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

Enzymes: Protein Catalysts of Cells
Mechanisms of Enzymatic Catalysis
Quantitative Transition State Theory

Back in the 1980s of the 19th century, Arrhenius demonstrated that the Temperature dependence of a chemical reaction rate constant (k) follows the same functional form as the temperature dependence of an Equilibrium Constant, namely

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[integrate equation (3-37) and compare the resulting expression with the one given above]. The quantity E_а is termed the activation energy, and the constant A is the pre-exponential factor. Arrhenius's observations, along with studies on The Effect of salts on reaction rates and the existence of a quantitative relationship between rate constants and equilibrium constants, suggested that the reaction rate constant can be expressed as the product of a temperature-independent constant term and a constant K^≠ possessing The properties of an equilibrium constant for The formation of the Transition State [46]. A quantitative relation of this type was derived by Eyring:

where k_В is the Boltzmann constant and h is Planck's constant. In deriving this relation, Eyring proceeded from the assumption that all transition states decay at the same rate constant, equal to k_BT/h. Equation (6-68) is sometimes used in a modified form containing a transmission coefficient ϰ as a multiplier on the right-hand side, which is believed to be close to unity for most reactions. Expression (6-68) for the rate constant can be rewritten as follows:

where ∆G^≠ is the Free energy of activation. (At 25°С, ∆G^≠ = 5.71 lgk + 73.0 kJ∙mol^-1, with k having the dimension of s^-1.) Relation (6-69) is approximate; a more rigorous relation can be obtained using Statistical Mechanics [46]. It follows from equations (6-68) and (3-14) that

This shows that the quantity ∆H^≠ is approximately equal to the activation energy E_а. For Reactions in Solution, the following more rigorous

¹⁾ The reader will undoubtedly be interested in an excellent collection of papers outlining general approaches to studying the mechanisms of biochemical processes based on Transition State Theory: Transition States of Biochemical Processes, Gandour, R. D. and Schowen, R. L. (Eds.), New York and London, Plenum Press, 1978.— relation holds: ∆H^≠ = E_а — RT. Since the RT term at 25 °С is only 2.5 kJ∙mol^-1, ∆H^≠ is frequently assumed to be equal to E_а. Relation (6-70) demonstrates that the pre-exponential factor in parentheses is governed primarily by ∆S^≠, the Entropy change accompanying the Formation of the transition state. For enzymatic reactions, The values of ∆G^≠, ∆H^≠, and ∆S^≠ are sometimes determined, though extracting useful insights from these values can be quite challenging.

Factors that lead to the stabilization of the transition state (i.e., a decrease in ∆G^≠) enhance the reaction rate. Generally speaking, the very role of a catalyst is to facilitate the formation of a transition state that possesses a lower energy (is more stable) than the transition state of the uncatalyzed reaction. The stabilization of a reaction's transition state by an enzyme implies that the enzyme exhibits a higher affinity for the transition state than for the substrate or products. This concept was first articulated by Pauling [48]: "I think that Enzymes are molecules that are complementary in Structure to the activated complexes to be formed in the reactions being catalyzed. The attraction of the enzyme molecule for the activated complex would thus lead to a decrease in its energy and consequently to a decrease in the activation energy of the reaction and an increase in its rate."