Generalized mass action model

To simulate the overall network behavior, the power-law formalism is applied in two different ways. Within a generalized mass-action model (GMA), each biochemical interconversion is modeled with a power-law term, resulting in a differential equation analogous to Eq. (5)... 

In contrast to generalized mass-action models, an S-system model is obtained by lumping (or aggregating) all synthesizing and consuming reactions of each metabolite into a single power-law term, respectively. The mathematical structure of a S-System is independent of the complexity of the network. For any metabolite. S, -, we obtain... 

Similar to generalized mass-action models, lin-log kinetics provide a concise description of biochemical networks and are amenable to an analytic solution, albeit without sacrificing the interpretability of parameters. Note that lin-log kinetics are already written in term of a reference state v° and S°. To obtain an approximate kinetic model, it is thus sometimes suggested to choose the reference elasticities according to simple heuristic principles [85, 89]. For example, Visser et al. [85] report acceptable result also for the power-law formalism when setting the elasticities (kinetic orders) equal to the stoichiometric coefficients and fitting the values for allosteric effectors to experimental data. 

The mass action model (MAM) for binary ionic or nonionic surfactants and the pseudo-phase separation model (PSM) which were developed earlier (I EC Fundamentals 1983, 22, 230 J. Phys. Chem. 1984, 88, 1642) have been extended. The new models include a micelle aggregation number and counterion binding parameter which depend on the mixed micelle composition. Thus, the models can describe mixtures of ionic/nonionic surfactants more realistically. These models generally predict no azeotropic micellization. For the PSM, calculated mixed erne s and especially monomer concentrations can differ significantly from those of the previous models. The results are used to estimate the Redlich-Kister parameters of monomer mixing in the mixed micelles from data on mixed erne s of Lange and Beck (1973), Funasaki and Hada (1979), and others. 

Generalizing, one could state that the mass-action model simulates a cooperative (all or nothing) process with respect to large n values. This model is somewhat... 

Horn, E, and Jackson R., General mass action kinetics. Arch. Rati. Mech. Anal. 47,81 (1972). Jackson, R., and Glasser, D., A general mixing model for steady flow chemical reactors. Chem. Eng. Comm. 42, 17 (1986). 

As mentioned earlier, studies of simple linear surfactants in a solvent (i.e, those without any third component) allow one to examine the sufficiency of coarse-grained lattice models for predicting the aggregation behavior of micelles and to examine the limits of applicability of analytical lattice approximations such as quasi-chemical theory or self-consistent field theory (in the case of polymers). The results available from the simulations for the structure and shapes of micelles, the polydispersity, and the cmc show that the lattice approach can be used reliably to obtain such information qualitatively as well as quantitatively. The results are generally consistent with what one would expect from mass-action models and other theoretical techniques as well as from experiments. For example. Desplat and Care [31] report micellization results (the cmc and micellar size) for the surfactant h ti (for a temperature of = ksT/tts = /(-ts = 1-18 and... 

In the literature on micelle formation two primary models have gained general acceptance as useful (although not necessarily accurate) models for understanding the energetics of the process of self-association. The two approaches are the mass-action model, in which the micelles and monomeric species are considered to be in a kind of chemical equilibrium... 

The alternative approach to modeling micelle formation is to think in terms of a phase separation model in which, at the cmc, the concentration of the free surfactant molecules becomes constant (like a solubihty limit or Ksp), and all additional molecules go into the formation of micelles. Analysis of the two approaches produces the same general result in terms of the energetics of micelle formation (with some slight differences in detail), so that the choice of model is really a matter of preference and circumstances. There is evidence that the activity of free surfactant molecules does increase above the cmc, which tends to support the mass-action model however, for most purposes, that detail is of little consequence. 

Two approaches are generally used for modeling the properties of surfactant solutions, pseudophase and the mass action models. The pseudophase model is easier to use and has wider applicability in experimental work. For example, the pseudophase model is the basis for current interpretations of association colloid effects on the rates and equilibria of chemical reactions (Section 6). 

An alternative general approach (20) is the mass action model where the transfer of the solute in the micelle is treated as a binding phenomenon through the equilibrium process ... 

To cast the model in general form, we begin with the basis shown in Equation 9.5 and write each sorption reaction in the form of Equation 9.7. The mass action equation corresponding to the reaction for each sorbed species Aq is... 

Empirical Models vs. Mechanistic Models. Experimental data on interactions at the oxide-electrolyte interface can be represented mathematically through two different approaches (i) empirical models and (ii) mechanistic models. An empirical model is defined simply as a mathematical description of the experimental data, without any particular theoretical basis. For example, the general Freundlich isotherm is considered an empirical model by this definition. Mechanistic models refer to models based on thermodynamic concepts such as reactions described by mass action laws and material balance equations. The various surface complexation models discussed in this paper are considered mechanistic models. 

The equivalent to the law of mass action, as encountered in the previous chapter (e.g. in equation (3.22)), are systems of differential equations, defined by the chemical model or the reaction mechanism and the corresponding rate constants. We start with a general chemical reaction, just to practise the notation — it is not a realistic example ... 

As it follows from the above-said, nowadays any study of the autowave processes in chemical systems could be done on the level of the basic models only. As a rule, they do not reproduce real systems, like the Belousov-Zhabotinsky reaction in an implicit way but their solutions allow to study experimentally observed general kinetic phenomena. A choice of models is defined practically uniquely by the mathematical formalism of standard chemical kinetics (Section 2.1), generally accepted and based on the law of mass action, i.e., reaction rates are proportional just to products of reactant concentrations. 

Irregular behaviour of concentrations and the correlation functions observed in the chaotic regime differ greatly from those predicted by law of mass action (Section 2.1.1). Following Nicolis and Prigogine [2], the stochastic Lotka-Volterra model discussed in this Section, could be considered as an example of generalized turbulence. 

The law of mass action is a traditional base for modelling chemical reaction kinetics, but its direct application is restricted to ideal systems and isothermal conditions. More general is the Marceline-de Donder kinetics examined by Feinberg [15], but this also is not always sufficient. Let us give the most general of the reasonable forms of kinetic law matched to thermodynamics. The rate of the reversible reaction eqn. (5) is... 

Note that the physicochemical mechanisms that enables us to perform the chromatographic bioseparations are not always adsorption-like but can involve ion exchange, ion exclusion, or size exclusion. Even if it is generally possible to fit experimental data with a mathematical function derived from the adsorption theory, it is strongly advisable to refer to the proper physicochemical process before modeling the separation. For instance, ion exchange can be modeled with selectivity coefficients (derived from the mass action law) that can be constant or not,18,19 ion-exclusion can be modeled thanks to theories based on the Donnan exclusion, etc. 

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