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 below for the graphics view of the angles). A special way for plotting protein torsion angles was also introduced by Ramachandran and co-authors, and was subsequently named the Ramachandran plot. The Ramachandran plot provides an easy way to view the distribution of torsion angles in a protein structure. It also provides an overview of excluded regions that show which rotations of the polypeptide are not allowed due to steric hindrance (collisions between atoms). The Ramachandran plot of a particular protein may also serve as an important indicator of the quality of its three-dimensional structures (see below).

Torsion angles are among the most important local structural parameters that control protein folding - essentially, if we would have a way to predict the Ramachandran angles for a particular protein, we would be able to predict its fold. The torsion angles
phi and psi provide the flexibility required for the polypeptide backbone to adopt a certain fold, since the third possible torsion angle within the protein backbone (called omega, ω) is essentially flat and fixed to 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 in one plane. Thus, rotation of the protein chain can be described as rotation of the peptide bond planes relative to each other.

Definition
Torsion angles are dihedral angles, which are defined by 4 points in space. In proteins the two torsion angles φ and ψ describe the rotation of the polypeptide chain around the two bonds on both sides of the Cα atom, as shown in the figure below:

torsion angle structure

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 figure: Positive angles correspond to clockwise rotation:

torsion angle, definition

The mentioned above, restriction of the Ramachandran angles in proteins to certain values is clearly visible in the Ramachandran plot below. The plot shows that each type of secondary structure elements occupies its characteristic range of φ and ψ angles, marked α is for α-helices and β is for β-sheet on the left (from J Richardson, Adv. Pro. Chem. 34, 174-175, 1981):

Ramachandran plot, glycine compared to whole-protein structure

The horizontal axis shows φ values, while the vertical shows ψ values. Each dot on the plot shows the angles for an amino acid. Notice that the counting starts in the left hand corner from -180 and extend to +180 for both the vertical and horizontal axes. This is a convenient presentation and allows clear distinction of the characteristic regions of α-helices and β-sheets. The regions on the plot with the highest density of dots are the so-called “allowed” regions, also called low-energy regions. Some values of φ and ψ are forbidden since the involved atoms will come too close to each other, resulting in a steric clash. For a high-quality and high resolution experimental structure these regions are usually empty or almost empty - very few amino acid residues in proteins have their torsion angles within these regions. But there are sometimes exclusions from this rule - such values can be found and they most probably will result in some strain in the polypeptide chain. In such cases additional interactions will be present to stabilize such structures. They may have functional significance and may be conserved within a protein family (Pal and Chakrabarti, 2002).

Another exception from the principle of clustering around the α- and β-regions can be seen on the right plot of the above figure. In this case the Ramachandran plot shows torsion angle distribution for one single residue, glycine. Glycine 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. This is also the reason for the high conservation of glycine residues in protein families, since the presence of turns at certain positions is a characteristic of a particular fold of a structure. Another residue with special properties is proline, which in contrast to glycine fixes the torsion angles at a certain value, very close to that of an extended β-strand. Proline is often found at the end of helices and functions as a “helix disruptor”.

Theoretically, the average phi and psi values for α-helices and β-sheets should be 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 in Acta Crystallographica (http://www.fos.su.se/%7Esvenh/Conformations.pdf), you can find detailed discussion of the fine structure of φ- and ψ-angle distribution in the Ramachandran plot.

The Ramachndran plot and the quality of a protein structure
In cases when the protein X-ray structure was not properly refined, and especially for bad or wrong homology models, we may find torsion angles in disallowed regions of the Ramachandran plot − this type of deviations usually indicates problems with the structure. Based on this, the Ramachandran plot is usually used in assessing the quality of experimental structures or homology models. The image below shows two Ramachandran plots for the same structure refined at different resolutions. The structure on the left was refined sometime at the early days of protein crystallography, while the one on the right was refined using more modern refinement programs. Red indicates low-energy regions, brown allowed regions, yellow the so-called generously-allowed regions and pale-yellow marks disallowed regions. On the left plot you may see many dots in the disallowed regions, but almost none on the right (the ones which are seen are for glycine residues). You may also notice that the torsion angles on the left plot lack real clustering around secondary structure regions and have a much wider distribution, compared to the plot on the right (also compare to the left plot on the figure above). Generally this is a result of bad geometry - high resolution structures generally tend to have better clustering within the allowed regions of the plot:

Ramachandran plot for well refined structure

Thus, torsion angles outside the low-energy regions, whenever observed, should be carefully examined. They may indicate problems in the structure, but they may also be true and may provide some interesting insights into the function of the protein.
There will be further discussions of other aspects related to the
quality of experimental protein structures, and the quality of homology models.
In the next section we will move to discussing protein
secondary structure.