On the method of quasi-steady-state approximation

Sufficient conditions for the validity of the quasi-steady-state approximation widely used in chemical kinetics are considered by means of the qualitative geometric theory of differential equations with small parameters.     

Single-substrate enzyme kinetics: the quasi-steady-state approximation and beyond

We analyze the standard model of enzyme-catalyzed reactions at various substrate-enzyme ratios by adopting a different scaling scheme and computational procedure. The regions of validity of the quasi-steady-state approximation are noted. Certain prevalent conditions are checked and compared against the actual findings. Efficacies of a few other measures, obtained from the present work, are highlighted. Some recent observations are rationalized, particularly at moderate and high enzyme concentrations.     

Dynamic properties of a heterogeneous catalytic reaction with several steady states

For the simplest catalytic mechanism (three-stage) permitting several steady states of the surface, a qualitative analysis of the properties of its kinetic model trajectories has been performed. Types of the steady states have been determined. Absence of oscillations (damped and undamped) has been shown.

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Для простейшего каталитического механизма (трехстадийного), допускающего несколько стационарных состояний поверхности, проведен качественный анализ свойств траекторий его кинетической модели. Определены типы стационарных состояний. Показано отсутствие осцилляций (затухающих и незатухающих).

     

Communication: limitations of the stochastic quasi-steady-state approximation in open biochemical reaction networks

It is commonly believed that, whenever timescale separation holds, the predictions of reduced chemical master equations obtained using the stochastic quasi-steady-state approximation are in very good agreement with the predictions of the full master equations. We use the linear noise approximation to obtain a simple formula for the relative error between the predictions of the two master equations for the Michaelis-Menten reaction with substrate input. The reduced approach is predicted to overestimate the variance of the substrate concentration fluctuations by as much as 30%. The theoretical results are validated by stochastic simulations using experimental parameter values for enzymes involved in proteolysis, gluconeogenesis, and fermentation.     

Theory of chemically activated unimolecular reactions. Weak collisions and steady states

The prediction of decomposition/stabilization ratios for chemically activated unimolecular reactions is usually based on the assumption of steady state and irreversible deactivation. The levels below the activation energy E_o are treated as a sink. In the present work we assume that the reaction can be described by a weak-collision master equation. An exact analysis shows that one can expect three timescales: an initial transient, an intermediate steady state, and an asymptotic steady state. The intermediate steady state is well defined only if the eigenvalues of the corresponding thermal rate matrix can be separated into two groups one of which contains eigenvalues of substantially smaller magnitude. An efficient and accurate method for the estimation of intermediate steady-state populations and branching ratios is described. It avoids the irreversible deactivation assumption which is shown to lead to errors particularly for weak collisions. The new analysis is illustrated in a series of calculations for a model of the sec-butyl radical system. The numerical solutions are readily obtained by use of a recently developed equilibrium finite-basis-set method.     

Dynamics of chemical reactions and nonphysical steady states

Data on the position of nonphysical (lying beyond the region of determination) steady states are shown to be of use for understanding the dynamic behavior of chemical reactions, in particular, the reasons for slow relaxations.

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Validity of the coupled states approximation for molecular collisions

The process of rotational excitation in the body-fixed frame is examined in detail. Physical arguments are presented to justify the coupled states approximation which involves a diagonalization of the body-fixed centrifugal potential. Cross sections for rotational transitions in HeHCN and in ArN₂ collisions are reported and are compared to exact close coupling cross sections. The coupled states method is found to maintain its high accuracy for the extremely strong-coupling HeHCN system and also for the large Δj transitions in the ArN₂ system even when many partial waves are required. General rules are given for the applicability of this approach in terms of the strength and the position of the anisotropy in the potential energy surface.     

On the multiplicity criterion for steady states in catalytic reactions

CNDO/2 studies of NH₃ adsorption in terms of cluster models of the-Al₂O₃ surface sites have revealed that coordinatively unsaturated aluminium ions occupying tetrahedral and octahedral sites are Lewis acids that are stronger in the former case. NH₃ molecule adsorption on these sites is accompanied by an increase in the H–N–H bond angle and a weakening of the N–H bond.

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/2 , -Al₂O₃ , , , , . NH₃ H–N–H N–H.

     

Temporal asymmetry of fluctuations in nonequilibrium steady states

Temporal asymmetries of fluctuation paths in nonequilibrium microscopic shearing systems are observed for the first time. Inspired by theories that predict asymmetry of fluctuation paths in stochastic dynamics, we focus on deterministic reversible particle models, which represent a small part of a macroscopic system. We have monitored and measured the asymmetry of the fluctuation paths of various observables as they go away from and towards the mean. The understanding of such asymmetries may scatter light on how irreversibility emerges from the microscopic reversible dynamics and on the behavior of mesoscopic (nanoscale) systems.     

Multiplicity of steady states in oxidative heterogeneous catalysis

Typical mechanisms of heterogeneous catalytic reactions and their kinetíc models that are characterized by a plurality of steady states are considered. Models of adsorption that lead to a plurality of equilibrium adsorption states are discussed.Translated from Teoreticheskaya i Éksperimental'naya Khimiya, Vol. 28, No. 4, pp. 285–300, July–August, 1992.     

Positions of steady states in two-component chemical systems

The general kinetic equation for two-component chemical systems is analyzed. It is shown that the positions of steady states in concentration spaces can be detected by a qualitative analysis of the chemical mechanism.

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Non-Born-Oppenheimer approximation for very weakly bound states of molecular anions

The influence of nuclear rotation on weak electron binding in the long range field of a linear polar molecule is treated in a way that leads ultimately, with suitable approximation, to the familiar equations for close coupling of electron-nuclear-rotational motions. Subsequently, a conventional pseudopotential approximation is invoked to examine the rotational spectra of HCN and DCN anions. It is shown that the number of rotationally excited anion states cannot be reliably predicted by assuming that zero binding occurs when the rotational energy equals the electron affinity obtained in the Born-Oppenheimer approximation. A method is suggested for combining accurate molecular orbital and parameterized pseudopotential methods to provide accurate electron affinities for very weakly bound anionic states.     

Localized impurity states in the Hartree-Fock,LCAO approximation. I.

One-electron energies and wave functions for deep trap impurity electrons in a crystal are calculated by the Hartree-Fock, single determinant method. The interactions arising from a many-electron single determinant crystal wave function, with automatic inclusion of exchange effects, are those which determine the one-electron functions and energies. The crystal plus impurity system has no translational symmetry and hence the Bloch theorem is not applicable for the solution of the essentially infinite Hartree-Fock eigenvalue matrix. Thus we develop a technique in which the Hamiltonian and overlap matrices are written in terms of bordered matrices, with the interaction of the impurity functions with the rest of the crystal environment contained in the bordering rows and columns. The resulting secular equation explicitly includes the effects of orthogonalization of the entire basis set, including the impurity functions. This technique could be used in an iterative calculation of the electronic structure of a small number of electrons, assuming that the rest of the electrons in the environment are fixed according to an initial estimate.     

Limiting phenomena and multiplicity of steady states in gas-phase oxidation processes

It was found that in a chain-branching oxidation reaction of hydrogen at low pressures, there is a concentration range of the reagents over which there are three steady states: the boundaries of this range determine the positions of the limiting transitions between the regions of slow and rapid occurrence of the process. The dependences of the position of the limiting transitions in noncatalytic oxidation reactions of hydrogen and of oxidation of CO on platinum metals on the rate constants of the elementary stages of these processes have been compared. At different ratios of the rate constants of the heterogeneous-homogeneous catalytic process, the dependences of the reaction rate on the concentration of the reagent may contain a break, or may have the form of a loop or hystereses, both clockwise and counterclockwise.Translated from Teoreticheskaya i Éksperimental'naya Khimiya, Vol. 23, No. 1, pp. 28–33, January–February, 1987.     

Analysis of fluctuations in chemical reactions with several steady states

The effect of fluctuations in nonequilibrium systems is treated in terms of the stochastic theory. A solution of the fundamental equation is analyzed using the Monte-Carlo method /1, 2/. The results are compared with the conclusions following from the equations corresponding to a phenomenological model.

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