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

Abstract

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.Despite substantial differences in their chemical composition, all linear polymers are presumed to share the same universal phase structure [1][2][3]. But the phase where a particular polymer resides depends on many factors including polymer concentration, the quality of solvent, ambient temperature and pressure. Three phases are commonly identified, each of them categorized by the manner how the polymer fills the space [1][2][3][4]: If the solvent is poor and the attractive interactions between monomers dominate, a single polymer chain is presumed to collapse into a space-filling conformation. In a good solvent environment or at sufficiently high temperatures, a single polymer chain tends to swell until its geometric structure bears similarity to a self-avoiding random walk (SARW). The collapsed phase and the SARW phase are separated by Θ-point where a polymer has the characteristics of an ordinary random walk (RW). Biologically active proteins are commonly presumed to reside in the space filling collapsed phase, under physiological conditions. We shall examine proteins as an important subset of polymers, for which a large amount of experimental data is available in PDB [5].The phase where a polymer resides can be determined from the value of the scaling exponent ν [1-3]. To define this quantity, we consider the asymptotic behavior of the radius of gyration when the number of monomers N is very large. With r i the coordinates of the skeletal atoms of the polymer, the radius of gyration becomes in this limit [1][2][3][6][7][8] The length scale R 0 is an effective Kuhn distance between the skeletal atoms in the large-N limit. It is in principle a computable quantity, that depends on all the atomic level details of the polymer and all...

Multiple scales and phases in discrete chains with application to folded proteins

“…The quantity (6), (7) bears resemblance to the radius of gyration (1), except that (6), (7) is dimensionless. We also note that (6), (7) relates to, but is quite different from, the concept of folding angle introduced in [24].…”