Kinetic equation catalytic cycle

Since the catalytic cycle operates with relatively rapid kinetics, E and ES will obtain a steady state governed by Equations (4.2) and (4.3) and the quasi-steady state concentrations of enzyme and complex will change rapidly in response to relatively slow changes in [S]. Thus the quasi-steady approximation is justified based on a difference in timescales between the catalytic cycle kinetics and the overall rate of change of biochemical reactions. 

The derivation of the Michaelis-Menten equation in the previous section differs from the standard treatment of the subject found in most textbooks in that the quasi-steady approximation is justified based on the argument that the catalytic cycle kinetics is rapid compared to the overall biochemical reactant kinetics. In... 

A kinetic model describing the HRP-catalyzed oxidation of PCP by H202 should account for the effects of the concentrations of HRP, PCP, and H202 on the reaction rate. To derive such an equation, a reaction mechanism involving saturation kinetics is proposed. Based on the reaction scheme described in Section 17.3.1, which implies that the catalytic cycle is irreversible, the three distinct reactions steps (Equations 17.2 to 17.4) are modified to include the formation of Michaelis-Menten complexes ... 

A standard kinetic analysis of the mechanism 4a-4e using the steady state approximation yields a rate equation consistent with the experimental observations. Thus since equations 4a to 4e form a catalytic cycle their reaction rates must be equal for the catalytic system to be balanced. The rate of H2 production... 

Another way of representing catalytic cycles is shown in Figure 4.1. Note that the reactions have been drawn as irreversible reactions while most of them are actually equilibria. If one wants to derive kinetic equations, we strongly recommend the use of reaction sequences as shown on the previous page, because cycles such as 4.1 often lead to mistakes. 

Crabtree and coworkers proposed a catalytic cycle for the reaction outUned in Equation 6.10. The mechanism is based on labeling and kinetic studies, and is outlined in Scheme 6.4 [25]. Adduct 36 was observed in nuclear magnetic resonance (NMR) spectra and appears to be a catalyst resting state. It should be noted that there is no change in the oxidation state of Ir, and that the key step is thought... 

Hydrolysis of 4-nitrophenyl acetate (NA) (0.5-2.0 mM) was catalyzed by 11 in 10% volume/volume (v/v) CH3CN aqueous solution under comicellar conditions with 10 mM Triton X-100 at pH 9.2 (20 mM CHES buffer) and 25°C (Scheme 7). The second-order dependence of the rate constant, obsd, on the concentration of NA (10-50 dM) and 11 (0.2-1.0 mM) at pH 10.2 (2 mM CAPS buffer) and 25°C with I = 0.10 (NaN03) fits the kinetic equation (5). No other reaction such as acetate transfer to Triton X-100 was observed, as confirmed by a H NMR experiment with a 10% D20 solution of 2.0 mM NA, 0.2 mM 3, and 10 mM Triton X-100. Since the second-order kinetics held after several catalytic cycles, it was concluded that the NA hydrolysis catalytic. In Equation (5), vobsd is the observed NA hydrolysis rate catalyzed by 3, as derived by subtraction of the buffer-promoted NA hydrolysis rate from total NA hydrolysis rate. 

More complex enzymatic reactions usually display Michaelis-Menten kinetics and can be described by Eq. (2). However, the forms of constants Km and Vm can be very complicated, consisting of many individual rate constants. King and Altman (7) have provided a method to readily derive the steady-state equations for enzymatic reactions, including the forms that describe Km and Vm. The advent of symbolic mathematics programs makes the implementation of these methods routine, even for very complex reaction schemes. The P450 catalytic cycle (Fig. 2) is an example of a very complicated reaction scheme. However, most P450-mediated reactions display standard hyperbolic saturation kinetics. Therefore, although the rate constants that determine Km and Vm are... 

Figure 4.2 Kinetic mechanism of a Michaelis-Menten enzyme. (A) The reaction mechanism for the irreversible case - Equation (4.1) - is based on a single intermediate-state enzyme complex (ES) and an irreversible conversion from the complex to free enzyme E and product P. (B) The reaction mechanism for the reversible case - Equation (4.7) - includes the formation of ES complex from free enzyme and product P. For both the irreversible and reversible cases, the reaction scheme is illustrated as a catalytic cycle.

Reactor control models for monoliths require a more detailed study of the time scales of all occurring subprocesses, because of their dynamic character. Under dynamic circumstances, the rates of the individual elementary steps of a catalytic cycle, such as adsorption, surface reaction, and desorption, are not equal to each other anymore, since the time scales of the corresponding processes may differ by many orders of magnitude. Therefore, accumulation effects on the catalyst surface have to be taken into account as well, which demands that continuity equations for surface species be included in the model. Such aspects may even play a role in the steady state if the kinetics depend on rate-determining steps that change according to the concentration level of the reactants... 

The application of graph theory methods for deriving kinetic equations of heterogeneous catalytic reactions is based upon the so-called Rule of Mason this is also known in American literature as the Shannon-Mason Rule of Cycles. Although established by Shannon in 1941 (23), the rule acquired great popularity after its rediscovery by Mason in 1955 (24.25). A strict proof for the validity of the Rule of Mason for multiroute linear mechanisms was presented only recently by Evstigneev and Yablonskii (26), where both an inductive proof and a proof based on the Rule of Krammer are set forth. 

When interfacial electron exchange rate(s) are sufficiently high and the response is free from mass transport hmitations, the catalytic current will be determined by the inherent activity of the enzyme. Variation of current (activity) with potential can be explained by an extension of the Michaelis-Menten description of enzyme kinetics that relates activity to oxidation state through incorporation of the Nemst equation." " The resulting expressions describe the catalytic cycle, and include rates of intramolecular electron exchange, chemical events, substrate binding and product release, together with the reduction potentials of centres in the enzyme, and the influence of... 

Organometallic catalysis is very diverse and many different mechanisms have been proposed in the literature for various reactions. Interestingly it is rather seldom that kinetic models are derived for the proposed catalytic cycles and the kinetic equations are even compared qualitatively with experimental observations. We will discuss several examples which represent typical cases in homogeneous catalysis by metall complexes. 

Catalysis by transition metals can often be represented by reaction mechanisms which correspond to the case of single catalytic cycles. Only one route exits but it could contain several steps. If the reaction mechanism can be simplified to only one intermediate besides the free catalyst form (the most abundunt surface intermediate or catalytic species) while other intermediates even if present are in inferior quantities, then the mechanism corresponds to the two-step sequence described in detail in section 4.3. As an example heterolytic oxidation can be considered (Figure 5.9). The catalyst can exist in two forms with different oxidation states and the catalytic cycle consists effectively of oxidation-reduction steps. The reaction rate in such a case is described by the kinetic equation (4.88). 

In the model proposed by Hyver Le Guyader (1990), two systems of equations are considered. In the first version, inactive p34 (i.e. cdc2 kinase) transforms into active p34, either spontaneously or in an auto-catalytic manner active p34 then combines with cyclin to yield active MPF. This situation is described by four differential equations, of a polynomial nature, in which the highest nonlinearities are of the quadratic type. In a second version of this model, governed by three kinetic equations of a similar form, the authors consider the effect of an activation of MPF by MPF itself as well as cyclin, and show that oscillations develop when the degradation of cyclin is brought about by the formation of a complex between cyclin and MPF. That study was the first to show the occurrence of sustained oscillations in a model based on the interactions between cyclin and cdc2 kinase. The type of kinetics considered for these interactions remained, however, remote from the actual kinetics of phosphorylation-dephosphorylation cycles. 

The development of a mechanistic model for a homogeneous reaction requires constructing a catalytic cycle, which is quite difficult. On the other hand, simple kinetic expressions of both the power law and hyperbolic types can be readily derived. These are usually adequate for reactor design. Thus in the analysis of homogeneous catalysis involving a gas-liquid reaction, the following general hyperbolic form of the rate equation may be used ... 

In the case of modified cobalt (29), (31) and modified rhodium catalysts (29), the above rate equation describes the observed experimental facts only partially because the product formation in these cases can occur in different catalytic cycles simultaneously which makes the kinetics more complex see, for example. Refs. 67 and (95-98). 

Catalytic kinetics in the twentieth century were dominated by rate equations.Rate constants, were and are, extracted from rate equations obtained by fitting kinetic data, usually obtained by adjusting the process parameters to enable linearity. A catalytic cycle, however, is a nonlinear dynamic system. Even with a fixed set of paramefers, turn-over limiting states may change with time and extent of fumover. Thus, depending on the portion of the catalytic reaction under study, the rate law may be different. Therefore, can statements as to the kinetic order of the overall catalytic reaction with respect to either substrate(s) consumption or product production obtained by traditional concentration kinetics always be universally assumed to be correct Even if they are, different reaction mechanisms may predict the same overall reaction rate. 

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