Adaptive free energy sampling in multidimensional collective variable space using boxed molecular dynamics

The past decade has seen the development of a new class of rare event methods in which molecular configuration space is divided into a set of boundaries/interfaces, and then short trajectories are run between boundaries. For all these methods, an important concern is how to generate boundaries. In this paper, we outline an algorithm for adaptively generating boundaries along a free energy surface in multi-dimensional collective variable (CV) space, building on the boxed molecular dynamics (BXD) rare event algorithm. BXD is a simple technique for accelerating the simulation of rare events and free energy sampling which has proven useful for calculating kinetics and free energy profiles in reactive and non-reactive molecular dynamics (MD) simulations across a range of systems, in both NVT and NVE ensembles. Two key developments outlined in this paper make it possible to automate BXD, and to adaptively map free energy and kinetics in complex systems. First, we have generalized BXD to multidimensional CV space. Using strategies from rigid-body dynamics, we have derived a simple and general velocity-reflection procedure that conserves energy for arbitrary collective variable definitions in multiple dimensions, and show that it is straightforward to apply BXD to sampling in multidimensional CV space so long as the Cartesian gradients ∇CV are available. Second, we have modified BXD to undertake on-the-fly statistical analysis during a trajectory, harnessing the information content latent in the dynamics to automatically determine boundary locations. Such automation not only makes BXD considerably easier to use; it also guarantees optimal boundaries, speeding up convergence. We have tested the multidimensional adaptive BXD procedure by calculating the potential of mean force for a chemical reaction recently investigated using both experimental and computational approaches - i.e., F + CD₃CN → DF + D₂CN in both the gas phase and a strongly coupled explicit CD₃CN solvent. The results obtained using multidimensional adaptive BXD agree well with previously published experimental and computational results, providing good evidence for its reliability.