Application of He’s variational iteration method in nonlinear boundary value problems in enzyme– substrate reaction diffusion processes: part 1. The steady-state amperometric response

A mathematical model of amperometric biosensors has been developed. In this paper, He’s variational iteration method is implemented to give approximate and analytical solutions of non-linear reaction diffusion equations containing a non linear term related to Michaelis–Menten kinetic of the enzymatic reaction. The variational iteration method which produces the solutions in terms of convergent series, requiring no linearization or small perturbation. These analytical results are compared with available limiting case result and are found to be in good agreement.     

Mathematical modelling of enzyme kinetics reaction mechanisms and analytical solutions of non-linear reaction equations

The boundary value problem in basic enzyme reactions is formulated and approximate expressions for substrate and product concentrations are presented. He’s variational iteration method is used to give approximate and analytical solutions of non-linear reaction equations containing a non-linear term related to enzymatic reaction. The relevant analytical solutions for the substrate, enzyme, substrate-enzyme and product concentration profiles are discussed in terms of dimensionless reaction diffusion parameters K, λ and e{\varepsilon}.     

Analytical expression of the steady-state catalytic current of mediated bioelectrocatalysis and the application of He’s Homotopy perturbation method

A mathematical model of mediated bioelectrocatalysis is restudied in this paper. Here He’s Homotopy perturbation method is implemented to give approximate and analytical solutions of steady-state non-linear reaction diffusion equation containing a non-linear term related to Michaelis–Menten kinetics of the enzymatic reaction. Approximate analytical expressions for mediator concentration and current have been derived for all values of saturation parameter α and reaction diffusion parameter k for the various types of boundary conditions. The Homotopy perturbation method which produces the solutions in terms of convergent series, requiring no linearization or small perturbation. These analytical results are compared with numerical results (Matlab program) and are found to be in good agreement.     

Slow manifold for a bimolecular association mechanism

Finding the slow manifold for two-variable ordinary differential equation (ODE) models of chemical reactions with a single equilibrium is generally simple. In such planar ODEs the slow manifold is the unique trajectory corresponding to the slow relaxation of the system as it moves towards the equilibrium point. One method of finding the slow manifold is to use direct iteration of a functional equation; another method is to obtain a series solution of the trajectory differential equation of the system. In some cases these two methods agree order-by-order in the singular perturbation parameter controlling the fast relaxation of the intermediate (complex). However, de la Llave has found a model ODE where the series method always diverges. Bimolecular association is an example of a chemical reaction where the series method for finding the slow manifold diverges but the iterative method converges. In this mechanism a complex is formed which can then undergo unimolecular decay, i.e., [reaction: see text]. The kinetics of this reaction are investigated and its properties compared with two other two-step mechanisms where series expansion and iteration methods are equivalent: the Michaelis-Menten mechanism for enzyme kinetics, and the Lindemann-Christiansen mechanism of unimolecular decay in gas kinetics.     

Reaction, diffusion and migration in conducting polymer electrodes: analysis of the steady-state amperometric response

The processes of reaction, diffusion and electromigration of a charged substrate within an electronically conducting polymer film deposited on an inert supporting electrode are examined in terms of a quantitative analysis and solution of the pertinent differential equations. An analytical expression for the concentration profiles of substrate within the polymer layer is derived and a theoretical expression for the corresponding steady-state amperometric current response is presented. The transport and kinetics of the substrate are discussed in terms of a dimensionless reaction/diffusion parameter γ and a migration/diffusion parameter β. Received: 13 March 1998 / Accepted: 17 July 1998     

Analytical expression of non-steady-state concentrations and current pertaining to compounds present in the enzyme membrane of biosensor

A mathematical model of trienzyme biosensor at an internal diffusion limitation for a non-steady-state condition has been developed. The model is based on diffusion equations containing a linear term related to Michaelis-Menten kinetics of the enzymatic reaction. Analytical expressions of concentrations and current of compounds in trienzyme membrane are derived. An excellent agreement with simulation data is noted. When time tends to infinity, the analytical expression of non-steady-state concentration and current approaches the steady-state value, thereby confirming the validity of the mathematical analysis. Furthermore, in this work we employ the complex inversion formula to solve the boundary value problem.     

Analysis of the approximate slow invariant manifold method for reactive flow equations

The Approximate Slow Invariant Manifold method of Singh, Powers and Paolucci is a useful method for addressing model reduction in systems of reactive flow equations. It exploits separations of time scales between slow and fast species, and it generalizes the Intrinsic Low-Dimensional Manifold method, which was developed for model reduction in the context of reaction kinetics, to systems with diffusive and active transport. In this article, we present a mathematical analysis of the Approximate Slow Invariant Manifold method in the context of systems of reaction–diffusion equations with slow and fast reaction kinetics. Beginning with systems of two species (one slow and one fast), and then treating general systems with multiple slow and fast species, we explicitly determine the accuracy of the Approximate Slow Invariant Manifold method. We find that it is correct up to and including the terms of first order in the small parameter that measures the separation of the kinetics time scales, and that it captures many of the terms at second order, as well. Our analysis includes precise statements of the errors at second order, and we find that these are proportional to the slow components of the reaction–diffusion equation, as well as to the curvature of the critical manifold. We illustrate the results analytically on two prototypical examples, the Michaelis–Menten–Henri model with diffusion of the slow species and the Davis–Skodje model in which both the slow and fast species diffuse.     

Two-Temperature Two-Dimensional Non Chemical Equilibrium Modeling of Ar–CO<Subscript>2</Subscript>–H<Subscript>2</Subscript> Induction Thermal Plasmas at Atmospheric Pressure

Here the authors developed a two-dimensional two-temperature chemical non-equilibrium (2T-NCE) model of Ar–CO₂–H₂ inductively coupled thermal plasmas (ICTP) around atmospheric pressure (760 torr). Assuming 22 different particles in this model and by solving mass conservation equations for each particle, considering diffusion, convection and net production terms resulting from 198 chemical reactions, chemical non-equilibrium effects were taken into account. Species density of each particle or simply particle composition was also derived from the mass conservation equation of each one taking the non chemical equilibrium effect into account. Transport and thermodynamic properties of Ar–CO₂–H₂ thermal plasmas were self-consistently calculated using the first order approximation of the Chapman–Enskog method at each iteration point implementing the local particle composition and temperature. Calculations at reduced pressure (500 and 300 torr) were also done to investigate the effect of pressure on non-equilibrium condition. Results obtained by the present model were compared with results from one temperature chemical equilibrium (1T-CE) model, one-temperature chemically non equilibrium (1T-NCE) model and finally with 2T-NCE model of Ar–N₂–H₂ plasmas. Investigation shows that consideration of non-chemical equilibrium causes the plasma volume radially wider than CE model due to particle diffusion. At low pressure with same input power, presence of diffusion is relatively stronger than at high pressure. Comparison of present reactive model with non-reactive Ar–N₂–H₂ plasmas shows that maximum temperature reaches higher in reactive C–H–O molecular system than non-reactive plasmas due to extra contribution of reaction heat.     

Computational modelling of amperometric biosensors in the case of substrate and product inhibition

In this paper the response of an amperometric biosensor at mixed enzyme kinetics and diffusion limitations is modelled in the case of the substrate and the product inhibition. The model is based on non-stationary reaction–diffusion equations containing a non-linear term related to non-Michaelis–Menten kinetics of an enzymatic reaction. A numerical simulation was carried out using a finite difference technique. The complex enzyme kinetics produced different calibration curves for the response at the transition and the steady-state. The biosensor operation is analysed with a special emphasis to the conditions at which the biosensor response change shows a maximal value. The dependence of the biosensor sensitivity on the biosensor configuration is also investigated. Results of the simulation are compared with known analytical results and with previously conducted researches on the biosensors.     

Dynamic disorder in single-molecule Michaelis-Menten kinetics: the reaction-diffusion formalism in the Wilemski-Fixman approximation

Single-molecule equations for the Michaelis-Menten [Biochem. Z. 49, 333 (1913)] mechanism of enzyme action are analyzed within the Wilemski-Fixman [J. Chem. Phys. 58, 4009 (1973); 60, 866 (1974)] approximation after the effects of dynamic disorder--modeled by the anomalous diffusion of a particle in a harmonic well--are incorporated into the catalytic step of the reaction. The solution of the Michaelis-Menten equations is used to calculate the distribution of waiting times between successive catalytic turnovers in the enzyme beta-galactosidase. The calculated distribution is found to agree qualitatively with experimental results on this enzyme obtained at four different substrate concentrations. The calculations are also consistent with measurements of correlations in the fluctuations of the fluorescent light emitted during the course of catalysis, and with measurements of the concentration dependence of the randomness parameter.     

Entrapment of living microbial cells in covalent polymeric networks

The kinetics of oxidative phenol degradation with microbial cellsCandida tropicalis, immobilized in a polyacrylamide and polymethacrylamide matrix, were mathematically simulated assuming zero-order and Michaelis-Menten rate equations. For zero-order kinetics an expanded equation for catalytic effectiveness as a function of the Thiele modulus, Biot number, and partition coefficients was derived and compared with numerical solutions for Michaelis-Menten kinetics. Errors with regard to the zero-order approximation become negligible ifc _o/K _M >2. Experimentally determined catalyst activities as a function of particle size and cell concentration were compared to calculated ones. Additional experiments to determine the diffusion and oxygen consumption ratios have been carried out in an effort to resolve the physical parameters to be used in the above mentioned calculations. Furthermore, experiments on cell growth during reincubation with nutrients and oxygen are reported; an increase in activity up to a factor of ten was observed. These experiments demonstrate that the microbial cells are entrapped in the polymer matrix in the living state.     

Finite-difference model for the reaction sintering of oxide ceramics by direct oxidation of intermetallic powder compacts

A kinetic model for the reaction sintering of oxide ceramics in the system Al₂O₃–SiO₂–ZrO₂ using mixtures of intermetallic compounds is presented. A 2D finite-difference model is developed to describe the exothermic gas-solid reactions taking place during the firing of ZrAl₃/ZrSi₂ powder compacts. The model accounts for the oxidation kinetics of the powder particles, as well as the consumption and diffusion of gaseous oxygen through the porous matrix. Additionally, possible changes in the pore structure of the green body due to the oxidation reactions and sintering effects are incorporated in the model. The resulting differential equations are coupled with a two-dimensional Fourier heat balance equation leading to a system of nonlinear partial differential equations, which is solved by the numerical method of lines. The influence of different processing parameters like sample composition and heating cycle on the reaction sintering process is investigated and the model-predicted reaction behaviour is compared to experimental results.     

Approximate analytical solutions for arbitrary l-state of the Hulthén potential with an improved approximation of the centrifugal term

An approximate analytical solution of the radial Schr?dinger equation for the generalized Hulthén potential is obtained by applying an improved approximation of the centrifugal term. The bound state energy eigenvalues and the normalized eigenfunctions are given in terms of hypergeometric polynomials. The results for arbitrary quantum numbers n _r and l with different values of the screening parameter δ are compared with those obtained by the numerical method, asymptotic iteration, the Nikiforov-Uvarov method, the exact quantization rule, and variational methods. The results obtained by the method proposed in this work are in a good agreement with those obtained by other approximate methods.      

用协同学方法研究化学反应速率和机理问题

用非平衡统计理论中的协同学方法分析了化学动力学中的反应速率和机理问题．对于任何给定的机理，通过引入一种非奇异变换，将相应的反应速率方程组直接地转换成Ｈａｋｅｎ形式，从而得到反应体系序参量以及其它变量的演化解析表达式．讨论了有机分子热分解的Ｒｉｃｅ－Ｈｅｒｚｆｅｌｄ机理．对醚的热分解链反应得到了一级动力学方程，理论结果与实验结果相符合．利用这种方法能解析地求解复杂反应的反应速率方程组，而不必引用在化学动力学中的“稳态近似”假设     

Analytical solution of nonlinear diffusion processes in modified electrode

The system of coupled nonlinear diffusion equations are solved analytically for the transport and kinetics of electrons and reactant in the layer of a modified electrode. Analytical expressions of concentrations of mediator and substrate are presented using Homotopy perturbation method. A simple expression of electrochemical rate constant K _ME⁺ is also obtained for all values of reaction parameters. The available limiting case results are compared with our results and are found to be in good agreement.     

Kinetics and mechanism of the oxidation of formic and oxalic acids by quinolinium fluorochromate

Kinetics and mechanism of oxidation of formic and oxalic acids by quinolinium fluorochromate (QFC) have been studied in dimethylsulphoxide. The main product of oxidation is carbon dioxide. The reaction is first-order with respect to QFC. Michaelis-Menten type of kinetics were observed with respect to the reductants. The reaction is acid-catalysed and the acid dependence has the form: k_obs =a +b[H⁺]. The oxidation of α-deuterioformic acid exhibits a substantial primary kinetic isotope effect (k_H/k_D = 6.01 at 303 K). The reaction has been studied in nineteen different organic solvents and the solvent effect has been analysed using Taft’s and Swain’s multiparametric equations. The temperature dependence of the kinetic isotope effect indicates the presence of a symmetrical cyclic transition state in the rate-determining step. Suitable mechanisms have been proposed.     

Mechanisms controlling the sensitivity of amperometric biosensors in flow injection analysis systems

This paper numerically investigates the sensitivity of an amperometric biosensor acting in the flow injection mode when the biosensor contacts an analyte for a short time. The analytical system is modelled by non-stationary reaction-diffusion equations containing a non-linear term related to the Michaelis-Menten kinetics of an enzymatic reaction. The mathematical model involves three regions: the enzyme layer where enzymatic reaction as well as the mass transport by diffusion takes place, a diffusion limiting region where only the diffusion takes place, and a convective region. The biosensor operation is analysed with a special emphasis to the conditions at which the biosensor sensitivity can be increased and the calibration curve can be prolonged by changing the injection duration, the permeability of the external diffusion layer, the thickness of the enzyme layer and the catalytic activity of the enzyme. The apparent Michaelis constant is used as a main characteristic of the sensitivity and the calibration curve of the biosensor. The numerical simulation was carried out using the finite difference technique.     

Modelling of solid state voltammetry of immobilized microcrystals assuming an initiation of the electrochemical reaction at a three-phase junction

The electrochemical reduction of a solid compound characterized by mixed ionic/electronic conductivity, immobilized on an electrode surface and in contact with an electrolyte solution, has been studied theoretically. The uptake or expulsion of electrons and electrolyte cation is coupled to maintain electroneutrality and is assumed to obey Fick's law of diffusion. Starting with the fully oxidized species, the simultaneous uptake of cations and electrons will be possible at the three-phase junction only, where electrode, solid and electrolyte solution meet. From this point, electrons and cations diffuse perpendicularly into the crystal lattice. The reaction zone grows owing to the formation of the electronically and ionically conducting reduced product. Two- and three-dimensional models have been utilized to simulate the diffusion and the current flow in response to an applied potential step. The resulting chronoamperometric curves have been analyzed with the help of fitting procedures. Under certain conditions, a transition of the three-phase reaction to a pure two-phase reaction occurs. This transition to a two-phase condition is the reason that a number of equations for the exhaustive conversion are similar to those known for planar diffusion, for example. To illustrate this, and for a better understanding of the phenomena, concentration profiles are presented for different degrees of the reaction and for varied simulation conditions. It is demonstrated how geometrical properties like crystal shape (cuboid with x ≠ y ≠ z) and crystal size as well as physical properties, e.g. the diffusion coefficients, govern the electrochemical behavior of mixed ionic/electronic conductors and form the basis of the current-time functions. The numerical simulation of a two-dimensional semi-infinite model of the reaction at the three-phase junction gives results comparable to an algebraical approach. The finite-difference method turned out to be suitable to solve the problems arising from the three-dimensional and finite diffusion conditions and from different crystal shapes. Received: 24 November 1999 / Accepted: 22 February 2000     

Theory and simulation of polymerization‐induced phase separation in polymeric media

A rigorous model of polymerization‐induced phase separation (PIPS), based on the non‐linear Cahn‐Hilliard (C‐H) and Flory‐Huggins (F‐H) theories combined with a second‐order polymerization reaction equation, has been formulated and its solutions characterized. The model describes phase separation in system consisting of a non‐reactive polymer and a monomer that undergoes condensation polymerization. The model consists of a balance equation for the low molecular weight polymerization regime and another balance equation for the high molecular weight entangled regime. The model equations are solved, and the solutions are characterized to identify the dynamical and morphological phenomena of the PIPS process. The extent of phase separation increases significantly with time during the early stage of phase separation, and slows down in the intermediate stage. The various types of phase‐separated morphologies are fully characterized using a novel morphological characterization techniques, known as the intensity and scale of segregation. Both the dynamical and morphological features of the PIPS method are sensitive to the magnitudes of the dimensionless diffusion coefficient D* and the dimensionless reaction rate constant K*. The scale of segregation and the droplet size decreases as D* and K* increase. On the other hand, the intensity of segregation increases with K*, but decreases with D*. The present results extend the present knowledge of the PIPS process by taking into account the effects arising from the presence of a non‐reactive polymer.     

Stability of solutions to a reaction diffusion system based upon chemical reaction kinetics

In this paper, we deal with the stability problem to some mathematical models that describe chemical reaction kinetics. One is a set of ordinary differential equations induced by one reversible chemical reaction mechanism containing three chemical species. The other is a set of reaction diffusion equations based on the same chemical reaction. We show that all solutions of the model are asymptotically stable by applying the Liapunov method. We thus find that the concentration of each species has certain limits as time proceeds.