Scaling regimes of molecular size and self-entanglements in very compact proteins

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Abstract
Establishing interrelations between size, compactness, and three-dimensional shape in biomolecules is important for a better understanding of the factors governing their folding features and their biological function. Whereas size and compactness can be characterized by parameters such as the radius of gyration, the description of folding features is less well defined. Recently, we have introduced the probability of overcrossings in two-dimensional projections of a rigid backbone as a descriptor of self-entanglements. This function provides a simple and intuitive characterization: the more complex the entanglements, the larger the mean number of overcrossings. In this work, we study relationships between size and entanglements on a special subclass of biomolecules with a global structural constraint: the family of native protein conformations which are the most compact within a range of amino acid residue numbers. Initially, we use a set of 373 experimental protein backbones exhibiting very diverse lengths, composition, and structural features. Within this set, we have located the proteins with the smallest radii of gyration for fixed ranges of monomer numbers. For this class of proteins, we observe power-law scaling behavior in size and entanglement complexity in terms of the residue number. The results suggest that there are two distinct regimes of scaling characterizing short compact proteins and long compact proteins, respectively. The change of regime appears to be localized roughly around 300 amino acid residues. We propose that this difference correlates with a change in the content of secondary structure for compact proteins: the content of β strands for short chains is almost twice as large as that of longer chains, the latter in turn being much richer in α helices. In summary, the work establishes that in forming a very compact polypeptide there are some constraints among the number of residues, the radius of gyration, the entanglement complexity, and the content of secondary structure of the molecular chain. Thus the degree of compactness in heteropolymers appears to exhibit more complex features than those found in homopolymers.