Equipartition and the Calculation of Temperature in Biomolecular Simulations

Since the behavior of biomolecules can be sensitive to temperature, the ability to accurately calculate and control the temperature in molecular dynamics (MD) simulations is important. Standard analysis of equilibrium MD simulations-even constant-energy simulations with negligible long-term energy drift-often yields different calculated temperatures for different motions, however, in apparent violation of the statistical mechanical principle of equipartition of energy. Although such analysis provides a valuable warning that other simulation artifacts may exist, it leaves the actual value of the temperature uncertain. We observe that Tolman's generalized equipartition theorem should hold for long stable simulations performed using velocity-Verlet or other symplectic integrators, because the simulated trajectory is thought to sample almost exactly from a continuous trajectory generated by a shadow Hamiltonian. From this we conclude that all motions should share a single simulation temperature, and we provide a new temperature estimator that we test numerically in simulations of a diatomic fluid and of a solvated protein. Apparent temperature variations between different motions observed using standard estimators do indeed disappear when using the new estimator. We use our estimator to better understand how thermostats and barostats can exacerbate integration errors. In particular, we find that with large (albeit widely used) time steps, the common practice of using two thermostats to remedy so-called hot solvent-cold solute problems can have the counter-intuitive effect of causing temperature imbalances. Our results, moreover, highlight the utility of multiple-time step integrators for accurate and efficient simulation.