At first it would seem that the Arrhenius equation (5.24) would be all that is needed to describe the temperature effect. In fact, it does hold, but only in the lower range of temperatures associated with life. At higher temperatures another reaction occurs: the dena-turation of the enzyme. One way to handle this is to treat the denaturation reaction as a simultaneous equilibrium. The dissociation equilibrium constant, Kd, is related to temperature by

where AGd, AHd, and ASd are Gibbs free energy, enthalpy, and entropy of deactivation, respectively. For example, the enthalpy and entropy of dissociation for trypsin are 68 kcal/ mol and 213 cal/mol • K, respectively, and the Gibbs free energy for the reaction is 1.97 kcal/mol. With this and a similar relationship for the rate constant, k, in equation (5.36) the following expression for maximum rate in the Michaelis-Menten equation can be derived:

a ~ k[EJi " 1 + exp(ASd/R) exp(-AHd/RT) (5'4/)

where p is a kinetic rate coefficient. This is curve (a) in Figure 5.4.

Figure 5.4 shows this relationship. Note that the side of this curve below the temperature optimum is slightly concave upward. This portion can be approximated empirically as a simple exponential, as shown in Figure 5.4b:

Figure 5.4 Effect of temperature on hydrogen peroxide decomposition by catalase; (a) is equation (5.47) with E = 3.5kcal/mol, AHd = 55.5kcal/mol, ASd = 168kcal/mol • K, p = 258mm3/min; (b) is equation (5.48) fitted to equation (5.47) at 20oC and 25°C; r20 = 185.8 mm3/min, 6 = 1.024. (Based on Bailey and Ollis, 1986.)

Figure 5.4 Effect of temperature on hydrogen peroxide decomposition by catalase; (a) is equation (5.47) with E = 3.5kcal/mol, AHd = 55.5kcal/mol, ASd = 168kcal/mol • K, p = 258mm3/min; (b) is equation (5.48) fitted to equation (5.47) at 20oC and 25°C; r20 = 185.8 mm3/min, 6 = 1.024. (Based on Bailey and Ollis, 1986.)

where T is the temperature in degrees Celsius, r20 the reaction rate at 20°C, and 0 an empirical coefficient. This expression is commonly used to describe the effect of temperature on biological growth rates in biological waste treatment processes. In the example of Figure 5.4, the denominator of equation (5.47) is very close to 1.0 up to a temperature of 40°C. Thus, the numerator carries most of the effect of temperature and the exponential form of (5.48) holds.

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