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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 π 1 and π 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.

A dihedral angle

π 1 is the plane uniquely defined by the first three atoms A i 2 , A i 1 , and A i . Similarly, π 2 is the plane uniquely defined by the last three atoms A i 1 , and A i , and A i 1 . The dihedral angle, θ, is defined as the smallest anglebetween these two planes. You can read more about the angle between two intersecting planes here .
In this module, because bond lengths and bond angles are being ignored as underlying degrees of freedom of a protein, the only remaining degrees of freedom are the dihedral rotations. Representing protein conformations with the dihedral anglesas the only underlying degrees of freedom is known as the idealized or rigid geometry model . Ignoring bond lengths and bond angles greatly reduces the number of degrees of freedom and therefore the computational complexity of representing and manipulating protein structures. Even more efficient representations which reduce the number of degrees of freedom even further exist , but these are beyond the scope of this introduction.

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 φ , is a consequence of the rotation about the bond between N and C α , and the other, ψ , which is a consequence of the rotation about the bond between C α 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.

Dihedral angles in arginine

The backbone atoms appear at the bottom of the illustration (the peptide bond is not rotatable). The sidechain dihedrals are conventionally designated by χ and a subscript.
One can generate different three dimensional structures of the same protein by varying the dihedral angles. There are 2N backbone dihedral DOFs for a protein with N amino acids, and up to 4N side chain dihedrals that one can vary to generate new protein conformations. Changes in backbone dihedral angles generally have a greater effect on the overall shape of the protein than changes in side chain dihedral angles. Think about why.

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.

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Source:  OpenStax, Geometric methods in structural computational biology. OpenStax CNX. Jun 11, 2007 Download for free at http://cnx.org/content/col10344/1.6
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