Kurz equation

The Kurz equation [Eq. (2.8)] has simplified and channeled many discussions in enzymology. Many other explanations of enzyme action and many phenomena observed with enzyme reactions can be reduced to the validity of Eq. (2.8), as shown in the following paragraphs. 

Many enzymatic reaction mechanisms are accelerated or even only made possible by the absence of water or other solvents. The enzyme supports desolvation by redirecting part of the possible binding energy liberated from the binding between enzyme and substrate. 

Upper boundary of rate constant of an enzyme reaction diffusion control 

The upper boundary of the reaction rate is reached when every collision between substrate and enzyme molecules leads to reaction and thus to product. In this case, the Boltzmann factor, exp(-EJRT), is equal to lin the transition-state theory equations and the reaction is diffusion-limited or diffusion-controlled (owing to the difference in mass, the reaction is controlled only by the rate of diffusion of the substrate molecule). The reaction rate under diffusion control is limited by the number of collisions, the frequency Z of which can be calculated according to the Smoluchowski equation [Smoluchowski, 1915 Eq. (2.9)]. 

R is the normalized radius of both participating particles (1 /rx + l/r2), D the normalized diffusion coefficient (1/Dt + 1/D2). The collision frequency Z is then calculated from Eq. (2.10). 

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Important milestones in the rationalization of enzyme catalysis were the lock-and-key concept (Fischer, 1894), Pauling s postulate (1944) and induced fit (Koshland, 1958). Pauling s postulate claims that enzymes derive their catalytic power from transition-state stabilization the postulate can be derived from transition state theory and the idea of a thermodynamic cycle. The Kurz equation, kaJkunat Ks/Kt, is regarded as the mathematical form of Pauling s postulate and states that transition states in the case of successful catalysis must bind much more tightly to the enzyme than ground states. Consequences of the Kurz equation include the concepts of effective concentration for intramolecular reactions, coopera-tivity of numerous interactions between enzyme side chains and substrate molecules, and diffusional control as the upper bound for an enzymatic rate. 

Origin of Enzymatic Activity Derivation of the Kurz Equation... 

Let us return one more time to the Kurz equation [Eq. (2.8)], which is regarded as the quantification of Pauling s postulate that transition-state stabilization, i.e., the tighter binding between enzyme and transition state, expressed by fCT, as compared to binding between enzyme and ground state, expressed by JCS, is the source of catalytic rate enhancement. 

Identifying Ks with KM, KT approximately equals the ratio (kuncat KM)/kcat, which is the inverse of termed the proficiency" which has been determined experimentally for a number of systems (see Section 2.3.4 below). If the Kurz equation captured the whole essence of enzyme catalysis, Kr should be proportional to KM)/kcat. 

A more stringent criterion emanates from the thermodynamic cycle already encountered in Chapter 2 (Figure 2.2). For an inhibitor, which necessarily is an imperfect mimic of the transition state, and thus for which clearly K XT, the relationship kcat = kuncat (XS/XT) [the Kurz equation Chapter 2, Eq. (2.8)], has to be modified to Eq. (9.12), reflecting the proportionality of X, and XT, and setting Xs = XM. 

Yamada el al. (1991) and Yamada and Matsumiya (1992) tried a similar modification but with the composition of the liquid being modified according to Clyne and Kurz (1981). In this case the concentration of the solute, i, in the liquid changes by dCi given by the equation... 

Webb, Occult Establishment, 281-83. For more historical analyses of this now commonplace staple of avant-garde conviction, see Kosinski, 63ff. She does not, however, cite some earlier statements by Eliphas Levi with the same emphatic effect. In any event, the equation Artist as Magician is truly ancient see Kris and Kurz, 69ff. 

The key equation resulting from the application of two known theories, the Bom-Haber cycle and the transition-state theory, was formulated for any catalyst and without reference to enzymes (Kurz, 1963) a good derivation can be found in the article by Kraut (1988). 

The activation parameters for the acid-catalyzed hydrolysis of long chain alkyl sulfates compared to those for non-micellar ethyl sulfate calculated from potentiometric data indicate that the rate acceleration accompanying micellization is primarily a consequence of a decrease in the enthalpy of activation rather than an increase in the entropy (Kurz, 1962). However, the activation energies for the acid-catalyzed hydrolysis of sodium dodecyl sulfate calculated from spectrophotometric data have been reported to be identical (Table 8) for micellar and non-micellar solutions, but the entropy of activation for the hydrolysis of the micellar sulfate was found to be 6 9 e.u. greater than that for the non-micellar system (Motsavage and Kostenbauder, 1963). This apparent discrepancy may be due to the choice of the non-micellar state as the basis of comparison, i.e. ethyl sulfate and non-micellar dodecyl sulfate, to temperature dependent errors in the values of the acid catalyzed rate constant determined potentiometrically, or to deviations in the rate constants from the Arrhenius equation. 

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