  We used statistical thermodynamics of conformational fluctuations and the
elements of algebraic graph theory together with data from 2000 protein crystal
structures, and showed that folded native proteins with harmonic interactions
exhibit distribution functions each of which appear to be universal across all
proteins. The three universal distributions are: (i) the eigenvalue spectrum of
the protein graph Laplacian, (ii) the B-factor distribution of residues, and
(iii) the vibrational entropy difference per residue between the unfolded and
the folded states. The three distributions, which look independent of each
other at first sight, are strongly associated with the Rouse chain model of a
polymer as the unfolded protein. We treat the folded protein as the strongly
perturbed state of the Rouse chain. We explain the underlying factors
controlling the three distributions and discuss the differences from those of
randomly folded structures.
