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.     

Methods of determining rate constants in single-substrate–single-product enzyme reactions. Use of induced transport: limitations of product inhibition

1. The calculation of the rate constants from steady-state kinetics of a single-substrate-single-product enzyme reaction in which there is an isomerization of the enzyme is described. 2. It is shown that even with the use of isotopically labelled substrates a set of solutions for the constants is obtained rather than a unique solution. However, limits are derived within which they must lie. 3. The most appropriate observations to determine the rate constants are measurements of V(max.) and K(m) for both substrate and product, and measurement of the degree of countertransport in an induced-transport test. 4. Experimental procedures for induced-transport tests and the quantitative interpretation of the results obtained are discussed. 5. Product inhibition is shown to be an ambiguous and imprecise means of determining the rate constants. Further, the absence of a [substrate]x[product] term in the denominator of the steady-state rate equation does not necessarily mean that the isomerization of the enzyme is rapid, since the term also disappears when the isomerization is very slow. 6. Similar considerations apply to carrier mechanisms.     

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.     

Alleviating product inhibition in cellulase enzyme Cel7A

Enzymes that degrade cellulose into glucose are one of the most expensive components of processes for converting cellulosic biomass to fuels and chemicals. Cellulase enzyme Cel7A is the most abundant enzyme naturally employed by fungi to depolymerize cellulose, and like other cellulases is inhibited by its product, cellobiose. There is thus great economic incentive for minimizing the detrimental effects of product inhibition on Cel7A. In this work, we experimentally generated 10 previously proposed site‐directed mutant Cel7A enzymes expected to have reduced cellobiose binding energies (the majority of mutations were to alanine). We then tested their resilience to cellobiose as well as their hydrolytic activities on microcrystalline cellulose. Although every mutation tested conferred reduced product inhibition (and abolished it for some), our results confirm a trade‐off between Cel7A tolerance to cellobiose and enzymatic activity: Reduced product inhibition was accompanied by lower overall enzymatic activity on crystalline cellulose for the mutants tested. The tempering effect of mutations on inhibition was nearly constant despite relatively large differences in activities of the mutants. Our work identifies an amino acid in the Cel7A product binding site of interest for further mutational studies, and highlights both the challenge and the opportunity of enzyme engineering toward improving product tolerance in Cel7A. Biotechnol. Bioeng. 2016;113: 330–338. © 2015 The Authors. Biotechnology and Bioengineering Published by Wiley Periodicals, Inc.     

The computerized derivation of rate equations for enzyme reactions on the basis of the pseudo-steady-state assumption and the rapid-equilibrium assumption.

A computer program is developed for the derivation of the rate equation for enzyme reactions on the basis of the pseudo-steady-state assumption and the combination of the pseudo-steady-state and the rapid-equilibrium assumptions. The program not only has an easy input method, but also can obtain a complete rate equation in itself on only one run. The usefulness of the program is demonstrated by deriving the rate equations for some typical enzyme reactions. Details of the program have been deposited as Supplementary Publication SUP 50141 (42 pages) at the British Library Lending Division, Boston Spa, Wetherby, West Yorkshire LS23 7QB, U.K., from whom copies may be obtained as indicated in Biochem. J. (1988), 249, 5.     

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.