Human Biochemistry, Volume 1 - Murray R. 1993

Structure and Functions of Proteins and Enzymes
Enzymes: Kinetics
Limitations of the Michaelis-Menten Equation

Some Enzymes and other Ligand-binding Proteins, such as Hemoglobin (see chapters 6 and 10), do not obey classical Michaelis—Menten saturation kinetics. In this case, the plot of v versus [S] is sigmoidal in shape (Fig. 8.17). This usually indicates cooperative substrate binding by multiple sites—binding to one site affects binding to another, as is the case with hemoglobin (chapter 6).

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Fig. 8.17. Sigmoidal saturation kinetic curve.

In this case, the graphical method described above for estimating the Substrate Concentration at which the reaction rate is half-maximal becomes inapplicable (the corresponding coordinate plot no longer yields a straight line). Instead, one should turn to the graphical representation of the Hill equation, originally proposed to describe the Cooperative binding of O₂ to hemoglobin (chapter 6). The Hill equation, transformed so that the plot in the appropriate coordinates is a straight line, has the form

where k' is a constant. It follows from the equation that under conditions where [S] is small compared to k', the reaction rate increases as the n-th power of [S]. Fig. 8.18 shows a Hill plot constructed from kinetic data for an enzyme characterized by cooperative substrate binding. The plot of log (v/V_max — v) versus log [S] is a straight line with a slope equal to n, where n is an empirical parameter depending on the number of substrate-binding sites and The Nature of the interaction between them. When n = 1, the binding sites are independent of each other. When n > 1, there is cooperative interaction between the sites; the larger n is, the higher the degree of cooperativity and the more pronounced the sigmoidity of the saturation curves. When n < 1, it is referred to as negative cooperativity.

When the reaction rate is equal to half the maximum (v = V_max/2), v/(V_max—v) = 1 and log [v/(V_max — v)] = 0. Thus, to determine the value of S₅₀ (the substrate concentration at which the rate is half the maximum), one must drop a perpendicular from the point on the line where log [v/(V_max—v)] = 0 onto the x-axis.

Fig. 8.18. Graphical METHOD FOR DETERMINING the substrate concentration at which the reaction rate is half-maximal from the Hill equation under conditions where the kinetic curve is sigmoidal in nature.