On the Numerical Computation of Enzyme Kinetic Parameters (original) (raw)
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A new approach to a century-old problem: Henri-Michaelis-Menten enzyme kinetics
AIP Conference Proceedings, 2012
A new approach to Henri-Michaelis-Menten (HMM) enzyme kinetics is presented. In the first part, a simplified but useful description that stresses the cyclic nature of the catalytic process is summarized. In particular, the timedependence of the substrate concentration is obtained in a simple way that dispenses the quasi-steady-state approximation. In the second part an exact one-dimensional formulation of HMM kinetics is presented. The whole problem is condensed in a second-order non-linear autonomous differential equation, and the exact solution of HMM kinetics is given as a set of Maclaurin series. From the same evolution equation, a number of approximate solutions, some known, some new, can be derived in a systematic way. The evolution equation obtained is also well suited for the numerical computation of the concentrations of all species as a function of time for any given combination of parameters.
Extending the kinetic solution of the classic Michaelis–Menten model of enzyme action
Journal of Mathematical Chemistry
The principal aim of studies of enzyme-mediated reactions has been to provide comparative and quantitative information on enzyme-catalyzed reactions under distinct conditions. The classic Michaelis–Menten model (Biochem Zeit 49:333, 1913) for enzyme kinetic has been widely used to determine important parameters involved in enzyme catalysis, particularly the Michaelis–Menten constant (K M ) and the maximum velocity of reaction (V max ). Subsequently, a detailed treatment of the mechanisms of enzyme catalysis was undertaken by Briggs–Haldane (Biochem J 19:338, 1925). These authors proposed the steady-state treatment, since its applicability was constrained to this condition. The present work describes an extending solution of the Michaelis–Menten model without the need for such a steady-state restriction. We provide the first analysis of all of the individual reaction constants calculated analytically. Using this approach, it is possible to accurately predict the results under new experimental conditions and to characterize and optimize industrial processes in the fields of chemical and food engineering, pharmaceuticals and biotechnology.
A different view of Henri-Michaelis-Menten (HMM) enzyme kinetics is presented. In the first part of the paper, a simplified but useful description that stresses the cyclic nature of the catalytic process is introduced. The time-dependence of the substrate concentration after the initial transient phase is derived in a simple way that dispenses with the mathematical technique known as quasi-steady-state approximation. In the second part of the paper an exact one-dimensional formulation of HMM kinetics is considered. The whole problem is condensed in a single one-variable evolution equation that is a second-order non-linear autonomous differential equation, and the control parameters are reduced to three dimensionless quantities: enzyme efficiency, substrate reduced initial concentration, and enzyme reduced initial concentration. The exact solution of HMM kinetics is obtained as a set of Maclaurin series. From the same equation, a number of approximate solutions, some known, some new, are derived in a systematic way that allows a precise evaluation of the respective level of approximation and conditions of validity. The evolution equation obtained is also shown to be well suited for the numerical computation of the concentrations of all species as a function of time for any given combination of parameters.
A novel method for analyzing enzyme kinetic systems
Applied Mathematics and Computation, 1997
The rate law governing the kinetics of a Single enzyme mediated reaction may be derived relatively easily by hand, given knowledge of the enzyme's mechanism. Such rate lows are typically non-linear in the concentrations of metabolites involved. When a number of enzymes interact, the composite rate law for the complete System involves the simultaneous Solution of the individual enzyme rate laws. We show how Computer algebra tan be used to solve this previously intractable Problem, using the method of Gröbner Bsses. We present an experimental example where kinetic Parameters for individual enzymes arc measured by making observations of a multi-enzyme System, and fitting these data to the rate law for the complete System.
17 Alternative Perspectives of Enzyme Kinetic Modeling
2012
The basis of enzyme kinetic modelling was established during the early 1900’s when the work of Leonor Michaelis and Maud Menten produced a pseudo-steady state equation linking enzymatic catalytic rate to substrate concentration (Michaelis & Menten, 1913). Building from the Michaelis-Menten equation, other equations used to describe the effects of modifiers of enzymatic activity were developed based on their effect on the catalytic parameters of the Michaelis-Menten equation. Initially, inhibitors affecting the substrate affinity were deemed competitive and inhibitors affecting the reaction rate were labelled non-competitive (McElroy 1947). These equations have persisted as the basis for inhibition studies and can be found in most basic textbooks dealing with the subject of enzyme inhibition. Here the functionality of the competitive and non-competitive equations are examined to support the development of a unified equation for enzymatic activity modulation. From this, a modular appr...
Parameter estimation of an enzyme kinetic system using computer algebra techniques
Applied Mathematics and Computation, 1999
A procedure for fitting enzyme kinetic data directly to the flux equation was described. It involves choosing parameters that minimize the sum of the squares of deviations due to errors in s, the substrate concentration at time t. Estimates of the standard errors of the parameters are provided using computer algebra and numerical analysis techniques.
Some Mathematical and Statistical Aspects of Enzyme Kinetics
Most calculus or differential equations courses utilize examples taken from physics, often discussing them in great detail. Chemistry, however, is seldom utilized to illustrate mathematical concepts. This tendency should be reversed because chemistry, especially chemical kinetics, provides the opportunity to apply mathematics readily. We will analyze some basic ideas behind enzyme kinetics, which allow us to deal with separable and linear differential equations as well as realize the need to use power series to approximate x e and) 1 ln(x close to the origin, and to apply the recently defined Lambert W function. The models studied in this context require the estimation of parameters based on experimental data, which in turn allows us to discuss simple and multiple linear regression, transformations and non-linear regression and their implementation using statistical software.
Journal of Mathematical Chemistry, 2007
This work presents an alternative analysis of the integrated rate equations corresponding to the simple Michaelis-Menten mechanism without product inhibition. The suggested new results are reached under a minimal set of assumptions and include, as a particular case, the classical integrated Michaelis-Menten equation. Experimental designs and a kinetic data analysis are suggested to the estimation of the maximum steady-state rate, V max , the Michaelis-Menten constant, K m , the initial enzyme * Corresponding author. 789 0259-9791/07/1100-0789/0 © 2006 Springer Science+Business Media, Inc. R. Varón et al. / Integrated form of the Michaelis-Menten Equation concentration, [E] 0 , and the catalytic constant, k 2 . The goodness of the analysis is tested with simulated time progress curves obtained by numerical integration.
2010
Mathematical models are widely used to create complex biochemical models. Model reduction in order to limit the complexity of a system is an important topic in the analysis of the model. A way to lower the complexity is to identify simple and recurrent sets of reactions and to substitute them with one or more reactions in such a way that the important properties are preserved but the analysis is easier. In this paper we consider the typical recurrent reaction scheme E + S − − ⇀ ↽ − − ES − − → E + P which describes the mechanism that an enzyme, E, binds a substrate, S, and the resulting substrate-bound enzyme, ES, gives rise to the generation of the product, P . If the initial quantities and the reaction rates are known, the temporal behaviour of all the quantities involved in the above reactions can be described exactly by a set of differential equations. It is often the case however that, as not all necessary information is available, only approximate analysis can be carried out. The most well-known approximate approach for the enzyme mechanism is provided by the kinetics of Michaelis-Menten. We propose, based on the concept of the flow-equivalent server which is used in Petri nets to model reduction, an alternative approximate kinetics for the analysis of enzymatic reactions. We evaluate the goodness of the proposed approximation with respect to both the exact analysis and the approximate kinetics of Michaelis and Menten. We show that the proposed new approximate kinetics can be used and gives satisfactory approximation not only in the standard deterministic setting but also in the case when the behaviour is modeled by a stochastic process.
World Journal Of Advanced Research and Reviews, 2022
For some time now, there has been growing interest in pre-steady-state (PSS) kinetic parameters for whatever reasons, the measurement of which needs high-tech equipment capable of transient timescale duration of assay. The proposition, however, is that all kinetic parameters, PSS and beyond, can be determined with appropriate PSS derivable equations and the usual Michaelis-Menten (MM) and Briggs-Haldane (BH) equations, respectively. The objectives of the research were: 1) To derive equations, for the determination of reverse rate constant when the substrate concentration, [S] « MM constant, KM, 2) determine by calculation, the reverse rate constant, forward rate constant, and consequently, show that it is possible to determine rate constant often seen to be masked within original MM cum BH mathematical formalism, and 3) validate corollaries from the derivation that justify procedural issue. Theoretical, experimental (Bernfeld method), and computational methods were explored. Pre-steady-state equations for the determination of kinetic parameters, the reverse rate constant, k-1, for the process ES E + S, the 2nd order rate constant, k1, and the rate, v1, for the formation of enzyme-substrate complex, ES, were derived. The derived originating equations with associated corollaries were validated and have been seen to be capable of reproducing experimental variables and kinetic parameters; rate constants that seemed masked in MM formalism were unmasked. Steady-state (SS) cum zero order kinetic parameters were » their PSS values. "Negative" catalytic efficiency (k-1/KM) was » "positive" catalytic efficiency, (kcat/KM), with lower [ET]. In conclusion, the equations for PSS kinetic parameters were derivable. Previously masked kinetic parameters in the MM/BB mathematical formalism can now be calculated using MM data; thus, all kinetic parameters can be determined regardless of the reaction pathway's state, PSS, and SS. PSS kinetic parameters were « SS/zero order values.