Steady-state master equation methods

This paper investigates the application of the steady-state master equation (SSME) to complex unimolecular reactions. It is shown how the steady-state approximation and the Boltzmann reservoir approximation can be applied, both together or separately, to improve the efficiency and the numerical robustness of simple master equation calculations. The case of a two-well isomerization is considered in detail, and two versions of the SSME are derived for it, one of which shows very clearly and analytically the relationship between the rate coefficients and the flux coefficients. It is shown for a reversible second order recombination reaction that the second order phenomenological equation can be regained in either version of the SSME. It is also shown how, under steady state conditions, the SSME may be used to determine rate coefficients for each reaction in a multiple well scheme.     

Bistability in the chemical master equation for dual phosphorylation cycles

Dual phospho/dephosphorylation cycles, as well as covalent enzymatic-catalyzed modifications of substrates are widely diffused within cellular systems and are crucial for the control of complex responses such as learning, memory, and cellular fate determination. Despite the large body of deterministic studies and the increasing work aimed at elucidating the effect of noise in such systems, some aspects remain unclear. Here we study the stationary distribution provided by the two-dimensional chemical master equation for a well-known model of a two step phospho/dephosphorylation cycle using the quasi-steady state approximation of enzymatic kinetics. Our aim is to analyze the role of fluctuations and the molecules distribution properties in the transition to a bistable regime. When detailed balance conditions are satisfied it is possible to compute equilibrium distributions in a closed and explicit form. When detailed balance is not satisfied, the stationary non-equilibrium state is strongly influenced by the chemical fluxes. In the last case, we show how the external field derived from the generation and recombination transition rates, can be decomposed by the Helmholtz theorem, into a conservative and a rotational (irreversible) part. Moreover, this decomposition allows to compute the stationary distribution via a perturbative approach. For a finite number of molecules there exists diffusion dynamics in a macroscopic region of the state space where a relevant transition rate between the two critical points is observed. Further, the stationary distribution function can be approximated by the solution of a Fokker-Planck equation. We illustrate the theoretical results using several numerical simulations.     

The finite state projection algorithm for the solution of the chemical master equation

This article introduces the finite state projection (FSP) method for use in the stochastic analysis of chemically reacting systems. One can describe the chemical populations of such systems with probability density vectors that evolve according to a set of linear ordinary differential equations known as the chemical master equation (CME). Unlike Monte Carlo methods such as the stochastic simulation algorithm (SSA) or tau leaping, the FSP directly solves or approximates the solution of the CME. If the CME describes a system that has a finite number of distinct population vectors, the FSP method provides an exact analytical solution. When an infinite or extremely large number of population variations is possible, the state space can be truncated, and the FSP method provides a certificate of accuracy for how closely the truncated space approximation matches the true solution. The proposed FSP algorithm systematically increases the projection space in order to meet prespecified tolerance in the total probability density error. For any system in which a sufficiently accurate FSP exists, the FSP algorithm is shown to converge in a finite number of steps. The FSP is utilized to solve two examples taken from the field of systems biology, and comparisons are made between the FSP, the SSA, and tau leaping algorithms. In both examples, the FSP outperforms the SSA in terms of accuracy as well as computational efficiency. Furthermore, due to very small molecular counts in these particular examples, the FSP also performs far more effectively than tau leaping methods.     

Master equation methods in gas phase chemical kinetics

In this article, we discuss the application of master equation methods to problems in gas phase chemical kinetics. The focus is on reactions that take place over multiple, interconnected potential wells and on the dissociation of weakly bound free radicals. These problems are of paramount importance in combustion chemistry. To illustrate specific points, we draw on our experience with reactions we have studied previously.     

Mixed quantum-classical Redfield master equation

Redfield master equation is derived from mixed quantum-classical Liouville equation using product initial conditions. Simple two-level system example is given and comparison with Fermi golden rule is made.     

Quantum chemical and master equation simulations of the oxidation and isomerization of vinoxy radicals

The vinoxy radical, a common intermediate in gas-phase alkene ozonolysis, reacts with O2 to form a chemically activated alpha-oxoperoxy species. We report CBS-QB3 energetics for O2 addition to the parent (*CH2CHO, 1a), 1-methylvinoxy (*CH2COCH3, 1b), and 2-methylvinoxy (CH3*CHCHO, 1c) radicals. CBS-QB3 predictions for peroxy radical formation agree with experimental data, while the G2 method systematically overestimates peroxy radical stability. RRKM/master equation simulations based on CBS-QB3 data are used to estimate the competition between prompt isomerization and thermalization for the peroxy radicals derived from 1a, 1b, and 1c. The lowest energy isomerization pathway for radicals 4a and 4c (derived from 1a and 1c, respectively) is a 1,4-shift of the acyl hydrogen requiring 19-20 kcal/mol. The resulting hydroperoxyacyl radical decomposes quantitatively to form *OH. The lowest energy isomerization pathway for radical 4b (derived from 1b) is a 1,5-shift of a methyl hydrogen requiring 26 kcal/mol. About 25% of 4a, but only approximately 5% of 4c, isomerizes promptly at 1 atm pressure. Isomerization of 4b is negligible at all pressures studied.     

Efficient computation of transient solutions of the chemical master equation based on uniformization and quasi-Monte Carlo

A quasi-Monte Carlo method for the simulation of discrete time Markov chains is applied to the simulation of biochemical reaction networks. The continuous process is formulated as a discrete chain subordinate to a Poisson process using the method of uniformization. It is shown that a substantial reduction of the number of trajectories that is required for an accurate estimation of the probability density functions (PDFs) can be achieved with this technique. The method is applied to the simulation of two model problems. Although the technique employed here does not address the typical stiffness of biochemical reaction networks, it is useful when computing the PDF by replication. The method can also be used in conjuncture with hybrid methods that reduce the stiffness.     

Quantum chemical and master equation studies of the methyl vinyl carbonyl oxides formed in isoprene ozonolysis

Methyl vinyl carbonyl oxide is an important intermediate in the reaction of isoprene and ozone and may be responsible for most of the (*)OH formed in isoprene ozonolysis. We use CBS-QB3 calculations and RRKM/master equation simulations to characterize all the pathways leading to the formation of this species, all the interconversions among its four possible conformers, and all of its irreversible isomerizations. Our calculations, like previous studies, predict (*)OH yields consistent with experiment if thermalized syn-methyl carbonyl oxides form (*)OH quantitatively. Natural bond order analysis reveals that the vinyl group weakens the C=O bond of the carbonyl oxide, making rotation about this bond accessible to this chemically activated intermediate. The vinyl group also allows one conformer of the carbonyl oxide to undergo electrocyclization to form a dioxole, a species not previously considered in the literature. Dioxole formation, which has a CBS-QB3 reaction barrier of 13.9 kcal/mol, is predicted to be favored over vinyl hydroperoxide formation, dioxirane formation, and collisional stabilization. Our calculations also predict that two dioxole derivatives, 1,2-epoxy-3-butanone and 3-oxobutanal, should be major products of isoprene ozonolysis.     

Reduction and solution of the chemical master equation using time scale separation and finite state projection

The dynamics of chemical reaction networks often takes place on widely differing time scales--from the order of nanoseconds to the order of several days. This is particularly true for gene regulatory networks, which are modeled by chemical kinetics. Multiple time scales in mathematical models often lead to serious computational difficulties, such as numerical stiffness in the case of differential equations or excessively redundant Monte Carlo simulations in the case of stochastic processes. We present a model reduction method for study of stochastic chemical kinetic systems that takes advantage of multiple time scales. The method applies to finite projections of the chemical master equation and allows for effective time scale separation of the system dynamics. We implement this method in a novel numerical algorithm that exploits the time scale separation to achieve model order reductions while enabling error checking and control. We illustrate the efficiency of our method in several examples motivated by recent developments in gene regulatory networks.     

Time-convolutionless master equation for mesoscopic electron-phonon systems

The time-convolutionless master equation for the electronic populations is derived for a generic electron-phonon Hamiltonian. The equation can be used in the regimes where the golden rule approach is not applicable. The equation is applied to study the electronic relaxation in several models with the finite number of normal modes. For such mesoscopic systems the relaxation behavior differs substantially from the simple exponential relaxation. In particular, the equation shows the appearance of the recurrence phenomena on a time scale determined by the slowest mode of the system. The formal results are quite general and can be used for a wide range of physical systems. Numerical results are presented for a two level system coupled to Ohmic and super-Ohmic baths, as well as for a model of charge-transfer dynamics between semiconducting organic polymers.     

Kinetics of bimolecular chemical reaction by a modified diffusion equation

In order to describe the kinematical behavior of the bimolecular chmical reaction in a dilute solution of species characterized by a single diffusion coefficient and by some nonuniform spatial distribution of the active site, a diffusion equation with a simple sink term is derived by reducing the many-body problem to a one-body problem. The equation is normalized with the concentration of unreacted particles. The so-called second-order reaction rate constant can be calculated from the solution of the equation. The equation is applied to the intermolecular termination reaction of polymer radicals on the assumption of free draining. The reaction rate constant gradually decreases with time.     

Solution of the master equation for unimolecular reactions

A method is given for computing the rate coefficient of a unimolecular reaction as an eigenvalue solution of an integral master equation, based on Nesbet's algorithm, which overcomes computational difficulties associated with this problem. An illustrative fit to pressure-dependent data on the pyrolysis of azoethane is presented.     

Sensitivity amplification in the phosphorylation-dephosphorylation cycle: nonequilibrium steady states, chemical master equation, and temporal cooperativity

A new type of cooperativity termed temporal cooperativity [Biophys. Chem. 105, 585 (2003); Annu. Rev. Phys. Chem. 58, 113 (2007)] emerges in the signal transduction module of phosphorylation-dephosphorylation cycle (PdPC). It utilizes multiple kinetic cycles in time, in contrast to allosteric cooperativity that utilizes multiple subunits in a protein. In the present paper, we thoroughly investigate both the deterministic (microscopic) and stochastic (mesoscopic) models and focus on the identification of the source of temporal cooperativity via comparing with allosteric cooperativity. A thermodynamic analysis confirms again the claim that the chemical equilibrium state exists if and only if the phosphorylation potential DeltaG=0, in which case the amplification of sensitivity is completely abolished. Then we provide comprehensive theoretical and numerical analysis with the first-order and zero-order assumptions in PdPC, respectively. Furthermore, it is interestingly found that the underlying mathematics of temporal cooperativity and allosteric cooperativity are equivalent, and both of them can be expressed by "dissociation constants," which also characterizes the essential differences between the simple and ultrasensitive PdPC switches. Nevertheless, the degree of allosteric cooperativity is restricted by the total number of sites in a single enzyme molecule that cannot be freely regulated, while temporal cooperativity is only restricted by the total number of molecules of the target protein that can be regulated in a wide range and gives rise to the ultrasensitivity phenomenon.     

Combined chemical equation for a multiple reaction

A simple procedure has been proposed for obtaining a combined chemical equation for a multiple reaction as defined by Temkin. The procedure is based on the concept of a normal basis for chemical equations which is introduced in the present paper. In some cases the proposed method makes it possible to obtain a form of combined chemical equation which is more convenient for practical use than that obtained by the familiar method.Translated from Teoreticheskaya i Éksperimental'naya Khimiya, Vol. 30, No. 5, pp. 289–292, September–October, 1994.     

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.     

Electronic excitation dynamics in multichromophoric systems described via a polaron-representation master equation

We derive a many-site version of the non-Markovian time-convolutionless polaron master equation [Jang et al., J. Chem Phys. 129, 101104 (2008)] to describe electronic excitation dynamics in multichromophoric systems. By treating electronic and vibrational degrees of freedom in a combined frame (polaron frame), this theory is capable of interpolating between weak and strong exciton-phonon coupling and is able to account for initial non-equilibrium bath states and spatially correlated environments. Besides outlining a general expression for the expected value of any electronic system observable in the original frame, we also discuss implications of the Markovian and Secular approximations highlighting that they need not hold in the untransformed frame despite being strictly satisfied in the polaron frame. The key features of the theory are illustrated using as an example a four-site subsystem of the Fenna-Mathews-Olson light-harvesting complex. For a spectral density including a localised mode, we show that oscillations of site populations may only be observed when non-equilibrium bath effects are taken into account. Furthermore, we illustrate how this formalism allows us to identify the electronic and vibrational components of the oscillatory dynamics.     

Numerical inversion of the V—V master equation

The information-theoretic approach to vibrational—vibrational (V—V) energy transfer is employed to invert numerically the V—V master equation for vibrational enregy exchange between CO molecules.     

On solving the master equation in spatially periodic systems

We present a new method for solving the master equation for a system evolving on a spatially periodic network of states. The network contains 2(ν) images of a "unit cell" of n states, arranged along one direction with periodic boundary conditions at the ends. We analyze the structure of the symmetrized (2(ν)n) × (2(ν)n) rate constant matrix for this system and derive a recursive scheme for determining its eigenvalues and eigenvectors, and therefore analytically expressing the time-dependent probabilities of all states in the network, based on diagonalizations of n × n matrices formed by consideration of a single unit cell. We apply our new method to the problem of low-temperature, low-occupancy diffusion of xenon in the zeolite silicalite-1 using the states, interstate transitions, and transition state theory-based rate constants previously derived by June et al. [J. Phys. Chem. 95, 8866 (1991)]. The new method yields a diffusion tensor for this system which differs by less than 3% from the values derived previously via kinetic Monte Carlo (KMC) simulations and confirmed by new KMC simulations conducted in the present work. The computational requirements of the new method are compared against those of KMC, numerical solution of the master equation by the Euler method, and direct molecular dynamics. In the problem of diffusion of xenon in silicalite-1, the new method is shown to be faster than these alternative methods by factors of about 3.177 × 10(4), 4.237 × 10(3), and 1.75 × 10(7), respectively. The computational savings and ease of setting up calculations afforded by the new method of master equation solution by recursive reduction of dimensionality in diagonalizing the rate constant matrix make it attractive as a means of predicting long-time dynamical phenomena in spatially periodic systems from atomic-level information.     

Analytical chemical methods as systems producing chemical information by inference

Summary Analytical chemical methods as systems produce chemical information about the material to be analyzed. Analytical chemical systems as semiosis consist of analytical signal production and analytical chemical signal interpretation and produce chemical information by inference in an indirect way through analytical information. From the logical point of view the chemical information produced by analytical chemical systems is only credible. Generalizing the results the idea of diagnostic systems can be introduced and the analytical chemical methods as systems are a special type of diagnostic systems.

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Chemisch-analytische Systeme zur Erlangung chemischer Informationen

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Presented at the First International Symposium on History and Philosophy in Analytical Chemistry, Vienna, November 22–23, 1985