Michaelis-Menten scheme

The Michaelis-Menten scheme nicely explains why a maximum rate, V"max, is always observed when the substrate concentration is much higher than the enzyme concentration (Figure 11.1). Vmax is obtained when the enzyme is saturated with substrate. There are then no free enzyme molecules available to turn over additional substrate. Hence, the rate is constant, Vmax, and is independent of further increase in the substrate concentration. 

The method of derivation used for the Michaelis-Menten scheme gives this rate law ... 

To recognize that Eq. (33) indeed is the overall reaction of the Michaelis Menten scheme, an additional requirement is that the concentration of enzyme-bound substrate is negligible compared to the total substrate concentration. The corresponding differential equations of the irreversible Michaelis Menten scheme can then be simplified to... 

For reversible enzymatic reactions, the Haldane relationship relates the equilibrium constant KeqsNith the kinetic parameters of a reaction. The equilibrium constant Keq for the reversible Michaelis Menten scheme shown above is given as... 

For the Michaelis-Menten scheme involving interaction of enzyme (E) with substrate (S) or product (P) ... 

Another instance where first-order kinetics applies is the conversion of reversible enzyme-substrate complex (ES) to regenerate free enzyme (E) plus product (P) as part of the Michaelis-Menten scheme ... 

This reaction cycle has more steps than the simple Michaelis-Menten scheme. Nonetheless, the steady-state rate equations describing these reaction cycles have indistinguishable functions, and one cannot determine the number of intermediary steps by steady-state kinetics alone. 

At this point, it is instructive to notice that the numerical example given above for the minimal Michaelis-Menten scheme will probably be very relevant to the situation for the majority of the enzymes considered so far. In consequence, it appears that the effects of organic solvents on the enantioselectivity are not restricted to their relative effects on the ground state system and the transition state. Instead, substantial contributions from the diffusional process parameters have to be taken into account as well. Since these contributions are probably better described by the Einstein-Smoluchovski relation,... 

The solution of scheme 3.10 is somewhat more complicated than the solution of the Michaelis-Menten scheme the steady state approximation is applied to the concentration of ES. That is, if the reaction rate measured is approximately constant over the time interval concerned, then [ES] is also constant ... 

The Michaelis-Menten scheme may be extended to cover a variety of cases in which additional intermediates, covalently or noncovalently bound, occur on the reaction pathway. It is found in all examples that the Michaelis-Menten equation still applies, although KM and cataTemqwcombinations of various rate and equilibrium constants. KM is always less than or equal to Ks in these cases. Suppose that, as for example in the following scheme, there are several intermediates and the final catalytic step is slow ... 

In Fig. 2.10, the boundary between the enzyme-containing layer and the transducer has been considered as having either a zero or a finite flux of chemical species. In this respect, amperometric enzyme sensors, which have a finite flux boundary, stand apart from other types of chemical enzymatic sensors. Although the enzyme kinetics are described by the same Michaelis-Menten scheme and by the same set of partial differential equations, the boundary and the initial conditions are different if one or more of the participating species can cross the enzyme layer/transducer boundary. Otherwise, the general diffusion-reaction equations apply to every species in the same manner as discussed in Section 2.3.1. Many amperometric enzyme sensors in the past have been built by adding an enzyme layer to a macroelectrode. However, the microelectrode geometry is preferable because such biosensors reach steady-state operation. 

The simplest mechanism for an enzyme-catalysed reaction (Michaelis-Menten scheme) is... 

Let us give some examples for the graphs of linear mechanisms. The simplest mechanism of an enzyme catalytic reaction is the Michaelis Menten scheme... 

Consider again the simple irreversible Michaelis-Menten scheme ... 

Figure 4.1 The cyclic transformations (A) and Gibbs energy (B) diagrams of the simplest catalytic reaction described by the Michaelis-Menten scheme (1.23)- 1.24).

That the TC group is positioned near the CD moiety in 42 was confirmed by measuring kinetics of deacylation of 43 promoted by 42 [M Cu(II), Ni(II), Cu(II)] and the analogue of 42 prepared by random functionalization. Kinetics of reactions catalyzed by the PEI derivatives followed the Michaelis-Menten scheme. Parameter MK is close to the formation constant for the most stable complex formed between the polymer and 43. For the PEI derivative prepared by site-directed functionalization, 1/AT was (3.7-6.4) x 10 M at 25 °C, being 5.8-S.7 times greater than those for the analogue prepared by random functionalization. This indicates that an extra binding force is present in the complex formed between 43... 

The term f(P)Z in (3.71), when f P) is given by (3.74), leads formally to the Michaelis-Menten dynamics (3.39), if Et is identified with the predator density and P with the substrate. This analogy has been elaborated in the literature. For example Real (1977) describes predator-prey dynamics with the Michaelis-Menten scheme (3.27), with S the prey, C the intermediate state of the prey when it is eaten, E is the predator searching for food and P is the new predator biomass produced during the consumption process, so that Et = E + P is the total amount of predator. This leads to a justification... 

The kinetic data were analyzed in terms of the Michaelis-Menten scheme (Eq. 1). Parameter stands for the reactivity of the catalyst (C) toward the... 

Figure 5. The plot of ka versus Q for cleavage of myoglobin (So = 4.7 pA/) by Co BU at pH 7.5 and 37°C (0.05 M buffer addition of 0.5 M NaCl did not affect the rate data appreciably). Straight lines a (Co < So) and b (Cq > Sq) stand for Vq/Sq (vq initial velocity) and k, respectively, predicted by Michaelis-Menten scheme under the condition of Co 3> Km- [Adapted from Ref. (126).]...

The depicted mechanism is of course much more complex than the simple Michaelis-Menten scheme, but it can be simplified by employing the steady state approximation. This may be done in a relatively simple manner by applying either the King-Altman method or the less well-known Christiansen formalism. Applying the King-Altman method to the catalytic cycle given in Scheme 4.3 leads to a rate equation that is equal to the substi-tuted-enzyme mechanism, the detailed derivation of which was debated by Cornish-Bowden. ... 

Inhibition by the product implies considering the reaction sequence depicted in Sch. 9 in place of Sch. 1. A 2 no longer stands for the one-step formation of the product, R, and the reduced form of the enzyme, RE, but rather the formation of the complex, RER, associating the reduced form of the enzyme with the product. To complete the Michaelis-Menten scheme pertaining to the product, the complex may... 

It is easy to see from this Figure how the Michaelis-Menten scheme could be violated (for example when there are too few empty micelles, or when the concentration of reagents is so low that collisions among micelles become the rate limiting step). Until now, however no deviations from the Michaelis-Menten kinetics have been reported. 

As studies have shown, the observed dependence of the initial rate of enzymatic destruction of the substrate concentration can be described within the Michaelis-Menten scheme. Figure 7.2 shows the dependence of the initial rate of CHT concentration in solution. 

The kinetic scheme is analogous to the Michaelis-Menten scheme for enzyme catalysis. The solvent-dependence of the rate-enhancement ratio can be explained in terms of different values for the initial interaction between substrate and micelle (Mic) measured by the partition coefficient K. The full scheme for a transformation of substrate (S) to product (P) is ... 

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