Integrated Rate Equation Research Articles

Optimization of Protease-Catalyzed Acyl Transfer Reactions. Determination of the Partition Constant Characterizing Nucleophile Efficiency

The partitioning of the acyl-enzyme between aminolysis by an added nucleophile and hydrolysis plays a key-role in protease-catalyzed acyl transfer reactions. It can be characterized by the partition constant, which is equal to the nucleophile concentration for which aminolysis and hydrolysis proceed at the same velocity. We describe a method for calculation of the partition constant from the product ratio which is based on the integrated rate equation. Therefore, it can be applied to reactions performed under synthesis-like conditions, i.e. a high degree of nucleophile consumption during the reaction. In principle, the dependence of the partition constant on nucleophile concentration can be determined from a single reaction. V8-protease-catalyzed acyl transfer reactions using Z-Glu-OMe as acyl donor and amino acid amides as nucleophiles were investigated as an application of the method. The central role of the partition constant in optimization of preparative protease-catalyzed acyl transfer reactions is ...

Analysis of progress curves for enzyme-catalysed reactions. Automatic construction of computer programs for fitting integrated rate equations.

The computer analysis of progress curves for enzyme-catalysed reactions involves a series of mathematical and computational tasks. The three most daunting of these are the derivation of an integrated rate equation, solving this equation so that the amount of product formed by the reaction at any time can be calculated, and incorporating this solution into a non-linear-regression computer program. This paper describes the basis of a computer program that greatly simplifies the problem. The proposed mechanism is specified in the familiar kinetic constant form, which is automatically translated into a program capable of fitting this mechanism to a series of experimental progress curves. The approach is illustrated for a reversible reaction with one substrate and one product, and tested with some data obtained for the fumarase reaction. A copy of the program has been deposited as Supplementary Publication SUP 50148 (13 pages) at the British Library Document Supply Centre, Boston Spa, Wetherby, West Yorkshire LS23 7BQ, U.K., from whom copies can be obtained on the terms indicated in Biochem. J. (1989) 257, 5.

A single-parameter family of adjustments for fitting enzyme kinetic models to progress-curve data.

Current methods for fitting integrated rate equations to enzyme progress curves treat each observation as if it were an independent measurement. When the data are obtained by taking several successive readings from each of a series of progress curves, the data will not be truly independent and will exhibit autocorrelation. Here we propose a simple pragmatic extension of integrated rate equations which takes account of first-order autocorrelations. The value of the method is assessed when applied to five sets of experimental data.

Anomalous linewidths and peak-height ratios in137Ba hyperfine lines

We have investigated several effects of optical pumping in the hyperfine spectrum of the6s6p1P1⇔6s21S0 transition in137Ba. Most of these effects are explained by absorption strength changes which occur because of redistribution of population among magnetic substates. At very low laser intensities, no redistribution effects are observed. At higher intensities, it is possible to either empty the magnetic substates that are accessible to optical excitation, or to redistribute the population among these states until a steady-state condition is achieved. The first case results in the familiar disappearance of a hyperfine line. The less familiar second case can result in peak-height ratios in the Ba1P1⇔1S0 hyperfine spectrum that differ by almost a factor of three from the low-intensity case. In this second case, the observed linewidth can either broaden or narrow, depending on whether redistribution decreases or increases absorption strength. At high intensities, saturation effects dominate and branching to intermediateD states becomes apparent. We report here the result of a numerically integrated rate equation model which shows good agreement with our experiments.

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 K r: ▪ 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/ K i) and K r can be obtained from a series of experiments with varying initial concentrations of inhibitor. Reliable estimates of k +2 and K i individually are dependent upon the relative magnitudes of the kinetic parameters describing inactivation. The special case, K r = K i, 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.

Kinetic mechanism and substrate specificity of glutathione peroxidase activity of ebselen (PZ51)

The glutathione peroxidase activity of ebselen (PZ51) was studied using different hydro-peroxidic substrates. The single progression curves obtained in the spectrophotometric test were processed by a computer to fit the integrated rate equation that describes the ping pong reaction of the Se glutathione peroxidases. Ebselen catalyzes the GSH peroxidase reaction with a mechanism that appears kinetically identical to the mechanism of the enzymes. The inactivation of the catalytic properties of ebselen by iodoacetate suggests that a selenol moiety is involved. Among the substrates tested, the best hydroperoxidic substrates are the hydroperoxy derivatives of phosphatidyl choline. Ebselen is active also on membrane hydroperoxides as does phospholipid hydroperoxide glutathione peroxidase but not glutathione peroxidase.

Nucleoside transporter of pig erythrocytes. Kinetic properties, isolation and reaction with nitrobenzylthioinosine and dipyridamole

Rapid kinetic techniques were used to measure the transport of uridine in pig erythrocytes in zero- trans entry and exit and equilibrium exchange protocols. The kinetic parameters were computed by fitting appropriate integrated rate equations to the time-courses of transmembrane equilibration of radiolabeled uridine. Transport of uridine conformed to the simple carrier model with directional symmetry, but differential mobility of substrate-loaded and empty carrier. At 5° C, the carrier moved about 30-times faster when loaded than when empty. Uridine transport was inhibited in a concentration-dependent manner by nitrobenzylthioinosine and dipyridamole and the inhibition correlated with the binding of the inhibitors to high-affinity binding sites on the cells ( K d about 1 and 10 nM, respectively). Thus, in its kinetic properties, differential mobility when empty and loaded, and sensitivity to inhibition by nitrobenzylthioinosine and dipyridamole, the transporter of pig erythrocytes is very similar to that of human erythrocytes. Also, the total number of high-affinity binding sites for nitrobenzylthioinosine and dipyridamole/cell were similar for the two cell types and the [ 3H]nitrobenzylthioinosine-labeled carrier of pig erythrocytes, just as that of human red cells, was mainly recovered in the band 4.5 protein fraction of Triton X-100-solubilized membranes. However, sodium dodecylsulfate-polyacrylamide gel electrophoresis of photoaffinity-labeled band 4.5 membrane proteins indicated a slightly higher molecular weight for the transporter from pig than human erythrocytes. We have also confirmed the lack of functional sugar transport in erythrocytes from adult pigs by measuring the uptake of various radiolabeled sugars. But in spite of the lack of functional sugar transport we recovered as much band 4.5 protein from pig as from human erythrocyte membranes.

Analytical methods for fitting integrated rate equations. A discontinuous assay.

The integrated rate equation for reactions with stoichiometry A----P + Q is: e0t = -Cf . ln(1-delta P/A0) + C1 delta P + 1/2C2(delta P)2 where the coefficients C are linear or quadratic functions of the kinetic constants and the initial substrate and product concentrations. I have used the 21 progress curves described in the accompanying paper [Cox & Boeker (1987) Biochem. J. 245, 59-65] to develop computer-based analytical and statistical techniques for extracting kinetic constants by fitting this equation. The coefficients C were calculated by an unweighted non-linear regression: first approximations were obtained from a multiple regression of t on delta P and were refined by the Gauss-Newton method. The procedure converged in six iterations or less. The bias in the coefficients C was estimated by four methods and did not appear to be significant. The residuals in the progress curves appear to be normally distributed and do not correlate with the amount of product produced. Variances for Cf, C1 and C2 were estimated by four resampling procedures, which gave essentially identical results, and by matrix inversion, which came close to the others. The reliability of C2 can also be estimated by using an analysis-of-variance method that does not require resampling. The final kinetic constants were calculated by standard multiple regression, weighting each coefficient according to its variance. The weighted residuals from this procedure were normally distributed.

Analysis of enzyme kinetics by using integrated rate equations. Arginine decarboxylase.

We have used an integrated rate equation to analyse the reaction catalysed by the inducible arginine decarboxylase from Escherichia coli B. The stoichiometry Arginine----agmatine + CO2 is the simplest of the multiple-substrate/multiple-product cases. Twenty-one time courses were carried out at various initial concentrations of arginine and agmatine, and were then fitted to the integrated equation by using appropriate analytical procedures. Values were obtained for six of the seven possible kinetic constants, corresponding to kcat, KArg, the terms for competitive inhibition by agmatine, by CO2 and by agmatine and CO2 together, and the term for uncompetitive inhibition by agmatine. The uncompetitive constant for CO2 was indeterminate. Our results indicate that it is both practical and experimentally economical to obtain kinetic constants from full time courses.

Integrated rate equation for free‐radical copolymerizations with monomer 2 in low concentration and a small r2‐value

Integrated rate equation for free‐radical copolymerizations with monomer 2 in low concentration and a small <i>r</i>₂‐value

ChemInform Abstract: The Reversed‐Flow Gas‐Chromatography Technique Applied to the Kinetics of the Methanation of Carbon Monoxide

The reversed-flow gas-chromatography technique has been applied to the methanation reaction of carbon monoxide over two Ni/Al2O3 catalysts. A continuous feed by diffusion of carbon monoxide was used, ensuring a continuous change in its concentration with time over the catalyst bed. An integrated rate equation has been derived, permitting the calculation from experimental data of the rate constant and the order of the reaction with respect to carbon monoxide. This was done for a wide range of fractional conversions to methane at each temperature. Arrhenius activation parameters were computed for each reaction order. The latter changes with temperature discontinuously, but does not change with CO concentration at the same temperature. This is explained by the fact that the method is a pulse technique under steady-state conditions. The findings are compared with recent results in the literature, agreeing with some of them but not with others. They are, however, consistent with a conventional reaction mechanism, assuming a non-homogeneous catalytic surface. A rate equation for the formation of methane, based on this mechanism, has been derived and is used to explain the experimental data, especially the change of reaction order with respect to carbon monoxide with temperature. The rate-determining step is the hydrogenation of CH or CH3 groups.

Computer simulations of the kinetics of irreversible enzyme inhibition by an unstable inhibitor.

Computer simulations of the irreversible inhibition of an enzyme by an unstable inhibitor are presented. Data obtained at the end point of reaction are shown to conform poorly in many situations with relationships derived from integrated rate equations by setting t = infinity, and the implications concerning the experimental use of this method to determine kinetic constants describing inactivation are considered. The alternative approach of conducting experiments under conditions of inhibitor excess over enzyme is further discussed, and a graphical procedure is suggested for the description of time courses of reaction of enzyme with unstable inhibitor when an enzyme-inhibitor adsorptive complex is involved.

Beta nerve growth factor binding to PC12 cells. Association kinetics and cooperative interactions.

The association kinetics of 125I beta nerve growth factor (NGF) binding to the PC12 clonal cell line have been examined in detail at 0.5 and 37 degrees C. These data were examined by utilizing a reversible second-order integrated rate equation, and the results were not consistent with a simple bimolecular process. Two association rates were required to explain the results adequately. At 37 degrees C, the faster component was estimated to have a second-order association rate constant of 1.4 X 10(7) M-1 s-1, while the rate constant for the slower component (3.8 X 10(6) M-1 s-1) was about 4-fold lower. As shown by others, the temperature dependence of the dissociation kinetics indicated that while the rapidly dissociating component was only slightly slowed by lowering the chase temperature to 0.5 degrees C, the second component was slowed by about 270-fold, from 8 X 10(-4) s-1 to 3 X 10(-6) s-1. The binding data that describe the slowly dissociating component were obtained by utilizing this differential temperature dependence and revealed a concave downward Scatchard plot. The binding parameters determined from computer analysis using a nonlinear fitting program (LIGAND) suggest that this component consists of (a) an interacting class of about 4000 sites/cell that have a first stoichiometric steady-state dissociation constant of 65 pM and a second stoichiometric interaction constant of 16 pM, indicative of positively cooperative interactions, and (b) a class of sites consistent with a ratio of sites/Kd of about 11.1 sites/(cell X pM). The steady-state binding results at 37 degrees C indicated only one class of binding sites (155,000 +/- 18,000 sites/cell) that had an apparent Kd of 0.52 +/- 0.03 nM. One class of sites was also observed at 0.5 degrees C, and the receptor concentration was found to be reduced (99,000 +/- 7600 sites/cell) while the Kd was increased (1.7 +/- 0.14 nM). A significant level of positively cooperative interactions was observed frequently at 37 degrees C that was not due to a failure to reach steady-state conditions, internalization, or degradation. Since cooperativity of binding was never observed at 0.5 degrees C, a membrane event may be involved. Determination of the contribution of the different classes of NGF receptors found on PC12 cells to the biological actions of NGF requires a clear understanding of their kinetic properties and their relationship to each other. The studies presented here indicate that their interactions are more complex than previously described.

Nozzle flows with nonequilibrium condensation

The compressible nozzle flow of a mixture of a condensible vapor and an inert carrier gas with nonequilibrium condensation is considered. An asymptotic method is used with respect to two disparate parameters appearing in the nonequilibrium condensation rate equation. A high activation limit and another limit consideration that signify a large nucleation time followed by a small droplet growth time are employed. Under the limits, the coupled integral rate equation is so controlled by the activation function of homogeneous nucleation over its six zones that there appears to be a one-to-one correspondence between these zones and the six characteristic condensation zones of the nozzle flow itself. The six zones are the initial growth of condensate, further growth, rapid growth, onset, nucleation with growth, and droplet growth zone. The last zone can be greatly simplified if terms that imply a reasonably sizable order of error are neglected. This results in the rectilinear, one-dimensional flow equations of a normal shock wave structured or resisted by droplet growth.

Progress-curve analysis in enzyme kinetics. Numerical solution of integrated rate equations.

Progress curves of enzyme-catalysed reactions are described by equations of a type that precludes direct calculation of the extent of reaction at any time. Previously, such equations have been solved by the Newton-Raphson method, but this procedure may fail when based upon the usual formulae. An alternative formulation is proposed that is both quicker and more robust.

Topochemistry of the Wood-Isccyanate Reaction an Analysis of Reaction Profiles

Abstract The rate of reaction of Scots pine (Pinus sylvestris) with n-butyl isocyanate was followed by measuring weight percent gain (WPG) as a function of time (t). A model was developed to predict the form that an equation representing the WPG versus t ‘reaction profile’ might take. A curve of that form was fitted to the data using standard commercial software. The resulting ‘whole wood’ integrated rate equation was split into its component equations to yield information about the rate and extent of reaction in the lignin and holocellulose fractions of the wood. Assignment of equations, based on defined assumptions, indicate that in the initial stages of the reaction lignin reacts faster and to a higher degree of substitution than the holocellulose.

The reversed-flow gas-chromatography technique applied to the kinetics of the methanation of carbon monoxide

The reversed-flow gas-chromatography technique has been applied to the methanation reaction of carbon monoxide over two Ni/Al2O3 catalysts. A continuous feed by diffusion of carbon monoxide was used, ensuring a continuous change in its concentration with time over the catalyst bed. An integrated rate equation has been derived, permitting the calculation from experimental data of the rate constant and the order of the reaction with respect to carbon monoxide. This was done for a wide range of fractional conversions to methane at each temperature. Arrhenius activation parameters were computed for each reaction order. The latter changes with temperature discontinuously, but does not change with CO concentration at the same temperature. This is explained by the fact that the method is a pulse technique under steady-state conditions. The findings are compared with recent results in the literature, agreeing with some of them but not with others. They are, however, consistent with a conventional reaction mechanism, assuming a non-homogeneous catalytic surface. A rate equation for the formation of methane, based on this mechanism, has been derived and is used to explain the experimental data, especially the change of reaction order with respect to carbon monoxide with temperature. The rate-determining step is the hydrogenation of CH or CH3 groups.

A kinetic study of tetracycline decomposition in acid solution

The individual rate constants of the reactions involved in the reversible epimerizations of tetracycline and anhydrotetracycline are reported together with those for the dehydration reactions of tetracycline and epitetracycline. These values, obtained by the initial rate method, are compared with literature values obtained by curve fitting procedures. At pH 1.5 dehydration is the faster process. The initial rate method is compared with the conventional integrated rate equation method for the decomposition of the parent tetracycline at different temperatures between 30 and 75°C and over a wider range of pH than has previously been reported. The advantages and limitations of the initial rate method are discussed.

Analysis of progress curves for a highly concentrated Michaelian enzyme in the presence or absence of product inhibition.

Methods are given for analysing the time course of an enzyme-catalysed reaction when the concentration of the enzyme itself is high, a situation which is often found in vivo. (1) The integrated form of the kinetic equation for a concentrated Michaelian enzyme in absence of product inhibition is given. Parameters are shown to be calculated easily using non-linear fitting procedures. (2) A general algorithm to analyse progress-curve data in more complex cases (i.e. when the analytical form of the integrated rate equation is not known or is exceedingly complex) is proposed. This algorithm may be used for any enzyme mechanism for which the differential form of the kinetic equation may be written analytically. We show that the method allows differentiation between the main types of product inhibition which may occur in the case of a highly concentrated Michaelian enzyme.

A standard formula for the determination of the initial rate of hydrolysis of carboxymethylcellulose

The integrated rate equation of Huang, originally used to describe the hydrolysis of insoluble acid treated cellulose, is shown equally applicable in describing the hydrolysis of carboxymethylcellulose (CMC) using a dilution series of Cellulomonas sp. ATCC 21399 crude cellulase as enzyme preparation. Interpretation of the progress curves of hydrolysis of CMC according to the integrated rate equation is used to calculate a standard formula for the conversion of the rate of hydrolysis into the initial velocity of hydrolysis. The validity of the standard formula is tested, using enzyme preparations from Cellulomonas grown under varied conditions, and enzyme preparations containing purified endoglucanases from Cellulomonas.