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.     

On the difficulties of fitting the double Michaelis-Menten equation to kinetic data

The double Michaelis-Menten equation describes the reaction kinetics of two independent, saturable uptake mechanisms. The use of this equation to describe drug uptake has been reported several times in the literature, and several methods have been published to fit the equation to data. So far, however, confidence intervals on the fitted kinetic parameters have not been provided. We present a grid-search method for fitting the double Michaelis-Menten equation to kinetic uptake data, and a Monte-Carlo procedure for estimating confidence intervals on the fitted parameters. We show that the fitting problem is extremely ill-conditioned, and that very accurate data are required before any confidence can be placed in the fitted parameters.     

The reliability of Michaelis constants and maximum velocities estimated by using the integrated Michaelis–Menten equation

1. Experimental progress curves were simulated for a reaction obeying Michaelis-Menten kinetics. 2. K(m) and V were estimated (a) by fitting the integrated Michaelis-Menten equation to the progress curves, and (b) from the initial slopes of the curves (i.e. from initial velocities). 3. The integrated equation could not be fitted successfully by a non-linear method, so it was transformed and fitted by a linear method. 4. Provided that the initial substrate concentration was greater than K(m) and the data were precise enough, the integrated equation gave parameter estimates which were unbiased and as reliable as those derived from initial velocities although based on fewer experiments. 5. The integrated equation could be used for progress curves of unknown origin.     

Analysis of the kinetic mechanism of halophilic NADP-dependent glutamate dehydrogenase

The amination of 2-oxoglutarate catalyzed by NADP-specific glutamate dehydrogenase (EC 1.4.1.4, L-glutamate:NADP+ oxidoreductase (deaminating)) from Halobacterium halobium has been analyzed by initial rate, graphical analysis, and product and competitive inhibition studies. Initial rate and graphical analysis reveal that a B term (representing 2-oxoglutarate) is not statistically necessary for an initial rate equation. However, the absence of a B term does not distinguish between ordered and random binding of NADPH and ammonia. The patterns of product inhibition by NADP+ and L-glutamate, and competitive inhibition by hydroxylamine and succinate permit deduction of the kinetic mechanism as ordered, with NADPH, 2-oxoglutarate and ammonia added in that order, and L-glutamate release preceding NADP+ release.     

Kinetic study of an enzyme-catalysed reaction in the presence of novel irreversible-type inhibitors that react with the product of enzymatic catalysis

In the present paper a kinetic study is made of the behaviour of a Michaelis-Menten enzyme-catalysed reaction in the presence of irreversible inhibitors rendered unstable in the medium by their reaction with the product of enzymatic catalysis. A general mechanism involving competitive, non-competitive, uncompetitive and mixed irreversible inhibition with one or two steps has been analysed. The differential equation that describes the kinetics of the reaction is non-linear and computer simulations of its dynamic behaviour are presented. The results obtained show that the systems studied here present kinetic co-operativity for a target enzyme that follows the simple Michaelis-Menten mechanism in its action on the substrate, except in the case of an uncompetitive-type inhibitor.     

A new multi-wavelength model-based method for determination of enzyme kinetic parameters

Lineweaver-Burk plot analysis is the most widely used method to determine enzyme kinetic parameters. In the spectrophotometric determination of enzyme activity using the Lineweaver-Burk plot, it is necessary to find a wavelength at which only the substrate or the product has absorbance without any spectroscopic interference of the other reaction components. Moreover, in this method, different initial concentrations of the substrate should be used to obtain the initial velocities required for Lineweaver-Burk plot analysis. In the present work, a multi-wavelength model-based method has been developed and validated to determine Michaelis-Menten constants for some enzyme reactions. In this method, a selective wavelength region and several experiments with different initial concentrations of the substrate are not required. The absorbance data of the kinetic assays are fitted by non-linear regression coupled to the numeric integration of the related differential equation. To indicate the applicability of the proposed method, the Michaelis-Menten constants for the oxidation of phenanthridine, 6-deoxypenciclovir and xanthine by molybdenum hydroxylases were determined using only a single initial concentration of the substrate, regardless of any spectral overlap.     

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.     

Improved non-parametric statistical methods for the estimation of Michaelis-Menten kinetic parameters by the direct linear plot.

The theoretical basis for the direct linear plot [Eisenthal & Cornish-Bowden (1974) Biochem. J. 139, 715-720], a non-parametric statistical method for the analysis of data-fitting the Michaelis-Menten equation, was reinvestigated in order to accommodate additional experimental designs and to provide estimates of precision more directly comparable with those obtained by parametric statistical methods. Methods are given for calculating upper and lower confidence limits for the estimated parameters, for accommodating replicate measurements and for comparing the results of two separate experiments. Factors that influence the proper design of experiments are discussed.     

A comparison of seven methods for fitting the Michaelis-Menten equation.

The Michaelis-Menten equation was fitted to simulated data containing different sorts of error by using the three linear transformations, and the methods of S. R. Cohen [Anal. Biochem. (1968) 22, 549-552], R. Eisenthal & A. Cornish-Bowden [Biochem. J. (1974) 139, 715-120], F. de M. Merino [Biochem. J. 143, 93-95] and G. N. Wilkinson [Biochem. J. (1961) 808 324-332). The best methods were those of Eisenthal & Cornish-Bowden (1974) and Wilkinson (1961).     

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.     

泉州湾河口湿地主要红树植物叶面积测算

植物叶面积的测算对于评价生态系统初级生产力具有重要意义.本研究分别选用“最大叶长”、“最大叶宽”以及“最大叶长×最大叶宽”等指标,利用不同类型的线性或非线性回归方程,对泉州湾河口湿地主要红树植物秋茄、桐花树和白骨壤的叶面积进行测算,从而确定各自最佳拟合回归方程.结果表明:二元非线性回归方程Y=0.7297X10.8698 X2.11600、幂指数方程Y=0.9740X0.9634和Y=0.7773X 0.9954分别为秋茄、桐花树和白骨壤叶面积的最佳拟合回归方程.进一步的0-1回归检验和相对误差值分析显示,以上回归方程均能精确地估算各自的叶面积,其中,白骨壤叶面积测算更为精确.     

Estimation of the initial velocity of enzyme-catalysed reactions by non-linear regression analysis of progress curves.

Most methods for studying the kinetic properties of an enzyme involve the determination of initial velocities. When the reaction progress curve shows significant curvature due to depletion of the substrate, accumulation of inhibitory products or instability of the enzyme, estimation of the initial velocity is a subjective and inexact process. Two methods have been suggested [Cornish-Bowden (1975) Biochem. J. 144, 305-312; Boeker (1982) Biochem J. 203, 117-123] that attempt to eliminate this subjective element. The present study offers a third alternative, which is based on fitting a reparameterized form of the integrated Michaelis-Menten equation to the progress curves by non-linear regression. This method yields estimates and standard errors of the initial velocity and of the time to reach 50% reaction. No prior knowledge of the apparent product concentration at zero time or infinite time is required, since both of these quantities are also estimated from the data. It is shown that this method yields reliable estimates of the initial velocity under a wide range of circumstances, including those where the two previously published methods perform poorly.     

Fatty acid synthetase. A steady state kinetic analysis of the reaction catalyzed by the enzyme from pigeon liver.

The kinetic mechanism of pigeon liver fatty acid synthetase action has been studied using steady state kinetic analysis. Initial velocity studies are consistent with an earlier suggestion that the enzyme catalyzes this reaction by a seven-site ping-pong mechanism. Although the range of substrate concentrations that could be used was limited by several factors, the initial velocity patterns showing the relationship between the substrates acetyl coenzyme CoA, malonyl-CoA, and NADPH appear to be a series of parallel lines, regardless of which substrate is varied at fixed levels of a second substrate. However, two of the substrates, acetyl-CoA and malonly-CoA, apparently exhibit a competitive substrate inhibition with respect to each other, but NADPH shows no inhibition of any kind. Product inhibition patterns suggest that free CoA is competitive versus acetyl-CoA and malonyl-CoA and is uncompetitive versus NADPH, and that NADP+ is competitive versus NADPH and uncompetitive versus acetyl-CoA or malonyl-CoA. These results are consistent with a seven-site ping-pong mechanism with intermediates covalently bound to 4'-phosphopantetheine (part of acyl carrier protein). Double competitive substrate inhibition by acetyl-CoA and malonyl-CoA is consistent with the rate equation derived for the over-all mechanism. The kinetic mechanism developed from these results is capable of explaining the formation of fatty acids from malonyl-CoA and NADPH alone (Katiyar, S. S., Briedis, A. V., and Porter, J. W. (1974) Arch. Biochem. Biophys. 162, 412-420) and also the formation of triacetic acid lactone from either malonyl-CoA alone or acetyl-CoA plus malonyl-CoA.     

Calculation of inhibitor Ki and inhibitor type from the concentration of inhibitor for 50% inhibition for Michaelis-Menten enzymes

The use of I50 (concentration of inhibitor required for 50% inhibition) for enzyme or drug studies has the disadvantage of not allowing easy comparison among data from different laboratories or under different substrate conditions. Modifications of the Michaelis-Menten equation for treatment of inhibitors can allow both the determination of the type of inhibition (competitive, noncompetitive, and uncompetitive) and the Ki for the inhibitor. For competitive and uncompetitive inhibitors when the assay conditions are [S] = Km, then Ki = I50/2. For different conditions of [S] there is a divergence between competitive and uncompetitive inhibitors that may be used to identify the type of inhibitor. The equation for Ki also differs. For noncompetitive inhibitors the Ki = I50 and this relationship is valid with changing [S]. The equations developed require a single substrate, reversible-type inhibitors, and kinetics of the Michaelis-Menten type. Examples of the use of the equations are illustrated with experimental data from scientific publications.     

A new graphical method for determining parameters in Michaelis-Menten-type kinetics for enzymatic lactose hydrolysis

A new graphical method was developed to determine the kinetic parameters in the Michaelis-Menten-type equation. This method was then applied to studying the kinetics of lactose hydrolysis by Aspergillus niger beta-galactosidase. In this study, the reaction temperature ranged between 8 and 60 degrees C, and the initial lactose concentration ranged between 2.5 and 20%. A kinetic model similar to the conventional Michaelis-Menten equation with competitive product inhibition by galactose was tested using this graphical method as well as a nonlinear computer regression method. The experimental data and the model fit together fairly well at 50 degrees C. However, a relative large disparity was found for reactions at 30 degrees C. A three-parameter integrated model derived from the reversible reaction mechanism simulates the experimental data very well at all temperatures studied. However, this reversible reaction model does not follow the Arrhenius temperature dependence. Nevertheless, reaction rate constants for the proposed model involving the enzyme-galactose complex (in addition to the Michaelis complex) as an intermediate in lactose hydrolysis follow the Arrhenius temperature dependence fairly well, suggesting that this model can be best used for describing the enzymatic lactose hydrolysis. The lack of fit between the model predictions and data may be largely attributed to the effects of galactose mutarotation and oligosaccharide formation during lactose hydrolysis.     

Optimization of maltodextrin hydrolysis by glucoamylase in a batch reactor

The hydrolysis of maltodextrins (10 DE) by glucoamylase was studied in a batch reactor at temperatures between 40 and 80 degrees C and substrate concentration range from 17 to 300 kg/m(-3). The experimental data were fitted to a model including thermal deactivation of the enzyme. In the model, the reaction rate was correlated with an extended Michaelis-Menten equation including inhibition by product, and the thermal deactivation of glucoamylase was fitted with a first-order reaction. The dependence of rate parameters on temperature was correlated using the Arrhenius equation. The differential equation of the model was integrated and the optimal enzyme demand and temperature were determined for isothermal operation.     

Evidence concerning a possible steady state rate equation for E. coli alkaline phosphatase

• 1.1. Steady state data was obtained for alkaline phosphatase over a wide range of experimental conditions using two substrates, four inhibitors, two modifiers and several pH, ionic strength and temperatures values.
• 2.2. The data was fitted by rational functions of degree 1:1, 2:2 and 3:3 using a non-linear regression program and then the F-test was used to assess the goodness of fit.
• 3.3. A proportion of the curves could only be fitted by 2:2 functions but many of them could be adequately fitted by 1:1 functions.
• 4.4. No statistically significant improvement in fit occurred with 3:3 functions.
• 5.5. Data was simulated using a computer program to see what sort of curves could be generated by a two sites mechanism proposed for alkaline phosphatase and this study showed that it is difficult to detect cubic terms in this rate equation.
• 6.6. It was concluded that alkaline phosphatase does not obey Michaelis-Menten kinetics. Rather, the steady state data require a mechanism of at least second degree but do not exclude a rate equation of third degree.

     

Predictive modelling approach applied to spoilage fungi: growth of Penicillium brevicompactum on solid media

Growth of Penicillium brevicompactum was examined on five solid media. Fungal growth was established by diameter measurements up to 50 days. Seventy experimental curves were fitted by Baranyi's primary predictive model. The growth rates were then analysed by non-parametric statistical methods. Penicillium brevicompactum could colonize the surface of solid media containing up to 700 g l-1 of sugar (50% glucose-50% fructose) with a growth rate of 0.9 mm day-1 (median values). Fitting curves by non-linear models followed by a non-parametric multiple comparison seems to be a convenient method for detecting differences in fungal growth on solid media. These two methods would be useful for studying fungal spoilage of bakery products with intermediate water activity.