Rate equations cooperative models

To account for positive cooperativity and sigmoidal rate equations, a number of theoretical models for allosteric regulation have been developed. Common to most models is the assumption (and requirement) that enzymes act as multimers and exhibit interactions between the units. We briefly mention the most... 

Al O revealing that a prominent activity of Mg-Al mixed oxide catalysts originates from the cooperative action of both the basic and acidic sites closely present on the surface. Furthermore, kinetic data of this addition reaction could be well adjusted with a rate equation based on the Langmuir-Hinshelwood model where the CO and epoxide are independently adsorbed on the different sites, i.e., basic sites and acidic sites, respectively. 

For example, the standard synergetic approach [52-54] denies the possibility of any self-organization in a system with with two intermediate products if only the mono- and bimolecular reaction stages occur [49] it is known as the Hanusse, Tyson and Light theorem. We will question this conclusion, which in fact comes from the qualitative theory of non-linear differential equations where coefficients (reaction rates) are considered as constant values and show that these simplest reactions turn out to be complex enough to serve as a basic models for future studies of non-equilibrium processes, similar to the famous Ising model in statistical physics. Different kinds of auto-wave processes in the Lotka and Lotka-Volterra models which serve as the two simplest examples of chemical reactions will be analyzed in detail. We demonstrate the universal character of cooperative phenomena in the bimolecular reactions under study and show that it is reaction itself which produces all these effects. 

The above treatment is based on the simplest model of the cooperative processes. In real cases, not all the portions of the cooperative assemble are identical and rate of elementary rearrangements are equal. Such cases require special consideration with the use modem theoretical methods. Nevertheless, equations of types 2.48-2.50 disclose, in principle, the physical meaning the physical meaning of experimentally determined activation parameters of enzymatic processes. 

The check the validity of the arguments a series of model calculations based on the reaction mechanism of Fig. 10 were performed [19], details to be published). They are based on a set of coupled differential equations and on the assumption that the phase transition between microtubule growth and shrinkage can be explained by cooperative interactions between tubulin subunits at microtubule ends. Two examples are shown in Fig. 13 where the rate of GTP hydrolysis following the incorporation of tubulin into microtubules was varied. When this rate is fast (Fig. 13 a) one finds pronounced oscillations (this is equivalent to an intermediate stability of microtubules, compare Fig. lib). When the rate of hydrolysis is reduced (Fig. 13 b) the oscillations disappear because microtubules are effectively stabilized, and they remain assembled in a steady state (compare Fig. 11 a). 

A homotropic effect occurs when the binding of one substrate molecule perturbs the rate of catalysis of a second molecule of the same substrate. It is possible for the homotropic effect to be either positive, that is, to give rise to an increased rate of catalysis (homotropic activation, positive cooperativity, or autoactivation), or a negative, that is, causing a decreased rate of catalysis (homotropic inactivation, negative cooperativity, or substrate inhibition). These circumstances cannot be adequately modeled by the simple Michaelis-Menten equation, and neither do direct plots of v... 

Piskiewicz [119] has developed a kinetic model of micellar catalysis, based on the Hill equation of enzyme kinetics, which assumes a cooperative interaction between reactants and surfactant to form reactive substrate-micelle complexes. This model is probably not applicable to systems in which the surfactant is in large excess over substrate, as in most micellar mediated reactions, but it gives a very reasonable explanation of the rate effects of very dilute surfactants. 

For an enzyme that follows MichaeHs—Menten kinetics, R = SI. For a regulatory enzyme that gives a sigmoidal rate plot, Rj < 81 if the enzyme is exhibiting positive cooperativity, a term that means that the substrate and enzyme bind in such a way that the rate increases to a greater extent with increasing [S] than the MichaeHs—Menten model predicts. Cases with R-s > 81 indicate negative cooperativity so that the catalytic effect becomes less than that found in MichaeHs—Menten kinetics. In these cases, kinetic analysis is usually carried out by means of the HiU equation. 

It is important to note, and should always be remembered, that all the rate and binding equations for cooperative and allosteric enzymes were derived under rapid equilibrium assumption. Therefore, all kinetic models for the cooperative phenomena can be equally well applied to enzyme reactions that are in the rapid equilibrium and to the binding of ligands to enzymes and proteins. 

A mathematical model for numerical simulation of onboard SCR monolithic converters has been developed in our group in cooperation with Daimler, as outlined in References (144-148). It represents an evolution of the fully transient, heterogeneous model of a single monolith channel (98,116), whose equations are summarized in Table 5. As opposite to many of the other literature models, this is a spatially 2D (ID -I- ID) model, accounting also for diffusion and reaction of reactants and products inside the porous walls of extruded honeycomb SCR catalysts. It should be emphasized that simplified surface reaction models omitting to describe the complex, rate-controlling dynamics of NH3 intraporous dif-fusion/adsorption/reaction may yield significant errors under certain conditions, especially in the case of extruded monolith catalysts. 

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