The integrated Michaelis-Menten equation

A critical consideration of the assumptions involved in the steady-state treatment of enzyme kinetics showed that these assumptions are only partially correct. A new theory is developed, from which it follows that an equation of the type of the integrated Michaelis-Menten equation can describe an enzyme-catalyzed reaction without assuming a steady state. This implies that such a reaction is determined by all reaction constants involved, which can be determined by analog computation.     

Statistical analysis of the Michaelis-Menten equation

An application of the method of maximum likelihood (ML) is described for analysing the results of enzyme kinetic experiments in which the Michaelis-Menten equation is obeyed. Accurate approximate solutions to the ML equations for the parameter estimates are presented for the case in which the experimental errors are of constant relative magnitude. Formulae are derived that approximate the standard errors of these estimates. The estimators are shown to be asymptotically unbiased and the standard errors observed in simulated data rapidly approach the theoretical lower bound as the sample size increases. The results of a large-scale Monte Carlo simulation study indicate that for data with a constant coefficient of variation, the present method is superior to other published methods, including the conventional transformations to linearity and the nonparametric technique proposed by Eisenthal and Cornish-Bowden (1974, Biochemical Journal 139, 715-720). Finally, the present results are extended to the analysis of simple receptor binding experiments using the general approach described by Munson and Rodbard (1980, Analytical Biochemistry 107, 220-239).     

Explicit analytic approximations for time-dependent solutions of the generalized integrated Michaelis-Menten equation

A single nucleotide polymorphism (SNP) is a common genetic variation when a single nucleotide differs between members of a species or paired chromosome. Due to its association with disease susceptibility and drug resistance, SNP detection is of great value in studying the variation in drug responses. Here we present two quantitative SNP detection methods for a single-base mismatch in RNA, based on nick-joining and nick-generating activities of T4 RNA ligase and DNAzyme, respectively. T4 RNA ligase successfully discriminated a one-base mismatch in the ligation junction, and the designed DNAzyme cleaved RNA by discerning a single-base mismatch in the cleaving site.     

Parametric analysis of the errors associated with the Michaelis-Menten equation

Based on the well-known mechanism describing Michaelis-Menten kinetics, three rate expressions may be developed: the exact solution (Model 1), a rate equation resulting from the pseudo-steady-state assumption (Model 2), and Model 2 with the additional assumption that the amount of free substrate is approximately equal to the total amount of substrate (Model 3). Although Model 1 is the most precise, it must be integrated numerically and it requires three experimentally determined parameters. Models 2 and 3, however, are simpler and require only two parameters. Using dimensionless forms of the three models, we have evaluated the errors in the two simplified models relative to the exact solution using a wide range of parameter values. The choice of model for reactor design depends on the initial substrate to enzyme ratio (alpha(0)), and on the ratio of the Michaelis-Menten constant to the enzyme concentration (sigma). Based on a 2% model error criteria, when alpha(0) > 15 or sigma >/= 100, Model 3 is adequate; if 5 < alpha(0) < 15, or if sigma >/= 10, then Model 2 may be used; and if alpha(0) < 5 and sigma < 10, then the exact solution (Model 1) is required.     

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).     

Comparison of several non-linear-regression methods for fitting the Michaelis-Menten equation.

A multicatalytic proteinase from rat skeletal muscle contains active site(s) catalysing the degradation of benzoyl-Val-Gly-Arg 4-methyl-7-coumarylamide, succinyl-Ala-Ala-Phe 4-methylcoumarylamide and [14C]methylcasein as well as benzyloxy-carbonyl-Leu-Leu-Glu 2-naphthylamide. These activities are 7-14-fold activated by 1 mM-sodium dodecyl sulphate. The activation leads to a higher susceptibility to the proteinase inhibitor chymostatin and to a lower ability to be inhibited and precipitated by antibodies raised against the non-activated enzyme. Since no changes in Mr or subunit composition were observed in the SDS-activated form, some conformational changes seem to occur during the activation step. More pronounced activation was observed in the presence of physiological concentrations of fatty acids; oleic acid at 100 microM concentrations stimulated the proteinase about 50-fold. In contrast with the non-activated proteinase, the activated enzyme considerably degrades muscle cytoplasmic proteins in vitro. Thus it is not unlikely that, in vivo, potential activators such as fatty acids can induce the multicatalytic proteinase to participate in muscle protein breakdown.     

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.     

Asymptotic analysis of the Michaelis-Menten reaction equations

The irreversible Michaelis-Menten reaction is studied by the use of the method of multiple scales. Three stages of the reaction are identified, one of which is studied in detail. The results are compared with those of two earlier analyses.     

A semi-integrated method for the determination of enzyme kinetic parameters and graphical representation of the Michaelis-Menten equation

A semi-integrated method for the determination of the enzyme kinetics parameters (Km and V) and graphical representation of the Michaelis-Menten equation is proposed as a variation of determination of initial reaction rate (v) as a function of initial substrate concentration ([S]0). The method is based on the determination of the time required to exhaust half of the initial substrate concentration as a function of the initial substrate concentration. The advantages and limitations of this method are discussed.     

Michaelis-Menten equation for an enzyme in an oscillating electric field.

The electric charges on an enzyme may move concomitantly with a conformational change. Such an enzyme will absorb energy from an oscillating electric field. If in addition the enzyme has a larger association constant for substrate than for product, as is often true, it can use this energy to drive the catalyzed reaction away from equilibrium. Approximate analytical expressions are given for the field-driven flux, electrical power absorbed, free-energy produced per unit time, thermodynamic efficiency, and zero-flux concentrations. The field-driven flux is written as a generalized Michaelis-Menten equation.     

Applicability of the integral form of the Michaelis-Menten equation for the kinetic study of transport ATPases

Applicability of the integrated form of the Michaelis-Menten equation to kinetic analysis of transport ATPases has been shown during continuous pH-metric recording of their activity. Two values of Km for both Na, K-ATPase and Ca-ATPase have been found to be consistent with the reported data. Both values of Km for Na, K-ATPase change with temperature, i. e. at 37 degrees, 26 degrees and 15 degrees C they are as follows: Km1--21, Km2--171; Km1--3.32, Km2--47; and Km1--1,2, Km2--20 microM, respectively. This method of determination of Km and V for transport ATPases compares favourably with the previously used methods in resulting efficiency.     

A computer program for the implicit regression of the Hill equation]

A computer program for implicit regression of the Hill equation. The fundamentals for the regression of implicit functions following the Gauss method are dealt with. Basing on these fundamentals a program for the regression of the Hill equation is described. The binding constant, the Hill coefficient and end extinction can be estimated. The mean errors of parameters were calculated in the linear model. The program was written in ALGOL 60 for the computer Robotron 300. The program is available on request.