Hydration dependence of the mass fractal dimension and anomalous diffusion of vibrational energy in proteins

Vibrational dynamics of proteins and energy flow depend on protein geometry as well as interactions of a protein molecule with the surrounding solvent. We compute the mass fractal dimension D of proteins ranging from 100 to over 10 000 amino acids comparing values for the bare protein with those computed when buried and hydration waters are included in the calculation. Including water in the calculation increases D by about 0.3 to 2.87 on average above D computed for the dehydrated protein. The mass fractal dimension of proteins that are partially unfolded by molecular dynamics (MD) simulation is also computed and found to vary little when the radius of gyration changes within about 10% of that for the Protein Data Bank structure. MD simulations of vibrational energy diffusion in proteins reveal that the exponent characterizing anomalous diffusion of vibrational energy does not change much with hydration, which is seen to be due to an increase in the spectral dimension with hydration by a factor similar to the increase in D.