10.torsion angles

torsion angles

1. Torsion angles
Protein Interactions

2. 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.

3. 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.

4. 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°.

5. 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.

6. 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.

9. 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

10. 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.

11. 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.

12. 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