Enzymatic Kinetic Isotope Effects from Path-Integral Free Energy Perturbation Theory

Abstract

Path-integral free energy perturbation (PI-FEP) theory is presented to directly determine the ratio of quantum mechanical partition functions of different isotopologs in a single simulation. Furthermore, a double averaging strategy is used to carry out the practical simulation, separating the quantum mechanical path integral exactly into two separate calculations, one corresponding to a classical molecular dynamics simulation of the centroid coordinates, and another involving free-particle path-integral sampling over the classical, centroid positions. An integrated centroid path-integral free energy perturbation and umbrella sampling (PI-FEP/UM, or simply, PI-FEP) method along with bisection sampling was summarized, which provides an accurate and fast convergent method for computing kinetic isotope effects for chemical reactions in solution and in enzymes. The PI-FEP method is illustrated by a number of applications, to highlight the computational precision and accuracy, the rule of geometrical mean in kinetic isotope effects, enhanced nuclear quantum effects in enzyme catalysis, and protein dynamics on temperature dependence of kinetic isotope effects.

Solvation Induction of Free Energy Barriers of Decarboxylation Reactions in Aqueous Solution from Dual-Level QM/MM Simulations

“… 51 We have developed an interaction energy decomposition scheme to divide the total electrostatic component of the Δ E Xs tr term ( eq 5 ) into permanent electrostatic (or vertical) interaction and polarization terms: 51 , 62 , 72 where the vertical interaction term E Xs o is the energy of transfer for the solute, obtained by keeping its wave function and charge density strictly the same as in the gas phase at the same molecular geometry: Equation 8 is equivalent to the first-order perturbation energy obtained by treating the solute–solvent interaction Hamiltonian as an external perturbation to the gas-phase wave function. 51 , 62 The remaining energy contributions all involve changes in the solute wave function due to intermolecular interactions and are collectively called polarization energy. The polarization term ( eq 9 ) includes a net gain in interaction energy ( ) and a work penalty needed to distort the solute wave function ( ) to create charge (polarization).…”

Dual QM and MM Approach for Computing Equilibrium Isotope Fractionation Factor of Organic Species in Solution

“…Although Equation (9) can be directly used in path integral molecular dynamics or Monte Carlo simulations, in practice, the increased computational costs severely limit the capability of sufficient sampling of phase space. This becomes a major problem for biological systems to compute kinetic and equilibrium isotope effects of enzymatic processes [ 21 , 28 ]. There are two main reasons for this challenge if Equation (9) was directly used.…”

“…Consequently, we obtain directly the ratio of the mass-dependent partition functions, i.e., the equilibrium constant, between the heavy ( H ) and light ( L ) isotopologues: where is the difference in the quantum-mechanical ( qm ) effective potential [ 19 ] between the light and heavy isotopes, indicates an ensemble average over the potential for species L , and with being Boltzmann’s constant and the absolute temperature. We emphasize that the present path integral simulations are carried out for one isotopologue [ 17 , 20 , 21 ]—typically the most abundant one—which is the light particle of oxygen ( 16 O) in the present study (of course, there is no difference if one chooses to carry out the simulation using the heavy particle). Note also that “ L ” and “ H ” in Equation (2) are not limited to a single isotope exchange, but multiple substitutions of different atoms can be simultaneously treated.…”

Deuterium Isotope Effects on Acid-Base Equilibrium of Organic Compounds

“…) through a single simulation, as [35] first described in [34]. This procedure relies on the use of free energy perturbation (FEP) theory [36] to treat the mass of the proton (H) as a perturbation variable to the heavier isotope (D) in path integral sampling (see below):…”