用麦夸方法最优拟合逻辑斯谛曲线

一般对非线性逻辑斯谛生长曲线的拟合是采用首先对原方程线性化以后用线性最小二乘的方法。此方法不是最优的。本文提出用麦夸方法对曲线进行拟合,并且比较了Gause、Andrewartha、May、Pearl、Krebs、万昌秀等人提出的方法与计算结果。麦夸方法对生物实验及生态学中诸多非线性曲线的参数估计具有普遍的意义。     

A master equation for a spatial population model with pair interactions

We derive a closed master equation for an individual-based population model in continuous space and time. The model and master equation include Brownian motion, reproduction via binary fission, and an interaction-dependent death rate moderated by a competition kernel. Using simulations we compare this individual-based model with the simplest approximation, the spatial logistic equation. In the limit of strong diffusion the spatial logistic equation is a good approximation to the model. However, in the limit of weak diffusion the spatial logistic equation is inaccurate because of spontaneous clustering driven by reproduction. The weak-diffusion limit can be partially analyzed using an exact solution of the master equation applicable to a competition kernel with infinite range. This analysis shows that in the case of a top-hat kernel, reducing the diffusion can increase the total population. For a Gaussian kernel, reduced diffusion invariably reduces the total population. These theoretical results are confirmed by simulation.     

Estimation of total water and fatness from weight and height: inaccurate for lean women

Mellits and Cheek have provided a regression equation which predicts total water, and by extension, fat, from weight and height of women. The equation has been used by many researchers as a convenient means to estimate fat when more complicated body composition estimation techniques are not possible. It is an essential component of evidence provide by Frisch for the critical fat hypothesis. The equation has not been validated on another sample, however, and has a large standard error. I test the Mellits and Cheek equation on five samples of young women, two of which show a normal range of heights and weights. The other three are athletic samples composed of lean women. In the normal samples mean fat percent determined by density (from underwater weighing) is closely approximated by the weight and height equation but not in the lean samples. When measured fat is regressed on estimated fat, correlations are low, and standard errors are high, suggesting that the Mellits and Cheek equation predicts poorly the body fat percentage of individual females. In summary, the Mellits and Cheek equation is moderately useful for the prediction of group means for body composition on samples with normal ranges of height and weight. It does not produce acceptable results on groups selected for extreme leanness or obesity. It is not recommended for fat prediction of individuals.     

Equations and calculations for fermentations of butyric acid bacteria

A stoichiometric equation has been derived which describes the interrelations among the various products and biomass in fermentations of butyric acid bacteria. The derivation of the equation is based on an assumed ATP yield, two biological regularities, and the biochemistry of product formation of the fermentations. The equation obeys the constraints imposed on growth and product formation by thermodynamics and the biochemical topology. The validity of the equation is tested using a variety of fermentation data from the literature. The uses, improvements, limitations, and extensions of the equation are also discussed in detail. For example, the fermentation equation is used to calculate the maximal possible yields of the main fermentation products.     

Experimental evidence for allosteric modifier saturation as predicted by the bi-substrate Hill equation

The cooperative enzyme reaction rates predicted by the bi-substrate Hill equation and the bi-substrate Monod-Wyman-Changeux (MWC) equation when allosterically inhibited are compared in silico. Theoretically, the Hill equation predicts that when the maximum inhibitory effect at a certain substrate condition has been reached, an increase in allosteric inhibitor concentration will have no effect on reaction rate, that is the Hill equation shows allosteric inhibitor saturation. This saturating inhibitory effect is not present in the MWC equation. Experimental in vitro data for pyruvate kinase, a bi-substrate cooperative enzyme that is allosterically inhibited, are presented. This enzyme also shows inhibitor saturation, and therefore serves as experimental evidence that the bi-substrate Hill equation predicts more realistic allosteric inhibitor behaviour than the bi-substrate MWC equation.     

Equations and calculations for fermentations of butyric acid bacteria. Reprinted from Biotechnology and Bioengineering, Vol. XXVI, Pp 174-187 (1984)

A stoichiometric equation has been derived which describes the interrelations among the various products and biomass in fermentations of butyric acid bacteria. The derivation of the equation is based on an assumed ATP yield, two biological regularities, and the biochemistry of product formation of the fermentations. The equation obeys the constraints imposed on growth and product formation by thermodynamics and the biochemical topology. The validity of the equation is tested using a variety of fermentation data from the literature. The uses, improvements, limitations, and extensions of the equation are also discussed in detail. For example, the fermentation equation is used to calculate the maximal possible yields of the main fermentation products.     

General proof of the validity of a new tensor equation of plant growth

Plant cell/organ growth may be partly described by a local tensor equation. We provide a mathematical proof that the Lockhart (global) equation is the diagonal component of this tensor equation.     

Protein Electrostatics: Rapid Multigrid-Based Newton Algorithm for Solution of the Full Nonlinear Poisson-Boltzmann Equation

Abstract

A new method for solving the full nonlinear Poisson-Boltzmann equation is outlined. This method is robust and efficient, and uses a combination of the multigrid and inexact Newton algorithms. The novelty of this approach lies in the appropriate combination of the two methods, neither of which by themselves are capable of solving the nonlinear problem accurately. Features of the Poisson-Boltzmann equation are fully exploited by each component of the hybrid algorithm to provide robustness and speed. The advantages inherent in this method increase with the size of the problem. The efficacy of the method is illustrated by calculations of the electrostatic potential around the enzyme Superoxide Dismutase. The CPU time required to solve the full nonlinear equation is less than half that needed for a conjugate gradient solution of the corresponding linearized Poisson-Boltzmann equation. The solutions reveal that the field around the active sites is significantly reduced as compared to that obtained by solving the corresponding linearized Poisson-Boltzmann equation. This new method for the nonlinear Poisson-Boltzmann equation will enable fast and accurate solutions of large protein electrostatics problems.     

Thermodynamics of the isothermal denaturation of lysozyme by guanidinium chloride.

A thermodynamic analysis of the isothermal denaturation of lysozyme by guanidinium chloride has been performed. The analysis is based on the equation which relates the equilibrium constant for denaturation to the preferential binding of denaturant. The equation has been derived previously by thermodynamic methods, whereas in this article a derivation based on statistical mechanics is given. By application of the equation the free energy of denaturation is first calculated and from it, by subtracting the calorimetrically-determined enthalpy of denaturation, the entropy of denaturation is determined.     

A New Generalized Logistic Sigmoid Growth Equation Compared with the Richards Growth Equation

A new sigmoid growth equation is presented for curve-fitting,analysis and simulation of growth curves. Like the logisticgrowth equation, it increases monotonically, with both upperand lower asymptotes. Like the Richards growth equation, itcan have its maximum slope at any value between its minimumand maximum. The new sigmoid equation is unique because it alwaystends towards exponential growth at small sizes or low densities,unlike the Richards equation, which only has this characteristicin part of its range. The new sigmoid equation is thereforeuniquely suitable for circumstances in which growth at smallsizes or low densities is expected to be approximately exponential,and the maximum slope of the growth curve can be at any value.Eleven widely different sigmoid curves were constructed withan exponential form at low values, using an independent algorithm.Sets of 100 variations of sequences of 20 points along eachcurve were created by adding random errors. In general, thenew sigmoid equation fitted the sequences of points as closelyas the original curves that they were generated from. The newsigmoid equation always gave closer fits and more accurate estimatesof the characteristics of the 11 original sigmoid curves thanthe Richards equation. The Richards equation could not estimatethe maximum intrinsic rate of increase (relative growth rate)of several of the curves. Both equations tended to estimatethat points of inflexion were closer to half the maximum sizethan was actually the case; the Richards equation underestimatedasymmetry by more than the new sigmoid equation. When the twoequations were compared by fitting to the example dataset thatwas used in the original presentation of the Richards growthequation, both equations gave good fits. The Richards equationis sometimes suitable for growth processes that may or may notbe close to exponential during initial growth. The new sigmoidis more suitable when initial growth is believed to be generallyclose to exponential, when estimates of maximum relative growthrate are required, or for generic growth simulations.Copyright1999 Annals of Botany Company Asymptote,Cucumis melo,curve-fitting, exponential growth, intrinsic rate of increase, logistic equation, maximum growth rate, model, non-linear least-squares regression, numerical algorithm, point of inflexion, relative growth rate, Richards growth equation, sigmoid growth curve.     

Dependence of cellular potential on ionic concentrations. Data supporting a modification of the constant field equation.

The resting potential in the squid axon has been measured at various concentrations of Cl, K, Na, and Ca ions. The results of these measurements are compared with the Goldman-Hodgkin-Katz (GHK) equation and a modified constant field equation. This modified equation was derived by including currents carried by divalent ions and the effects of the unstirred layer and the periaxonal space. It is shown that, although the GHK equation can fit the V vs. [K]o data well, it has difficulty explaining the observed dependence of V on [Na]o when the axon is bathed in K-free artificial sea water. The use of the modified constant field equation removes this difficulty.     

求解分形介质中分数阶扩散模型的相似方法(英文)

分形介质中输运现象的分数阶扩散方程是一个积分-偏微分方程,含有由分形Hausdorff维数d_f和反常扩散指数d_w确定的参数.对于这类方程的求解问题,给出了尺度变换群的不变子并且导出了关于尺度不变解的积分-常微分方程.最后利用Mellin变换和Fox函数得到尺度不变解.     

An approximation diffusion equation for a long narrow channel with varying cross-sectional area

The diffusion equation for long narrow channels which lie parallel to a rectangular coordinate and have varying cross-sectional areas may be approximated by an equation which involves only one space variable and the average concentration at each value of this space variable. This equation is derived and is discussed along with its assumptions.     

Mixtures of tight-binding enzyme inhibitors. Kinetic analysis by a recursive rate equation.

When two or more tight-binding inhibitors are present in an enzyme assay, the equation that relates the initial velocity v to the concentration of reactants cannot be written in an algebraically explicit form. Rather, for n inhibitors it is an implicit polynomial equation of degree n + 1 with respect to v. The complexity of the polynomial coefficients dramatically increases with each added inhibitor. Solving the transcendental rate equation by traditional methods of numerical mathematics has proven tedious because of the sensitivity of these methods to initial estimates and because of the existence of multiple roots. However, the equation can be rearranged into a convenient recursive form, one in which the velocity appears on both sides and the solution is found iteratively. The algebraic form of the recursive rate equation is remarkably simple and differs from the rate equation for classical rather than tight-binding inhibition only by an added term. The numerical stability and the speed of convergence were tested on the case of two competitive inhibitors. Initial estimates of velocity that spanned 12 orders of magnitude converged within five iterations. The velocities computed with the recursive method for a single tight-binding inhibitor were identical with the values predicted by the Morrison equation. The method is used to analyze experimental data for the inhibition of rat liver dihydrofolate reductase by mixtures of the anticancer drug methotrexate and its metabolic precursor form, methotrexate-alpha-aspartate (a prodrug).     

Analysis of negative cooperativity for glutamate dehydrogenase

The empirical equation, which describes negative cooperativity in the enzyme kinetics, has been proposed. The equation is obtained from the Michaelis-Menten equation where the Michaelis constant is replaced by the effective Michaelis constant, which is a linear function of the v/Vmax ratio (v is the rate of the enzymatic reaction and Vmax is the limiting value of v at saturating concentrations of substrate). The equation allows the limiting values of the Michaelis constant at v/Vmax --> 0 and V/Vmax --> 1 to be estimated, K0 and Klim, respectively. The Klim/K0 ratio is considered as a quantitative characteristic of negative cooperativity. The applicability of the equation has been demonstrated for the kinetic data obtained for glutamate dehydrogenases from various sources (negative kinetic cooperativity for coenzyme). The negative cooperativity for the functions of saturation of protein by ligand is also analyzed. The data on binding of spin-labeled NAD, NADH, and NADPH by beef liver glutamate dehydrogenase are used as examples.     

激光与DNA作用方程的稳定态分析

本文在庞小峰改进后的Ｙｏｍｏｓａ模型基础上,引进了激光与ＤＮＡ分子系统作用的非线性方程。     

Desorption of Nitroaromatic Compounds From Explosive-contaminated Soil by Washing

The soil contaminated by explosive production wastewater was treated by washing using water as solvent. The effect of contact time and temperature, water/soil ratio and washing steps on desorption efficiency was investigated. Six kinetic models—parabolic diffusion model, zero-order equation, pseudo-first-order equation, pseudo-second-order equation, power function equation and Elovich equation—were used to study the desorption kinetics of nitroaromatic compounds from contaminated soil to water. The eluent of contaminated soil before and after washing was characterized by UV–vis analysis. The results showed that the removal rate was fast at the initial stage and then slowed down after 60 min. The desorption of contaminants from soil to water is endothermic. Washing with small quantities of water in high frequency is preferred when water volume is limited. The pseudo-second-order model can be used to describe the desorption process. Soil washing can remove most of the contaminants from the contaminated soil.     

Age structure in predator-prey systems: Intraspecific carnivore interaction,passive diffusion,and the paradox of enrichment

An existing arthropod predator-prey model incorporating age structure in the carnivore through the use of the von Foerster equation is extended to include the effects of intraspecific carnivore interaction and passive diffusion or migration. A linear stability analysis of the community equilibrium point of that differential-integral equation system is performed and the resulting secular equation analyzed by the method of D-partitions. These stability results are then compared to those obtained by employing an analogous differential equation model without age structure, in particular as they relate to the so-called paradox of enrichment. In the absence of passive diffusion, it is shown that, unlike for a differential equation model, the paradox of enrichment can occur even with a carnivore which exhibits intraspecific competition. This destabilizing effect of age structure is seen to occur most dramatically when interspecific interactions are large, while the effect of passive diffusion is to offset that tendency and restabilize the system. These predictions are in accordance with relevant experimental evidence involving mites.     

A non-ideal replacement for the Boyle van't Hoff equation

A non-ideal osmotic equilibrium equation is proposed as a replacement for the Boyle van’t Hoff equation to describe the equilibrium volume of a living cell as a function of external osmolality. Contrary to common understanding, the Boyle van’t Hoff equation is only thermodynamically correct for ideal, dilute solutions. However, the Boyle van’t Hoff equation is commonly used to determine the osmotically inactive fraction of the cell. This involves extrapolating to infinite osmolality, which violates the ideal, dilute solution constraint. It has been noted that the osmotically inactive fractions obtained from the Boyle van’t Hoff equation for human erythrocytes are markedly larger than measured values of the dry volume fraction of the cell. Using the new osmotic equilibrium equation to analyze experimental osmotic equilibrium data reduces the inferred osmotically inactive fraction of human erythrocytes by approximately 20%.     

Microbial Life and Temperature: A Semi Empirical Approach

Many groups have examined the effect of temperature on the survival of microorganisms, resulting in the development of several models. Some of these models are based on the Arrhenius equation and the others are based on multidimensional response surface equations. We argue that the former are inadequate and the latter lack biological meaning. We show that an equation (the GLE equation) deduced from the Theory of Rate Processes is more accurate than the Arrhenius equation. The excellent standard deviation values of the apparent free energy of activation obtained with the GLE equation for microbial growth, embryogenic and other processes show that this equation is more suitable than the Arrhenius equation. The GLE equation shows how temperature affects survival. Thus, organisms survive longer at low temperatures than at normal temperatures. The recent discovery of microorganisms in Siberian permafrost samples that are several million years old, in deep oil fields, mines and other extreme habitats appears to be consistent with the GLE equation. Another example, the enhanced resistance of spores at extreme temperatures can be easily explained by their high apparent free energy of activation We also examined the implications of the GLE equation on food sterilization practices and on exobiology.