Principles of Protein Structure - G. Schultz 1982

Structural Role of the Peptide Bond
Conformational Energy

The hard-sphere model is a good approximation. The data obtained by Ramachandran et al. [28, 29], represented as a —ψ map (often referred to as the "Ramachandran map", see Fig. 2.3, b), are supported by studies of crystalline Globular Proteins. Figure 2.4 summarizes all — and ψ angles found in 13 proteins. The highest density of experimental data points is observed near (—60°, —60°) corresponding to the right-handed a-helix, which reflects the high a-helix content in globular proteins. Another distribution maximum is located near (—90°, +120°) and corresponds to an extended chain with residues forming a ß-pleated sheet. Since the density near (—90°, 0°) is also fairly high, the steric repulsion between N_i and is not as significant as suggested by the hard-sphere model.

The region corresponding to the left-handed a_L-helix is rather sparsely populated compared to adjacent areas. The left-handed a_L-helix has not yet been observed experimentally. About 10% of the points fall into regions that are forbidden for all residues except Gly, located in the right and lower-left PARTS OF THE diagram. Based on the frequency of Gly indicated in Table 1.1, and assuming a uniform distribution of Gly residues over their accessible conformational space, one would expect only half of these points to appear. Consequently, about 5% of all residues containing С_β atoms reside in the forbidden region. This is corroborated by more precise distribution patterns for eight proteins, where —, ψ values are plotted separately for each residue type [31]. It should be borne in mind, however, that the experimental data used may also contain errors arising from the misinterpretation of electron density maps.

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Fig. 2.4. Main-chain dihedral angles for approximately 2,500 residues from 13 proteins [30]. This map is sometimes called the Ramachandran map.

Potential energy maps reveal Steric hindrances more accurately. The hard-sphere model represents a rather crude approach for describing steric constraints. Obviously, it can be improved by replacing hard spheres with a set of potential Structure/4.html">Energy Functions. Potential energy maps have been calculated by numerous authors [32–34] for various residues. As an example, Figure 2.5 shows the potential energy map for Alanine. There are no qualitative differences compared to the hard-sphere map. The potential energies of the right-handed a_R-helix and the left-handed a_L-helix are higher than the energy of the extended chain by approximately 0.5 and 2.5 kcal/mol, respectively. In the case of the right-handed a_R-helix, this energy increase appears to be outweighed by the thermodynamic advantage of strong intramolecular hydrogen bonding within the a-helix. The left-handed a_L-helix possesses no obvious structural advantages over the right-handed one. Both helices are cylinders of identical shape. Therefore, if a cylindrical structure is required, there is no reason to employ the energetically unfavorable left-handed a_L-helix if the alternative helix can successfully fulfill the same function.

Fig. 2.5. Potential energy distribution in the (—, ψ) plane for a pair of peptide units with an intervening Ala residue [34].

The map was constructed using the Pauling–Corey Peptide bond parameters (Fig. 2.1, a). Isoenergy lines are drawn at 1 kcal/mol intervals into the negative region from zero. The zero equipotential is indicated by dashed lines. THE POSITION OF the twisted ß-sheet is marked. The potential surface will change if Hydrogen Bonds involving peptide units distant along the chain (e.g., in an a-helix) are taken into account.

The potential energy map reveals a low-energy bridge between the extended chain region (—, ψ) = (—120°, +120°) and the a-helix region (—60°, —60°). Such a bridge is absent in the hard-sphere model, even when employing extremely small contact radii. Since this so-called bridge region is heavily populated according to data on crystalline proteins (Fig. 2.4), it is indeed readily accessible in reality. The hard-sphere model fails in this instance. Steric hindrances between the N_i and H_i+1 atoms (Fig. 2.3, a) arising at these angles are compensated by dipole-dipole interactions corresponding to a weak Hydrogen bond. Similar interactions have been found in crystals of model compounds [35].

The peptide bond exhibits a certain degree of flexibility. The conformational maps in Figures 2.3 and 2.5 were constructed under the assumption of a rigid peptide bond with Pauling–Corey parameters (Fig. 2.1, a). Further refinement of the model requires the Introduction of potentials for bond-angle deformations, bond-length variations, and torsional rotation around the peptide bond. Naturally, this renders the conformational space of a single residue multidimensional, making any direct application or exhaustive description rather difficult. For estimation purposes, we list the deviations corresponding to a 1 kcal/mol increase in potential energy:

In addition, out-of-plane atomic displacements must also be taken into account [36]. In principle, flexibility can be incorporated into a two-dimensional energy map by permitting all flexible deformations and calculating the minimum energy for each given pair of —, ψ angles in every case. It should be noted that if The flexibility of all backbone units is taken into account, the backbone conformation cannot be fully described solely by The values of the — and ψ angles.

Thus, if the peptide bond possesses significant flexibility, the conformational map constructed for fixed parameters of this bond (Fig. 2.5) may prove inaccurate in regions associated with mild steric hindrances—such as those permitted only for Gly residues—since these hindrances can be relieved by minor deviations in Bond Angles, torsion angles, and Bond Lengths. This explains the 5% of cases in Figure 2.4 where —, ψ values fall into regions that are forbidden According to the hard-sphere model due to steric interactions with the С_β atom.