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.
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