Biochemical Engineering Fundamentals, Part 1 - Bailey J., Ollis D. 1989

Kinetics of Enzyme-Catalyzed Reactions
Kinetics of Simple Single- and Two-Substrate Enzyme Reactions
Determination of Parameters in the Michaelis-Menten Equation

In its original form, the Michaelis–Menten Equation (3.3) is inconvenient for determining the kinetic parameters v_mах and K_m. As shown in Fig. 3.7, it is quite difficult to determine the exact value of vmax from the plot of v versus s. However, through a series of simple transformations of equation (3.3), it is easy to obtain the equations below, which are more suitable for presenting results in graphical form and for the graphical determination of parameters:

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Each of the above equations represents a linear relationship between two variables. At the same time, when determining kinetic parameters from such relationships, several inherent limitations and disadvantages must be considered. In the graphical representation of equation (3.14) as a plot of 1/v versus 1/s (known as the Lineweaver–Burk plot), the dependent and independent variables are clearly separated (Fig. 3.11, a). The most accurately determined reaction rates, which are close to v_max, cluster near the origin, whereas the less accurately determined values are far from it and thus have the greatest influence on the slope of the line Km/v_mах; therefore, the method of least squares should not be applied in such cases. The second equation (3.15) provides a more even distribution of large v values and, consequently, a more accurate Determination of the slope of the line, 1/v_mах; however, the intercept on the coordinate axis is often too small, making it difficult to determine Кm with sufficient accuracy by this method. The third method uses the so-called Eadie–Hofstee plot, which is a plot of v versus v/s (Fig. 3.11, b); here, the least accurately determined variable, v, is part of both the function and the argument.

FIG. 3.11. a — experimental results of the enzymatic activity of Pepsin in Lineweaver–Burk coordinates; b — experimental data on the Chymotrypsin-catalyzed Hydrolysis of methyl dihydrocinnamate in Eadie–Hofstee coordinates. [Reproduced with permission from: Laidler K. J., The Chemical Kinetics of Enzyme Action, pp. 65—66, The Clarendon Press, Oxford, 1958.]

The above considerations suggest the following strategy for determining v_max and К_m: first, v_mах is determined from the plot of equation (3.14) (by finding the exact value of the intercept on the 1/v axis) or equation (3.15) (by accurately determining the slope of the line). Then, returning to the plot of v versus s, one finds s¹/₂, i.e., the Substrate Concentration at which v equals v_maх/2. As previously mentioned in the Structure/133.html">Discussion of equation (3.3), Кm is equal in magnitude and dimension to s ¹/₂.

It is important to have an idea of the ranges in which these kinetic parameters can vary. Table 3.4 lists A number of parameters characteristic of various Enzymes. Note that the value of k₂ can vary over a very wide range, whereas К_m, by contrast, typically has values of 2∙10^-3—10∙10^-3 M. Almost all the data presented in Table 3.4 were obtained at moderate temperatures and near-neutral pH. The exception is pepsin, whose primary biological function is the Hydrolysis of Proteins in the acidic environment of The Stomach, and which is therefore most active at low pH values; these values, naturally, were also used in the experimental determination of its kinetic parameters. Models reflecting The Effect of pH and Temperature on the Kinetics of Enzymatic reactions will be discussed in Section 3.6.