Kinetic analysis of enzyme systems with suicide substrate in the presence of a reversible competitive inhibitor, tested by simulated progress curves

The use of suicide substrates remains a very important and useful method in enzymology for studying enzyme mechanisms and designing potential drugs. Suicide substrates act as modified substrates for the target enzymes and bind to the active site. Therefore the presence of a competitive reversible inhibitor decreases the rate of substrate-induced inactivation and protects the enzyme from this inactivation. This lowering on the inactivation rate has evident physiological advantages, since it allows the easy acquisition of experimental data and facilitates kinetic data analysis by providing another variable (inhibitor concentration). However despite the importance of the simultaneous action of a suicide substrate and a competitive reversible inhibition, to date no corresponding kinetic analysis has been carried out. Therefore we present a general kinetic analysis of a Michaelis-Menten reaction mechanism with double inhibition caused by both, a suicide substrate and a competitive reversible inhibitor. We assume rapid equilibrium of the reversible reaction steps involved, while the time course equations for the reaction product have been derived with the assumption of a limiting enzyme. The goodness of the analytical solutions has been tested by comparison with the 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. In conclusion, we present a complete kinetic analysis of an enzyme reaction mechanism as described above in an attempt to fill a gap in the theoretical treatment of this type of system.     

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.     

Kinetic analysis of enzyme reactions with slow-binding inhibition.

In this paper we present a general kinetic study of slow-binding inhibition processes, i.e. enzyme reactions that do not respond instantly to the presence of a competitive inhibitor. The analysis that we present is based on the equation that describes the formation of products with time in each case on the experimental progress curve. It is carried out under the condition of limiting enzyme concentration and allows the discrimination between the different cases of slow-binding inhibition. The mechanism in which the formation of complex enzyme-inhibitor is a single or two slow steps or follow a rapid equilibrium, has been considered. The corresponding explicit equations of each case have been obtained and checked by numerical integration. A kinetic data analysis to evaluate the corresponding kinetic parameters is suggested. We illustrate the method, numerically by computer simulation, of the reaction and present some numerical examples that demonstrate the applicability of our procedure.     

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.     

The origins of enzyme kinetics

The equation commonly called the Michaelis–Menten equation is sometimes attributed to other authors. However, although Victor Henri had derived the equation from the correct mechanism, and Adrian Brown before him had proposed the idea of enzyme saturation, it was Leonor Michaelis and Maud Menten who showed that this mechanism could also be deduced on the basis of an experimental approach that paid proper attention to pH and spontaneous changes in the product after formation in the enzyme-catalysed reaction. By using initial rates of reaction they avoided the complications due to substrate depletion, product accumulation and progressive inactivation of the enzyme that had made attempts to analyse complete time courses very difficult. Their methodology has remained the standard approach to steady-state enzyme kinetics ever since.     

Progress curve analysis for enzyme and microbial kinetic reactions using explicit solutions based on the Lambert W function

We present a simple method for estimating kinetic parameters from progress curve analysis of biologically catalyzed reactions that reduce to forms analogous to the Michaelis-Menten equation. Specifically, the Lambert W function is used to obtain explicit, closed-form solutions to differential rate expressions that describe the dynamics of substrate depletion. The explicit nature of the new solutions greatly simplifies nonlinear estimation of the kinetic parameters since numerical techniques such as the Runge-Kutta and Newton-Raphson methods used to solve the differential and integral forms of the kinetic equations, respectively, are replaced with a simple algebraic expression. The applicability of this approach for estimating Vmax and Km in the Michaelis-Menten equation was verified using a combination of simulated and experimental progress curve data. For simulated data, final estimates of Vmax and Km were close to the actual values of 1 microM/h and 1 microM, respectively, while the standard errors for these parameter estimates were proportional to the error level in the simulated data sets. The method was also applied to hydrogen depletion experiments by mixed cultures of bacteria in activated sludge resulting in Vmax and Km estimates of 6.531 microM/h and 2.136 microM, respectively. The algebraic nature of this solution, coupled with its relatively high accuracy, makes it an attractive candidate for kinetic parameter estimation from progress curve data.     

Determination of enzyme kinetic parameters by continuous addition of substrate to a single reaction mixture and analysis by a tangent-slope procedure: I. Analysis of the method using computed progress curves

A new method has been developed which provides reliable estimates of enzyme kinetic constants from single reaction progress curves recorded under conditions of continuously increasing substrate concentration. Equally spaced data points simulating such progress curves and containing known amounts of superimposed random noise were fit to the Hill equation by (i) direct nonlinear curve-fitting of raw data, and (ii) a tangent-slope technique in which the raw data are numerically differentiated, transformed into substrate versus velocity data, and then analyzed as linear plots. Both integral and differential procedures provided accurate and precise estimates of the Hill parameters (S_0.5, V, and n) from single reaction mixtures. However, the tangent-slope method was at least 10-fold faster to compute and was not dependent on accurate initial guesses of the Hill parameters or integration of the rate equation. With the tangent-slope method, the optimal number of data points used in calculating tangent slopes was found to be 9 or 11. The reliability of the Hill parameters determined with the tangent-slope method was relatively insensitive to the maximum substrate concentration over a range of $\text{S}{}_{\mathrm{<mtext>max</mtext>}}\text{S}{}_{0.5}$ of 1.5 to 10; the optimal value was 3. Through further analysis of simulated data, it was found that slow enzyme inactivation (<4% loss during the assay), or product competitive inhibition (maximum product concentration < 30% of the inhibitor dissociation constant) does not produce serious errors in the Hill parameters. Methods are presented to detect and distinguish enzyme inactivation and product competitive inhibition. It is suggested that continuous addition methodology combined with tangent-slope analysis provides the basis for a flexible system for kinetic characterization of enzymes which has wider applicability and other advantages over multicuvette or conventional progress curve methodology. A major advantage in contrast to the progress curve approach is that product accumulation and associated product effects are lowest at lower substrate concentrations.     

Deriving the rate equations for product inhibition patterns in bisubstrate enzyme reactions

In this work, the full rate equations for 17 completely reversible bisubstrate enzyme kinetic mechanisms, with two substrates in the forward and two in the reverse direction, have been presented; among these are rapid equilibrium, steady-state, and mixed steady-state and rapid equilibrium mechanisms. From each rate equation eight product inhibition equations were derived, four for the forward and four for the reverse direction. All the corresponding product inhibition equations were derived in full; thus a total of 17 × 8 = 136 equations, were presented. From these equations a list of product inhibition patterns were constructed and presented in a tabular form, both for the primary plots (intercept effects) and the secondary plots (slope effects).

The purpose of this work is to help investigators in practical work, especially biologists working with enzymes, to choose quickly an appropriate product inhibition pattern for the identification of the kinetic mechanism. The practical application of above product inhibition analysis was illustrated with three examples of yeast alcohol dehydrogenase-catalyzed reactions.     

Deriving the rate equations for product inhibition patterns in bisubstrate enzyme reactions

In this work, the full rate equations for 17 completely reversible bisubstrate enzyme kinetic mechanisms, with two substrates in the forward and two in the reverse direction, have been presented; among these are rapid equilibrium, steady-state, and mixed steady-state and rapid equilibrium mechanisms. From each rate equation eight product inhibition equations were derived, four for the forward and four for the reverse direction. All the corresponding product inhibition equations were derived in full; thus a total of 17 x 8 = 136 equations, were presented. From these equations a list of product inhibition patterns were constructed and presented in a tabular form, both for the primary plots (intercept effects) and the secondary plots (slope effects). The purpose of this work is to help investigators in practical work, especially biologists working with enzymes, to choose quickly an appropriate product inhibition pattern for the identification of the kinetic mechanism. The practical application of above product inhibition analysis was illustrated with three examples of yeast alcohol dehydrogenase-catalyzed reactions.     

The kinetic effect of product instability in a Michaelis-Menten mechanism with competitive inhibition

In most kinetic studies it is assumed that both the reactant and the products are stable. However, under certain conditions spontaneous decomposition or deterioration caused by one of the participating species occurs. Studies, in which a species (the free enzyme, the enzyme-substrate complex, an inhibitor or the product of the reaction) is unstable, have appeared in the literature. However, to our knowledge, the enzymatic systems, in which a competitive inhibition and a decomposition or transformation of the products take place simultaneously, have not been studied so far. In this paper, we present a kinetic analysis of an enzyme reaction that follows a Michaelis-Menten mechanism, in which the free enzyme suffers a competitive inhibition simultaneously with the decomposition of the immediate product. In this study, we have linearised the differential equations that describe the kinetics of the process. Under the assumption of limiting concentration of enzyme, we have obtained and tested the explicit equation describing the time dependence of the product concentration using numerical calculus. With this equation and the experimental progress curve of the product, we constructed an easy procedure for the evaluation of the principal kinetic parameters of the process.     

A general solution for the steady-state kinetics of immobilized enzyme systems

A power series solution is presented which describes the steady-state concentration profiles for substrate and product molecules in immobilized enzyme systems. Diffusional effects and product inhibition are incorporated into this model. The kinetic consequences of diffusion limitation and product inhibition for immobilized enzymes are discussed and are compared to kinetic behavior characteristic of other types of effects, such as substrate inhibition and substrate activation.     

Kinetic studies on the enzyme (S)-hydroxynitrile lyase from hevea brasiliensis using initial rate methods and progress curve analysis

(S)-Hydroxynitrile lyase (EC 4.1.2.39) from Hevea brasiliensis(rubber tree) catalyzes the reversible cleavage of cyanohydrins to aldehydes or ketones and prussic acid (HCN). Enzyme kinetics in both directions was studied on a model system with mandelonitrile, benzaldehyde, and HCN using two different methods-initial rate measurements and progress curve analysis. To discriminate between possible mechanisms with the initial rate method, product inhibition was studied. Benzaldehyde acts as a linear competitive inhibitor against mandelonitrile whereas HCN shows S-linear I-parabolic mixed-type inhibition. These results indicate an Ordered Uni Bi mechanism with the formation of a dead-end complex of enzyme, (S)-mandelonitrile and HCN. Prussic acid is the first product released from the enzyme followed by benzaldehyde. For progress curve analysis, a kinetic model of an Ordered Uni Bi mechanism including a dead-end complex, enzyme inactivation, and the chemical parallel reaction was set up, which described the experimental values very well. From the reaction rates obtained the kinetic constants were calculated and compared with the ones obtained from the initial rate method. Good agreement could be achieved between the two methods supporting the suggested mechanism. Copyright 1999 John Wiley & Sons, Inc.     

Redox potential dependence of photophosphorylation and electron transfer in continuous illumination of Rhodopseudomonas sphaeroides chromatophores

The rate of ethanol elimination in fed and fasted rats can be predicted based on the liver content of alcohol dehydrogenase (EC 1.1.1.1), the steady-state rate equation, and the concentrations of substrates and products in liver during ethanol metabolism. The specific activity, kinetic constants, and multiplicity of enzyme forms are similar in fed and fasted rats, although the liver content of alcohol dehydrogenase falls 40% with fasting. The two major forms of the enzyme were separated and found to have very similar kinetic properties. The rat alcohol dehydrogenase is subject to substrate inhibition by ethanol at concentrations above 10 mM and follows a Theorell-Chance mechanism. The steady-state rate equation for this mechanism predicts that the in vivo activity of the enzyme is limited by NADH product inhibition at low ethanol concentrations and by both NADH inhibition and substrate inhibition at high ethanol concentrations. When the steady-state rate equation and the measured concentrations of substrates and products in freeze-clamped liver of fed and fasted rats metabolizing alcohol are employed to calculate alcohol oxidation rates, the values agree very well with the actual rates of ethanol elimination determined in vivo.     

A generalized numerical approach to steady-state enzyme kinetics: Applications to protein kinase inhibition

A generalized numerical treatment of steady-state enzyme kinetics is presented. This new approach relies on automatic computer derivation of the underlying mathematical model (a system of simultaneous nonlinear algebraic equations) from a symbolic representation of the reaction mechanism (a system of biochemical equations) provided by the researcher. The method allows experimental biochemists to analyze initial-rate enzyme kinetic data, under the steady-state approximation, without having to use any mathematical equations. An illustrative example is based on the inhibition kinetics of p56^lck kinase by an ATP competitive inhibitor. A computer implementation of the new method, in the modified software package DYNAFIT [Kuzmi?, P. (1996) Anal. Biochem. 237, 260–273], is freely available to all academic researchers.     

The purification and steady-state kinetic behaviour of rabbit heart mitochondrial NAD(P)+ malic enzyme.

The mitochondrial NAD(P)+ malic enzyme [EC 1.1.1.39, L-malate:NAD+ oxidoreductase (decarboxylating)] was purified from rabbit heart to a specific activity of 7 units (mumol/min)/mg at 23 degrees C. A study of the reductive carboxylation reaction indicates that this enzymic reaction is reversible. The rate of the reductive carboxylation reaction appears to be completely inhibited at an NADH concentration of 0.92 mM. A substrate saturation curve of this reaction with NADH as the varied substrate describes this inhibition. The apparent kinetic parameters for this reaction are Ka(NADH) = 239 microM and Vr = 1.1 mumol/min per mg at 23 degrees C. The steady-state product-inhibition patterns for pyruvate and NADH indicate a sequential binding of the substrates: NAD+ followed by L-malate. These data also indicate that NADH is the last product released. A steady-state kinetic model is proposed that incorporates NADH-enzyme dead-end complexes.     

Polyphosphate kinase from M. tuberculosis: an interconnect between the genetic and biochemical role

The enzyme Polyphosphate Kinase (PPK) catalyses the reversible transfer of the terminal γ-Pi of ATP to form a long chain Polyphosphate (PolyP). Using an IPTG inducible mycobacterial vector, the vulnerability of this gene has been evaluated by antisense knockdown experiments in M. tuberculosis. Expression profiling studies point to the fact that down regulation of PPK caused cidality during the late phase in contrast to its bacteriostatic mode immediately following antisense expression. PPK thus seems to be a suitable anti-tubercular drug target. The enzyme which is a tetramer has been cloned in E. coli and purified to homogeneity. An enzyme assay suitable for High Throughput Screening was optimized by using the statistical Taguchi protocol and the kinetic parameters determined. The enzyme displayed a strong product inhibition by ADP. In order to accurately estimate the product inhibition, progress curve analysis of the enzyme reaction was monitored. The kinetic equation describing the progress curve was suitably modified by taking into account the product inhibition. The reversible nature of the enzyme indicated a possibility of a two way ATP↔ADP switch operating in the bacteria as a response to its growth requirement.     

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.     

Regulation of human lymphocyte S-adenosylmethionine synthetase by product inhibition

A steady-state kinetic analysis of human S-adenosylmethionine synthetase indicates that the reaction is Bi Ter with ordered addition of ATP and L-methionine and release of S-adenosylmethionine as the first product. Pyrophosphate and phosphate are then released randomly. I-Parabolic inhibition by phosphate with respect to ATP indicates that this product must bind to more than one site. A model in which phosphate binds to the pyrophosphate site gives a rate equation that is consistent with the kinetic data. Values have been determined for those constants in the equation that are large enough to evaluate, and the in vitro kinetic behavior of S-adenosylmethionine synthetase can be predicted at substrate and product concentrations that are expected intracellularly. Inhibition by combinations of products, especially pyrophosphate and phosphate, is synergistic. Of particular interest is the ability of pyrophosphate and phosphate to increase the sensitivity of the enzyme to inhibition by S-adenosylmethionine. This phenomenon may play a role in regulating steady-state cellular concentrations of S-adenosylmethionine.     

Identification of enzyme inhibitory mechanisms from steady-state kinetics

Enzyme inhibitors are used in many areas of the life sciences, ranging from basic research to the combat of disease in the clinic. Inhibitors are traditionally characterized by how they affect the steady-state kinetics of enzymes, commonly analyzed on the assumption that enzyme-bound and free substrate molecules are in equilibrium. This assumption, implying that an enzyme-bound substrate molecule has near zero probability to form a product rather than dissociate, is valid only for very inefficient enzymes. When it is relaxed, more complex but also more information-rich steady-state kinetics emerges. Although solutions to the general steady-state kinetics problem exist, they are opaque and have been of limited help to experimentalists.Here we reformulate the steady-state kinetics of enzyme inhibition in terms of new parameters. These allow for assessment of ambiguities of interpretation due to kinetic scheme degeneracy and provide an intuitively simple way to analyze experimental data. We illustrate the method by concrete examples of how to assess scheme degeneracy and obtain experimental estimates of all available rate and equilibrium constants. We suggest simple, complementary experiments that can remove ambiguities and greatly enhance the accuracy of parameter estimation.     

Alternative algebraic rate‐integration approach for progress‐curve analysis of enzyme kinetics

A recent article of Zavrel et al. in this journal (Eng. Life Sci. 2010, 10, 191–200) described a comparison of several computer programs for progress‐curve analysis with respect to different computational approaches for parameter estimation. The authors applied both algebraic and dynamic parameter estimations, although they omitted time‐course analysis through the integrated rate equation. Recently, it was demonstrated that progress‐curve analysis through the integrated rate equation can be considered a simple and useful alternative for enzymes that obey the generalized Michaelis–Menten reaction mechanism. To complete this gap, the time‐dependent solution of the generalized Michaelis–Menten equation is here fitted to the progress curves from the Zavrel et al. reference article. This alternative rate‐integration approach for determining the kinetics parameters of Michaelis–Menten‐type enzymes yields the values with the greatest accuracy, as compared with the results obtained by other (algebraic or dynamic) parameter estimations.