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.     

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).     

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).     

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.     

The comparison of the estimation of enzyme kinetic parameters by fitting reaction curve to the integrated Michaelis-Menten rate equations of different predictor variables

The estimation of enzyme kinetic parameters by nonlinear fitting reaction curve to the integrated Michaelis-Menten rate equation ln(S(0)/S)+(S(0)-S)/K(m)=(V(m)/K(m))xt was investigated and compared to that by fitting to (S(0)-S)/t=V(m)-K(m)x[ln(S(0)/S)/t] (Atkins GL, Nimmo IA. The reliability of Michaelis-Menten constants and maximum velocities estimated by using the integrated Michaelis-Menten equation. Biochem J 1973;135:779-84) with uricase as the model. Uricase reaction curve was simulated with random absorbance error of 0.001 at 0.075 mmol/l uric acid. Experimental reaction curve was monitored by absorbance at 293 nm. For both CV and deviation <20% by simulation, K(m) from 5 to 100 micromol/l was estimated with Eq. (1) while K(m) from 5 to 50 micromol/l was estimated with Eq. (2). The background absorbance and the error in the lag time of steady-state reaction resulted in negative K(m) with Eq. (2), but did not affect K(m) estimated with Eq. (1). Both equations gave better estimation of V(m). The computation time and the goodness of fit with Eq. (1) were 40-fold greater than those with Eq. (2). By experimentation, Eq. (1) yielded K(m) consistent with the Lineweaver-Burk plot analysis, but Eq. (2) gave many negative parameters. Apparent K(m) by Eq. (1) linearly increased, while V(m) were constant, vs. xanthine concentrations, and the inhibition constant was consistent with the Lineweaver-Burk plot analysis. These results suggested that the integrated rate equation that uses the predictor variable of reaction time was reliable for the estimation of enzyme kinetic parameters and applicable for the characterization of enzyme inhibitors.     

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.     

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

We have investigated the rapid phosphorylation of proteins in B-lymphocytes incubated with the tumour-promoting phorbol ester 12-O-tetradecanoylphorbol 13-acetate (TPA), anti-Ig and combinations of TPA and the Ca2+ ionophore ionomycin. Two-dimensional electrophoretic analysis was used to identify the proteins phosphorylated in cells preincubated with [32P]Pi. TPA induced a characteristic pattern of labelled proteins, four of which (pp85, pp76, pp66 and pp63) showed a dose-dependent incorporation of 32P on serine residues. The phosphorylation of pp63 and pp66, in particular, correlated with the mitogenic dose-response curve. Addition of the Ca2+ ionophore ionomycin to B-cells also stimulated a characteristic incorporation of 32P into proteins, which included pp63 and pp66. With combined doses of TPA and ionomycin, these two proteins show an enhanced phosphorylation, which correlated well with the synergistic enhancement of proliferation shown by this combination of agents. Protein kinase C (PKC) was partially purified from B-cells and separated into alpha and beta subtypes. The activation of both PKCs was assessed with increasing doses of TPA and concentrations of Ca2+ of 0.1 microM and 2 microM. For both forms of PKC, in particular the beta form, higher concentrations of Ca2+ shifted the dose-response curve for TPA to the left and increased the maximum activation. Anti-Ig, which stimulated B-cells by cross-linking surface immunoglobulin and causing hydrolysis of PtdIns(4,5)P2, also caused increased phosphorylation of several proteins, which again included pp63 and pp66. These data suggest that PKC, particularly the beta form, is involved in the early part of the proliferation cascade for human B-lymphocytes. It is most probably activated in a synergistic manner by the increased Ca2+ and diacylglycerol levels which result from the earlier hydrolysis of PtdIns(4,5)P2.     

FitSpace Explorer: An algorithm to evaluate multidimensional parameter space in fitting kinetic data

Fitting several sets of kinetic data directly to a model based on numerical integration provides the best method to extract kinetic parameters without relying on the simplifying assumptions required to achieve analytical solutions of rate equations. However, modern computer programs make it too easy to enter an overly complex model, and standard error analysis grossly underestimates errors when a system is underconstrained and fails to reveal the full degree to which multiple parameters are linked through the complex relationships common in kinetic data. Here we describe the application of confidence contour analysis obtained by measuring the dependence of the sum square error on each pair of parameters while allowing all remaining parameters to be adjusted in seeking the best fit. The confidence contours reveal complex relationships between parameters and clearly outline the space over which parameters can vary (the “FitSpace”). The utility of the method is illustrated by examples of well-constrained fits to published data on tryptophan synthase and the kinetics of oligonucleotide binding to a ribozyme. In contrast, analysis of alanine racemase clearly refutes claims that global analysis of progress curves can be used to extract the free energy profiles of enzyme-catalyzed reactions.     

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.     

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 computer analysis of the validity of the integrated Michaelis-Menten equation

The validity of assumptions made in integrating the Michaelis-Menten equation (i.e. the steady-state assumption) was examined by simulating the model on the digital computer. Time courses thus obtained by numerical integration were compared with data generated by the three most common forms of the integrated equation. Agreement was good within specified limits, the chief exception being the early transient phase of the reaction. The observed differences were very much less than theoretical estimates of the maximum error, even when rate constants were chosen that should exaggerate that error.     

Application of jackknife procedures to inter-experiment comparisons of parameter estimates for the Michaelis-Menten equation.

The jackknife procedure is introduced as a means of making comparisons among Michaelis-Menten parameter estimates for six different experimental conditions. In addition to providing a solution to the general inter-experimental comparison problem, the jackknife procedure will provide valid parameter estimates even when some of the assumptions usually required for statistical analysis are violated, e.g., the random errors are not normally distributed and the variances are not homogeneous. Other recent variations of the jackknife have also been introduced and briefly investigated: (i) the linear jackknife, which is more efficient computationally, and (ii) the weighted jackknife, which reduces the influence of design points (substrate concentrations) that have an excessive influence on the precision of parameter estimates.     

Exact and user-friendly kinetic analysis of the two-step rapid equilibrium Michaelis-Menten mechanism

Most enzyme kinetic experiments are carried out under pseudo-first-order conditions, that is, when one of the reactant species (the enzyme or the substrate) is in a large excess of the other species. More accurate kinetic information about the system can be gained without the restrictions of the pseudo-first-order conditions. We present a practical and general method of analysis of the common two-step rapid equilibrium Michaelis-Menten mechanism. The formalism is exact in that it does not involve any other approximations such as the steady-state, limitations on the reactant concentrations or on reaction times. We apply this method to the global analysis of kinetic progress curves for bovine alkaline phosphatase assays carried out under both pseudo-first-order and pseudo-second-order conditions.     

The fitting of dynamic models to observed data

The question of how to fit a general cubic model of a multicomponent, interactive growth system to observed data is addressed. A multidimensional-polynomial type of regression analysis is used, with a least-squares criterion. By testing the scheme on a problem with known solution, the way in which the accuracy of the results varies with the number of datum points used is investigated in an heuristic manner.     

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.