A generalized theoretical treatment of the kinetics of an enzyme-catalysed reaction in the presence of an unstable irreversible modifier

Kinetic behavior of the general modifier mechanism of Botts and Morales with non-equilibrium binding

In this paper, we perform a complete analysis of the kinetic behavior of the general modifier mechanism of Botts and Morales in both equilibrium steady states and non-equilibrium steady states (NESS). Enlightened by the non-equilibrium theory of Markov chains, we introduce the net flux into discussion and acquire an expression of the rate of product formation in NESS, which has clear biophysical significance. Up till now, it is a general belief that being an activator or an inhibitor is an intrinsic property of the modifier. However, we reveal that this traditional point of view is based on the equilibrium assumption. A modifier may no longer be an overall activator or inhibitor when the reaction system is not in equilibrium. Based on the regulation of enzyme activity by the modifier concentration, we classify the kinetic behavior of the modifier into three categories, which are named hyperbolic behavior, bell-shaped behavior, and switching behavior, respectively. We show that the switching phenomenon, in which a modifier may convert between an activator and an inhibitor when the modifier concentration varies, occurs only in NESS. Effects of drugs on the Pgp ATPase activity, where drugs may convert from activators to inhibitors with the increase of the drug concentration, are taken as a typical example to demonstrate the occurrence of the switching phenomenon.

The influence of product instability on slow-binding inhibition

We present a kinetic study of an enzyme reaction that takes place with slow-binding inhibition where the immediate product undergoes a spontaneous or induced process of decomposition. A kinetic study of an enzyme process, in which a slow-binding inhibition process and a decomposition of the immediate product of the reaction take place simultaneously is performed. The corresponding explicit concentration-time equations were obtained. Using the analytical solutions obtained, which were tested numerically, we suggest a procedure that allows the discrimination between the particular cases considered and the evaluation of the principal kinetic parameters of the reaction.

Expressions for the fractional modification in different monocyclic enzyme cascade systems: analysis of their validity tested by numerical integration

This paper presents the derivation, under a minimal set of assumptions, of a general expression for the steady-state fractional modification of an interconvertible protein involved in four different schemes of monocyclic enzyme cascade systems. From this general expression we derive, as particular cases, other, simpler expressions by applying additional assumptions and which have, therefore, a smaller range of validity. Some of these particular expressions coincide with those already obtained in previous contributions on individualised analyses. We discuss the relationships between the kinetic parameters and the concentrations needed for the fulfilment of the additional assumptions. The goodness of the analysis was tested by reference to the shape in the steady-state of the simulated time progress curves obtained by numerical integration.

Kinetic analysis of the general modifier mechanism of Botts and Morales involving a suicide substrate

Suicide substrates are widely used in enzymology for studying enzyme mechanisms and designing potential drugs. The presence of a reversible modifier decreases or increases the rate of substrate-induced inactivation, with evident physiological and experimental consequences. To date, only the action of a competitive or uncompetitive inhibitor of an enzyme system involving suicide substrate has been reported. In this paper, we analyse the kinetics of enzyme-catalysed reactions which evolve in accordance with the general modifier mechanisms of Botts and Morales in which enzyme inactivation is induced by suicide substrate. Rapid equilibrium of all of the reversible reaction steps involved is assumed and the time course equations for the residual enzyme activity, the inactive enzyme forms and the reaction product are derived. Partition ratios giving the relative weight of the product and inactive enzyme concentrations, and the relative contribution to the product formation of each of the unmodified and modified catalytic routes, are studied. New indices pointing to the conditions under which the modifier acts as inhibitor or as activator are suggested. The goodness of the analytical solutions is tested by comparison with the simulated curves obtained by numerical integration. An experimental design and kinetic data analysis to evaluate the kinetic parameters from the time progress curves of the product are proposed. From these results, those corresponding to several reaction mechanisms involving both a suicide substrate and a modifier, and which can be regarded as particular cases of the general case analysed here, can be directly and easily derived.

Kinetic analysis of multisite phosphorylation using analytic solutions to Michaelis-Menten equations

Phosphorylation-induced expression or modulation of a functional protein is a common signal in living cells. Many functional proteins are phosphorylated at multiple sites and it is frequently observed that phosphorylation at one site enhances or suppresses phosphorylation at another site. Therefore, characterizing such cooperative phosphorylation is important. In this study, we determine a temporal progress curve of multisite phosphorylation by analytically integrating the Michaelis-Menten equations in time. Using this theoretical progress curve, we derive the useful criterion that an intersection of two progress curves implies the presence of cooperativity. Experiments generally yield noisy progress curves. We fit the theoretical progress curves to noisy progress curves containing 4% Gaussian noise in order to determine the kinetics of the phosphorylation. This fitting correctly identifies the sites involved in cooperative phosphorylation.

On a nonelementary progress curve equation and its application in enzyme kinetics

The analytical equation describing progress curves of an enzyme catalyzed reaction acting upon the Michaelis-Menten mechanism has been known for the case in which only the free enzyme incurs a loss of its activity, either spontaneously or as a result of an irreversible inhibitor action. The solution of differential equations which defines the rates of enzyme inactivation and substrate utilization is expressed by a nonelementary function in equation of an implicit type that precludes direct calculation of the extent of reaction at any time. Previously, the implicit equations have been rearranged to the alternative formulas and solved by the Newton-Raphson method, but this procedure may fail when used upon the presented equation. For this reason the other root-finding numerical method was applied, and the enzyme kinetic parameters of such numerically solved implicit equation for the reaction mechanism of irreversibly inhibited acetylcholinesterase were fitted to the experimental data by a nonlinear regression computer program.

The kinetic effect of product instability in a Michaelis-Menten mechanism with competitive inhibition

In most kinetic studies it is assumed that both the reactant and the products are stable. However, under certain conditions spontaneous decomposition or deterioration caused by one of the participating species occurs. Studies, in which a species (the free enzyme, the enzyme-substrate complex, an inhibitor or the product of the reaction) is unstable, have appeared in the literature. However, to our knowledge, the enzymatic systems, in which a competitive inhibition and a decomposition or transformation of the products take place simultaneously, have not been studied so far. In this paper, we present a kinetic analysis of an enzyme reaction that follows a Michaelis-Menten mechanism, in which the free enzyme suffers a competitive inhibition simultaneously with the decomposition of the immediate product. In this study, we have linearised the differential equations that describe the kinetics of the process. Under the assumption of limiting concentration of enzyme, we have obtained and tested the explicit equation describing the time dependence of the product concentration using numerical calculus. With this equation and the experimental progress curve of the product, we constructed an easy procedure for the evaluation of the principal kinetic parameters of the process.

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 enzyme systems with unstable suicide substrates

This paper deals with kinetic studies of enzyme reaction mechanisms with enzyme inactivation induced by an unstable suicide substrate. An initial steady-state of the catalytic route is assumed and the time course equations for the total active enzyme forms and the reaction product have been derived. 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 is suggested and the time course equations of an important reaction mechanisms involving a stable suicide substrate and which can be regarded as particular case of that under study has also been derived from the corresponding equations. The simplicity of our method allows its systematic application to more complex mechanisms.

Inactivation kinetics of the reduced spinach chloroplast fructose-1,6-bisphosphatase by subtilisin

The course of inactivation of the reduced spinach chloroplast fructose-1,6-bisphosphatase by digestion with subtilisin has been followed by the progress curve method [Tsou, C. L. (1988) Adv. Enzymol. 61, 381-436] and found to follow first-order kinetics. On the basis of the hydrolysis of the substrate, fructose 1,6-bisphosphate, at different concentrations during proteolysis by subtilisin, the first-order inactivation rate constants for the free enzyme and the enzyme-substrate complex can both be determined. The ratio between the inactivation rate constants for the free enzyme and the enzyme-substrate complex indicates strong protection against subtilisin proteolysis by the substrate. It is proposed that the above ratio can be used as a quantitative measure of substrate protection for enzyme inactivation generally. As it has been found that the site of proteolysis is located in a loop region near the N-terminus and well away from the active site, the substrate protection indicates a conformation change of the enzyme away from the substrate binding site.

Kinetics of inactivation of bovine pancreatic ribonuclease A by bromopyruvic acid

The kinetic theory of substrate reaction during the modification of enzyme activity [Duggleby (1986) J. Theor. Biol. 123, 67-80; Wang and Tsou (1990) J. Theor. Biol. 142, 531-549] has been applied to a study of the inactivation kinetics of ribonuclease A by bromopyruvic acid. The results show that irreversible inhibition belongs to a non-competitive complexing type inhibition. On the basis of the kinetic equation of substrate reaction in the presence of the inhibitor, all microscopic kinetic constants for the free enzyme, the enzyme-substrate complex and the enzyme-product complex have been determined. The non-competitive inhibition type indicates that neither the substrate nor the product affects the binding of bromopyruvic acid to the enzyme and that the ionization state of His-119 may be the same in both the enzyme-substrate and the enzyme-product complexes.

Kinetics of inactivation of aminoacylase by 2-chloromercuri-4-nitrophenol: a new type of complexing inhibitor

The kinetic theory of the substrate reaction during modification of enzyme activity previously described [Tsou (1988) Adv. Enzymol. 61, 381-436] has been applied to a study of the inactivation kinetics of aminoacylase by 2-chloromercuri-4-nitrophenol (MNP). The results indicate that the mechanism of reaction between MNP and aminoacylase is a special type of irreversible inhibition. The main features of this type of inhibitor are as follows: (i) the reaction kinetics of inhibitor with enzyme is a single exponential process; (ii) inhibition shows a noncompetitive, complexing behavior; (iii) the first inhibitor-enzyme complex, EI, still has some enzyme activity, and hence the plot of [P]infinity versus the reciprocal of inhibitor concentration gives a straight line with a positive intercept at the ordinate. On the basis of the kinetic equation of substrate reaction in the presence of the inhibitor, a plotting method has been developed for determining the inhibition kinetic constants. As an example, all reaction kinetic constants of aminoacylase with 2-chloromercuri-4-nitrophenol have been determined. The results of the present study suggest that the essential thiol group at the active site of aminoacylase may have a significant effect on the catalytic step but is not involved in substrate binding.

Kinetic study of an enzyme-catalysed reaction in the presence of novel irreversible-type inhibitors that react with the product of enzymatic catalysis

In the present paper a kinetic study is made of the behaviour of a Michaelis-Menten enzyme-catalysed reaction in the presence of irreversible inhibitors rendered unstable in the medium by their reaction with the product of enzymatic catalysis. A general mechanism involving competitive, non-competitive, uncompetitive and mixed irreversible inhibition with one or two steps has been analysed. The differential equation that describes the kinetics of the reaction is non-linear and computer simulations of its dynamic behaviour are presented. The results obtained show that the systems studied here present kinetic co-operativity for a target enzyme that follows the simple Michaelis-Menten mechanism in its action on the substrate, except in the case of an uncompetitive-type inhibitor.

Experimental approach to the kinetic study of unstable site-directed irreversible inhibitors: kinetic origin of the apparent positive co-operativity arising from inactivation of trypsin by p-amidinophenylmethanesulphonyl fluoride

Experimental characterization of enzyme inactivation by unstable irreversible inhibitors has only previously been carried out by using discontinuous methods involving preincubation, removal of samples and further residual activity assays. A continuous method for the kinetic study of these inhibitors in the presence of an auxiliary substrate was recently proposed in a theoretical study. This method was based on approximate expressions for the evolution of the product concentration, which contained series expansions with five or more exponential terms, seriously complicating their use in practice. In the present paper, a new experimental method has been developed for the kinetic study of unstable and site-directed irreversible inhibitors, considering two different ranges of inhibitor concentration. Thus at low inhibitor concentrations, the system evolves from an initial to a final steady state, the rates of which are described by exact analytical equations. At high inhibitor concentrations, however, the product accumulation can be described by an exact uniexponential equation. This simple and efficient method has been applied to the kinetic study of trypsin inactivation by p-amidinophenylmethanesulphonyl fluoride, near the optimum pH of the enzyme. The dependence of the final steady-state rate on the substrate concentration shows apparent positive co-operativity which has not previously been reported. The kinetic origin of this type of co-operativity is predicted by one of the exact analytical equations derived here. The instability of new protein and non-protein irreversible inhibitors has to be carefully characterized to prevent true unstable irreversible inhibitors being wrongly described as allosteric reversible inhibitors.

In defence of the general validity of the Cha method of deriving rate equations. The importance of explicit recognition of the thermodynamic box in enzyme kinetics

1. The suggestion by Segel & Martin [(1988) J. Theor. Biol. 135, 445-453] that the well-known schematic method for the derivation of rate equations for combined quasi-equilibrium-steady-state models proposed by Cha [(1968) J. Biol. Chem. 243, 820-825] may not be generally applicable was shown to be incorrect. By contrast, Cha's method was shown (a) to yield correct initial-rate equations that are exact and (b) not to require any constraints on the relative magnitudes of rate constants for slow steps outside the quasi-equilibrium segments of the kinetic model, including those suggested by Segel & Martin. 2. Examination of residual King-Altman patterns for the general modifier model of Botts & Morales [(1953) Trans. Faraday Soc. 49, 696-707] revealed the reasons for the erroneous conclusions reached by Segel & Martin. The errors arise from the failure to take account of fluxes in parallel pathways that connect the two isolated groups of enzyme species existing in quasi-equilibrium with the modifier. 3. A similar failure to take explicit account of parallel pathways in a thermodynamic box that delayed proper appreciation of the form of pH-dependence of kcat/Km is briefly discussed.