A Simple Mathematical Model of Second-Messenger Mediated Slow Excitatory Postsynaptic Potentials

We have developed a novel and simple mathematical model of a slow excitatory postsynaptic potential (EPSP) based on an abstraction of the processes of activation, inactivation, and summation of a cAMP, protein kinase A (PKA)-dependent second-messenger cascade. The model describes the activation of receptors, G-proteins, and production of cAMP as the first stage and uses first-order, non-rate-limited kinetics. The second stage corresponds to the release of active, PKA catalytic subunit and can use first- or higher-order kinetics. The third stage represents simple phosphorylation of ion channels and is limited by the number of channels available. The decay of each stage is based on first-order, mass-action kinetics. These equations and some variations were solved numerically and values of the parameters were determined by fitting to a variety of experimental data from myenteric neurons of the guinea-pig ileum. The model produced a slow EPSP with a nonlinear stimulus-response relationship that resulted from the underlying kinetics of the signaling cascade. This system of equations is suitable for incorporation into a large-scale computer simulation, and the methodology should be generalizable to other pathways.     

Global Kinetic Explorer: A new computer program for dynamic simulation and fitting of kinetic data

We describe a new dynamic kinetic simulation program that allows multiple data sets to be fit simultaneously to a single model based on numerical integration of the rate equations describing the reaction mechanism. Unlike other programs that allow fitting based on numerical integration of rate equations, in the dynamic simulation rate constants, output factors, and starting concentrations of reactants can be scrolled while observing the change in the shape of the simulated reaction curves. Fast dynamic simulation facilitates the exploration of initial parameters that serve as the starting point for nonlinear regression in fitting data and facilitates exploration of the relationships between individual constants and observable reactions. The exploration of parameter space by dynamic simulation provides a powerful tool for learning kinetics and for evaluating the extent to which parameters are constrained by the data. This feature is critical to avoid overly complex models that are not supported by the data.     

Use of nonlinear regression to analyze enzyme kinetic data: application to situations of substrate contamination and background subtraction

In a recent publication, A. Lundin, P. Arner, and J. Hellmér [Anal. Biochem. 177, 125-131 (1989)] describe a method whereby kinetic substrate assays can be performed when the assay mixture includes a significant contaminating levels of substrate. Their method requires various rearrangements of the data, and involves three separate linear regression calculations. We show how the same data may be analyzed directly, and far more simply, by nonlinear regression. Unlike the linear regression method, nonlinear regression allows direct calculation of the actual values for Km, Vmax, and the concentration of contaminating substrate (as well as estimates of their standard errors); the former method gives only apparent values. The nonlinear regression technique is also statistically a more valid means of analysis, as the rearrangements required to give linearized equations will considerably distort the error distribution and render simple unweighted linear regression inappropriate. The ease of incorporating extra parameters into standard equations when nonlinear regression is used is further illustrated by fitting enzyme reaction data which describe a first-order process when a significant nonspecific background is present. For this equation no simple rearranged linear plot is possible, but nonlinear regression is easily applied to determine the kinetic parameters.     

Fitting curves to data using nonlinear regression: a practical and nonmathematical review

Many types of data are best analyzed by fitting a curve using nonlinear regression, and computer programs that perform these calculations are readily available. Like every scientific technique, however, a nonlinear regression program can produce misleading results when used inappropriately. This article reviews the use of nonlinear regression in a practical and nonmathematical manner to answer the following questions: Why is nonlinear regression superior to linear regression of transformed data? How does nonlinear regression differ from polynomial regression and cubic spline? How do nonlinear regression programs work? What choices must an investigator make before performing nonlinear regression? What do the final results mean? How can two sets of data or two fits to one set of data be compared? What problems can cause the results to be wrong? This review is designed to demystify nonlinear regression so that both its power and its limitations will be appreciated.     

Cometabolism of chlorinated solvents by nitrifying bacteria: kinetics, substrate interactions, toxicity effects, and bacterial response

Pure cultures of ammonia-oxidizing bacteria, Nitrosomonas europaea, were exposed to trichloroethylene (TCE), 1,1-dichloroethylene (1,1-DCE), chloroform (CF), 1,2-dichloroethane (1,2-DCA), or carbon tetrachloride (CT), in the presence of ammonia, in a quasi-steady-state bioreactor. Estimates of enzyme kinetics constants, solvent inactivation constants, and culture recovery constants were obtained by simultaneously fitting three model curves to experimental data using nonlinear optimization techniques and an enzyme kinetics model, referred to as the inhibition, inactivation, and recovery (IIR) model, that accounts for inhibition of ammonia oxidation by the solvent, enzyme inactivation by solvent product toxicity, and respondent synthesis of new enzyme (recovery). Results showed relative enzyme affinities for ammonia monooxygenase (AMO) of 1,1-DCE approximately TCE > CT > NH(3) > CF > 1,2-DCA. Relative maximum specific substrate transformation rates were NH(3) > 1,2-DCA > CF > TCE approximately 1,1-DCE > CT (=0). The TCE, CF, and 1,1-DCE inactivated the cells, with 1,1-DCE being about three times more potent than TCE or CF. Under the conditions of these experiments, inactivating injuries caused by TCE and 1,1-DCE appeared limited primarily to the AMO enzyme, but injuries caused by CF appeared to be more generalized. The CT was not oxidized by N. europaea while 1,2-DCA was oxidized quite readily and showed no inactivation effects. Recovery capabilities were demonstrated with all solvents except CF. A method for estimating protein yield, the relationship between the transformation capacity model and the IIR model, and a condition necessary for sustainable cometabolic treatment of inactivating substrates are presented. (c) 1997 John Wiley & Sons, Inc. Biotechnol Bioeng 54: 520-534, 1997.     

Flash-induced oxygen evolution in photosynthesis: simple solution for the extended S-state model that includes misses, double-hits, inactivation, and backward-transitions

Flash-induced oxygen evolution in higher plants, algae, and cyanobacteria exhibits damped period-four oscillations. To explain such oscillations, Kok suggested a simple phenomenological S-state model, in which damping is due to empirical misses and double-hits. Here we developed an analytical solution for the extended Kok model that includes misses, double-hits, inactivation, and backward-transitions. The solution of the classic Kok model (with misses and double-hits only) can be obtained as a particular case of this solution. Simple equations describing the flash-number dependence of individual S-states and oxygen evolution in both cases are almost identical and, therefore, the classic Kok model does not have a significant advantage in its simplicity over the extended version considered in this article. Developed equations significantly simplify the fitting of experimental data via standard nonlinear regression analysis and make unnecessary the use of many previously developed methods for finding parameters of the model. The extended Kok model considered here can provide additional insight into the effect of dark relaxation between flashes and inactivation.     

Model Testing in Radioligand/Receptor Interaction by Monte Carlo Simulation

Abstract

Monte Carlo simulations have been applied for evaluating the reliability of parameter estimates as well as for testing models in radioligand saturation binding experiments. Scatchard analysis was compared to the nonlinear least-square curve fitting method for one-site saturation binding curves. It was found that linear regression analysis from the transformed data in the Scatchard plot yielded generally less accurate parameter estimates than nonlinear regression analysis of untransformed data. The advantage of the nonlinear least-squares curve fitting method was especially pronounced in cases where the scatter and number of data points, as well as the radioligand concentration range, were chosen similar to less optimal experimental conditions. Under such circumstances, several K_D and B_max values derived by Scatchard analysis led to physically impossible negative values whereas the same data analyzed by nonlinear regression yielded reasonable parameter estimates. Furthermore, it was found that for both means of analysis, K_D and B_max correlated positively. In another set of Monte Carlo experiments, saturation binding curves involving two receptor sites were generated and subsequently analyzed according to both a one-site and a two-site model. The confidence with which one is able to distinguish the two-site model from nonlinear least-squares curve fitting was then estimated for optimal, as well as for, less ideal experimental condigions.     

Allometric equations for predicting body mass of dinosaurs

We use data from the literature to compare two statistical procedures for estimating mass (or size) of quadrupedal dinosaurs and other extraordinarily large animals in extinct lineages. Both methods entail extrapolation from allometric equations fitted to data for a reference group of contemporary animals having a body form similar to that of the dinosaurs. The first method is the familiar one of fitting a straight line to logarithmic transformations, followed by back-transformation of the resulting equation to a two-parameter power function in the arithmetic scale. The second procedure entails fitting a two-parameter power function directly to arithmetic data for the extant forms by nonlinear regression. In the example presented here, the summed circumferences for humerus plus femur for 33 species of quadrupedal mammals was the predictor variable in the reference sample and body mass was the response variable. The allometric equation obtained by back-transformation from logarithms was not a good fit to the largest species in the reference sample and presumably led to grossly inaccurate estimates for body mass of several large dinosaurs. In contrast, the allometric equation obtained by nonlinear regression described data in the reference sample quite well, and it presumably resulted in better estimates for body mass of the dinosaurs. The problem with the traditional analysis can be traced to change in the relationship between predictor and response variables attending transformation, thereby causing measurements for large animals not to be weighted appropriately in fitting models by least squares regression. Extrapolations from statistical models obtained by back-transformation from lines fitted to logarithms are unlikely to yield reliable predictions for body size in extinct animals. Numerous reports on the biology of dinosaurs, including recent studies of growth, may need to be reconsidered in light of our findings.     

A working guide to boosted regression trees

1. Ecologists use statistical models for both explanation and prediction, and need techniques that are flexible enough to express typical features of their data, such as nonlinearities and interactions. 2. This study provides a working guide to boosted regression trees (BRT), an ensemble method for fitting statistical models that differs fundamentally from conventional techniques that aim to fit a single parsimonious model. Boosted regression trees combine the strengths of two algorithms: regression trees (models that relate a response to their predictors by recursive binary splits) and boosting (an adaptive method for combining many simple models to give improved predictive performance). The final BRT model can be understood as an additive regression model in which individual terms are simple trees, fitted in a forward, stagewise fashion. 3. Boosted regression trees incorporate important advantages of tree-based methods, handling different types of predictor variables and accommodating missing data. They have no need for prior data transformation or elimination of outliers, can fit complex nonlinear relationships, and automatically handle interaction effects between predictors. Fitting multiple trees in BRT overcomes the biggest drawback of single tree models: their relatively poor predictive performance. Although BRT models are complex, they can be summarized in ways that give powerful ecological insight, and their predictive performance is superior to most traditional modelling methods. 4. The unique features of BRT raise a number of practical issues in model fitting. We demonstrate the practicalities and advantages of using BRT through a distributional analysis of the short-finned eel (Anguilla australis Richardson), a native freshwater fish of New Zealand. We use a data set of over 13 000 sites to illustrate effects of several settings, and then fit and interpret a model using a subset of the data. We provide code and a tutorial to enable the wider use of BRT by ecologists.     

Half-time analysis of the kinetics of irreversible enzyme inhibition by an unstable site-specific reagent

The half-time method for the determination of Michaelis parameters from enzyme progress-curve data (Wharton, C.W. and Szawelski, R.J. (1982) Biochem. J. 203, 351-360) has been adapted for analysis of the kinetics of irreversible enzyme inhibition by an unstable site-specific inhibitor. The method is applicable to a model in which a product (R) of the decomposition of the site-specific reagent, retaining the chemical moiety responsible for inhibitor specificity, binds reversibly to the enzyme with dissociation constant Kr: (formula; see text). Half-time plots of simulated enzyme inactivation time-course data are shown to be unbiased, and excellent estimates of the apparent second-order rate constant for inactivation (k +2/Ki) and Kr can be obtained from a series of experiments with varying initial concentrations of inhibitor. Reliable estimates of k +2 and Ki individually are dependent upon the relative magnitudes of the kinetic parameters describing inactivation. The special case, Kr = Ki, is considered in some detail, and the integrated rate equation describing enzyme inactivation shown to be analogous to that for a simple bimolecular reaction between enzyme and an unstable irreversible inhibitor without the formation of a reversible enzyme-inhibitor complex. The half-time method can be directly extended to the kinetics of enzyme inactivation by an unstable mechanism-based (suicide) inhibitor, provided that the inhibitor is not also a substrate for the enzyme.     

Determination of molecular parameters by fitting sedimentation data to finite-element solutions of the Lamm equation.

A method for fitting experimental sedimentation velocity data to finite-element solutions of various models based on the Lamm equation is presented. The method provides initial parameter estimates and guides the user in choosing an appropriate model for the analysis by preprocessing the data with the G(s) method by van Holde and Weischet. For a mixture of multiple solutes in a sample, the method returns the concentrations, the sedimentation (s) and diffusion coefficients (D), and thus the molecular weights (MW) for all solutes, provided the partial specific volumes (v) are known. For nonideal samples displaying concentration-dependent solution behavior, concentration dependency parameters for s(sigma) and D(delta) can be determined. The finite-element solution of the Lamm equation used for this study provides a numerical solution to the differential equation, and does not require empirically adjusted correction terms or any assumptions such as infinitely long cells. Consequently, experimental data from samples that neither clear the meniscus nor exhibit clearly defined plateau absorbances, as well as data from approach-to-equilibrium experiments, can be analyzed with this method with enhanced accuracy when compared to other available methods. The nonlinear least-squares fitting process was accomplished by the use of an adapted version of the "Doesn't Use Derivatives" nonlinear least-squares fitting routine. The effectiveness of the approach is illustrated with experimental data obtained from protein and DNA samples. Where applicable, results are compared to methods utilizing analytical solutions of approximated Lamm equations.     

Kinetics of Chemical Modification of Arginine Residues in Mitochondrial Creatine Kinase from Bovine Heart: Evidence for Negative Cooperativity

The kinetics of chemical modification of arginine residues in mitochondrial creatine kinase (mit-CK) from beef heart by 4-hydroxy-3-nitrophenylglyoxal (HNPG) have been studied with simultaneous registration of enzyme inactivation. Experiments showed that complete inactivation of mit-CK corresponded to modification of two arginine residues per mit-CK monomer. The data on the modification kinetics can be described by the sum of two exponential terms and suggest strong negative cooperativity in the binding of HNPG to arginine residues. The rate constants for the fast and slow phases of modification differ by a factor of about 50. The corresponding rate constants for inactivation differ by a factor of about 30. The rate constant for the slow stage of inactivation is twice as large as that for the rate constant for the slow stage of modification, i.e., the inactivation process is ahead of the modification process.     

The impact of temperature on the inactivation of enteric viruses in food and water: a review

Temperature is considered as the major factor determining virus inactivation in the environment. Food industries, therefore, widely apply temperature as virus inactivating parameter. This review encompasses an overview of viral inactivation and virus genome degradation data from published literature as well as a statistical analysis and the development of empirical formulae to predict virus inactivation. A total of 658 data (time to obtain a first log(10) reduction) were collected from 76 published studies with 563 data on virus infectivity and 95 data on genome degradation. Linear model fitting was applied to analyse the effects of temperature, virus species, detection method (cell culture or molecular methods), matrix (simple or complex) and temperature category (<50 and ≥50°C). As expected, virus inactivation was found to be faster at temperatures ≥50°C than at temperatures <50°C, but there was also a significant temperature-matrix effect. Virus inactivation appeared to occur faster in complex than in simple matrices. In general, bacteriophages PRD1 and PhiX174 appeared to be highly persistent whatever the matrix or the temperature, which makes them useful indicators for virus inactivation studies. The virus genome was shown to be more resistant than infectious virus. Simple empirical formulas were developed that can be used to predict virus inactivation and genome degradation for untested temperatures, time points or even virus strains.     

Parameter estimation for ligand binding systems kinetics applied to 1,25-dihydroxycholecalciferol

A mathematical analysis of the kinetics of the hormone-receptor interaction was applied to the 1,25-dihydroxycholecalciferol-intestinal receptor system. The exact analytical solution and the numerical integration of the kinetic equation were installed in a Statistical Analysis System (SAS) computer program to estimate the rate constants of the reaction. Estimates of the parameters obtained by these two methods are similar, demonstrating that the numerical integration can be combined with the nonlinear regression procedure for least-squares parameter fitting using a simple SAS program. This enables estimation of kinetics rate constants when the kinetic equation cannot be solved analytically. The ratio of the rate constants (ka/kd) found by the nonlinear procedure is close to the independently determined equilibrium (Scatchard) constant in the nonlinear analysis.     

Gating current harmonics. III. Dynamic transients and steady states with intact sodium inactivation gating.

Internally perfused squid giant axons with intact sodium inactivation gating were prepared for gating current experiments. Gating current records were obtained in sinusoidally driven dynamic steady states and as dynamic transients as functions of the mean membrane potential and the frequency of the command sinusoid. Controls were obtained after internal protease treatment of the axons that fully removed inactivation. The nonlinear analysis consisted of determining and interpreting the harmonic content in the current records. The results indicate the presence of three kinetic processes, two of which are associated with activation gating (the so-called primary and secondary processes), and the third with inactivation gating. The dynamic steady state data show that inactivation gating does not contribute a component to the gating current, and has no direct voltage-dependence of its own. Rather, the inactivation kinetics appear to be coupled to the primary activation kinetics, and the coupling mechanism appears to be one of reciprocal steric hindrance between two molecular components. The mechanism allows the channel to become inactivated without first entering the conducting state, and will do so in about 40 percent of depolarizing voltage-clamp steps to 0 mV. The derived model kinetics further indicate that the conducting state may flicker between open and closed with the lifetime of either state being 10 microseconds. Dynamic transients generated by the model kinetics (i.e., the behavior of the harmonic components as a function of time after an instantaneous change in the mean membrane potential from a holding potential of -80 mV) match the experimental dynamic transients in all details. These transients have a duration of 7-10 ms (depending on the level of depolarization), and are the result of the developing inactivation following the discontinuous voltage change. A detailed hypothetical molecular model of the channel and gating machinery is presented.     

Linear and Nonlinear Regression Analyses for Binary Sorption Kinetics of Methylene Blue and Safranin onto Pretreated Rice Husk

This article extends the study of the comparison between linear and nonlinear forms of the two widely used kinetic models, namely, pseudo-first-order and pseudo-second-order, by considering the binary biosorption of the basic dyes methylene blue and safranin onto pretreated rice husk in a batch system. The present investigation showed that nonlinear forms of pseudo-first-order and pseudo-second-order models were more suitable than the linear forms for fitting the experimental data. The sorption process was found to follow the pseudo-second-order kinetics. The results suggest that it is not appropriate to use the linear method in determining the kinetic parameters of a particular kinetic model. The nonlinear method is a better way to obtain the kinetic parameters than the linear method and thus it should be primarily adopted to determine the kinetic parameters.     

Kriging with nonparametric variance function estimation

A method for fitting regression models to data that exhibit spatial correlation and heteroskedasticity is proposed. It is well known that ignoring a nonconstant variance does not bias least-squares estimates of regression parameters; thus, data analysts are easily lead to the false belief that moderate heteroskedasticity can generally be ignored. Unfortunately, ignoring nonconstant variance when fitting variograms can seriously bias estimated correlation functions. By modeling heteroskedasticity and standardizing by estimated standard deviations, our approach eliminates this bias in the correlations. A combination of parametric and nonparametric regression techniques is used to iteratively estimate the various components of the model. The approach is demonstrated on a large data set of predicted nitrogen runoff from agricultural lands in the Midwest and Northern Plains regions of the U.S.A. For this data set, the model comprises three main components: (1) the mean function, which includes farming practice variables, local soil and climate characteristics, and the nitrogen application treatment, is assumed to be linear in the parameters and is fitted by generalized least squares; (2) the variance function, which contains a local and a spatial component whose shapes are left unspecified, is estimated by local linear regression; and (3) the spatial correlation function is estimated by fitting a parametric variogram model to the standardized residuals, with the standardization adjusting the variogram for the presence of heteroskedasticity. The fitting of these three components is iterated until convergence. The model provides an improved fit to the data compared with a previous model that ignored the heteroskedasticity and the spatial correlation.     

A robust curve-fitting procedure for the analysis of plasmid DNA strand break data from gel electrophoresis

A robust method for fitting to the results of gel electrophoresis assays of damage to plasmid DNA caused by radiation is presented. This method makes use of nonlinear regression to fit analytically derived dose-response curves to observations of the supercoiled, open circular and linear plasmid forms simultaneously, allowing for more accurate results than fitting to individual forms. Comparisons with a commonly used analysis method show that while there is a relatively small benefit between the methods for data sets with small errors, the parameters generated by this method remain much more closely distributed around the true value in the face of increasing measurement uncertainties. This allows for parameters to be specified with greater confidence, reflected in a reduction of errors on fitted parameters. On test data sets, fitted uncertainties were reduced by 30%, similar to the improvement that would be offered by moving from triplicate to fivefold repeats (assuming standard errors). This method has been implemented in a popular spreadsheet package and made available online to improve its accessibility.     

A cometabilic kinetics model incorporating enzyme inhbition, inactivation, and recovery: I. Model development, analysis, and testing

Cometabolic biodegradation prcesses are important for bioremediation of hazardous waste sites. However, these proceeses are not well understood and have not been modeled thoroughly. Traditional Michaelis-Menten kinetics models often are used, but toxic effects and bacterial responses to toxicity may cause changes in enzyme levels, rendering such models inappropriate. In this article, a conceptual and mathematical model of cometabolic enzyme kinetics i described. Model derivation is based on enzyme/growth-substrate/nongrowth-substrate interaction and incorporates enzyme inhibition (caused by the presence of a cometabolic compound), inactivation (resulting from toxicity of a cometabolic product), and recovery (associated with bacterial synthesis of new enbzyme in response to inactivation). The mathematical model consists of a system of two, nonlinear ordinary differential equations that can be solved implicitly using numerical methods, providing estimates of model parameters. Model analysis shows that growth substraate adn nongrowth substrate oxidation rates are related by a dimensionless constant. Reliability of tehy model solution prcedure is verifiedl by abnalyzing data ses, containing random error, from simulated experimentss with trichhloroethyylene (TCE) degradation by ammonia-oxidizing bacterialunder various conditions. Estimation of the recovery rate contant is deterimined to be sensitive to intial TCE concentration. Model assumptions are evaluated in a companion article using data from TCE degradation experiments with amoniaoxidizing bacteria. (c) 1995 John Wiley & Sons, Inc.