Monte Carlo Calculations on Polypeptide Chains. IV. Hard-Sphere Models for Randomly Coiling Polysarcosine and Poly-N-Methyl-L-Alanine

Abstract

Monte Carlo studies of the average dimensions of randomly coiling polysarcosine and poly‐N‐methyl‐L‐alanine were carried out using hard‐sphere models for these polypeptide chains similar to those used previously in this laboratory for polyglycine and poly‐L‐alanine. Non‐self‐intersecting chains of various chain‐lengths up to 272 amino acid residues for polyscarcosine and 320 for poly‐N‐methyl‐L‐alanine were generated, using the sample enrichment technique of Wall and Erpenbeck, and then used to calculate average chain dimensions. Unperturbed chains for both polypeptides up to 300 amino acid residues in length were also generated by ignoring hard‐sphere conflicts and were then used to calculate average unperturbed dimensions. As in previous work reported from this laboratory chain attrition could be represented by the empirical equation λ N =λ ∞ [ (N−c)/(N+d) ] e , with the attrition constants λ∞ having the values 0.0453 and 0.0195 for, respectively, polysarcosine and poly‐N‐methyl‐L‐alanine. These values are lower than those obtained previously for polyglycine and poly‐L‐alanine. The mean square end‐to‐end distance and mean square radius of gyration for both the non‐self‐intersecting and unperturbed chains could be represented at long chain lengths by the equations 〈 r 2 〉=aN b and 〈 s 2 〉=a′ N b{′ } and over the entire range of chainlengths by the equations 〈 r 2 〉=aN b exp [ (c 1 /N)+(c 2 /N 2 )+(c 3 /N 3 ) ] and 〈 s 2 〉=a′N b{′ } exp [ (c 1 ′ /N)+(c 2 ′ /N 2 )+(c 3 ′ /N 3 ) ] . The latter equations were also used to reanalyze the data from earlier calculations on multistate per residue models of polyglycine and poly‐L‐alanine. The parameters b and b′ for unperturbed chains in all cases were found to be consistent, within the statistical reliability of the data, with the theoretical value of 1.00 and for non‐self‐interesting chains with the theoretical value of 1.20, except for poly‐N‐methyl‐L‐alanine which had lower values. It is proposed that these low values are valid and not just statistical deviations and that b and b′ should not in all cases have the value 1.20 although values near 1.20 are found for all previous polypeptide models studied to date. It is further proposed that there should be a continuum of possible values between 1.00, characteristic of unperturbed chains, and 1.20, often found for non‐self‐intersecting chains, and including values above 1.20. As found in earlier studies the ratio 〈 r 2 〉/〈 s 2 〉 for the models in this study passes through a maximum as a function of increasing chain length for both non‐self‐intersecting chains and unperturbed chains and then approaches at long chain lengths, within the statistical reliability of the data, the theoretical value of 6.00 for unperturbed chains and a slightly larger value for non‐self‐intersecting chains. The entire range of the data for 〈 r 2 〉/〈 s 2 〉 could also be well represented by an equation of the form 〈 r 2 〉/〈 s 2 〉=a′ ′N b{′ ′ } exp [ (c 1 ′ ′ /N)+(c 2 ′ ′ /N 2 )+(c 3 ′ ′ /N 3 ) ] with b′ ′≈ 0 .