Mechanisms and Free Energies of Enzymatic Reactions

Abstract

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.

An Integrated Path Integral and Free-Energy Perturbation−Umbrella Sampling Method for Computing Kinetic Isotope Effects of Chemical Reactions in Solution and in Enzymes

“…The nuclear quantum effects are non-negligible even for bond cleavage involving two carbon atoms, which reduce the free energy barrier by 0.45 kcal/mol. The intrinsic 13 C primary KIE has been determined by the formula of eq 25, and the PI-UM method yields a computed KIE of 1.0319 ( 0.8773 at 25°C for the decarboxylation of N-methyl picolinate in water (Table 1). For comparison, the use of the PI-FEP/UM method resulted in a similar KIE (1.0318 ( 0.0028), but the associated statistical error is much smaller.…”

Uma perspectiva computacional sobre catálise enzimática

“…A velocidade máxima de uma reação reflete a diferença entre as energias livres do ponto mais alto (ES ‡ ) e do mais baixo (E+S, no caso do perfil C) ao longo da coordenada de reação. A quantidade de energia livre disponível para catálise pela desestabilização do reagente (∆G des ) também é pequena em comparação com a diferença entre as barreiras da reação catalisada e em solução (∆G ‡ sol 28 e a lisozima são dois exemplos em que o substrato ligado à enzima é forçado a adotar uma conformação diferente daquela mais estável em solução aquosa. A diferença de energia livre entre os confôrmeros pode chegar a 5 kcal/mol, mas esta energia não é necessariamente usada na diminuição de ∆G ‡ enz 18 .…”

Enzymatic catalysis and transfers in solution. I. Theory and computations, a unified view

“…In a reaction between a donor DH and an acceptor A, DH+ A → D + HA, where H can be a H • , H + , or H − , the difference of bond lengths of the dissociating ͑DH͒ bond and the newly forming ͑HA͒ bond is used as a reaction coordinate, e.g., by Gao et al 1 The free energy of formation of the transition state from the reactants is calculated and, with it, the reaction rate. A nuclear tunneling effect on the rate is added.…”

“…This effect of the merging of the two charge distributions in the vicinity of n =1/2 does not appear to have been explored in existing high level quantum chemistry H + , H − transfer calculations, The change of charge distribution along the coordinate n in the vicinity of n =1/2 was noted, in a different language, in Ref. 1. A test of approximation ͑iii͒ is discussed in Sec.…”

“…For example, if one energy "surface" of the reactants has the form 1 2 ͑k a q 2 + k b Q 2 ͒ and products' surface is displaced from it but with the same force constants, 1 2 ͓k a ͑q − a͒ 2 + k b ͑Q − b͒ 2 + ⌬͔, then the energy difference ⌬E total * is k a q a + k b Q b + const. A new coordinate perpendicular to this plane in ͑q , Q͒ space can be chosen, leading from the bottom of reactants' potential energy well to the plane where ⌬E total =0 ͑the TS͒ and then to the well of the products.…”