New relations for steady-state enzyme kinetics and their application to rapid equilibrium assumption

With the use of graph theory, new relations for steady-state enzyme kinetics are derived and strictly proved for an arbitrary mechanism of enzyme-catalyzed reaction containing a reversible segment. Using these relations, a general principle for rapid equilibrium assumption is formulated and proved: the reversible bound segment can be considered as an equilibrium segment only when the values of the base trees that are not proper to this segment can be neglected (within prescribed accuracy) in relation to the values of the base trees that belong to this segment. In contrast with the foreign base trees, the base trees that are proper to the segment have the following properties: the tree that is directed to the base within this segment does not contain edges leaving this segment; and the tree that is directed to the base outside the segment contains only one edge leaving this segment. Equilibrium variations are assessed for steady-state concentrations of intermediates in the equilibrium segment, numerical expressions are obtained for the accuracy of determination of intermediate concentrations as well as for the accuracy of determination of the rate of enzyme-catalyzed reaction under rapid equilibrium assumption.     

Quasi-equilibrium assumption for arbitrary mechanism of enzymatic reaction. criteria for existence of equilibrium segments

The possible application of the quasi-equilibrium assumption for an arbitrary mechanism of enzymatic reaction is considered. It is shown at what ratios of kinetic constants a segment consisting of two, three, and four intermediates may be considered as an equilibrium one. Expressions for evaluation of accuracy of distribution of intermediate concentrations inside the equilibrium segment and accuracy of determination of intermediate concentrations inside and outside the equilibrium segment as a function of the ratio of kinetic constants are derived. A method for determination of the limitations on the ratio of rate constants for an equilibrium segment of arbitrary structure is suggested.     

Quasi-equilibrium assumption in enzyme kinetics. Necessary and sufficient conditions and accuracy of its application for single-substrate reactions

Steady-state kinetics of compulsory-ordered single-substrate irreversible and reversible enzyme reactions with two, three, and arbitrary number of intermediates were observed. Necessary and sufficient conditions for application of the quasi-equilibrium assumption and restrictions of this assumption were found in cases of two and three intermediates in the equilibrium segment. For all cases, accuracy of the quasi-equilibrium assumption was evaluated.     

Determination of half-reaction equilibrium in a ping-pong enzyme mechanism

Substituted enzyme (or ping-pong) mechanisms usually involve enzymes that exist in two forms that alternate during the catalytic reaction. A method is described here for determining the position of the equilibrium of a half reaction in a ping-pong enzyme mechanism that is based on the kinetics of the burst reaction which occurs upon addition of reactants that recycle the enzyme from one form to another. The theoretical basis for the analysis is developed, and the method is applied to the half reaction of the aldimine form of aspartate transaminase with difluoro-oxaloacetate. Special issue dedicated to Herman Bachelard     

The computerized derivation of steady-state rate equations for enzyme kinetics.

A FORTRAN 77 program is described for the derivation of steady-state rate equations for enzyme kinetics. Input is very simple and consists of the two enzyme forms and the two rate constants for each step in the mechanism. The program may be run interactively or off-line. The results are produced after collecting together the algebraic coefficients of like concentration terms, taking account of sign. A fully interactive BASIC version running on a BBC Microcomputer is also available. Details of the programs have been deposited as Supplementary Publication SUP 50126 (45 pages) with the British Library Lending Division, Boston Spa, Wetherby, West Yorkshire LS23 7BQ, U.K., from whom copies may be obtained as indicated in Biochem. J. (1984) 217, 5.     

Plasmid copy number control: a case study of the quasi-steady-state assumption.

Perelson & Brendel (1989, J. molec. Biol. 208, 245-255) have proposed kinetic models for the control of plasmid copy number, based on experiments by J. Tomizawa and his associates. The quasi-steady-state assumptions (QSSA) made in the analysis of these models are justified in the present paper, thereby providing an example of how QSSA can provide a powerful and reliable tool in the analysis of biological kinetics.     

The equilibrium assumption in estimating the parameters of metapopulation models

1.  The construction of a predictive metapopulation model includes three steps: the choice of factors affecting metapopulation dynamics, the choice of model structure, and finally parameter estimation and model testing.
2.  Unless the assumption is made that the metapopulation is at stochastic quasi-equilibrium and unless the method of parameter estimation of model parameters uses that assumption, estimates from a limited amount of data will usually predict a trend in metapopulation size.
3.  This implicit estimation of a trend occurs because extinction-colonization stochasticity, possibly amplified by regional stochasticity, leads to unequal numbers of observed extinction and colonization events during a short study period.
4.  Metapopulation models, such as those based on the logistic regression model, that rely on observed population turnover events in parameter estimation are sensitive to the implicit estimation of a trend.
5.  A new parameter estimation method, based on Monte Carlo inference for statistically implicit models, allows an explicit decision about whether metapopulation quasi-stability is assumed or not.
6. Our confidence in metapopulation model parameter estimates that have been produced from only a few years of data is decreased by the need to know before parameter estimation whether the metapopulation is in quasi-stable state or not.
7. The choice of whether metapopulation stability is assumed or not in parameter estimation should be done consciously. Typical data sets cover only a few years and rarely allow a statistical test of a possible trend. While making the decision about stability one should consider any information about the landscape history and species and metapopulation characteristics.     

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.     

Analysis of the mechanism of chloramphenicol acetyltransferase by steady-state kinetics. Evidence for a ternary-complex mechanism.

The mechanism of the enzymic reaction responsible for chloramphenicol resistance in bacteria was examined by steady-state kinetic methods. The forward reaction catalysed by chloramphenicol acetyltransferase leads to inactivation of the antibiotic. Use of alternative acyl donors and acceptors, as well as the natural substrates, has yielded data that favour the view that the reaction proceeds to the formation of a ternary complex by a rapid-equilibrium mechanism wherein the addition of substrates may be random but a preference for acetyl-CoA as the leading substrate can be detected. Chloramphenicol and acetyl-CoA bind independently, but the correlation between directly determined and kinetically derived dissociation constants is imperfect because of an unreliable slope term in the rate equation. The reverse reaction, yielding acetyl-CoA and chloramphenicol, was studied in a coupled assay involving citrate synthase and malate dehydrogenase, and is best described by a rapid-equilibrium mechanism with random addition of substrates. The directly determined dissociation constant for CoA is in agreement with that derived from kinetic measurements under the assumption of an independent-sites model.     

The sucked-flow analyser: A simple apparatus for automatic determination of steady-state enzyme kinetics

The sucked-flow analyser is a modified stopped-flow apparatus for automatic measurements of initial rates of enzyme reactions at varying concentrations of substrate or inhibitor. The flow through the system is driven by a water suction pump and regulated by magnetic valves. Sample and reagent solution are aspired through tubes whose resistances to laminar flow determine a precise ratio of mixing in the observation cell. A minicomputer controls all operations, collects the data and calculates the results. Measurements on teh β-galactosidase and the cytochome P-450 reactions are presented.     

An investigation of basic assumption in enzyme kinetics using results of the geometric theory of differential equations

Up to the present time, the following property of the product component in the reversible one substrate-one intermediate-one product enzymic mechanism has been taken only as anassumption, viz., during the course of the reaction, the time-rate of change of product concentration is never negative and the product concentration never exceeds its equilibrium value. Applying the methods of the geometric theory of ordinary differential equations it is shown that this result follows as a direct deduction from the differential equations governing the mechanism together with the initial conditions. Further, the nature of the equilibrium point as a stable node is established.     

Sedimentation equilibrium in a solution containing an arbitrary number of solute species at arbitrary concentrations: theory and application to concentrated solutions of ribonuclease

Simple expressions are derived describing the equilibrium concentration gradient of each species in a solution containing an arbitrary number of solute species at arbitrary concentration, as a function of the concentration of all species. Quantitative relationships between the species gradients and experimentally observable signal gradients are presented. The expressions are model-free and take into account both attractive and repulsive interactions between all species. In order to analyze data obtained from strongly nonideal solutions, a statistical thermodynamic model for repulsive solute-solute interactions is required. The relations obtained are utilized to analyze the dependence of the equilibrium gradient of ribonuclease A in phosphate-buffered saline, pH 7.4, upon total protein concentration. Experimental results are interpreted in the context of a model for weak self-association leading to the formation of significant amounts of oligomers at total protein concentrations exceeding 25 g/l.     

Evaluation by steady-state enzyme kinetics of the role of tightly bound nucleotides during photophosphorylation

The ATP synthetase of chloroplast membranes binds ADP and ATP with high affinity, and the binding becomes quasi-irreversible under certain conditions. One explanation of the function of these nucleotides is that they are transiently tightly bound during ATP synthesis as part of the catalytic process, and that the release of tightly bound ATP from one catalytic site is promoted when ADP and P(i) bind to a second catalytic site on the enzyme. Alternatively, it is possible that the tightly bound nucleotides are not catalytic, but instead have some regulatory function. We developed steady-state rate equations for both these models for photophosphorylation and tested them with experiments where two alternative substrates, ADP and GDP, were phosphorylated simultaneously. It was impossible to fit the results to the equations that assumed a catalytic role for tightly bound nucleotides, whether we assumed that both ADP and GDP, or only ADP, are phosphorylated by a mechanism involving substrate-induced release of product from another catalytic site. On the other hand, the equations derived from the regulatory-site model that we tested were able to fit all the results relatively well and in an internally consistent manner. We therefore conclude that the tightly bound nucleotides most likely do not derive from catalytic intermediates of ATP synthesis, but that substrate (and possibly also product) probably bind both to catalytic sites and to noncatalytic sites. The latter may modulate the transition of the ATP-synthesizing enzyme complex between its active and inactive states.     

The kinetics of the cytochrome-c-azurin redox equilibrium

A preliminary analysis of the temperature-jump relaxation spectrum of the cytochrome-c-azurin redox system from $\text{Pseudomonas fluorescens}$ revealed two chemical relaxation times. These are explained in terms of two association steps and an electron transfer between the associated species.     

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.     

A 'branched' mechanism of the reverse reaction of yeast glutathione reductase. An estimation of the enzyme standard potential values from the steady-state kinetics data

The reduced glutathione-linked NADP+ reduction, catalyzed by yeast glutathione reductase, follows a 'sequential' or 'ping-pong' mechanism at high or low NADP+ concentrations, respectively. The pattern of the NADPH and NADP+ cross-inhibition reflects not only the competition for the binding site, but the shift of the reaction equilibrium as well. A 'branched' scheme of the glutathione reductase reaction is presented. The enzyme standard potential (-255 mV, pH 7.0) was estimated from the ratio of the NADPH and NADP+ rate constants corresponding to the ping-pong mechanism.