Fluctuations of an α‐helix

Fluctuations in backbone dihedral angles in the α‐helical conformation of homopolypeptides are studied based on an assumption that the conformational energy function of a polypeptide consisting of n amino‐acid residues can be approximated by a 2n‐dimensional parabola around the minimum point in the range of fluctuations. A formula is derived that relates 〈ΔθiΔθj〉, the mean value of the product of deviations of dihedral angles ϕi and ψi (collectively designated by θi) from their energy minimum values, with a matrix inverse to the second derivative matrix F,n of the conformational energy function at the minimum point. A method of calculating the inverse matrix Fn−1 explicitly is given. The method is applied to calculating 〈ΔθiΔθj〉 for the α‐helices of poly(L‐alanine) and polyglycine. The autocorrelations 〈(Δϕi)2〉 and 〈(Δψi)2〉 at 300°K are found to be about 66 deg2 and 49 deg2, respectively, for poly(L‐alanine), and 84 deg2 and 116 deg2, respectively, for polyglycine. The length of correlations of fluctuations along the chain is found for both polypeptides to be about eight residues long.