Molecular Theory of the Helix-Coil Transition in Polyamino Acids. II. Numerical Evaluation of s and σ for Polyglycine and Poly-l-alanine in the Absence (for s and σ) and Presence (for σ) of Solvent

The Zimm–Bragg parameters $s$ and $\sigma$ for the helix‐coil transition in polyglycine and poly‐l‐alanine are calculated in terms of molecular quantities, based on our earlier formulation which is modified here to take effects at junctions between helical and coil sequences into account more properly. The statistical weight of a helical sequence is computed for both regular and nonregular helices. In the latter, the dihedral angles $\phi$ and $\psi$ (and $\chi$ for alanine) near the ends of helical sequences differ from those in the interior; this diffuseness, which extends over about five residues at each end, reduces the values of $\sigma$ by a factor of 10^0.0−10^0.8. The dihedral angles and the corresponding conformational energies of both regular and nonregular helices are obtained by energy minimization. The entropy factor in the statistical weight of a helical sequence consisting of $j$ residues [$\text{(j\hspace{0.17em}+\hspace{0.17em}1)}$ peptide units] is obtained from the second derivatives of the energy surface in the 2j‐dimensional space for both regular and nonregular helices. The calculation of the statistical weight of a coil sequence involves the determination of the largest eigenvalue of a certain nonsymmetric real matrix. In calculating the entropic parts of the statistical weights of both helical and coil sequences, the dihedral angle $\chi$ is not varied (but is kept at its minimum‐energy value) since this source of entropy cancels when $s$ and $\sigma$ are obtained (as is shown numerically). The computations were carried out for three different sets of parameters (dielectric constant $D$ and radius of hydrogen atom $R_H$ ): (A) $\text{D\hspace{0.17em}=\hspace{0.17em}4.0}$ , $R_H\text{\hspace{0.17em}=\hspace{0.17em}1.200 Å}$ , (B) $\text{D\hspace{0.17em}=\hspace{0.17em}4.0}$ , $R_H\text{\hspace{0.17em}=\hspace{0.17em}1.275 Å}$ , and (C) $\text{D\hspace{0.17em}=\hspace{0.17em}1.0}$ , $R_H\text{\hspace{0.17em}=\hspace{0.17em}1.275 Å}$ . Since solvent effects are not included specifically in this paper (except for the use of $\text{D\hspace{0.17em}=\hspace{0.17em}4.0}$ in parameters sets A and B), the calculated values of $s$ (and hence of the transition temperature) cannot be compared with experimental ones; despite the omission of solvent effects (which will be included in the next paper), the calculation of $s$ still provides much insight into the influence of the various molecular parameters on the helix‐coil transition. On the other hand, a comparison of calculated and experimental values of $\sigma$ is possible because this quantity is not affected by the binding of solvent molecules to free NH and CO groups. For poly‐l‐alanine, $s$ and $\sigma$ are computed for both right‐ and left‐handed $\alpha$ helices; a transition from the right‐ to the left‐handed $\alpha$ helix of poly‐l‐alanine (before melting to coil) is found to take place as the temperature is raised, for all three sets of parameters examined. Variations in the values of $D$ and $R_H$ do not affect $s$ markedly, but they have a striking effect on $\sigma$ . The results for parameter set A are: for polyglycine, the helix‐coil transition temperature is 210°C, at which ${\mathrm{\Delta H}}_8\text{\hspace{0.17em}=\hspace{0.17em}−\hspace{0.17em}3.97}\mathrm{kcal}/\mathrm{mole}$ residue, ${\mathrm{\Delta S}}_8\text{\hspace{0.17em}=\hspace{0.17em}−\hspace{0.17em}8.19}\mathrm{eu}$ , and ${\text{σ\hspace{0.17em}=\hspace{0.17em}10}}^{\text{−5.1}}$ ; for poly‐l‐alanine, the transition temperature from the right‐ to the left‐handed $\alpha$ helix is 90°C, and the transition temperature from the left‐handed $\alpha$ helix to the random coil is 460°C, (for the latter transition, ${\mathrm{\Delta H}}_8\text{\hspace{0.17em}=\hspace{0.17em}−\hspace{0.17em}4.49}\mathrm{kcal}/\mathrm{mole}$ residue, ${\mathrm{\Delta S}}_8\text{\hspace{0.17em}=\hspace{0.17em}−\hspace{0.17em}6.19}\mathrm{eu}$ , and ${\text{σ\hspace{0.17em}=\hspace{0.17em}10}}^{\text{−4.1}}$ ). For $\text{D\hspace{0.17em}=\hspace{0.17em}4.0}$ , the agreement between the calculated and experimental values of $\sigma$ is reasonable, while, for $\text{D\hspace{0.17em}=\hspace{0.17em}1.0}$ , the calculated values become too small. Both entropic and enthalpic terms are found to contribute to the value of $\sigma$ , i.e., the calculated values of $\sigma$ are found to be temperature dependent. Finally, the computer programs developed for these calculations are discussed.