Theory of the cooperative transition between two ordered conformations of poly(L-proline). II. Molecular theory in...

Phenomenological theories of the transition between helical form I (cis peptide bond) and helical form II (trans peptide bond) of poly(-L-proline), which is a typical order in equilibrium order transition, have been presented by Schwarz (using the parameters s, sigma, beta', and beta'' in a 2 X 2 matrix formulation) and by the present authors (using the parameters s, sigma betaC, and betaN in a 4 X 4 matrix formulation). A molecular theory of the same transition has been formulated to account for the phenomenological parameters. The statistical weights of regular helical sequences with and without junctions between the two forms were computed from empirical potential energy functions. Two puckering conformations of the pyrrolidine ring, i.e., with the Cgamma atom down and up, were allowed, and the free energy was computed for chains with four types of puckering, viz., regular down, regular up, random A, and random B, in the latter two of which the up and down puckerings were randomly distributed. The random A and random B chains have higher energy than those with regular down or up puckering, in both forms I and II. From both an energetical and a free energetical point of view, form I is more stable than form II under vacuum at room temperature. The dependence of the relative stabilities of form I and form II under vacuum on chain length was examined from both an energy and free energy point of view. The four parameters, s, sigma, beta', and beta'', which describe the transitions in Schwarz's theory, were calculated from the statistical weights of various types of sequences. It was found that the thermally induced transition between form I and II under vacuum occurs with the pyrrolidine rings remaining in the down conformation. The calculated values of s suggest that form I is more stable than form II in the regular down chain, while form II is more stable than form I in the regular up chain under vacuum at room temperature. The calculated values of sigma for regular down and regular up pyrrolidine ring puckering are in good agreement with experimental observations, whereas those for random A and random B puckering are much smaller than the experimental values. A theory for the effect of solvent on the parameters s, sigma, beta', and beta'' (at constant temperature) is developed, and the computations involving solvent effects are described in the next paper.