MIR: a versatile program for the statistical analysis of enzyme kinetic data

We have developed a package program for the estimation of Michaelis-Menten parameters for enzymes that conform to different kinetic mechanisms. Data from different experimental schemes can be fitted with appropriate weighing factors to any of 6 mathematical models, corresponding to 5 kinetic mechanisms: ordered bi-bi, Theorell-Chance, rapid equilibrium random bi-bi, rapid equilibrium ordered bi-bi and ping pong bi-bi. The program also performs a significance test to discriminate between different candidate models. To illustrate the performance of the program, real data from kinetic experiments with glucose 6-phosphate from Leuconostoc mesenteroides have been fitted to different mathematical models, and the results are discussed. The program can be easily implemented for the fitting of kinetic data to any other model.     

The elementary mass action rate constants of P-gp transport for a confluent monolayer of MDCKII-hMDR1 cells

The human multi-drug resistance membrane transporter, P-glycoprotein, or P-gp, has been extensively studied due to its importance to human health and disease. Thus far, the kinetic analysis of P-gp transport has been limited to steady-state Michaelis-Menten approaches or to compartmental models, neither of which can prove molecular mechanisms. Determination of the elementary kinetic rate constants of transport will be essential to understanding how P-gp works. The experimental system we use is a confluent monolayer of MDCKII-hMDR1 cells that overexpress P-gp. It is a physiologically relevant model system, and transport is measured without biochemical manipulations of P-gp. The Michaelis-Menten mass action reaction is used to model P-gp transport. Without imposing the steady-state assumptions, this reaction depends upon several parameters that must be simultaneously fitted. An exhaustive fitting of transport data to find all possible parameter vectors that best fit the data was accomplished with a reasonable computation time using a hierarchical algorithm. For three P-gp substrates (amprenavir, loperamide, and quinidine), we have successfully fitted the elementary rate constants, i.e., drug association to P-gp from the apical membrane inner monolayer, drug dissociation back into the apical membrane inner monolayer, and drug efflux from P-gp into the apical chamber, as well as the density of efflux active P-gp. All three drugs had overlapping ranges for the efflux active P-gp, which was a benchmark for the validity of the fitting process. One novel finding was that the association to P-gp appears to be rate-limited solely by drug lateral diffusion within the inner monolayer of the plasma membrane for all three drugs. This would be expected if P-gp structure were open to the lipids of the apical membrane inner monolayer, as has been suggested by recent structural studies. The fitted kinetic parameters show how P-gp efflux of a wide range of xenobiotics has been maximized.     

Fitting of enzyme kinetic data without prior knowledge of weights.

A method is described for fitting equations to enzyme kinetic data that requires minimal assumptions about the error structure of the data. The dependence of the variances on the velocities is not assumed, but is deduced from internal evidence in the data. The effect of very bad observations ('outliers') is mitigated by decreasing the weight of observations that give large deviations from the fitted equation. The method works well in a wide range of circumstances when applied to the Michaelis-Menten equation, but it is not limited to this equation. It can be applied to most of the equations in common use for the analysis of steady-state enzyme kinetics. It has been implemented as a computer program that can fit a wide variety of equations with two, three or four parameters and two or three variables.     

A model of the sodium dependence of dopamine uptake in rat striatal synaptosomes

Initial velocity of uptake of dopamine (DA) has been measured in rat striatal synaptosomes as a function of both [DA] and [Na]. Carrier mediated uptake is totally dependent on external sodium. The data were fitted to a rapid equilibrium model which has been found in previous studies to fit, with appropriate simplification, uptake data for glutamate, GABA, and choline in several brain regions under varying conditions. This model also gives a good fit to the dopamine data. The minimal best fit simplification of this model allows for DA uptake along with two sodium ions and predicts that apparent maximal velocity of uptake should increase with [Na], while the Michaelis-Menten constant should decrease. The minimal best fit model for DA, and a number of kinetic parameters which quantitate the model, are compared to those for the GABA, glutamate, and choline transporters. The results are consistent with a symmetrical, rapid equilibrium model, which has been presented previously for other neurotransmitters and precursors (18). This model offers a unifying basis for understanding the sodium and membrane potential dependence of neurotransmitter transport and the possible participation of transporters in depolarization induced release throughout the CNS.     

Simultaneous estimation ofV max,Km, and the rate of endogenous substrate production (R) from substrate depletion data

The nonlinear and 3 linearized forms of the integrated Michaelis-Menten equation were evaluated for their ability to provide reliable estimates of uptake kinetic parameters, when the initial substrate concentration (S₀) is not error-free. Of the 3 linearized forms, the one where t/(S₀–S) is regressed against ln(S₀/S)/(S₀–S) gave estimates ofV _max and K_m closest to the true population means of these parameters. Further, this linearization was the least sensitive of the 3 to errors (±1%) in S₀. Our results illustrate the danger of relying on r² values for choosing among the 3 linearized forms of the integrated Michaelis-Menten equation. Nonlinear regression analysis of progress curve data, when S₀ is not free of error, was superior to even the best of the 3 linearized forms. The integrated Michaelis-Menten equation should not be used to estimateV _max and K_m when substrate production occurs concomitant with consumption of added substrate. We propose the use of a new equation for estimation of these parameters along with a parameter describing endogenous substrate production (R) for kinetic studies done with samples from natural habitats, in which the substrate of interest is an intermediate. The application of this new equation was illustrated for both simulated data and previously obtained H₂ depletion data. The only means by whichV _max, K_m, and R may be evaluated from progress curve data using this new equation is via nonlinear regression, since a linearized form of this equation could not be derived. Mathematical components of computer programs written for fitting data to either of the above nonlinear models using nonlinear least squares analysis are presented.     

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.     

Kinetic analysis of enzyme systems with suicide substrate in the presence of a reversible, uncompetitive inhibitor.

We present a general kinetic analysis of enzyme catalyzed reactions evolving according to a Michaelis-Menten mechanism, in which an uncompetitive, reversible inhibitor acts. Simultaneously, enzyme inactivation is induced by an unstable suicide substrate, i.e. it is a Michaelis-Menten mechanism with double inhibition: one originating from the substrate and another originating from the reversible inhibitor. Rapid equilibrium of the reversible reaction steps involved is assumed and the time course equations for the reaction product have been derived under the assumption of limiting enzyme. The goodness of the analytical solutions has been tested by comparison with simulated curves obtained by numerical integration. A kinetic data analysis to determine the corresponding kinetic parameters from the time progress curve of the product is suggested.     

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.     

Locating the rate-determining step(s) for three-step hydrolase-catalyzed reactions with DYNAFIT

Hydrolytic reactions of oligopeptide 4-nitroanilides catalyzed by human-alpha-thrombin, human activated protein C and human factor Xa were studied at pH 8.0-8.4 and 25.0+/-0.1 degrees C by the progress curve method and individual rate constants were calculated mostly within 10% internal error using DYNAFITV. A systematic strategy has been developed for fitting a three-step consecutive mechanism to eighteen hundred to six thousand time-course data points polled from two to four independent kinetic experiments. Enzyme and substrate concentrations were also calculated. Individual rate constants well reproduce published values obtained under comparable conditions and the Michaelis-Menten kinetic parameters calculated from these elementary rate constants are also within reasonable limits of published values. For comparison, the integrated Michaelis-Menten equation was also fitted to data from twelve sets. Both the k(cat) and k(cat)/K(m) values are within 15% agreement with those calculated using the elementary rate constants obtained with DYNAFITV. Rate constants for the second and third consecutive steps are within 3-4 fold indicating that both determine the overall rate. The Factor Xa-catalyzed hydrolysis of N-alpha-Z-D-Arg-Gly-Arg-pNA.2HCl at pH 8.4 in a series of buffers containing increasing fractions of deuterium at 25.0+/-0.1 degrees C shows a very strong dependence of k(3) and a moderate dependence of k(2) on D content in the buffer: the fractionation factors are: 0.49+/-0.03 for K(1,) 0.70+/-0.05 for k(2), and (0.32+/-0.03)(2) for k(3).     

A comparison of methods for estimating the kinetic parameters of two simple types of transport process

Sets of experimental data, with known characteristics and error structures, have been simulated for the Michaelis-Menten equation plus a second term, either for linear transport or for competitive inhibition. The Michaelis-Menten equation plus linear term was fitted by several methods and the accuracy and the precision of the parameter estimates from the several methods were compared. The model-fitting methods were: three for least-squares non-linear regression, computer versions of two graphical methods and of two non-parametric methods. The most precise and accurate method was that of D.W. Marquardt (J. Soc. Ind. Appl. Math. 11 (1963) 431–441). The Michaelis-Menten equation with competitive inhibition was also fitted by several methods, viz., two for least-squared non-linear regression, a non-parametric method and four variants of the Preston-Schaeffer-Curran plot (Preston, R.L. et al. (1974) J. Gen. Physiol. 64, 443–467). The most precise and accurate of these was the non-linear regression method of W.W. Cleland (Adv. Enzymol. 29 (1967) 1–32). For both these models, the various graphical methods and non-parametric methods gave poor results and are not recommended.     

A Model for Single-Substrate Trimolecular Enzymatic Kinetics

We developed a kinetic model for a single-substrate trimolecular enzymatic system, where a receptor binds and stretches a substrate to expose its cleavage site, allowing an enzyme to bind and cleave it into product. We demonstrated that the general kinetics of the trimolecular enzymatic system is more complex than the Michaelis-Menten kinetics. Under a limiting condition when the enzyme-substrate binding is in fast equilibrium, the enzymatic kinetics of the trimolecular system reduces to the Michaelis-Menten kinetics. In another limiting case when the receptor dissociates negligibly slowly from the substrate, the trimolecular system is simplified to a bimolecular system, which follows the Michaelis-Menten equation if and only if there is no enzyme-substrate complex initially. We applied this model to a particular trimolecular system important to hemostasis and thrombosis, consisting of von Willebrand factor (substrate), platelet glycoprotein Ibα (receptor), and ADAMTS13 (enzyme). Using parameters from independent experiments, our model successfully predicted published data from two single-molecule experiments and fitted/predicted published data from an ensemble experiment.     

gamma-Glutamylcyclotransferase: inhibition by D-beta-aminoglutaryl-L-alanine and analysis of the solvent kinetic isotope effect

Spin-echo NMR spectroscopy was used to record the cleavage of a gamma-glutamyl--amino-acid by (5-L-glutamyl)-L-amino-acid 5-glutamyltransferase (cyclizing) (gamma-glutamylcyclotransferase) in human erythrocyte hemolysates. The Michaelis-Menten steady-state kinetic parameters were obtained by fitting the integrated Michaelis-Menten equation to the reaction time curves. The product, L-5-oxoproline, was shown to be an inhibitor of the reaction. The active site of the enzyme was probed by studies of the inhibition by D- and L-beta-aminoglutaryl-L-alanine which are the beta-amino-acid isomers of D- and L-gamma-glutamyl-L-alanine (the latter being a natural substrate of the enzyme); the D-isomer was the more potent inhibitor (Ki = 0.30 +/- 0.02 mmol/l water). When the alanyl alpha-carboxyl of the inhibitor was reduced to a hydroxyl (i.e. to give D-beta-aminoglutaryl-L-alaninol) the potency of inhibition was reduced. The previously reported kinetic isotope effect of solvent 2H2O on the enzyme-catalyzed reaction has been further studied using a proton inventory. We propose that the solvent kinetic isotope effect is due to an intramolecular proton transfer between the glutamyl amino group and the peptide bond nitrogen.     

Statistical considerations in the estimation of enzyme kinetic parameters by the direct linear plot and other methods

The statistical implications of the direct linear plot for enzyme kinetic data, described in the preceding paper (Eisenthal & Cornish-Bowden, 1974), are discussed for the case of the Michaelis-Menten equation. The plot is shown to lead directly to non-parametric confidence limits for the kinetic parameters, V and K(m), which depend on far less sweeping assumptions about the nature of experimental error than those implicit in the method of least squares. Median estimates of V and K(m) can also be defined, which are shown to be more robust than the least-squares estimates in a wide variety of experimental situations.     

Analysis of covariance with incomplete data via semiparametric model transformations

We propose a method for fitting semiparametric models such as the proportional hazards (PH), additive risks (AR), and proportional odds (PO) models. Each of these semiparametric models implies that some transformation of the conditional cumulative hazard function (at each t) depends linearly on the covariates. The proposed method is based on nonparametric estimation of the conditional cumulative hazard function, forming a weighted average over a range of t-values, and subsequent use of least squares to estimate the parameters suggested by each model. An approximation to the optimal weight function is given. This allows semiparametric models to be fitted even in incomplete data cases where the partial likelihood fails (e.g., left censoring, right truncation). However, the main advantage of this method rests in the fact that neither the interpretation of the parameters nor the validity of the analysis depend on the appropriateness of the PH or any of the other semiparametric models. In fact, we propose an integrated method for data analysis where the role of the various semiparametric models is to suggest the best fitting transformation. A single continuous covariate and several categorical covariates (factors) are allowed. Simulation studies indicate that the test statistics and confidence intervals have good small-sample performance. A real data set is analyzed.     

Use of an oxygen microsensor for the determination of intrinsic kinetic parameters of an immobilized oxygen reducing enzyme

An oxygen microsensor was used to measure internal oxygen profiles in biocatalyst particles of different diameter and activity. The particles were made of agarose gel and contained an oxygen reducing enzyme, L-lactate mono-oxygenase. The kinetics of the enzyme could be well described by the Michaelis-Menten equation. From the internal substrate concentration profile the intrinsic kinetic parameters were determined by means of fitting a simulated profile to the measurements, using Marquardt's algorithm. The intrinsic kinetic parameters found following this procedure appeared to be independent of particle radius or enzyme loading used, proving the method to be reliable. These parameters were also compared with the kinetic parameters of the free enzyme which were determined in a biological oxygen monitoring system. The intrinsic kinetic parameters showed a decrease with a factor 2.3 for V(m) value and with a factor 2.7 for the K(m) value compared to the parameters for the free enzyme. From this the conclusion can be drawn that the immobilization as such or the carrier material not only can have an effect on the maximum intrinsic conversion rate (V(m)) but also on the affinity of the enzyme (K(m)) for oxygen.     

The use of model-fitting in the interpretation of 'dual' uptake isotherms

Abstract. Published data of the concentration dependence of the uptake rate (uptake isotherms) of K⁺ Na⁺, Cl^?, SO2^?₄, and L-lysine in barley roots, and glucose and 3-O-methylglucose in potato tuber tissue, were re-examined. In as much as these isotherms yield non-linear, concave upward Eadie-Hofstee plots, they might have been termed ‘dual’ isotherms. In addition, all these isotherms have been considered to display discontinuous transitions in gradient. The following models that yield continuous isotherms were fitted to the isotherms: (1) the sum of a Michaelis-Menten term and a linear term; (2) the sum of two Michaelis-Menten terms; (3) the sum of two Michaelis-Menten terms and a linear one. Goodness of fit was judged from: (i) the weighted mean square of deviates; (ii) the standard errors of the kinetic parameters; (iii) the algebraic significance of the terms; (iv) a Rankits plot of the residuals; (v) a Runs test on the residuals. For the precise and detailed isotherms of SO^2? uptake, only model (3) gave a fit that was satisfactory in all respects. There appeared to be no reason to consider these isotherms as multiphasic. The same conclusion was reached for the L-lysine uptake isotherms. For the other isotherms the results were less conclusive. Thai for K⁺ and Na+ could, at any rate, be described satisfactorily by a continuous model, the best fit being obtained with model (2). The uptake isotherms of Cl^? and 3-0-methylglucose could be best described by model (2), and that of glucose by model (3), only the result of the Runs test being unsatisfactory. It is concluded that there is hardly any evidence that the presumed ‘jumps’ or discontinuities or inflections in the gradient of uptake isotherms are not due to experimental error in the data. It is suggested that many uptake isotherms may be described by model (3), although the reason for this is still incompletely understood.     

Enzymatic hydrolysis of soluble starch with an alpha-amylase from Bacillus licheniformis

The enzymatic hydrolysis of soluble starch with an alpha-amylase from Bacillus licheniformis (commercial enzyme Termamyl 300 L Type DX) have been experimentally studied at pH 7.5, within the temperature range of 37-75 degrees C, at initial substrate concentrations of between 0.25 and 2.00 g/L, and enzyme concentrations of between 0.575 x 10(-4) and 13.8 x 10(-4) g/L. To follow the reaction a procedure based on the iodometric method for measuring alpha-amylase activity was used. The kinetics of the enzymatic hydrolysis was fitted to the Michaelis-Menten equation using the integral method, taking into account that the thermal deactivation of the enzyme follows a second-order kinetic. These parameters were fitted to the Arrhenius equation obtaining activation energies of 24.4 and 41.7 kJ/mol and preexponential factors of 734.9 g/L and 1.74 x 10(8) min(-1) for K(M) and k, respectively.     

Real time kinetics of restriction endonuclease cleavage monitored by fluorescence resonance energy transfer.

The kinetics of PaeR7 endonuclease-catalysed cleavage reactions of fluorophor-labeled oligonucleotide substrates have been examined using fluorescence resonance energy transfer (FRET). A series of duplex substrates were synthesized with an internal CTCGAG PaeR7 recognition site and donor (fluorescein) and acceptor (rhodamine) dyes conjugated to the opposing 5' termini. The time-dependent increase in donor fluorescence resulting from restriction cleavage of these substrates was continuously monitored and the initial rate data was fitted to the Michaelis-Menten equation. The steady state kinetic parameters for these substrates were in agreement with the rate constants obtained from a gel electrophoresis-based fixed time point assay using radiolabeled substrates. The FRET method provides a rapid continuous assay as well as high sensitivity and reproducibility. These features should make the technique useful for the study of DNA-cleaving enzymes.     

A transport kinetic concept for ion uptake by plants

Summary A previously presented transport kinetic model for ion uptake by plants was tested by using data for copper uptake by barley grown in soil or water culture and corresponding data for copper concentration in the soil solution or culture solution. As no significant differences were obtained between experimentally determined values for copper uptake and those calculated by using the model, it was concluded that the experimental evidence was consistent with the proposed transport kinetic model. The model allows the Michaelis-Menten Constant of ion uptake and the Mean Maximal Rate of ion uptake to be calculated for various time intervals of the growth period of plants. The Michaelis-Menten Constant for copper uptake by barley grown in soil was the same as that for plants grown in water culture, provided that the soil solution and the water culture solution had the same chemical composition. The concentration of copper in the soil solution, at which the rate of copper uptake was zero, was 2–4 times larger than corresponding values obtained in water culture solution. re]19751028