Michaelis-Menten type

Under physiological conditions, [S] is seldom saturating, and itself is not particularly informative. That is, the in vivo ratio of [S]/A , usually falls in the range of 0.01 to 1.0, so active sites often are not filled with substrate. Nevertheless, we can derive a meaningful index of the efficiency of Michaelis-Menten-type enzymes under these conditions by employing the following equations. As presented in Equation (14.23), if... 

Each of the processes shown in Figure 2.8 can be described by a Michaelis-Menten type of biochemical reaction, a standard generalized mathematical equation describing the interaction of a substrate with an enzyme. Michaelis and Men ten realized in 1913 that the kinetics of enzyme reactions differed from the kinetics of conventional... 

As with the Langmuir adsorption isotherm, which in shape closely resembles Michaelis-Menten type biochemical kinetics, the two notable features of such reactions are the location parameter of the curve along the concentration axis (the value of Km or the magnitude of the coupling efficiency factor) and the maximal rate of the reaction (Vmax). In generic terms, Michaelis-Menten reactions can be written in the form... 

The operational model allows simulation of cellular response from receptor activation. In some cases, there may be cooperative effects in the stimulus-response cascades translating activation of receptor to tissue response. This can cause the resulting concentration-response curve to have a Hill coefficient different from unity. In general, there is a standard method for doing this namely, reexpressing the receptor occupancy and/or activation expression (defined by the particular molecular model of receptor function) in terms of the operational model with Hill coefficient not equal to unity. The operational model utilizes the concentration of response-producing receptor as the substrate for a Michaelis-Menten type of reaction, given as... 

Shimidzu etal.111 studied the catalytic activity of poly (4(5)-vinylimidazole-co-acrylic add) 60 (PVIm AA) in hydrolyses of 3-acetoxy-N-trimethylanilinium iodide 61 (ANTI) and p-nitrophenylacetate 44 (PNPA). The hydrolyses of ANTI followed the Michaelis-Menten-type kinetics, and that of PNPA followed the second-order kinetics. Substrate-binding with the copolymer was strongest at an imidazole content of 30 mol%. The authors concluded that the carboxylic acid moiety not... 

The remarkable inertness of dialkyl ethers to one-equivalent oxidants is good evidence that the readier oxidation of alcohols involves more than simple electron abstraction. Di-isopropyl ether is oxidised by Co(III) in CH3CN-H2O mixtures with complicated kinetics individual runs show first-order decay of Co(III) but the rate coefficients increase with increasing [Co(III)], and the order with respect to substrate is less than one but is neither fractional nor of a Michaelis-Menten type. The main product is acetone and the following reaction sequence is proposed... 

Cyclodextrins as catalysts and enzyme models It has long been known that cyclodextrins may act as elementary models for the catalytic behaviour of enzymes (Breslow, 1971). These hosts, with the assistance of their hydroxyl functions, may exhibit guest specificity, competitive inhibition, and Michaelis-Menten-type kinetics. All these are characteristics of enzyme-catalyzed reactions. 

Similarly, for the steady-state situation with one Michaelis-Menten type of uptake site, the Best equation (17) still applies, now with a bioconversion capacity given by ... 

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. 

Most problems associated with approximate kinetics are avoided when Michaelis Menten-type rate equations are utilized. Though this choice sacrifices the possibility of analytical treatment, reversible Michaelis Menten-type equations are straightforwardly consistent with fundamental thermodynamic constraints, have intuitively interpretable parameters, are computationally no more demanding than logarithmic functions, and are well known to give an excellent account of biochemical kinetics. Consequently, Michaelis Menten-type kinetics are an obvious choice to translate large-scale metabolic networks into (approximate) dynamic models. It should also be emphasized that simplified Michaelis Menten kinetics are common in biochemical practice almost all rate equations discussed in Section III.C are simplified instances of more complicated rate functions. 

The immobilization of flavin in the cationic polysoaps [57] also facilitates flavin-mediated oxidation reactions (Shinkai et al., 1978b,c, 1980b). Interestingly, [57] oxidizes NADH according to Michaelis-Menten-type kinetics... 

The expression for the effectiveness factor q in the case of zero-order kinetics, described by the Michaelis-Menten equation (Eq. 8) at high substrate concentration, can also be analytically solved. Two solutions were combined by Kobayashi et al. to give an approximate empirical expression for the effectiveness factor q [9]. A more detailed discussion on the effects of internal and external mass transfer resistance on the enzyme kinetics of a Michaelis-Menten type can be found elsewhere [10,11]. 

A Michaelis-Menten type graph for an allosteric enzyme shows not the usual hyperbolic shape as shown in Section 1.4, but a sigmoidal relationship between [S] and activity. 

Figure 4. Michaelis-Menten type plot of percent of contractile activity versus the histamine substrate concentration and the toxin concentration in the saline solution.

Determine whether these data can be reasonably fitted by a kinetic equation of the Michaelis-Menten type, or... 

In all cases an enzymic process is composed of several consecutive reaction steps. Even the simplest Michaelis-Menten type rapid equihbrium mechanism involves two steps, the binding of the substrate, S, to a specific site in the active centre, and the chemical transformation of the bound S to product P, during which the enzyme becomes free again. The Michaelis constant characterizes the affinity of the enzyme to its... 

The analytical solutions of these equations are easily obtained for first-order or constant-order reactions, but numerical solutions are required for Michaelis-Menten type reactions. The above equations, in these cases, are usually rewritten in terms of dimensionless variables. For a spherical pellet this equation is ... 

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. 

Example 5.5 Fitting a Michaelis - Menten type kinetic model... 

A. Holmberg, On the practical identifiability of microbial growth models incorporating Michaelis-Menten type Nonlinearities,... 

Similarly, for enzyme-catalyzed reactions of the Michaelis-Menten type, we can derive Equation 7.3 from Equation 3.31. 

For the irreversible first-order reaction and the Michaelis-Menten type reac-... 

Figure 7.2 Volume ratios of CSTR to PFR for first, second order, and Michaelis-Menten type reactions.

Apparent reaction rates with immobilized enzyme particles also decrease due to the mass transfer resistance of reactants (substrates). The Thiele modulus of spherical particles of radius R for the Michaelis-Menten type reactions is given as... 

Consider an idealized simple case of a Michaelis-Menten type bioreaction taking place in a vertical cylindrical packed-bed bioreactor containing immobilized enzyme particles. The effects of mass transfer within and outside the enzyme particles are assumed to be negligible. The reaction rate per dilfcrential packed height (m) and per unit horizontal cross-sectional area of the bed (m ) is given as (cf. Equation 3.28) ... 

These kinetic expressions now enable us to understand what chemical and biological information is imbedded within the Michaelis-Menten type formulation typically used to describe such enzyme kinetics (Eq. 17-79 or 17-81 Fig. 17.16). First, we see that K,MM is given by ... 

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 ... 

Consider a chemical in a completely mixed reactor with a constant water exchange rate Q and volume V. The chemical is produced by a Michaelis-Menten type of auto-catalytic reaction (see Eq. 12-26) ... 

The total transit time is longer because of the third step. If the reciprocals of equation 3.68 are taken, a Michaelis-Menten-type equation is generated. 

In general, the concentration of dNTPf follows Michaelis-Menten-type kinetics. If the partitioning ratio equals the fraction F of inserted intermediate giving products, then F is of the form... 

In concomitance with the displacement observed by i.r., an evolution of the catalytic activity has been observed while studying the liquid-phase epoxidation of cyclohexene in the presence of (EGDA)- Mo(VI), freshly prepared or after four months of conditioning at room temperature under inert atmosphere. As usual, the appearance of epoxide was followed by gas chromatographic analyses or by direct titration of oxirane oxygen and the disappearance of hydroperoxide was monitored by iodometric titration. In figure we report concentration-time for typical runs in ethylbenzene at 80°C obtained with the experimental procedure already described (ref. 9). It may be seen that with a freshly prepared catalyst an induction period is observed which lowers the initial catalytic activity. Our modified Michaelis-Menten type model equation (ref. 9) cannot adequately fit the kinetic curves obtained due to the absence of kinetic parameters which account for the apparent initial induction period (see Figure). 

Even this scheme represents a complex situation, for ES can be arrived at by alternative routes, making it impossible for an expression of the same form as the Michaelis-Menten equation to be derived using the general steady-state assumption. However, types of non-competitive inhibition consistent with the Michaelis-Menten type equation and a linear Linweaver-Burk plot can occur if the rapid-equilibrium assumption is valid (Appendix S.A3). In the simplest possible model, involving simple linear non-competitive inhibition, the substrate does not affect the inhibitor binding. Under these conditions, the reactions... 

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