How to calculate Dihedral Angle using Cartesian Coordinates or using RasMol

Dihedral angle is an angle between two planes.

Protein backbone dihedral angles are called φ (phi), involving the backbone atoms CO_n-1 - N_n - C^α_n - CO_n , ψ (psi) involving the backbone atoms N_n - C^α_n - CO_n - N_n+1) and ω (omega) involving the backbone atoms C^α_n - CO_n - N_n+1 - C^α_n+1). Thus, φ controls the CO_n-1 - CO_n distance, ψ controls the N_n - N_n+1 distance and ω controls the C^α_n - C^α_n+1 distance. Where n is the current atom for which you want to calculate the dihedral angles. Dihedral angle varies between -180 to +180. It is positive if rotation between atoms is clockwise and negative if rotation is anticlockwise.

To calculate phi for a residue "n" we need CO atom of n-1 residue thus we can not calculate phi for first residue of protein backbone similarly to calculate psi of residue "n" we need N atom of residue n+1 thus psi for last atom in protein backbone can not be calculated.

Atom arrangement is shown in figure 1 and 2. Figure 2 is newman projection of atom A, B, C, D, angle between A and D if we see arrangement through B-C axis is dihedral angle Ɵ. x,y,z are cartesian coordinates of atoms.

Thus using cartesian coordinates of these atoms these angles can be calculated by following method -

Using atom A, B, C, D we can define two planes. Plane 1 will have atom A-B-C and plane 2 will have atom B-C-D atoms. The angle between these two planes is dihedral angle.

Using cartesian coordinates of four atoms we can calculate 3 vectors v1, v2 and v3 as shown in figure 1.

Eg.-   v1 = ( x2-x1 )  , ( y2-y1 )  , ( z2-z1 ) likewise v2 and v3.

Get normal vectors to both of planes, first normal vector will be cross product of v1 and v2 

(v1 X v2), second normal vector will be cross product of vector v2 and v3 (v2 X v3). Name these normal vectors v4 and v5.

The angle between these two vectors v4 and v5 will be dot product of these vectors. This is dihedral angle.

To get rotation use vector v2. Vector produced by cross product of v4 and v5 will be either parallel or anti-parallel to v2 that can be used to determine rotation clockwise or anticlockwise.

Finding out Dihedral angle using RasMol - 

To find out dihedral angle using RasMol first choose CPK color by selecting "Colours" -> "CPK" and label atoms by selecting "Settings" -> "Pick Label", then clicking on individual atom or by selecting "Options" -> "Labels".

then select "Settings" -> "Pick Torsion" and choose four atoms for which we have to find out dihedral angle,
To find out phi of a residue "n" select atom CO of residue n-1 then N of n then C^α of n then CO of n
To find out psi of a residue "n" select N of n then C^α of n then CO of n then N of residue n+1.

Find out Dihedral Angle using RasMol

Find out Dihedral Angle using RasMol

N, CA, CO atoms in PDB Atom record will be represented as "N", "CA", "C" (column number 3)