![Dihedral angle](https://upload.wikimedia.org/wikipedia/commons/3/3c/Dihedral_angle.svg)
A dihedral angle is the angle between two intersecting planes or
half-planes. In
chemistry, it is the clockwise angle between half-planes through two sets of three atoms, having two atoms in common. In
solid geometry
In mathematics, solid geometry or stereometry is the traditional name for the geometry of three-dimensional, Euclidean spaces (i.e., 3D geometry).
Stereometry deals with the measurements of volumes of various solid figures (or 3D figures), inc ...
, it is defined as the
union
Union commonly refers to:
* Trade union, an organization of workers
* Union (set theory), in mathematics, a fundamental operation on sets
Union may also refer to:
Arts and entertainment
Music
* Union (band), an American rock group
** ''Un ...
of a
line and two half-planes that have this line as a common
edge
Edge or EDGE may refer to:
Technology Computing
* Edge computing, a network load-balancing system
* Edge device, an entry point to a computer network
* Adobe Edge, a graphical development application
* Microsoft Edge, a web browser developed ...
. In
higher dimension
In physics and mathematics, the dimension of a mathematical space (or object) is informally defined as the minimum number of coordinates needed to specify any point within it. Thus, a line has a dimension of one (1D) because only one coordin ...
s, a dihedral angle represents the angle between two
hyperplanes.
The planes of a flying machine are said to be at positive dihedral angle when both starboard and port main planes (commonly called wings) are upwardly inclined to the lateral axis. When downwardly inclined they are said to be at a negative dihedral angle.
Mathematical background
When the two intersecting planes are described in terms of
Cartesian coordinates by the two equations
:
:
the dihedral angle,
between them is given by:
:
and satisfies
Alternatively, if and are
normal vector to the planes, one has
:
where is the
dot product
In mathematics, the dot product or scalar productThe term ''scalar product'' means literally "product with a scalar as a result". It is also used sometimes for other symmetric bilinear forms, for example in a pseudo-Euclidean space. is an alge ...
of the vectors and is the product of their lengths.
The absolute value is required in above formulas, as the planes are not changed when changing all coefficient signs in one equation, or replacing one normal vector by its opposite.
However the
absolute values can be and should be avoided when considering the dihedral angle of two
half plane
In geometry, a half-space is either of the two parts into which a plane divides the three-dimensional Euclidean space.
If the space is two-dimensional, then a half-space is called a half-plane (open or closed). A half-space in a one-dimensional ...
s whose boundaries are the same line. In this case, the half planes can be described by a point of their intersection, and three vectors , and such that , and belong respectively to the intersection line, the first half plane, and the second half plane. The ''dihedral angle of these two half planes'' is defined by
:
,
and satisfies
In this case, switching the two half-planes gives the same result, and so does replacing
with
In chemistry (see below), we define a dihedral angle such that replacing
with
changes the sign of the angle, which can be between and .
In polymer physics
In some scientific areas such as
polymer physics, one may consider a chain of points and links between consecutive points. If the points are sequentially numbered and located at positions , , , etc. then bond vectors are defined by =−, =−, and =−, more generally.
This is the case for
kinematic chains or
amino acid
Amino acids are organic compounds that contain both amino and carboxylic acid functional groups. Although hundreds of amino acids exist in nature, by far the most important are the alpha-amino acids, which comprise proteins. Only 22 alpha a ...
s in a
protein structure
Protein structure is the three-dimensional arrangement of atoms in an amino acid-chain molecule. Proteins are polymers specifically polypeptides formed from sequences of amino acids, the monomers of the polymer. A single amino acid monom ...
. In these cases, one is often interested in the half-planes defined by three consecutive points, and the dihedral angle between two consecutive such half-planes. If , and are three consecutive bond vectors, the intersection of the half-planes is oriented, which allows defining a dihedral angle that belongs to the interval . This dihedral angle is defined by
:
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. Both operations do not change the cosine, but change the sign of the sine. Thus, together, they do not change the angle.
A simpler formula for the same dihedral angle is the following (the proof is given below)
:
or equivalently,
:
This can be deduced from previous formulas by using the
vector quadruple product
In mathematics, the quadruple product is a product of four vectors in three-dimensional Euclidean space. The name "quadruple product" is used for two different products, the scalar-valued scalar quadruple product and the vector-valued vector qua ...
formula, and the fact that a
scalar triple product
In geometry and algebra, the triple product is a product of three 3-dimensional vectors, usually Euclidean vectors. The name "triple product" is used for two different products, the scalar-valued scalar triple product and, less often, the vector- ...
is zero if it contains twice the same vector:
:
Given the definition of the
cross product, this means that
is the angle in the clockwise direction of the fourth atom compared to the first atom, while looking down the axis from the second atom to the third. Special cases (one may say the usual cases) are
,
and
, which are called the ''trans'', ''gauche
+'', and ''gauche
−'' conformations.
In stereochemistry
![Dihedral angles of Butane](https://upload.wikimedia.org/wikipedia/commons/4/41/Dihedral_angles_of_Butane.svg)
In
stereochemistry, a torsion angle is defined as a particular example of a dihedral angle, describing the geometric relation of two parts of a molecule joined by a
chemical bond
A chemical bond is a lasting attraction between atoms or ions that enables the formation of molecules and crystals. The bond may result from the electrostatic force between oppositely charged ions as in ionic bonds, or through the sharing of ...
. Every set of three non-colinear atoms of a
molecule
A molecule is a group of two or more atoms held together by attractive forces known as chemical bonds; depending on context, the term may or may not include ions which satisfy this criterion. In quantum physics, organic chemistry, and bioche ...
defines a half-plane. As explained above, when two such half-planes intersect (i.e., a set of four consecutively-bonded atoms), the angle between them is a dihedral angle. Dihedral angles are used to specify the
molecular conformation.
Stereochemical
Stereochemistry, a subdiscipline of chemistry, involves the study of the relative spatial arrangement of atoms that form the structure of molecules and their manipulation. The study of stereochemistry focuses on the relationships between stereois ...
arrangements corresponding to angles between 0° and ±90° are called ''syn'' (s), those corresponding to angles between ±90° and 180° ''anti'' (a). Similarly, arrangements corresponding to angles between 30° and 150° or between −30° and −150° are called ''clinal'' (c) and those between 0° and ±30° or ±150° and 180° are called ''periplanar'' (p).
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 example, with ''n''-
butane two planes can be specified in terms of the two central carbon atoms and either of the methyl carbon atoms. The ''syn''-conformation shown above, with a dihedral angle of 60° is less stable than the ''anti''-conformation with a dihedral angle of 180°.
For macromolecular usage the symbols T, C, G
+, G
−, A
+ and A
− are recommended (ap, sp, +sc, −sc, +ac and −ac respectively).
Proteins
A
Ramachandran plot
In biochemistry, a Ramachandran plot (also known as a Rama plot, a Ramachandran diagram or a †,ψplot), originally developed in 1963 by G. N. Ramachandran, C. Ramakrishnan, and V. Sasisekharan, is a way to visualize energetically allowed regions ...
(also known as a Ramachandran diagram or a
'φ'',''ψ''plot), originally developed in 1963 by
G. N. Ramachandran, C. Ramakrishnan, and V. Sasisekharan, is a way to visualize energetically allowed regions for backbone dihedral angles ''ψ'' against ''φ'' of
amino acid
Amino acids are organic compounds that contain both amino and carboxylic acid functional groups. Although hundreds of amino acids exist in nature, by far the most important are the alpha-amino acids, which comprise proteins. Only 22 alpha a ...
residues in
protein structure
Protein structure is the three-dimensional arrangement of atoms in an amino acid-chain molecule. Proteins are polymers specifically polypeptides formed from sequences of amino acids, the monomers of the polymer. A single amino acid monom ...
.
In a
protein
Proteins are large biomolecules and macromolecules that comprise one or more long chains of amino acid residues. Proteins perform a vast array of functions within organisms, including catalysing metabolic reactions, DNA replication, res ...
chain three dihedral angles are defined:
* ω (omega) is the angle in the chain C
α − C' − N − C
α,
* φ (phi) is the angle in the chain C' − N − C
α − C'
* ψ (psi) is the angle in the chain N − C
α − C' − N (called ''φ′'' by Ramachandran)
The figure at right illustrates the location of each of these angles (but it does not show correctly the way they are defined).
The planarity of the
peptide bond usually restricts ''ω'' to be 180° (the typical ''
trans
Trans- is a Latin prefix meaning "across", "beyond", or "on the other side of".
Used alone, trans may refer to:
Arts, entertainment, and media
* Trans (festival), a former festival in Belfast, Northern Ireland, United Kingdom
* ''Trans'' (fil ...
'' case) or 0° (the rare ''
cis
Cis or cis- may refer to:
Places
* Cis, Trentino, in Italy
* In Poland:
** Cis, Świętokrzyskie Voivodeship, south-central
** Cis, Warmian-Masurian Voivodeship, north
Math, science and biology
* cis (mathematics) (cis(''θ'')), a trigonome ...
'' case). The distance between the C
α atoms in the ''trans'' and ''cis''
isomers
In chemistry, isomers are molecules or polyatomic ions with identical molecular formulae – that is, same number of atoms of each element – but distinct arrangements of atoms in space. Isomerism is existence or possibility of isomers.
...
is approximately 3.8 and 2.9 Ã…, 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.
The side chain dihedral angles are designated with ''χ
n'' (chi-''n''). They tend to cluster near 180°, 60°, and −60°, which are called the ''trans'', ''gauche
−'', and ''gauche
+'' conformations. The stability of certain sidechain dihedral angles is affected by the values ''φ'' and ''ψ''. For instance, there are direct steric interactions between the C''γ'' of the side chain in the ''gauche
+'' rotamer and the backbone nitrogen of the next residue when ''ψ'' is near -60°. This is evident from statistical distributions in
backbone-dependent rotamer libraries.
Converting from dihedral angles to Cartesian coordinates in chains
It is common to represent polymers backbones, notably proteins, in
internal coordinates
In chemistry, the Z-matrix is a way to represent a system built of atoms. A Z-matrix is also known as an internal coordinate representation. It provides a description of each atom in a molecule in terms of its atomic number, bond length, bond ...
; that is, a list of consecutive dihedral angles and bond lengths. However, some types of
computational chemistry instead use
cartesian coordinates. In computational structure optimization, some programs need to flip back and forth between these representations during their iterations. This task can dominate the calculation time. For processes with many iterations or with long chains, it can also introduce cumulative numerical inaccuracy. While all conversion algorithms produce mathematically identical results, they differ in speed and numerical accuracy.
Geometry
Every polyhedron has a dihedral angle at every edge describing the relationship of the two faces that share that edge. This dihedral angle, also called the ''face angle'', is measured as the
internal angle
In geometry, an angle of a polygon is formed by two sides of the polygon that share an endpoint. For a simple (non-self-intersecting) polygon, regardless of whether it is convex or non-convex, this angle is called an interior angle (or ) if ...
with respect to the polyhedron. An angle of 0° means the face normal vectors are
antiparallel and the faces overlap each other, which implies that it is part of a
degenerate
Degeneracy, degenerate, or degeneration may refer to:
Arts and entertainment
* Degenerate (album), ''Degenerate'' (album), a 2010 album by the British band Trigger the Bloodshed
* Degenerate art, a term adopted in the 1920s by the Nazi Party i ...
polyhedron. An angle of 180° means the faces are parallel, as in a
tiling
Tiling may refer to:
*The physical act of laying tiles
* Tessellations
Computing
*The compiler optimization of loop tiling
*Tiled rendering, the process of subdividing an image by regular grid
*Tiling window manager
People
*Heinrich Sylvester T ...
. An angle greater than 180° exists on concave portions of a polyhedron.
Every dihedral angle in an
edge-transitive
In geometry, a polytope (for example, a polygon or a polyhedron) or a tiling is isotoxal () or edge-transitive if its symmetries act transitively on its edges. Informally, this means that there is only one type of edge to the object: given t ...
polyhedron has the same value. This includes the 5
Platonic solid
In geometry, a Platonic solid is a convex, regular polyhedron in three-dimensional Euclidean space. Being a regular polyhedron means that the faces are congruent (identical in shape and size) regular polygons (all angles congruent and all e ...
s, the 13
Catalan solid
In mathematics, a Catalan solid, or Archimedean dual, is a dual polyhedron to an Archimedean solid. There are 13 Catalan solids. They are named for the Belgian mathematician Eugène Catalan, who first described them in 1865.
The Catalan s ...
s, the 4
Kepler–Poinsot polyhedra, the two quasiregular solids, and two quasiregular dual solids.
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:
:
See also
*
Atropisomer
References
{{Reflist
External links
The Dihedral Angle in Woodworking at Tips.FMgives a step-by-step derivation of these exact values.
Stereochemistry
Protein structure
Euclidean solid geometry
Angle