The appropriate use of regression equations for the estimation of segmental inertia parameters

Linear regression equations are commonly used in conjunction with experimental data to provide linear relationships between quantities which are dimensionally distinct. In many cases theoretical relationships between such quantities are known and can be used as a basis for non-linear regression equations. This study compares linear and non-linear approaches for estimating the segmental moments of inertia from anthropometric measurements using the data of Chandler et al. [Chandler et al. (1975) Investigation of inertial properties of the human body. AMRL Technical Report 74-137, Wright Patterson Air Force Base. OH.] Right limb data were used to derive the equations while left limb data were used as a cross-validation sample to evaluate the inertia estimates calculated from the equations. For the limb segments the standard error estimates had average values of 21% for the linear equations and 13% for the non-linear equations. Data on a 10 yr-old boy was used to compare the two approaches outside the sample range. The mean percentage residuals were 286% for the linear equations and 20% for the non-linear equations. A set of non-linear equations is provided.     

A comparison of methods for fitting allometric equations to field metabolic rates of animals

We re-examined data for field metabolic rates of varanid lizards and marsupial mammals to illustrate how different procedures for fitting the allometric equation can lead to very different estimates for the allometric coefficient and exponent. A two-parameter power function was obtained in each case by the traditional method of back-transformation from a straight line fitted to logarithms of the data. Another two-parameter power function was then generated for each data-set by non-linear regression on values in the original arithmetic scale. Allometric equations obtained by non-linear regression described the metabolic rates of all animals in the samples. Equations estimated by back-transformation from logarithms, on the other hand, described the metabolic rates of small species but not large ones. Thus, allometric equations estimated in the traditional way for field metabolic rates of varanids and marsupials do not have general importance because they do not characterize rates for species spanning the full range in body size. Logarithmic transformation of predictor and response variables creates new distributions that may enable investigators to perform statistical analyses in compliance with assumptions underlying the tests. However, statistical models fitted to transformations should not be used to estimate parameters of equations in the arithmetic domain because such equations may be seriously biased and misleading. Allometric analyses should be performed on values expressed in the original scale, if possible, because this is the scale of interest.     

Equations of Substrate-inhibition Kinetics Applied to Pig Kidney Diamine Oxidase (DAO,E.C. 1.4.3.6)

Pig kidney diamine oxidase (DAO) and other semicarbazide-sensitive amine oxidases (SSAO) show clear substrate-inhibition kinetics and a reaction-scheme mechanism based on two substrate binding sites. We evaluated several reaction scheme mechanisms with a non-linear regression program (NCSS), estimating R², the constants of the equations and their standard errors and we determined the deviation of experimental data from theoretical equations. The best fit was obtained with a “dead end” mechanism with two binding sites. Based on this scheme, other schemes for a two-substrate reaction and for mechanisms of inhibition were constructed. These reaction schemes, even at low substrate concentration, fitted experimental data better than Michaelis-Menten kinetics, and provided information on the mechanisms of action of inhibitors. The presence of two substrate-binding sites on pig kidney DAO was confirmed by all experimental data.     

Equations of substrate-inhibition kinetics applied to pig kidney diamine oxidase (DAO, E.C. 1.4.3.6)

Pig kidney diamine oxidase (DAO) and other semicarbazide-sensitive amine oxidases (SSAO) show clear substrate-inhibition kinetics and a reaction-scheme mechanism based on two substrate binding sites. We evaluated several reaction scheme mechanisms with a non-linear regression program (NCSS), estimating R2, the constants of the equations and their standard errors and we determined the deviation of experimental data from theoretical equations. The best fit was obtained with a "dead end" mechanism with two binding sites. Based on this scheme, other schemes for a two-substrate reaction and for mechanisms of inhibition were constructed. These reaction schemes, even at low substrate concentration, fitted experimental data better than Michaelis-Menten kinetics, and provided information on the mechanisms of action of inhibitors. The presence of two substrate-binding sites on pig kidney DAO was confirmed by all experimental data.     

Fitting of enzyme kinetic data without prior knowledge of weights.

A method is described for fitting equations to enzyme kinetic data that requires minimal assumptions about the error structure of the data. The dependence of the variances on the velocities is not assumed, but is deduced from internal evidence in the data. The effect of very bad observations ('outliers') is mitigated by decreasing the weight of observations that give large deviations from the fitted equation. The method works well in a wide range of circumstances when applied to the Michaelis-Menten equation, but it is not limited to this equation. It can be applied to most of the equations in common use for the analysis of steady-state enzyme kinetics. It has been implemented as a computer program that can fit a wide variety of equations with two, three or four parameters and two or three variables.     

运用改进单纯形法拟合Logistic曲线的研究

Ｌｏｇｉｓｔｉｃ方程是研究有限空间内种群增长规律的重要工具之一本文运用改进单纯形法最优拟合Ｌｏｇｉｓｔｉｃ曲线,结果表明改进单纯形法具有较强的拟合非线性方程的能力,对生物实验及生态、生理学中诸多非线性曲线的参数估计具有普遍意义．     

A network model of successive partitioning-limited solute diffusion through the stratum corneum

As the most exposed point of contact with the external environment, the skin is an important barrier to many chemical exposures, including medications, potentially toxic chemicals and cosmetics. Traditional dermal absorption models treat the stratum corneum lipids as a homogenous medium through which solutes diffuse according to Fick's first law of diffusion. This approach does not explain non-linear absorption and irregular distribution patterns within the stratum corneum lipids as observed in experimental data. A network model, based on successive partitioning-limited solute diffusion through the stratum corneum, where the lipid structure is represented by a large, sparse, and regular network where nodes have variable characteristics, offers an alternative, efficient, and flexible approach to dermal absorption modeling that simulates non-linear absorption data patterns. Four model versions are presented: two linear models, which have unlimited node capacities, and two non-linear models, which have limited node capacities. The non-linear model outputs produce absorption to dose relationships that can be best characterized quantitatively by using power equations, similar to the equations used to describe non-linear experimental data.     

A method for solution of the multi-objective inverse problems under uncertainty

We describe a method to solve multi-objective inverse problems under uncertainty. The method was tested on non-linear models of dynamic series and population dynamics, as well as on the spatiotemporal model of gene expression in terms of non-linear differential equations. We consider how to identify model parameters when experimental data contain additive noise and measurements are performed in discrete time points. We formulate the multi-objective problem of optimization under uncertainty. In addition to a criterion of least squares difference we applied a criterion which is based on the integral of trajectories of the system spatiotemporal dynamics, as well as a heuristic criterion CHAOS based on the decision tree method. The optimization problem is formulated using a fuzzy statement and is constrained by penalty functions based on the normalized membership functions of a fuzzy set of model solutions. This allows us to reconstruct the expression pattern of hairy gene in Drosophila even-skipped mutants that is in good agreement with experimental data. The reproducibility of obtained results is confirmed by solution of inverse problems using different global optimization methods with heuristic strategies.     

Qualitative behavior of predator-prey communities

We examine a set of n-species predator-prey models which incorporate functional responses of the predators to their prey and non-linear intraspecific interactions. We review the limitations of a linear analysis around equilibrium states and provide an extension of the Routh-Hurwitz criteria to the non-linear regime by using Birkhoff's normal forms of differential equations. Qualitative properties like the orbit structure around isolated singularities become clear in this method. It is possible to obtain the radius of the torus and the renormalized frequencies when the eigenvalues of the community matrix have small positive real parts, and to classify different topological structures near bifurcation values of a convenient set of control parameters. Examples of two, three, and four species are analyzed in the context of normal forms.We conclude with some suggestions concerning the coupling of a small subsystem to a large community, and the relations between the graphical method of isoclines and normal forms. This correspondence indicates a road to generalize the study of many-dimensional systems when the intuition provided by the graphical method fails.     

Parameter identification of the calvin photosynthesis cycle

Summary This article is concerned with the determination of kinetic parameters of the Calvin photosynthesis cycle which is described by seventeen nonlinear ordinary differential equations. It is shown that the task requires dynamic data for several sets of initial conditions. The numerical technique is based upon an algorithm for non-linear optimization and Gear's numerical integration scheme for stiff systems of differential equations. The sensitivity of the parameters to noise in the data is tested with a method adapted from Rosenbrook and Storey. A preliminary set of parameters has been obtained from a preliminary set of experimental data. The numerical methods are then tested with synthetic data derived from these parameters. The mathematical model and the results obtained in the simulation are used as an aid in designing new experiments.     

Numerical bifurcation analysis of ecosystems in a spatially homogeneous environment

The dynamics of single populations up to ecosystems, are often described by one or a set of non-linear ordinary differential equations. In this paper we review the use of bifurcation theory to analyse these non-linear dynamical systems. Bifurcation analysis gives regimes in the parameter space with quantitatively different asymptotic dynamic behaviour of the system. In small-scale systems the underlying models for the populations and their interaction are simple Lotka-Volterra models or more elaborated models with more biological detail. The latter ones are more difficult to analyse, especially when the number of populations is large. Therefore for large-scale systems the Lotka-Volterra equations are still popular despite the limited realism. Various approaches are discussed in which the different time-scale of ecological and evolutionary biological processes are considered together.     

A survey of non-linear optimization techniques

Optimization means the provision of a set of numerical parameter values which will give the best fit of an equation, or series of equations, to a set of data. For simple systems this can be done by differentiating the equations with respect to each parameter in turn, setting the set of partial differential equations to zero, and solving this set of simultaneous equations (as for exwnple in linear regression). In more complicated cases, however, it may be impossible to differentiate the equations, or very difficultly soluble non-linear equations may result. Many numerical optimization techniques to overcome these difficulties have been developed in the least ten years, and this review explains the logical basis of most of them, without going into the detail of computational procedures.The methods fall naturally into two classes - direct search methods, in which only values of the function to be minimized (or maximized) are used - and gradient methods, which also use derivatives of the function. The author considers all the accepted methods in each class, although warning that gradient methods should not be used unless the analytical differentiation of the function to be minimized is possible.If the solution is constrained, that is, certain values of the parameters are regarded as impossible or certain relations between the parameter values must be obeyed, the problem is more difficult. The second part of the review considers methods which have been proposed for the solution of constrained optimization problems.     

Progress-curve equations for reversible enzyme-catalysed reactions inhibited by tight-binding inhibitors.

The rate equation for a tight-binding inhibitor of an enzyme-catalysed first-order reversible reaction was used to derive two integrated equations. One of them covers the situations in which competitive, uncompetitive or non-competitive inhibition occurs and the other refers to the special non-competitive case where the two inhibition constants are equal. For these equations, graphical and non-linear regression methods are proposed for distinguishing between types of inhibition and for calculating inhibition constants from progress-curve data. The application of the non-linear regression to the analysis of stimulated progress curves in the presence of a tight-binding inhibitor is also presented. The results obtained are valid for any type of 'dead-end'-complex-forming inhibitor and can be used to characterize an unknown inhibitor on the basis of progress curves.     

Methods for parameter identification in oscillatory networks and application to cortical and thalamic 600 Hz activity.

Directed information transfer in the human brain occurs presumably by oscillations. As of yet, most approaches for the analysis of these oscillations are based on time-frequency or coherence analysis. The present work concerns the modeling of cortical 600 Hz oscillations, localized within the Brodmann Areas 3b and 1 after stimulation of the nervus medianus, by means of coupled differential equations. This approach leads to the so-called parameter identification problem, where based on a given data set, a set of unknown parameters of a system of ordinary differential equations is determined by special optimization procedures. Some suitable algorithms for this task are presented in this paper. Finally an oscillatory network model is optimally fitted to the data taken from ten volunteers.     

Maximum likelihood prediction of lake acidity based on sedimented diatoms

Abstract. As an example of ecological gradient analysis, Gaussian response functions, with Poisson or quasi-Poisson error distribution, were fitted for diatom taxa on a pH gradient. It is possible to predict or infer the pH of lake water from the fitted curves using the method of maximum likelihood, which is easily implemented in standard non-linear regressionprograms. Due to overdis-persion with respect to the Poisson distribution, moment estimates forthe negative binomial distribution were also applied, both in estimating the species response curves and in prediction. Simulations indicated that the theoretical maximum precision (measuredby standard deviation of prediction errors) in our data set was 0.17 pH units. The observed errors were much greater (SD 0.35 to 0.43). It seems that roughly equal proportions of the excess error were caused (1) by systematic differences between the training (estimation) data and the validation (prediction) data, and (2) from a misspecified model. It is suggested that the error due to model misspecification consists of inadequacy of the presumed error distribution and of inadequacy of the simple Gaussian response function.     

Multiple inhibition of glutathione S-transferase A from rat liver by glutathione derivatives: kinetic analysis supporting a steady-state random sequential mechanism.

Glutathione derivatives inhibit glutathione S-transferase A [cf. Biochem. J. (1975) 147, 513--522]. The steady-state kinetics of this inhibition have been investigated in detail by using S-octyglutathione, glutathione disulphide and S-(2-chloro-4-nitrophenyl)glutathione: the last compound is a product of the enzyme-catalused reaction. Interpreted in terms of generalized denotations of inhibition patterns, the compounds were found to be competitive with the substrate glutathione. Double-inhibition experiments involving simultaneous use of two inhibitors indicated exclusive binding of the inhibitors to the enzyme. The discrimination between alternative rate equations has been based on the results of weighted non-linear regression analysis. The experimental error was determined by replicate measurements and was found to increase with velocity. The established error structure was used as a basis for weighting in the regression and to construct confidence levels for the judgement of goodness-of-fit of rate equations fitted to experimental data. The results obtained support a steady-state random model for the mechanism of action of glutathione S-transferase A and exclude a number of simple kinetic models.     

A versatile digital computer program for non-linear regression analysis

1.

1. A general purpose digital computer program is described for application to biological experiments that require a non-linear regression analysis. The mathematical function, or model, to be fitted to a given set of experimental data is written as a section within the program. Given initial estimates for the parameters of the function, the program uses an iterative procedure to adjust the parameters until the sum of squares of residuals has converged to a minimum.     

Analysis of progress curves by simulations generated by numerical integration.

A highly flexible computer program written in FORTRAN is presented which fits computer-generated simulations to experimental progress-curve data by an iterative non-linear weighted least-squares procedure. This fitting procedure allows kinetic rate constants to be determined from the experimental progress curves. Although the numerical integration of the rate equations by a previously described method [Barshop, Wrenn & Frieden (1983) Anal. Biochem. 130, 134-145] is used here to generate predicted curves, any routine capable of the integration of a set of differential equations can be used. The fitting program described is designed to be widely applicable, easy to learn and convenient to use. The use, behaviour and power of the program is explored by using simulated test data.     

三参数增长模型拟合:以季风常绿阔叶林中两个优势乔木种群为例

Logistic、Mitscherlich、Gompertz方程是一类三参数饱和增长曲线模型,广泛地应用于许多学科领域．本文基于logistic方程饱和值K估计的三点法、四点法,推导出Mitscherlich、Gompertz方程K值的三点法、四点法估计公式,并以南亚热带季风常绿阔叶林中两种优势乔木厚壳桂、黄果厚壳桂种群为例,先用三点法或四点法估计出K值,再通过线性回归与非线性回归相结合的方法,可获得三个增长模型中三个参数的最优无偏估计．实例研究表明,两个优势种群增长数据均符合三个增长模型,但更符合增长曲线呈S形的logistic、Gompertz方程,且以logistic方程最适合于观察;黄果厚壳桂种群增长快于厚壳桂种群．