In Michaelis-Menten enzyme kinetics

As in Michaelis-Menten enzyme kinetics, the steady-state assumption of ML is given by... 

The initial reaction rate of a catalyzed reaction versus the concentration of the substrate [>q (Eq. (9.39), where K, =k, /ki). The catalytic reaction could be homogeneous, heterogeneous or enzyme catalysis so long as it follows the simple catalytic mechanism. The substrate concentration, [X]. at a tate of half the maximum reaction rate, V, I2, defines in Michaelis-Menten enzyme kinetics. 

Microbial Biotransformation. Microbial population growth and substrate utilization can be described via Monod s (35) analogy with Michaelis-Menten enzyme kinetics (36). The growth of a microbial population in an unlimiting environment is described by dN/dt = u N, where u is called the "specific growth rate and N is microbial biomass or population size. The Monod equation modifies this by recognizing that consumption of resources in a finite environment must at some point curtail the rate of increase (dN/dt) of the population ... 

An exponential function that describes the increase in product during a first-order reaction looks a lot like a hyperbola that is used to describe Michaelis-Menten enzyme kinetics. It s not. Don t get them confused. If you can t keep them separated in your mind, then just forget all that you ve read, jump ship now, and just figure out the Michaelis-Menten description of the velocity of enzyme-catalyzed reaction—it s more important to the beginning biochemistry student anyway. 

In Box 12.2, a simple model for a special kind of catalyzed reaction, the Michaelis-Menten enzyme kinetics, is presented, which leads to the following kinetic expression ... 

KiMM is given the subscript, MM, to remind us that it reflects Michaelis-Menten enzyme kinetics as distinguished from KiM used above to model microbial growth kinetics (see Monod cases above). Note, is the same as KE in Box 12.2 when it s value represents the reciprocal of the equilibrium constant for the binding step. 

Figure 17.16 Relationships of biodegradation rate, v, to substrate concentration, [/], when Michaelis-Menten enzyme kinetics is appropriate (a) when plotted as hyperbolic relationship (Eq. 17-79 in text), or (b) when plotted as inverse equation, Vv =...

Assuming again that the cycle kinetics are rapid and maintain enzyme and complex in a rapid quasi-steady state, we can obtain the steady state velocity for the reversible Michaelis-Menten enzyme kinetics ... 

A more cogent mathematical treatment of this problem was given in the 1970s by several mathematical biologists. For details see books by Lin and Segel [130] and Murray [146], Here we provide a brief account of this approach. The approach uses the somewhat advanced mathematical method of singular perturbation analysis, but does provides a deep appreciation of the Michaelis-Menten enzyme kinetics. 

Recall that in the standard Michaelis-Menten enzyme kinetics we approximate the kinetics of substrate and product using Equation (3.32) or (4.26) for the essentially irreversible case ... 

A plot of D versus [H+] is shown in Figure 3.4. The graph is hyperbolic, and upon careful examination, it would appear to resemble the Michaelis-Menten enzymes kinetics found in biochemistry (10). The plot in Figure 3.4 as well as Eq. (3.12) show that in the limit as the hydronium ion concentration gets very large, K becomes small in comparison to [H ], and in the limit of a very large hydronium ion concentration, the following can be stated In the limit as... 

In addition to the laws of enzyme microkinetics, the kinetic equations from chemical reactions based on the type 1 situation shown in Fig. 4.12 also provide a suitable approach. The power law equations with various reaction orders, n, differ in the cjt relationship of their reaction components. In Fig. 5.15, the time course of substrate concentration is compared for n = 0,1/2,1, and 2 and for Michaelis-Menten enzyme kinetics. The substrate disappearance and the oxygen utilization in biological waste water treatment may be cited as realistic examples. For the simple case, integration is possible (Levenspiel, 1972). The integrated solutions for various reaction orders are... 

There was proved that all these three conditions may be regarded as equivalent in reducing Haldane-Radic to Michaelis-Menten enzymic kinetics, yet, being the last one a new one added, specific to some mutant in vivo enzyme kinetics of cholinesterases, though the present detailed theoretical and fitting analysis show its sufficiency the envisaged reduction taking place even when the first two conditions do not apply. 

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