The Estimation of a Multivariate Fitness Function from Several Samples Taken from a Population

The situation is considered where the multivariate distribution of certain variables X₁, X₂, …, X_p is changing with time in a population because natural selection related to the X's is taking place. It is assumed that random samples taken from the population at times t₁, t₂, …, t_s are available and it is desirable to estimate the fitness function w_t(x₁, x₂,…,x_p) which shows how the number of individuals with X_i = x_i, i = 1, 2, …, p at time t is related to the number of individuals with the same X values at time zero. Tests for population changes are discussed and indices of the selection on the population dispersion and the population mean are proposed. The situation with a multivariate normal distribution is considered as a special case. A maximum likelihood method that can be applied with any form of population distribution is proposed for estimating w_t. The methods discussed in the paper are illustrated with data on four dimensions of male Egyptian skulls covering a time span from about 4500 B.C. to about 300 A.D. In this case there seems to have been very little selection on the population dispersion but considerable selection on means.     

On the study of two interacting populations one of which acts independently of the other

Many ecological and biological systems can be studied in terms of a bivariate stochastic branching process, {X ₁ (t), X ₂ (t)}, each of whose components (or populations) varies in magnitude according to the laws of a generalized birth-death process. Of particular interest is such a model in which the birth and death rates of the first population,X ₁, are constant while those of the second population,X ₂, exhibit a functional dependence upon the magnitude of the first. It is shown, first, that the existence of the stochastic mean of a birth death process implies the existence of all higher moments. The values of all the factorial moments of such a process are then determined. The moments of the dependent population of the bivariate process are given in terms of its expectation and the joint probability density function of the process is determined. It is possible, therefore, to use Bayesian techniques to infer conclusions about the independent population, given information about the variation of the dependent one.     

On a Characterization of the Exponential Distribution by Weak Homoscedasticity of Record Values

A sequence {X_n, n≤1} of independent and identically distributed random values with continuous cumulative distribution function F(x) is considered. X_j is a record value of this sequence if X_j ≤ max (X₁, X₂, …, X_j?1). We define L(n)=min.(j!j>L(n?1.), X_j<X_L(n?1)), with L(0) = 1. Let Z_n=X_L(n)? X_L(n?1), n ≤ 1. We will show that the conditional variance of Z_n given X_L(n?1)=x does not depend on × if and only if F(x) is exponential.     

Clock gene variation is associated with breeding phenology and maybe under directional selection in the migratory barn swallow

Background

In diverse taxa, photoperiodic responses that cause seasonal physiological and behavioural shifts are controlled by genes, including the vertebrate Clock orthologues, that encode for circadian oscillator mechanisms. While the genetic network behind circadian rhythms is well described, relatively few reports exist of the phenological consequences of and selection on Clock genes in the wild. Here, we investigated variation in breeding phenology in relation to Clock genetic diversity in a long-distance migratory bird, the barn swallow (Hirundo rustica).

Methodology/Principal Findings

In a sample of 922 adult barn swallows from a single population breeding in Italy we found one very common (Q₇) and three rare (Q₅, Q₆, Q₈) length variants of a functionally significant polyglutamine repeat. Rare (2.9%) Q₇/Q₈ heterozygous females, but not males, bred significantly later than common (91.5%) Q₇/Q₇ females, consistent with the expectation that ‘long’ alleles cause late breeding, as observed in a resident population of another bird species. Because breeding date depends on arrival date from migration, present results suggest that the association between breeding date and Clock might be mediated by migration phenology. In addition, fecundity selection appears to be operating against Q₇/Q₈ because late migrating/breeding swallows have fewer clutches per season, and late breeding has additional negative selection effects via reduced offspring longevity. Genotype frequencies varied marginally non-significantly with age, as Q₇/Q₈ frequency showed a 4-fold reduction in old individuals. This result suggests negative viability selection against Q₇/Q₈, possibly mediated by costs of late breeding.

Conclusions/Significance

This is the first study of migratory birds showing an association between breeding phenology and Clock genotype and suggesting that negative selection occurs on a phenologically deviant genotype. Low polymorphism at Clock may constrain microevolutionary phenological response to changing climate, and may thus contribute to the decline of barn swallow populations.     

The Point Evaluation Method for Approximating Function Means,Variances and Covariances

A perennial problem in statistics is the determination of biases, variances and covariances for functions of random variables X₁, X₂, …, X_n which themselves have a known distribution. A common approach is through equations based upon Taylor series approximations but a “point evaluation” method may sometimes be a useful alternative. This involves approximating the multivariate distribution of the X variables by the 2ⁿ points given by X₁=μ₁±₁, X₂ = μ₂ ±₂, …, X_n = = μ_n μ_n, where μ_i is the mean and σ_i the standard deviation of Xi, with appropriate point weights. An advantage over the Taylor series approach is that function derivatives do not have to be explicitely calculated. The point evaluation method is particularly useful in cases where the X variables are uncorrelated. Then the evaluation of the 2ⁿ points can be replaced by the evaluation of 2n points. The point evaluation method is illustrated with powers of a normally distributed variable, and with estimation of gene frequencies from ABO blood group frequencies.     

重庆市蝴蝶多样性环境健康指示作用和环境监测评价体系构建

在重庆市生态功能区蝴蝶多样性参数研究的基础上,进行区系相似性分析;选取22个蝴蝶多样性参数和5项主要环境因子指标,进行回归和主成分分析,进而探讨监测指标、监测种类和类群的选取与评价体系的建立。分析结果表明:当进行区域级的监测时,在22个指标中,种、科、属的多样性指数和均匀度指数可以忽略;5项环境指标中,人口压力(X22)因子重于面积比例(X18)、人口密度(X19)、林地比例(X20)、土地载有量(X21)等4项因子。选出物种数(X1),蝴蝶属(X2)、科级(X3)单元的多少,种类优势度(X9),各数量等级的物种数贡献率(X10—X13),人口压力(X22)及4个主成分的综合得分等10个因子,作为重庆市生态功能区环境健康状况的10个评价指标;和24个蝴蝶监测种类,12个蝴蝶监测类群一起,探讨评价方法,共同构筑了重庆市生态功能区环境监测评价体系。在此基础上,对各功能区环境健康状况的评价表明,功能区1的环境健康评价为非常健康,5、9、10区为健康功能区;2、3、6、7、14区为亚健康功能区;不健康功能区是4、8、11、12、13区。功能区13、14是城市化区域,它们的分值在14个区中,居第十二位和第七位,不是最低的,这表明城镇化与环境质量并不完全是因果关系,即通过绿化和环境保护,城镇环境是可能得到很好改善的。期望此监测评价体系在实践中进一步完善。     

Age- and density-dependent population dynamics in static and variable environments

We consider a general model of a single-species population with age- and density-dependent per capita birth and death rates. In a static environment we show that if the per capita death rate is independent of age, then the local stability of any stationary state is guaranteed by the requirement that, in the region of the steady state, the density dependence of the birth rate should be negative and that of the death rate positive. In a variable environment we show that, provided the system is locally stable, small environmental fluctuations will give rise to small age structure and population fluctuations which are related to the driving environmental fluctuations by a simple “transfer function.” We illustrate our general theory by examining a model with a per capita death rate which is age and density independent and a per capita birth rate which is zero up to some threshold age a₀, adopts a finite density-dependent value up to a maximum age a_o + α, and is zero thereafter. We conclude from this model that resonance due specifically to single-species age-structure effects will only be of practical importance in populations whose members have a life cycle consisting of a long immature phase followed by a short burst of intense reproductive effort (α a_o).     

Characterizing the effects of the protein environment on the reduction potentials of metalloproteins

The reduction potentials of electron transfer proteins are critically determined by the degree of burial of the redox site within the protein and the degree of permanent polarization of the polypeptide around the redox site. Although continuum electrostatics calculations of protein structures can predict the net effect of these factors, quantifying each individual contribution is a difficult task. Here, the burial of the redox site is characterized by a dielectric radius R _p (a Born-type radius for the protein), the polarization of the polypeptide is characterized by an electret potential ? _p (the average electrostatic potential at the metal atoms), and an electret-dielectric spheres (EDS) model of the entire protein is then defined in terms of R _p and ? _p. The EDS model shows that for a protein with a redox site of charge Q, the dielectric response free energy is a function of Q ², while the electret energy is a function of Q. In addition, R _p and ? _p are shown to be characteristics of the fold of a protein and are predictive of the most likely redox couple for redox sites that undergo different redox couples.     

Approximate Confidence Intervals for Contrasts in the Balanced Three Factor Mixed Model

When conducting a statistical analysis of data from a designed experiment, an investigator is often interested in confidence intervals for contrasts of the fixed effects. If the analysis involves a mixed linear model, exact confidence intervals for contrasts of the fixed effects are not always available. In such cases, confidence intervals with approximate coverage probabilities must be used. As will be shown, this problem may be generalized to that of constructing a confidence interval for the parameter μ, where X is a normal random variable with mean μ and variance ∑ a_qθ_q, where a₁…,a_Q are known constants, U_q = n_qS/θ_q is a chi-squared random variable with n_q degrees of freedom, for each q = 1,…, Q, and X,U₁,…, U_Q are mutually independent. In this paper, we consider the case where Q = 3 and a₃ ≤0.     

Stability of the analytical solution of Penna model of biological aging

There are some analytical solutions of the Penna model of biological aging; here, we discuss the approach by Coe et al. (Phys. Rev. Lett. 89, 288103, 2002), based on the concept of self-consistent solution of a master equation representing the Penna model. The equation describes transition of the population distribution at time t to next time step (t + 1). For the steady state, the population n(a, l, t) at age a and for given genome length l becomes time-independent. In this paper we discuss the stability of the analytical solution at various ranges of the model parameters—the birth rate b or mutation rate m. The map for the transition from n(a, l, t) to the next time step population distribution n(a + 1, l, t + 1) is constructed. Then the fix point (the steady state solution) brings recovery of Coe et al. results. From the analysis of the stability matrix, the Lyapunov coefficients, indicative of the stability of the solutions, are extracted. The results lead to phase diagram of the stable solutions in the space of model parameters (b, m, h), where h is the hunt rate. With increasing birth rate b, we observe critical b ₀ below which population is extinct, followed by non-zero stable single solution. Further increase in b leads to typical series of bifurcations with the cycle doubling until the chaos is reached at some b _c. Limiting cases such as those leading to the logistic model are also discussed.     

Binding and release kinetics of inhibitors of Q−A oxidation in thylakoid membranes

Oxygen flash yield patterns of dark adapted thylakoid membranes as measured with a Joliot-type O₂-electrode indicate that inhibitors that block the oxidation of the reduced primary quinone Q^?_A of Photosystem II vary greatly in the rate of binding to and release from the inhibitor / Q_B binding environment. The ‘classical’ Photosystem-II herbicides like diuron and atrazine exhibit slow binding and release kinetics, whereas, for example, phenolic inhibitors, o-phenanthroline and synthetic quinones are exchanging quite rapidly with Q_B (about once per second or faster at inhibitor concentrations causing about 50% inhibition of O₂ evolution). No general relationship between the efficiency of the inhibitor and the exchange rate is observed; it depends mainly on the type of inhibitor. Based on the classical Kok model, equations are derived in order to calculate oxygen yields evolved by thylakoids in single-turnover flashes as a function of the rate constants of inhibitor binding to and release from the inhibitor / Q_B binding environment in the presence of an oxidized or semireduced Q_A · Q_B or Q_A · inhibitor complex. Fitting of theoretical and experimental values yields that o-phenanthroline binds much faster to an oxidized than to a semireduced Q_A · Q_B complex. This fits very well with the hypothesis that the Q^?_B affinity to the site is much higher than that of Q_B. In the case of i-dinoseb, however, inhibitor / quinone exchange seems to occur mainly in the semiquinone state. Possibilities to explain this result are discussed.     

A stochastic model for the economic management of a renewable animal resource

An optimal economic harvesting policy, which maximizes the present value of an animal population, capable of renewing itself, is discussed. It is assumed that, unhindered, the successive population levels, X_n, form a Markov chain, with transitions

$\text{X}{}_{\mathrm{n+1}}\text{=?(X}{}_{\mathrm{n}}\text{) + ?}{}_{\mathrm{n}}\text{?(X}{}_{\mathrm{n}}\text{)}$

, where f is the recruitment function, and {?_n} is an iid sequence of random shocks. When a positive set-up cost is present an optimal policy is of the (S,s) type. The optimal population level is compared with that of an equivalent deterministic model. Bioeconomic conditions, which imply the optimality of conservation, or extinction are investigated.     

A Note on the Characterization of the Normal Distribution

Suppose X₁, X₂,…, X_n are independent and identically distributed random variables with absolutely continuous distribution function F. It is known that if F is standard normal distribution then (i)∑X²_i is a chi-square with n degrees of freedom and (ii)nX¯² is a chi-square with 1 degrees of freedom where X¯=1/n ∑X_i. Here the above two properties are utilized to characterize the normal distribution.     

The sufficient-component discrete-cause model and its extension to several risk factors

This paper is concerned with Koopman's (1981) discussion of using the ICDR as a measure of deviation from an additive model of no interaction. This measure has been shown to be zero or slightly negative when the assumptions of no interaction hold in the sufficient-component discrete-cause model. The discussion centers around a theoretical structure of sufficient causes for juvenile-onset diabetes where there is a component cause common to all sufficient causes and two binary factors or risks X and Y are measured (X: Coxsackie B₄ infection; Y: HLA4). In this paper, we have extended Koopman's discussion to the case where three or more binary factors X, Y, Z,… are measured. We have considered interaction between factors in every subset and have proved that the ICDR is again either zero or negative when the assumptions of no interaction hold in the sufficient-component cause model. It is supposed that these factors in conjunction will inevitably result in beta-cell destruction together with a set of immunologic conditions that permit dissemination of the enterovirus infection and permit an autoimmune response.     

Variational models of life-histories: When do solutions exist?

Techniques are presented for analyzing Euler-Lotka models in which age-specific birth and death rates depend on a collection v = (v₁,…, v_n) of trait parameters. Sufficient conditions are established for the existence of a v such that the growth rate r(v) is locally optimal. We analyze a general age-at-maturity model, and give criteria for the existence of such variational solutions within each of several sub-models.     

Multivariate Probability in Terms of Marginal Probability and Correlation Coefficient

This paper is motivated by a practical problem relating to student performance in a number of subjects of equal standing. Its mathematical formulation is to find an approximation to a multivariate probability of the form Pr {X₁⩾a, X₂⩾a, …, X_N⩾a} for arbitrary a and N, in terms of p = Pr {X₁⩾a} and q = Corr (X_i, X_j), i≠j, where X_i, i = 1, …, N are exchangeable random variables with mean 0 and variance unity.     

Probability density functions for axial ratios of sectioning profiles of anisotropically arranged elliptical microvessels

The article theoretically regards probability density functions (PDFs) for axial ratio (X/Y) of sectioning profiles of elliptical microvessels (MVs) arranged with anisotropy in a biological tissue volume. A technique for the PDF_X/Y calculations in anisotropy of the elliptical MVs is described. The essence of this technique is introducing anisotropy in PDF(α,φ), i.e. the function of the joint distribution of the polar and planar angles α and φ, which define mutual orientation of the elliptical MVs and sectioning planes. With the aid of this technique, the anisotropy cases are studied with PDF(α,φ) given by pair combinations of the following distributions: (i) a uniform distribution of the angles α and/or φ, (ii) the angle α distribution with , and (iii) Gaussian distributions of the α or φ values. Specifically, PDF_X/Y curves are obtained for MVs with the true, or three-dimensional, axial ratio X₀/Y₀=2.0, and the anisotropy effects on the X/Y expected frequencies are analysed. Conclusions of this analysis, the PDF_X/Y calculation technique, and the PDF_X/Y curves obtained are useful for stereological reconstruction of anisotropically organised microcirculatory networks, with an ellipticity of their MVs being taken into consideration.     

Mathematical models for cellular systems the von Foerster equation. Part I

P. B. M. Walker (1954) and H. C. Longuet-Higgins (quoted by Walker), as well as O. Scherbaum and G. Rasch (1957), made the first attempts towards a mathematical study of the age distribution in a cellular population. It was H. Von Foerster (1959), however, who derived the complete differential equation for the age density function,n(t, a). His equation is obtained from an analysis of the infinitesimal changes occurring during a time elementdt in a group of cells with ages betweena anda+da. The behavior of the population is determined by a quantity λ which we call the loss function. In this paper a rigorous discussion of the Von Foerster equation is presented, and a solution is given for the special case when λ depends, ont (time) anda (age) but not on other variables (such asn itself). It is also shown that the age density,n(t, a), is completely known only if the birth rate,α(t), and the initial age distribution, β(a), are given as boundary conditions. In Section II the steady state solution and some plausible forms of intrinsic loss functions (depending ona only) are discussed in view of later applications.     

A Semiparametric Estimator of the Crossing Point in the Two‐Sample Linear Shift Function: Application to Crossing Lifetime Distributions

Let X and Y be two random variables with continuous distribution functions F and G. Consider two independent observations X₁, … , X_m from F and Y₁, … , Y_n from G. Moreover, suppose there exists a unique x* such that F(x) > G(x) for x < x* and F(x) < G(x) for x > x* or vice versa. A semiparametric model with a linear shift function (Doksum, 1974) that is equivalent to a location‐scale model (Hsieh, 1995) will be assumed and an empirical process approach (Hsieh, 1995) is used to estimate the parameters of the shift function. Then, the estimated shift function is set to zero, and the solution is defined to be an estimate of the crossing‐point x*. An approximate confidence band of the linear shift function at the crossing‐point x* is also presented, which is inverted to yield an approximate confidence interval for the crossing‐point. Finally, the lifetime of guinea pigs in days observed in a treatment‐control experiment in Bjerkedal (1960) is used to demonstrate our procedure for estimating the crossing‐point. (© 2004 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

The Relationship between Corvis ST Tonometry Measured Corneal Parameters and Intraocular Pressure,Corneal Thickness and Corneal Curvature

The purpose of the study was to investigate the correlation between Corneal Visualization Scheimpflug Technology (Corvis ST tonometry: CST) parameters and various other ocular parameters, including intraocular pressure (IOP) with Goldmann applanation tonometry. IOP with Goldmann applanation tonometry (IOP-G), central corneal thickness (CCT), axial length (AL), corneal curvature, and CST parameters were measured in 94 eyes of 94 normal subjects. The relationship between ten CST parameters against age, gender, IOP-G, AL, CST-determined CCT and average corneal curvature was investigated using linear modeling. In addition, the relationship between IOP-G versus CST-determined CCT, AL, and other CST parameters was also investigated using linear modeling. Linear modeling showed that the CST measurement ‘A time-1’ is dependent on IOP-G, age, AL, and average corneal curvature; ‘A length-1’ depends on age and average corneal curvature; ‘A velocity-1’ depends on IOP-G and AL; ‘A time-2’ depends on IOP-G, age, and AL; ‘A length-2’ depends on CCT; ‘A velocity-2’ depends on IOP-G, age, AL, CCT, and average corneal curvature; ‘peak distance’ depends on gender; ‘maximum deformation amplitude’ depends on IOP-G, age, and AL. In the optimal model for IOP-G, A time-1, A velocity-1, and highest concavity curvature, but not CCT, were selected as the most important explanatory variables. In conclusion, many CST parameters were not significantly related to CCT, but IOP usually was a significant predictor, suggesting that an adjustment should be made to improve their usefulness for clinical investigations. It was also suggested CST parameters were more influential for IOP-G than CCT and average corneal curvature.