Mass fractal dimension and the compactness of proteins

Vibrational dynamics and energy flow in a protein are related by Alexander-Orbach theory to the protein's mass fractal dimension $D$ and spectral dimension $\overline{d}$. Burioni et al. [Proteins: Struct., Funct. Bioinf. 55, 529 (2004)] recently proposed a relation between $\overline{d}$ and protein size based on their computational analysis of a set of proteins ranging from about 100 to several thousand amino acids. We report here values for $D$ computed for 200 proteins from the Protein Data Bank (PDB) ranging from about 100 to over 10 000 amino acids and examine variation of $D$ with protein size. The average $D$ is found to be 2.5, significantly smaller than a completely compact three-dimensional collapsed polymer. Indeed, we find that on average a protein in its PDB configuration fills about three-quarters of the volume within the protein surface. Protein mass is also found to scale with radius of gyration with an exponent of 2.5 for this set of proteins.