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the term Vmax may be substituted for k2[E]0; in addition, the constant Km is used to represent (k 1 k2)/k1, so that Eq. 2.16 simpli es to the standard form of the Michaelis Menten Equation (Eq. 2.17): n Vmax S = Km S 2:17
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Note that the Michaelis Menten constant, Km , is related to the dissociation constant for the enzyme substrate complex, Kd ; in fact, Km will always be >Kd , but will approach Kd as k2 approaches 0. It is also noteworthy that when S Km , n Vmax =2; in other words, the reaction rate is half-maximal when the substrate concentration is equal to Km . There are two simpli cations of the Michaelis Menten equation that are of tremendous analytical importance. The rst occurs at low substrate concentration: if S ( Km , then Km S % Km . Under these conditions, Eq. 2.17 simpli es to Eq. 2.18: n Vmax S =Km Constant S 2:18
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The initial rate of reaction is therefore directly proportional to initial substrate concentration at low [S], and can be used to quantitate substrate. This is shown in the initial linear region of the plot of reaction rate against substrate concentration in Figure 2.5, where the slope in this region is equal to Vmax =Km . The second simpli cation of Eq. 2.17 occurs at high substrate concentration. If the substrate concentration greatly exceeds Km , then S = Km S % 1 and n % Vmax . Under these conditions, n Vmax k2 E 0 2:19
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The initial reaction rate is now independent of [S], as shown in the plateau region of Figure 1.5. The rate now depends linearly on enzyme concentration, and a plot of initial rate against total enzyme concentration will have a slope of k2 . Thus, at high substrate concentrations, initial reaction rates can be used to determine enzyme concentrations. In practice, Eq. 2.18 may be used if S < 0:1Km , while Eq. 2.19 is useful if S > 10 Km . Remember that the steady-state assumption underlying these expressions is only valid if S ) E 0 , and that, in practice, substrate concentrations in excess of 103 E 0 are used. 2.5.2. Experimental Determination of Michaelis Menten Parameters Equations 2.18 and 2.19 show that the two regions of analytical utility in the rate versus substrate concentration pro le of an enzyme occur at S < 0:1 Km (for substrate quantitation) and S > 10 Km (for enzyme quantitation). In developing a new assay, or in adapting an established assay to new conditions, it is thus important to establish the Km value of the enzyme. It is of practical importance to also establish
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Figure 2.7. Eadie Hofstee plot for n measurements with varying [S].
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and adjust Vmax , so that an assay may be accomplished in the minimum time required to yield a given precision. Four graphical methods may be used to establish Km and Vmax values under given experimental conditions, called the Eadie Hofstee, Hanes, Lineweaver Burk, and Cornish Bowden Eisenthal methods. All four involve the measurement of initial rates of reaction as a function of initial substrate concentration, at constant enzyme concentration. 2.5.2.1. Eadie Hofstee Method.5,6 The Michaelis Menten equation (Eq. 2.17) can be algebraically rearranged to yield Eq. 2.20. n Vmax Km n= S 2:20
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A plot of n (as y) against n/[S] (as x) will yield, after linear regression, a y intercept of Vmax and a slope of Km (Fig. 2.7). This plot is the preferred linear regression method for determining Km and Vmax , since precision and accuracy are somewhat better than those obtained using the Hanes plot, and much better than those found using the Lineweaver Burk method. 2.5.2.2. Hanes Method.7 A different rearrangement of the Michaelis Menten equation yields the form that is used in the Hanes plot: S =n Km =Vmax S =Vmax 2:21
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In the Hanes plot (Fig. 2.8), S =n (as y) is plotted against [S], yielding a slope of 1=Vmax and a y intercept of Km =Vmax following linear regression. The parameter Km is then calculated as the y intercept divided by the slope.
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