How accurate are the nonlinear chemical Fokker-Planck and chemical Langevin equations?

The chemical Fokker-Planck equation and the corresponding chemical Langevin equation are commonly used approximations of the chemical master equation. These equations are derived from an uncontrolled, second-order truncation of the Kramers-Moyal expansion of the chemical master equation and hence their accuracy remains to be clarified. We use the system-size expansion to show that chemical Fokker-Planck estimates of the mean concentrations and of the variance of the concentration fluctuations about the mean are accurate to order Ω(-3∕2) for reaction systems which do not obey detailed balance and at least accurate to order Ω(-2) for systems obeying detailed balance, where Ω is the characteristic size of the system. Hence, the chemical Fokker-Planck equation turns out to be more accurate than the linear-noise approximation of the chemical master equation (the linear Fokker-Planck equation) which leads to mean concentration estimates accurate to order Ω(-1∕2) and variance estimates accurate to order Ω(-3∕2). This higher accuracy is particularly conspicuous for chemical systems realized in small volumes such as biochemical reactions inside cells. A formula is also obtained for the approximate size of the relative errors in the concentration and variance predictions of the chemical Fokker-Planck equation, where the relative error is defined as the difference between the predictions of the chemical Fokker-Planck equation and the master equation divided by the prediction of the master equation. For dimerization and enzyme-catalyzed reactions, the errors are typically less than few percent even when the steady-state is characterized by merely few tens of molecules.     

Formulation of magnetically perturbed time-dependent density functional theory

A formulation of time-dependent density functional theory (TDDFT) in the presence of a static imaginary perturbation is derived. A perturbational approach is applied leading to corrections to various orders in the quantities of interest, namely, the excitation energies and transition densities. The perturbed TDDFT equations are relatively straightforward to derive but the resulting expressions are rather cumbersome. Simplifications of these equations are suggested. Both the simplified and full expressions are used to obtain equations for first- and second-order corrections to the excitation energy, the first-order correction to the transition density, and the corrections for both quantities to first-order in two different perturbations. This formulation, called magnetically perturbed TDDFT, details how conventional TDDFT calculations can be corrected to allow for the inclusion of a static magnetic field and/or spin-orbit coupling.     

Kinetics of free-radical decay reactions in polymeric solids

Kinetic equations for the decay of the free radicals in polymeric solids are given for the following assumptions on which they are based: (1) two simultaneous first-order but physically separated decay reactions; (2) two simultaneous noninteracting second-order decay reactions; (3) combined simultaneous but intermingled first- and second-order decay reactions; (4) the same but for independent, i.e., not intermingled, first- and second-order decay reactions; (5) a second-order decay reaction in the presence of some free radicals that do not decay; and (6) a first-order decay reaction in the presence of some free radicals that do not decay. In all of the above physical systems the total concentration only can be measured. Hence the above kinetic equations refer to the change of the total concentration with time. It is found that the data for the decay of the free radicals in irradiated isotactic polypropylene and 61% styrene-39% butadiene block copolymer agree best with the equations for the second-order decay in the presence of a fraction of nondecaying free radicals.     

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.     

Effect of scaling of a quantum mechanical force field on the frequencies and forms of molecular vibrations

The effect of scaling of an ab initio quantum mechanical force field on the frequencies and forms of normal vibrations are studied in terms of first- and second-order perturbation theory. Scaling the force constant matrix according to Pulay using certain assumptions in first-order perturbation theory is equivalent to scaling vibration frequencies and does not modify the form of vibrations. In this case, the second-order corrections to the frequencies and forms of vibrations become zero. The first-order perturbation theory formulas are used to verify the assumptions by calculating the frequencies and matrices of transition to perturbed forms of vibrations of ethane, propane, ethylene, cyclopropene, and isobutene molecules from quantum mechanical force fields found with the 6-31G basis set. It is shown that the vibration frequencies calculated by the formulas of first-order perturbation theory are in good agreement with exact values; the matrix of transition to perturbed eigenvectors is rarefied, with only ≈1% of its elements being markedly nonzero. Moscow State University. Translated fromZhurnal Strukturnoi Khimii, Vol. 39, No. 2, pp. 210–216, March–April, 1998. This work was supported by RFFR grant No. 96-03-34085.     

An integration method of enzyme analysis

The application of an integration method of kinetic analysis to first-order and second-order reactions is discussed with particular emphasis on enzyme analyses. Transducer signals or concentrations of products or substrates are integrated for a Fixed time and the net integral of the increased or decreased signal or concentration is related to the initial substrate or enzyme concentration. Equations are developed describing the dependence of the integrals on enzyme and substrate concentrations for first- and second-order reactions, and examples are presented illustrating different cases. The application of this method to complicated enzymatic systems is discussed.     

Restoring detailed balance in the Landau-Teller probabilities for collision-induced vibrational transitions

The general quasi-classical treatment for collision-induced vibrational transitions in diatomic molecules, under near-adiabatic conditions, is used to derive quantum corrections for probabilities, calculated in the external field approximation originally used by Landau and Teller. The quantum corrections are expressed through the Landau-Teller classical collision time. The first-order correction to the classical exponent restores detailed balance for up- and down-transitions and does not depend on the properties of the bath except for its temperature. The limits of applicability of the first-order correction are discussed.     

Central limit theorem for chemical kinetics in complex systems

The prevalence of apparently first-order kinetics of reactant disappearance in complex systems with many possible reaction pathways is usually attributed to the dominance of a single rate limiting step. Here, we investigate another possible explanation: that apparently first-order kinetics might arise because the aggregate behavior of many processes, with varying order of reaction and rate constant, approaches a central limit that is indistinguishable from first-order behavior. This hypothesis was investigated by simulating systems of increasing complexity and deriving relationships between the apparent reaction order of such systems and various measures of their complexity. Transformation of a chemical species by parallel irreversible reactions that are zero-, first-, or second-order is found to converge to a central limit as the number of parallel reactions becomes large. When all three reaction orders are represented, on average, in equal proportions, this central limit is experimentally indistinguishable from first-order. A measure of apparent reaction order was used to investigate the nature of the convergence both stochastically and by deriving theoretical limits. The range of systems that exhibit a central limit that is approximately first-order is found to be broad. First-order like behavior is also found to be favored when the distribution of material among the parallel processes (due to differences in rate constants for the individual reactions) is more complex. Our results show that a first-order central limit exists for the kinetics of chemical systems and that the variable controlling the convergence is the physical complexity of reaction systems.     

From Langmuir kinetics to first- and second-order rate equations for adsorption

So far, the first- and second-order kinetic equations have been most frequently employed to interpret adsorption data obtained under various conditions, whereas the theoretical origins of these two equations still remain unknown. Using the Langmuir kinetics as a theoretical basis, this study showed that the Langmuir kinetics can be transformed to a polynomial expression of dtheta t /d t = k 1(theta e - theta t ) + k 2(theta e - theta t ) (2), a varying-order rate equation. The sufficient and necessary conditions for simplification of the Langmuir kinetics to the first- and second-order rate equations were put forward, which suggested that the relative magnitude of theta e over k 1/ k 2 governs the simplification of the Langmuir kinetics. In cases where k 1/ k 2 is greater than theta e or k 1/ k 2 is very close to theta e, adsorption kinetics would be reasonably described by the first-order rate equation, whereas the Langmuir kinetics would be reduced to the second-order equation only at k 1/ k 2 < theta e. It was further demonstrated that both theta e and k 1/ k 2 are the function of initial adsorbate concentration ( C 0) at a given dosage of adsorbent, indicating that simplification of the Langmuir kinetics indeed is determined by C 0. Detailed C 0-depedent boundary conditions for simplifying the Langmuir kinetics were also established and were verified by experimental data.     

Computational study of cyclohexanone-monomer co-initiation mechanism in thermal homo-polymerization of methyl acrylate and methyl methacrylate

This paper presents a systematic computational study of the mechanism of cyclohexanone-monomer co-initiation in high-temperature homopolymerization of methyl acrylate (MA) and methyl methacrylate (MMA). Previous experimental studies of spontaneous thermal homopolymerization of MA and MMA showed higher monomer conversion in the presence of cyclohexanone than xylene. However, these studies did not reveal the initiation mechanism(s) or the initiating species. To identify the initiation mechanism and the initiating species, we explore four different mechanisms, (1) Kaim, (2) Flory, (3) α-position hydrogen transfer, and (4) Mayo, using first-principles density functional theory (DFT) and second-order M?ller-Plesset perturbation theory (MP2) calculations. Transition-state geometries for each mechanism are determined using B3LYP/6-31G* and assessed with MP2/6-31G*. Activation energies and rate constants are calculated using transition-state theory. The harmonic oscillator approximation and tunneling corrections are applied to compute the reaction rate constants. This study indicates that α-position hydrogen transfer and Mayo mechanisms have comparable barriers and are capable of generating monoradicals for initiating polymerization of MA and MMA; these two mechanisms can cause cyclohexanone-monomer co-initiation in thermal polymerization of MA and MMA.     

First-order gradient correction for the exchange-energy density functional for atoms

Summary Spurred by earlier discoveries by Deb and others that a first-order correction in the kinetic energy functional leads to an improved kinetic energy, a first-order gradient term is studied as a component of the gradient-corrected functional for atomic exchange energy. This term is shown to improve the local density approximation to the exchange energy more than does the usual second-order gradient correction. The imperative of systematically deriving this gradient correction is discussed but not resolved.     

The use of digital simulation to improve the cyclic voltammetric determination of rate constants for homogeneous chemical reactions following charge transfers

Cyclic voltammetry (CV) is a very useful electrochemical tool used to study reaction systems that include chemical steps that are coupled to electron transfers. This type of system generally involves the chemical reaction of an electrochemically generated free radical. Published methods exist that are used to determine the kinetics of electrochemically initiated chemical reactions from the measurements of the peak current ratio (i_pa/i_pc) of a cyclic voltammogram. The published method requires working curves to relate a kinetic parameter to the peak current ratio.In the presented work, a digital simulation package was used to obtain improved working curves for specific working conditions. The curves were compared with the published results for the first- and second-order chemical reactions following the charge transfer step mechanisms.According to the presented results, the previously published working curve is reliable for a mechanism with a first-order chemical reaction; however, a change in the switching potential requires a recalculation of the curve. In the case of mechanisms with a second-order step (dimerisation and disproportionation), several different views exist on how the second-order chemical term should be expressed so that different values of the constant are obtained. Parameters such as electrode type, electrode area, electroactive species concentration, switching potential, scan rate and method for peak current ratio calculation modify the working curves and must always be specified.We propose a standardised method to obtain the most reliable kinetic constant values.The results of this work will permit researchers who handle simulation software to construct their own working curves. Additionally, those who do not have the simulation software could use the working curves described here.The revelations of the presented experiments may be useful to a broad chemistry audience because this study presents a simple and low-cost procedure for the study of free radicals that otherwise should be studied with more sophisticated and expensive techniques, such as ESR or pulse radiolysis.     

Effect of pressure on the radiation-induced polymerization of ethylene in tert-butyl alcohol

The effects of pressure of the radiation-induced polymerization of ethylene in tert-butyl alcohol were studied. The reaction was carried out by use of a reactor with a capacity of 100 ml under the following conditions; pressure, 60–400 kg/cm²; temperature, 24 ± 3°C; dose rate, 2.0 × 10⁴?1.6 × 10⁵ rad/hr; amount of medium (tert-butyl alcohol containing 5 vol-% water), 70 ml. The results of polymerization were analyzed by a kinetical treatment based on a reaction mechanism with both first- and second-order terminations for the concentration of propagating, radical. On the basis of the kinetical treatment the rate constants of each elementary reaction at several pressures were determined, and the activation volumes of elementary reactions were obtained and are discussed in connection with the reaction mechanism. Consequently, the rate constants of propagation, first-order termination, and second-order termination at pressure p and at 24°C were expressed by,      

Kalman filtering of data from first- and second-order kinetics

The Kalman filter algorithm was used to process data obtained in the simultaneous determination of species following first- and second-order kinetics. The performance of the algorithm in the quantification of chemical components from simulated data was assessed, and the influence of various parameters involved was estimated. The algorithm was applied to the resolution of cysteine-ascorbic acid and glutathione-ascorbic acid mixtures where the ascorbic acid followed pseudo first-order kinetics and the amino acids second-order kinetics in the reaction with the copper(II)-neocuproine system. Some features of the determinations (namely, afforded concentration ratios, accuracy and precision) are discussed.     

绝热式热动力学的研究I:可逆反应动力学的数学模型

本文提出了几类可逆反应通用的微分动力学方程.并应用无量纲参数法,建立了可逆反应动力学的数学模型.还给出了一个解析实例.     

A closed-shell coupled-cluster treatment of the Breit-Pauli first-order relativistic energy correction

First-order relativistic corrections to the energy of closed-shell molecular systems are calculated, using all terms in the two-component Breit-Pauli Hamiltonian. In particular, we present the first implementation of the two-electron Breit orbit-orbit integrals, thus completing the first-order relativistic corrections within the two-component Pauli approximation. Calculations of these corrections are presented for a series of small and light molecules, at the Hartree-Fock and coupled-cluster levels of theory. Comparisons with four-component Dirac-Coulomb-Breit calculations demonstrate that the full Breit-Pauli energy corrections represent an accurate approximation to a fully relativistic treatment of such systems. The Breit interaction is dominated by the spin-spin interaction, the orbit-orbit interaction contributing only about 10% to the total two-electron relativistic correction in molecules consisting of light atoms. However, the relative importance of the orbit-orbit interaction increases with increasing nuclear charge, contributing more than 20% in H(2)S.     

SCF perturbation theory and intermolecular interactions

A self-consistent perturbation theory is derived in the framework of Roothaan's MOLCAO procedure for closed shell systems. Contrary to previous investigations which have considered only one particle perturbations, two particle perturbation operators are considered. Expressions for the first-order density matrix and first- and second-order energy corrections are obtained. A diagram formulation of the complete perturbation expansion is presented. The results are applied to the treatment of the intermolecular interaction problem. The interaction energy is represented as a sum of several contributions: Coulomb, exchange, resonance, polarization and exchange repulsion. A semi-empirical version of the theory is suggested which explicitly involves all the physically significant energy terms and may be useful for the investigation of complex systems.     

多胺型阴离子交换纤维吸附铬(VI)的动力学

以聚丙烯腈纤维为原料, 采用化学改性法, 制备了多胺型阴离子交换纤维. 研究该纤维对Cr(VI)的吸附特性. 在研究的温度及浓度范围内, 该纤维对Cr(VI)吸附的平衡数据符合Langmuir和Freundlich吸附等温方程, 对Cr(VI)有较强的亲和力, 吸附反应易于进行. 重点研究了该纤维对Cr(VI)的吸附动力学特性, 分别采用Lagergren一级动力学方程、修正伪一级动力学方程、伪二级动力学方程和颗粒内扩散方程进行拟合, 计算相应的速率常数. 研究表明, 该吸附是一个快速吸附过程, 20 min即可接近吸附平衡, 吸附过程符合伪二级动力学方程, 以化学吸附为主, 该纤维能够多次反复对Cr(VI)进行吸附.     

Spectroscopic Parameters of X3∑-, a1△, and A'3△ Electronic States of SO Radical

The potential energy curves (PECs) of three low-lying electronic states (X³∑^-, a¹△, and A^'3△) of SO radical have been studied by ab initio quantum chemical method. The calcula-tions were carried out with the full valence complete active space self-consistent field method followed by the highly accurate valence internally contracted multireference configuration in-teraction (MRCI) approach in combination with correlation-consistent basis sets. Effects of the core-valence correlation and relativistic corrections on the PECs are taken into account. The core-valence correlation correction is carried out with the cc-pCVDZ basis set. The way to consider the relativistic correction is to use the second-order Douglas-Kroll Hamiltonian approximation, and the correction is performed at the level of cc-pV5Z basis set. To obtain more reliable results, the PECs determined by the MRCI calculations are also corrected for size-extensivity errors by means of the Davidson modification (MRCI+Q). These PECs are extrapolated to the complete basis set limit by the two-point energy extrapolation scheme. With these PECs, the spectroscopic parameters are determined.     

Radiation chemistry of linear low-density polyethylene. II. Kinetics of alkyl and allyl free-radical decay reactions

Quenched and annealed samples of linear low-density polyethylene (LLDPE) were γ irradiated in vacuo at 77 K; the kinetics of the alkyl free-radical decay reactions were studied at room temperature, and of the allyl free-radical reactions at 60, 70, and 80°C. The ESR signals saturate at a slightly higher microwave power in the LLDPE than in high-density polyethylene (HDPE), and the alkyl radicals start decaying at a lower temperature in the LLDPE than in the HDPE. As in the HDPE the decay of the alkyl free radicals at room temperature in the LLDPE follows the kinetic equation for two simultaneous first-order reactions with the fraction of the faster-decaying component being slightly greater in the quenched than in the annealed samples. In the case of the allyl free radicals the decay at 60°C follows the equation based on one fraction of the radicals decaying according to second-order kinetics in the presence of other nondecaying radicals. At higher temperatures the data are best understood in terms of a second-order rate equation with a continuously variable time-dependent rate constant as suggested by Hamill and Funabashi.