Chemical kinetics, stochastic description

An interesting, but probably incorrect, application of the probabilistic master equation is the description of chemical kinetics in a dilute gas.5 Instead of using the classical deterministic theory, several investigators have introduced single time functions of the form P(n1,n2,t) where P(nu n2, t) is the probability that there are nl particles of type 1 and n2 particles of type 2 in the system at time t. They use the transition rate A(nt, n2 n2, n2, t) from the state with particles of type 1 and n2 particles of type 2 to the state with nt and n2 particles of types 1 and 2, respectively, at time t. The rates that are used are obtained by assuming that only uncorrelated binary collisions occur in the system. These rates, however, are only correct in the thermodynamic limit for a low density system. In this limit, the Boltzmann equation is valid from which the deterministic theory follows. Thus, there is no reason to attach any physical significance to the differences between the results of the stochastic theory and the deterministic theory.6... 

Its obvious peculiarity as compared with the standard chemical kinetics, equation (2.1.10), is the emergence of the fluctuational second term in r.h.s. The stochastic reaction description by means of equation (2.2.37) permits us to obtain the equation for dispersions crjj which, however, contains higher-order momenta. It leads to the distinctive infinite set of deterministic equations describing various average quantities, characterizing the fluctuational spectrum. 

A class of continuous-time Markov processes with integer state space and with transitions allowed only between adjacent states plays a central role in the stochastic description of chemical kinetics.122 134 135... 

Ray Kapral came to Toronto from the United States in 1969. His research interests center on theories of rate processes both in systems close to equilibrium, where the goal is the development of a microscopic theory of condensed phase reaction rates,89 and in systems far from chemical equilibrium, where descriptions of the complex spatial and temporal reactive dynamics that these systems exhibit have been developed.90 He and his collaborators have carried out research on the dynamics of phase transitions and critical phenomena, the dynamics of colloidal suspensions, the kinetic theory of chemical reactions in liquids, nonequilibrium statistical mechanics of liquids and mode coupling theory, mechanisms for the onset of chaos in nonlinear dynamical systems, the stochastic theory of chemical rate processes, studies of pattern formation in chemically reacting systems, and the development of molecular dynamics simulation methods for activated chemical rate processes. His recent research activities center on the theory of quantum and classical rate processes in the condensed phase91 and in clusters, and studies of chemical waves and patterns in reacting systems at both the macroscopic and mesoscopic levels. 

Chemical process rate equations involve the quantity related to concentration fluctuations as a kinetic parameter called chemical relaxation. The stochastic theory of chemical kinetics investigates concentration fluctuations (Malyshev, 2005). For diffusion of polymers, flows through porous media, and the description liquid helium, Fick s and Fourier s laws are generally not applicable, since these laws are based on linear flow-force relations. A general formalism with the aim to go beyond the linear flow-force relations is the extended nonequilibrium thermodynamics. Polymer solutions are highly relevant systems for analyses beyond the local equilibrium theory. 

Interestingly, Eq. (5.21) is the differential equation commonly employed in the context of chemical kinetics to model the dynamics of a chemical species that is produced via a catalytic reaction and degraded linearly (Houston 2001). In particular, the first term on the right-hand side of Eq. (5.21) corresponds to Michaelis-Menten equation, which is commonly used to model the velocity of enzymatic reactions (Houston 2001 Lehninger et al. 2005). Let us close this section by stating that knowing how Eq. (5.21) can be deduced from a stochastic chemical dynamics approach, allows us to better understand its range of validity and its connection with the relevant quantities of the stochastic-description. 

Over the last years it has become clear that the dynamics of most biological phenomena can be studied via the techniques of either nonlinear dynamics or stochastic processes. In either case, the biological system is usually visualized as a set of interdependent chemical reactions and the model equations are derived out of this picture. Deterministic, nonlinear dynamic models rely on chemical kinetics, while stochastic models are developed from the chemical master equation. Recent publications have demonstrated that deterministic models are nothing but an average description of the behavior of unicellular stochastic models. In that sense, the most detailed modeling approach is that of stochastic processes. However, both the deterministic and the stochastic approaches are complementary. The vast amount of available techniques to analytically explore the behavior of deterministic, nonlinear dynamical models is almost completely inexistent for their stochastic counterparts. On the other hand, the only way to investigate biochemical noise is via stochastic processes. 

This selection process is then iterated, beginning from an initial state of the system, as defined by species populations, to simulate a chemical evolution. A statistical ensemble is generated by repeated simulation of the chemical evolution using different sequences of random numbers in the Monte Carlo selection process. Within limits imposed by computer time restrictions, ensemble population averages and relevant statistical information can be evaluated to any desired degree of accuracy. In particular, reliable values for the first several moments of the distribution can be obtained both inexpensively and efficiently via a computer algorithm which is incredibly easy to implement (21, 22), especially in comparison to now-standard techniques foF soTving the stiff ordinary differential equations (48, 49) which may arise in the deterministic description of chemical kinetics (53). Now consider briefly the essential features of a simple chemical model which illustrates well the attributes of stochastic chemical simulations. 

This paper has focused on two recent computer methods for discrete simulation of chemical kinetics. Beginning with the realization that truly microscopic computer experiments are not at all feasible, I have tried to motivate the development of a hierarchy of simulations in studies of a class of chemical problems which best illustrate the absolute necessity for simulation at levels above molecular dynamics. It is anticipated (optimistically ) that the parallel development of discrete event simulations at different levels of description may ultimately provide a practical interface between microscopic physics and macroscopic chemistry in complex physicochemical systems. With the addition to microscopic molecular dynamics of successively higher-level simulations intermediate between molecular dynamics at one extreme and differential equations at the other, it should be possible to examine explicitly the validity of assumptions invoked at each stage in passing from the molecular level to the stochastic description and finally to the macroscopic formulation of chemical reaction kinetics. 

Stochastic models are also able to capture complicated pattern formation seen in chemically reacting media and can be used to study the effects of fluctuations on chemical patterns and wave propagation. The mesoscopic dynamics of the FHN model illustrates this point. In order to formulate a microscopically based stochastic model for this system, it is first necessary to provide a mechanism whose mass action law is the FHN kinetic equation. Some features of the FHN kinetics seem to preclude such a mechanistic description for example, the production of u is inhibited by a term linear in V, a contribution not usually encountered in mass action kinetics. However, if each local region of space is assumed to be able to accommodate only a maximum number m of each chemical species, then such a mechanism may be written. We assume that the chemical reactions depend on the local number of molecules of the species as well as the number of vacancies corresponding to each species, in analogy with the dependence of some surface reactions on the number of vacant surface sites or biochemical reactions involving complexes of allosteric enzymes that depend on the number of vacant active sites. 

In several theoretical models for small systems [77,79,80-81] the stochastic nature of the biochemical processes under certain circumstances reveals itself as the apparent violation of the Second Law of Thermodynamics. We consider below some of these models, demonstrating that the description of such processes in small systems based on averaging (nonlocal) formalism can often be misleading. These models demonstrate several unusual thermodynamic and kinetic properties that contradict the conventional laws of chemical thermodynamics and kinetics, described as a rule in terms of the average concentrations of the reagents and chemical intermediates. 

Gillespie s algorithm numerically reproduces the solution of the chemical master equation, simulating the individual occurrences of reactions. This type of description is called a jump Markov process, a type of stochastic process. A jump Markov process describes a system that has a probability of discontinuously transitioning from one state to another. This type of algorithm is also known as kinetic Monte Carlo. An ensemble of simulation trajectories in state space is required to accurately capture the probabilistic nature of the transient behavior of the system. 

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