THE Michaelis-Menten Kinetics

In this section the basic kinetic model for enzyme-catalyzed bioconversions is presented. Understanding this model is the foundation for deriving more complex models. In their theory of enzyme catalysis, Michaelis and Menten 113 postulated the existence of an enzyme substrate complex (ES), which is built up in a reversible 

If the rate constant k2 is much smaller than the rate constant k., of the enzyme, the substrate and the enzyme-substrate complex are in equilibrium, which is not disturbed by the decomposition of ES into E and P ( rapid equilibrium-assumption ). Based on this assumption, Michaelis and Menten derived the following rate equation (Eq. (17))  

Umax (mmol L-1min-1) Maximum reaction rate 

This is the classic form of the Michaelis-Menten equation. Eq. (17) can be rearranged to 

The curve v = /[S] belonging to the Michaelis-Menten function is shown in Fig. 7-18. 

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The values of the Michaelis-Menten kinetic parameters, Vj3 and C,PP characterise the kinetic expression for the micro-environment within the porous structure. Kinetic analyses of the immobilised lipase in the membrane reactor were performed because the kinetic parameters cannot be assumed to be the same values as for die native enzymes. 

Let us consider the determination of two parameters, the maximum reaction rate (rITOIX) and the saturation constant (Km) in an enzyme-catalyzed reaction following Michaelis-Menten kinetics. The Michaelis-Menten kinetic rate equation relates the reaction rate (r) to the substrate concentrations (S) by... 

Quite often the asymptotic behavior of the model can aid us in determining sufficiently good initial guesses. For example, let us consider the Michaelis-Menten kinetics for enzyme catalyzed reactions,... 

MS is lower than that of M the system is in the regime of substrate saturation addition of more S does not lead to a rate increase. The behaviour of the reaction rate in case B is typical of enzymes and in biochemistry this is referred to as Michaelis-Menten kinetics. The success of the application of the Michaelis-Menten kinetics in biochemistry is based on the fact that indeed only two reactions are involved the complexation of the substrate in the pocket of the enzyme and the actual conversion of the substrate. Usually the exchange of the substrate in the binding pocket is very fast and thus we can ignore the term k2[H2] in the denominator. Complications arise if the product binds to the binding site of the enzyme, product inhibition, and more complex kinetics result. 

From the usual expression of the Michaelis -Menten kinetics... 

In order to predict the effect of a mixture of chemicals with the same target receptor, but with different nonlinear dose-effect relationships, either physiological or mathematical modeling can be applied. For interactions between chemicals and a target receptor or enzyme, the Michaelis-Menten kinetics (first order kinetics but with saturation) are often applicable. This kind of action can then be considered a special case of similar combined action (dose addition). 

Let us consider some frequently used nonlinear functions. The Langmuir isotherm and the Michaelis-Menten kinetics are of the form... 

Figure 11.17 Comparison of a CSTR and a PFR, for the Michaelis-Menten kinetics.

One of the simplest cases of two consequtive reation steps is the Michaelis-Menten kinetics ... 

The data for the ferricenium half-reaction deserve several comments. The Michaelis-Menten kinetics obtained in the UV-vis experiment supports the formation of the GO-ferricenium intermediates postulated in Scheme 5. The ratio k2i)/K which corresponds to the bimolecular interaction of GO(red) with the ferricenium ion, equals ca. 1 x 105 M-1 s 1 (79) and this must be compared with the observed rate constants of 1.4 x 105 M-1 s 1 found for ferrocene using the electrochemical technique under similar conditions (87). 

Data in Fig. 9 show that ferrocene is the most reactive in this family of HRP substrates. The rate constant k7 of 2 x 105 M-1 s 1 at pH 7 and 25 °C indicates that its reactivity is comparable with the frequently used electron donors of HRP. Ferrocene follows first-order kinetics and k7 should be compared with the ratio cat/ 4Vb where kc t and Km are the catalytic and the Michaelis constants for substrates obeying the Michaelis-Menten kinetics, respectively. Such are iodide, guaiacol, and ABTS (2,2 -azino-bis(3-ethylbenzothiazoline-6 -sulfonic acid) (128). The available ratios of 0.15 x 105, 1.3 x 105, and 34 x 105 M 1 s-1, respectively (129), indicate that ferrocene is more reactive than iodide and comparable with guaiacol. High reactivity of ferrocene makes it a convenient analytical reagent for routine assays of H2O2 in the presence of HRP by monitoring the enzymically produced ferricenium dye at 617 nm (113). 

The Fc-HRP activity was quantified using two different substrates of HRP, i.e., ABTS and water-soluble ferrocene derivatives. Rate laws and kinetic parameters for native HRP and Fc-HRP have been compared. The native and the reconstituted enzymes catalyze the oxidation of ABTS in accordance with the Michaelis-Menten kinetics the inverse rate versus [ABTS]-1 plots are linear and the values of the maximum rates Vm and the Michaelis constant Km are summarized... 

In contrast to ABTS, the kinetic behavior of native HRP and Fc HRP toward water-soluble ferrocenes HOOCFc and Me2NCH2Fc is remarkably different. Instead of first-order kinetics observed for native HRP, the reaction rate levels off on increasing the ferrocene concentration for Fc HRP, Fig. 13. The Michaelis Menten kinetics holds for both ferrocene substrates, and the inverse rate vs. [ferrocene]-1 plots are linear. The values of Vm for HOOCFc and... 

As in the ferrocene case, the reaction is first order in both HRP and a metal electron donor suggesting kinetic insignificance of the enzyme-electron donor intermediate (181). It should be mentioned that the first order in Mn is observed for complexes of moderate reactivity. Very reactive complexes such as [0s(bpy)2(py)(H20)]+ in Table VIII obey the Michaelis-Menten kinetics because the formation of Compound I [Eq. (5)] starts to slow down the rate (see below). The dependence of the rate of reaction 44 on the H202 concentration shown in Fig. 20 resembles the ferrocene case (119). The decline in rate after reaching a maximum has routinely been rationalized in terms of an inactivation of HRP. The true nature of this phenomenon has recently been underscored in the course of detailed spectroscopic and electrochemical studies of the HRP-catalyzed oxidation of [OsCl(bpy)2(py)]+ (121). Thus, transition metal electron donors are convenient probes for solving fundamental problems of enzymology. 

Analytic solution of the Michaelis-Menten kinetic equation The simplest mechanism of enzyme reactions is of the form... 

It would be distinctly arrogant to say that we understand how enzymes work. At best we catch glimpses of their action. One model involves the key-and-lock concept—an attempt to rationalize their specificity. A much simplified presentation is shown in Fig. 7.115. The idea is that certain shapes in the enzyme structure are precise fits for a part of the reactant molecule. A famous formulation of this is the Michaelis-Menten kinetics. If E is the enzyme and R is some part of a reactant (a complex biomolecule),... 

Allosteric enzymes do not follow the Michaelis-Menten kinetic relationships between substrate concentration Fmax and Km because their kinetic behaviour is greatly altered by variations in the concentration of the allosteric modulator. Generally, homotrophic enzymes show sigmoidal behaviour with reference to the substrate concentration, rather than the rectangular hyperbolae shown in classical Michaelis-Menten kinetics. Thus, to increase the rate of reaction from 10 per cent to 90 per cent of maximum requires an 81-fold increase in substrate concentration, as shown in Fig. 5.34a. Positive cooperativity is the term used to describe the substrate concentration-activity curve which is sigmoidal an increase in the rate from 10 to 90 per cent requires only a nine-fold increase in substrate concentration (Fig. 5.346). Negative cooperativity is used to describe the flattening of the plot (Fig. 5.34c) and requires requires over 6000-fold increase to increase the rate from 10 to 90 per cent of maximum rate. 

In relation to enzymic cytochrome P-450 oxidations, catalysis by iron porphyrins has inspired many recent studies.659 663 The use of C6F5IO as oxidant and Fe(TDCPP)Cl as catalyst has resulted in a major improvement in both the yields and the turnover numbers of the epoxidation of alkenes. 59 The Michaelis-Menten kinetic rate, the higher reactivity of alkyl-substituted alkenes compared to that of aryl-substituted alkenes, and the strong inhibition by norbornene in competitive epoxidations suggested that the mechanism shown in Scheme 13 is heterolytic and presumably involves the reversible formation of a four-mernbered Fev-oxametallacyclobutane intermediate.660 Picket-fence porphyrin (TPiVPP)FeCl-imidazole, 02 and [H2+colloidal Pt supported on polyvinylpyrrolidone)] act as an artificial P-450 system in the epoxidation of alkenes.663... 

In conclusion, the values of the Michaelis-Menten kinetic parameters, rmax and KM, can be estimated, as follows ... 

To determine the Michaelis-Menten kinetic parameters based on initial-rate reactions in a series of batch runs... 

Determine the Michaelis-Menten kinetic parameters as described in this chapter. 

Evaluate the Michaelis-Menten kinetic parameters by employing (a) the Langmuir plot, (b) the Lineweaver-Burk plot, (c) the Eadie-Hofstee plot, and (d) non-linear regression procedure. 

A carbohydrate (S) decomposes in the presence of an enzyme (E). The Michaelis-Menten kinetic parameters were found to be as follows ... 

The Km value of an enzyme is known to be 0.01 mol/L. To measure the maximum reaction rate catalyzed by the enzyme, you measured the initial rate of the reaction and found that 10 percent of the initial substrate was consumed in 5 minutes. The initial substrate concentration is 3.4 times lO mol/L. Assume that the reaction can be expressed by the Michaelis-Menten kinetics. 

A substrate is converted to a product by the catalytic action of an enzyme. Assume that the Michaelis-Menten kinetic parameters for this enzyme reaction are ... 

The Michaelis-Menten kinetic parameters for a soluble enzyme for a certain substrate are found to be... 

There is an interesting parallel between substrate binding and adsorption. Since each enzyme molecule has one active site, and since these active sites all have the same structure, we can think of enzyme molecules in solution as a surface with many equivalent adive sites. In this case, k2 in the Michaelis-Menten kinetics (Eq. 5.1 see Chapter 2 for a detailed discussion) represents the rate of adsorption, k x the rate of desorption, and k2 the rate of the surface readion followed by fast product desorption. Moreover, this system fits the assumptions of the Langmuir isotherm (all sites identical, one molecule per site, no lateral interadions) even better than the adive sites on some real solid catalysts ... 

The value of km can be obtained according, for example, to the Michaelis-Menten kinetics as follows ... 

Figure 14.6 (a) Concentration vs. membrane thickness applying the Michaelis-Menten kinetics (continuous lines) and its limiting... 

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