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Collective structural vibrational modes of proteins lay in the terahertz frequency range. We discuss several measurements exploring picosecond dynamics in hen egg white lysozyme suggesting that terahertz time domain spectroscopy can provide unique insight to critical protein dynamics. We find a rapid increase in terahertz absorbance and index at both a critical hydration point and separately a critical temperature. Specifically, we discuss the hydration and temperature transitions and suggest that these are possibly linked. We discuss the possibility that the terahertz response is mainly determined by relaxational response of side chains within the protein and the energy barriers for these motions are hydration dependent. Finally we show that simple normal mode calculations do not reproduce the strong hydration dependence, further suggesting that resonant vibrational response is not sufficient to describe the terahertz dielectric response.