Statistical mechanical studies of polypeptides. I. Theory of the helix–coil transition including right and left‐handed helical states

The Lifson‐Roig and Zimm‐Bragg theories of the helix–coil transition in polypeptides are generalized to include both right‐ and left‐handed α‐helical states. The partition functions for these more general theories are formulated in terms of the parameters u , v R , V L , W R , and w L for the generalized Lifson‐Roig theory and σ R , σ L , s R , and s L for the generalized Zimm‐Bragg theory. Matrix equations are derived for calculating such average molecular properties as the fraction of the amino acid residues hydrogen bonded into right‐ and left‐handed α‐helices, the average number of right‐ and left‐handed helical sequences per molecule, the number‐average length (in residues) of the right‐ and left‐handed helical sequences, and the degree of solvent binding to the peptide NH and CO groups. These equations are shown to be conveniently adaptable to machine methods of calculation, thus avoiding the difficulty of solving an eigenvalue problem where the secular equation is of a high order. A discussion is given of the various energetic and entropic effects which determine the screw sense and stability of helices and of the extent to which it is valid to interpret experimental data by adjustment of the parameters of these statistical mechanical theories which include in their formulation only near‐neighbor interactions between residues.