Deterministic kinetic modeling

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. 

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. 

Several studies showed that CO2 pressure has a very low impact on the oxidation rate but increases the carburization rate [48,62]. Therefore, after exposure under CO2 at 550°C for 1000 h, the amount of carbon transferred into T91 steel grade was multiplied by 1.5 if CO2 pressure was increased from 1 to 250 bar [48]. A deterministic kinetic model for oxidation based on the oxidation model presented previously was developed and allowed to predict flie low influence of CO2 pressure [36,56]. This kinetic model does not rely on any fitted parameters but only on thermodynamic data such as the equilibrium oxygen pressures between the inner oxide and iron and between magnetite and haematite and kinetic data such as the diffusion coefficient of iron through the oxide layer. 

In this book we mainly discuss deterministic kinetic models based on differential equations. The stochastic simulation of chemical kinetic models was only mentioned briefly in Sect. 2.1.3. We note that it is also possible to investigate stochastic models by sensitivity analysis and we refer the readers to the articles of Gunawan et al. (2005), Degasperi and Gilmore (2008), Charzyhska et al. (2012) and Pantazis et al. (2013). 

La Verne and Pimblott [19] and Pimblott and La Verne [43] refined Eq. (15) and developed an analytical description of these effects of scavengers using the deterministic diffusion kinetic model outlined in Section 2. For a single scavenger, they showed that the dependence of the amount of scavenging reaction on the concentration of S could be better described by [22] ... 

The kinetic model just described is a compromise that affords a realistic possibility of making quantitative estimates with available data and yet preserves a level of deterministic rigor by requiring that the problem be formulated in terms of specific redox-active species. 

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-... 

From the above it can be concluded that in many instances the introduction of an artificial radionuclide into the environment provides us with a natural tracer experiment. Indeed, this is the basis for the application of deterministic compartmental models, based on tracer kinetics, to radioecology (Whicker and Schultz, 1982). This approach is largely based on the assumption that radionuclide movements will exhibit first order kinetics although the existence of naturally-occurring tracees (stable isotopes) at relatively high abundance may result in more complex concentration-dependent kinetics. Furthermore, nutrient analogues may exert even more complex effects on processes such as radioion absorption across root plasma membranes this will become evident later in the chapter. 

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]. 

At least eight different kinetic models can be defined, depending on the specification of time (X), state-space (T) and nature of determination (Z). As was explained earlier, time can be discrete (D) or continuous (C), the state-space can be also discrete (D) or continuous (C), and the nature of determination is deterministic (D) or stochastic (S). 

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. 

This paper is focused on the deterministic 3D modelling approach that will be applied to a three-enzymes metabolic pathway as a case study. In Sect. 2, the three-enzymes metabolic pathway will be introduced and its time evolution in a bulk solution will be reproduced by a simple kinetic model. In Sect. 3, dilferent possible scenarios of compart-mentalization will presented. The 3D model will be illustfated in its computational details in Sect. 4, while the computational results will be discussed in Sect. 5. Finally, some conclusion will be drawn in the last section. 

On the other hand, if the numbers of reacting molecules are very small, for example in the order of O(10 iV ), then integer numbers of molecules must be modeled along with discrete changes upon reaction. Importantly, the reaction occurrences can no longer be considered deterministic, but probabilistic. In this chapter we present the theory to treat reacting systems away from the thermodynamic limit. We first present a brief overview of continuous-deterministic chemical kinetics models and then discuss the development of stochastic-discrete models. 

It should be realized that unlike the study of equilibrium thermodynamics for which a model is often mapped onto Ising system, elementary mechanism of atomic motion plays a deterministic role in the kinetic study. In an actual alloy system, diffusion of an atomic species is mainly driven by vacancy mechanism. The incorporation of the vacancy mechanism into PPM formalism, however, is not readily achieved, since the abundant freedom of microscopic path of atomic movement demands intractable number of variational parameters. The present study is, therefore, limited to a simple spin kinetics, known as Glauber dynamics [14] for which flipping events at fixed lattice points drive the phase transition. Hence, the present study for a spin system is regarded as a precursor to an alloy kinetics. The limitation of the model is critically examined and pointed out in the subsequent sections. 

Model formulation. After the objective of modelling has been defined, a preliminary model is derived. At first, independent variables influencing the process performance (temperature, pressure, catalyst physical properties and activity, concentrations, impurities, type of solvent, etc.) must be identified based on the chemists knowledge about reactions involved and theories concerning organic and physical chemistry, mainly kinetics. Dependent variables (yields, selectivities, product properties) are defined. Although statistical models might be better from a physical point of view, in practice, deterministic models describe the vast majority of chemical processes sufficiently well. In principle model equations are derived based on the conservation law ... 

The different theoretical models for analyzing particle deposition kinetics from suspensions can be classified as either deterministic or stochastic. The deterministic methods are based on the formulation and solution of the equations arising from the application of Newton s second law to a particle whose trajectory is followed in time, until it makes contact with the collector or leaves the system. In the stochastic methods, forces are freed of their classic duty of determining directly the motion of particles and instead the probability of finding a particle in a certain place at a certain time is determined. A more detailed classification scheme can be found in an overview article [72]. 

When applied to spatially extended dynamical systems, the PoUicott-Ruelle resonances give the dispersion relations of the hydrodynamic and kinetic modes of relaxation toward the equilibrium state. This can be illustrated in models of deterministic diffusion such as the multibaker map, the hard-disk Lorentz gas, or the Yukawa-potential Lorentz gas [1, 23]. These systems are spatially periodic. Their time evolution Frobenius-Perron operator... 

If thermal fluctuations were taken into account, the regular patterns selected by this kinetic mechanism would be expected to be less sharp. In particular, when wjwa, is not so small, the effects of mass conservation are spread out over many terraces and several terraces in front of the step bunch become larger than These would be particularly advantageous sites where thennal nucleation could occur, even before the induced width of the terrace as predicted by the deterministic models would exceed Wc. Thus nucleation sites and times are less precisely determined in this case, and we... 

A number of different techniques have been developed for studying nonhomogeneous radiolysis kinetics, and they can be broken down into two groups, deterministic and stochastic. The former used conventional macroscopic treatments of concentration, diffusion, and reaction to describe the chemistry of a typical cluster or track of reactants. In contrast, the latter approach considers the chemistry of simulated tracks of realistic clusters using probabilistic methods to model the kinetics. Each treatment has advantages and limitations, and at present, both treatments have a valuable role to play in modeling radiation chemistry. 

In the following two sections, deterministic (prescribed diffusion and FACSIMILE) and stochastic (random flights and IRT) approaches for the modeling of radiation chemical kinetics will be described. Then representative calculations for simple aqueous systems will be shown The stochastic approach to modeling radiolysis kinetics is more physically realistic than the primitive deterministic models however, it is also more conceptually advanced, requiring a more detailed (fuller) knowledge of the system under consideration. 

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