Enzyme-kinetic-type model

Experimentally determined rate constants for various micellar-mediated reactions show either a monotonic decrease (i.e., micellar rate inhibition) or increase (i.e., micellar rate acceleration) with increase in [Suifl CMC, where [Surf]T represents total micelle-forming surfactant concentration (Figure 3.1). Menger and Portnoy obtained rate constants — [SurfJx plots — for hydrolysis of a few esters in the presence of anionic and cationic surfactants, which are almost similar to those plots shown in Figure 3.1. These authors explained their observations in terms of a proposed reaction mechanism as shown in Scheme 3.1 which is now called Menger s phase-separation model, enzyme-kinetic-type model, or preequilibrium kinetic (PEK) model for micellar-mediated reactions. In Scheme 3.1, Kj is the equilibrium... 

The values of rate constants for oxidation of rosaniline hydrochloride in mixed micellar solutions (CTABr + Tween-20, CTABr + Tween-80) are less than those in pure CTABr solution but are higher than those in pure nonionic surfactant solutions, whereas the values of rate constants in mixed micellar solutions (SDS + Triton X-100, SDS + Triton X-102) are less than those in pure anionic as well as in nonionic surfactants. The effects of mixed micelles of cationic-cationic, cationic-nonionic, and anionic-nonionic surfactants on the rate of alkaline hydrolysis of A-phenylbenzohydroxamic acid have been studied at 55°C where the addition of cationic surfactant to nonionic surfactant accelerates the rate of hydrolysis, and the kinetic data have been analyzed by the Monger s enzyme-kinetic-type model. ... 

Effects of cationic (cetylpyridinium chloride, CPC) and anionic (SDS) micelles on the rate of reaction of chromium(VI) oxidation of formaldehyde have been studied in the presence and absence of picolinic acid. Cationic micelles (CPC) inhibit whereas anionic micelles (SDS) catalyze the reaction rates that could be attributed to electrostatic interactions between reactants (cationic metal ions and catalyst H+) and ionic head groups of ionic micelles. Experimentally determined kinetic data on these metaUomicellar-mediated reactions have been explained by different kinetic models such as pseudophase ion-exchange (PIE) model, Monger s enzyme-kinetic-type model, and Piszkiewicz s cooperativity model (Chapter 3). The rate of oxidation of proline by vanadium(V) with water acting as nucleophile is catalyzed by aqueous micelles. Effects of anionic micelles (SDS) on the rate of A-bromobenzamide-catalyzed oxidation of ethanol, propanol, and n-butanol in acidic medium reveal the presence of premicellar catalysis that has been rationalized in light of the positive cooperativity model. ... 

The minimum value of /Jdf/v required for a reliable model depends on the quality of the determination of the data to be correlated. The smaller the experimental error in the data, the smaller the value of /Jdf/v required for dependable results. Experience indicates that in the case of chemical reactivity data /Jdf/v should be not less than 3. For bioactivity studies /Jdf/v depends heavily on the type of data for rate and equilibrium constants obtained from enzyme kinetics a value of not less than 3 is reasonable while for toxicity studies on mammals at least 7 is required. 

There are still other causes of nonlinearities than (apparent or real) higher-order transformation kinetics. In Section 12.3 we discussed catalyzed reactions, especially the enzyme kinetics of the Michaelis-Menten type (see Box 12.2). We may also be interested in the modeling of chemicals which are produced by a nonlinear autocatalytic reaction, that is, by a production rate function, p(Q, which depends on the product concentration, C,. Such a production rate can be combined with an elimination rate function, r(C,), which may be linear or nonlinear and include different processes such as flushing and chemical transformations. Then the model equation has the general form ... 

Often the key entity one is interested in obtaining in modeling enzyme kinetics is the analytical expression for the turnover flux in quasi-steady state. Equations (4.12) and (4.38) are examples. These expressions are sometimes called Michaelis-Menten rate laws. Such expressions can be used in simulation of cellular biochemical systems, as is the subject of Chapters 5, 6, and 7 of this book. However, one must keep in mind that, as we have seen, these rates represent approximations that result from simplifications of the kinetic mechanisms. We typically use the approximate Michaelis-Menten-type flux expressions rather than the full system of equations in simulations for several reasons. First, often the quasi-steady rate constants (such as Ks and K in Equation (4.38)) are available from experimental data while the mass-action rate constants (k+i, k-i, etc.) are not. In fact, it is possible for different enzymes with different detailed mechanisms to yield the same Michaelis-Menten rate expression, as we shall see below. Second, in metabolic reaction networks (for example), reactions operate near steady state in vivo. Kinetic transitions from one in vivo steady state to another may not involve the sort of extreme shifts in enzyme binding that have been illustrated in Figure 4.7. Therefore the quasi-steady approximation (or equivalently the approximation of rapid enzyme turnover) tends to be reasonable for the simulation of in vivo systems. 

The reaction rate for the hydrolysis of starch (Eq. 2.1 in Table 3) is a Michaelis-Menten type model, which considers competitive product inhibition of glucose and substrate inhibition of starch. The hydrolysis of maltose (Eq. 2.2 in Table 3) is represented by a Michaelis-Menten type model with competitive product inhibition. These equations were tested by Lopez et al. [3] for hydrolysis of chestnut puree by an alpha and glucoamylase mixture. As the enzyme STARGEN also contains amormts of alpha and glucoamylase, it was not surprising that they (Eqs. 2.1-2.2 in Table 3) fit the hydrolysis data better than non-inhibitory Michaelis-Menten kinetics. 

Common to most of these models is the primary role played by autocatalysis in the origin of sustained oscillations in the mitotic control system. In the absence of self-activation by cdc2 kinase, oscillations would not occur in this class of model. Let us consider next a model that bypasses the absolute requirement for such a positive feedback. In contrast to the above-described models which rely on polynomial kinetic equations, the model examined in the following section is based on enzyme kinetics of the Michaelis-Menten type, closer to the phosphorylation-dephosphorylation nature of the reactions that control the activation of cdc2 kinase. 

The rates of enzyme-catalyzed reactions do not lit simple models for first- or second-order kinetics. Typically, the rate is a nonlinear function of concentration, as shown in Figure 1.9. At low substrate concentrations, the reaction appears first order, but the rate changes more slowly at more moderate concentrations, and the reaction is nearly zero order at high concentrations. A model to explain this behavior was developed in 1913 by L. Michaelis and M. L. Menton [6], and their names are still associated with this type of kinetics. The model presented here is for the simple case of a... 

In the practical work with enzymes, a relatively limited number of different types of titration curves is encountered for monobasic and dibasic adds. Equations (14.47) and (14.48) show the major types of equations for monobasic adds which are encountered in the technical and scientific literature (Cleland, 1977, 1982 Giimshaw et al, 1981 Purich Allison, 2000). Naturally, a rare occurrence of some rate equations and some pH profiles in the literature is in no correlation whatsoever with their importance in enzymology. A specific problem in enzyme kinetics will always raise toe need for a specific kinetic model and the corresponding rate equations. 

The examples to be presented illustrate the diversity of fields of applications, but they are mentioned in outline form only. Many biological phenomena used to be modelled by real or formal kinetic models. A biochemical control theory that is partially based on non-mass-action-type enzyme kinetics seems to be under elaboration, and certain aspects will be illustrated. A few specific models of fluctuation and oscillation phenomena in neurochemical systems will be presented. The formal structure of population dynamics is quite similar to that of chemical kinetics, and models referring to different hierarchical levels from elementary genetics to ecology are well-known examples. Polymerisation, cluster formation and recombination kinetics from the physical literature will be mentioned briefly. Another question to be discussed is how electric-circuit-like elements can be constructed in terms of chemical kinetics. Finally, kinetic theories of selection will be mentioned. 

The latter method, called the PI-FEP/UM approach, allows accurate primary and secondary kinetic isotope effects to be computed for enzymatic reactions. These methods are illustrated by applications to three enzyme systems, namely, the proton abstraction and reprotonation process catalyzed by alanine race-mase, the enhanced nuclear quantum effects in nitroalkane oxidase catalysis, and the temperature (in)dependence of the wild-type and the M42W/G121V double mutant of dihydrofolate dehydrogenase. These examples show that incorporation of quantum mechanical effects is essential for enzyme kinetics simulations and that the methods discussed in this chapter offer a great opportunity to more accurately model the mechanism and free energies of enzymatic reactions. 

K. Tummler, T. Lubitz, M. Schelker, E. Klipp, New types of experimental data shape the use of enzyme kinetics for dynamic network modeling, FEBS J. 281 (2014) 549-571. 

In conclusion, the steady-state kinetics of mannitol phosphorylation catalyzed by II can be explained within the model shown in Fig. 8 which was based upon different types of experiments. Does this mean that the mechanisms of the R. sphaeroides II " and the E. coli II are different Probably not. First of all, kinetically the two models are only different in that the 11 " model is an extreme case of the II model. The reorientation of the binding site upon phosphorylation of the enzyme is infinitely fast and complete in the former model, whereas competition between the rate of reorientation of the site and the rate of substrate binding to the site gives rise to the two pathways in the latter model. The experimental set-up may not have been adequate to detect the second pathway in case of II " . The important differences between the two models are at the level of the molecular mechanisms. In the II " model, the orientation of the binding site is directly linked to the state of phosphorylation of the enzyme, whereas in the II" model, the state of phosphorylation of the enzyme modulates the activation energy of the isomerization of the binding site between the two sides of the membrane. Steady-state kinetics by itself can never exclusively discriminate between these different models at the molecular level since a condition may be proposed where these different models show similar kinetics. The II model is based upon many different types of data discussed in this chapter and the steady-state kinetics is shown to be merely consistent with the model. Therefore, the II model is more likely to be representative for the mechanisms of E-IIs. 

One-step hydroxylation of aromatic nucleus with nitrous oxide (N2O) is among recently discovered organic reactions. A high eflSciency of FeZSM-5 zeolites in this reaction relates to a pronounced biomimetic-type activity of iron complexes stabilized in ZSM-5 matrix. N2O decomposition on these complexes produces particular atomic oj gen form (a-oxygen), whose chemistry is similar to that performed by the active oxygen of enzyme monooxygenases. Room temperature oxidation reactions of a-oxygen as well as the data on the kinetic isotope effect and Moessbauer spectroscopy show FeZSM-5 zeolite to be a successfiil biomimetic model. 

Although not all facets of the reactions in which complexes function as catalysts are fully understood, some of the processes are formulated in terms of a sequence of steps that represent well-known reactions. The actual process may not be identical with the collection of proposed steps, but the steps represent chemistry that is well understood. It is interesting to note that developing kinetic models for reactions of substances that are adsorbed on the surface of a solid catalyst leads to rate laws that have exactly the same form as those that describe reactions of substrates bound to enzymes. In a very general way, some of the catalytic processes involving coordination compounds require the reactant(s) to be bound to the metal by coordinate bonds, so there is some similarity in kinetic behavior of all of these processes. Before the catalytic processes are considered, we will describe some of the types of reactions that constitute the individual steps of the reaction sequences. 

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