Fluctuations of an ?-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.