A Quadrupolar Fullerene Model System for Benchmarking Enhanced Sampling of Trapped Waters in Free Energy Calculations

Enhanced sampling methods, such as hybrid Monte Carlo/molecular dynamics (MC/MD), grand canonical MC/MD, nonequilibrium candidate MC/MD, etc., are widely used to sample the slow rearrangement of interfacial water molecules in binding free energy calculations. However, direct comparison of the accuracies of these methods is oftentimes challenging due to differences in studied systems, simulation parameters, and system setup across studies. To overcome this challenge, we introduce a novel and well-defined model system for benchmarking water sampling methods: a closed, custom-defined C₉₀ fullerene. Our C₉₀ fullerene inherently adopts a distorted oblate-spheroidal geometry rather than being spherical; essentially, C₉₀ is large enough to be almost a miniature, capped nanotube, unlike its C₆₀ "buckyball" sibling. Unlike conventional fullerenes with nonpolar, hydrophobic cavities, our custom-defined fullerene incorporates some level of polarity in the form of modest partial charges, creating a quadrupolar cavity that energetically favors water binding. In our quadrupolar fullerene, positive and negative partial charges are distributed around the polar and the equatorial regions, respectively, on its approximate oblate spheroidal geometry. Due to a significant free energy barrier imposed by the fullerene wall, water exchange between the cavity and bulk solvent is essentially impossible on MD time scales. To allow solvent water to equilibrate in the cavity, we introduce a solvent inlet, by performing Hamiltonian replica exchange (i.e., HREX) simulations, which reveal that the quadrupolar fullerene cavity can accommodate a maximum of two waters. From these HREX simulations, we also identify the water binding sites inside the cavity and their occupancies, which show that fullerene conformations with 2 or 1 waters inside the cavity are dominant, with nearly equal free energies. We further validate a nonequilibrium switching (NES) protocol for computing water displacement free energies, by removing a water from the conformations with 2 waters inside the cavity and comparing with the relevant free energies from the HREX simulations. Our NES free energies agree within statistical error with those obtained from HREX. Our findings establish the quadrupolar fullerene as a well-defined, reproducible test system for rigorously benchmarking current as well as future enhanced sampling techniques targeting slow water sampling in confined environments.