Empirical power laws for the radii of gyration of protein oligomers

The radius of gyration is a fundamental structural parameter that is particularly useful for describing polymers. It has been known since Flory's seminal work in the mid-20th century that polymers show a power-law dependence, where the radius of gyration is proportional to the number of residues raised to a power. The power-law exponent has been measured experimentally for denatured proteins and derived empirically for folded monomeric proteins using crystal structures. Here, the biological assemblies in the Protein Data Bank are surveyed to derive the power-law parameters for protein oligomers having degrees of oligomerization of 2-6 and 8. The power-law exponents for oligomers span a narrow range of 0.38-0.41, which is close to the value of 0.40 obtained for monomers. This result shows that protein oligomers exhibit essentially the same power-law behavior as monomers. A simple power-law formula is provided for estimating the oligomeric state from an experimental measurement of the radius of gyration. Several proteins in the Protein Data Bank are found to deviate substantially from power-law behavior by having an atypically large radius of gyration. Some of the outliers have highly elongated structures, such as coiled coils. For coiled coils, the radius of gyration does not follow a power law and instead scales linearly with the number of residues in the oligomer. Other outliers are proteins whose oligomeric state or quaternary structure is incorrectly annotated in the Protein Data Bank. The power laws could be used to identify such errors and help prevent them in future depositions.