Kinetics Michaelis-Menten approach

Michaelis-Menten approach (Michaelis and Menten, 1913) It is assumed that the product-releasing step, Eq. (2.6), is much slower than the reversible reaction, Eq. (2.5), and the slow step determines the rate, while the other is at equilibrium. This is an assumption which is often employed in heterogeneous catalytic reactions in chemical kinetics.3 Even though the enzyme is... 

The kinetic parameters for a free enzyme in solution are readily derived using the Michaelis-Menten approach describing pseudo-steady-state conversions. Consider Equation (31.1) representing the conversion of a substrate S into a product P, catalyzed by an enzyme E. The rate of formation of an enzyme/substrate complex, ES, is denoted as ku the reverse reaction by and the rate of subsequent conversion to the free product by k2. 

The Eadie-Hofstee plot does a betterjob than the Line-weaver-Burke plot in evenly distributing the data points over the entire substrate concentration range, and can be a useful visual technique for ascertaining whether enzyme kinetics are typical (as shown) or atypical (see Figure 8.18, B and G). The Michaelis-Menten approach basically assumes that enzymes present a single binding site to each substrate. Estimates of V x of drug... 

The above rate equation is in agreement with that reported by Madhav and Ching [3]. Tliis rapid equilibrium treatment is a simple approach that allows the transformations of all complexes in terms of [E, [5], Kls and Kjp, which only deal with equilibrium expressions for the binding of the substrate to the enzyme. In the absence of inhibition, the enzyme kinetics are reduced to the simplest Michaelis-Menten model, as shown in Figure 5.21. The rate equation for the Michaelis-Menten model is given in ordinary textbooks and is as follows 11... 

Kinetic data fitting the rate equation for catalytic reactions that follow the Michaelis-Menten equation, v = k A]/(x + [A]), with[A]0 = 1.00 X 10 J M, k = 1.00 x 10 6 s 1, and k = 2.00 X 10-J molL1. The left panel displays the concentration-time profile on the right is the time lag approach. 

This equation is fundamental to all aspects of the kinetics of enzyme action. The Michaelis-Menten constant, KM, is defined as the concentration of the substrate at which a given enzyme yields one-half of its maximum velocity. is the maximum velocity, which is the rate approached at infinitely high substrate concentration. The Michaelis-Menten equation is the rate equation for a one-substrate enzyme-catalyzed reaction. It provides the quantitative calculation of enzyme characteristics and the analysis for a specific substrate under defined conditions of pH and temperature. KM is a direct measure of the strength of the binding between the enzyme and the substrate. For example, chymotrypsin has a Ku value of 108 mM when glycyltyrosinylglycine is used as its substrate, while the Km value is 2.5 mM when N-20 benzoyltyrosineamide is used as a substrate... 

At high substrate concentrations relative to Km ([S] Km), The Michaelis-Menten equation reduces to v = Vmax, substrate concentration disappears, and the dependence of velocity on substrate concentration approaches a horizontal line. When the reaction velocity is independent of the concentration of the substrate, as it is at Vmax, it s given the name zero-order kinetics. 

The nonlinear form of the Michaelis-Menten equation, 10.2-9, does not permit simple estimation of the kinetic parameters (Km and V ). Three approaches may be adopted ... 

This relationship corresponds to the simplest Michaelis-Menten kinetics (Eq. (3)). In addition to the equation derived earlier by Halpern et al. for the simplest model case of a C2-symmetric ligand without intramolecular exchange [21b], every other possibility of reaction sequence corresponding to Scheme 10.3 can be reduced to Eq. (13). Only the physical content of the values of kobs and Km, which must be determined macroscopically, differs depending upon the approach (see [59] for details). Nonetheless, the constants k0bs and KM allow conclusions to be made about the catalyses ... 

Results have generally been disappointing. It can be difficult to remove the TSA from the polymer, but a more fundamental problem concerns the efficiency of the catalysis observed. The most efficient systems catalyze the hydrolysis of carboxylate and reactive phosphate esters with Michaelis-Menten kinetics and accelerations (koAJKM)/kunoJ approaching 103,1661 but the prospects for useful catalysis of more complex reactions look unpromising. Apart from the usual difficulties the active sites produced are relatively inflexible, and the balance between substrate binding and product inhibition is particularly acute. 

The necessity of developing approximate kinetics is unclear. It is sometimes argued that uncertainties in precise enzyme mechanisms and kinetic parameters requires the use of approximate schemes. However, while kinetic parameters are indeed often unknown, the typical functional form of generic rate equations, namely a hyperbolic Michaelis Menten-type function, is widely accepted. Thus, rather than introducing ad hoc functions, approximate Michaelis Menten kinetics can be utilized an approach that is briefly elaborated below. 

It has been found experimentally that in most cases v is directly proportional to the concentration of enzyme [.E0] and that v generally follows saturation kinetics with respect to the concentration of substrate [limiting value called Vmax. This is expressed quantitatively in the Michaelis-Menten equation originally proposed by Michaelis and Menten. Km can be seen as an apparent dissociation constant for the enzyme-substrate complex ES. The maximal velocity Vmax = kcat E0. ... 

Analyses of enzyme reaction rates continued to support the formulations of Henri and Michaelis-Menten and the idea of an enzyme-substrate complex, although the kinetics would still be consistent with adsorption catalysis. Direct evidence for the participation of the enzyme in the catalyzed reaction came from a number of approaches. From the 1930s analysis of the mode of inhibition of thiol enzymes—especially glyceraldehyde-phosphate dehydrogenase—by iodoacetate and heavy metals established that cysteinyl groups within the enzyme were essential for its catalytic function. The mechanism by which the SH group participated in the reaction was finally shown when sufficient quantities of purified G-3-PDH became available (Chapter 4). 

For an enzyme with typical Michaelis-Menten kinetics, the value of e ranges from about 1 at substrate concentrations far below Km to near 0 as Vmax is approached. Allosteric enzymes can have elasticities greater than 1.0, but not larger than their Hill coefficients (p. 167). 

There are methods used Lo study enzymes other than those of chemical instrumental analysis, such as chromatography, that have already been mentioned. Many enzymes can be crystallized, and their structure investigated by x-ray or electron diffraction methods. Studies of the kinetics of enzyme-catalyzed reactions often yield useful data, much of this work being based on the Michaelis-Menten treatment. Basic to this approach is the concept (hat the action of enzymes depends upon the formation by the enzyme and substrate molecules of a complex, which has a definite, though transient, existence, and then decomposes into the products, of the reaction. Note that this point of view was the basis of the discussion of the specilicity of the active sites discussed abuve. 

The result of this approach was a 100-fold increase in the hydrolytic activity of the imprinted polymer compared with the background at pH = 7.6. As a control, another polymer was made using a complex between amidine and benzoate, showing a surprisingly 20-fold increase in the hydrolysis of the substrate. The authors also reported a kinetic investigation of the TSA-imprinted and the benzoate-imprinted polymers, in addition to the free catalyst in solution. Although the ratio substrate/catalyst is not specified, and therefore the steady-state conditions could not be verified, the authors claimed for the two polymers a Michaelis-Menten kinetic behaviour, with a higher profile for the TSA-imprinted polymer. On the other hand, the free catalyst in solution showed, as expected, a linear dependence of the rate from the substrate concentration. The TSA also showed a moderate selectivity towards its own substrate. 

Another approach for the determination of the kinetic parameters is to use the SAS NLIN (NonLINear regression) procedure (SAS, 1985) which produces weighted least-squares estimates of the parameters of nonlinear models. The advantages of this technique are that (1) it does not require linearization of the Michaelis-Menten equation, (2) it can be used for complicated multiparameter models, and (3) the estimated parameter values are reliable because it produces weighted least-squares estimates. 

Michaelis-Menten kinetic parameters can also be estimated by running a series of steady-state CSTR runs with various flow rates and plotting Cs versus (Cst)/(CSq- Cs). Another approach is to use the Langmuir plot (Csr vs Cs) after calculating the reaction rate at different flow rates. The reaction rate can be calculated from the relationship r = F (CSq - Cs)/V. However, the initial rate approach described in Section 2.2.4 is a better way to estimate the kinetic parameters than this method because steady-state CSTR runs are much more difficult to make than batch runs. 

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