The mechanistic interpretation of initial velocity and product inhibition data arising from enzyme catalysed reactions whose catalytic cycle is unbranched

A procedure is described which determines the mechanisms of unbranched enzyme catalysed reactions from a knowledge of the experimentally determined kinetic constants. The basis of complementarity in enzymic mechanisms is also discussed.     

An evolutionary approach to enzyme kinetics: Optimization of ordered mechanisms

A theoretical investigation is presented which allows the calculation of rate constants and phenomenological parameters in states of maximal reaction rates for unbranched enzymic reactions. The analysis is based on the assumption that an increase in reaction rates was an important characteristic of the evolution of the kinetic properties of enzymes. The corresponding nonlinear optimization problem is solved taking into account the constraint that the rate constants of the elementary processes do not exceed certain upper limits. One-substrate-one-product reactions with two, three and four steps are treated in detail. Generalizations concern ordered uni-uni-reactions involving an arbitrary number of elementary steps. It could be shown that depending on the substrate and product concentrations different types of solutions can be found which are classified according to the number of rate constants assuming in the optimal state submaximal values. A general rule is derived concerning the number of possible solutions of the given optimization problem. For high values of the equilibrium constant one solution always applies to a very large range of the concentrations of the reactants. This solution is characterized by maximal values of the rate constants of all forward reactions and by non-maximal values of the rate constants of all backward reactions. Optimal kinetic parameters of ordered enzymic mechanisms with two substrates and one product (bi-uni-mechanisms) are calculated for the first time. Depending on the substrate and product concentrations a complete set of solutions is found. In all cases studied the model predicts a matching of the concentrations of the reactants and the corresponding Michaelis constants, which is in good accordance with the experimental data. It is discussed how the model can be applied to the calculation of the optimal kinetic design of real enzymes.     

Optimal design of feedback control by inhibition

The local stability of unbranched biosynthetic pathways is examined by mathematical analysis and computer simulation using a novel nonlinear formalism that appears to accurately describe biochemical systems. Four factors affecting the stability are examined: strength of feedback inhibition, equalization of the values among the corresponding kinetic parameters for the reactions of the pathway, pathway length, and alternative patterns of feedback interactions. The strength of inhibition and the pattern of feedback interactions are important determinants of steady-state behavior. The simple pattern of end-product inhibition in unbranched pathways may have evolved because it optimizes the steady-state behavior and is temporally most responsive to change. Stability in these simple systems is achieved by shortening pathway length either physically or, in the case of necessarily long pathways, kinetically by a wide devergence in the values of the corresponding kinetic parameters for the reactions of the pathway. These conclusions are discussed in the light of available experimental evidence.     

On the stability of the kinetics of a certain class of biomolecular reactions

Summary By means of a formal kinetic model, an analysis of the behavior of a certain type of unbranched sequences of biomolecular reactions is made. The main results are in (i) the characterization of the steady-state, (ii) the specification of a condition under which the largest physically admissible invariant set, containing the steady-state as invariant subset, can be obtained, and (iii) the deduction of parameter restrictions sufficient to assure asymptotic stability in the large in the given invariant set with respect to the steady-state.     

Kinetic model for bifunctional multisubstrate enzymes. steady-state approximation

The steady-state approximation of the generalized model of a bifunctional multisubstrate enzyme is considered. Cases when the reactions catalyzed by the bifunctional enzyme are independent of one another and when one reaction influences the other are considered. Shunting of the kinetic mechanisms and activation of one reaction by the second reaction catalyzed by the bifunctional enzyme are analyzed. The kinetic equations are derived and qualitatively analyzed. It is shown in all cases discussed that the functional relationships between the kinetic parameters observed and the concentrations of the substrates participating in the reactions can be unambiguously defined on the basis of the concept of relationship of the intermediate enzyme forms in the enzymatic process.     

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.     

Determination the rate constants of some biexponential reactions

Reactions that are described by biexponential functions are typical for many biological processes. The kinetics of these reactions is described by transcendental irrational equations interconnecting the reagent concentrations, time and rate constants. Meantime, their graphical representation in the semi-logarithmic coordinates can be decomposed into two straight lines that intercept at some angle. New simple methods for asymptotic numerical solution of the equations describing these reactions are suggested. These methods permit determining the rate constants using the kinetic data of initial substance concentration, which transform into final product according to a two-component model, a sequential model or a competitive model.     

Application of a graphic method for analyzing the kinetics of multienzyme reactions

A new graphic method is proposed to solve kinetic equations for polyenzymic reactions. Each graph apex is corresponded by the transmitting function deduced from kinetic equations by means of Laplas transformation. Application of this procedure allows to simplify the solution of kinetic equations and its analysis. The procedure suggested makes it possible to use the methods of automatic control when solving theoretical problems of enzymology.     

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).     

Isotope exchange in biochemical networks application of an electric circuit technique to a biological reaction system

Electric circuit techniques are employed to derive the kinetic equations for a class of reactions occurring in complex networks as encountered in cellular systems, namely, exchange at equilibrium. The equations connecting equilibrium velocities with kinetic parameters of individual steps of an enzymatic reaction are derived in three ways : by direct chemical considerations, by showing that a form of Kirchhoff's rules is valid for exchange at equilibrium, and by employing the device known in electricity as the delta-star transformation.     

Velocity of Ellman's reaction and its implication for kinetic studies in the millisecond time range

A detailed study of the velocity of the reaction between Ellman's reagent and thiocholine was undertaken, in order to test the possibilities of this reaction as a detection method for the earlier stages of cholinesterases reactions. Experiments were carried out on a stopped-flow apparatus with a built-in spectrophotometer. The obtained experimental data were analyzed by fitting the data to theoretical kinetic equations derived for the reaction. In this way, a complete kinetic characterization of the reaction was obtained. An important practical result derived from our investigations is the finding that, under most experimental conditions, the Ellman's reactions is more than sufficiently rapid as a detection method. However, in the case of reactions in the time scale of 200 milliseconds or less, this being 5 times the half life of Ellman's reaction at standard conditions, one has to consider the interference of this reaction with the enzyme reaction itself.     

Kinetic model for DBT desulphurization by resting whole cells of Pseudomonas putida CECT5279

Bio-desulphurization kinetics of dibenzothiophene (DBT) using Pseudomonas putida CECT 5279, a genetically modified micro-organism (GMO), is studied. A kinetic model describing the 4S route of DBT desulphurization is proposed. Bio-desulphurization experiments have been carried out using resting whole cells of P. putida CECT 5279 obtained at different growth times as biocatalysts. The kinetic equations proposed for each reaction have been previously checked by studying each reaction of the 4S route individually, employing different substrates in different experiments. Finally, simple Michaelis–Menten kinetic equations for the three first reactions catalyzed by two mono-oxygenases (DszC and DszA) and a kinetic equation taking into account competitive inhibition due to product for the final reaction catalyzed by a desulfinase (DszB) have been adopted. DBT has been desulphurized using cells obtained at different growth times (5, 10, 23, 30 and 45 h). The overall kinetic model proposed involving the four reactions of the 4S route was fitted to all the experimental data yielding a set of kinetic parameters able to describe the system evolution. Cell age has influence on the rates of all the reactions: reactions (1), (2) and (3) present maximum rates for cell grown during 30 h, while reaction (4) shows a maximum rate for cells with around 10 h of growth time. However, affinities of each substrate and the inhibition constant of the last reaction are not influenced by the time of growth.     

Irreversibility in unbranched pathways: preferred positions based on regulatory considerations

It has been observed experimentally that most unbranched biosynthetic pathways have irreversible reactions near their beginning, many times at the first step. If there were no functional reasons for this fact, then one would expect irreversible reactions to be equally distributed among all positions in such pathways. Since this is not the case, we have attempted to identify functional consequences of having an irreversible reaction early in the pathway. We systematically varied the position of the irreversible reaction in model pathways and compared the resulting systemic behavior according to several criteria for functional effectiveness, using the method of mathematically controlled comparisons. This technique minimizes extraneous differences in systemic behavior and identifies those that are fundamental. Our results show that a pathway with an irreversible reaction located at the first step, and with all other reactions reversible, is on average better than an otherwise equivalent pathway with all reactions reversible, which in turn is on average better than an otherwise equivalent pathway with an irreversible reaction located at any step other than the first. Pathways with an irreversible first reaction and low concentrations of intermediates (one of the primary criteria for functional effectiveness) exhibit the following profile when compared to fully reversible pathways: changes in the concentration of intermediates in response to changes in the level of initial substrate are equally low, the robustness of the intermediate concentrations and of the flux is similar, the margins of stability are similar, flux is more responsive to changes in demand for end product, intermediate concentrations are less responsive to changes in demand for end product, and transient times are shorter. These results provide a functional rationale for the positioning of irreversible reactions at the beginning of unbranched biosynthetic pathways.     

A method for defining steady-state unidirectional fluxes through branched chemical, osmotic and chemiosmotic reactions

A general mathematical technique is described for deriving analytical expressions and obtaining numerical solutions for the steady-state unidirectional fluxes between two chemical states via any set of intermediate states present within any hypothetical system of unbranched or branched and overlapping elementary processes. The technique is a restricted application of the theory of Markov processes with conditional probabilities being assigned to the chemical state transitions constituting the system of reactions. While, in principle, the technique requires the summation of an infinite power series of a matrix defining the conditional probabilities of single state transitions, the power series is evaluated by means of the Taylor series expansion for matrices. As this technique allows isotopic exchange velocity equations to be derived from systems of reactions in which no distinction between the labelled and unlabelled species is required it provides a distinct and independent alternative to previously proposed methods. The technique is illustrated by application to a mechanism for second-order carrier-mediated transport.     

Understanding glucose transport by the bacterial phosphoenolpyruvate:glycose phosphotransferase system on the basis of kinetic measurements in vitro

The kinetic parameters in vitro of the components of the phosphoenolpyruvate:glycose phosphotransferase system (PTS) in enteric bacteria were collected. To address the issue of whether the behavior in vivo of the PTS can be understood in terms of these enzyme kinetics, a detailed kinetic model was constructed. Each overall phosphotransfer reaction was separated into two elementary reactions, the first entailing association of the phosphoryl donor and acceptor into a complex and the second entailing dissociation of the complex into dephosphorylated donor and phosphorylated acceptor. Literature data on the K(m) values and association constants of PTS proteins for their substrates, as well as equilibrium and rate constants for the overall phosphotransfer reactions, were related to the rate constants of the elementary steps in a set of equations; the rate constants could be calculated by solving these equations simultaneously. No kinetic parameters were fitted. As calculated by the model, the kinetic parameter values in vitro could describe experimental results in vivo when varying each of the PTS protein concentrations individually while keeping the other protein concentrations constant. Using the same kinetic constants, but adjusting the protein concentrations in the model to those present in cell-free extracts, the model could reproduce experiments in vitro analyzing the dependence of the flux on the total PTS protein concentration. For modeling conditions in vivo it was crucial that the PTS protein concentrations be implemented at their high in vivo values. The model suggests a new interpretation of results hitherto not understood; in vivo, the major fraction of the PTS proteins may exist as complexes with other PTS proteins or boundary metabolites, whereas in vitro, the fraction of complexed proteins is much smaller.     

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.     

Kinetic method for differentiating mechanisms for ligand exchange reactions: application to test for substrate channeling in glycolysis.

We have derived analytical expressions for the kinetics of the two mechanisms involved in ligand substitution reactions. These mechanisms are (i) a dissociative mechanism in which the leaving ligand is first dissociated prior to the binding of the incoming ligand and (ii) an associative mechanism where a ternary complex is formed between the incoming ligand and the complex containing the leaving ligand. The equations obtained provide the theoretical basis for differentiating these two mechanisms on the basis of their kinetic patterns of the displacement reactions. Analysis of these equations shows that an associative mechanism can only generate an increasing kinetic pattern for the observed pseudo-first-ordered rate constants as a function of increasing concentration of the incoming ligand and plateaus, in most cases, at a value higher than the off-rate constant of the leaving ligand. However, a dissociative mechanism can generate either an increasing or a decreasing (kapp decreases with increasing concentrations of the incoming ligand) kinetic pattern, depending on the magnitudes of the individual rate constants involved, and, in either case, it will plateau at kapp equal to the koff of the leaving ligand. Therefore, the decreasing kinetic pattern is a hallmark for a dissociative mechanism. This general method was used to settle the dispute of whether NADH is transferred directly via the enzyme-enzyme complex between glycerol-3-phosphate dehydrogenase (GPDH; EC 1.1.1.8) and L-lactate dehydrogenase (LDH; EC 1.1.1.27).(ABSTRACT TRUNCATED AT 250 WORDS)     

Assessment and parameter identification of simplified models to describe the kinetics of semi-continuous biomethane production from anaerobic digestion of green and food waste

Biochemical reactions occurring during anaerobic digestion have been modelled using reaction kinetic equations such as first-order, Contois and Monod which are then combined to form mechanistic models. This work considers models which include between one and three biochemical reactions to investigate if the choice of the reaction rate equation, complexity of the model structure as well as the inclusion of inhibition plays a key role in the ability of the model to describe the methane production from the semi-continuous anaerobic digestion of green waste (GW) and food waste (FW). A parameter estimation method was used to investigate the most important phenomena influencing the biogas production process. Experimental data were used to numerically estimate the model parameters and the quality of fit was quantified. Results obtained reveal that the model structure (i.e. number of reactions, inhibition) has a much stronger influence on the quality of fit compared with the choice of kinetic rate equations. In the case of GW there was only a marginal improvement when moving from a one to two reaction model, and none with inclusion of inhibition or three reactions. However, the behaviour of FW digestion was more complex and required either a two or three reaction model with inhibition functions for both ammonia and volatile fatty acids. Parameter values for the best fitting models are given for use by other authors.