Biochemistry - The Chemical Reactions of Living Cells Volume 2 - D. Metzler 1980

Enzymes: Protein Catalysts of Cells
Fundamentals of Enzyme Kinetics
Linear Forms of Rate Equations

To determine the parameters K_М and V_mах from experimentally obtained velocity values, equation (6-15) can be transformed into one of its linear forms, for example

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Plotting the double-reciprocal graph (1/[S]; 1/v) (the Lineweaver–Burk plot; Fig. 6-3, A) allows The values of K_М/V_max and 1/V_max to be determined, as they correspond to the slope of the line and its intercept on the ordinate axis, respectively. It is preferable to use the v/[S] versus v plot (the Eadie–Hofstee plot), which is similar to the Scatchard plot (Fig. 4-3).

Although the point with coordinates [S] = 0, v = 0 cannot be plotted on the graph, the ratio v/[S] as v→0 approaches a well-defined value equal to V_max/K_M. Note the distribution of points on the lines shown in Fig. 6-3. Substrate concentrations were chosen such that the increase in velocity from point to point was relatively uniform (which is the goal in experimental design). In the Eadie–Hofstee plot, as in the original v versus [S] plot, the points are distributed quite evenly, whereas in the Lineweaver–Burk plot they are heavily clustered near the origin. A second advantage of the Eadie–Hofstee plot is that it allows points spanning the entire range of possible substrate concentrations—from zero to infinity—to be plotted on a single line.

FIG. 6-3. Linear transformations of the Michaelis–Menten Equation. A. Plot of 1/v versus 1/[S] (double-reciprocal, or Lineweaver–Burk, plot). The intercept on the ordinate axis is equal to 1/V_maх, and the slope is K_M/V_mах. B. Plot of v/[S] versus v (Eadie–Hofstee plot). The slope of the line is equal to –1/K_M, and the intercepts on the ordinate and abscissa axes are equal to V_mах/К_M and V_mах, respectively.

When analyzing data on Metabolic control via the REGULATION OF ENZYMATIC Activity, it is common to use the plot of v versus lg[S] (Fig. 6-4). This plot also covers the full range of possible substrate concentrations (disregarding the point corresponding to [S] = 0, i.e., lg[S] = –∞). Therefore, a single scale can be used for all Enzymes, regardless of the concentration range over which saturation occurs. The dependence of v on lg[S] is sigmoidal both for simple cases described by equation (6-15) and for enzymes exhibiting cooperative substrate binding (Sec. B, 5). In this case, the traditional Classification of curves ("hyperbolic" and "sigmoidal") must be abandoned. Although the degree of cooperativity can be estimated directly from the slope of the v versus lg[S] curve at its midpoint (Ch. 4, Sec. C, 7), determining V_max from this type of plot is inconvenient ¹⁾. Consequently, kinetic parameters are best evaluated using the linear transformation in {v; v/[S]} coordinates (Fig. 6-3, B). Alternatively, computer-based statistical Methods can be employed to calculate both parameters (K_M and V_mах) and generate the curve of best fit through the experimental points (Fig. 6-4). For describing cooperative binding data, computers are virtually indispensable (Sec. B, 5).

¹⁾ More reliable values of K_М and V_mах than those obtained from the Lineweaver–Burk plot are also given by the following linear transformation:

(see Cornish-Bowden A. Introduction/14.html">Principles of Enzyme kinetics, Butterworths, London, 1976, pp. 26, 182; Ainsworth S. Steady-state enzyme kinetics, Macmillan Press, London, 1977, p. 176). — Translator's Note.

FIG. 6-4. Plot of v versus lg[S] for an enzymatic reaction. According to equation (6-44), in the presence of a competitive inhibitor, the entire curve shifts to the right along the abscissa axis, meaning that higher substrate concentrations are required to fully saturate the enzyme.