Deterministic kinetics

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... 

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. 

The time evolution of the mean value of the number of reactants [Eq. (4.70)] coincides with the one given by the deterministic kinetic equation, that is,... 

Lente proposed a discrete-state stochastic modeling approach in which chiral amplification could be described by a quadratic autocatalytic model without considering cross-inhibition [67,68]. However, the discrepancy between the usually employed deterministic kinetic approach, which reinforces the need for cross-inhibition, and the discrete-state stochastic approach is only apparent. The discrete approach considers the repetitive reproduction of single molecules which, in the case of a chiral system, obviously are individually all enantiomerically pure. Hence, basically no amplification of the ee occurs at all during the discrete scenario. It has been indicated that deter-... 

The chemical master equation (CME) for a given system invokes the same rate constants as the associated deterministic kinetic model. Yet the CME is more fundamental than the deterministic kinetic view. Just as Schrodinger s equation is the fundamental equation for modeling motions of atomic and subatomic particle systems, the CME is the fundamental equation for reaction systems. Remember that Schrodinger s equation is not a model for a specific mechanical system. Rather, it is a theoretical framework upon which models for particular systems can be developed. In order to write down a model for an atomic system based on Schrodinger s equation, one needs to know how to write down the Hamiltonian a priori. Similarly, the CME is not a model for a specific biochemical reaction system it is a theoretical framework. To determine the CME model for a reaction system, one must know what are the possible elementary reactions and the associated rate constants. 

The CME is the equation for the probability function p, or equivalently if the system s volume is constant, for the probability function p(ni, n9, , jv, t)where is the number of molecules of species i. With given concentrations (ci, c2, , c v) at a time t, deterministic kinetic models give precisely what the concentrations will be at time t + St. According to the stochastic CME, however, the concentrations at t + St can take many different values, each with certain probability. 

The loss of stabihty and transfer to the so called nonthermodynamic (i.e., described by deterministic kinetic or other dynamic equations) branch happens at a = a, when, for example, the excess energy dissipa tion at a > a turns negative ... 

We consider once again the deterministic kinetic Eqn. (II.4) with the uniform error rate expression Eqn. (A 1.5) relating mutation rates to replication rates ... 

We consider a complex chemical system and focus on a set of S species M , u = 1,..., 5, which can carry one or more identical molecular fragments that are unchanged during the process in the following we refer to these species as carriers. For simplicity, in this section we limit ourselves to the case of isothermal, well-stirred, homogeneous systems, for which the concentrations c = c (t), u = 1,..., 5, of the chemicals M , m = 1,..., 5, are space independent and depend only on time. Later on we consider the more complicated case of reaction-diffusion systems. The deterministic kinetic equations of the process can be expressed in the following form ... 

To describe the dynamic system behavior, deterministic kinetic rate equations of the form... 

Here X, Y are internal and A external components, andO is the zero complex. The deterministic kinetic equation is... 

The kinetics for catalytic systems can be modeled by one of two general methods. The first is based on continuum concentrations and uses deterministic kinetics whereas the second approach follows the temporal fate of individual molecules over the smface via stochastic kinetics. Both approaches have known advantages and disadvantages, as will be discussed. B These methods provide the constructs for simulating the elementary kinetics. However, in order to do so, they require an accurate and comprehensive initial kinetic database that contains parameters for the full spectra of elementary surface processes that make up the catalytic cycle. The ultimate goal for both approaches would be to call upon quantum mechanics calculations in situ in order to establish the potential energy surface as the simulation proceeds. This, however, is still well beyond our computational capabilities. 

Deterministic kinetic modeling approaches are mean-field approaches, whereby the molecules experience only an averaged interaction of the others. These models are reasonable if the lateral interactions between reactant molecules, reagents or products are absent or if diffusional effects maintain a state of ideal mixing. In the latter case, the kinetic parameters will also be concentration dependent. 

Studies of the development of reactions within hot spots have been sparse but a recent report by Green, Pilling and Robertson (33) suggests a possible approach. They compare a simple deterministic kinetic model with a stochastic model. The latter may be applicable where hot spots are small so that the numbers of reactive species involved are also small. It is shown that the stochastic model can predict hot spot quenching under conditions where the deterministic model would suggest that reaction might propagate. 

Then for a given initial value of px a transient change in px occurs imtil a nonequilibrium state is reached. The pressure at that stationary state must be determined from the kinetic equations of the system. For mass action kinetics the deterministic kinetic equations (neglect of fluctuations in the pressures or... 

In Chap. 2 we obtained a thermodynamic state function d>, (2.13), valid for single variable non-linear systems, and (2.6), valid for single variable linear systems. We shall extend the approach used there to multi-variable systems in Chap. 4 and use the results later for comparison with experiments on relative stability. However, the generalization of the results in Chap. 2 for multi-variable linear and non-linear systems, based on the use of deterministic kinetic equations, does not yield a thermodynamic state function. In order to obtain a thermodynamic state function for multi-variable systems we need to consider fluctuations, and now turn to this analysis [1]. 

The macroscopic, the deterministic, kinetic equations for this system are... 

As is the cases in earlier chapters, the function in (4.17) is zero at stationary states, increases on removal from stable stationary states and decreases from any initial given state on its approach to the nearest stable stationary state along a deterministic kinetic trajectory. These specifications make a Liapunov function in the vicinity of stable stationary states, which indicates the direction of the deterministic motion. Hence for every variation from a stable stationary state we have... 

In Chap. 5 we discussed reaction diffusion systems, obtained necessary and sufficient conditions for the existence and stability of stationary states, derived criteria of relative stability of multiple stationary states, all on the basis of deterministic kinetic equations. We began this analysis in Chap. 2 for homogeneous one-variable systems, and followed it in Chap. 3 for homogeneous multi-variable systems, but now on the basis of consideration of fluctuations. In a parallel way, we now follow the discussion of the thermod3mamics of reaction diffusion equations with deterministic kinetic equations, Chap. 5, but now based on the master equation for consideration of fluctuations. 

Now we impose a flux of a chemical species present in the system and inquire on the effect of that imposition on the stochastic potential of the system. For that we need to go from the deterministic kinetic equations to a stochastic... 

The deterministic kinetic equations for this system are given in [13], but we need not repoduce them here they are nonlinear and for experimental values of the rate coefficients represent a damped oscillator. 

We introduced the concepts of fluctuations and dissipation in Chap. 2, where we discussed the approach of a chemical system to a nonequilibrium stationary state we recommend a review of that chapter. We restricted there the analysis to linear and nonhnear one-variable chemical systems and shall do so again in this chapter, except for a brief referral to extensions to multivariable systems at the end of the chapter. In Chap. 2 we gave some connections between deterministic kinetics, with attending dissipation, and fluctuations, see for example (2.33), which equates the probability of a fluctuation in the concentration X to the deterministic kinetics, see (2.8, 2.9). Here we enlarge on the relations between dissipative, deterministic kinetics, and fluctuations for the purpose of an introduction to the interesting topic of fluctuation dissipation relations. This subject has a long history, more than 100 years [1,2] Reference [1] is a classical review with many references to fundamental earlier work. A brief reminder of one of the early examples, that of Brownian motion, may be helpful. 

What are the advantages of the formulation, (18.8) The term t x) is the net flux of the deterministic kinetics, (18.6), and the derivative of the state function (j> is the species specific affinity, (/tx — for the linear case, or (/Xx — /x x) for the nonlinear case, the driving force for the reaction toward a stationary state. Thus we have a flux-driving force relation. Second, the formulation is symmetric with respect to /+(x) and t (x), which is not the case with other formulations. Third, the state function 4> determines the probability distribution of fluctuations in x from its value at the stationary state, see (2.34). Further, as we shall show shortly, the term D x) is a measure of the strength... 

The fluctuations of this type are called non-intermittent they are commonly encountered in statistical mechanics and have a negligible contribution to the behaviour of macroscopic systems. For these types of processes in the limit of large volumes, the average values of concentrations computed by taking fluctuations into account are practically identical to the values computed by neglecting the fluctuations and solving the deterministic kinetic equations. 

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