Stochastic kinetics

Burns and Curtiss (1972) and Burns et al. (1984) have used the Facsimile program developed at AERE, Harwell to obtain a numerical solution of simultaneous partial differential equations of diffusion kinetics (see Eq. 7.1). In this procedure, the changes in the number of reactant species in concentric shells (spherical or cylindrical) by diffusion and reaction are calculated by a march of steps method. A very similar procedure has been adopted by Pimblott and La Verne (1990 La Verne and Pimblott, 1991). Later, Pimblott et al. (1996) analyzed carefully the relationship between the electron scavenging yield and the time dependence of eh yield through the Laplace transform, an idea first suggested by Balkas et al. (1970). These authors corrected for the artifactual effects of the experiments on eh decay and took into account the more recent data of Chernovitz and Jonah (1988). Their analysis raises the yield of eh at 100 ps to 4.8, in conformity with the value of Sumiyoshi et al. (1985). They also conclude that the time dependence of the eh yield and the yield of electron scavenging conform to each other through Laplace transform, but that neither is predicted correctly by the diffusion-kinetic model of water radiolysis. 

The procedures discussed so far take as fundamental variables the species concentration and specific rates, the latter obtained from homogeneous experiments. Such procedures are called deterministic—that is, admitting no fluctuation in the number of reactant species—as opposed to stochastic methods where statistical variation is built in. 

The difference is clearly seen for a spur initially containing two dissociations of AB molecules into radicals A and B (Pimblott and Green, 1995). Considering the same reaction radii for the reactions A + A, A + B, and B + B and the same initial distributions of radicals, the statistical ratio of the products should be 1 4 1 for A2 AB B2, since there is one each of A-A and B-B distances but there are four A-B distances. For n dissociations in the spur, this combinatorial ratio is n(n - l)/2 n2 n(n - l)/2, whereas deterministic kinetics gives this ratio always as 1 2 1. Thus, deterministic kinetics seriously underestimates cross-recombination and overestimates molecular products, although the difference tends to diminish for bigger spurs. Since smaller spurs dominate water radiolysis (Pimblott and Mozumder, 1991), many authors stress the importance of stochastic kinetics in principle. Stochasticity enters in another form in 

Chapter 7 Spur Theory of Radiation Chemical Yields 

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Nearly all computations of radiation-chemical yields use either diffusion kinetics (see, e.g., Schwarz, 1969) or stochastic kinetics (Zaider et ah, 1983 Clifford et al, 1987 Pimblott, 1988 Paretzke et ah, 1991 Pimblott et ah, 1991). Diffusion kinetics uses deterministic rate laws and considers the reactions to be (partially) diffusion controlled while the reactants are also diffusing... 

In Sect. 4.9.1, experimental rationalization was provided for the W value of ionization in gaseous and liquid water, giving respectively 30.0 and 20.8 eV. The corresponding ionization potentials are respectively 12.6 and 8.3 eV. For the purpose of diffusion and stochastic kinetics, one often requires the statistical distribution P(i,j) of the number of ionizations i and excitations j, conditioned on i ionizations, for a spur of energy . Pimblott and Mozumder (1991) write P(i, j) = r(i) 2(j i), where F(i) is the probability of having i ionizations and 2(j i) is the probability of having j excitations conditioned on i ionizations. These probabilities are separately normalized to unity. 

Stochastic kinetics requires details of individual particle reactions. It is computer-intensive and produces a huge volume of output. In this sense, it is overparameterized. However, stochastic kinetics can be made consistent with the statistics of energy deposition and reaction. 

Using deterministic kinetics, one can force-fit the time evolution of one species—for example, eh but then those of other yields (e.g., OH) will be inconsistent. Stochastic kinetics can predict the evolutions of radicals correctly and relate these to scavenging yields via Laplace transforms. 

Raap IA, Grollmann U (1983) The stochastic kinetics of intramolecular reactions in linear polymers. Macromol Chem 184 123-134... 

Zheng, Q. and Ross, J., Comparison of deterministic and stochastic kinetics for nonlinear systems, J. Chem. Phys., 1991, 94, 3644—3648. 

Mathematical models are widely applied in biosciences and different modeling routes can be taken to describe biological systems. The type of model to use depends completely on the objective of the study. Models can be dynamic or static, deterministic or stochastic. Kinetic models are commonly used to study transient states of the cell such as the cell cycle [101] or signal transduction pathways [102], whereas stoichiometric models are generally used when kinetics parameters are unknown and steady state systems is assumed [48, 103]. 

Arkin, A., J. Ross, and H. H. McAdams. 1998. Stochastic kinetic analysis of developmental pathway bifurcation in phage -infected Escherichia coli cells. Genetics 149 1633 18. 

Thermal and acid-catalyzed deprotection kinetics of PBOCST and PTBMA was monitored by UV and IR spectroscopy, respectively [515], and compared very favorably with models based on a stochastic kinetics simulator (CKS)... 

The simulation model development is divided into three sections. The first discusses the probabilistic modelling of lignin structure, and the use of probability distribution functions to generate representative lignin moieties. The second section details the depolymerization of lignin using stochastic kinetics. The final portion describes the combination of these elements into a Monte Carlo simulation and also presents representative predictions... 

Recently, a non-equilibrium statistical thermodynamic theory based on stochastic kinetics has been formulated which has been applied to isothermal non-equilibrium steady state for biological systems [4]. Rate equations in terms of the probabilities of enzyme concentration are used instead of concentration. Expressions for the Gibbs free energy and entropy for the isothermal system are obtained in terms of dynamic cyclic reaction. 

A DFT study of the reaction has probed the roles of aldehyde and diethyl- 0 A newly developed stochastic kinetic method confirms that a slow 0... 

Fluctuation phenomena occurring in consequence of transport processes through biological membranes can be readily studied. Membranes are two-dimensional, and 10 molecules are contained in a 500 pm membrane surface, which is available to experimental techniques (Neher Stevens, 1977). (Membrane noise can be analysed by using the language of stochastic kinetics, as will be shown in Subsection 5.5.3, as well as in Chapter 7.)... 

Noise analysis, in particular the evaluation of experimental spectra, has been extensively used for interpreting the kinetics of transport processes in membranes. (See, for example Frehland (1982) and Section 5.5 of this book.) In Chapter 7 an example will be given of how to use stochastic kinetics to obtain stoichiometric and kinetic information referring to the transmitter-receptor interaction at the surface of the postsynaptic membrane of nerve and muscle cells. 

While the first term is described according to the rules of homogeneous stochastic kinetics, the second term is as follows (Nicolis Malek-Mansour, 1980) ... 

Erdi, P. Ropolyi, L. (1979). Investigation of transmitter-receptor interactions by analyzing postsynaptic membrane noise using stochastic kinetics. Biol. Cyb., 32, 41-5. 

Toth, J. Structure of the state space in stochastic kinetics. Proc. 5th Pann. Symp. on Math, Stat. (submitted a). 

Toth, J. Torok, T. L. (1980). Poissonian stationary distribution a degenerate case of stochastic kinetics. React. Kinet. Catal. Lett., 13, 167-71. 

Many problems on stochastic kinetics have been debated with Professor Michel Moreau (Paris). One of us (P.E.) enjoyed his hospitality in June 1985 the main part of Chapter 5 was written during this time. Some short visits to Bordeaux were useful for both of us. 

Lateral interactions influence the reactants, products, intermediates and even transition states for a reaction. Reactant molecules likely adsorb in different local environments and are therefore exposed to different lateral interactions depending upon the relative number, type and position of neighboring adsorbates. Stochastic kinetic methods provide the best hope of capturing these molecular differences. Traditional deterministic modeling of catalytic systems average over the smface coverage and thus provide only a mean field description. Individual smface sites, as well as intermolecular interactions, however, can be... 

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