Competitive and uncompetitive inhibitors simultaneously acting on an autocatalytic zymogen activation reaction

The time course of the residual enzyme activity of a general model consisting of an autocatalytic zymogen activation process inhibited by an irreversible competitive inhibitor and an irreversible uncompetitive inhibitor has been studied. Approached analytical expressions which furnish the time course of the residual enzyme activity from the onset of the reaction depending on the rate constants and initial concentration have been obtained. The goodness and limitations of the analytical equations were checked by comparing with the results obtained from the numerical integration, i.e. with the simulated progress curves. A dimensionless parameter giving the relative contributions of both the activation and the inhibitions routes is suggested, so that the value of this parameter determines whether the activation or the inhibitions routes prevail or if both processes are balanced during the time for which the analytical expressions are valid. The effects of the initial zymogen, free enzyme and inhibitors concentrations are analysed. Finally an experimental design and kinetic data analysis is proposed to evaluate simultaneously the kinetic parameters involved and to discriminate between different zymogen activation processes which can be considered particular cases of the general model.     

Linear mixed irreversible inhibition of the autocatalytic activation of zymogens. Kinetic analysis checked by simulated progress curves

Autocatalytic zymogen activation is a phenomenon of great importance for understanding some fundamental physiological processes involved in the enzyme regulation of gastrointestinal-tract enzymes, blood coagulation, fibrinolysis and the complement system. Examples of such processes are the activation of prekallikrein, trypsinogen and pepsinogen, all of which are controlled by natural proteinase inhibitors. This work studies the kinetics of a general autocatalytic zymogen activation process overlapped by two two-step irreversible inhibitions, i.e. a linear mixed irreversible inhibition. The kinetic equations for the whole course of the reaction are derived for this mechanism. In addition, we determine the corresponding kinetics for a number of particular cases of the general model analyzed, i.e. for reversible and irreversible non-competitive, competitive and uncompetitive inhibition systems which are considered particular cases of the general mechanism studied. The kinetic behavior of the system is related to a parameter, a dimensionless quantity, which shows whether the inhibition or the activation route prevails, in a similar way to that which we have previously carried out for other mechanisms. Finally, based on the kinetic equations obtained, a procedure for discriminating between the different mechanisms considered is suggested. The results of this contribution can be directly applied to most physiological autocatalytic zymogen activations in the presence of an inhibitor, allowing their complete kinetic characterization and suggesting procedures for varying the relative weight of the catalytic and inhibition routes or for changing the predominant route.     

Kinetic analysis of a general model of activation of aspartic proteinase zymogens

Starting from a simple general reaction mechanism of activation of aspartic proteinase zymogens involving an uni- and a bimolecular simultaneous route, the time course equation of the concentration of the zymogen and of the activated enzyme have been derived. From these equations, an analysis quantifying the relative contribution to the global process of the two routes has been carried out for the first time. This analysis suggests a way to predict the time course of the relative contribution as well as the effect of the initial zymogen and activating enzyme concentrations, on the relative weight. An experimental design and kinetic data analysis is suggested to estimate the kinetic parameters involved in the reaction mechanism proposed. Finally, we apply some of our results to experimental data obtained by other authors in experimental studies of the activation of some aspartic proteinase zymogens.     

Kinetic analysis of a general model of activation of aspartic proteinase zymogens involving a reversible inhibitor. I. Kinetic analysis

Starting from a simple general reaction mechanism of activation of aspartic proteinases zymogens involving a uni- and a bimolecular simultaneous activation route and a reversible inhibition step, the time course equation of the zymogen, inhibitor and activated enzyme concentrations have been derived. Likewise, expressions for the time required for any reaction progress and the corresponding mean activation rates as well as the half-life of the global zymogen activation have been derived. An experimental design and kinetic data analysis is suggested to estimate the kinetic parameters involved in the reaction mechanism proposed.     

Kinetic behaviour of proenzymes activation in the presence of different inhibitors for both activating and activated enzymes

In the present paper, a kinetic analysis of a general model for proenzyme activation, where the activating enzyme and also the activated one are reversibly inhibited in two steps by two different inhibitors, has been performed. The cases in which both inhibitors are the same, or in which the inhibition is irreversible (only one or the two inhibition routes) are treated as particular cases of the general model. In addition, the kinetic behaviour of many other proenzyme activation systems involving inhibition, particular cases of the reaction scheme under study, can be obtained. The total number of particular cases for the general model under study is 370, so this approach offers to the scientific community working in limited proteolysis regulation for the first time a method based on general solutions which only needs to be specified to their concrete problem of zymogen activation. Finally, new adimensional parameters are introduced, allowing the knowledgement, in the case that any of the inhibition routes is irreversible, the relative weight of both activation and irreversible inhibition routes.     

Kinetic analysis of a general model of activation of aspartic proteinase zymogens involving a reversible inhibitor. II. Contribution of the uni- and bimolecular activation routes

From the kinetic study carried out in part I of this series (preceding article) an analysis quantifying the relative contribution to the global process of the uni- and bimolecular routes has been carried out. This analysis suggests a way to predict the time course of the relative contribution as well as the effect on this relative weight of the initial zymogen, inhibitor and activating enzyme concentrations.     

Kinetics of autocatalytic zymogen activation measured by a coupled reaction: pepsinogen autoactivation

A kinetic study was performed of a model for an autocatalytic zymogen activation process involving both intra- and intermolecular routes, to which a chromogenic reaction in which the active enzyme acts upon one of its substrates was coupled to continuously monitor the reaction. Kinetic equations describing the evolution of species involved in the system with time were obtained. These equations are valid for any zymogen autocatalytic activation process under the same initial conditions. Experimental design and kinetic data analysis procedures to evaluate the kinetic parameters, based on the derived kinetic equations, are suggested. In addition, a dimensionless distribution coefficient was defined, which shows mathematically whether the intra- or the intermolecular route prevails once the kinetic parameters involved in the system are known. The validity of the results obtained was checked using simulated curves for the species involved. As an example of application of the method, the system is experimentally illustrated by the continuous monitoring of pepsinogen transformation to pepsin.     

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.     

The kinetics of enzyme systems involving activation of zymogens

A general model of zymogen activation is proposed and explicit kinetic equations for the time courses of the various species and products involved are given. These equations are valid for the whole course of the reaction and therefore for both the transient phase and the steady state. This model is sufficiently general to include mechanisms possessing one or more steps of zymogen activation besides possible steps of inhibition (reversible or irreversible) or inactivation.     

Contribution of the intra- and intermolecular routes in autocatalytic zymogen activation: application to pepsinogen activation

Taking as the starting point a recently suggested reaction scheme for zymogen activation involving intra- and intermolecular routes and the enzyme-zymogen complex, we carry out a complete analysis of the relative contribution of both routes in the process. This analysis suggests the definition of new dimensionless parameters allowing the elaboration, from the values of the rate constants and initial conditions, of the time course of the contribution of the two routes. The procedure mentioned above related to a concrete reaction scheme is extrapolated to any other model of autocatalytic zymogen activation involving intra- and intermolecular routes. Finally, we discuss the contribution of both of the activating routes in pepsinogen activation into pepsin using the values of the kinetic parameters given in the literature.     

Kinetics of an autocatalytic zymogen reaction in the presence of an inhibitor coupled to a monitoring reaction

A global kinetic analysis of a model consisting of an autocatalytic zymogen-activation process, in which an irreversible inhibitor competes with the zymogen for the active site of the proteinase, and a monitoring coupled reaction, in which the enzyme acts upon one of its substrates, is presented. This analysis is based on the progress curves of any of the two products released in the monitoring reaction. The general solution is applied to an important particular case in which rapid equilibrium conditions prevail. Finally, we suggest a procedure to predict whether the inhibition or activation route dominates in the steady state of the system. These results generalize our previous analysis of simpler mechanisms.     

Kinetic analysis of zymogen autoactivation in the presence of a reversible inhibitor.

Limited proteolysis is a highly specific irreversible process, which can serve to initiate physiological function by converting a precursor protein into a biologically active form. When the activating enzyme and the activated enzyme coincide, the process is an autocatalytic zymogen activation (i.e. reactions in which the zymogens serves as a substrate for the corresponding active enzyme). The activity of proteases is frequently regulated by the binding of specific protease inhibitors. Thus, to understand the biological regulation of proteolysis, one must understand the role of protease inhibitors. In the present study, a detailed kinetic analysis of autocatalytic reaction modulated by a reversible inhibitor is represented. On the basis of the kinetic equation, a novel procedure is developed to evaluate the kinetic parameters of the reaction. As an example of the application of this method, effects of acetamidine, p-amidinobenzamidine and benzamidine on the autoactivation of trypsinogen by trypsin were studied.     

Kinetic analysis of enzyme systems with suicide substrate in the presence of a reversible, uncompetitive inhibitor.

We present a general kinetic analysis of enzyme catalyzed reactions evolving according to a Michaelis-Menten mechanism, in which an uncompetitive, reversible inhibitor acts. Simultaneously, enzyme inactivation is induced by an unstable suicide substrate, i.e. it is a Michaelis-Menten mechanism with double inhibition: one originating from the substrate and another originating from the reversible inhibitor. Rapid equilibrium of the reversible reaction steps involved is assumed and the time course equations for the reaction product have been derived under the assumption of limiting enzyme. The goodness of the analytical solutions has been tested by comparison with simulated curves obtained by numerical integration. A kinetic data analysis to determine the corresponding kinetic parameters from the time progress curve of the product is suggested.     

Optimal temperature policy for immobilized enzyme packed bed reactor performing reversible Michaelis-Menten kinetics using the disjoint policy.

The optimal temperature policy that maximizes the time-averaged productivity of a continuous immobilized enzyme packed bed reactor is determined. This optimization study takes into consideration the enzyme thermal deactivation with substrate protection during the reactor operation. The general case of reversible Michaelis-Menten kinetics under constant reactor feed flow rate is assumed. The corresponding nonlinear optimization problem is solved using the calculus of variations by applying the disjoint policy. This policy reduces the optimization problem into a differential-algebraic system, DAE. This DAE system defines completely the optimal temperature-time profiles. These profiles depend on the kinetic parameters, feed substrate concentration, operating period, and the residence time and are characterized by increasing form with time. Also, general analytical expressions for the slopes of the temperature and residual enzyme activity profiles are derived. An efficient solution algorithm is developed to solve the DAE system, which results into a one-dimensional optimization problem with simple bounds on the initial feed temperature. The enzymatic isomerization of glucose into fructose is selected as a case study. The computed productivities are very close to that obtained by numerical nonlinear optimization with simpler problem to solve. Moreover, the computed conversion profiles are almost constant over 90% of the operating periods, thus producing a homogeneous product.     

Analysis of the thermal unfolding of porcine procarboxypeptidase A and its functional pieces by differential scanning calorimetry

Porcine pancreatic procarboxypeptidase A and its tryptic peptides, carboxypeptidase A and the activation segment, have been studied by high-sensitivity differential scanning calorimetry (DSC). The thermal denaturation of the zymogen and the active enzyme has been carried out at two pH values, 7.5 and 9.0, at different ionic strengths and at different scan rates. The endothermic transitions for these two proteins were always irreversible under all conditions investigated. The denaturation behaviour of both proteins seems to fit very well with the kinetic model for the DSC study of irreversible unfolding of proteins recently proposed by one of our groups. From this model, the activation energies obtained for the denaturation of the pro- and carboxypeptidase were 300 +/- 20 kJ mol-1 and 250 +/- 14 kJ mol-1 respectively. On the other hand, the isolated activation segment appears as a thermostable piece with a highly reversible thermal unfolding which follows a two-state process. The denaturation temperature observed for the isolated segment was always at least 15 K higher than those of the zymogen and the active enzyme.     

Quantitative analysis of a fully generalized four-state kinetic scheme

To describe the macroscopic behavior of many ion channels, at a minimum a four-state kinetic scheme is needed to provide for three processes: a delay in activation development, the activation process, and inactivation. I present here an analytical solution for a fully generalized four-state kinetic scheme in which every state can transit to every other state and any initial conditions can be specified. The solution describes the time courses of the probabilities of occupancy of each state during a step change in the rate constants of the scheme and includes closed-form expressions for the relaxation time constants and steady-state probabilities of occupancy as functions of the rate constants. Solutions for several relevant special cases are also included along with demonstrations that the general solution yields the correct behavior for several reduced or special cases where the result is independently known.     

Kinetic analysis of a simplified scheme of autocatalytic zymogen activation.

Proteolytic enzymes are usually biosynthesized as somewhat larger inactive precursors known as zymogens. These zymogens must undergo an activation process, usually a limited proteolysis, to attain their catalytic activity. When the activating enzyme and the activated enzyme coincide, the process is an autocatalytic zymogen activation. In the present study, a kinetic analysis of the entire progress curve for the autocatalytic zymogen activation reactions is presented. On the basis of the kinetic equations, a novel procedure is developed to evaluate the kinetic parameters of the reactions. This procedure is particularly useful for the fast zymogen autoactivation reactions. As two examples, the novel procedure is used to analyse the autocatalytic activation of bovine trypsinogen and human blood coagulation factor XII (Hageman factor).     

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

A generalized theoretical treatment of the kinetics of an enzyme-catalysed reaction in the presence of an unstable irreversible inhibitor (or activator) is presented. Analytical expressions describing the time-dependence of product formation have been derived in coefficient form amenable to non-linear regression analysis for two operationally distinct types of reaction mechanism dependent on whether the reaction of the unstable modifier (X) with either or both the free enzyme (E) and enzyme-substrate complex (ES) occurs as a simple bimolecular process, or proceeds through the intermediacy of either or both adsorptive enzyme-modifier (EX) and enzyme-modifier-substrate (EXS) complexes in what may be considered as an extension of the Botts-Morales general modifier mechanism for (stable) reversible enzyme inhibitors and activators. Special cases of both models are classified in an analogous way to the traditional naming of reversible enzyme modifications, and guidelines concerning tests of mechanism and determination of kinetic parameters are given. In particular, it has been shown that kinetic constants describing enzyme inactivation by an unstable site-specific inhibitor forming a reversible EX complex prior to covalent modification step may be determined from a single progress curve. Kinetic analysis of the extended Botts-Morales mechanism describing irreversible enzyme inactivation has demonstrated that analytical expressions describing the time-course of product formation may be derived for a stable modifier by retaining the usual steady-state assumptions regarding the fluxes around ES and EXS provided quasi-equilibrium modifier binding to E and ES is assumed, but for unstable modifiers all of the binding steps must be assumed to be at quasi-equilibrium in the steady-state, except under restrictive circumstances.     

Irreversible inhibition of trypsin by TLCK. A continuous method for kinetic study of irreversible enzymatic inhibitors in the presence of substrate

This work presents a kinetic study on irreversible inhibition of trypsin by TLCK, using a new experimental approach. The method consists of the incubation of the enzyme with an irreversible inhibitor in the presence of a substrate which allows enzyme turnover as well as continuous measurement of the appearance of the product, a simultaneous change in the initial concentrations of the irreversible inhibitor and enzyme being undertaken, though a constant ratio between the latter, is maintained. This new approach enables the kinetic constants for TLCK, k2 and K1, to be determined.