Dimensional Change of Polypeptide Molecules in the Helix-Coil Transition Region. II

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Abstract
The mean‐square end‐to‐end distance 〈R2〉 and radius of gyration 〈S2〉 are calculated for polypeptide molecules in the helix‐coil transition region. The helical content, the sequence length distributions of helical and randomly coiled sections, etc., are also calculated for illustration. The statistical mechanical model on which the present calculation is based, is essentially equal to the second models in Gibbs‐DiMarzio's and Zimm‐Bragg's theories, respectively, though formulation is made on a more general model for the two purposes, i.e., for some convenience in calculating various quantities and for an examination of the degree of approximation of the above model. As a result, this model is found to be very satisfactory in spite of its simplicity. In order to calculate chain dimensions, two configurational models are examined. The one (model I), is somewhat rigorous, account being taken of the effects of bond lengths, bond angles, and hindered rotations. Two kinds of residues, i.e., those involved in helical sections, and those in randomly coiled sections, are distinguished by assigning different internal rotational potentials. The other (model II), is more approximate but more tractable. It can be obtained by replacing a helical section by a rod with an appropriate length and a randomly coiled section by an appropriate random flight chain. Numerical computations show that the two models give very close results for the infinitely long chain in the region where essential dimensional change occurs. By using model II, the effects of statistical mechanical parameters on 〈R2〉 are examined in detail. As a result, for example, it is shown that the larger the mean number of residues involved in a helical section becomes, the larger 〈R2〉 becomes, even if the helical content remains unchanged. For large N (e.g., N>300), as chain configurations go from randomly coiled to rodlike, 〈R2〉 and 〈S2〉 first decrease, pass through a minimum, and then increase until they reach the values for the perfect α helix. The appearance of this minimum is due to the considerably extended local configuration of randomly coiled residues. Namely, the appearance of a short helical section, turns the extended configuration to the folded one of the α helix and hence the polymer chain shrinks.