Torsion Angles and the Ramachandran Plot

The two torsion angles of the polypeptide chain, also called Ramachandran angles (after the Indian physicist who first introduced the Ramachandran plot), describe the rotations of the polypeptide backbone around the bonds between N-Cα (called Phi, φ) and Cα-C (called Psi, ψ). The Ramachandran plot provides an easy way to view the distribution of torsion angles of a protein structure (RAMACHANDRAN GN, RAMAKRISHNAN C, SASISEKHARAN V., J Mol Biol., 7:95-99). It also provides an overview of allowed and disallowed regions of torsion angle values, serving as an important indicator of the quality of protein 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 3D folding. The reason is that these angles 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 (see below for the definition of the angles). This is due to the partial double-bond character of the peptide bond, which restricts rotation around the C-N bond, placing two successive alpha-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.

Torsion angles are dihedral angles, which are defined by 4 points in space. In proteins the two torsion angles phi and psi describe the rotation of the polypeptide chain around the two bonds on both sides of the Ca atom:
Protein torsion angles phi and psi
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 deviation of the A-B and C-D bonds from each-other is measured by the deviation of D from A (rotation around the B-C bond): positive angles correspond to clockwise rotation.
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The Ramachandran angles in proteins are restricted to certain values, since some angles will result in sterical clashes between main chain and side chain atoms in the polypeptide. In addition, for each type of the secondary structure elements there is a characteristic range of torsion angle values, which can clearly be seen on the Ramachnadran plot: on the left plot the region marked alpha is for alpha-helices and beta is for beta-sheet.
Ramachandran plot
The horizontal axis on the plot shows phi value, while the vertical shows psi values. Each dot on the Ramachandran plot provides the phi and psi values for an amino acid in a protein. Notice that the counting in the left hand corner starts from -180 and extends to +180 for both 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 of the Ramachandran plot, also called low-energy regions. Some values of phi and psi are forbidden since some atoms will come too close to each other, resulting in a so-called "steric clash". You probably know that when two atoms are too close to each other the energy of the system gets too high. For a high-quality 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 exclusions from this rule - sometimes 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 and, as mentioned earlier in the discussion of the basics principles of
protein structure, it allows high flexibility in the polypeptide chain, as well as torsion angles, which are normally not allowed for other amino acids residues. That is why glycine is often found in loop regions, where the polypeptide chain makes 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 protein structure. As mentioned above, another residue with special properties is proline. Proline, in contrast to glycine, fixes the torsion angles at values, which are very close to those of an extended conformation of the polypeptide (like in a beta-sheet). Proline is often found at the end of helices and functions as a helix disruptor.

Theoretically, the average phi and psi values for alpha-helices and beta-sheets should be clustered around -57, -47 and -80, +150, respectively. However, for real experimental structures these values were found to be different. A detailed discussion of the fine structure of phi- and psi-angle distribution in the Ramachandran plot can be found in the work by
Hovmöller et al., 2002.

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 will find torsion angles in disallowed regions of the Ramachandran plot - this type of deviations indeed indicates problems with the structure. Based on this the Ramachandran plot is widely used in assessing the quality of experimental structures or structures built using homology modeling. The figure below shows two Ramachandran plots for the same protein 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. You may notice that the torsion angles on the left plot lack real clustering around secondary structure regions and show a much wider distribution, compared to the plot on the right (also compare to the left plot on the figure above). Generally this is often observed for low resolution structures with bad geometry - high resolution structures tend to have better clustering within the allowed regions of the plot. On the left plot you may also see many dots in the disallowed regions, but almost none on the right (the ones which are seen are for glycine residues):

High and low resolution Ramachandran plots
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.

Basics of Protein Structure