Temperature–Energy-space Sampling Molecular Dynamics: Deterministic, Iteration-free, and Single-replica Method utilizing Continuous Temperature System

We developed coupled Nosé–Hoover (NH) molecular dynamics equations of motion (EOM), wherein the heat-bath temperature for the physical system (PS) fluctuates according to an arbitrary predetermined weight. The coupled NH is defined by suitably jointing the NH EOM of the PS and the NH EOM of the temperature system (TS), where the inverse heat-bath temperature β is a dynamical variable. In this study, we define a method to determine the effective weight for enhanced sampling of the PS states. The method, based on ergodic theory, is reliable, and eliminates the need for time-consuming iterative procedures and resource-consuming replica systems. The resulting TS potential in a two dimensional (β, ϵ)-space forms a valley, and the potential minimum path forms a river flowing through the valley. β oscillates around the potential minima for each energy ϵ, and the motion of β derives a motion of ϵ and receives the ϵ’s feedback, which leads to a mutual boost effect. Thus, it also provides a specific dynamical mechanism to explain the features of enhanced sampling such that the temperature-space “random walk” enhances the energy-space “random walk.” Surprisingly, these mutual dynamics between β and ϵ naturally arise from the static probability theory formalism of double density dynamics that was previously developed, where the Liouville equation with an arbitrarily given probability density function is the fundamental polestar. Numerical examples using a model system and an explicitly solvated protein system verify the reliability, simplicity, and superiority of the method.