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.     

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.     

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.     

Generalized rate equation for single-substrate enzyme catalyzed reactions

The most widely used rate expression for single-substrate enzyme catalyzed reactions, namely the Michaelis-Menten kinetics is based upon the assumption that enzyme concentration is in excess of the substrate in the medium and the rate is mainly limited by the substrate concentration according to saturation kinetics. However, this is only a special case and the actual rate expression varies depending on the initial enzyme/substrate ratio (E₀/S₀). When the substrate concentration exceeds the enzyme concentration the limitation is due to low enzyme concentration and the rate increases with the enzyme concentration according to saturation kinetics. The maximum rate is obtained when the initial concentrations of the enzyme and the substrate are equal. A generalized rate equation was developed in this study and special cases were discussed for enzyme catalyzed reactions.     

A general method for the analysis of random bisubstrate enzyme mechanisms

In the present communication, a general method for the kinetic analysis of random bisubstrate mechanisms is described. The method comprises a stepwise application of the following kinetic and ligand-binding experiments: determination of steady-state kinetic constants, product inhibition patterns, maximum rate relationships, application of alternate substrates, application of dead-end inhibitors, direct binding of substrates, kinetic isotope effects, and isotope exchange studies. This general method was applied to a practical example: a yeast alcohol dehydrogenase-catalyzed oxidation of 2-propanol by NAD⁺ at pH 7.0, 25°C. It was found that this fully reversible reaction proceeds by a steady-state random Bi-Bi mechanism, whereby both dead-end complexes are formed.     

Corrected equations for calculation of constants in enzyme inhibition and activation

Equations for calculation of the constants of biparametrical types of enzyme inhibition and activation were obtained that take into account a ratio of the lengths of L vector projections representing such reactions in the three-dimensional K (m)V I coordinate system. This allows higher accuracy of calculation and is more correct for comparison of these constants. Examples of data analysis of enzyme inhibition and activation by using the traditional equations (they do not take into account the lengths of vector projections) and corrected ones (they take into account the lengths of vector projections) are given. The corrected and traditional equations are used for calculation of the constants of biparametrical types of enzyme inhibition and activation.     

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.     

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.     

Angiotensin-converting enzyme inhibition reduces neuroblastoma cell growth rate

Because of the known capacity of angiotensin II to serve as a growth factor in multiple tissues, we elected to study the effects of renin-angiotensin system inhibition on the growth of human SH-SY5Y neuroblastoma cells. Cells were treated with captopril (0.05-5 mg/ml), enalapril, or enalaprilat (0.02-5 mg/ml) or saralasin (0.1-0.25 mg/ml). In all cases, statistically significant reductions in cell growth were seen over 5 days of culture. In additional experiments, captopril and enalaprilat significantly decreased thymidine incorporation into DNA in these cells. The administration of angiotensin II in the presence of captopril partially offset these suppressive effects.     

Kinetic analysis of enzyme reactions with slow-binding inhibition.

In this paper we present a general kinetic study of slow-binding inhibition processes, i.e. enzyme reactions that do not respond instantly to the presence of a competitive inhibitor. The analysis that we present is based on the equation that describes the formation of products with time in each case on the experimental progress curve. It is carried out under the condition of limiting enzyme concentration and allows the discrimination between the different cases of slow-binding inhibition. The mechanism in which the formation of complex enzyme-inhibitor is a single or two slow steps or follow a rapid equilibrium, has been considered. The corresponding explicit equations of each case have been obtained and checked by numerical integration. A kinetic data analysis to evaluate the corresponding kinetic parameters is suggested. We illustrate the method, numerically by computer simulation, of the reaction and present some numerical examples that demonstrate the applicability of our procedure.     

Kinetics of enzymes subject to very strong product inhibition: Analysis using simplified integrated rate equations and average velocities

Enzymes which catalyze energetically unfavorable reactions in the physiological direction are likely to be strongly inhibited by the reaction products. (Some energetically favorable reactions may also display strong "product inhibition" when assayed in the reverse direction.) In some cases, the inhibition caused by an accumulating product is so potent that true initial velocities cannot be directly determined using conventional assay methods. Continuous removal of the inhibitory product may be mitigated against by the nature of the assay or the unavailability of the appropriate coupling enzyme. It can be shown that if (a) only one inhibitory product is allowed to accumulate and (b) the substrate concentrations remain essentially constant over the assay period (i.e. Kproduct less than or equal to 10(-2)Ksubstrate, so that the decreasing reaction rate stems only from progressive product inhibition), then plots of reciprocal average (apparent) velocity (i.e. 1/v = t/[P]) versus [P] are linear and extrapolate to 1/v0, the reciprocal of the initial uninhibited velocity at the fixed substrate concentrations. Intercept replots give the usual initial velocity reciprocal plot patterns and permit Vmax and the substrate Km's to be determined. Slope replots are diagnostic of the type of inhibition exerted by the accumulating product and permit the inhibition constants to be determined. If all the appropriate coupling enzymes are available, some kinetic mechanisms can be diagnosed using data derived from the reaction progress curves in the presence of one accumulating product at a time.