From continuum Fokker-Planck models to discrete kinetic models

Two theoretical formalisms are widely used in modeling mechanochemical systems such as protein motors: continuum Fokker-Planck models and discrete kinetic models. Both have advantages and disadvantages. Here we present a "finite volume" procedure to solve Fokker-Planck equations. The procedure relates the continuum equations to a discrete mechanochemical kinetic model while retaining many of the features of the continuum formulation. The resulting numerical algorithm is a generalization of the algorithm developed previously by Fricks, Wang, and Elston through relaxing the local linearization approximation of the potential functions, and a more accurate treatment of chemical transitions. The new algorithm dramatically reduces the number of numerical cells required for a prescribed accuracy. The kinetic models constructed in this fashion retain some features of the continuum potentials, so that the algorithm provides a systematic and consistent treatment of mechanical-chemical responses such as load-velocity relations, which are difficult to capture with a priori kinetic models. Several numerical examples are given to illustrate the performance of the method.     

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).     

Evaluation of the kinetic parameters of the activation of trypsinogen by trypsin

Kinetic analysis of the mechanism of trypsinogen activation by trypsin under rapid equilibrium conditions and certain relationships between the rate constants are presented. The kinetic equations are valid from the beginning of the reaction. In addition, we suggest a procedure, based on the above equations, for the evaluation of the kinetic parameters of the reaction. This procedure is applied to a set of experimental data collected during the activation of bovine trypsinogen by trypsin at 30 degrees C (pH 8.1) in 0.01 M CaCl2. In this system, the amount of active enzyme increases exponentially, as expected from an autocatalytic process. The apparent rate constant, delta, governing this increase would vary linearly with the trypsinogen concentration, [Z]0, if no Michaelis complex was detectable. However, the increase in delta with [Z]0 is clearly non-linear and fits a hyperbola (delta = k2[Z]0/(Kz + [Z]0)) well.     

Analysis of progress curves by simulations generated by numerical integration.

A highly flexible computer program written in FORTRAN is presented which fits computer-generated simulations to experimental progress-curve data by an iterative non-linear weighted least-squares procedure. This fitting procedure allows kinetic rate constants to be determined from the experimental progress curves. Although the numerical integration of the rate equations by a previously described method [Barshop, Wrenn & Frieden (1983) Anal. Biochem. 130, 134-145] is used here to generate predicted curves, any routine capable of the integration of a set of differential equations can be used. The fitting program described is designed to be widely applicable, easy to learn and convenient to use. The use, behaviour and power of the program is explored by using simulated test data.     

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.     

Assessment of formal and low structured kinetic modeling of polyhydroxyalkanoate synthesis from complex substrates

A formal kinetic mathematical model for poly-(3-hydroxybutyrate-co-3-hydroxyvalerate-co-4-hydroxybutyrate) [P(3HB-co-3HV-co-4HB)] terpolyester synthesis from glucose and galactose derived from whey permeate supplemented with gamma-butyrolactone by the archaeon Haloferax mediterranei was created. Further, a low structured mathematical model for poly-3-hydroxybutyrate synthesis from whey permeate by Pseudomonas hydrogenovora was developed. In both cases, biosyntheses for obtaining the experimental data used for compiling the models were performed via fed-batch cultivations. The model developed for H. mediterranei consists of 10 differential and 11 algebraic equations, including 27 kinetic constants. The model compiled for P. hydrogenovora encompasses 10 differential and 3 algebraic equations, including 36 kinetic constants. Both models were solved by Runge-Kuta variable step numerical integration with Monte Carlo parameter optimization procedure. Difficulties arising from the modeling of redirection of metabolic fluxes from biomass growth toward polyhydroxyalkanoate synthesis and byproducts are discussed.     

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 reversible reactions on linear lattices with neighbor effects

As a model for a variety of reaction processes on long chain molecules, for example, helix formation, a kinetic theory on a linear lattice is presented. Each reaction site can undergo reversible transitions between two states (0 and 1) with rates depending on the slate of its nearest neighbors. The system of coupled rate equations for the frequencies of specified runs of 0's and 1's is infinite for an infinite chain. In contrast to the case of irreversible processes, the system cannot, be written down by inspection. A procedure for the systematic derivation of the rate equations is developed which can be programmed on a computer. Explicit expressions for runs up to length four, involving runs up to length five are obtained without recourse to the computer. For the solution of the rate equations a closure must necessarily be imposed, and a possible procedure is pointed out. Furthermore the equilibrium relations following from the model are considered. The well-known equilibrium results for nearest-neighbor interactions represent a special case of these equations.     

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.     

Analysis of the mechanism and kinetics of thermal inactivation of enzymes: Critical assessment of isothermal inactivation experiments

The study deals with information obtained from conventional isothermal inactivation experiments. When the lumped character of enzyme activity was analysed, identical activity-time profiles could be obtained from several mechanisms when the number of equivalent mechanisms increased exponentially with the number of parameters in the corresponding kinetic equations. The assessment of isothermal evaluation of inactivation kinetics was demonstrated using selected experiments from the literature exhibiting a deviation from first-order kinetics. Experiments were classified, according to the shape of the inactivation curve, into biphasic, grace-period and activation-period inactivations. A unified evaluation procedure, based on the discrimination among a few kinetic models derived from a general unimolecular series mechanism, was employed for all experiments. It was shown that regardless of the shape of the inactivation curve, only a few experiments needed more than a simple series mechanism, represented by a biexponential curve, for a satisfactory fit. Carrying out inactivation experiments at several temperatures was shown to be very effective in checking the consistency of kinetic equations obtained by isothermal evaluation. The results of isothermal evaluation could always be questioned in this way and no consequent conclusions on inactivation mechanism could be drawn.     

Kinetics of the trypsinogen activation by enterokinase and trypsin

A global kinetic analysis of the mechanisms of the trypsinogen activation by enterokinase and trypsin is presented. The kinetic equations of both the transient-phase and the steady-state of these mechanisms are presented. In addition, we here derive the corresponding kinetic equations for the case in which the condition of rapid equilibrium prevails and we propose a kinetic data analysis. The significance of this approach to the treatment of other zymogen activation processes is discussed.     

The use of steady-state rate equations to analyse progress curve data.

Analysis of progress curves for enzyme-catalyzed reactions has been made by using a procedure that does not require the derivation of complex integrated rate equations. The method involves conversion of progress curve data to reaction velocities that are then fitted to the appropriate differential rate equation. Application of the procedure to data obtained for the reaction catalyzed by aspartate aminotransferase (L-aspartate:2-oxoglutarate aminotransferase, EC 2.6.1.1), showed that the resulting values for the kinetic parameters agreed well with those obtained by conventional progress curve analysis (Duggleby, R.G. and Morrison, J.F. (1978) Biochim. Biophys. Acta 526, 398--409).     

Kinetics of a model of autocatalysis, coupling of a reaction in which the enzyme acts on one of its substrates.

A global kinetic analysis is presented of a model of an enzyme autocatalytic process, to which a reaction is coupled, in which the enzyme acts upon one of its substrates. The kinetic equations of both the transient phase and the steady state are derived for this mechanism. In addition, we determine the corresponding kinetic equations for several particular cases which are characterized by certain relations between the rate constants. Finally, a kinetic data analysis is proposed for one of these particular cases. It can easily be extended to any of the other cases.     

Three steady state situation in an open chemical reaction system. I

In an isothermal continuously stirred tank reactor (open chemical reaction system) fed by sulphuric acid solutions of bromate, bromide and cerium)(III) bistability (three steady state situation) is experimentally observed. This remarkable behavior, based on the instability of one steady state, has important consequences for the understanding of excitability and biochemical control mechanisms. The mass-balance equations for the reactor and the chemical mechanism of the reaction are combined into a simple mathematical model. The behavior of the resulting nonlinear differential equations is examined analytically and by a graphical integration procedure (method of isoclines). Using realistic kinetic data, the model shows the same behavior as observed in the experiment.     

Kinetic study in the transient phase of the suicide inactivation of frog epidermis tyrosinase

This paper deals with the kinetic study of a multisubstrate mechanism with enzyme inactivation induced by a suicide substrate. A transient phase approach has been developed that enables the deduction of explicit equations of product concentration vs. time. From these equations kinetic constants which characterize the suicide substrate can be obtained. This study with tyrosinase enzyme, which acts on L-dopa and catechol allowed us to determine the corresponding kinetic parameters, indicating that catechol is about 8-times more powerful as a suicide substrate than is L-dopa.     

Active site-directed and allosteric effectors of regulatory enzymes: the activation of aspartate transcarbamylase by substrate and transition state analogues

An extension of the two-state theory of Monod, Wyman &; Changeux (1965) has been used to explain quantitatively the effects of substrate and transition state analogues on the kinetic and physcco-chemical properties of aspartate transcarbamylase. The derived equations accurately predict the observed activation at low concentrations followed by inhibition at higher concentrations of such analogues. They also predict the binding, kinetic and physical behaviour of the enzyme in the presence of both active site-directed effectors, such as succinate, and allosteric effectors, such as cytidine triphosphate, both of which the enzyme is subjected to physiologically. The equations are applicable to other enzymes provided that the system behaves as though there is only one substrate and that binding equations can be converted to kinetic equations by replacing a dissociation constant by a microscopic Michaelis constant.