EFFECTS OF TEMPERATURE: ENZYMES 757 



the rate of formation of EI is given by: 



d{El) 



dt 



= ^■(E)(I) (15-9) 



Changes in the rates of inhibition with temperature will thus depend on the 

 variations of the rate constant, the enzyme concentration, and the inhibitor 

 concentration with temperature. In most cases the concentrations will 

 not change appreciably with temperature, at least over a physiological 

 range, and only the rate constant need be considered. The rate constant 

 is expressed in transition state theory by: 



k = e-^^H'iRT) e (jS'aR) (15-10) 



Nh 



where h is Planck's constant and J^* and J*S* are the enthalpy and en- 

 tropy of activation for the formation of the EI complex. The transmission 

 coefficient x is assumed to be unity and hence is not included in the equa- 

 tion. 



In order to determine the change in rate constant with temperature we 

 may write: 



ln^,=ln — +lnT, +--^-^^ (15-11) 



7? A S* A 77* 



ln..=ln^+lnr..^-^ (15-12) 



where h-^ and h^ are the rate constants at temperatures T^ and T^- Thus: 



, k, AH* 

 In -^ = 



k^ R 



1 1 



t7 "t; 



-f In ^ (15-13) 



J- 1 



It may be seen that the temperature dependence of the rate constant is 

 related to J//*, the enthapy of activation, just as the temperature depend- 

 ence of the equilibrium constant is related to AH, the over-all enthalpy 

 change in the formation of the EI complex. In the physiological range, an 

 increase in the temperature of 10° will double the rate constant when 

 AH* is about 12 kcal/mole. 



The enthalpy of activation may be obtained by a simple plotting pro- 

 cedure resulting from rewriting Eq. 15-10 in the following form: 



1 ^ 1 ^ ZIS* AH* ,,_,,. 



In = In 1 (15-14) 



T Nh ^ R RT 



when log (k/T) is plotted against 1/T, a straight line with a slope of 

 — (J^*/2.303i?) will be obtained if JH* does not change appreciably 



