The Backbone as a Series of Rotatable Bonds
Every peptide backbone consists of a repeating sequence of three bonds per residue: N–Cα, Cα–C, and C–N. As established in Articles 2.2 and 2.5, the C–N peptide bond is restricted to near-planarity by its partial double-bond character, leaving only the N–Cα and Cα–C bonds with genuine rotational freedom. The conformational state of a peptide backbone is therefore fully described by two variable torsion angles per residue, φ and ψ, plus the near-constant ω that specifies whether the peptide bond is trans or cis. Understanding how these three angles are defined, measured, and constrained is the geometrical foundation for everything that follows in this chapter.
Defining the Three Angles
Torsion angles are defined by four sequentially bonded atoms and measured about the central bond. By the IUPAC-IUB convention, the angle is positive when rotation about the central bond is clockwise as viewed from the first atom toward the last, and zero when the first and fourth atoms are eclipsed. [1]
The φ angle describes rotation about the N–Cα bond. It is defined by the four atoms C(i−1)–N(i)–Cα(i)–C(i), where (i) identifies the current residue and (i−1) identifies the preceding one. At φ = 180°, the carbonyl carbon of the preceding residue is anti to the carbonyl carbon of the current residue. At φ = 0°, these atoms are eclipsed, a highly strained conformation that is essentially never observed. The range of energetically accessible φ values is restricted by steric clashes between the side chain, the preceding carbonyl oxygen, and adjacent backbone atoms.
The ψ angle describes rotation about the Cα–C bond. It is defined by the four atoms N(i)–Cα(i)–C(i)–N(i+1). At ψ = 180°, the backbone nitrogen of the current residue is anti to the backbone nitrogen of the following residue. Like φ, the range of accessible ψ values is limited by steric interactions, primarily with the following residue and with the side chain of the current residue.
The ω angle describes rotation about the C–N peptide bond. It is defined by Cα(i)–C(i)–N(i+1)–Cα(i+1). As established in Article 2.2, the partial double-bond character of the C–N bond restricts ω to near 180° in the trans conformation (the strong preference for all non-proline residues) or near 0° in the rare cis conformation. Deviations of more than five degrees from these values are uncommon in well-determined structures.
Sign Convention and Historical Note
The current IUPAC-IUB sign convention was established in 1970 to resolve an inconsistency in Ramachandran's original 1963 notation, which defined torsion angles differently. [1] Ramachandran's original convention placed 0° at the extended conformation rather than the eclipsed conformation, shifting all angle values by 180° relative to the current standard. Literature published before approximately 1970 uses the older convention. When reading older structural papers, the coordinate system in use must be identified to interpret torsion angle values correctly. All modern structural databases, including the Protein Data Bank, and all current software use the IUPAC-IUB convention.
Torsion Angle Values in Common Secondary Structures
Each regular secondary structure occupies a specific region of (φ, ψ) space. The values in the table below are idealized; real structures deviate from these means by several degrees, and the degree of deviation is itself informative about structural quality and strain.
| Secondary Structure | φ (°) | ψ (°) |
|---|---|---|
| Right-handed α-helix | −57 | −47 |
| 310-helix | −49 | −26 |
| π-helix | −57 | −70 |
| Antiparallel β-sheet | −139 | +135 |
| Parallel β-sheet | −119 | +113 |
| Polyproline II helix | −75 | +145 |
| Left-handed α-helix | +57 | +47 |
| β-turn type I (central residues) | −60, −90 | −30, 0 |
| β-turn type II (central residues) | −60, +80 | +120, 0 |
Values shown are idealized means. Real structures deviate from these values by several degrees; the Ramachandran plot (Article 4.2) maps the full distribution of observed values in high-resolution crystal structures.
The Physical Basis of Torsion Angle Restrictions
The restrictions on φ and ψ arise entirely from steric interactions: van der Waals clashes between atoms that come too close when certain angle combinations are adopted. The amide oxygen of the preceding residue and the amide hydrogen of the following residue are the primary actors in restricting φ and ψ respectively, together with the side chain at Cα. The detailed geometry of these interactions determines which regions of (φ, ψ) space are allowed, which defines the Ramachandran plot, and which of those allowed regions are populated in practice, which defines the observable secondary structures. Understanding why specific angle combinations are forbidden requires nothing more than visualizing the atoms that would collide if those combinations were adopted.
Side Chain Torsion Angles
The backbone torsion angles φ, ψ, and ω describe only the main chain conformation. Side chain conformations are described by additional torsion angles designated χ1, χ2, χ3, and so on, defined about successive bonds from the Cα outward. χ1 is defined by N–Cα–Cβ–Cγ and describes the rotation about the Cα–Cβ bond. The preferred values of side chain torsion angles, the side chain rotamers, are non-uniform: gauche+ (χ1 ≈ −60°), trans (χ1 ≈ 180°), and gauche− (χ1 ≈ +60°) are the three most common, with frequencies depending on the residue identity and the local backbone conformation. Rotamer libraries derived from high-resolution crystal structures are used in structure refinement, homology modeling, and computational design. Side chain torsion angles are addressed further in the context of structure determination in Chapter 13.
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