Use and abuse of the quasi-steady-state approximation

The transient kinetic behaviour of an open single enzyme, single substrate reaction is examined. The reaction follows the Van Slyke-Cullen mechanism, a spacial case of the Michaelis-Menten reaction. The analysis is performed both with and without applying the quasi-steady-state approximation. The analysis of the full system shows conditions for biochemical pathway coupling, which yield sustained oscillatory behaviour in the enzyme reaction. The reduced model does not demonstrate this behaviour. The results have important implications in the analysis of open biochemical reactions and the modelling of metabolic systems.     

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.     

Mechanistic origin of the sigmoidal rate behaviour of glucokinase.

Model studies are presented which demonstrate that reactions proceeding by a random ternary-complex mechanism may exhibit most pronounced deviations from Michaelis-Menten kinetics even if the reaction is effectively ordered with respect to net reaction flow. In particular, the kinetic properties and reaction flow characteristics of glucokinase can be accounted for in such terms. It is concluded that insufficient evidence has been presented to support the idea that glucokinase operates by a 'mnemonical' type of mechanism involving glucose binding to distinct conformational states of free enzyme. The sigmoidal rate behaviour of glucokinase can presently be more simply explained in terms of glucose binding to differently ligated states of the enzyme.     

Phase-plane analyses of the Michaelis-Menten reaction equations

The irreversible Michaelis-Menten scheme may be reduced to a pair of autonomous first-order differential equations. The phase-plane behaviour of these is investigated.     

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.     

Steady state kinetics of an enzyme reaction with one substrate and one modifier

This study examines the steady state kinetics of a reaction involving an enzyme, a substrate and a modifier when the reaction follows Michaelis-Menten kinetics. Conditions for Michaelis-Menten kinetics are deduced, and it is shown that an analogue of detailed balance determines the complexity of the rate equations in these cases. A scheme to distinguish many cases of Michaelis-Menten kinetics is presented. It is shown that steady state kinetics are, in general, insufficient to specify the mechanism of a reaction, since different effects of a modifier can give identical steady state kinetic data.     

The catalase–hydrogen peroxide system. Kinetics of catalatic action at high substrate concentrations

1. A re-examination of the catalase-hydrogen peroxide reaction at high substrate concentrations, by using the quenched-flow technique, reveals a more complex kinetic behaviour than that previously reported. At constant reaction time the catalatic process obeys Michaelis-Menten kinetics, but the apparent Michaelis constant is markedly time-dependent, whereas the conventional catalase activity is independent of time. 2. The kinetics of the ;time effect' were analysed and it is suggested that the effect derives from the formation of an inactive species (thought to be catalase Compound II). The process shows Michaelis-Menten kinetics, with a Michaelis constant equal to that for the catalatic reaction in the limit of zero reaction time. 3. It has been confirmed that certain buffer components have marked inhibitory effects on the catalatic reaction and that, in unbuffered systems, catalatic activity is substantially independent of pH in the range 4.7-10.5.     

The total quasi-steady-state approximation is valid for reversible enzyme kinetics

The Briggs-Haldane approximation of the irreversible Michaelis-Menten scheme of enzyme kinetics is cited in virtually every biochemistry textbook and is widely considered the classic example of a quasi-steady-state approximation. Though of similar importance, the reversible Michaelis-Menten scheme is not as well characterized. This is a serious limitation since even enzymatic reactions that go to completion may be reversible. The current work derives a total quasi-steady-state approximation (tQSSA) for the reversible Michaelis-Menten and delineates its validity domain. The tQSSA allows the derivation of uniformly valid approximations for the limit of low enzyme concentrations, ET相似文献   

Mathematical modelling of dynamics and control in metabolic networks. II. Simple dimeric enzymes

The dynamics of enzyme cooperativity are examined by studying a homotropic dimeric enzyme with identical reaction sites, both of which follow irreversible Michaelis-Menten kinetics. The problem is approached via scaling and linearization of the governing mass action kinetic equations. Homotropic interaction between the two sites are found to depend on three dimensionless groups, two for the substrate binding step and one for the chemical transformation. The interaction between the two reaction sites is shown capable of producing dynamic behavior qualitatively different from that of a simple Michaelis-Menten system; when the two sites interact to increase enzymatic activity over that of two independent monomeric enzymes (positive cooperativity) damped oscillatory behavior is possible, and for negative cooperativity in the chemical transformation step a multiplicity of steady states can occur, with one state unstable and leading to runaway behavior. Linear analysis gives significant insight into system dynamics, and their parametric sensitivity, and a way to identify regions of the parameter space where the approximate quasi-stationary and quasi-equilibrium analyses are appropriate.     

On the difficulties of fitting the double Michaelis-Menten equation to kinetic data

The double Michaelis-Menten equation describes the reaction kinetics of two independent, saturable uptake mechanisms. The use of this equation to describe drug uptake has been reported several times in the literature, and several methods have been published to fit the equation to data. So far, however, confidence intervals on the fitted kinetic parameters have not been provided. We present a grid-search method for fitting the double Michaelis-Menten equation to kinetic uptake data, and a Monte-Carlo procedure for estimating confidence intervals on the fitted parameters. We show that the fitting problem is extremely ill-conditioned, and that very accurate data are required before any confidence can be placed in the fitted parameters.     

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 of enzymatic reactions with unstable species

We present a general kinetic analysis of the Michaelis-Menten mechanism for the case in which the enzyme, the enzyme-substrate complex and the product are unstable. The kinetic data analysis which we suggest is based on the time progress curves of the product and/or on the time progress curve of the species into which the immediate product is transformed. This analysis, carried out under conditions of limiting enzyme concentration, allows the determination of the rate and equilibrium constants if adequate experimental results are available. We illustrate the method numerically by computer simulation of the reaction with added experimental errors.     

Kinetic behaviour of proenzymes activation in the presence of different inhibitors for both activating and activated enzymes

In the present paper, a kinetic analysis of a general model for proenzyme activation, where the activating enzyme and also the activated one are reversibly inhibited in two steps by two different inhibitors, has been performed. The cases in which both inhibitors are the same, or in which the inhibition is irreversible (only one or the two inhibition routes) are treated as particular cases of the general model. In addition, the kinetic behaviour of many other proenzyme activation systems involving inhibition, particular cases of the reaction scheme under study, can be obtained. The total number of particular cases for the general model under study is 370, so this approach offers to the scientific community working in limited proteolysis regulation for the first time a method based on general solutions which only needs to be specified to their concrete problem of zymogen activation. Finally, new adimensional parameters are introduced, allowing the knowledgement, in the case that any of the inhibition routes is irreversible, the relative weight of both activation and irreversible inhibition routes.     

Regulation of enzyme activity in adsorptive enzyme systems

The kinetic behaviour of adsorptive enzyme systems with free and adsorbed enzyme forms in rapid equilibrium has been analysed. It has been shown that the dependences of enzymic reaction rate on substrate or “adsorptive effector” concentrations reveal the deviations from simple kinetic laws of Michaelis-Menten type (positive or negative kinetic co-operativity). Such kinetic anomalies should be observed when adsorption of the enzyme results in the changing catalytic properties and when the state of the equilibrium between free and bound enzyme forms depends on the presence of low molecular substances (substrates, coenzymes and various cellular metabolites). The physiological significance of adsorption-desorption processes for the enzyme activity regulation has been emphasized.     

Asymptotic analysis of the Michaelis-Menten reaction equations

The irreversible Michaelis-Menten reaction is studied by the use of the method of multiple scales. Three stages of the reaction are identified, one of which is studied in detail. The results are compared with those of two earlier analyses.     

Effect of pH on the production of lipolyzed butter oil by a calf pregastric esterase immobilized in a hollow-fiber reactor: I. uniresponse kinetics

A calf pregastric esterase immobilized in a hollow-fiber reactor was employed to hydrolyze milkfat, thereby producing a lipolyzed butteroil. The reaction kinetics can be modeled by a two-parameter model of the general Michaelis-Menten form based on a ping-pong bi-bi mechanism; the rate of enzyme deactivation can be modeled as a first-order reaction. The initial concentration of accessible glyceride bonds, [G](O), was estimated by complete saponification of the substrate butteroil as 2400 mM. An extra sum of squares test indicated that not only the parameters of the kinetic generalized Michaelis-Menten model, but also the deactivation-rate constant varied significantly with pH. The optimum pH, for lypolysis is near 6.0 at a temperature of 40 degrees C because at this pH the rate of deactivation of the esterase is minimized.     

Kinetic behaviour of free lipase and mica-based immobilized lipase catalyzing the synthesis of sugar esters

The utilization of natural mica as a biocatalyst support in kinetic investigations is first described in this study. The formation of lactose caprate from lactose sugar and capric acid, using free lipase (free-CRL) and lipase immobilized on nanoporous mica (NER-CRL) as a biocatalyst, was evaluated through a kinetic study. The apparent kinetic parameters, K(m) and V(max), were determined by means of the Michaelis-Menten kinetic model. The Ping-Pong Bi-Bi mechanism with single substrate inhibition was adopted as it best explains the experimental findings. The kinetic results show lower K(m) values with NER-CRL than with free-CRL, indicating the higher affinity of NER-CRL towards both substrates at the maximum reaction velocity (V(max,app)>V(max)). The kinetic parameters deduced from this model were used to simulate reaction rate data which were in close agreement with the experimental values.     

Kinetic behaviour of soluble and mitochondrial bound lactate dehydrogenase

Rabbit liver mitochondrial fraction shows lactate dehydrogenase activity. The kinetic behaviour of mitochondrial bound enzyme fits a bibi sequential type mechanism as well as the cytosolic rabbit liver lactate dehydrogenase. The bound enzyme has greater values of Km(NADH) and Km(pyruvate) than the soluble one, suggesting that binding induces a decrease in the affinity of both substrates. The behaviour of the free and the mitochondrial-bound enzyme is of the Michaelis-Menten type, but the kinetics of a mixture of rabbit liver cytosolic and mitochondrial-bound lactate dehydrogenase is sigmoidal, suggesting that a cooperative phenomenon takes place.     

Mass-transfer limitations for immobilized enzyme-catalyzed kinetic resolution of racemate in a fixed-bed reactor

A mathematical model has been developed for immobilized enzyme-catalyzed kinetic resolution of racemate in a fixed-bed reactor in which the enzyme-catalyzed reaction (the irreversible uni-uni competitive Michaelis-Menten kinetics is chosen as an example) was coupled with intraparticle diffusion, external mass transfer, and axial dispersion. The effects of mass-transfer limitations, competitive inhibition of substrates, deactivation on the enzyme effective enantioselectivity, and the optical purity and yield of the desired product are examined quantitatively over a wide range of parameters using the orthogonal collocation method. For a first-order reaction, an analytical solution is derived from the mathematical model for slab-, cylindrical-, and spherical-enzyme supports. Based on the analytical solution for the steady-state resolution process, a new concise formulation is presented to predict quantitatively the mass-transfer limitations on enzyme effective enantioselectivity and optical purity and yield of the desired product for a continuous steady-state kinetic resolution process in a fixed-bed reactor.