random matting system, equilibrium process is a process with the increase of the information entropy, that is a evolution process among the same species, and is a process the living things keep it’s diversity.3. The follow conclusion is made by the research on the informatics model in population linkage heredity in QTL analysis:(1) Information entropy S(c) in back cross population will gradually get larger from completely linked In2 to un-linked 2In2 as decreases in intensity of link, and about recombination frequency C is a increased convex function。(2).On the estimation of recombination frequency c in information theory in back cross population, no matter c is estimated by observation value in their ancestor genotype frequency and recombined genotype frequency or by the two combination observation value of genotype frequency .The estimation value c? is the square of difference between S ( p)and S (c), S ( p)means the information entropy of the estimation value in back cross population and S (c) means the information entropy of the expectation value. f ( c)= [S(p)?S(c)]2 As f (c) is in the minimal value,the minimal value is zero. And [ f (c?)]min =0(3).Ifθ=(1 ?c)2, S (θ)is the information entropy of the phenotype expectation value in generation F2 , if 14≤θ≤1, S (θ)is not only monotone increasing convex function, but also(4) With population of generation F2 ,on the estimation of recombination frequency c in information theory , the recombination frequency c ,is the correspondent value ofθwhen the function f (θ)= [S(p)?S(θ)]2 is in the lowest . The estimation value of recombination frequency c equals c? =1?θ?.the minimal value of function equals zero. and [ f (θ?)]min =0更多还原