Steady-state analysis, Michaelis-Menten equation

Although the Michaelis-Menten equation is applicable to a wide variety of enzyme catalyzed reactions, it is not appropriate for reversible reactions and multiple-substrate reactions. However, the generalized steady-state analysis remains applicable. Consider the case of reversible decomposition of the enzyme-substrate complex into a product molecule and enzyme with mechanistic equations. 

Here k2 and also /c4 and k5, are second-order rate constants. The release of product, as determined by /c4 and k5, may be rate-limiting. At zero time the reverse reactions may be ignored, and steady-state analysis shows that the Michaelis-Menten equation (Eq. 9-16b) will be replaced by Eq. 9-39. Here, D is a constant and A is also constant if X is present at a fixed concentration. 

The linear response range of the glucose sensors can be estimated from a Michaelis-Menten analysis of the glucose calibration curves. The apparent Michaelis-Menten constant KMapp can be determined from the electrochemical Eadie-Hofstee form of the Michaelis-Menten equation, i = i - KMapp(i/C), where i is the steady-state current, i is the maximum current, and C is the glucose concentration. A plot of i versus i/C (an electrochemical Eadie-Hofstee plot) produces a straight line, and provides both KMapp (-slope) and i (y-intercept). The apparent Michaelis-Menten constant characterizes the enzyme electrode, not the enzyme itself. It provides a measure of the substrate concentration range over which the electrode response is approximately linear. A summary of the KMapp values obtained from this analysis is shown in Table I. 

This is an important question that has been addressed by many enzyme kineticists over the years. For the correct application of the Briggs-Haldane steady-state analysis, in a closed system, [S]0 must be >[E](), where the > sign implies a factor of at least 1,000. M. F. Chaplin in 1981 noted that the expression v0= V max[S]c/(Km + [S]0 + [E]0) yields, for example, only a 1 percent error in the estimate of v() for [S]0 = 10 x [E]0 and [S]0 = 0.1 Km the expression thus applies under much less stringent conditions than does the simple Michaelis-Menten equation. In open systems [S]0 can approximate [E]0 and a steady state of enzyme-substrate complexes can pertain computer simulation of both types of system is the best way to gain insight into the conditions necessary for a steady state of the complex. 

The Michaelis-Menten equation (8.8) and the irreversible Uni Uni kinetic scheme (Scheme 8.1) are only really applicable to an irreversible biocatalytic process involving a single substrate interacting with a biocatalyst that comprises a single catalytic site. Hence with reference to the biocatalyst examples given in Section 8.1, Equation (8.8), the Uni Uni kinetic scheme is only really directly applicable to the steady state kinetic analysis of TIM biocatalysis (Figure 8.1, Table 8.1). Furthermore, even this statement is only valid with the proviso that all biocatalytic initial rate values are determined in the absence of product. Similarly, the Uni Uni kinetic schemes for competitive, uncompetitive and non-competitive inhibition are only really applicable directly for the steady state kinetic analysis for the inhibition of TIM (Table 8.1). Therefore, why are Equation (8.8) and the irreversible Uni Uni kinetic scheme apparently used so widely for the steady state analysis of many different biocatalytic processes A main reason for this is that Equation (8.8) is simple to use and measured k t and Km parameters can be easily interpreted. There is only a necessity to adapt catalysis conditions such that... 

The mechanisms of enzyme inhibition fall into three main types, and they yield particular forms of modified Michaelis-Menten equations. These can be derived for single-substrate/single-product enzymic reactions using the steady-state analysis of Sec. 5.10, as follows. 

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