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Long range correlations and folding angle in polymers with applications to α-helical proteins

The conformational complexity of linear polymers far exceeds that of point-like atoms and molecules. Polymers can bend, twist, even become knotted. Thus they may also display a much richer phase structure than point particles. But it is not very easy to characterize the phase of a polymer. Essentially, the only attribute is the radius of gyration. The way how it changes when the degree of polymerization becomes different, and how it evolves when the ambient temperature and solvent properties change, discloses the phase of the polymer. Moreover, in any finite length chain there are corrections to scaling, that complicate the detailed analysis of the phase structure. Here we introduce a quantity that we call the folding angle, a novel tool to identify and scrutinize the phases of polymers. We argue for a mean-field relationship between its values and those of the scaling exponent in the radius of gyration. But unlike in the case of the radius of gyration, the value of the folding angle can be evaluated from a single structure. As an example we estimate the value of the folding angle in the case of crystallographic α-helical protein structures in the Protein Data Bank (PDB). We also show how the value can be numerically computed using a theoretical model of α-helical chiral homopolymers.