Michaelis-Menten Reaction Mechanism Research Articles

The Michaelis–Menten Reaction at Low Substrate Concentrations: Pseudo-First-Order Kinetics and Conditions for Timescale Separation

We demonstrate that the Michaelis–Menten reaction mechanism can be accurately approximated by a linear system when the initial substrate concentration is low. This leads to pseudo-first-order kinetics, simplifying mathematical calculations and experimental analysis. Our proof utilizes a monotonicity property of the system and Kamke’s comparison theorem. This linear approximation yields a closed-form solution, enabling accurate modeling and estimation of reaction rate constants even without timescale separation. Building on prior work, we establish that the sufficient condition for the validity of this approximation is s0≪K\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$s_0 \\ll K$$\\end{document}, where K=k2/k1\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$K=k_2/k_1$$\\end{document} is the Van Slyke–Cullen constant. This condition is independent of the initial enzyme concentration. Further, we investigate timescale separation within the linear system, identifying necessary and sufficient conditions and deriving the corresponding reduced one-dimensional equations.

Algorithmic criteria for the validity of quasi-steady state and partial equilibrium models: the Michaelis-Menten reaction mechanism.

We present "on the fly" algorithmic criteria for the accuracy and stability (non-stiffness) of reduced models constructed with the quasi-steady state and partial equilibrium approximations. The criteria comprise those introduced in Goussis (Combust Theor Model 16:869-926, 2012) that addressed the case where each fast time scale is due to one reaction and a new one that addresses the case where a fast time scale is due to more than one reactions. The development of these criteria is based on the ability to approximate accurately the fast and slow subspaces of the tangent space. Their validity is assessed on the basis of the Michaelis-Menten reaction mechanism, for which extensive literature is available regarding the validity of the existing various reduced models. The criteria predict correctly the regions in both the parameter and phase spaces where each of these models is valid. The findings are supported by numerical computations at indicative points in the parameter space. Due to their algorithmic character, these criteria can be readily employed for the reduction of large and complex mathematical models.

Natural Parameter Conditions for Singular Perturbations of Chemical and Biochemical Reaction Networks

We consider reaction networks that admit a singular perturbation reduction in a certain parameter range. The focus of this paper is on deriving "small parameters" (briefly for small perturbation parameters), to gauge the accuracy of the reduction, in a manner that is consistent, amenable to computation and permits an interpretation in chemical or biochemical terms. Our work is based on local timescale estimates via ratios of the real parts of eigenvalues of the Jacobian near critical manifolds. This approach modifies the one introduced by Segel and Slemrod and is familiar from computational singular perturbation theory. While parameters derived by this method cannot provide universal quantitative estimates for the accuracy of a reduction, they represent a critical first step toward this end. Working directly with eigenvalues is generally unfeasible, and at best cumbersome. Therefore we focus on the coefficients of the characteristic polynomial to derive parameters, and relate them to timescales. Thus, we obtain distinguished parameters for systems of arbitrary dimension, with particular emphasis on reduction to dimension one. As a first application, we discuss the Michaelis-Menten reaction mechanism system in various settings, with new and perhaps surprising results. We proceed to investigate more complex enzyme catalyzed reaction mechanisms (uncompetitive, competitive inhibition and cooperativity) of dimension three, with reductions to dimension one and two. The distinguished parameters we derive for these three-dimensional systems are new. In fact, no rigorous derivation of small parameters seems to exist in the literature so far. Numerical simulations are included to illustrate the efficacy of the parameters obtained, but also to show that certain limitations must be observed.

Stochastic time-dependent enzyme kinetics: Closed-form solution and transient bimodality.

We derive an approximate closed-form solution to the chemical master equation describing the Michaelis-Menten reaction mechanism of enzyme action. In particular, assuming that the probability of a complex dissociating into an enzyme and substrate is significantly larger than the probability of a product formation event, we obtain expressions for the time-dependent marginal probability distributions of the number of substrate and enzyme molecules. For delta function initial conditions, we show that the substrate distribution is either unimodal at all times or else becomes bimodal at intermediate times. This transient bimodality, which has no deterministic counterpart, manifests when the initial number of substrate molecules is much larger than the total number of enzyme molecules and if the frequency of enzyme-substrate binding events is large enough. Furthermore, we show that our closed-form solution is different from the solution of the chemical master equation reduced by means of the widely used discrete stochastic Michaelis-Menten approximation, where the propensity for substrate decay has a hyperbolic dependence on the number of substrate molecules. The differences arise because the latter does not take into account enzyme number fluctuations, while our approach includes them. We confirm by means of a stochastic simulation of all the elementary reaction steps in the Michaelis-Menten mechanism that our closed-form solution is accurate over a larger region of parameter space than that obtained using the discrete stochastic Michaelis-Menten approximation.

A theory of reactant-stationary kinetics for a mechanism of zymogen activation

A theoretical analysis is performed on the nonlinear ordinary differential equations that govern the dynamics of a reaction mechanism of zymogen activation. The reaction consists of a primary non-observable zymogen activation reaction that it is coupled to an indicator (observable) reaction. The product of the first reaction is the enzyme of the indicator reaction, and both reactions are governed by the Michaelis–Menten reaction mechanism. Using singular perturbation methods, we derive asymptotic solutions that are valid under the quasi-steady-state and reactant-stationary assumptions. In particular, we obtain closed form solutions that are analogous to the Schnell–Mendoza equation for Michaelis–Menten type reactions. These closed-form solutions approximate the evolution of the observable reaction and provide the mathematical link necessary to measure the enzyme activity of the non-observable reaction. Conditions for the validity of the asymptotic solutions are also derived, and we demonstrate that these asymptotic expressions are applicable under reactant-stationary kinetics.

New insights into the catalytic mechanism of human glycine N-acyltransferase.

Even though the glycine conjugation pathway was one of the first metabolic pathways to be discovered, this pathway remains very poorly characterized. The bi-substrate kinetic parameters of a recombinant human glycine N-acyltransferase (GLYAT, E.C. 2.3.1.13) were determined using the traditional colorimetric method and a newly developed HPLC-ESI-MS/MS method. Previous studies analyzing the kinetic parameters of GLYAT, indicated a random Bi-Bi and/or ping-pong mechanism. In this study, the hippuric acid concentrations produced by the GLYAT enzyme reaction were analyzed using the allosteric sigmoidal enzyme kinetic module. Analyses of the initial rate (v) against substrate concentration plots, produced a sigmoidal curve (substrate activation) when the benzoyl-CoA concentrations was kept constant, whereas the plot with glycine concentrations kept constant, passed through a maximum (substrate inhibition). Thus, human GLYAT exhibits mechanistic kinetic cooperativity as described by the Ferdinand enzyme mechanism rather than the previously assumed Michaelis-Menten reaction mechanism.

An algebraic model to determine substrate kinetic parameters by global nonlinear fit of progress curves

We propose that the time course of an enzyme reaction following the Michaelis-Menten reaction mechanism can be conveniently described by a newly derived algebraic equation, which includes the Lambert Omega function. Following Northrop's ideas [Anal. Biochem.321, 457–461, 1983], the integrated rate equation contains the Michaelis constant (KM) and the specificity number (kS≡kcat/KM) as adjustable parameters, but not the turnover number kcat. A modification of the usual global-fit approach involves a combinatorial treatment of nominal substrate concentrations being treated as fixed or alternately optimized model parameters. The newly proposed method is compared with the standard approach based on the “initial linear region” of the reaction progress curves, followed by nonlinear fit of initial rates to the hyperbolic Michaelis-Menten equation. A representative set of three chelation-enhanced fluorescence EGFR kinase substrates is used for experimental illustration. In one case, both data analysis methods (linear and nonlinear) produced identical results. However, in another test case, the standard method incorrectly reported a finite (50–70 μM) KM value, whereas the more rigorous global nonlinear fit shows that the KM is immeasurably high.

On series approximations of Michaelis-Menten kinetics

The Michaelis-Menten reaction mechanism yields a system of nonlinear differential equations that have recently been approximated using various series approaches and, in some cases, [ L / L ] Pade approximants. The coefficients of the series obtained from these different techniques are identical, so any differences in perfomance of the approximations are determined by the number of terms or the form of the approximant employed. Furthermore, the approximations successfully represent only the initial phase of the reaction and do not describe the entire timecourse. Some improvement can be obtained using Pade approximants of a different form and, from these, we provide an estimate of the time at which the steady-state occurs.

Realistic enzymology for post-translational modification: Zero-order ultrasensitivity revisited

Unlimited ultrasensitivity in a kinase/phosphatase “futile cycle” has been a paradigmatic example of collective behaviour in multi-enzyme systems. However, its analysis has relied on the Michaelis–Menten reaction mechanism, which remains widely used despite a century of new knowledge. Modifying and demodifying enzymes accomplish different biochemical tasks; the donor that contributes the modifying group is often ignored without the impact of this time-scale separation being taken into account; and new forms of reversible modification are now known. We exploit new algebraic methods of steady-state analysis to reconcile the analysis of multi-enzyme systems with single-enzyme biochemistry using zero-order ultrasensitivity as an example. We identify the property of “strong irreversibility”, in which product re-binding is disallowed. We show that unlimited ultrasensitivity is preserved for a class of complex, strongly irreversible reaction mechanisms and determine the corresponding saturation conditions. We show further that unlimited ultrasensitivity arises from a singularity in a novel “invariant” that summarises the algebraic relationship between modified and unmodified substrate. We find that this singularity also underlies knife-edge behaviour in allocation of substrate between modification states, which has implications for the coherence of futile cycles within an integrated tissue. When the enzymes are irreversible, but not strongly so, the singularity disappears in the form found here and unlimited ultrasensitivity may no longer be preserved. The methods introduced here are widely applicable to other reversible modification systems.

A Graphical User Interface for a Method to Infer Kinetics and Network Architecture (MIKANA)

One of the main challenges in the biomedical sciences is the determination of reaction mechanisms that constitute a biochemical pathway. During the last decades, advances have been made in building complex diagrams showing the static interactions of proteins. The challenge for systems biologists is to build realistic models of the dynamical behavior of reactants, intermediates and products. For this purpose, several methods have been recently proposed to deduce the reaction mechanisms or to estimate the kinetic parameters of the elementary reactions that constitute the pathway. One such method is MIKANA: Method to Infer Kinetics And Network Architecture. MIKANA is a computational method to infer both reaction mechanisms and estimate the kinetic parameters of biochemical pathways from time course data. To make it available to the scientific community, we developed a Graphical User Interface (GUI) for MIKANA. Among other features, the GUI validates and processes an input time course data, displays the inferred reactions, generates the differential equations for the chemical species in the pathway and plots the prediction curves on top of the input time course data. We also added a new feature to MIKANA that allows the user to exclude a priori known reactions from the inferred mechanism. This addition improves the performance of the method. In this article, we illustrate the GUI for MIKANA with three examples: an irreversible Michaelis–Menten reaction mechanism; the interaction map of chemical species of the muscle glycolytic pathway; and the glycolytic pathway of Lactococcus lactis. We also describe the code and methods in sufficient detail to allow researchers to further develop the code or reproduce the experiments described. The code for MIKANA is open source, free for academic and non-academic use and is available for download (Information S1).

The oxidation of ascorbate at copolymeric sulfonated poly(aniline) coated on glassy carbon electrodes

Self-doped poly(aniline)s as electrode coatings to catalyze ascorbate oxidation are revisited in this article. Sulfonated poly(aniline) (SPAN) was deposited on glassy carbon electrodes as a copolymer of aniline and its sulfonated derivative, 2-aminobenzenesulfonic acid (2-ABSA). The resulting deposits are reproducible and show good stability and electroactivity at pH > 7, enabling studies at typical physiological pH values. Calibration curves were obtained using a rotating disc electrode at a sampling potential of 0.2 V, displaying linear dependence in the region 0–20 mM ascorbate. A kinetic model based on the Michaelis–Menten reaction mechanism, previously validated for poly(aniline) composites, was used to analyse the form of the calibration curve leading to values of the effective reaction constants K ME and k′ ME. New calibration curves constructed for different sampling potentials were used to elucidate the rate limiting step at saturated kinetics. Rotating disc voltammetry performed at increasing pH (from pH 2 to 9) showed a dramatic decrease in the limiting current, without any evidence for a change in the reaction mechanism.

Rational polynomial equation helps to select among homeomorphic kinetic models for cholinesterase reaction mechanism

The hydrolysis of substrates by cholinesterases does not follow the Michaelis-Menten reaction mechanism. In addition to the inhibition by excess substrate, these enzymes often show an unexpectedly high activity at low substrate concentrations. It seems that these deviations are the consequence of an unusual architecture of the active site, buried deep inside the core of the molecule. Kinetic data and structural evidence allow for a detailed prediction of the events during a very fast substrate turnover. Recently, we presented a procedure which provides an unbiased framework for mathematical modelling of the complex cholinesterase reaction [J. Stojan, M. Goličnik, D. Fournier, Rational polynomial equation as an unbiased approach for the kinetic studies of Drosophila melanogaster acetylcholinesterase reaction mechanism, Biochim. Biophys. Acta 1703 (2004) 53–61]. It is based on regression analysis of a rational polynomial using classical initial rate data. Here, we extend the use of the rational polynomial rate equation for finding and comparing some selected homeomorphic reaction schemes useful for the mechanistic interpretation of cholinesterase kinetic data.

Conversion of a Group II Intron into a New Multiple-Turnover Ribozyme that Selectively Cleaves Oligonucleotides: Elucidation of Reaction Mechanism and Structure/Function Relationships

The self-splicing ai5g group II intron was transformed into a three-part ribozyme that site-specifically cleaves small oligonucleotide substrates with multiple turnover. The ribozyme is composed of intron domain 1 (D1, 425 nucleotides), with catalytically essential domain 5 (D5, 58 nucleotides) provided separately as a reaction cofactor. Together, the D1/D5 complex cleaves small substrates analogous in sequence to the 5'-splice site of the intron. Activity of the ribozyme was studied using a combination of single- and multiple-turnover experiments in which the concentrations of the RNA components were varied in order to probe their individual role in the overall mechanism. Values for kcat, Km, and kcat/Km were the same within experimental error for the two enzymological approaches. These kinetic analyses reveal that the ribozyme utilizes a classic Michaelis-Menten reaction mechanism in which the chemical step of catalysis (kcat = kchem = approximately 0.03 min-1 at full saturation) is rate limiting for the overall reaction. The D1/D5 complex binds tightly to the substrate (Km = 6.3 nM) and specifically recognizes sequences both 5' and 3' to the ribozyme cleavage site. These studies represent the first quantitative analysis of group II recognition and affinity for the 5'-splice site. As observed in previous studies on the role of D5 RNA, D5 binds tightly to the ternary complex (Km = 870 nM). The second-order rate constant for RNA cleavage (kcat/Km = 3.3 x 10(6)) is an order of magnitude slower than that observed for other ribozymes in this mechanistic class, all of which are rate-limited by steps other than chemistry. That kcat equals kchem in this ribozyme is supported by the overall kinetic analysis and by the observation that an Rp phosphorothioate is cleaved approximately 3-fold more slowly than a phosphate at the cleavage site. These studies represent a preliminary examination of stereochemical preference by a group II intron active site in the transition state. The substrate specificity, reaction conditions, and mutational sensitivity of this ribozyme are consistent with a reaction analogous to the first step of group II intron self-splicing, although its stereochemical preference is analogous to a second-step reversal.

On the dynamics of the irreversible Michaelis-Menten reaction mechanism

The mass action kinetic model of the irreversible Michaelis-Menten reaction mechanism is mathematically intractable: an explicit analytical solution cannot be obtained. This difficulty is overcome by applying simplifying kinetic assumptions but a full understanding of their dynamic implications and applicability is not readily available. This paper shows how simple modal analysis can provide both a conceptually appealing insight into the reaction dynamics and justification of the commonly used quasi-steady-state and quasi-equilibrium assumptions. The key results are that the quasi-steady-state assumption is applicable when the initial enzyme concentration, e 0, is much smaller than the Michaelis constant, K m, or when the initial substrate concentration, s 0, is much greater than K m. These results show that the commonly accepted criterion e 0 < < s 0 is incomplete and should be decomposed into e 0 < < K m and K m << s 0. The quasi-equilibrium assumption is valid when e 0 ⪢⪢ K m and when the rate of product formation is much slower than reversion to the substrate from the intermediate state, or k 2 ⪢⪢ k −1. The important dimensionless parameter ratios characterizing the reaction dynamics are e 0/ K m, s 0/ K m and k 2/ k −1.

Mathematical modelling of dynamics and control in metabolic networks. I. On michaelis-menten kinetics

As a starting point for modeling of metabolic networks this paper considers the simple Michaelis-Menten reaction mechanism. After the elimination of diffusional effects a mathematically intractable mass action kinetic model is obtained. The properties of this model are explored via scaling and linearization. The scaling is carried out such that kinetic properties, concentration parameters and external influences are clearly separated. We then try to obtain reasonable estimates for values of the dimensionless groups and examine the dynamic properties of the model over this part of the parameter space. Linear analysis is found to give excellent insight into reaction dynamics and it also gives a forum for understanding and justifying the two commonly used quasi-stationary and quasi-equilibrium analyses. The first finding is that there are two separate time scales inherent in the model existing over most of the parameter space, and in particular over the regions of importance here. Full modal analysis gives a new interpretation of quasi-stationary analysis, and its extension via singular perturbation theory, and a rationalization of the quasi-equilibrium approximation. The new interpretation of the quasi-steady state assumption is that the applicability is intimately related to dynamic interactions between the concentration variables rather than the traditional notion that a quasi-stationary state is reached, after a short transient period, where the rates of formation and decomposition of the enzyme intermediate are approximately equal. The modal analysis reveals that the generally used criterion for the applicability of quasi-stationary analysis that total enzyme concentration must be much less than total substrate concentration, et much less than St, is incomplete and that the criterion et much less than Km much less than St (Km is the well known Michaelis constant) is the appropriate one. The first inequality (et much less than Km) guarantees agreement over the longer time scale leading to quasi-stationary behavior or the applicability of the zeroth order outer singular perturbation solution but the second half of the criterion (Km much less than St) justifies zeroth order inner singular perturbation solution where the substrate concentration is assumed to be invariant. Furthermore linear analysis shows that when a fast mode representing the binding of substrate to the enzyme is fast it can be relaxed leading to the quasi-equilibrium assumption. The influence of the dimensionless groups is ascertained by integrating the equations numerically, and the predictions made by the linear analysis are found to be accurate.(ABSTRACT TRUNCATED AT 400 WORDS)