Dihedral angle
When the two intersecting planes are described in terms of Cartesian coordinates by the two equations the dihedral angle,
Alternatively, if nA and nB are normal vector to the planes, one has where nA · nB is the dot product of the vectors and |nA| |nB| is the product of their lengths.
However the absolute values can be and should be avoided when considering the dihedral angle of two half planes whose boundaries are the same line.
The dihedral angle of these two half planes is defined by and satisfies
[2] This is the case for kinematic chains or amino acids in a protein structure.
If u1, u2 and u3 are three consecutive bond vectors, the intersection of the half-planes is oriented, which allows defining a dihedral angle that belongs to the interval (−π, π].
This dihedral angle is defined by[3] or, using the function atan2, This dihedral angle does not depend on the orientation of the chain (order in which the point are considered) — reversing this ordering consists of replacing each vector by its opposite vector, and exchanging the indices 1 and 3.
[4][5] Every set of three non-colinear atoms of a molecule defines a half-plane.
The two types of terms can be combined so as to define four ranges of angle; 0° to ±30° synperiplanar (sp); 30° to 90° and −30° to −90° synclinal (sc); 90° to 150° and −90° to −150° anticlinal (ac); ±150° to 180° antiperiplanar (ap).
The synperiplanar conformation is also known as the syn- or cis-conformation; antiperiplanar as anti or trans; and synclinal as gauche or skew.
For macromolecular usage the symbols T, C, G+, G−, A+ and A− are recommended (ap, sp, +sc, −sc, +ac and −ac respectively).
A Ramachandran plot (also known as a Ramachandran diagram or a [φ,ψ] plot), originally developed in 1963 by G. N. Ramachandran, C. Ramakrishnan, and V. Sasisekharan,[7] is a way to visualize energetically allowed regions for backbone dihedral angles ψ against φ of amino acid residues in protein structure.
The distance between the Cα atoms in the trans and cis isomers is approximately 3.8 and 2.9 Å, respectively.
The vast majority of the peptide bonds in proteins are trans, though the peptide bond to the nitrogen of proline has an increased prevalence of cis compared to other amino-acid pairs.
[9] The side chain dihedral angles are designated with χn (chi-n).
The stability of certain sidechain dihedral angles is affected by the values φ and ψ.
[11] For instance, there are direct steric interactions between the Cγ of the side chain in the gauche+ rotamer and the backbone nitrogen of the next residue when ψ is near -60°.
[12] This is evident from statistical distributions in backbone-dependent rotamer libraries.
An angle greater than 180° exists on concave portions of a polyhedron.
Given 3 faces of a polyhedron which meet at a common vertex P and have edges AP, BP and CP, the cosine of the dihedral angle between the faces containing APC and BPC is:[13] This can be deduced from the spherical law of cosines, but can also be found by other means[14].
