最大信息熵原理与群体遗传平衡

建立了用最大信息熵原理推导群体遗传平衡定律的统一数学模型,并给出了模型的统一解,此解正是Hardy-Weinberg定律所给出的平衡群体的基因型频率,说明当群体信息熵达到最大时,群体基因型频率不再变化,即达到“平衡”。这证明了最大熵分布就是Hardy-Weinberg平衡分布。Hardy-Weinberg平衡定律与最大信息熵原理的内在一致性说明,杂交和随机交配是一个不可逆过程,使群体基因型信息熵增大,无序性增,是选择和近亲交配使群体的信息熵降低,有序性增加,育种过程实际就是调节群体信息熵的过程。过程信息熵的含义是表示一个概率分布的不确定性,最大熵原理意味着在一定的约束条件,选择具有最大不确定性的分布,从而其分布是最为随机的。最大熵原理在信息,工程,天文,地理,图像处理,模式识别等自然科学和社会科学领域都有广泛的成功应用,本文从群体遗传学角度证明了这一原理具有普遍适用性。熵是描述系统状态的函数,而最大熵原理则表明了系统发展变化的趋势,系统的最终状态必然是熵增加至最大值的状态,对于任何系统都是如此。因此,群体遗传系统的平衡定律可以统一用最大熵原理进行判定和描述;任意群体的基因型信息熵在随机交配世代传递时有不断增加的趋势;在一定约束条件下基因型信息熵达到最大值时,就称之为达到遗传平衡。本文将信息论原理应用于群体遗传学研究,揭示了基因信息熵的生物学意义,并表明可以用信息学和控制论的原理和方法来研究群体遗传学问题。     

A symmetric two locus model with viability and fertility selection

A two locus deterministic population genetic model is analysed. One locus is under viability selection, the other under fertility selection with both forms of selection completely symmetric. It is shown that linkage equilibrium may occur at two different equilibrium points. For a two-locus polymorphism to be stable, it is necessary that the viability locus be overdominant but not necessary that the fertility locus, considered separately, be able to support a stable polymorphism. The overlaps in stability are not as complex as under two locus symmetric fertilities, but considerably more complex than with symmetric viabilities. Extensions of the analysis for the central linkage equilibrium point with multiple viability and fertility loci are indicated.Research supported in part by NIH grants GM 28106 and GM 10452     

Perl and Python codes to estimate Hardy–Weinberg‐ and linkage‐disequilibrium‐related parameters

We have developed two pieces of Perl and Python codes to calculate the natural log of lambda, a quantity that expresses deviations from Hardy–Weinberg equilibrium, and R, a weighted correlation coefficient between alleles of locus pairs as a quantity that expresses linkage disequilibrium. These quantities can be used in an estimation approach as tools to facilitate the characterization and the comparison of populations.     

The effect of single and concomitant pathogen infections on condition and fecundity of the wild rabbit (Oryctolagus cuniculus)

We assessed the effect of two pathogens (myxoma virus and Eimeria stiedae) and five macroparasites (gastrointestinal helminth species) of the wild rabbit (Oryctolagus cuniculus) upon total host body mass and abdominal fat level. Additionally, we assessed the effects of these organisms on the number of foetuses in adult females during the peak breeding period. Both mass of abdominal fat and total body mass of the rabbit were negatively associated with myxoma virus infection and increasing helminth species richness. Total body mass was also negatively associated with the protozoan parasite E. steidae. No relationship was found between any of the parasites/pathogens and the number of foetuses in adult females, although only relatively small sample sizes were available for this section of the analysis. Increasing host body mass was positively associated with number of foetuses and we propose that mass reduction caused by the pathogen and parasite species could also have the consequence of reducing foetal number.     

A general population genetic framework for antagonistic selection that accounts for demography and recurrent mutation

Antagonistic selection--where alleles at a locus have opposing effects on male and female fitness ("sexual antagonism") or between components of fitness ("antagonistic pleiotropy")--might play an important role in maintaining population genetic variation and in driving phylogenetic and genomic patterns of sexual dimorphism and life-history evolution. While prior theory has thoroughly characterized the conditions necessary for antagonistic balancing selection to operate, we currently know little about the evolutionary interactions between antagonistic selection, recurrent mutation, and genetic drift, which should collectively shape empirical patterns of genetic variation. To fill this void, we developed and analyzed a series of population genetic models that simultaneously incorporate these processes. Our models identify two general properties of antagonistically selected loci. First, antagonistic selection inflates heterozygosity and fitness variance across a broad parameter range--a result that applies to alleles maintained by balancing selection and by recurrent mutation. Second, effective population size and genetic drift profoundly affect the statistical frequency distributions of antagonistically selected alleles. The "efficacy" of antagonistic selection (i.e., its tendency to dominate over genetic drift) is extremely weak relative to classical models, such as directional selection and overdominance. Alleles meeting traditional criteria for strong selection (N(e)s > 1, where N(e) is the effective population size, and s is a selection coefficient for a given sex or fitness component) may nevertheless evolve as if neutral. The effects of mutation and demography may generate population differences in overall levels of antagonistic fitness variation, as well as molecular population genetic signatures of balancing selection.     

Population genetics of polymorphism and divergence for diploid selection models with arbitrary dominance

We develop a Poisson random-field model of polymorphism and divergence that allows arbitrary dominance relations in a diploid context. This model provides a maximum-likelihood framework for estimating both selection and dominance parameters of new mutations using information on the frequency spectrum of sequence polymorphisms. This is the first DNA sequence-based estimator of the dominance parameter. Our model also leads to a likelihood-ratio test for distinguishing nongenic from genic selection; simulations indicate that this test is quite powerful when a large number of segregating sites are available. We also use simulations to explore the bias in selection parameter estimates caused by unacknowledged dominance relations. When inference is based on the frequency spectrum of polymorphisms, genic selection estimates of the selection parameter can be very strongly biased even for minor deviations from the genic selection model. Surprisingly, however, when inference is based on polymorphism and divergence (McDonald-Kreitman) data, genic selection estimates of the selection parameter are nearly unbiased, even for completely dominant or recessive mutations. Further, we find that weak overdominant selection can increase, rather than decrease, the substitution rate relative to levels of polymorphism. This nonintuitive result has major implications for the interpretation of several popular tests of neutrality.     

Two locus models of selection and mutation within and among full-sib lines

Summary General models for continued full-sib mating with two diallelic autosomal loci taking account of linkage, mutation and selection within and among lines are considered. The problems are first approached by deriving the full probability transition matrix, taking account of linkage, mutation and within-line selection. Exact solutions to the equilibrium system are possible, but the computational effort is prohibitive, and this is exacerbated by the introduction of between-line selection. A second approach is based on decomposing the transition matrix into blocks whose properties suggest approximations that lead to a rapid iterative solution of the equilibrium system. Extensive numerical analysis of models of within-line selection and of combined within- and between-line selection were made. The results show that equilibrium values are essentially independent of the degree of linkage under models of within-line selection. This is because mutation plays a dominant role in determining equilibrium structure. Results from models of combined within- and between-line selection show that between-line selection has the dominant influence on gene frequency equilibrium. Both within-line and between-line selection produce appreciable linkage disequilibrium only when selection is disruptive. The results also suggest that much of the twolocus equilibrium structure can be predicted from a knowledge of single-locus equilibria.     

One- and multi-locus multi-allele selection models in a random environment

Summary We deduce conditions for stochastic local stability of general perturbed linear stochastic difference equations widely applicable in population genetics. The findings are adapted to evaluate the stability properties of equilibria in classical one- and multi-locus multi-allele selection models influenced by random temporal variation in selection intensities. As an example of some conclusions and biological interpretations we analyse a special one-locus multi-allele model in more detail.This work was supported in part by Stiftung Volkswagenwerk.     

Evolution of virulence: interdependence, constraints, and selection using nested models

Natural selection acts on virus populations at two distinct but interrelated levels: within individual hosts and between them. Studies of the evolution of virulence typically focus on selection acting at the epidemiological or between-host level and demonstrate the importance of trade-offs between disease transmission and virulence rates. Within-host studies reach similar conclusions regarding trade-offs between transmission and virulence at the level of individual cells. Studies which examine selection at both scales assume that between- and within-host selection are necessarily in conflict. We explicitly examine these ideas and assumptions using a model of within-host viral dynamics nested within a model of between-host disease dynamics. Our approach allows us to evaluate the direction of selection at the within- and between-host levels and identify situations leading to conflict and accord between the two levels of selection.     

Equilibria and dynamics of selection at a diallelic autosomal locus in the Nagylaki-Crow continuous model of a monoecious population

Let birth rates and death rates be constant, birth rates positive, fertilities additive, and each birth rate not larger than twice any other birth rate. Global convergence to equilibria is proved for the model in the title. There is at most one polymorphic equilibrium or there are a continuum of equilibria. The phase portraits are given. If there is a polymorphic equilibrium, then the largest negatively invariant set in the state space is a continuous curve connecting the two fixation equilibria. This curve coincides with the Hardy-Weinberg manifold exactly when the death rate is additive. Disregarding extinction, the polymorphic equilibria are the same for the continuous model as for the corresponding discrete model exactly when the death rate is additive.     

Nonlinear frequency-dependent selection at a single locus with two alleles and two phenotypes

 The paper investigates the discrete frequency dynamics of two phenotype diploid models where genotypic fitness is an exponential function of the expected payoff in the matrix game. Phenotypic and genotypic equilibria are defined and their stability compared to frequency-dependent selection models based on linear fitness when there are two possible phenotypes in the population. In particular, it is shown that stable equilibria of both types can exist in the same nonlinear model. It is also shown that period-doubling bifurcations emerge when there is sufficient selection in favor of interactions between different phenotypes. Received: 22 October 1998     

Weak convergence of discrete time non-markovian processes related to selection models in population genetics

We consider discrete time stochastic processes defined by solutions to some non-linear difference equations whose coefficients are autocorrelated random sequences. It is proved that these processes converge weakly in D[0, T] to diffusion processes, under the assumption that the random sequences satisfy some mixing condition. Diffusion approximation for stochastic selection models in population genetics is discussed, as the application of this limit theorem.     

Hybridization and selection for improving seed protein in barley

Summary Hybridization followed by continuous selection of lines of barley from different cross-combinations involving high protein-high lysine genotypes and the agronomically superior strains resulted in breaking the negative correlations between 1000 grain weight and high protein content and high DBC values. The methodology of DBC-Kjeldahl protein adopted in the present study is likely to be useful in identifying high lysine lines. The present study has shown considerable variability with respect to protein content and grain weight and has provided interesting genotypes which can be used in synthesising lines with improved nutritional quality and productivity in barley. The success in breaking the undesirable linkages to factors that impair the endosperm development is due to sufficient genetic variability in the initial breeding material as well as the use of suitable breeding procedures like the full-sib mating in the early segregating generations.     

Convergence of genotypic frequencies for differential selfing and positive assortative mating at a biallelic locus

For a biallelic model of differential self-fertilization and differential positive assortative mating based on genotype, it is shown that the genotypic frequencies converge for all sets of mating system parameters. Overdominance and underdominance with respect to the parameters are necessary but not sufficient conditions for global convergence to a polymorphic equilibrium and local attractiveness of both the fixation states, respectively. There are cases of overdominance and underdominance for which one fixation state is globally attractive. The relationship of the result to those known from the classical viability selection model are briefly discussed. For the multiallelic version, it is shown that after the first generation all of the homozygote frequencies are always in excess of the corresponding Hardy-Weinberg proportions if at least one homozygote rate of self-fertilization or assortment probability is positive.     

Equilibria and dynamics of selection at a diallelic autosomal locus in the Nagylaki-Crow continuous model of a monoecious population with random mating

Global convergence to equilibria is proved for the model in the title under the assumption that death rates (and fertilities) are constant and fertilities are additive and positive. The phase portraits are determined. There is at most one polymorphic equilibrium or there are a continuum of equilibria.     

A theoretical method for evaluating the relative importance of positive selection and neutral drift from observed base changes

To evaluate the relative importance of positive selection and neutral drift from the nucleotide base changes observed in the homologous alignment of genes, a theoretical equation of base changes is formulated by including both the influence of selection and the base substitutions due to mutations. Under the assumption that the average rate of base substitutions estimated from synonymous changes is the ``true' mutation rate applicable at all positions, this method is applied to the vertebrate globin gene family, and evaluates the departures of base change rates from the ``true' mutation rate at the first and second codon positions as a consequence of preferential selection for the conservation of important function. In addition to the strong effect of selection on the amino acid residues in the internal region mostly common to myoglobin and hemoglobin chains, the distinctive directions of selective parameter values are seen at sites on the globin surface, distinguishing the subunit contact residues of hemoglobins from the polar residues on the surface of myoglobins. Moreover, this effect of selection distinguishing between the myoglobin and hemoglobin chain genes becomes weaker in cold-blooded vertebrates, especially in fish, strongly suggesting the possibility that the clear distinction between these globins is a result of selection out of the changes regarded as neutral ones in an ancestor of vertebrates. Thus, the present method may also serve to investigate the homology of many other proteins from the aspect of molecular evolution, mainly focusing on the evolution of their biological functions. Received: 2 January 1996 / Accepted: 20 February 1997     

Global convergence properties in multilocus viability selection models: the additive model and the Hardy-Weinberg law

A natural coordinate system is introduced for the analysis of the global stability of the Hardy-Weinberg (HW) polymorphism under the general multilocus additive viability model. A global convergence criterion is developed and used to prove that the HW polymorphism is globally stable when each of the loci is diallelic, provided the loci are overdominant and the multilocus recombination is positive. As a corollary the multilocus Hardy-Weinberg law for neutral selection is derived.Research supported in part by NIH grants GM 39907-01, GM 10452-26 and NSF Grant DMS 86-06244Research supported in part by a US-Israel Binational Science Foundation grant 85-00021 and NIH grant GM 28016     

Phenotypic response to selection for traits with direct and maternal components when generations overlap

Summary Prediction of response to selection for traits with direct and maternal components is described for discrete and overlapping generations. Expected phenotypic response is the sum of direct and maternal contributions, the latter having a genetic and an environmental component. With overlapping generations the selection differentials achieved on these components are added to respective updated vectors containing age-sex distributions with values of previous selection rounds. An example demonstrates that in the early stages, results may be considerably affected by environmental correlations between direct and maternal effects. The method could be helpful in interpreting phenotypic changes in a population selected for traits with maternal effects.