Anomalous diffusion of vibrational energy in proteins

The evolution of vibrational wave packets built from the normal modes of cytochrome c, myoglobin and green fluorescent protein is investigated. Vibrational energy flow in these proteins is found to exhibit anomalous subdiffusion, a consequence of trapping of energy by spatially localized normal modes contained in the wave packet. Anomalous subdiffusion is characterized by an exponent, ν, that is related to the spectral dimension, $\mathrm{d̄,}$ and fractal dimension, D, of the protein. The dispersion relation describing variation of the protein’s normal mode frequencies with wave number is also characterized by an exponent, a, that is related to $\mathrm{d̄}$ and D. Values of the exponent, a, computed for the three proteins are consistent with the computed values for ν. The values of D obtained from ν, a, and $\mathrm{d̄}$ for each protein are the same within computational error, and close to the mass fractal dimension computed for each protein, all values falling in the range $\text{D=2.3±0.2.}$ We find also that relaxation of the center of energy of a wave packet computed in terms of normal modes follows stretched exponential kinetics with an exponent equal to 2ν.