Dihedral angle

In aerospace engineering, the dihedral is the angle between the two wings; see dihedral.

In geometry, the angle between two planes is called their dihedral angle.

Figure 1: Dihedral angle of three vectors, defined as an exterior spherical angle. The longer and shorter black segments are arcs of the great circles passing through $\mathbf {b} _{1}$ and $\mathbf {b} _{2}$ and through $\mathbf {b} _{2}$ and $\mathbf {b} _{3}$, respectively.

Figure 2: Dihedral angle defined by three bond vectors (shown in red, green and blue) connecting four atoms.

Figure 3: Dihedral angle defined by three bond vectors (shown in red, green and blue) connecting four atoms. From this perspective, the second bond vector (green) is coming out of the page.

Figure 4: The backbone dihedral angles of proteins.

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 $\phi _{AB}$ between two planes denoted A and B is the angle between their two normal unit vectors $\mathbf {n} _{A}$ and $\mathbf {n} _{B}$

$\cos \phi _{AB}=\mathbf {n} _{A}\cdot \mathbf {n} _{B}$

A dihedral angle can be signed; for example, the dihedral angle $\phi _{AB}$ 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. Thus, $\phi _{AB}=-\phi _{BA}$. For precision, one should specify the angle or its supplement, since both rotations will cause the planes to coincide.

Alternative definitions

Since a plane can be defined in several ways (e.g., by vectors or points in them, or by their normal vectors), there are several equivalent definitions of a dihedral angle.

Any plane can be defined by two non-collinear vectors lying in that plane; taking their cross product and normalizing yields the normal vector to the plane. Thus, a dihedral angle can be defined by four, pairwise non-collinear vectors.

We may also define the dihedral angle of three non-collinear vectors $\mathbf {b} _{1}$, $\mathbf {b} _{2}$ and $\mathbf {b} _{3}$ (shown in red, green and blue, respectively, in Figure 1). The vectors $\mathbf {b} _{1}$ and $\mathbf {b} _{2}$ define the first plane, whereas $\mathbf {b} _{2}$ and $\mathbf {b} _{3}$ define the second plane. The dihedral angle corresponds to an exterior spherical angle (Figure 1), which is a well-defined, signed quantity.

$\phi =\mathrm {atan2} \left(|\mathbf {b} _{2}|\mathbf {b} _{1}\cdot [\mathbf {b} _{2}\times \mathbf {b} _{3}],[\mathbf {b} _{1}\times \mathbf {b} _{2}]\cdot [\mathbf {b} _{2}\times \mathbf {b} _{3}]\right)$

where the two-argument atan2 takes care of the sign.

Dihedral angles in polyhedra

Every polyhedron, regular and irregular, convex and concave, has a dihedral angle at every edge.

A dihedral angle (also called the face angle) is the internal angle at which two adjacent faces meet. An angle of zero degrees means the face normal vectors are antiparallel and the faces overlap each other (Implying part of a degenerate polyhedron). An angle of 180 degrees means the faces are parallel (like a tiling). An angle greater than 180 exists on concave portions of a polyhedron.

Every dihedral angle in an edge-transitive polyhedron has the same value. This includes the 5 Platonic solids, the 4 Kepler-Poinsot solids, the two quasiregular solids, and two quasiregular dual solids.

See Table of polyhedron dihedral angles.

Dihedral angles of four atoms

To a good approximation, the bond lengths and bond angles of most molecules do not change between synthesis and degradation. Hence, the structure of a molecule can be defined with high precision by the dihedral angles between three successive chemical bond vectors (Figure 2). The dihedral angle $\phi$ varies only the distance between the first and fourth atoms; the other interatomic distances are constrained by the chemical bond lengths and bond angles.

To visualize the dihedral angle of four atoms, it's helpful to look down the second bond vector (Figure 3). 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. The second bond vector is coming out of the page. The dihedral angle $\phi$ is the counterclockwise angle made by the vectors $\mathbf {b} _{1}$ (red) and $\mathbf {b} _{3}$ (blue). When the fourth atom eclipses the first atom, the dihedral angle is zero; when the atoms are exactly opposite (as in Figure 2), the dihedral angle is 180°.

Dihedral angles of biological molecules

The backbone dihedral angles of proteins are called $\phi$ (involving the backbone atoms $\mathrm {C'-N-C^{\alpha }-C'}$), $\psi$ (involving the backbone atoms $\mathrm {N-C^{\alpha }-C'-N}$) and $\omega$ (involving the backbone atoms $\mathrm {C^{\alpha }-C'-N-C^{\alpha }}$). Thus, $\phi$ controls the $\mathrm {C'-C'}$ distance, $\psi$ controls the $\mathrm {N-N}$ distance and $\omega$ controls the $\mathrm {C^{\alpha }-C^{\alpha }}$ distance.

The planarity of the peptide bond usually restricts $\omega$ to be 180° (the typical trans case) or 0° (the rare cis case). The distance between the $\mathrm {C^{\alpha }}$ atoms in the trans and cis isomers is approximately 3.8 and 2.9 Å, respectively. The cis isomer is mainly observed in Xaa-Pro peptide bonds (where Xaa is any amino acid).

The sidechain dihedral angles of proteins are denoted as $\chi _{1}$-$\chi _{5}$, depending on the distance up the sidechain. The $\chi _{1}$ dihedral angle is defined by atoms $\mathrm {N-C^{\alpha }-C^{\beta }-C^{\gamma }}$, the $\chi _{2}$ dihedral angle is defined by atoms $\mathrm {C^{\alpha }-C^{\beta }-C^{\gamma }-C^{\delta }}$, and so on.

The sidechain dihedral angles tend to cluster near 180°, 60°, and -60°, which are called the $trans$, $\mathrm {gauche} ^{+}$ and $\mathrm {gauche} ^{-}$ conformations. The choice of sidechain dihedral angles is affected by the neighboring backbone and sidechain dihedrals; for example, the $\mathrm {gauche} ^{+}$ conformation is rarely followed by the $\mathrm {gauche} ^{+}$ conformation (and vice versa) because of the increased likelihood of atomic collisions.

Dihedral angles have also been defined by the IUPAC for other molecules, such as the nucleic acids (DNA and RNA) and for polysaccharides.

See also

• Ramachandran plot

External links

• Analysis of the 5 Regular Polyhedra gives a step-by-step derivation of these exact values.
• Olshevsky, George. "Dihedral angle". Glossary for Hyperspace. Archived from the original on 4 February 2007.