Monte Carlo Calculations on Polypeptide Chains. III. Multistate per Residue Hard Sphere Models for Randomly Coiling Polyglycine and Poly-l-alanine

Abstract

Monte Carlo studies of randomly coiling polyglycine and poly-l-alanine were carried out using the same hard-sphere model for these polypeptides that was used previously in this laboratory in which the bond lengths and bond angles are held fixed, the peptide bond is fixed in the planar trans configuration, free internal rotation is assumed about the backbone N–Cα and Cα–C′ single bonds, and the individual atoms or groups of atoms interact pairwise by hard-sphere potential functions. In these studies however, all the allowed conformational states per residue from the dipeptide maps taken in 10° increments in the rotational angles φ and ψ for rotation, respectively, about the backbone N–Cα and Cα–C′ single bonds are used whereas formerly only three states per residue had been used for poly-l-alanine and four states per residue for polyglycine. Non-self-intersecting chains of various chainlengths up to 180 amino acid residues were generated for both polyglycine and poly-l-alanine using the sample enrichment technique of Wall and Erpenbeck and then were used to calculate the average chain dimensions. Chains were also generated in which the hard-sphere potential functions were omitted and from which the unperturbed average chain dimensions were calculated. As found previously for the four-state per residue model of polyglycine and the three-state per residue model of poly-l-alanine, the point by point attrition parameter λN was found to be an increasing function of chainlength N over the entire range of N included in the calculations, and the data could be well fitted by the empirical equation λN=λ∞[(N−c)/(N+d)]e, where λ∞ was 0.0500 for polyglycine and 0.0518 for poly-l-alanine. These values for λ∞ are the best estimates of the traditional attrition constants for these models. The mean square end-to-end distance and mean square radius of gyration for both non-self-intersecting and unperturbed chains were found to obey the equations 〈 r2〉=aNb and 〈s2〉=a′Nb′ for large N. The parameters b and b′ were found to be very close to the exact theoretical value of 1.00 for unperturbed chains and to the value 1.20 for non-self-intersecting chains. As found in previous studies for the four-state per residue model for polyglycine and the three-state per residue model for poly-l-alanine, the ratio 〈r2〉/〈s2〉 for the models used in this study passes through a maximum as a function of chainlength in the vicinity of chainlength 10–15 for both non-self-intersecting and unperturbed polyglycine and poly-l-alanine chains and then approaches, within the statistical reliability of the data, the exact theoretical value of 6 for the longest generated unperturbed chains and a value of about 6.4–6.6 for the longest generated non-self-intersecting chains. The results of calculations of the conformational entropy per residue are presented and the results of this study are briefly discussed in terms of Windwer's concepts of intrinsic excluded volume and excess excluded volume.