Doubly Polarized QM/MM with Machine Learning Chaperone Polarizability

Abstract

A major shortcoming of semiempirical (SE) molecular orbital methods is their severe underestimation of molecular polarizability compared with experimental and ab initio (AI) benchmark data. In a combined quantum mechanical and molecular mechanical (QM/MM) treatment of solution-phase reactions, solute described by SE methods therefore tends to generate inadequate electronic polarization response to solvent electric fields, which often leads to large errors in free energy profiles. To address this problem, here we present a hybrid framework that improves the response property of SE/MM methods through high-level molecular-polarizability fitting. Specifically, we place on QM atoms a set of corrective polarizabilities (referred to as chaperone polarizabilities), whose magnitudes are determined from machine learning (ML) to reproduce the condensed-phase AI molecular polarizability along the minimum free energy path. These chaperone polarizabilities are then used in a machinery similar to a polarizable force field calculation to compensate for the missing polarization energy in the conventional SE/MM simulations. Because QM atoms in this treatment host SE wave functions as well as classical polarizabilities, both polarized by MM electric fields, we name this method doubly polarized QM/MM (dp-QM/MM). We demonstrate the new method on the free energy simulations of the Menshutkin reaction in water. Using AM1/MM as a base method, we show that ML chaperones greatly reduce the error in the solute molecular polarizability from 6.78 to 0.03 Å 3 with respect to the density functional theory benchmark. The chaperone correction leads to ∼10 kcal/mol of additional polarization energy in the product region, bringing the simulated free energy profiles to closer agreement with the experimental results. Furthermore, the solute−solvent radial distribution functions show that the chaperone polarizabilities modify the free energy profiles through enhanced solvation corrections when the system evolves from the charge-neutral reactant state to the charge-separated transition and product states. These results suggest that the dp-QM/MM method, enabled by ML chaperone polarizabilities, provides a very physical remedy for the underpolarization problem in SE/MM-based free energy simulations.

Facilitating ab initio QM/MM free energy simulations by Gaussian process regression with derivative observations

“…For error estimation of free energy along the string MFEP, we used a procedure developed by Zhu and Hummer, 70 slightly modified for non-uniform CV grids. 69…”

“…Long-range electrostatics for MM/MM and QM/MM interactions were treated by the particle mesh Ewald (PME) 67 and QM/MM-PME 68 methods, respectively. For the setup of the MD and free energy simulations, we followed our previous procedures, 43,46,69 where we used the bond breaking (C-Cl) and forming (N-C) distances as the CVs to represent the string MFEP; additional simulation details can be found in Section S1 (ESI †). For error estimation of free energy along the string MFEP, we used a procedure developed by Zhu and Hummer, 70 slightly modified for nonuniform CV grids.…”

“…For error estimation of free energy along the string MFEP, we used a procedure developed by Zhu and Hummer, 70 slightly modified for nonuniform CV grids. 69 We implemented the extended-kernel QM-GPR/MM method formulated in Section 2 for MD-based free energy simulations using a combination of GPflow 71 (a GP python library based on Tensorflow 72 ) and CHARMM 59 (see ESI † for additional implementation details). Because Cartesian coordinates cannot be directly used as GPR input features without the loss of rotational and translational invariance of potential energy, we composed our input features using a set of internal coordinates transformed by a scaling function to form the input vector x.…”

Machine-Learning-Assisted Free Energy Simulation of Solution-Phase and Enzyme Reactions

“…Inspired by these approaches to include an MM environment in the development of the MLP, in this work we aimed to develop a more robust protocol for training MLPs/ΔMLPs for free energy simulations of enzyme reactions by incorporating the effects of long-range MM electrostatic interactions, such as under periodic boundary conditions. (In a separate manuscript, a different ai -QM/MM-based machine learning approach was proposed in which the ANN was trained to produce a set of chaperone polarizabilities that augment the insufficient polarizability of the semiempirical Hamiltonian.) Anticipating some amino acid side chains and/or solvent molecules to move in and out of the cutoff boundary with the progression of the reaction, we opted not to follow the “cutoff” approach because the number of MM atoms retained in the cutoff might change, for example, between neighboring umbrella-sampling windows, and thus, additional smoothing , might be needed to ensure a continuous potential energy surface.…”

Affordable Ab Initio Path Integral for Thermodynamic Properties via Molecular Dynamics Simulations Using Semiempirical Reference Potential

“…If a larger QM region is necessary, for instance, when the solvent plays a more critical role than pure background charges polarizing the reaction center, more accurate reference potential is indispensable. Some recent studies have looked into this issue, and some practical solutions have been proposed, including optimizing the semiempirical methods and fitting the (delta) energy using machine learning techniques. ,,,− It is worth emphasizing that with the uncertainties in the free-energy profiles in the present study, it is difficult to quantitatively predict the magnitude of nuclear quantum effect (see Figure S4), and more extended simulations are required.…”