Quantum mechanics/molecular mechanics minimum free-energy path for accurate reaction energetics in solution and enzymes: Sequential sampling and optimization on the potential of mean force surface

Abstract

To accurately determine the reaction path and its energetics for enzymatic and solution-phase reactions, we present a sequential sampling and optimization approach that greatly enhances the efficiency of the ab initio quantum mechanics/molecular mechanics minimum free-energy path (QM/MM-MFEP) method. In the QM/MM-MFEP method, the thermodynamics of a complex reaction system is described by the potential of mean force (PMF) surface of the quantum mechanical (QM) subsystem with a small number of degrees of freedom, somewhat like describing a reaction process in the gas phase. The main computational cost of the QM/MM-MFEP method comes from the statistical sampling of conformations of the molecular mechanical (MM) subsystem required for the calculation of the QM PMF and its gradient. In our new sequential sampling and optimization approach, we aim to reduce the amount of MM sampling while still retaining the accuracy of the results by first carrying out MM phase-space sampling and then optimizing the QM subsystem in the fixed-size ensemble of MM conformations. The resulting QM optimized structures are then used to obtain more accurate sampling of the MM subsystem. This process of sequential MM sampling and QM optimization is iterated until convergence. The use of a fixed-size, finite MM conformational ensemble enables the precise evaluation of the QM potential of mean force and its gradient within the ensemble, thus circumventing the challenges associated with statistical averaging and significantly speeding up the convergence of the optimization process. To further improve the accuracy of the QM/MM-MFEP method, the reaction path potential method developed by Lu and Yang [Z. Lu and W. Yang, J. Chem. Phys. 121, 89 (2004)] is employed to describe the QM/MM electrostatic interactions in an approximate yet accurate way with a computational cost that is comparable to classical MM simulations. The new method was successfully applied to two example reaction processes, the classical SN2 reaction of Cl-+CH3Cl in solution and the second proton transfer step of the reaction catalyzed by the enzyme 4-oxalocrotonate tautomerase. The activation free energies calculated with this new sequential sampling and optimization approach to the QM/MM-MFEP method agree well with results from other simulation approaches such as the umbrella sampling technique with direct QM/MM dynamics sampling, demonstrating the accuracy of the iterative QM/MM-MFEP method.

Free Energies of Chemical Reactions in Solution and in Enzymes with Ab Initio Quantum Mechanics/Molecular Mechanics Methods

“…It is even possible to further simplify this QM ESP charge model by truncating the polarization effects of the QM ESP charges [61,62]. Applications with these simplified QM ESP charge models in the ab initio QM/MM simulation of reaction processes have been successful.…”

“…After the perturbation of the QM geometry and the external MM electrostatic potential, the polarized charges are determined through two response kernels, namely [81,82], (7) and [81] (8) The second kernel can now be computed analytically with a recently introduced ESP fitting method [83]. Just as in the polarization effects introduced in the term , we also expand the term E 1 (r QM , r MM ) in a similar manner [81,62].…”

“…Beyond this distance, the MM point charges merely contribute by providing a static electrostatic potential. Therefore a simple technical scheme has been adopted in our simulations with periodic boundary condition [61,62,60] in which a cutoff of 9 ~ 14 Å was used for selecting MM charges for the QM SCF calculation. The QM calculation yielded polarized QM ESP charges that were in turn used to represent the QM subsystem during MD simulations with the long range electrostatic interactions treated by the Particle-mesh Ewald Method [63].…”

Exploring chemical reaction mechanisms through harmonic Fourier beads path optimization

“…Free energy profiles or potentials of mean force (PMFs) along the path are preferred to the potential energy profiles because they can be directly compared with experiment. [58][59][60][61] This is particularly true of the large systems with rugged energy landscapes. However, computing a valid reaction PMF requires conformational sampling and a proper choice of reaction coordinate.…”

“…[58][59][60][61] Importantly, the indirect QM/MM PMF calculations critically depend on reaction paths optimized on QM/MM potential energy surfaces. [59][60][61] For a given configuration of the system, the paths optimized with the combined CG-HFB method should give the best estimate of the reaction barrier and identify the best reaction coordinate.…”