A nonlinear integral equation for visual impedance

The Hartline-Ratliff equation is a linear integral equation of the second kind and is employed in modeling inhibitory networks. Saturation of the inhibiting elements is commonly modeled as a function whose form is sigmoid; however, the resulting integral equation is nonlinear. Whenever the unknown function within the integral is hypothesized to be a nondecreasing nonlinear function, the Hartline-Ratliff equation becomes a nonlinear integral equation of the Hammerstein type. We present existence and uniqueness theorems for a Hammerstein equation which represents a further generalization of the Hartline-Ratliff equation.     

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。     

Kinetic theory of a gas-discharge positive column and wall sheath

The positive column and wall sheath in a gas discharge are studied with allowance for ion collisions in a plasma and ion reflection from a solid surface under conditions of incomplete ion neutralization. The kinetic equation for ions in a positive column is reduced to a Fredholm equation of the second kind. This makes it possible to solve the kinetic equation using a resolvent and thereby derive a single integrodifferential equation for the potential, which is referred to as a generalized plasma-sheath equation. Specific versions of the plasma-sheath equation are obtained that take into account charge exchange of the ions in a plasma and the thermal spread in velocities of the ionization-produced ions.     

Technical-methodological report: a nomogram for peak leg power output in the vertical jump

Leg power is an important component in assessing both performance-related and health-related fitness. The Lewis equation and nomogram have been used for years to estimate leg power. A recent evaluation of the Lewis equation and further research led to the development of the Sayers equation. This equation provides an estimate of peak leg power, which has greater relevance than average power. Our purpose was to provide a simple and effective nomogram for calculating peak leg power output. The Sayers equation was transformed to an alignment nomogram and evaluated for facility of use and accuracy. The resultant alignment nomogram is easy to use and generates values for peak leg power in the vertical jump, which are well within the precision of the regression equation (r > 0.9999, CV < 0.2%). Interobserver error was less than 0.3% with a correlation of 0.9999. The Keir nomogram provides a simple and effective representation of the Sayers equation for use in both performance-related and health-related fitness assessments.     

A Generalized Simplest Equation Method and Its Application to the Boussinesq-Burgers Equation

In this paper, a generalized simplest equation method is proposed to seek exact solutions of nonlinear evolution equations (NLEEs). In the method, we chose a solution expression with a variable coefficient and a variable coefficient ordinary differential auxiliary equation. This method can yield a Bäcklund transformation between NLEEs and a related constraint equation. By dealing with the constraint equation, we can derive infinite number of exact solutions for NLEEs. These solutions include the traveling wave solutions, non-traveling wave solutions, multi-soliton solutions, rational solutions, and other types of solutions. As applications, we obtained wide classes of exact solutions for the Boussinesq-Burgers equation by using the generalized simplest equation method.     

广义Schumacher模型的改进及其应用

通过对前人提出的生长方程的具体分析,提出了一种改进的Schumacher生长方程．该模型包含了Gompenz函数、Schumacher方程及广义Schumacher方程,具有很强的自适应性和实用性．采用遗传算法。利用该模型对珍稀植物长苞铁杉和侧柏生长资料分别进行了拟合．结果表明,改进的Schumacher方程的拟合精度明显优于Schumache,方程和广义Schumacher方程,也优于经典的Logistic模型和李新运等自适应模型。可以在林木生长动态模拟及种群增长动态研究中广泛应用．     

The Nernst equation applied to oxidation-reduction reactions in myoglobin and hemoglobin. Evaluation of the parameters

Analyses of the binding of oxygen to monomers such as myoglobin employ the Mass Action equation. The Mass Action equation, as such, is not directly applicable for the analysis of the binding of oxygen to oligomers such as hemoglobin. When the binding of oxygen to hemoglobin is analyzed, models incorporating extensions of mass action are employed. Oxidation-reduction reactions of the heme group in myoglobin and hemoglobin involve the binding and dissociation of electrons. This reaction is described with the Nernst equation. The Nernst equation is applicable only to a monomeric species even if the number of electrons involved is greater than unity. To analyze the oxidation-reduction reaction in a molecule such as hemoglobin a model is required which incorporates extensions of the Nernst equation. This communication develops models employing the Nernst equation for oxidation-reduction reactions analogous to those employed for hemoglobin in the analysis of the oxygenation (binding of oxygen) reaction.     

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%.     

Characteristics of the upper airway pressure-flow relationship during sleep

In examining the mechanical properties of the respiratory system during sleep in healthy humans, we observed that the inspiratory pressure-flow relationship of the upper airway was often flow limited and too curvilinear to be predicted by the Rohrer equation. The purposes of this study were 1) to describe a mathematical model that would better define the inspiratory pressure-flow relationship of the upper airway during sleep and 2) to identify the segment of airway responsible for the sleep-related flow limitation. We measured nasal and total supralaryngeal pressure and flow during wakefulness and stage 2 sleep in five healthy male subjects lying supine. A right rectangular hyperbolic equation, V = (alpha P)/(beta + P), where V is flow, P is pressure, alpha is an asymptote for peak flow, and beta is pressure at a flow of alpha/2, was used in its linear form, P/V = (beta/alpha) + (P/alpha). The goodness of fit of the new equation was compared with that for the linearized Rohrer equation P/V = K1 + K2V. During wakefulness the fit of the hyperbolic equation to the actual pressure-flow data was equivalent to or significantly better than that for the Rohrer equation. During sleep the fit of the hyperbolic equation was superior to that for the Rohrer equation. For the whole supralaryngeal airway during sleep, the correlation coefficient for the hyperbolic equation was 0.90 +/- 0.50, and for the Rohrer equation it was 0.49 +/- 0.25. The flow-limiting segment was located within the pharyngeal airway, not in the nose.(ABSTRACT TRUNCATED AT 250 WORDS)     

Mathematical representation of batch culture data

A generalized logistic equation is proposed for the mathematical representation of batch culture kinetic data. Properties of the equation are discussed. A computer program is used to fit the generalized equation to both artificial and actual batch culture data. The equation is shown to be capable of fitting data exhibiting lag, exponential, deceleration, stationary, and death phases, as well as diauxic growth. The fitted equation is useful for differentiation, interpolation, and other manipulations of the data, and it is a convenient means of data storage.     

Clarke-Carbon公式的推导

Clarke-Carbon公式用于估计一个完整的基因文库所需的最少克隆数。本文通过建立基因克隆的统计模型,利用二项分布原理对该公式进行了推导,从而有利于在教学过程中对这一公式的确切理解。 Abstract:Clarke-Carbon equation can give a good estimation for the necessary number of recombinants in a gene library.By establishing a statistical model of process of cloning, a deduction and interpretation of this equation can be obtained according to the principle of binomial distribution.This will help to understand the equation more accurately for the teachers and learners.     

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.     

A Jump-Growth Model for Predator–Prey Dynamics: Derivation and Application to Marine Ecosystems

This paper investigates the dynamics of biomass in a marine ecosystem. A stochastic process is defined in which organisms undergo jumps in body size as they catch and eat smaller organisms. Using a systematic expansion of the master equation, we derive a deterministic equation for the macroscopic dynamics, which we call the deterministic jump-growth equation, and a linear Fokker–Planck equation for the stochastic fluctuations. The McKendrick–von Foerster equation, used in previous studies, is shown to be a first-order approximation, appropriate in equilibrium systems where predators are much larger than their prey. The model has a power-law steady state consistent with the approximate constancy of mass density in logarithmic intervals of body mass often observed in marine ecosystems. The behaviours of the stochastic process, the deterministic jump-growth equation, and the McKendrick–von Foerster equation are compared using numerical methods. The numerical analysis shows two classes of attractors: steady states and travelling waves.     

黑曲霉过氧化氢酶发酵过程的数学模型

研究了黑曲霉发酵生产过程氧化氢酶的分批发酵动力学,并建立了发酵过程菌体生长,基质消耗及酶合成的随时间变化的数学模型。Logistic方程,Luedekin-Piret方程及与Luedeking-Piret方程相似的基质消耗方程能够很好地分别描述黑曲霉细胞的生长,发酵产酶过程及葡萄糖的消耗,过氧化氢酶的发酵合成是生长耦联的,研究中还将3个动力学模型的预测值和实验值进行了比较。     

A New Modified CKD-EPI Equation for Chinese Patients with Type 2 Diabetes

Objective

To improve the performance of glomerular filtration rate (GFR) estimating equation in Chinese type 2 diabetic patients by modification of the CKD-EPI equation.

Design and patients

A total of 1196 subjects were enrolled. Measured GFR was calibrated to the dual plasma sample ^99mTc-DTPA-GFR. GFRs estimated by the re-expressed 4-variable MDRD equation, the CKD-EPI equation and the Asian modified CKD-EPI equation were compared in 351 diabetic/non-diabetic pairs. And a new modified CKD-EPI equation was reconstructed in a total of 589 type 2 diabetic patients.

Results

In terms of both precision and accuracy, GFR estimating equations all achieved better results in the non-diabetic cohort comparing with those in the type 2 diabetic cohort (30% accuracy, P≤0.01 for all comparisons). In the validation data set, the new modified equation showed less bias (median difference, 2.3 ml/min/1.73 m² for the new modified equation vs. ranged from −3.8 to −7.9 ml/min/1.73 m² for the other 3 equations [P<0.001 for all comparisons]), as was precision (IQR of the difference, 24.5 ml/min/1.73 m² vs. ranged from 27.3 to 30.7 ml/min/1.73 m²), leading to a greater accuracy (30% accuracy, 71.4% vs. 55.2% for the re-expressed 4 variable MDRD equation and 61.0% for the Asian modified CKD-EPI equation [P = 0.001 and P = 0.02]).

Conclusion

A new modified CKD-EPI equation for type 2 diabetic patients was developed and validated. The new modified equation improves the performance of GFR estimation.     

A quantitative method for the analysis of mammalian cell proliferation in culture in terms of dividing and non-dividing cells

The application of the exponential growth equation is the standard method employed in the quantitative analyses of mammalian cell proliferation in culture. This method is based on the implicit assumption that, within a cell population under study, all division events give rise to daughter cells that always divide. When a cell population does not adhere to this assumption, use of the exponential growth equation leads to errors in the determination of both population doubling time and cell generation time. We have derived a more general growth equation that defines cell growth in terms of the dividing fraction of daughter cells. This equation can account for population growth kinetics that derive from the generation of both dividing and non-dividing cells. As such, it provides a sensitive method for detecting non-exponential division dynamics. In addition, this equation can be used to determine when it is appropriate to use the standard exponential growth equation for the estimation of doubling time and generation time.     

Kinetic Theory Model for Ion Movement through Biological Membranes: III. Steady-State Electrical Properties with Solution Asymmetry

An electrodiffusion model for plasma membrane ion transport, which takes into account the influence of high electric field strengths and ion-membrane molecule interactions, is presented and analyzed. A generalized Nernst-Planck equation for steady-state situations is derived which has electric field-dependent mobility and diffusion coefficients. Under the assumption of a constant electric field within the membrane, this equation is integrated to give a more general form of the Goldman equation. Based on this equation numerical computations of ionic chord conductance as a function of applied electric field strength were carried out for several permeant ion concentration ratios. The model is capable of yielding significantly larger rectification ratios than is the Goldman equation. Further, high field asymptotes to the current vs. electric field strength curve do not generally intersect at the origin.     

Formulas for intrinsic noise evaluation in oscillatory genetic networks

The linear noise approximation is a useful method for stochastic noise evaluations in genetic regulatory networks, where the covariance equation described as a Lyapunov equation plays a central role. We discuss the linear noise approximation method for evaluations of an intrinsic noise in autonomously oscillatory genetic networks; in such oscillatory networks, the covariance equation becomes a periodic differential equation that provides generally an unbounded covariance matrix, so that the standard method of noise evaluation based on the covariance matrix cannot be adopted directly. In this paper, we develop a new method of noise evaluation in oscillatory genetic networks; first, we investigate structural properties, e.g., orbital stability and periodicity, of the solutions to the covariance equation given as a periodic Lyapunov differential equation by using the Floquet-Lyapunov theory, and propose a global measure for evaluating stochastic amplitude fluctuations on the periodic trajectory; we also derive an evaluation formula for the period fluctuation. Finally, we apply our method to a model of circadian oscillations based on negative auto-regulation of gene expression, and show validity of our method by comparing the evaluation results with stochastic simulations.     

Cancer chemotherapy: Optimal control using the Verhulst-Pearl equation

General (deterministic) ordinary differential equations for the representation of cancer growth are presented when the growth is perturbed due to the action of a chemotherapeutic agent. The Verhulst-Pearl equation is introduced as a particular example of a growth equation applicable to human tumors. An optimal control problem with general performance criterion and state equation is formulated and shown to possess a novel feedback control relationship. This relationship is used in two continuous drug delivery problems involving the Verhulst-Pearl equation.