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.     

Three mechanisms and rapid-equilibrium rate equations for a type of reductase reaction

Rapid-equilibrium rate equations for enzyme-catalyzed reactions are especially useful when the mechanism involves a number of pKs, but they are also useful when some reactants have stoichiometric numbers greater than one or hydrogen ions are produced or consumed in the rate-determining step. The pH dependencies of limiting velocities, Michaelis constants, and reaction velocities for the forward reaction are discussed for two examples of reductase reactions of the type mR + O -> products, where R is the reductant and O is the oxidant. For the nitrate reductase reaction (EC 1.9.6.1), m = 2 and two hydrogen ions are consumed. For the nitrite-ferredoxin reductase reaction (EC 1.7.7.1), m = 6 and eight hydrogen ions are consumed. The expressions for the limiting velocities, Michaelis constants, and rate equations for the forward reaction are derived for two ordered mechanisms and the random mechanism. Three Mathematica programs are used to make plots of kinetic parameters as functions of pH and three-dimensional plots of rapid-equilibrium velocities as functions of [O] and [R] for arbitrary sets of input parameters.     

A steady-state kinetic method for the verification of the rapid-equilibrium assumption in allosteric enzymes.

A method for testing the validity of the rapid-equilibrium assumption as it might apply to allosteric enzymes using exclusively steady-state kinetic data is presented. The method is based upon a recognition that the ratio of apparent dissociation constants for the allosteric ligand, obtained under conditions of limiting and saturating substrate concentration, must yield the thermodynamic value for the coupling parameter between the substrate and allosteric ligand even in the general steady-state case. If this value is found to be equal to the apparent coupling parameter determined from the ratio of limiting values of the Michaelis constant for substrate obtained in the absence and saturating presence of the allosteric ligand, then the substrate can be correctly viewed as effectively achieving a binding equilibrium with the enzyme in the steady-state. The utility and limitations of this method are demonstrated by examining the ADP activation of beef heart mitochondrial NAD-dependent isocitrate dehydrogenase.     

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.     

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.     

Calculation of steady-state rate equations and the fluxes between substrates and products in enzyme reactions.

1. Two methods are described for deriving the steady-state velocity of an enzyme reaction from a consideration of fluxes between enzyme intermediates. The equivalent-reaction technique, in which enzyme intermediates are systematically eliminated and replaced by equivalent reactions, appears the most generally useful. The methods are applicable to all enzyme mechanisms, including three-substrate and random Bi Bi Ping Pong mechanisms. Solutions are obtained in algebraic form and these are presented for the common random Bi Bi mechanisms. The steady-state quantities of the enzyme intermediates may also be calculated. Additional steps may be introduced into enzyme mechanisms for which the steady-state velocity equation is already known. 2. The calculation of fluxes between substrates and products in three-substrate and random Bi Bi Ping Pong mechanisms is described. 3. It is concluded that the new methods may offer advantages in ease of calculation and in the analysis of the effects of individual steps on the overall reaction. The methods are used to show that an ordered addition of two substrates to an enzyme which is activated by another ligand will not necessarily give hyperbolic steady-state-velocity kinetics or the flux ratios characteristic of an ordered addition, if the dissociation of the ligand from the enzyme is random.     

On the pseudo-steady-state approximation and Tikhonov theorem for general enzyme systems

The effectiveness of the Tikhonov theorem for justifying the so-called pseudo-steady-state approximation (PSSA) in general closed enzyme systems is proved under a widely satisfied sufficient condition on the ligand stoichiometry, which also allows us to state that all slow variables of the degenerate system are driven by a single scalar function (the pseudo- steady-state rate equation). Finally, remarks on the perturbative small parameter and open enzyme systems are made.     

A slide rule for deriving the rate equations of enzyme catalysed reactions with unbranched mechanisms

It has been proposed that mRNA stability in Escherichia coli is enhanced by association with ribosomes and that failure of ribosome initiation into polysomes results in message inactivation. This hypothesis is examined with the aid of a simple steady queuing model from which mRNA lifetimes and other cell parameters may be calculated. Agreement with experimentally determined lifetimes is good.     

Integrated rate equations for enzyme-catalysed first-order and second-order reactions.

Generalized rate equations covering all mechanisms giving hyperbolic initial-rate kinetics with stoichiometry A in equilibrium P, A in equilibrium P + Q, A + B in equilibrium P and A + B in equilibrium P + Q were integrated. The results are regular and reasonably economical.     

Computer program for the expression of the kinetic equations of enzyme reactions as functions of the rate constants and the initial concentrations.

A versatile computer program with an easy input method has been developed for the construction of the terms in kinetic equations of enzyme reactions. It allows the expression of the time-dependence of the concentrations of all of the species involved as functions of the kinetic parameters. The mathematical theory used in this paper, the program and examples of its use have been deposited as Supplementary Publication SUP 50159 (41 pages) at the British Library Document Supply Centre, Boston Spa, Wetherby, West Yorkshire LS23 7BQ, U.K., from whom copies can be obtained on the terms indicated in Biochem. J. (1990) 265, 5.     

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.     

Time course equations of the amount of substance in a linear compartmental system and their computerized derivation

In this contribution, we present the symbolic time course equations corresponding to a general model of a linear compartmental system, closed or open, with or without traps and with zero input. The steady state equations are obtained easily from the transient phase equations by setting the time --> infinity. Special attention has been given to the open systems, for which an exhaustive kinetic analysis has been developed to obtain important properties. Besides, the results have been particularized to open systems without traps and an alternative expression for the distribution function of exit times has been provided. We have implemented a versatile computer program, that is easy to use and with a user-friendly format of the input of data and the output of results. This computer program allows the user to obtain all the information necessary to derive the symbolic time course equations for closed or open systems as well as for the derivation of the distribution function of exit times.     

The equilibrium assumption is valid for the kinetic treatment of most time-dependent protein-modification reactions.

To facilitate mechanistic interpretation of the kinetics of time-dependent inhibition of enzymes and of similar protein modification reactions, it is important to know when the equilibrium assumption may be applied to the model: formula: (see text). The conventional criterion of quasi-equilibrium, k + 2 less than k-1, is not always easy to assess, particularly when k + 2 cannot be separately determined. It is demonstrated that the condition k + 2 less than k-1 is necessarily true, however, when the value of the apparent second-order rate constant for the modification reaction is much smaller than the value of k + 1. Since k + 1 is commonly at least 10(7)M-1.S-1 for substrates, it is probable that the equilibrium assumption may be properly applied to most irreversible inhibitions and modification reactions.