  There have been several studies suggesting that protein structures solved by
NMR spectroscopy and x-ray crystallography show significant differences. To
understand the origin of these differences, we assembled a database of
high-quality protein structures solved by both methods. We also find
significant differences between NMR and crystal structures---in the
root-mean-square deviations of the C$_{\alpha}$ atomic positions, identities of
core amino acids, backbone and sidechain dihedral angles, and packing fraction
of core residues. In contrast to prior studies, we identify the physical basis
for these differences by modelling protein cores as jammed packings of
amino-acid-shaped particles. We find that we can tune the jammed packing
fraction by varying the degree of thermalization used to generate the packings.
For an athermal protocol, we find that the average jammed packing fraction is
identical to that observed in the cores of protein structures solved by x-ray
crystallography. In contrast, highly thermalized packing-generation protocols
yield jammed packing fractions that are even higher than those observed in NMR
structures. These results indicate that thermalized systems can pack more
densely than athermal systems, which suggests a physical basis for the
structural differences between protein structures solved by NMR and x-ray
crystallography.
