Multiscale Free Energy Simulations: An Efficient Method for Connecting Classical MD Simulations to QM or QM/MM Free Energies Using Non-Boltzmann Bennett Reweighting Schemes

Abstract

THE RELIABILITY OF FREE ENERGY SIMULATIONS (FES) IS LIMITED BY TWO FACTORS: (a) the need for correct sampling and (b) the accuracy of the computational method employed. Classical methods (e.g., force fields) are typically used for FES and present a myriad of challenges, with parametrization being a principle one. On the other hand, parameter-free quantum mechanical (QM) methods tend to be too computationally expensive for adequate sampling. One widely used approach is a combination of methods, where the free energy difference between the two end states is computed by, e.g., molecular mechanics (MM), and the end states are corrected by more accurate methods, such as QM or hybrid QM/MM techniques. Here we report two new approaches that significantly improve the aforementioned scheme; with a focus on how to compute corrections between, e.g., the MM and the more accurate QM calculations. First, a molecular dynamics trajectory that properly samples relevant conformational degrees of freedom is generated. Next, potential energies of each trajectory frame are generated with a QM or QM/MM Hamiltonian. Free energy differences are then calculated based on the QM or QM/MM energies using either a non-Boltzmann Bennett approach (QM-NBB) or non-Boltzmann free energy perturbation (NB-FEP). Both approaches are applied to calculate relative and absolute solvation free energies in explicit and implicit solvent environments. Solvation free energy differences (relative and absolute) between ethane and methanol in explicit solvent are used as the initial test case for QM-NBB. Next, implicit solvent methods are employed in conjunction with both QM-NBB and NB-FEP to compute absolute solvation free energies for 21 compounds. These compounds range from small molecules such as ethane and methanol to fairly large, flexible solutes, such as triacetyl glycerol. Several technical aspects were investigated. Ultimately some best practices are suggested for improving methods that seek to connect MM to QM (or QM/MM) levels of theory in FES.

Figure 1

Workflow for using non-Boltzmann Bennett in the hybrid QM/MM free energy simulation approach.

Scheme 1. Typical Thermodynamic Cycle Used in Indirect Free Energy Calculations

Figure 2

Illustration of the QM-NBB scheme applied to indirect alchemical FES (i.e., reweighting only the end states). Gray nodes represent simulated states, and white nodes are virtual states that are generated through reweighting (thin arrows). Except for the first and the last free energy step, all free energy calculations are performed with regular BAR (eq 3); i.e., without reweighting. The first and the last free energy calculation use NBB to calculate the free energy difference between a virtual QM state and a simulated MM state.

Figure 3

Dual topology setup of a mutation from ethane to methanol. Starting from the hybrid molecule (middle), it is possible to calculate the potential energy of both end states by ignoring all atoms corresponding to the other end state. The system is divided into three groups: The common environment that is present in both end states (black); atoms that only exist in the ethane initial state (blue); and atoms that only exist in the methanol final state (red). The last two groups do not interact with each other.

Figure 4

Sampling of butane’s conformational space. The x-axis is butane’s central dihedral angle while the y-axis is probability.