Ab initio study of alanine polypeptide chain twisting

We have investigated the potential energy surfaces for alanine chains consisting of three and six amino acids. For these molecules we have calculated potential energy surfaces as a function of the Ramachandran angles $\phi$ and $\psi$, which are widely used for the characterization of the polypeptide chains. These particular degrees of freedom are essential for the characterization of the proteins folding process. Calculations have been carried out within the ab initio theoretical framework based on the density functional theory and accounting for all the electrons in the system. We have determined stable conformations and calculated the energy barriers for transitions between them. Using a thermodynamic approach, we have estimated the times of characteristic transitions between these conformations. The results of our calculations have been compared with those obtained by other theoretical methods and with the available experimental data extracted from the Protein Data Base. This comparison demonstrates a reasonable correspondence of the most prominent minima on the calculated potential energy surfaces to the experimentally measured angles $\phi$ and $\psi$ for alanine chains appearing in native proteins. We have also investigated the influence of the secondary structure of polypeptide chains on the formation of the potential energy landscape. This analysis has been performed for the sheet and the helix conformations of chains of six amino acids.

Figure 1

(Color online) Dihedral angles $\phi$ and $\psi$ used to characterize the potential energy surface of the polypeptide chain.

Figure 2

(Color online) Optimized geometries of alanine polypeptide chains calculated by the $\mathrm{B3LYP}∕6–31++\mathrm{G}(\mathrm{d},\mathrm{p})$ method: (a) alanine tripeptide; (b) alanine hexapeptide (sheet conformation); (c) alanine hexapeptide (helix conformation).

Figure 3

(Color online) Dependence of alanine tripeptides energy on angle $\omega$ calculated by the $\mathrm{B3LYP}∕6–31\mathrm{G}\left(\mathrm{d}\right)$ method at different values of angles $\phi$ and $\psi$.

Figure 4

(Color online) Potential energy surface for the alanine tripeptide calculated by the $\mathrm{B3LYP}∕6–31\mathrm{G}(2\mathrm{d},\mathrm{p})$ method. Energies are given in eV, kcal/mol, and kelvin. Numbers mark the energy minima on the potential energy surface. Arrows show transition paths between different conformations of the molecule.

Figure 5

(Color online) Optimized conformations of the alanine tripeptide. Different geometries correspond to different minima on the potential energy surface (see contour plot in Fig. 4). Below each image we present angles $\phi$ and $\psi$, which have been obtained with accounting for relaxation of all degrees of freedom in the system. Values in brackets give the angles calculated without accounting for relaxation. Above each image the energy of the corresponding conformation is given in eV. The energies are counted from the energy of conformation 1 (the energy of conformation 1 is given in a.u.). Values in parentheses correspond to the energies obtained without relaxation of all degrees of freedom in the system. Dashed lines show the strongest hydrogen bonds. Their lengths are given in angstroms.

Figure 6

(Color online) Transition barriers for between conformations $1\leftrightarrow 2$ of the alanine tripeptide. Circles and squares correspond to the barriers calculated without and with relaxation of all degrees of freedom in the system.

Figure 7

(Color online) Transition barriers for between conformations $1\leftrightarrow 2$ of the alanine dipeptide analog calculated by the $\mathrm{B3LYP}∕6–31+\mathrm{G}(2\mathrm{d},\mathrm{p})$ method accounting for the relaxation of all degrees of freedom in the system. Structure of the conformations 1 and 2 is shown near each minimum.

Figure 8

(Color online) Potential energy surface for the alanine hexapeptide with the sheet secondary structure [part (a)] and with the helix secondary structure [part (b)] calculated by the $\mathrm{B3LYP}∕6–31\mathrm{G}(2\mathrm{d},\mathrm{p})$ method. The energy scale is given in Fig. 4. Numbers mark the energy minima on the potential energy surface. Images of optimized conformations of the alanine hexapeptide are shown near the corresponding energy landscape. Values of angles $\phi$ and $\psi$, as well as the relative energies of the conformations, are given analogously to that in Fig. 5.

Figure 9

(Color online) Transitions barriers between conformations $1\leftrightarrow 2$ of the alanine hexapeptide with the sheet secondary structure. Circles and squares correspond to the barriers calculated without and with relaxation of all degrees of freedom in the system.

Figure 10

(Color online) Comparison of angles $\phi$ and $\psi$ of alanine residues in protein structures selected from the Brookhaven Protein Data Bank [9, 44] with the steric diagram for polyalanine [45] [part (a)]. Comparison of angles $\phi$ and $\psi$ of alanine residues in protein structures selected from the Brookhaven Protein Data Bank [9, 44] with the minima on the calculated potential energy surfaces for alanine tripeptide (b); alanine hexapeptide in sheet conformation (c); alanine hexapeptide in helix conformation (d). Transparent rhomboids correspond to alanines surrounded with alanines, while filled circles correspond to alanines surrounded by other amino acids. Dashed ellipses mark the regions of higher concentration of the observed angles.