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.     

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

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.     

Analytic resolution of the equilibrium equation of growth]

A binomial bioenergetical balance growth equation is considered. The analytical solution of this equation is proposed under certain limitations of parameters. This equation was shown to describe both the mass accumulation and loss in cases of negative energetical balance, thus allowing to widen the domain of application of the 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。     

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.     

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.     

Kinetics of microbial cell growth

As a rate equation of microbial cell growth, the Monod equation is widely used. However, this equation cannot fully correspond to real courses of microbial cell growth in many batch cultivations. Especially, predicted values based on this equation do not agree with observed values in many continuous cultivations. In this paper, which introduces new concepts of critical concentration and coefficient of consumption activity, the growth rate equation which corresponds to the whole period including lag period is newly derived and characteristics of microbial cell growth in batch cultivation are clarified. Further, applying the new rate equation to continuous cultivation, a general equation with which to calculate cell concentration is derived and characteristics of microbial cell growth in continuous cultivation are clarified. The calculated values of cell concentration based on the new theory showed quite good agreement with the observed values in both batch and continuous cultivation.     

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.     

Comparison of logistic equations for population growth.

Two different forms of the logistic equation for population growth appear in the ecological literature. In the form of the logistic equation that appears in recent ecology textbooks the parameters are the instantaneous rate of natural increase per individual and the carrying capacity of the environment. In the form of the logistic equation that appears in some older literature the parameters are the instantaneous birth rate per individual and the carrying capacity. The decision whether to use one form or the other depends on which form of the equation is biologically more realistic. In this study the form of the logistic equation in which the instantaneous birth rate per individual is a parameter is shown to be more realistic in terms of the birth and death processes of population growth. Application of the logistic equation to calculate yield from an exploited fish population also shows that the parameters must be the instantaneous birth rate per individual and the carrying capacity.     

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.     

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.     

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)     

The mathematical theory of lateral impedance of visual activity and an application of Markov processes

The mathematical relationship describing recurrent lateral inhibition is expressed as a linear operator equation. Under quite general conditions, the operator equation is shown to have a unique nonnegative solution. It is also shown that the linear operator for recurrent impedance is representable as an integral operator and that, when in application to physiological models it is interpreted as recurrent inhibition, the corresponding linear equation assumes a form more general than the well known Hartline-Ratliff equation. Finally, we introduce a class of impedance operators based on the probabilistic theory of Markov processes, solve the corresponding linear integral equation, and apply the theoretical properties of the solution to the analysis of physiological and psychophysical phenomena.     

A three--dimensional dyadic Walburn-Schneck constitutive equation for blood

A three-dimension dyadic form of the Walburn-Schneck constitutive equation for blood is presented. The dyadic equation is demonstrated to have the symmetries of material frame indifference and flow reversal and to be consistent with the scalar equation in Couette flow. The problem of flow in a tube or circular cross section is solved as an example.     

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.     

The hypercycle,traveling waves,and Wright's equation

A formal relation between the hypercycle equation and the delay differential equation of E. M. Wright is exhibited using a traveling waves approach. Several unsolved questions in either problem can be related and interpreted, in particular new motivation for the study of Wright's equation is obtained.     

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.     

Evaluation of a simplified generic bi-substrate rate equation for computational systems biology

The evaluation of a generic simplified bi-substrate enzyme kinetic equation, whose derivation is based on the assumption of equilibrium binding of substrates and products in random order, is described. This equation is much simpler than the mechanistic (ordered and ping-pong) models, in that it contains fewer parameters (that is, no K(i) values for the substrates and products). The generic equation fits data from both the ordered and the ping-pong models well over a wide range of substrate and product concentrations. In the cases where the fit is not perfect, an improved fit can be obtained by considering the rate equation for only a single set of product concentrations. Due to its relative simplicity in comparison to the mechanistic models, this equation will be useful for modelling bi-substrate reactions in computational systems biology.     

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.