Rate Equations and Product Yields of Serine Proteinase Catalysed Transfer Reactions

The steady state rate equations of transfer reactions catalysed by enzymes that follow the serine proteinase reaction mechanism in their hydrolysis reactions, have been solved and integrated. The integrated equations allow for calculations of maximal yields of product and of the time, t_max, at which that yield is present in a given reaction mixture. These important quantities have not been dealt with in previous theoretical studies of such systems.     

Initial rates. A new plot.

Excellent estimations of initial rates can be obtained from plots of delta P/t versus product formed (where P is the instantaneous concentration of the product). delta P/t is the chord from P0,t0 to P,t on an ordinary P-versus-t plot. When the chord is plotted as a function of product, the intercept at P0 of the resulting curve is necessarily dP/dt0. This curve approximates to a straight line extremely closely in all cases tested thus far. If delta P/t versus product is calculated from the integrated rate equation for a first-order reaction, and if a straight line is fitted through points representing the first 50% of the reaction, the discrepancy between the true initial rate and dP/dt0 estimated from the plot is 0.68%. For the most common form of the integrated rate equation for catalysed reactions the discrepancy varies between 0 and 0.90%. Because of the complexities of the integrated rate equations, catalysed second-order reactions have not been evaluated directly; uncatalysed reactions have been done instead. For a reaction with one reactant and two products, the discrepancy varies from 0.68 to 2.02%. For two reactants and one product, it varies from 0 to 0.68%; for two and two, 0 to 2.02%. The larger discrepancies occur only when unfavourable equilibrium constants are being overcome by the initial conditions.     

Explicit analytic approximations for time-dependent solutions of the generalized integrated Michaelis–Menten equation

Various explicit reformulations of time-dependent solutions for the classical two-step irreversible Michaelis–Menten enzyme reaction model have been described recently. In the current study, I present further improvements in terms of a generalized integrated form of the Michaelis–Menten equation for computation of substrate or product concentrations as functions of time for more real-world, enzyme-catalyzed reactions affected by the product. The explicit equations presented here can be considered as a simpler and useful alternative to the exact solution for the generalized integrated Michaelis–Menten equation when fitted to time course data using standard curve-fitting software.     

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).     

Calculation of biochemical net reactions and pathways by using matrix operations.

Pathways for net biochemical reactions can be calculated by using a computer program that solves systems of linear equations. The coefficients in the linear equations are the stoichiometric numbers in the biochemical equations for the system. The solution of the system of linear equations is a vector of the stoichiometric numbers of the reactions in the pathway for the net reaction; this is referred to as the pathway vector. The pathway vector gives the number of times the various reactions have to occur to produce the desired net reaction. Net reactions may involve unknown numbers of ATP, ADP, and Pi molecules. The numbers of ATP, ADP, and Pi in a desired net reaction can be calculated in a two-step process. In the first step, the pathway is calculated by solving the system of linear equations for an abbreviated stoichiometric number matrix without ATP, ADP, Pi, NADred, and NADox. In the second step, the stoichiometric numbers in the desired net reaction, which includes ATP, ADP, Pi, NADred, and NADox, are obtained by multiplying the full stoichiometric number matrix by the calculated pathway vector.     

Microfabricated reaction and separation systems

Over the past year there have been a number of recent advances in the fields of miniaturized reaction and separation systems, including the construction of fully integrated 'lab-on-a-chip' systems. Microreactors, which initially targeted DNA-based reactions such as the polymerase chain reaction, are now used in several other chemical and biochemical assays. Miniaturized separation columns are currently employed for analyzing a wide variety of samples including DNA, RNA, proteins and cells. Although significant advances have been made at the component level, the realization of an integrated analysis system still remains at the early stages of development.     

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.     

Integrated steady state rate equations and the determination of individual rate constants.

Integrated steady state rate equations have been used to determine the kinetic constants (Vs, Ks, Vp, and Kp) and rate constants (k1, k2, k3, and k4) of the reversible enzyme mechanism: (see article). The fumarase reaction has been used as a model to illustrate the procedures for determining these constants. In contrast to initial velocity studies, the values of the constants have been obtained by examining the enzyme reaction in only one direction rather than in both forward and reverse directions. To accomplish this, a new procedure is described for fitting data to integrated rate equations which eliminates problems encountered when data are analyzed graphically. The advantages of examining on enzyme reaction in one direction with these new procedures allow this method to be extended to the examination of enzymes with simple mechanisms where initial velocities are difficult to measure because either the substrate or product is not readily available, or because the reaction is not readily reversible.     

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.     

A stochastic analysis of first-order reaction networks

A stochastic model for a general system of first-order reactions in which each reaction may be either a conversion reaction or a catalytic reaction is derived. The governing master equation is formulated in a manner that explicitly separates the effects of network topology from other aspects, and the evolution equations for the first two moments are derived. We find the surprising, and apparently unknown, result that the time evolution of the second moments can be represented explicitly in terms of the eigenvalues and projections of the matrix that governs the evolution of the means. The model is used to analyze the effects of network topology and the reaction type on the moments of the probability distribution. In particular, it is shown that for an open system of first-order conversion reactions, the distribution of all the system components is a Poisson distribution at steady state. Two different measures of the noise have been used previously, and it is shown that different qualitative and quantitative conclusions can result, depending on which measure is used. The effect of catalytic reactions on the variance of the system components is also analyzed, and the master equation for a coupled system of first-order reactions and diffusion is derived. All authors contributed equally to this work.     

Computer simulation of sustained oscillations in peroxidase-oxidase reaction

A system of differential equations of second order exhibiting transitional behaviour and sustained oscillations has been obtained for a complete scheme of the peroxidase-oxidase reaction. The concentrations of hydrogen peroxide and of hydrogen donor radicals are slow variables of the system. The most essential reactions responsible for oscillations have been selected. Analysis of the system in phase plane and in parameter space has been carried out. The dependence of oscillation period and amplitude on the parameter values has been investigated.     

Kinetics of reactivation during refolding of guanidine-denatured pancreatic ribonuclease A

The method aforementioned (Liu, W. and Tsou, C.L. (1987) Biochim. Biophys. Acta 916, 455-464) for the study of the kinetics of irreversible modification of enzyme activity has been applied to the reactivation of guanidine-denatured ribonuclease A, by following the hydrolysis of cyclic CMP during refolding upon diluting a guanidine-denatured enzyme with a substrate-containing buffer. Appropriate equations have been derived to deal with the kinetics of the substrate reaction during the course of activation, while the product formed, 3'CMP, is a competitive inhibitor. When the overall process consists of multiple first-order reactions, the individual rate constants could be obtained by suitable semilogarithmic plots. Moreover, in certain cases, it can be distinguished from the shapes of the plots, whether the overall process consists of parallel or consecutive first-order reactions. The kinetics for the reactivation reaction has been compared to that for the refolding of the substrate binding site, as indicated by complex formation with the competitive inhibitor, 2'CMP, and for the refolding of the molecule as a whole. At pH 6.0 and 25 degrees C, only monophasic first-order reactions could be detected by manual mixing for both the reactivation and the refolding processes. At lower temperatures (0-10 degrees C), both processes consist of two first-order reactions. In all cases, the same rate constants have been obtained for the refolding and reactivation reactions.     

Study on the kinetics of enzymatic interesterification of triglycerides for biodiesel production with methyl acetate as the acyl acceptor

A new route for biodiesel production using methyl acetate instead of methanol as the acyl acceptor was proposed in our previous research, and it has been found that this novel route could enhance the stability of the immobilized lipase greatly. In this paper, the kinetics of lipase-catalyzed interesterification of triglycerides for biodiesel production with methyl acetate as the acyl acceptor was further studied. First, a simplified model based on Ping Pong Bi Bi with substrate competitive inhibition mechanism was proposed to describe the reaction kinetics of the interesterification. During our further study, it was observed that three consecutive and reversible reactions occurred in the interesterification of triglycerides and methyl acetate. So, a kinetic model based on mass balance of three second-order reversible reactions was developed and the reaction rate constant, k, was determined by solving the differential rate equations of the reaction system. The results showed that k_DG–MG (0.1124) and k_MG–TA (0.1129) were much higher than k_TG–DG (0.0311), which indicated that the first step reaction was the limit step for the overall interesterification.     

Model Studies of the Iron-Catalysed Haber-Weiss Cycle and the Ascorbate-Driven Fenton Reaction

Complementary hydroxylation assays and stopped-flow e.s.r. techniques have been employed in the investigation of the effect of various iron chelators (of chemical, biological and clinical importance) on hydroxyl-radical generation via the Haber-Weiss cycle and the ascorbate-driven Fenton reaction.

Chelators have been identified which selectively promote or inhibit various reactions involved in hydroxyl-radical generation (for example, NTA and EDTA promote all the reactions of both the Haber-Weiss cycle and the ascorbate-driven Fenton reaction, whereas DTPA and phytate inhibit the recycling of iron in these reactions). The biological chelators succinate and citrate are shown to be relatively poor catalysts of the Haber-Weiss cycle, whereas they are found to be effective catalysts of ·OH generation in the ascorbate-driven Fenton reaction.

It is also suggested that continuous redox-cycling reactions between iron, oxygen and ascorbate may represent an important mechanism of cell death in biological systems.     

A Kinetic Analysis of Coupled (or Auxiliary) Enzyme Reactions

As a case study, we consider a coupled (or auxiliary) enzyme assay of two reactions obeying the Michaelis–Menten mechanism. The coupled reaction consists of a single-substrate, single-enzyme non-observable reaction followed by another single-substrate, single-enzyme observable reaction (indicator reaction). In this assay, the product of the non-observable reaction is the substrate of the indicator reaction. A mathematical analysis of the reaction kinetics is performed, and it is found that after an initial fast transient, the coupled reaction is described by a pair of interacting Michaelis–Menten equations. Moreover, we show that when the indicator reaction is fast, the quasi-steady-state dynamics are governed by three fast variables and one slow variable. Timescales that approximate the respective lengths of the indicator and non-observable reactions, as well as conditions for the validity of the Michaelis–Menten equations, are derived. The theory can be extended to deal with more complex sequences of enzyme-catalyzed reactions.     

The role of hydrophobic properties of chemicals in promoting allosteric reactions

An allosteric reaction has been found in a variety of instances where an inverted parabolic relationship between biological activity and hydrophobicity is apparent, that is the activity first decreases as hydrophobicity increases and after a certain point, activity begins to increase. This could be attributed to the ligands causing a change in the receptor structure. In this report, the role of hydrophobic properties of chemicals in promoting allosteric reactions have been discussed in term of hydrophobicity (logP) by the formulation of a total number of 50 QSAR equations. The QSAR model of this type may be represented by Eq. I.     

Thermodynamic calculations for biochemical transport and reaction processes in metabolic networks

Thermodynamic analysis of metabolic networks has recently generated increasing interest for its ability to add constraints on metabolic network operation, and to combine metabolic fluxes and metabolite measurements in a mechanistic manner. Concepts for the calculation of the change in Gibbs energy of biochemical reactions have long been established. However, a concept for incorporation of cross-membrane transport in these calculations is still missing, although the theory for calculating thermodynamic properties of transport processes is long known. Here, we have developed two equivalent equations to calculate the change in Gibbs energy of combined transport and reaction processes based on two different ways of treating biochemical thermodynamics. We illustrate the need for these equations by showing that in some cases there is a significant difference between the proposed correct calculation and using an approximative method. With the developed equations, thermodynamic analysis of metabolic networks spanning over multiple physical compartments can now be correctly described.     

Immobilized enzymes: electrokinetic effects on reaction rates in a porous medium

The effect of the internal diffusion and electrical surface charge on the overall rate of a reaction catalyzed by an enzyme immobilized on a porous medium are examined. Effectiveness factors have been calculated which compare the global reaction rate to that existing in the absence of the internal diffusion and/or the electrical field. The surface charge, assumed to arise from the dissociation equilibria of the acidic and basic surface groups of the enzyme, generates an electrical double layer at the pore surface. The double-layer potential is governed by the Poisson-Boltzmann equation. It is shown that the diffusion potential can be characterized by a modulus which depends upon the surface reaction rate, the charges and diffusivities of the substrate and products, the ionic strength, and the pore dimensions. The flux of a charged species in the pore occurs under the influences of the concentration gradient and the electrical potential gradient. The governing equations are solved by an iterative numerical method. The effects of pH, enzyme concentration, and substrate concentration on the rates of two different hydrolysis reactions catalyzed by immobilized papain are examined. The release of H(+) in one of the reactions causes the lowering of internal pH, and also a constancy of the internal pH when the external pH in creases beyond a certain value. The latter reaction also shows a maximum in the reaction rate with respect to enzyme concentration. The reaction not involving H(+) as a product shows a maximum in the reaction rate with respect to external pH, but a monotonic increase in the reaction rate as the enzyme concentration increases.     

Monitoring of artificial enzyme membrane systems by electric currents: I. Analytical treatment with direct current and buffered conductivity

Continuous electric fields (E) modify the transport flows and the intramembrane concentration profiles of protons or of ionic substrates or cofactors (inhibitors). These ‘mediators’ induce variations in enzyme activity, quantifiable by a generalized Damköhler group II Ψ distinguishing electrotransport reactions from diffusion reactions. For three typical reaction schemas, using only one mediator, the steady-state equations have been established. Depending on boundary conditions, the direction of electric current (for asymmetrical systems) and the value of Ψ. activations, inhibitions or activations followed by inactivations have been found. With buffered conductivity (supporting electrolyte), the limiting concentration profiles (E → ∞) are uniformly equal to the boundary values; i.e., diffusion constraints are suppressed and the regime is controlled by the reaction. The calculations give the relative activity variations for partially suppressed transport controls.