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We report results on the translational dynamics of the hydration water of the lysozyme protein upon cooling obtained by means of molecular dynamics simulations. The self van Hove functions and the mean square displacements of hydration water show two different temperature activated relaxation mechanisms, determining two dynamic regimes where transient trapping of the molecules is followed by hopping phenomena to allow to the structural relaxations. The two caging and hopping regimes are different in their nature. The low-temperature hopping regime has a time scale of tenths of nanoseconds and a length scale on the order of 2-3 water shells. This is connected to the nearest-neighbours cage effect and restricted to the supercooling, it is absent at high temperature and it is the mechanism to escape from the cage also present in bulk water. The second hopping regime is active at high temperatures, on the nanoseconds time scale and over distances of nanometers. This regime is connected to water displacements driven by the protein motion and it is observed very clearly at high temperatures and for temperatures higher than the protein dynamical transition. Below this temperature, the suppression of protein fluctuations largely increases the time-scale of the protein-related hopping phenomena at least over 100 ns. These protein-related hopping phenomena permit the detection of translational motions of hydration water molecules longly persistent in the hydration shell of the protein.

Protein-Water and Water-Water Long-Time Relaxations in Protein Hydration Water upon Cooling—A Close Look through Density Correlation Functions