Derivation of the Michaelis-Menten equation

For most enzymes, the rate of reaction can be described by the Michaelis-Menten equation which was originally derived in 1913 by Michaelis and MENTEN 21 . Its derivation can be achieved by making one of two assumptions, one of which is a special case of the more general Briggs-Haldane scheme, whilst the alternative is the rapid-equilibrium method given in Appendix 5.3(2 ). 

The steady-state assumption is valid when the concentration ES of the enzyme-substrate complex ES is constant, and when the total enzyme concentration, tot, is small relative to that of the substrate, i.e. tot S. In most cases this assumption holds over a long period of the reaction, as illustrated in Fig. 5.10, which shows the significance of this assumption during reaction processes. 

Since the concentration of the enzyme-substrate complex is constant, its rate of change is zero. That is  

As the enzyme-substrate is formed by one pathway but disappears by two (reversal of formation and product formation), then  

The composite term ((kr] + kf2)/kft) in the above equation is the Michaeiis constant 

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The Km and Vj1iax of the Michaelis-Menten equation are actually made up of sums and products of little k s. You only have to look in most biochemistry texts to see a description of the derivation of the Michaelis-Menten equation in terms of little k s. The little k s are like quarks and leptons—you ve heard the names, but you re not quite sure what they are and even less sure about how they work. There s a section later (actually last) in the book if you haven t heard or can t remember about rate constants. 

In the Briggs-Haldane derivation of the Michaelis-Menten equation, the concentration of ES is assumed to be at steady state, i.e., its rate of production [Eq. (3.12)] is exactly counterbalanced by its rate of dissociation [Eq. (3.13)]. Since the rate of formation of ES from E -(- P is vanishingly small, it is neglected. Equating the two equations and rearranging yields Eq. (3.14), where KM replaces (k2 + h)/k and is known as the Michaelis-Menten... 

Here we develop the basic logic and the algebraic steps in a modern derivation of the Michaelis-Menten equation, which includes the steady-state assumption introduced by Briggs and Haldane. The derivation starts with the two basic steps of the formation and breakdown of ES (Eqns 6-7 and 6-8). Early in the reaction, the concentration of the product, [P], is negligible, and we make the simplifying assumption that the reverse reaction, P—>S (described by k 2), can be ignored. This assumption is not critical but it simplifies our task. The overall reaction then reduces to... 

The initial rate of the enzyme-catalyzed reaction is directly proportional to [S] (Equation El 1.3). Most clinical assays using enzymes are performed under the conditions of Equation El 1.3. From further study of this equation, you will note that also depends on enzyme concentration, since there is an enzyme concentration term hidden in Vmax. (If you have forgotten this, review the derivation of the Michaelis-Menten equation in your biochemistry textbook.) This can be used to advantage, because if a reaction used for a clinical analysis is very slow (it probably will be, since [S] is low), extra enzyme can be used so that the reaction will proceed to completion in a reasonable period of time. 

Appendix 5.3. Derivation of the Michaelis-Menten Equation using the Rapid Equilibrium Assumption... 

Which consideration or assumption does not enter into the derivation of the Michaelis-Menten equation ... 

The Michaelis-Menten equation is often employed in soil-water systems to describe kinetics of ion uptake by plant roots and microbial cells, as well as microbial degradation-transformation of organics (e.g., pesticides, industrial organics, nitrogen, sulfur, and natural organics) and oxidation or reduction of metals or metalloids. Derivation of the Michaelis-Menten equation(s) is demonstrated below. 

The derivation of the Michaelis-Menten equation in the previous section differs from the standard treatment of the subject found in most textbooks in that the quasi-steady approximation is justified based on the argument that the catalytic cycle kinetics is rapid compared to the overall biochemical reactant kinetics. In... 

Reproduce the derivation of the Michaelis-Menten equation in the text. Relate the Michaelis-Menten equation to experimentally derived plots of velocity (V) versus substrate concentration [S]. List the assumptions underlying the derivation. 

Derivation of the Michaelis-Menten equation using the suits of the Mathcad symbolic redactor is given in Fig. 2.24. Here, Km is the Michaelis constant whose physical meaning corresponds to the dissociation constant for the enzyme-substrate complex. Km = k jk. ... 

FIGURE 6.5 Derivation of the Michaelis-Menten equation using the steady-state assuntption... 

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