Torsion angles

Protein Interactions

Geometry

● A dihedral or torsion angle is the angle between two planes.

● The dihedral angle of two planes can be seen by looking at the planes "edge on", i.e., along their line of intersection.

● The dihedral angle can be defined as the angle through which plane A must be rotated (about their common line of intersection) to align it with plane B.

For 4 atoms

● The structure of a molecule can be defined with high precision by the dihedral angles between three successive chemical bond vectors.

● The dihedral angle varies only the distance between the first and fourth atoms; the other interatomic distances are constrained by the chemical bond lengths and bond angles.

For 4 atoms

● To visualize the dihedral angle of four atoms, it's helpful to look down the second bond vector

● The first atom is at 6 o'clock, the fourth atom is at roughly 2 o'clock and the second and third atoms are located in the center.

● When the fourth atom eclipses the first atom, the dihedral angle is zero; when the atoms are exactly opposite the dihedral angle is 180°.

Biological molecules● The backbone dihedral angles of

proteins are called – φ (phi, involving the backbone atoms C'-N-

Cα-C'),

– ψ (psi, involving the backbone atoms N-Cα-C'-N) and

– ω (omega, involving the backbone atoms Cα-C'-N-Cα).

● Thus, φ controls the C'-C' distance, ψ controls the N-N distance and ω controls the Cα-Cα distance.

Computation

● The dihedral angle between two planes relies on being able to efficiently generate a normal vector to each of the planes.

● One approach is to use the cross product. ● If A1, A2, and A3 are three non-collinear points on plane

A, and B1, B2, and B3 are three non-collinear points on plane B,

● then UA = (A2−A1) × (A3−A1) is orthogonal to plane A and UB = (B2−B1) × (B3−B1) is orthogonal to plane B.

Torsion angles and Ramchandran plot

● The two torsion angles of the polypeptide chain, also called Ramachandran angles, 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

Torsion angles and Ramchandran plot

● 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.

Torsion angles and Ramchandran plot

● 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.

Torsion angles and Ramchandran plot

● 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