element, namely, the choice of the reaction coordinate for determining the free energy of activation to characterize the mechanism of enzymatic processes. Then, we illustrate a variety of factors that have been found to contribute to catalysis in specific enzymatic reactions by lowering the free energy of activation relative to that for the uncatalyzed process in aqueous solution. Finally, we provide a summary of the major conclusions. 2. Methods for Computational Studies of Enzymatic Reactions in Aqueous Solution In this section, we present a brief summary of the theory and key computational techniques that we use for studying chemical reactions catalyzed by enzymes and the corresponding uncatalyzed reactions, both in aqueous solution. 2.1. Generalized Transition State Theory Generalized transition state theory (TST) provides a theoretical framework for understanding chemical reactions in the gas phase, in solution, and in enzymes. Conventional 24,25 and generalized 26 transition state theory were originally developed for gas-phase reactions, but transition state theory is readily generalized to liquid-phase reactions, 27 and it has become the framework for both qualitative and quantitative studies of reactions catalyzed by enzymes. The rate constant for a reaction at temperature T can be conveniently expressed as follows: (1) where β = 1/(k B T), k B being Boltzmann's constant, h is Planck's constant, and k TST is the transition state theory rate constant. The transmission coefficient, γ(T), which has a value of unity in simple transition state theory, has three components, 7 (2) which account for, respectively, dynamical recrossing of the transition state hypersurface that separates the reactants and products, quantum mechanical tunneling in the reaction coordinate, and nonequilibrium distributions in phase space. Note that γ(T), κ(T), and Γ(T) are called, respectively, the transmission coefficient, the tunneling transmission coefficient, and the recrossing transmission coefficient. In eq 1, ΔG ‡ (T) is the molar standard-state quasithermodynamic free energy of activation, which is related to the potential of mean force, W(T,q), also called the PMF, by eq 3, 28,29 (3) where q ‡ and q R are values of the reaction coordinate, q, at the transition state and reactant state, respectively, G R (q) corresponds to the free energy of the mode in the reactant state, R, which correlates with the reaction coordinate, and C(T,q) is a correction term that is due to the Jacobian of the transformation from a locally rectilinear reaction coordinate to the Gao et al.

The H/D primary kinetic isotope effect (KIE) for the hydride transfer reaction catalyzed by Escherichia coli dihydrofolate reductase (ecDHFR) is calculated as a function of temperature employing ensemble-averaged variational transition-state theory with multidimensional tunneling. The calculated KIEs display only a small temperature dependence over the temperature range of 5 to 45 °C. We identify two key features that contribute to canceling most of the temperature dependence of the KIE that would be expected on the basis of simpler models. Related issues such as the isotope effects on Arrhenius preexponential factors, large differences between free energies of activation and Arrhenius activation energy, and fluctuations of effective barriers are also discussed. This paper presents a theoretical explanation of the unusual temperature dependence 1 of the kinetic isotope effect (KIE) for the hydride transfer chemical step of the reaction catalyzed by E. coli dihydrofolate reductase (ecDHFR). The explanation identifies two general features that may be important in interpreting enzymatic kinetic isotope effects more generally.Many enzyme reactions involve hydron transfer (transfer of H + , H − , or H • ). Such reactions have significant contributions from zero-point energy and tunneling; KIEs have been extensively used to study these effects. [1][2][3][4][5][6][7] Surprisingly, a number of enzymes have been found to display almost temperature-independent KIEs, 1-4 which are contrary to experience with small-molecule chemistry or simple tunneling models. Recent multidimensional tunneling calculations have been successfully used to study tunneling effects on enzymatic KIEs, but the most accurate methods have so far been applied only at a single temperature. 8 DHFR catalyzes the reduction of 7,8-dihydrofolate (DHF) to 5,6,7, with the key chemical step being the transfer of a hydride ion from the nicotinamide ring of the cofactor nicotinamide adinine dinucleotide phosphate (NADPH). At pH ≅ 7, product release is partly rate-limiting 9 and the H/D kinetic isotope effect is about 1.1-1.3, but the intrinsic KIE on the hydride transfer step is >3. 10 Furthermore, at 25 °C, the phenomenological free energy of activation derived from the rate constant 9 by using © 2005 American Chemical Society HHS Public Access Author Manuscript Author ManuscriptAuthor ManuscriptAuthor Manuscript transition-state theory is 13.4 kcal/mol, 7 and the free energy of reaction calculated from the experimental equilibrium constant 9 is −4.4 kcal/mol. 7Here, we report a dynamical simulation of the ecDHFR system using a combined quantum mechanical and molecular mechanical (QM/MM) approach. As depicted in Figure 1, the reactive fragment that involves transferring a hydride ion from the C4 position of NADPH (the cofactor) to the C6 in N5-preprotonated DHF (the substrate) to form THF (the product) is treated quantum mechanically. The hydride transfer KIEs at 5 and 45 °C and 1 atm are calculated using ensemble-averaged variational transition-stat...

It has been suggested that the magnitudes of secondary kinetic isotope effects (2 degrees KIEs) of enzyme-catalyzed reactions are an indicator of the extent of reaction-center rehybridization at the transition state. A 2 degrees KIE value close to the corresponding secondary equilibrium isotope effects (2 degrees EIE) is conventionally interpreted as indicating a late transition state that resembles the final product. The reliability of using this criterion to infer the structure of the transition state is examined by carrying out a theoretical investigation of the hybridization states of the hydride donor and acceptor in the Escherichia coli dihydrofolate reductase (ecDHFR)-catalyzed reaction for which a 2 degrees KIE close to the 2 degrees EIE was reported. Our results show that the donor carbon at the hydride transfer transition state resembles the reactant state more than the product state, whereas the acceptor carbon is more productlike, which is a symptom of transition state imbalance. The conclusion that the isotopically substituted carbon is reactant-like disagrees with the conclusion that would have been derived from the criterion of 2 degrees KIEs and 2 degrees EIEs, but the breakdown of the correlation with the equilibrium isotope effect can be explained by considering the effect of tunneling.

Molecular dynamics simulations using a combined QM/MM potential have been performed to study the catalytic mechanism of human cathepsin K, a member of the papain family of cysteine proteases. We have determined the two-dimensional free energy surfaces of both acylation and deacylation steps to characterize the reaction mechanism. These free energy profiles show that the acylation step is rate limiting with a barrier height of 19.8 kcal/mol in human cathepsin K and of 29.3 kcal/mol in aqueous solution. The free energy of activation for deacylation step is 16.7 kcal/mol in cathepsin K and 17.8 kcal/mol in aqueous solution. The reduction of free energy barrier is achieved by stabilization of the oxyanion in the transition state. Interestingly, although the "oxyanion hole" has been formed in the Michaelis complex, the amide units do not donate hydrogen bonds directly to the carbonyl oxygen of the substrate, but they stabilize the thiolate anion nucleophile. Hydrogen bonding interactions are induced as the substrate amide group approaches the nucleophile, moving more than 2 angstroms and placing the oxy-anion in contact with Gln19 and the backbone amide of Cys25. The hydrolysis of peptide substrate shares a common mechanism both for the catalyzed reaction in human cathepsin K and the uncatalyzed reaction in water. Overall, the nucleophilic attack by Cys25 thiolate and the proton transfer reaction from His162 to the amide nitrogen are highly coupled, whereas a tetrahedral intermediate is formed along the nucleophilic reaction pathway.