Dihedral angle

Dihedral angle

Overview
In geometry
Geometry
Geometry arose as the field of knowledge dealing with spatial relationships. Geometry was one of the two fields of pre-modern mathematics, the other being the study of numbers ....

, a dihedral or torsion angle is the angle
Angle
In geometry, an angle is the figure formed by two rays sharing a common endpoint, called the vertex of the angle.Angles are usually presumed to be in a Euclidean plane with the circle taken for standard with regard to direction. In fact, an angle is frequently viewed as a measure of an circular arc...

 between two plane
Plane (mathematics)
In mathematics, a plane is a flat, two-dimensional surface. A plane is the two dimensional analogue of a point , a line and a space...

s.




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 between two planes denoted A and B is the angle
between their two normal
Surface normal
A surface normal, or simply normal, to a flat surface is a vector that is perpendicular to that surface. A normal to a non-flat surface at a point P on the surface is a vector perpendicular to the tangent plane to that surface at P. The word "normal" is also used as an adjective: a line normal to a...

 unit vectors and


A dihedral angle can be signed
Sign (mathematics)
In mathematics, the word sign refers to the property of being positive or negative. Every nonzero real number is either positive or negative, and therefore has a sign. Zero itself is signless, although in some contexts it makes sense to consider a signed zero...

; for example, the dihedral angle 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, .
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Encyclopedia
In geometry
Geometry
Geometry arose as the field of knowledge dealing with spatial relationships. Geometry was one of the two fields of pre-modern mathematics, the other being the study of numbers ....

, a dihedral or torsion angle is the angle
Angle
In geometry, an angle is the figure formed by two rays sharing a common endpoint, called the vertex of the angle.Angles are usually presumed to be in a Euclidean plane with the circle taken for standard with regard to direction. In fact, an angle is frequently viewed as a measure of an circular arc...

 between two plane
Plane (mathematics)
In mathematics, a plane is a flat, two-dimensional surface. A plane is the two dimensional analogue of a point , a line and a space...

s.




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 between two planes denoted A and B is the angle
between their two normal
Surface normal
A surface normal, or simply normal, to a flat surface is a vector that is perpendicular to that surface. A normal to a non-flat surface at a point P on the surface is a vector perpendicular to the tangent plane to that surface at P. The word "normal" is also used as an adjective: a line normal to a...

 unit vectors and


A dihedral angle can be signed
Sign (mathematics)
In mathematics, the word sign refers to the property of being positive or negative. Every nonzero real number is either positive or negative, and therefore has a sign. Zero itself is signless, although in some contexts it makes sense to consider a signed zero...

; for example, the dihedral angle 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, . 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
Cross product
In mathematics, the cross product, vector product, or Gibbs vector product is a binary operation on two vectors in three-dimensional space. It results in a vector which is perpendicular to both of the vectors being multiplied and normal to the plane containing them...

 and normalizing yields the normal unit 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 , and (shown in red, green and blue, respectively, in Figure 1). The vectors and define the first plane, whereas and define the second plane. The dihedral angle corresponds to an exterior spherical angle
Spherical angle
A spherical angle is a particular dihedral angle; it is the angle between two intersecting arcs on a sphere, and is measured by the angle between the planes containing the arcs ....

 (Figure 1), which is a well-defined, signed quantity.


where the two-argument atan2
Atan2
In trigonometry, the two-argument function atan2 is a variation of the arctangent function. For any real arguments and not both equal to zero, is the angle in radians between the positive -axis of a plane and the point given by the coordinates on it...

 takes care of the sign.

Dihedral angles in polyhedra


Every polyhedron, regular
Regular polyhedron
A regular polyhedron is a polyhedron whose faces are congruent regular polygons which are assembled in the same way around each vertex. A regular polyhedron is highly symmetrical, being all of edge-transitive, vertex-transitive and face-transitive - i.e. it is transitive on its flags...

 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
Antiparallel (mathematics)
-Definitions:Given two lines m_1 \, and m_2 \,, lines l_1 \, and l_2 \, are anti-parallel with respect to m_1 \, and m_2 \, if \angle 1 = \angle 2 \,....

 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 solid
Platonic solid
In geometry, a Platonic solid is a convex polyhedron that is regular, in the sense of a regular polygon. Specifically, the faces of a Platonic solid are congruent regular polygons, with the same number of faces meeting at each vertex; thus, all its edges are congruent, as are its vertices and...

s, the 4 Kepler–Poinsot polyhedra, the two quasiregular solids, and two quasiregular dual solids.

Dihedral angles of four atoms


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 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), which is equivalent to the Newman projection
Newman projection
A Newman projection, useful in alkane stereochemistry, visualizes chemical conformations of a carbon-carbon chemical bond from front to back, with the front carbon represented by a dot and the back carbon as a circle . The front carbon atom is called proximal, while the back atom is called distal...

 in chemistry. 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 is the counterclockwise angle made by the vectors (red) and (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 protein
Protein
Proteins are biochemical compounds consisting of one or more polypeptides typically folded into a globular or fibrous form, facilitating a biological function. A polypeptide is a single linear polymer chain of amino acids bonded together by peptide bonds between the carboxyl and amino groups of...

s are called φ (phi, involving the backbone atoms C'-N-Cα-C'), ψ (psi, involving the backbone atoms N-Cα-C'-N) and ω (omega, involving the backbone atoms Cα-C'-N-Cα). Thus, φ controls the C'-C' distance, ψ controls the N-N distance and ω controls the Cα-Cα distance.

The planarity of the peptide bond
Peptide bond
This article is about the peptide link found within biological molecules, such as proteins. A similar article for synthetic molecules is being created...

 usually restricts to be 180° (the typical trans case) or 0° (the rare cis case). The distance between the Cα atoms in the trans and cis isomers
Geometric isomerism
In organic chemistry, cis/trans isomerism or geometric isomerism or configuration isomerism or E/Z isomerism is a form of stereoisomerism describing the orientation of functional groups within a molecule...

 is approximately 3.8 and 2.9 Å, respectively. The cis isomer is mainly observed in Xaa-Pro
Proline
Proline is an α-amino acid, one of the twenty DNA-encoded amino acids. Its codons are CCU, CCC, CCA, and CCG. It is not an essential amino acid, which means that the human body can synthesize it. It is unique among the 20 protein-forming amino acids in that the α-amino group is secondary...

 peptide bond
Peptide bond
This article is about the peptide link found within biological molecules, such as proteins. A similar article for synthetic molecules is being created...

s (where Xaa is any amino acid
Amino acid
Amino acids are molecules containing an amine group, a carboxylic acid group and a side-chain that varies between different amino acids. The key elements of an amino acid are carbon, hydrogen, oxygen, and nitrogen...

).

The sidechain dihedral angles of protein
Protein
Proteins are biochemical compounds consisting of one or more polypeptides typically folded into a globular or fibrous form, facilitating a biological function. A polypeptide is a single linear polymer chain of amino acids bonded together by peptide bonds between the carboxyl and amino groups of...

s are denoted as χ15, depending on the distance up the sidechain. The χ1 dihedral angle is defined by atoms
N-Cα-Cβ-Cγ, the χ2 dihedral angle is defined by atoms
Cα-Cβ-Cγ-Cδ, and so on.

The sidechain dihedral angles tend to cluster near 180°, 60°, and −60°, which are called the trans, gauche+, and gauche- conformations. The choice of sidechain dihedral angles is affected by the neighbouring backbone and sidechain dihedrals; for example, the gauche+ conformation is rarely followed by the 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 acid
Nucleic acid
Nucleic acids are biological molecules essential for life, and include DNA and RNA . Together with proteins, nucleic acids make up the most important macromolecules; each is found in abundance in all living things, where they function in encoding, transmitting and expressing genetic information...

s (DNA
DNA
Deoxyribonucleic acid is a nucleic acid that contains the genetic instructions used in the development and functioning of all known living organisms . The DNA segments that carry this genetic information are called genes, but other DNA sequences have structural purposes, or are involved in...

 and RNA
RNA
Ribonucleic acid , or RNA, is one of the three major macromolecules that are essential for all known forms of life....

) and for polysaccharides.

Methods of computation


The dihedral angle between two planes relies on being able to efficiently generate a normal vector to each of the planes. One approach is to use the cross product
Cross product
In mathematics, the cross product, vector product, or Gibbs vector product is a binary operation on two vectors in three-dimensional space. It results in a vector which is perpendicular to both of the vectors being multiplied and normal to the plane containing them...

. If A1, A2, and A3 are three non-collinear points on plane A, and B1, B2, and B3 are three non-collinear points on plane B, then is orthogonal to plane A and is orthogonal to plane B. The (unsigned) dihedral angle can therefore be computed with either

Another approach to computing the dihedral angle is first to pick an arbitrary vector V that is not tangent to either of the two planes. Then applying the Gram–Schmidt process to the three vectors (A2A1, A3A1, V) produces an orthonormal basis
Orthonormal basis
In mathematics, particularly linear algebra, an orthonormal basis for inner product space V with finite dimension is a basis for V whose vectors are orthonormal. For example, the standard basis for a Euclidean space Rn is an orthonormal basis, where the relevant inner product is the dot product of...

 of space, the third vector of which will be normal to plane A. Doing the same with the vectors (B2B1, B3B1, V) yields a vector normal to plane B. The angle between the two normal vectors can then be computed by any method desired. This approach generalizes to higher dimensions, but does not work with flats
Flat (geometry)
In geometry, a flat is a subset of n-dimensional space that is congruent to a Euclidean space of lower dimension. The flats in two-dimensional space are points and lines, and the flats in three-dimensional space are points, lines, and planes....

 that have a codimension
Codimension
In mathematics, codimension is a basic geometric idea that applies to subspaces in vector spaces, and also to submanifolds in manifolds, and suitable subsets of algebraic varieties.The dual concept is relative dimension.-Definition:...

 greater than 1.

To compute the dihedral angle between two flats, it is additionally necessary to ensure that each of the two normal vectors is selected to have a minimal projection onto the other flat. The Gram–Schmidt process does not guarantee this property, but it can be guaranteed with a simple eigenvector technique. If
is a matrix of orthonormal basis vectors for flat A, and is a matrix of orthonormal basis vectors for flat B, and the eigenvector with the smallest corresponding eigenvalue of , and the eigenvector with the smallest corresponding eigenvalue of ,

then, the angle between and is the dihedral angle between A and B, even if A and B have a codimension greater than 1.

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