Statistical mechanics of helical wormlike chains. I. Differential equations and moments

A statistical-mechanical theory of wormlike chains with helical conformations arising from bending and torsional energies is developed. A differential equation for the trivariate distribution function of the end-to-end distance, the unit tangent vector, and the unit curvature vector is derived from the path integral formulation, and several moments are evaluated. The results show that the characteristic ratio for the end-to-end distance or the radius of gyration as a function of chain length t exhibits a maximum with a swelling at some t greater than the maximum point under certain conditions. It is applied to atactic and syndiotactic poly(methylmethacrylate) chains, and their helix parameters are determined reasonably. The theory is also applied to the helix–coil transition in polypeptide chains.