Torsion angles

Many aspects of the geometry of proteins have a fixed character. Bond lengths, bond angles and the planarity of certain groups are all geometric characteristics with very limited freedom. Nevertheless, proteins can fold in a near infinite number of different structures which is caused by two facts:

• There are 20 different protein side chains (actually 19 + no side chain for glycine), and each has a different influence on the local backbone.
• Two of the three backbone torsion angles (1 in proline) have the freedom to adapt different values, and thus to adjust themselves to better match with the own side chain and the rest of the protein.

What is a torsion angle

Figure 24. If you look from the N (blue) via the two carbons (green) to the oxygen (red) you see that a clockwise rotation of about 120 degrees is needed to rotate the N on top of the O. This rotation is clock-wise rotation and therefore this torsion angle is called positive: +120 degrees. The funny thing is, that if you look from O to N, and try to rotate the O on top of the N the rotation is also clockwise. So, torsion angles are order-independent.

Question 18: Draw a square and give the X-axis as title φ and the Y-axis ψ. Make sure that (φ,ψ)=(0,0) in in the middle of the square. Let both axes run from -180^o  to +180^o. Now, use Yasara to measure the backbone torsion angles φ and ψ of the residues Thr-2 and Ser-11 in crambin. Place these two residues in your square. Now do this also for the residues 3,7,15,20,28,30 (and please do this the smart way, the bioinformatics way).

Definition of Φ and Ψ (phi, psi) and the Ramachandran plot

Once upon a time, many many years ago in a far away country called India a bioinformatician avant la lettre called Ramachandran worked without a computer but nevertheless computed nearly correct what torsion angles of amino acids should be.

In the next plot you find the combinations of Phi and Psi that are favourable for helix, strand, and left handed helix or some residues in a β-turn:

Figure 26. φ-ψ ~ (-60,-40) Helix φ-ψ ~ (-150,150) Strand φ-ψ ~ (40,40) Turn (and left handed helix, but those are very rare)

However, if you click on this figure, you see where we observe residues in the plot (i.e. which φ-ψ combinations are really observed). So Ramachandran was rather accurate, but a bit off-target with his helix.

Additionally, we know that omega (rotation about the peptide bond) is almost always around 180 degrees.

Definition of φ-ψ (phi-psi), atoms involved

Now that you know the values for φ and ψ in, for example, a helix, and you know the handedness of residues, you can look at any helix (take for example one helix from crambin) and figure out which atoms define φ and which define ψ.

The chirality around C_α

There are three dozen rules of thumb to remember the handedness of amino acids. This is just one of them:

Figure 27. This configuration can be remembered as the CORN law. Imagine looking along the H-Cα  bond with the H atom closest to you. When reading clockwise, the groups attached to the Cα  spell the word CORN. (Bioinformaticians commonly use R to indicate the generic side chain).

Most amino acids in our world are L (that is an indication of chirality). The amino acid shown in the previous picture is L. the opposite chirality, D, amino acids do occur in nature, but most times when you find one in a PDB file they are the result of an error in the structure determination.

Figure 28. In the small molecule twoD.pdb you find two D amino acids.

Question 19: Find the two D amino acids. And the second question is: Why does replacement of an L amino acid by a D amino acid lead to changes in the structure?

Question 20:

• Which atoms define φ
• Which atoms define ψ
• Which atoms define ω