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Figure 2.8. Hanes plot for n measured as a function of [S].
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2.5.2.3. Lineweaver Burk Method.8 The Michaelis Menten equation may be algebraically rearranged to Eq. 2.22, yielding a third linear plot, called the Lineweaver Burk or double-reciprocal plot: 1=n 1=Vmax Km = Vmax S 2:22
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The reciprocal rate 1=n is plotted against the reciprocal substrate concentration 1/ [S], yielding a straight line with a positive slope of Km =Vmax and a y intercept of 1=Vmax . The parameter Km is then calculated from the linear regression values of the slope divided by the y intercept (Fig. 2.9). While the Lineweaver Burk plot is
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Figure 2.9. Lineweaver Burk plot for n measured as a function of [S].
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the most commonly used graphical method for the determination of Michaelis Menten parameters, it yields the poorest accuracy and precision in these values. The main reason for this is that, since it is a reciprocal plot, the smallest [S] and n values yield the largest values on the plot. These small values have the greatest relative imprecision, yet standard linear regression programs assume constant uncertainty in all y values (1=n) plotted. One approach to minimizing this problem uses weighted linear regression, where these imprecise points obtained at lower concentrations are given less signi cance in the calculation of slope and intercept values. The popularity of the Lineweaver Burk plot results from its use as a diagnostic plot for enzyme inhibitors, a topic discussed in a Section 2.7. 2.5.2.4. Cornish Bowden Eisenthal Method.9 This method is distinct from the previous three linear regression methods, in that each pair of n; S values is used to construct a separate line on a plot in which Vmax and Km form the y and x axes, respectively. Beginning with another version of the Michaelis Menten equation, in which Vmax is the y value and Km is the x value as shown in Eq. 2.23, Vmax n n= S Km ; 2:23
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it can be seen that each n; S data point will yield a unique y intercept (n) and slope (n/[S]). These values de ne a unique line on the Vmax versus Km plot. Each (n/[S]) pair will de ne a different line, but in the absence of experimental uncertainty, all of these lines will intersect at an identical point that de nes the Vmax and Km for the enzyme studied. This type of plot is shown in Figure 2.10. In practice, the lines do not pass through a single unique point as shown in Figure 2.10, but instead a cluster of intersection points is observed. All of the intersection points are then used to calculate average Km and Vmax values. This unusual method has been shown to yield the best accuracy and precision for statistically treated model data sets.
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Figure 2.10. Cornish Bowden Eisenthal plots obtained with four n; S data pairs.
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TABLE 2.3. Error-Free Michaelis Menten Data for Method of Comparison [S] 0.25 0.50 0.75 1.00 1.25 1.50 1.75 n 0.200 0.333 0.428 0.500 0.556 0.600 0.636
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2.5.3. Comparison of Methods for the Determination of Km Values10 The precisions and accuracies of the four methods described above for the determination of Km values were compared statistically by rst generating an error-free data set of n and [S] data, and then introducing different types of error to the n values to generate a total of 100 individual data sets. The error-free data set was generated using Eq. 2.17 with Vmax Km 1, and the seven individual values are given below in Table 2.3. Fifty data sets containing absolute errors were then generated by adding random numbers of mean zero and standard deviation 0.05 to each n value. Another 50 data sets were generated to contain relative error, by multiplying each n value by a random number of mean one and standard deviation 0.10. These 100 data sets were then analyzed individually by each of the four methods, to generate 100 Km values for each method. The average Km value and its uncertainty for each method and each error type are summarized in Table 2.4. The values summarized in Table 2.4 show that, while all four methods give accurate Km values (in all cases, Km 1:00 is within the uncertainties reported), the
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TABLE 2.4. Mean Estimates of Km (True Value 1.00) for Data with Different Error Types Error Type Absolute Relative 1.13 0.82 1.05 0.46 0.93 0.46 0.79 0.31 1.09 0.43 1.05 0.33 0.88 0.31 0.94 0.29
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Method Lineweaver Burk (1=n vs. 1/[S]) Hanes ( S =n vs. [S]) Eadie Hofstee (n vs. n/[S]) Cornish Bowden Eisenthal (Vmax vs. Km )
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