Characterization of unstable enzyme systems which evolve according to a three-exponential equation

The time course of an enzyme catalyzed reaction is normally followed either by monitoring the instantaneous concentration or velocity of an enzyme species or a product. In many enzyme catalyzed reactions these time variations are multi-exponential. The accurate fit of the relevant curves to obtain the kinetic parameters involved can be difficult using conventional methods (Galvez et al. in J Theor Biol 89:37–44, 1981; Garcia-Canovas et al. in Biochim Biophys Acta 912:417–423, 1987; Tudela et al. in Biochim Biophys Acta 912:408–416, 1987; Teruel et al. in Biochim Biophys Acta 911:256–260, 1987; Garrido del Solo et al. in Biochem J 294:459–464, 1993; Varon et al. in Int J Biochem 25:1889–1895, 1993; Garrido del Solo et al. in An Quim 89:319–324, 1993; Varon et al. in J Mol Catal 83:273–285, 1993; Garrido del Solo et al. in Biochem J 303(Pt 2):435–440, 1994; Garrido del Solo and Varon in An Quim 91:13–18, 1995; Garrido del Solo et al. in Biosystems 38:75–86, 1996; Garrido del Solo et al. in Int J Biochem Cell Biol 28:1371–1379, 1996; Garrido del Solo et al. in Int J Biochem Cell Biol 30:735–743, 1998; Varon et al. in J Mol Catal 59:97–118, 1990). In order to circumvent such difficulties Arribas et al. (J Math Chem 44:379–404, 2008) proposed an evaluation method which is applicable regardless of the complexity of the kinetic equation. This procedure is based on the numerical determination of statistical moments from experimental time progress curves. The fitting of these experimentally obtained moments to the corresponding theoretical symbolic expressions allows, in most cases, all the individual rate constants involved to be evaluated. In this paper we perform a general analysis that can be applied to any unstable enzyme system described by a three-exponential equation and apply it to a substrates induced enzyme inactivation process that is described by this type of equation. To verify the goodness of the method we have simulated time progress curves and applied the suggested procedure to these curves, obtaining kinetic parameters values very close to those used to obtain simulated curves. Finally, we compare our results with those obtained in previous contributions in which other procedures were used.     

Thermal inactivation of immobilized enzymes: A kinetic study

The thermal inactivation kinetics of chymotrypsin, trypsin, and—-amylase bound to silica, polyacrylamide, and polystyrene were studied at different temperatures. The inactivation curves were analyzed by a kinetic model, assuming a first-order reaction of differently stable enzyme fractions on the matrix. In all cases the assumption of two enzyme fractions with distinctly different inactivation constants was sufficient for describing the inactivation progress (standard deviations between experimental and calculated inactivation curves, 1-4%). Both the inactivation constants as well as the relative concentrations of the enzyme fractions were found to change in dependence on temperature.     

Linear compartmental systems. III. Application to enzymatic reactions

In this paper we extend to enzyme systems the results previously obtained in paper I of this series for linear compartmental systems. We obtain the time course equations for both the enzyme and ligand species involved in the reaction mechanisms, which fit a general enzyme system model when the connections between the different enzyme species are of first or pseudofirst order. The kinetic equations obtained here for a given species, enzyme or ligand have the advantage over all previous equations described in the literature, in that they are in the most simplified form possible, since they only contain the kinetic parameters and initial concentrations of the enzymatic reaction which really have some influence on the time progress curves of the species under study. These kinetic equations are denominated optimized equation to distinguish them from the others, which shall call non-optimized equations. We discuss those cases when both types of equation coincide and we show how, when they do not coincide, the non-optimized equations can be simplified to the optimized ones. Therefore, we show that the optimized equations could be used in all cases to avoid the need of subsequent simplifications to eliminate the parameters that play no role in the corresponding time equations. To illustrate the use of this procedure we will apply it to two simple examples of enzymatic reactions.     

Impact of carbon dioxide on the surface tension of 1-hexanol aqueous solutions

Effects of carbon dioxide presence on the surface tension and adsorption kinetics of 1-hexanol solutions were investigated. Experiments were performed at a range of carbon dioxide vapor pressures and varying concentrations of 1-hexanol aqueous solution. Both dynamic and steady-state surface tensions of 1-hexanol aqueous solution were found to decrease with carbon dioxide pressure, and a linear relationship was observed between the steady-state surface tension and carbon dioxide pressure. To explain the experiments, adsorption and desorption of the two species (1-hexanol and carbon dioxide) from two sides of the vapor-liquid interface were considered. A modified Langmuir isotherm, the modified Langmuir equation of state and the modified kinetic transfer equation were developed. The resulting steady-state and dynamic surface tension data were modeled using the modified Langmuir equation of state and the modified kinetic transfer equation, respectively. Equilibrium constants and adsorption rate constants of 1-hexanol and carbon dioxide were evaluated through a minimization procedure for CO₂ pressures ranging from 0 to 690 kPa. From the steady-state modeling, the equilibrium parameters for 1-hexanol and carbon dioxide adsorption from vapor phase and liquid phase were found unchanged at different pressures of carbon dioxide. From the dynamic modeling, the adsorption rate constants for 1-hexanol and carbon dioxide from vapor phase and liquid phase were found to decrease with carbon dioxide pressure. Some fluctuations in the fitting parameters of the dynamic modeling (adsorption rate constants) were observed. These fluctuations may be due to experimental errors, or more likely the limitations of the model used. A major limitation of the model is related to large differences in adsorption/desorption between initial and final stages of the process, and a single set of property parameters cannot describe both initial and final states of the system. Variations may occur depending on which set of data, of initial or final states, is used in the model predictions over the entire time range.     

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.     

Effect of brownian diffusion on chemical kinetics

A new method of theoretical prediction of the kinetic rate constants of fast chemical reactions in solutions is presented. It takes into account the effect of finite diffusive displacements of the reacting molecules. The approach is based on the solution of the steady-state Fokker–Planck equation by the moments method of Grad developed in the theory of coagulation of aerosol particles. A comparison of the predicted rate constants with the experimental data provided by Schuh and Fischer for the self-reaction of tert-butyl radicals in n-alkanes shows a good correspondence.     

Analytic solutions to sets of first-order rate equations with up to six rate constants using a symbolic computer language SMP and application to biochemical kinetics

A symbolic computer language SMP* is employed to analytically solve sets of first-order linear differential equations which occur in kinetic rate-reaction studies. The solutions studied are fully analytic functions of time and the rate constants. Two typical systems are studied: the first contains four species and four rate constants, corresponding to four parallel and consecutive first-order reactions; and the second contains four species and six rate constants, including two additional reverse reactions. These analytic functions allow insight into the mechanism, analytic expressions for the rate constants, and more rapid and precise solutions for the species concentrations than a completely numerical solution of the differential rate equations themselves. The results of the first system are applied to a recent experimental study of enzyme kinetics in which constituent amino acid residues of an enzyme are photooxidized and the corresponding catalytic activity measured with time. A second application of the SMP gives rise to a rapid semianalytic method for obtaining the values of the four and six exponentially nonlinear rate constants from the experimental data.     

Alanine racemase free energy profiles from global analyses of progress curves

Free energy profiles for alanine racemase from Bacillus stearothermophilus have been determined at pH 6.9 and 8.9 from global analysis of racemization progress curves. This required a careful statistical design due to the problems in finding the global minimum in mean square for a system with eight adjustable parameters (i.e., the eight rate constants that describe the stepwise chemical mechanism). The free energy profiles obtained through these procedures are supported by independent experimental evidence: (1). steady-state kinetic constants, (2). solvent viscosity dependence, (3). spectral analysis of reaction intermediates, (4). equilibrium overshoots for progress curves measured in D(2)O, and (5). the magnitudes of calculated intrinsic kinetic isotope effects. The free energy profiles for the enzyme are compared to those of the uncatalyzed and the PLP catalyzed reactions. At pH 6.9, PLP lowers the free energy of activation for deprotonation by 8.4 kcal/mol, while the inclusion of apoenzyme along with PLP additionally lowers it by 11 kcal/mol.     

Semiempirical equations for modeling solid-state kinetics based on a Maxwell-Boltzmann distribution of activation energies: applications to a polymorphic transformation under crystallization slurry conditions and to the thermal decomposition of AgMnO4 crystals

Many solid-state reactions and phase transformations performed under isothermal conditions give rise to asymmetric, sigmoidally shaped conversion-time (x-t) profiles. The mathematical treatment of such curves, as well as their physical interpretation, is often challenging. In this work, the functional form of a Maxwell-Boltzmann (M-B) distribution is used to describe the distribution of activation energies for the reagent solids, which, when coupled with an integrated first-order rate expression, yields a novel semiempirical equation that may offer better success in the modeling of solid-state kinetics. In this approach, the Arrhenius equation is used to relate the distribution of activation energies to a corresponding distribution of rate constants for the individual molecules in the reagent solids. This distribution of molecular rate constants is then correlated to the (observable) reaction time in the derivation of the model equation. In addition to providing a versatile treatment for asymmetric, sigmoidal reaction curves, another key advantage of our equation over other models is that the start time of conversion is uniquely defined at t = 0. We demonstrate the ability of our simple, two-parameter equation to successfully model the experimental x-t data for the polymorphic transformation of a pharmaceutical compound under crystallization slurry (i.e., heterogeneous) conditions. Additionally, we use a modification of this equation to model the kinetics of a historically significant, homogeneous solid-state reaction: the thermal decomposition of AgMnO4 crystals. The potential broad applicability of our statistical (i.e., dispersive) kinetic approach makes it a potentially attractive alternative to existing models/approaches.     

An alternative analysis of enzyme systems based on the whole reaction time: evaluation of the kinetic parameters and initial enzyme concentration

This work presents an alternative analysis of the integrated rate equations corresponding to the simple Michaelis-Menten mechanism without product inhibition. The suggested new results are reached under a minimal set of assumptions and include, as a particular case, the classical integrated Michaelis–Menten equation. Experimental designs and a kinetic data analysis are suggested to the estimation of the maximum steady-state rate, V _max, the Michaelis–Menten constant, K _m, the initial enzyme concentration, [E]₀, and the catalytic constant, k ₂. The goodness of the analysis is tested with simulated time progress curves obtained by numerical integration.     

Curing degree prediction for S-TBBS-DPG natural rubber by means of a simple numerical model accounting for reversion and linear interaction

The paper presents a simple numerical model able to provide directly kinetic constants and reliable numerical rheometer curves for S-TBBS-DPG natural rubber. The approach is suitable to calculate the kinetic constants and maximum torque (MH) at any S-TBBS-DPG concentration, following a 3D mathematical interpolation/extrapolation procedure, when kinetic constants on few grid points of S-TBBS-DPG concentrations are available. In particular, the possibility to estimate with sufficient accuracy the behavior of natural rubber at any intermediate concentration of S-TBBS-DPG having engineering relevance has been proved, calibrating the model by means of simple closed form standard best fitting on few experimental data. The model used is a three kinetic parameters one, derived from the well known Han's and co-workers approach, where constants have been evaluated normalizing experimental rheometers curves following the commonly accepted Sun and Isayev method. The procedure has been validated against experimentally obtained rheometer curves by means of inverse analysis, exhibiting excellent prediction capabilities. The approach may be used for practical purposes in order to avoid expensive and cumbersome experimental investigations.     

Competitive ion complexation to polyelectrolytes: determination of the stepwise stability constants. The Ca2+/H+/polyacrylate system

This work presents a new methodology aimed at obtaining the stepwise stability constants corresponding to the binding of ions (or other small molecules) to macromolecular ligands having a large number of sites. For complexing agents with a large number of sites, very simple expressions for the stepwise stability constants arise. Such expressions are model-independent; that is, they allow the determination of the stepwise stability constants without making any previous assumption of the detailed complexation mechanism. The formalism is first presented for a single complexing ion and further extended to competitive systems where the competing ions can display, in general, different stoichiometric relationships. These ideas are applied to the analysis of experimental titrations corresponding to competitive binding of calcium ions to poly(acrylic acid) for different pH values and ionic strengths. Intrinsic stability constants were estimated from the stepwise stability constants (by removing the corresponding statistical factor), and split into specific and electrostatic contributions (by means of the Poisson-Boltzmann equation). After this treatment, the specific proton binding energies showed almost no dependence on the coverage and ionic strength. Likewise, for the range of concentrations studied, the specific component of the intrinsic stability constants of the calcium ions, calculated assuming bidentate binding of Ca to neighboring groups of a linear chain, is almost independent of the calcium and proton coverage and ionic strength.     

Transient analysis of electrolytically initiated polymerization

Electrochemically initiated polymerization at a planar electrode is treated theoretically and an approximate solution to the full time-dependent problem is found assuming the electrode process to be reversible (so that the Nernst relation holds). The homogeneous kinetics lead to a second-order coupled non-linear initial-boundary value problem, and the method developed for its solution is general and applicable to other problems. The approximate solution is converted to an equation for the transient current-versus-time curve, for electrode initiated polymerization, which can be used for the experimental determination of the polymerization and termination rate constants. An analysis procedure for such determination is outlined.     

Flexibility of enzymatic transitions as a hallmark of optimized enzyme steady-state kinetics and thermodynamics

We investigate the relations between the enzyme kinetic flexibility, the rate of entropy production, and the Shannon information entropy in a steady-state enzyme reaction. All these quantities are maximized with respect to enzyme rate constants. We show that the steady-state, which is characterized by the most flexible enzymatic transitions between the enzyme conformational states, coincides with the global maxima of the Shannon information entropy and the rate of entropy production. This steady-state of an enzyme is referred to as globally optimal. This theoretical approach is then used for the analysis of the kinetic and the thermodynamic performance of the enzyme triose-phosphate isomerase. The analysis reveals that there exist well-defined maxima of the kinetic flexibility, the rate of entropy production, and the Shannon information entropy with respect to any arbitrarily chosen rate constant of the enzyme and that these maxima, calculated from the measured kinetic rate constants for the triose-phosphate isomerase are lower, however of the same order of magnitude, as the maxima of the globally optimal state of the enzyme. This suggests that the triose-phosphate isomerase could be a well, but not fully evolved enzyme, as it was previously claimed. Herein presented theoretical investigations also provide clear evidence that the flexibility of enzymatic transitions between the enzyme conformational states is a requirement for the maximal Shannon information entropy and the maximal rate of entropy production.     

脂肪酶催化乳酸与乙醇合成乳酸乙酯的反应动力学

对脂肪酶催化乳酸与乙醇合成乳酸乙酯反应的动力学进行了研究,根据乒乓机制和双底物抑制的特性建立了反应速率方程.反应时间常数(tR)和扩散时间常数(tD)的计算结果表明,酯化反应速率未受到明显的限制.反应速率方程可以很好地预测实验结果,由非线性拟合得到的动力学参数中,乳酸(A)和乙醇(B)的抑制常数分别为KiA=10.7mmol/L和KiB=275.0mmol/L.这说明乳酸作为短链极性脂肪酸,对酶的失活作用远大于乙醇.乳酸在微液层中聚集并产生了使酶失活的低pH值环境,同时在酯化反应中存在竞争性抑制作用.     

Estimation of spectra of individual species in a multisolute solution

Spectra of a series of solutions containing different concentrations of four solutes were simulated, assuming either normal or non-normal distribution of experimental errors. These synthetic data were then analyzed by linear least-squares procedures in order to estimate the spectra of individual solutes. Analysis of regression residuals and the estimated spectra showed that a correct set of statistical weights can be found by an iterative procedure, if the qualitative form of the weighting function is known in advance. When ever a least-squares estimation was successful, the estimates were practically unbiased and the moments of the distribution of regression residuals were close to their expectances. However, the distribution moments of the regression estimates were found to be far from normal in all cases, although the estimates were not necessarily biased.     

Extensive theoretical studies on the low-lying electronic states of indium monochloride cation, InCl+

The global potential energy curves for the 14 low-lying doublet and quartet Lambda-S states of InCl+ are calculated at the scalar relativistic MR-CISD+Q (multireference configuration interaction with single and double excitations, and Davidson's correction) level of theory. Spin-orbit coupling is accounted for via the state interaction approach with the full Breit-Pauli Hamiltonian, which leads to 30 Omega states. The computed spectroscopic constants of nine bound Lambda-S states and 17 bound Omega states are in good agreement with the available experimental data. The transition dipole moments and Franck-Condon factors of selected transitions are also calculated, from which the corresponding radiative lifetimes are derived.     

KINMODEL (AGDC): a multipurpose computational method for kinetic treatment

KINMODEL (AGDC) is a kinetic computational methodology that is valid for the treatment of any reaction mechanism and that allows the determination of different kinetic and non-kinetic parameters from the experimental data acquired by monitoring absorbance at one or several different wavelengths. It is a numerical computational model that can be applied to any reaction mechanism, with the advantage that on changing the treatment from one mechanism to another it is not necessary to modify even a single line of the program code since it automatically establishes and solves the set of differential rate equations. It is able to treat a broad set of reaction mechanisms in the individual and joint determination of the following groups of parameters (a) the individual rate constants of the different reaction mechanisms; (b) the values of the molar absorption coefficients (which are very valuable in the case of intermediate species and their identification) of all the species involved in the mechanism, and (c) the concentrations of the species participating in the mechanism. The program can be used by non-experts in the field and it is able to treat mechanisms involving ambiguities in the solutions and in the identification of parameters when kinetic constants and molar absorption coefficients are optimized together, and it allows a discrimination to be made between the possible mechanisms responsible for the course of the reaction after the residuals have been analyzed statistically, automatically choosing the one that best fits the kinetic data.     

Multiple‐Well,multiple‐path unimolecular reaction systems. I. MultiWell computer program suite

Unimolecular reaction systems in which multiple isomers undergo simultaneous reactions via multiple decomposition reactions and multiple isomerization reactions are of fundamental interest in chemical kinetics. The computer program suite described here can be used to treat such coupled systems, including the effects of collisional energy transfer (weak collisions). The program suite consists of MultiWell, which solves the internal energy master equation for complex unimolecular reactions systems; DenSum, which calculates sums and densities of states by an exact‐count method; MomInert, which calculates external principal moments of inertia and internal rotation reduced moments of inertia; and Thermo, which calculates equilibrium constants and other thermodynamics quantities. MultiWell utilizes a hybrid master equation approach, which performs like an energy‐grained master equation at low energies and a continuum master equation in the vibrational quasicontinuum. An adaptation of Gillespie's exact stochastic method is used for the solution. The codes are designed for ease of use. Details are presented of various methods for treating weak collisions with virtually any desired collision step‐size distribution and for utilizing RRKM theory for specific unimolecular rate constants. © 2001 John Wiley & Sons, Inc. Int J Chem Kinet 33: 232–245, 2001     

Computational techniques applied to the study of the oxidation kinetics of iron and molybdenum cyanocomplexes by peroxynitrous acid

The oxidation kinetics of iron and molybdenum cyanocomplexes by peroxynitrous acid were studied by computational techniques. The rate constants of ONOOH decomposition and substrate oxidation were calculated by fitting the experimental data to the solution of the ordinary differential equations mechanism. There is a linear relationship between the rate constants which is solved by varying the initial concentration of one of the reactive species. The rate constants of the steps involved in ONOOH decomposition were also determined following the same procedure and considering a linear relationship between the rate constants. The results are consistent with the values previously found using the conventional method of integration, and they validate the prior steady state assumption. Copyright © 2008 John Wiley & Sons, Ltd.