Torsion Angles in Proteins & the Ramachandran Plot
The torsion angles (Phi and Psi angles) and the Ramachandran Plot in proteins play a central role in defining secondary structure elements, protein folding, and in evaluating the quality of protein three-dimensional structure. Here, we discuss the definition of the torsion angle and the Ramachandran plot, as well as the importance of the Ramachandran plot in protein structure validation.
Definition of the Torsion Angle in Proteins
The torsion angle (or, more generally, the dihedral angle) describes the relative rotation of two segments of the polypeptide chain around a chemical bond. The two primary torsion angles responsible for rotation within the polypeptide chain and essential in defining its general folding are known as the Ramachandran angles, named after the Indian physicist who studied the interactions in polypeptide chains (Ramachandran, G N et al., 1963, J Mol Biol, 7:95-99). In proteins, the Ramachandran angles represent the rotations of the polypeptide backbone around the bonds between N-Cα (referred to as Phi, φ) and Cα-C (known as Psi, ψ). An image on the right illustrates the concept. Other torsion angles in proteins describe the conformation of the amino acid side chains.
By using three consecutive bonds connecting four atoms to define the torsion angle, we can illustrate the rotation of the polypeptide chain around the central bond. The image below demonstrates this concept.
The illustration presents the standard IUPAC definition of a dihedral angle: A, B, C, and D indicate the positions of the four atoms used to define the dihedral angle. The rotation occurs around the central B-C bond. On the right, the atoms are viewed along the B-C bond, with atom A positioned at 12 o’clock. The A-B-D angle on the right describes the rotation around the B-C bond, with positive angles corresponding to clockwise rotation.
A section of a polypeptide chain illustrates the rotation around the bonds that define the protein’s Phi and Psi (φ and ψ) torsion angles. The φ and ψ angles are depicted with rounded arrows. Carbon atoms are shown in green, and the central carbon atom in the image is Cα.
The Ramachandran Plot
Ramachandran and co-workers introduced a special method for plotting and visualizing protein torsion angles and has since been referred to as the Ramachandran plot. In the Ramachandran plot, we can observe the distribution of torsion angles in a protein structure. The horizontal axis of the plot displays φ values, while the vertical axis shows ψ values. Both the horizontal and vertical axes start from -180 and extend to +180 (images below). Each dot in the Ramachandran plot represents an amino acid, with its φ and ψ angles serving as coordinates.
The images below show two Ramachandran plots for two X-ray structures of the same protein solved at a low resolution (on the left) and high resolution (on the right). On the plots, the regions corresponding to the φ and ψ angles of α-helices (red regions, lower cluster) and β-sheets (red regions, upper cluster) are the most populated and clearly separated from each other. These regions are referred to as the energetically most favored and correspond to energetically favored conformations of the polypeptide chain. In these conformations of the polypeptide chain, the interaction energy between the atoms that define the torsion angle is at its optimum. The other colors (ranging from brown to pale yellow; see figure legend) highlight groups of angles for which the energy progressively becomes less favorable (i.e., higher). The two Ramachandran plots show apparent differences in the distribution of torsion angles. The dots are significantly better clustered in the high resolution structure (on the right), with 99,6% of the dots in the most favorable and additionally allowed regions (see figure legend). For the low-resolution structure, only 68% of the dots are in the most favorable region, and 26.8% in the additional allowed regions. Notably, 2.5% of the torsion angles lie within the disallowed regions. What accounts for this difference? It is, of course, the resolution of the structure that almost automatically guarantees better quality. This is discussed further in the section on structure validation and assessment.
Theoretically, the average phi and psi values for α-helices and β-sheets were predicted to cluster around -57, -47, and -80, +150, respectively. However, experimental structures revealed different values. For those interested in a detailed analysis of the distribution of torsion angles of experimental structures, I recommend the paper by Hovmöller et al., 2002, which provides an excellent discussion of the subject.
Below is a Ramachandran plot showing the distribution of the torsion angles of a protein refined at two different resolutions.
The image on the left (PDB code 6adh) corresponds to a structure at approximately 2.9 Å resolution, while on the right, the structure (1het) is at 1.15 Å resolution (a significantly higher resolution). The dots for the high-resolution structure (1het) are well clustered within the energetically most favorable (red, 90.4%) and additionally allowed (brown,9.2%) regions of secondary structure (α-helices, β-sheets, and left-handed α-helix L-α). Conversely, at the lower resolution, many dots are located in energetically less favorable regions (yellow, generously allowed with 2.7% and disallowed with 2.5%). This difference is attributed to the higher quality of the high-resolution structure (see text above). The horizontal axis on the plot represents φ values, while the vertical axis represents ψ values. Both axes range from -180 to +180. Images generated at PDBsum.
The Ramachandran Plot and Protein Structure Validation
Generally, the higher the resolution of the X-ray data, the higher the quality of the three-dimensional structure. This is reflected, as mentioned above, in better clustering of the Phi and Psi angles in the secondary structure regions of the Ramachandran plot . This indicates that the Ramachandran plot can serve as a critical indicator in the quality validation of a three-dimensional structure. A well-refined high-resolution structure is expected to have the majority, if not all, of its torsion angles within the allowed regions of the plot (right image above). However, sometimes, we may find amino acids with “wrong” torsion angles on the plot, which may be for a good reason. The strain (high energy) created in a structure by such conformations may have functional significance (Pal & Chakrabarti, 2002) and may even be conserved within a protein family!
Another exception to the principle of clustering around the α- and β-regions is provided by glycine. Glycine residues lack a side chain, which allows for greater flexibility in regions of the polypeptide chain where Gly is found. Typically, it is surface loops, which explains the relatively high solvent exposure of Gly in protein structures. Consequently, these regions may adopt chain conformations that would otherwise be forbidden. As mentioned in the section on amino acids, proline, unlike glycine, adds rigidity to the polypeptide chain and fixes the torsion angles at a specific value, very close to that of an extended β-strand. The significance of glycine and proline residues in protein three-dimensional structures is reflected in their high conservation across protein sequences, which can be easily revealed through sequence alignment.