Monte Carlo calculations on polypeptide chains. VIII. Distribution functions for the end-to-end distance and radius of gyration for hard-sphere models of randomly coiling poly(glycine) and poly(L-alanine).

A Monte Carlo study of the distribution functions for the end-to-end distance and radius of gyration for hard-sphere models of poly(glycine) and poly(L-alanine) random coils has been conducted in the chain-length range n = 3 to 100 monomer units for both unperturbed chains and chains perturbed by long-range interactions (excluded volume effects). The distribution functions for the radius of gyration in all cases have been very precisely calculated, those for the perturbed end-to-end distance less precisely, and those for the unperturbed end-to-end distance least precisely. Empirical distribution functions of the form W(p) = ap-b exp(-cp-d) for the reduced end-to-end distance p = r/"r-2"-one-half and a similar form for the reduced radius of gyration could be least-squares fit to the Monte Carlo data. The expansion factors alpha-r and alpha-s were calculated vs. chain length and were used to test various versions of the two-parameter theory of the excluded volume effect. To be consistent with the chain-length dependence of alpha-r and alpha-s as determined by the Monte Carlo calculations, each of these theories required two different binary cluster integrals, a beta-r based on alpha-r and a beta-s based on alpha-s, both of which were strongly chain-length dependent. Both of these results suggest that the two-parameter theory is not applicable to the models used in this study. It was also found that, except for very short chain lengths, plots of ln alphs-r vs. ln n were linear, and thus that alpha-r could be estimated for long chain lengths. Comparison of these estimates with the experimental data on four polypeptide chains in one-earth solvents that the hard-sphere models used in this study yield expansion factors that do not seriously overestimate the magnitude of the excluded volume effect.