On approximate analytical solutions of differential equations in enzyme kinetics using homotopy perturbation method

Homotopy perturbation method is used to extend the approximate analytical solutions of non-linear reaction equations describing enzyme kinetics for combinations of parameters for which solutions obtained in previous works are not valid. Also, by constructing a new homotopy, alternative approximate analytical expressions for substrate, substrate-enzyme complex and product concentrations are found. These first-order approximate solutions give more accurate results than the second-order approximations derived in previous works.     

A modified homotopy perturbation method and the axial secular frequencies of a non-linear ion trap

In this paper, a modified version of the homotopy perturbation method, which has been applied to non-linear oscillations by V. Marinca, is used for calculation of axial secular frequencies of a non-linear ion trap with hexapole and octopole superpositions. The axial equation of ion motion in a rapidly oscillating field of an ion trap can be transformed to a Duffing-like equation. With only octopole superposition the resulted non-linear equation is symmetric; however, in the presence of hexapole and octopole superpositions, it is asymmetric. This modified homotopy perturbation method is used for solving the resulting non-linear equations. As a result, the ion secular frequencies as a function of non-linear field parameters are obtained. The calculated secular frequencies are compared with the results of the homotopy perturbation method and the exact results. With only hexapole superposition, the results of this paper and the homotopy perturbation method are the same and with hexapole and octopole superpositions, the results of this paper are much more closer to the exact results compared with the results of the homotopy perturbation method.     

A probability generating function method for stochastic reaction networks

In this paper we present a probability generating function (PGF) approach for analyzing stochastic reaction networks. The master equation of the network can be converted to a partial differential equation for PGF. Using power series expansion of PGF and Padé approximation, we develop numerical schemes for finding probability distributions as well as first and second moments. We show numerical accuracy of the method by simulating chemical reaction examples such as a binding-unbinding reaction, an enzyme-substrate model, Goldbeter-Koshland ultrasensitive switch model, and G(2)/M transition model.     

Optimal homotopy analysis method for the non-isothermal reaction–diffusion model equations in a spherical catalyst

We consider the Lane–Emden boundary value problems which appear in chemical applications, biochemical applications, and scientific disciplines. The Lane–Emden problem is transformed into an equivalent integral equation. The optimal homotopy analysis method is used to solve two specific models. The first problem models reaction–diffusion equation in a spherical catalyst, while the second problem models the reaction–diffusion process in a spherical biocatalyst. We obtain reliable analytical solutions of the concentrations and the effectiveness factors. Numerical results and graphs show the reliability and efficiency applicability of the employed method.     

A reduction method for multiple time scale stochastic reaction networks

In this paper we develop a reduction method for multiple time scale stochastic reaction networks. When the transition-rate matrix between different states of the species is available, we obtain systems of reduced equations, whose solutions can successively approximate, to any degree of accuracy, the exact probability that the reaction system be in any particular state. For the case when the transition-rate matrix is not available, one needs to rely on the chemical master equation. For this case, we obtain a corresponding reduced master equation with first-order accuracy. We illustrate the accuracy and efficiency of both approaches by simulating several motivating examples and comparing the results of our simulations with the results obtained by the exact method. Our examples include both linear and nonlinear reaction networks as well as a three time scale stochastic reaction-diffusion model arising from gene expression.     

A numerical characteristic method for probability generating functions on stochastic first-order reaction networks

We propose an efficient and accurate numerical scheme for solving probability generating functions arising in stochastic models of general first-order reaction networks by using the characteristic curves. A partial differential equation derived by a probability generating function is the transport equation with variable coefficients. We apply the idea of characteristics for the estimation of statistical measures, consisting of the mean, variance, and marginal probability. Estimation accuracy is obtained by the Newton formulas for the finite difference and time accuracy is obtained by applying the fourth order Runge–Kutta scheme for the characteristic curve and the Simpson method for the integration on the curve. We apply our proposed method to motivating biological examples and show the accuracy by comparing simulation results from the characteristic method with those from the stochastic simulation algorithm.     

Sign patterns for chemical reaction networks

Most differential equations found in chemical reaction networks (CRNs) have the form:

A reduction method for multiple time scale stochastic reaction networks with non-unique equilibrium probability

We develop a reduction method for general closed multiple time scale stochastic reaction networks for which the fast subsystem may have non-unique equilibrium probability. We obtain a reduced ODE system with nonhomogeneous terms whose solutions can approximate the solutions of the full system accurately. We then apply this reduction method to general linear network and nonlinear networks for which the state diagram can be constructed. We illustrate the accuracy of the reduction method by comparing computational results of the full systems with the reduced ODE systems for several examples. Finally, we show how the reduction method may be extended to three or more time scale reaction networks.     

Computation of stationary points via a homotopy method

A homotopy method is presented that locates both minimizers and saddle points of energy functions in an efficient manner. In contrast to other methods, it makes possible the exploration of large parts of potential energy surfaces. Along a homotopy path stationary points of odd and even order occur alternately. A path tracing procedure requiring only gradients and at most one evaluation of the Hessian matrix is given. Test results on a model potential and three MINDO/3 potentials are reported. Received: 6 May 1996 / Accepted: 2 April 1998 / Published online: 23 June 1998     

An approximate variational perturbation method

The Goldhammer—Feenberg variational procedure is applied to the perturbation treatment of the configuration interaction matrix of Diner, Malrieu and Claverie (PCILO) to yield a scheme (PVCILO) that appears more accurate for the small systems investigated. In addition, PVCILO shows promise as executing as an N² method, where N is the number of basis bonds.     

Calculations of first- and second-order nonlinear molecular hyperpolarizabilities by perturbation methods: I. An efficient method for evaluating time-independent hyperpolarizabilities

From a suitable reorganisation of the sum-over-states (SOS) equations of the usual time-independent perturbation theory, recurrent expressions for static polarizability (α) and second- (beta;) and third-(γ) order hyperpolarizabilities are obtained. These expressions are given in a well-adapted way for computer implementation and lead to an efficient algorithm reducing the computing time by a factor of 50 with respect to a “brute-force” translation of the standard SOS equations.     

Ladder operators in commutator perturbation method

The present work is a version of Van Vleck–Primas perturbation method in terms of generalized ladder operators. Using the superoperator approach, a fully general, self-consistent and a totally free scheme from representation is developed. © 1998 John Wiley & Sons, Inc. Int J Quant Chem 68: 79–90, 1998     

Kinetic determination of silver ion by its perturbation on Belousov-Zhabotinskii oscillating chemical reaction using potentiometric method.

Silver ion can perturb the Belousov-Zhabotinskii (B-Z) oscillating chemical reaction. Therefore, the B-Z oscillating system was applied in the determination of silver ion by using a platinum wire as an indicator electrode in the potentiometric method. The amplitude of the potentiometric oscillation increased linearly in proportion to [Ag+] in the range of 9.42 x 10(-6) M to 2.54 x 10(-4) M, with a correlation coefficient of 0.999 under the optimum conditions. The obtained LOD (2sigma) was 8.85 x 10(-6) M and the relative standard deviation (RSD) for five measurements of 1 x 10(-4) M silver ion was 5%. The influence of some potentially interference was also investigated.     

Adaptive approach for nonlinear sensitivity analysis of reaction kinetics

We present a unified approach for linear and nonlinear sensitivity analysis for models of reaction kinetics that are stated in terms of systems of ordinary differential equations (ODEs). The approach is based on the reformulation of the ODE problem as a density transport problem described by a Fokker-Planck equation. The resulting multidimensional partial differential equation is herein solved by extending the TRAIL algorithm originally introduced by Horenko and Weiser in the context of molecular dynamics (J. Comp. Chem. 2003, 24, 1921) and discussed it in comparison with Monte Carlo techniques. The extended TRAIL approach is fully adaptive and easily allows to study the influence of nonlinear dynamical effects. We illustrate the scheme in application to an enzyme-substrate model problem for sensitivity analysis w.r.t. to initial concentrations and parameter values.     

Synthesis and characterization of interpenetrating polymer networks for nonlinear optics

Two-component simultaneous interpenetrating networks (IPN) of thepoly(4′-[[2-(methylacryloxy)ethyl]ethylamino]-4-nitroazobenzene-co-methyl meth-acrylate) (PDR1MA-co-MMA)/poly(2,6-dimethyl-1,4-phenylene oxide) (PPO) system, the PDR1MA/PPO system and 4′-[[2-(acetoxy)ethyl]ethylamino]-4-nitroazo benzene (ACDR1) doped MMA/PPO system were synthesized and characterized. As studied by differential scanning calorimetry (DSC) the full IPNs of the PDR1MA-co-MMA/PPO system and the PDR1MA/PPO system showed a single T_g varying with the PPO composition. A single-phase morphology of the full IPNs was also indicated by scanning electron microscopy (SEM). Transparent films were cast onto clean microscopic glass slides and poled at 190°C for 1 h. The UV-VIS absorption spectra of the three IPN systems before and after curing and corona poling showed a shift in the maximum absorption due to the induced alignment of the nonlinear optical chromophores in the IPN systems. The absorption intensity of the full IPNs remained same after heating at 120°C for 72 h, indicating that the electric field-induced alignment is stable in the full IPN materials. Preliminary second harmonic generation (SHG) data on these IPNs are presented. © 1998 John Wiley & Sons, Inc. J Polym Sci A: Polym Chem 36: 553–561, 1998     

Solution of fractional-order integro-differential equations using optimal homotopy asymptotic method

Journal of Thermal Analysis and Calorimetry - In this article, a semi-analytical technique called optimal homotopy asymptotic method (OHAM) is presented for numerical solutions of fractional-order...