Torsion Angles and the Ramachandran Plot

Two torsion angles in the polypeptide chain, also called Ramachandran angles (after the Indian physicist who worked on modeling the interactions in polypeptide chains, Ramachandran, GN, et al., J Mol Biol, 7:95-99) describe the rotations of the polypeptide backbone around the bonds between N-Cα (called Phi, φ) and Cα-C (called Psi, ψ, see image for the graphics view of the angles).

torsion angle definition

A fragment of a polypeptide chain showing the torsion angles φ and ψ as rounded arrows.
The angle (also called dihedral angle) is defined by 3 consecutive bonds involving 4 atoms. The angle describes the rotation of the chain around the middle bond. In proteins the two torsion angles φ and ψ (also called Ramachandran angles) describe the rotation around N-Cα and Cα-C bonds, respectively.

The standard IUPAC definition of a dihedral angle is illustrated in the figure below. A, B, C and D illustrate the position of the 4 atoms used to define the dihedral angle. The rotation takes place around the central B-C bond. The view on the right is along the B-C bond with atom A placed at 12 o'clock. The rotation around the B-C bond is described by the A-B-D angle shown of the right image.

torsion angle, definition 2

The range of the Phi & Psi Ramachandran angles accessible to a polypeptide chain defines the flexibility of the backbone and its ability to adopt a certain fold.
The third possible torsion angle within the protein backbone (called omega, ω) describes the rotation at the peptide bond and is mostly flat and fixed to around 180 degrees. This is due to the partial double-bond character of the peptide bond, which restricts rotation around the C-N bond, placing two successive α-carbons and C, O, N and H between them on one plane.

The Ramachandran plot
A special way for plotting protein torsion angles was introduced by Ramachandran and co-authors and since then is called the Ramachandran plot. The Ramachandran plot provides a way to view the distribution of torsion angles in a protein structure and shows that the torsion angles corresponding to the two major secondary structure elements (α-helices and β-sheets) are clearly clustered within separate regions. The images below correspond to two different structures of the same protein. Each dot in the plot corresponds to an amino acid, with its φ and ψ angles. On the left is a structure at low resolution and on the right is a high-resolution structure.

Ramachandran plot for low resolution structure
Ramachandran plot for well refined structure

The Ramachandran plot shows the distribution of the torsion angles of a protein within certain regions.
The horizontal axis on the plot shows φ values, while the vertical shows ψ values. Both horizontal and vertical axes start from -180 and extend to +180. The images also show that φ and ψ angles of α-helices and β-sheets are separated and occupy different regions of the plot (marked as α and β).

The Ramachandran plot and structure quality
The higher resolution of the X-ray data usually gives higher quality three-dimensional structure. From the plots it is easy to see that some regions contain many more dots than others. These are called allowed regions, and the corresponding angles are called allowed, or favorable angles (red and brown regions). Other regions are less favorable and are poorly populated in good-quality structures (yellow color). This is a result of steric hindrance – certain rotations around the polypeptide chain will bring atoms too close to each other, creating steric repulsion. For this reason, Ramachandran plot serves as an important indicator of the quality of three-dimensional structures – a good quality structure is expected to have all its torsion angles within the allowed regions of the plot (image on the right).

However, sometimes we may find amino acids with “wrong” torsion angles for a good reason – the strain (high energy) created in a structure by some residues within unfavorable angles may be used by the protein for certain purposes and may have functional significance (
Pal and Chakrabarti, 2002). Another exception from the principle of clustering around the α- and β-regions is provided by glycine. Gly does not have a side chain, which allows high flexibility in the polypeptide chain, making otherwise forbidden rotation angles accessible. That is why glycine is often found in loop regions, where the polypeptide chain needs to make a sharp turn. As mentioned above, proline in contrast to glycine fixes the torsion angles at a certain value, very close to that of an extended β-strand.

Theoretically, the average phi and psi values for α-helices and β-sheets are clustered around -57, -47 and -80, +150, respectively. However, for real experimental structures these values were found to be different. In a paper by
Hovmöller et al., 2002, download pdf, a detailed discussion of the fine structure of φ- and ψ-angle distribution in the Ramachandran plot is presented (if asked for login on the page, just cancel and it will still work).