Before going in detail of dihedral angles, we need to understand the conformation of peptide bond and its rigidity.

Conformation of Peptide

In the peptide bond, the π-electrons from the carbonyl are delocalized between oxygen and the nitrogen. So, it means that the peptide bond has ~40% double bond character. As we know that the length of a normal C–N single bond is 1.45 Å and a C=N double bond is 1.25 Å, while the peptide C–N bond length is 1.33 Å. So, it well seems that this partial double bond character is obvious in the shortened bond length of the C–N bond. Also, due to its partial double bond character, rotation around the C–N bond is restricted. The peptide bond permits rotation about the bonds from the α-carbon, but not the amide C–N bond. Only the Φ and Ψ dihedral angles (see explanations in dihedral angle) can vary reasonably freely. Also, the six atoms in the peptide bond (the two α-carbons, the amide O, and the amide N and H) are coplanar. As a mesomeric system, the peptide bond is planar. Rotation around the C–N bond would only be possible at the expense of large amounts of energy, and the bond is therefore not freely rotatable.

Finally, we can conclude that the peptide bond has a dipole, with the O having a partial negative charge, and the N_amide having a partial positive charge. This permits the peptide bond to participate easily in electrostatic interactions, and contributes to the hydrogen bond strength between the backbone carbonyl and the N _amide proton

Dihedral angles

The major properties of the covalent bonds to hold proteins together are their bond distances and bond angles. The bond angles between two adjacent bonds on either side of a single atom, or the dihedral angles between three contiguous bonds and two atoms control the geometry of the protein folding. The preferred dihedral angles for different secondary structural elements will be discussed in this section. The peptide bond contains three sets of torsion angles (also known as dihedral angles).

i. ω angle

ii. Φ and ψ

i. ω angle

The ω angle, is the dihedral angle around the amide bond and is the least variable of all torsion angles. As discussed above, this angle is fixed due to orbital overlap between the carbonyl double bond and the N_amide lone pair orbital. In figure 2.1.5, the omega-angle is the bold one and the carbonyl oxygen and amide hydrogen are trans to each other. This trans conformation is favored energetically because it results in less steric hindrance between non bonded atoms in neighboring amino acids. But as discussed previously due to the restricted rotation around the C–N bond (ω angle), rotations are only possible around the N–C_α and C_α–C bonds. This means that nearly all peptide bonds in a protein will have a ω angle of 180°.

*   Proline is an exception; the tertiary amide formed by an X-Pro peptide bond does not have strong energetic preference for cis or trans configurations. As a result, both are observed in proteins. Interconversion between cis and trans X-Pro peptide bonds requires an enzyme (prolyl isomerase), which plays a role in folding of many proteins.

Φ and ψ

Dihedral angles that can vary more easily are important in considering protein structure. These angles are the Φ (phi, C_α–N_amide) and Ψ ( psi, C_α–C_amide) angles. The fully extended conformation of a protein corresponds to 180° for both Φ and Ψ (*Note that 180° = –180°).

When you look away from alpha carbon, the numeric values of angles increases in clockwise direction.

Importantly,

1. Φ = 0° when the C_amide-N_amide and C_amide-C_α bonds are in the same plane, and
2. Ψ = 0° when the N_amide-C_amide and N_amide-C_α bonds are in the same plane.

The (+) direction is clockwise while looking away from the C_α.

The dihedral angles (Φ / Ψ pairs) that the atoms of the peptide bond can assume are limited by steric hindrance. Some Φ / Ψ pairs will result in atoms being closer than allowed by the van der waals radii of the atoms, and are therefore due to steric hindrance they are extremely unlikely to be observed. For example: 0°:0°, 180°:0°, and 0°:180° are almost never observed because of backbone atom clashes. For tetrahedral carbons, the substituents are typically found in staggered conformations (see figure, below). Peptide bonds are much more complicated, because while the α-carbon is tetrahedral, the two other backbone atom types have planar arrangements. However, while a true staggered conformation is not possible, the same principle applies: the preferred conformations for peptide bond atoms have the substituent atoms at maximal distances from one another.

It becomes important to know that:-

1. Ψ angle of 180° results in an alignment of the N_amide with the carbonyl oxygen from the same residue. This is allowed, although not especially favored.
2.  Ψ angle of 0° places the N_amide from one residue very close to the N_amide from the previous residue; this results in a steric collision that results in an unfavorable electrostatic interaction, because both N_amide have partial positive charges.

The residue side-chains also impose steric constraints. Glycine, because of its very small side chain, has a much large ranger of possible Φ / Ψ pairs than any other amino acid residue. It is interesting to note that proline has a very limited range of Φ angles because its side-chain is covalently linked to its N_amide, while its accessible Ψ angles are similar to those of most other residues. Most other residues are limited to relatively few Φ / Ψ pairs (although more than proline). This is especially true for the β-branched residues like threonine, valine, and isoleucine, which are the most restricted, because these residue sidechains have more steric bulk due to the presence of two groups attached to their β- carbons.

Therefore, summarizing the dihedral angle discussed, one can deduce that every possible set of one value each of Φ and Ψ angles in the -180° to +180° range will signify one conformation of the protein.  We also came to know that for steric reasons, only specific combinations of the dihedral angles are possible. Most combinations of Φ and ψ are sterically “forbidden”.  These relationships can be demonstrated clearly by a so-called Φ /Ψ diagram.