Locally implicit solution of a reaction-diffusion system with stiff kinetics

The Tyson-Fife reaction-diffusion equations are solved numerically using a locally implicit approach. Since the variables evolve at very different time scales, the resulting system of equations is stiff. The reaction term is responsible for the stiffness and the time step is increased by using an implicit method. The diffusion operator is evaluated explicitly and the system of implicit nonlinear equations is decoupled. The method is particularly useful for parameter values in which the equations are very stiff, such as the values obtained directly from the experimental reaction rate constants. Previous efforts modified the parameters on the equations to avoid stiffness. The equations then become a simplified model of excitable media and, for those cases, the locally implicit method gives a faster although less accurate solution. Nevertheless, since the modified equations no longer represent a particular chemical system an accurate solution is not as important. The algorithm is applied to observe the transition from simple motion to compound motion of a spiral tip.     

A combined explicit-implicit method for high accuracy reaction path integration

We present the use of an optimal combined explicit-implicit method for following the reaction path to high accuracy. This is in contrast to most purely implicit reaction path integration algorithms, which are only efficient on stiff ordinary differential equations. The defining equation for the reaction path is considered to be stiff, however, we show here that the reaction path is not uniformly stiff and instead is only stiff near stationary points. The optimal algorithm developed in this work is a combination of explicit and implicit methods with a simple criterion to switch between the two. Using three different chemical reactions, we combine and compare three different integration methods: the implicit trapezoidal method, an explicit stabilized third order algorithm implemented in the code DUMKA3 and the traditional explicit fourth order Runge-Kutta method written in the code RKSUITE. The results for high accuracy show that when the implicit trapezoidal method is combined with either explicit method the number of energy and gradient calculations can potentially be reduced by almost a half compared with integrating either method alone. Finally, to explain the improvements of the combined method we expand on the concepts of stability and stiffness and relate them to the efficiency of integration methods.     

A Concise Description of an Old Problem: Application of Matrices to Obtain the Balancing Coefficients of Chemical Equations

It is not difficult to balance chemical equations and thus it is hardly given more thoughts. However, there is a mathematical principle for the balancing of chemical equations and this principle may be used for automation. The balancing of chemical equation is carried out, by formulating a reaction matrix and the latter is used in a matrix equation. The matrix equation is then solved to obtain the balancing coefficients. The conventional matrix inverse method cannot always be used and hence the solution is obtained by row reduced operations. These operations and any matrix manipulation are carried out with Matlab. Further this novel method can be used to classify chemical equations as nonfeasible, unique and nonunique.     

A method,based on statistical moments,to evaluate the kinetic parameters involved in unstable enzyme systems

The evaluation of individual rate constants involved in any reaction mechanism of an enzymatic systems first requires experimental monitoring of the time course of the concentration or product rate creation or of any enzyme species. The experimental progress curves obtained must then be fitted to the corresponding theoretical symbolic equation. Nevertheless, in some cases, e.g. when the equation involves two or more exponential terms, this fit is not easy and sometimes impossible. Simplification of the equation is usually required by assuming, for example, that the system has reached the steady-state, assuming an initial steady-state of a segment in the scheme of the reaction mechanism or assuming rapid equilibrium in one or more of the reversible steps, if there are any. But, obviously, simplified equations produce either fewer individual rate constants or global constants consisting of algebraic associations of individual rate constants or individual rate constants or global constants that might considerably differ from the real ones due to the approaches made. In this contribution, we suggest an alternative procedure for evaluating the rate constants of enzyme reactions corresponding to enzyme systems where one or more of the species involved is unstable or where one or more of the enzyme species is irreversibly inhibited, or both. The 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 expressions allows us, in most cases, to evaluate of all of the rate constants involved, with only a small error. To verify the goodness of the suggested procedure, it was applied to an unstable enzyme system which had previously been analysed with other methods. Finally, it is indicated how this procedure could also be extrapolated for application to any stable or unstable enzyme system.     

CALCULATION METHOD OF EFFECTIVENESS FACTOR FOR NON-LINEAR COMPLEX REACTION SYSTEM

A calculation procedure of the effectiveness factor for very complex multiple reaction system is developed by using the concept of continuous lumping. The calculation procedure proposed in this paper enables not only the reaction-diffusion equations to be solved as an initial value problem, but also the number of the nonlinear algebraic equation which have to be solved in each iterative step to be reduced considerably.     

Generalized theoretical and practical treatment of the kinetics of an enzyme-catalyzed reaction in the presence of an enzyme equimolar irreversible inhibitor

We revisit a previous analysis of the classical Michaelis-Menten enzyme reaction for the case in which the free enzyme incurs the loss of its activity by an irreversible inhibitor concentration dependent but time unaltered rate constant (see Golicnik, M. J. Chem. Inf. Comput. Sci. 2002, 42, 157-161). We study the kinetic model of an enzyme-catalyzed reaction in the presence of an equimolar irreversible inhibitor showing a time dependent inactivation rate constant because of considerable inhibitor amount depletion during the course of the reaction. We show that an analytical solution containing the nonelementary Gauss hypergeometric function can be found for the reactants in equation Phi of an implicit type that precludes direct calculation of the extent of reaction at any time. The transformation theory of the hypergeometric function is used to obtain rapidly convergent power series, and for the root calculation of equation Phi the divergence-proof root bracketing algorithm according to Van Wijngaarden-Dekker-Brent is performed. Numerically generated data are analyzed according to this mathematical procedures, and the results are compared with ones obtained by the numerical integration treatment.     

On the steady-state assumption and its application to the rotating disk voltammetry of adsorbed enzymes

Rotating disk voltammetry is routinely used to study electrochemically driven enzyme catalysis because of the assumption that the method produces a steady-state system. This assumption is based on the sigmoidal shape of the voltammograms. We have introduced an electrochemical adaptation of the King-Altman method to simulate voltammograms in which the enzyme catalysis, within an immobilized enzyme layer, is steady-state. This method is readily adaptable to any mechanism and provides a readily programmable means of obtaining closed form analytical equations for a steady-state system. The steady-state simulations are compared to fully implicit finite difference (FIFD) simulations carried out without any steady-state assumptions. On the basis of our simulations, we conclude that, under typical experimental conditions, steady-state enzyme catalysis is unlikely to occur within electrode-immobilized enzyme layers and that typically sigmoidal rotating disk voltammograms merely reflect a mass transfer steady state as opposed to a true steady state of enzyme intermediates at each potential.     

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.     

Stochastic theory of large-scale enzyme-reaction networks: finite copy number corrections to rate equation models

Chemical reactions inside cells occur in compartment volumes in the range of atto- to femtoliters. Physiological concentrations realized in such small volumes imply low copy numbers of interacting molecules with the consequence of considerable fluctuations in the concentrations. In contrast, rate equation models are based on the implicit assumption of infinitely large numbers of interacting molecules, or equivalently, that reactions occur in infinite volumes at constant macroscopic concentrations. In this article we compute the finite-volume corrections (or equivalently the finite copy number corrections) to the solutions of the rate equations for chemical reaction networks composed of arbitrarily large numbers of enzyme-catalyzed reactions which are confined inside a small subcellular compartment. This is achieved by applying a mesoscopic version of the quasisteady-state assumption to the exact Fokker-Planck equation associated with the Poisson representation of the chemical master equation. The procedure yields impressively simple and compact expressions for the finite-volume corrections. We prove that the predictions of the rate equations will always underestimate the actual steady-state substrate concentrations for an enzyme-reaction network confined in a small volume. In particular we show that the finite-volume corrections increase with decreasing subcellular volume, decreasing Michaelis-Menten constants, and increasing enzyme saturation. The magnitude of the corrections depends sensitively on the topology of the network. The predictions of the theory are shown to be in excellent agreement with stochastic simulations for two types of networks typically associated with protein methylation and metabolism.     

Hamilton-Jacobi equation for the least-action/least-time dynamical path based on fast marching method

Classical dynamics can be described with Newton's equation of motion or, totally equivalently, using the Hamilton-Jacobi equation. Here, the possibility of using the Hamilton-Jacobi equation to describe chemical reaction dynamics is explored. This requires an efficient computational approach for constructing the physically and chemically relevant solutions to the Hamilton-Jacobi equation; here we solve Hamilton-Jacobi equations on a Cartesian grid using Sethian's fast marching method. Using this method, we can--starting from an arbitrary initial conformation--find reaction paths that minimize the action or the time. The method is demonstrated by computing the mechanism for two different systems: a model system with four different stationary configurations and the H+H(2)-->H(2)+H reaction. Least-time paths (termed brachistochrones in classical mechanics) seem to be a suitable chioce for the reaction coordinate, allowing one to determine the key intermediates and final product of a chemical reaction. For conservative systems the Hamilton-Jacobi equation does not depend on the time, so this approach may be useful for simulating systems where important motions occur on a variety of different time scales.     

A New Singular Matrix Method for Balancing Chemical Equations and Their Stability

In this work is given a new singular matrix method for balancing new classes of chemical equations which reduce to an n × n matrix. The method offered here is founded by virtue of the solution of a homogeneous matrix equation by using of Drazin pseudoinverse matrix. The method has been tested on many typical chemical equations and found to be very successful for the all equations in our extensive balancing research. This method works successfully without any limitations. Chemical equations treated here possess atoms with fractional oxidation numbers. Also, in the present work are analyzed some necessary and sufficient criteria for stability of chemical equations over stability of their reaction matrices.     

A digital simulation study of two-dimensional convection diffusion for channel electrode

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Methanol and ethanol oxidation by chromium(VI) in aqueous perchloric acid

Summary Oxidation of short-chain alcohols (MeOH and EtOH) by Cr^VI in concentrated aqueous HClO₄ does not fit pseudo-first order rate equations, but results can be expressed in a two-exponential equation. A reaction mechanism based upon a chromate ester intermediate in equilibrium with the protonated alcohol and the chromic acid, suitable for a longer chain alcohol, does not account for kinetic results. A new mechanism is considered involving an intermediate reaction between chromic and perchloric acids.     

Comparison between theoretical values and simulation results of viscosity for the dissipative particle dynamics method

In order to investigate the validity of the dissipative particle dynamics method, which is a mesoscopic simulation technique, we have derived an expression for viscosity from the equation of motion of dissipative particles. In the concrete, we have shown the Fokker-Planck equation in phase space, and macroscopic conservation equations such as the equation of continuity and the equation of momentum conservation. The basic equations of the single-particle and pair distribution functions have been derived using the Fokker-Planck equation. The solutions of these distribution functions have approximately been solved by the perturbation method under the assumption of molecular chaos. The expressions of the viscosity due to momentum and dissipative forces have been obtained using the approximate solutions of the distribution functions. Also, we have conducted nonequilibrium dynamics simulations to investigate the influence of the parameters, which have appeared in defining the equation of motion in the dissipative particle dynamics method. The theoretical values of the viscosity due to dissipative forces in the Hoogerbrugge-Koelman theory are in good agreement with the simulation results obtained by the nonequilibrium dynamics method, except in the range of small number densities. There are restriction conditions for taking appropriate values of the number density, number of particles, time interval, shear rate, etc., to obtain physically reasonable results by means of dissipative particle dynamics simulations.     

A user-friendly programme 'SIMKINERSQUO; for simulation of kinetics involving complex reaction mechanisms

A programme, 'SIMKINERSQUO; is developed using a semi-implicit extrapolation method (SIEM) which uses the implicit midpoint rule and extrapolation to simulate complex mechanisms based on the kinetics of homogeneous chemical systems. The chemical kinetics pre-processor code is designed to translate a user-specified system of chemical rate equations into a system of chemical kinetic differential equations. The developed programme is applied to the 13-step mechanism of the reaction between Nile blue and acidic bromate. The results obtained compare well with the curves drawn using the other method, reported in literature.     

Calculation of homogeneous azeotropes in reactive and non-reactive mixtures using a stochastic optimization approach

In this paper we introduce an alternative strategy to find homogeneous azeotropes in reactive and non-reactive mixtures which is based upon the Simulated Annealing optimization technique. This stochastic optimization method is used to robustly solve a system of non-linear equations that results from the equalities of the orthogonal derivatives of the Gibbs energy and the Gibbs energy of mixing in the vapor and the liquid phases. For non-reactive systems, this equation system is solved by considering conventional composition variables while for reactive cases we use the transformed composition variables proposed by Ung and Doherty. Numerical performance of our approach is illustrated using several examples previously reported in the literature and results show that it is a suitable and robust strategy for the calculation of homogeneous azeotropes in mixtures with or without chemical reactions.     

An electrochemical platform for acetylcholinesterase activity assay and inhibitors screening based on Michael addition reaction between thiocholine and catechol-terminated SAMs

An electrochemical platform for acetylcholinesterase (AChE) activity assay and its inhibitors screening is developed based on the Michael addition reaction of thiocholine, the hydrolysis product of acetylthiocholine (AsCh) in the presence of AChE, with the electrogenerated o-quinone of catechol-terminated SAMs on a gold electrode. For understanding and confirming the mechanism of the reaction, the electrochemical behaviors of Michael addition reaction of two model compounds, cysteine (CYS) and glutathione (GSH), towards the catechol-terminated SAMs have been studied. The enzyme kinetics and the inhibition effects of three types of AChE inhibitors, which are tacrine, carbofuran and parathion-methyl, have been investigated using an amperometric method. Among these three inhibitors, tacrine exhibits the strongest inhibiting effect, which is reinforced by the resulting data of kinetic studies on each inhibitor's influence upon the enzyme activity.     

Numerical modeling of Joule heating-induced temperature gradient focusing in microfluidic channels

Temperature gradient focusing (TGF) is a recently developed technique for spatially focusing and separating ionic analytes in microchannels. The temperature gradient required for TGF can be generated either by an imposed temperature gradient or by Joule heating resulting from an applied electric field that also drives the flow. In this study, a comprehensive numerical model describing the Joule heating induced temperature development and TGF is developed. The model consists of a set of governing equations including the Poisson-Boltzmann equation, the Laplace equation, the Navier-Stokes equations, the energy equations and the mass transport equation. As the thermophysical and electrical properties including the liquid dielectric constant, viscosity, and electric conductivity are temperature-dependent, these governing equations are coupled, and therefore the coupled governing equations are solved numerically by using a CFD-based numerical method. The numerical simulations agree well with the experimental results, suggesting the valid mathematical model presented in this study.     

Kinetics of the Inactivation of Creatine Kinase During Inhibition by Planar Anions

The kinetic theory of the substrate reaction during the modification of enzyme activity previously described by Tsou has been applied to a study on the kinetics of slow irreversible inhibition of creatine kinase by planar anions. Kinetic equation of substrate reaction was derived according to the theoretical analysis and experiment data, and then was simplified. From the simplified equation for the substrate reaction in the presence of the inhibitors, the microscopic rate constants for the reaction of the inhibitors with enzyme were obtained. The mechanism of inhibition of enzyme activity was discussed.     

Stochastic simulation of catalytic surface reactions in the fast diffusion limit

The master equation of a lattice gas reaction tracks the probability of visiting all spatial configurations. The large number of unique spatial configurations on a lattice renders master equation simulations infeasible for even small lattices. In this work, a reduced master equation is derived for the probability distribution of the coverages in the infinite diffusion limit. This derivation justifies the widely used assumption that the adlayer is in equilibrium for the current coverages and temperature when all reactants are highly mobile. Given the reduced master equation, two novel and efficient simulation methods of lattice gas reactions in the infinite diffusion limit are derived. The first method involves solving the reduced master equation directly for small lattices, which is intractable in configuration space. The second method involves reducing the master equation further in the large lattice limit to a set of differential equations that tracks only the species coverages. Solution of the reduced master equation and differential equations requires information that can be obtained through short, diffusion-only kinetic Monte Carlo simulation runs at each coverage. These simulations need to be run only once because the data can be stored and used for simulations with any set of kinetic parameters, gas-phase concentrations, and initial conditions. An idealized CO oxidation reaction mechanism with strong lateral interactions is used as an example system for demonstrating the reduced master equation and deterministic simulation techniques.