  We present a pressure dependence study of the dynamics of lysozyme protein
powder immersed in deuterated $\alpha$,$\alpha$-trehalose environment via
quasi-elastic neutron scattering (QENS). The goal is to assess the
baro-protective benefits of trehalose on bio-molecules by comparing the
findings with those of a trehalose-free reference study. While the mean-square
displacement of the trehalose-free protein (hydrated to $d_{D_2O}\simeq$40 w\%)
as a whole, is reduced by increasing pressure, the actual observable relaxation
dynamics in the pico-(ps) to nano-seconds (ns) time range remains largely
unaffected by pressure - up to the maximum investigated pressure of 2.78(2)
Kbar. Our observation is independent of whether or not the protein is mixed
with the deuterated sugar. This suggests that the hydrated protein's
conformational states at atmospheric pressure remain unaltered by hydrostatic
pressures, below 2.78 Kbar. We also found the QENS response to be totally
recoverable after ambient pressure conditions are restored. Circular dichroism
and neutron diffraction measurements confirm that the protein structural
integrity is conserved and remains intact, after pressure is released. We
observe however a clear narrowing of the quasi-elastic neutron (QENS) response
as the temperature is decreased from 290 K to 230 K in both cases, which we
parametrize using the Kohlrausch-Williams-Watts (KWW) stretched exponential
model. Only the fraction of protons that are immobile on the accessible time
window of the instrument, referred to as the elastic incoherent structure
factor or (EISF) is observably sensitive to pressure, increasing only
marginally but systematically with increasing pressure.
