Scaling behavior of some molecular shape descriptors of polymer chains and protein backbones

Some global folding features of macromolecular chains are characterized using a family of molecular shape descriptors. These descriptors are derived from the following basic notion: the probability of observing N ‘‘crossings’’ between bonds (i.e., double points or overcrossings) when a rigid placement of a polymer backbone is projected onto two dimensions. The approach combines simple elements of geometry and topology of linear polymers, and it quantifies the compactness and the complexity of chain entanglements in three-space. The asymptotic behavior of the shape descriptors has been determined as a function of the chain length. It is found that the configurational averages of the most probable number of overcrossings ${\mathit{N}}^{\mathrm{*}}$, the mean number of overcrossings N¯, and the largest probability of overcrossings ${\mathit{A}}^{\mathrm{*}}$, obey power laws in terms of the number of monomers. The critical exponents have been estimated numerically for random-walk polymers with excluded-volume, as well as for a large number of experimental protein backbones. The results indicate that the scaling behavior is little affected by the configurational state of the polymers, since virtually the same exponents are obtained for both ‘‘swollen’’ and ‘‘compact’’ structures. The same scaling behavior is found in polymers with various excluded-volume interactions and in a set of 197 proteins. The mean number of overcrossings in proteins is well described by a simple law: N¯≊0.045${\mathit{n}}^{1.4}$, where n is the number of amino acid residues. The shape descriptors for proteins show little dispersion away from the asymptotic regime, whereas a less uniform behavior is found in the radius of gyration. The results complement the analyses based on other more familiar (geometrical) descriptors, and provide some insights into the large-scale folding structure of polymer chains and proteins.