0.1 Representing proteins in silico and protein forward kinematics  (Page 3/7)

Dihedral angles

In most organic molecules, including proteins, the most important internal degree of freedom is rotation about dihedral (torsional) angles . A dihedral angle is defined by four consecutively bonded atoms. Given four consecutive atoms $A_{i-2}$ , $A_{i-1}$ , $A_i$ , and $A_{i+1}$ , the dihedral angle is defined as the smallest angle between the planes ${\pi }_1$ and ${\pi }_2$ , as shown in the figure. Variation of the dihedral angle is a consequence of rotation of the two outer bonds about the central bond.

Dihedral representation of protein conformations

All amino acids share the same core of one nitrogen, two carbon, and one oxygen atoms. This shared core makes up the backbone of the protein. There are two freely rotatable backbone dihedral angles per amino acid residue in the protein chain: the first, designated $\phi$ , is a consequence of the rotation about the bond between $N$ and $C_{\alpha }$ , and the other, $\psi$ , which is a consequence of the rotation about the bond between $C_{\alpha }$ and $C$ . The peptide bond between C of one residue and N of the adjacent residue is not rotatable.

The number of backbone dihedrals per amino acid is 2, but the number of side chain dihedrals varies with the length of the side chain. Its value ranges from 0, in the case of glycine, which has no sidechain, to 5 in the case of arginine.

Protein forward kinematics

Kinematics is a branch of mechanics concerned with how objects move in the absence of mass (inertia) and forces. You can imagine that varying the dihedral angles will move a protein's atoms relative to each other in space. The problem of computing the new spatial locations of the atoms given a set of dihedral rotations is known as the forward kinematics problem.