Michaelis-Menten structure

The kinetics proposed for local oxygen uptake is of a simple irreversible Michaelis-Menten structure which has been verified for the terminal cytochrome oxidase of the respiration chain (iC, pM). The distribution of oxygen in... 

This is referred to as the Michaelis-Menten structure. All the reaction equations of this system can be represented by... 

We substitute (E. 167) into (E. 165), and Anally the reaction rate of Michaelis-Menten structure is determined as... 

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. 

The dependence (1 TP of v, on ATP is modeled as in the previous section, using an interval C [—00,1] that reflects the dual role of the cofactor ATP as substrate and as inhibitor of the reaction. All other reactions are assumed to follow Michaelis Menten kinetics with ()rs E [0, 1], No further assumption about the detailed functional form of the rate equations is necessary. Given the stoichiometry, the metabolic state and the matrix of saturation parameter, the structural kinetic model is fully defined. An explicit implementation of the model is provided in Ref. [84],... 

A key point should be to identify the rate-limiting step of the polymerization. Several studies indicate that the formation of the activated open monomer is the rate-limiting step. The kinetics of polymerization obey the usual Michaelis-Menten equation. Nevertheless, all experimental data cannot be accounted for by this theory. Other studies suggest that the nature of the rate-limiting step depends upon the structure of the lactone. Indeed, the reaction of nucleophilic hydroxyl-functionalized compounds with activated opened monomers can become the rate-limiting step, especially if stericaUy hindered nucleophilic species are involved. 

Satisfaction of kinetic order. Carriers follow Michaelis-Menten-type saturation kinetics or first-order kinetics. Ion channels follow the type of respective structure—unimolecular transmembrane channels and bimolecular half-channels follow first- and second-order kinetics, respectively. The kinetic order of supramolecular channels depends on the assembly number. However, this principle can be applied only when the association constants are small. If the association becomes strong, the kinetic order decreases down to zero. Then the validity becomes dubious in view of the absolute criterion of the mechanism. Decreased activation energy compared to the carrier transport mechanism and competitive inhibition by added other cations stand as criteria. 

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),... 

Now we consider situations in which transformation of the organic compound of interest does not cause growth of the microbial population. This may apply in many engineered laboratory and field situations (e.g., Semprini, 1997 Kim and Hao, 1999 Rittmann and McCarty, 2001). The rate of chemical removal in such cases may be controlled by the speed with which an enzyme catalyzes the chemical s structural change (e.g., steps 2, 3 and 4 in Fig. 17.1). This situation has been referred to as co-metabolism, when the relevant enzyme, intended to catalyze transformations of natural substances, also catalyzes the degradation of xenobiotic compounds due to its imperfect substrate specificity (Horvath, 1972 Alexander, 1981). Although the term, co-metabolism, may be used too broadly (Wackett, 1996), in this section we only consider instances in which enzyme-compound interactions limit the overall substrate s removal. Since enzyme-mediated kinetics were characterized long ago by Michaelis and Menten (Nelson and Cox, 2000), we will refer to such situations as Michaelis-Menten cases. 

Now we can see the types of biochemical factors that determine the rate constant, fcbio for Michaelis-Menten cases the ability of the enzyme to catalyze the transformation as reflected by the quotient, kE/KiMM, and the presence of enzyme in the microorganism population involved, as quantified by [Enz]tot/[B], In the following section, we develop some detailed kinetic expressions for one case of enzyme-mediated transformations. Examination of these results will help us to see how structural features of xenobiotic compounds may affect rates. Finally, we will improve our ability to understand the relative rates for structurally related chemicals that are transformed by the same mechanism and are limited at the same biodegradation step. 

The most-studied enzyme in this context is chymotrypsin. Besides being well characterized in both its structure and its catalytic mechanism, it has the advantage of a very broad specificity. Substrates may be chosen to obey the simple Michaelis-Menten mechanism, to accumulate intermediates, to show nonproductive binding, and to exhibit Briggs-Haldane kinetics with a change of rate-determining step with pH. 

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. 

In 1997 the same group developed the first imprinted polymer able to catalyse a Diels-Alder reaction between tetrachlorothiophene dioxide (43) and maleic anhydride (44) to give the product (45). The imprinting strategy was inspired by previous work carried out by Hilvert et al. in 1989 for the development of catalytic antibodies with Diels-Alder capabilities [26]. The chlorendic anhydride (46) was used as a template because of its structural analogy with the transition state of the reaction (TSA). The resulting imprinted polymer showed a Michaelis-Menten behaviour and a ratio kcal/kunca equal to 270 (Scheme 8). 

Another interesting example of C-C bond formation can be found in reports from Li and co-workers who, in 2004, described the preparation of an MIP with peroxidase-like activity capable of dimerising the homovanillic acid (HVA) (73) [49]. In this case a polymer was prepared by using the HVA substrate, instead of a TSA, as a template and a haemin unit as catalytic centre (74). The polymerisation was carried out in the presence of acrylamide and vinyl-pyridine in order to add extra functionalities aiding substrate recognition. The imprinted polymer showed an enzyme-like activity, as confirmed by adherence to the Michaelis-Menten saturation model, and it was inhibited by ferulic acid (75), a structural analogue of the substrate, which is also capable of inhibiting the natural peroxidase. 

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 structural submodel describes the central tendency of the time course of the antibody concentrations as a function of the estimated typical pharmacokinetic parameters and independent variables such as the dosing regimen and time. As described in Section 3.9.3, mAbs exhibit several parallel elimination pathways. A population structural submodel to mechanistically cover these aspects is depicted schematically in Fig. 3.14. The principal element in this more sophisticated model is the incorporation of a second elimination pathway as a nonlinear process (Michaelis-Menten kinetics) into the structural model with the additional parameters Vmax, the maximum elimination rate, and km, the concentration at which the elimination rate is 50% of the maximum value. The addition of this second nonlinear elimination process from the peripheral compartment to the linear clearance process usually significantly improves the fit of the model to the data. Total clearance is the sum of both clearance parts. The dependence of total clearance on mAb concentrations is illustrated in Fig. 3.15, using population estimates of the linear (CLl) and nonlinear clearance (CLnl) components. At low concentra-... 

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