Movatterモバイル変換

Dihedral angle

From Wikipedia, the free encyclopedia

Angle between two planes in space

This article is about the geometry concept. For the aeronautics concept, seeDihedral (aeronautics). For other uses, seeDihedral.

Angle between two half-planes (α, β, pale blue) in a third plane (red) perpendicular to line of intersection.

Adihedral angle is theangle between twointersecting planes orhalf-planes. It is a plane angle formed on a third plane, perpendicular to theline of intersection between the two planes or the commonedge between the two half-planes. Inhigher dimensions, a dihedral angle represents the angle between twohyperplanes. Inchemistry, it is the clockwise angle between half-planes through two sets of threeatoms, having two atoms in common.

Mathematical background

[edit]

When the two intersecting planes are described in terms ofCartesian coordinates by the two equations

a_{1}x+b_{1}y+c_{1}z+d_{1}=0

a_{2}x+b_{2}y+c_{2}z+d_{2}=0

the dihedral angle, $\varphi$ between them is given by:

\cos \varphi ={\frac {\left\vert a_{1}a_{2}+b_{1}b_{2}+c_{1}c_{2}\right\vert }{{\sqrt {a_{1}^{2}+b_{1}^{2}+c_{1}^{2}}}{\sqrt {a_{2}^{2}+b_{2}^{2}+c_{2}^{2}}}}}

and satisfies $0\leq \varphi \leq \pi /2.$ It can easily be observed that the angle is independent of $d_{1}$ and $d_{2}$ .

Alternatively, ifn_A andn_B arenormal vector to the planes, one has

\cos \varphi ={\frac {\left\vert \mathbf {n} _{\mathrm {A} }\cdot \mathbf {n} _{\mathrm {B} }\right\vert }{|\mathbf {n} _{\mathrm {A} }||\mathbf {n} _{\mathrm {B} }|}}

wheren_A · n_B is thedot product of the vectors and|n_A| |n_B| is the product of their lengths.^[1]

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 theabsolute values can be and should be avoided when considering the dihedral angle of twohalf planes whose boundaries are the same line. In this case, the half planes can be described by a pointP of their intersection, and three vectorsb₀,b₁ andb₂ such thatP +b₀,P +b₁ andP +b₂ belong respectively to the intersection line, the first half plane, and the second half plane. Thedihedral angle of these two half planes is defined by

\cos \varphi ={\frac {(\mathbf {b} _{0}\times \mathbf {b} _{1})\cdot (\mathbf {b} _{0}\times \mathbf {b} _{2})}{|\mathbf {b} _{0}\times \mathbf {b} _{1}||\mathbf {b} _{0}\times \mathbf {b} _{2}|}}

and satisfies $0\leq \varphi <\pi .$ In this case, switching the two half-planes gives the same result, and so does replacing $\mathbf {b} _{0}$ with $-\mathbf {b} _{0}.$ In chemistry (see below), we define a dihedral angle such that replacing $\mathbf {b} _{0}$ with $-\mathbf {b} _{0}$ changes the sign of the angle, which can be between−π andπ.

In polymer physics

[edit]

In some scientific areas such aspolymer physics, one may consider a chain of points and links between consecutive points. If the points are sequentially numbered and located at positionsr₁,r₂,r₃, etc. then bond vectors are defined byu₁=r₂−r₁,u₂=r₃−r₂, andu_i=r_i+1−r_i, more generally.^[2] This is the case forkinematic chains oramino acids in aprotein structure. 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. Ifu₁,u₂ andu₃ 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]

{\begin{aligned}\cos \varphi &={\frac {(\mathbf {u} _{1}\times \mathbf {u} _{2})\cdot (\mathbf {u} _{2}\times \mathbf {u} _{3})}{|\mathbf {u} _{1}\times \mathbf {u} _{2}|\,|\mathbf {u} _{2}\times \mathbf {u} _{3}|}}\\\sin \varphi &={\frac {\mathbf {u} _{2}\cdot ((\mathbf {u} _{1}\times \mathbf {u} _{2})\times (\mathbf {u} _{2}\times \mathbf {u} _{3}))}{|\mathbf {u} _{2}|\,|\mathbf {u} _{1}\times \mathbf {u} _{2}|\,|\mathbf {u} _{2}\times \mathbf {u} _{3}|}},\end{aligned}}

or, using the functionatan2,

\varphi =\operatorname {atan2} (\mathbf {u} _{2}\cdot ((\mathbf {u} _{1}\times \mathbf {u} _{2})\times (\mathbf {u} _{2}\times \mathbf {u} _{3})),|\mathbf {u} _{2}|\,(\mathbf {u} _{1}\times \mathbf {u} _{2})\cdot (\mathbf {u} _{2}\times \mathbf {u} _{3})).

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)

{\begin{aligned}\cos \varphi &={\frac {(\mathbf {u} _{1}\times \mathbf {u} _{2})\cdot (\mathbf {u} _{2}\times \mathbf {u} _{3})}{|\mathbf {u} _{1}\times \mathbf {u} _{2}|\,|\mathbf {u} _{2}\times \mathbf {u} _{3}|}}\\\sin \varphi &={\frac {|\mathbf {u} _{2}|\,\mathbf {u} _{1}\cdot (\mathbf {u} _{2}\times \mathbf {u} _{3})}{|\mathbf {u} _{1}\times \mathbf {u} _{2}|\,|\mathbf {u} _{2}\times \mathbf {u} _{3}|}},\end{aligned}}

or equivalently,

\varphi =\operatorname {atan2} (|\mathbf {u} _{2}|\,\mathbf {u} _{1}\cdot (\mathbf {u} _{2}\times \mathbf {u} _{3}),(\mathbf {u} _{1}\times \mathbf {u} _{2})\cdot (\mathbf {u} _{2}\times \mathbf {u} _{3})).

This can be deduced from previous formulas by using thevector quadruple product formula, and the fact that ascalar triple product is zero if it contains twice the same vector:

(\mathbf {u} _{1}\times \mathbf {u} _{2})\times (\mathbf {u} _{2}\times \mathbf {u} _{3})=[(\mathbf {u} _{2}\times \mathbf {u} _{3})\cdot \mathbf {u} _{1}]\mathbf {u} _{2}-[(\mathbf {u} _{2}\times \mathbf {u} _{3})\cdot \mathbf {u} _{2}]\mathbf {u} _{1}=[(\mathbf {u} _{2}\times \mathbf {u} _{3})\cdot \mathbf {u} _{1}]\mathbf {u} _{2}

Given the definition of thecross product, this means that $\varphi$ 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 $\varphi =\pi$ , $\varphi =+\pi /3$ and $\varphi =-\pi /3$ , which are called thetrans,gauche⁺, andgauche⁻ conformations.

In stereochemistry

[edit]

This sectionduplicates the scope of other articles, specificallyTorsion angle. Pleasediscuss this issue and help introduce asummary style to the section by replacing the section with a link and a summary or bysplitting the content into a new article.

Main article:Torsion angle


Configuration names according to dihedral angle	synn-Butane in the gauche⁻ conformation (−60°) Newman projection	synn-Butane sawhorse projection

Free energy diagram ofn-butane as a function of dihedral angle.

Atorsion angle, found instereochemistry, is a particular example of a dihedral angle describing the geometric relation of two parts of a molecule joined by achemical bond.^[4]^[5] Every set of three non-colinear atoms of amolecule 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 themolecular conformation.^[6]Stereochemical arrangements corresponding to angles between 0° and ±90° are calledsyn (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 calledclinal (c) and those between 0° and ±30° or ±150° and 180° are calledperiplanar (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 thesyn- orcis-conformation; antiperiplanar asanti ortrans; and synclinal asgauche orskew.

For example, withn-butane two planes can be specified in terms of the two central carbon atoms and either of the methyl carbon atoms. Thesyn-conformation shown above, with a dihedral angle of 60° is less stable than theanti-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

[edit]

ARamachandran plot (also known as a Ramachandran diagram or a [φ,ψ] plot), originally developed in 1963 byG. N. Ramachandran, C. Ramakrishnan, and V. Sasisekharan,^[7] is a way to visualize energetically allowed regions for backbone dihedral anglesψ againstφ ofamino acid residues inprotein structure.

In aprotein 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 (calledφ′ 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).^[8]

The planarity of thepeptide bond usually restrictsω to be 180° (the typicaltrans case) or 0° (the rarecis case). The distance between the C^α atoms in thetrans andcisisomers is approximately 3.8 and 2.9 Å, respectively. The vast majority of the peptide bonds in proteins aretrans, though the peptide bond to the nitrogen ofproline has an increased prevalence ofcis compared to other amino-acid pairs.^[9]

The side chain dihedral angles are designated withχ_n (chi-n).^[10] They tend to cluster near 180°, 60°, and −60°, which are called thetrans,gauche⁻, andgauche⁺ conformations. The stability of certain sidechain dihedral angles is affected by the valuesφ andψ.^[11] For instance, there are direct steric interactions between the Cγ of the side chain in thegauche⁺ rotamer and the backbone nitrogen of the next residue whenψ is near −60°.^[12] This is evident from statistical distributions inbackbone-dependent rotamer libraries.

Dihedral angles have also been defined by theIUPAC for other molecules, such as thenucleic acids (DNA andRNA) and forpolysaccharides.

In polyhedra

[edit]

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 theface angle, is measured as theinternal angle with respect to the polyhedron. An angle of 0° means the face normal vectors areantiparallel and the faces overlap each other, which implies that it is part of adegenerate polyhedron. An angle of 180° means the faces are parallel, as in atiling. An angle greater than 180° exists on concave portions of a polyhedron.

Every dihedral angle in a polyhedron that isisotoxal and/orisohedral has the same value. This includes the 5Platonic solids, the 13Catalan solids, the 4Kepler–Poinsot polyhedra, the 2 convexquasiregular polyhedra, and the 2 infinite families ofbipyramids andtrapezohedra.

Law of cosines for dihedral angle

[edit]

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]

\cos \varphi ={\frac {\cos(\angle \mathrm {APB} )-\cos(\angle \mathrm {APC} )\cos(\angle \mathrm {BPC} )}{\sin(\angle \mathrm {APC} )\sin(\angle \mathrm {BPC} )}}

This can be deduced from thespherical law of cosines, but can also be found by other means.^[14]

Higher dimensions

[edit]

Inm-dimensional Euclidean space, the dihedral angle⁠ $\varphi$ ⁠ between the twohyperplanes defined by the equations $\mathbf {n} _{\mathrm {A} }\cdot \mathbf {x} =c_{A}$ $\mathbf {n} _{\mathrm {B} }\cdot \mathbf {x} =c_{B}$ for vectorsn_A,n_B,x ∈R^m and constantsc_A andc_B, is given by $\cos \varphi ={\frac {\left\vert \mathbf {n} _{\mathrm {A} }\cdot \mathbf {n} _{\mathrm {B} }\right\vert }{|\mathbf {n} _{\mathrm {A} }||\mathbf {n} _{\mathrm {B} }|}}\,.$

References

[edit]

^"Angle Between Two Planes".TutorVista.com. Archived fromthe original on 2020-10-28. Retrieved2018-07-06.
^Kröger, Martin (2005).Models for polymeric and anisotropic liquids. Springer.ISBN 3540262105.
^Blondel, Arnaud; Karplus, Martin (7 Dec 1998). "New formulation for derivatives of torsion angles and improper torsion angles in molecular mechanics: Elimination of singularities".Journal of Computational Chemistry.17 (9):1132–1141.doi:10.1002/(SICI)1096-987X(19960715)17:9<1132::AID-JCC5>3.0.CO;2-T.
^IUPAC,Compendium of Chemical Terminology, 5th ed. (the "Gold Book") (2025). Online version: (2006–) "Torsion angle".doi:10.1351/goldbook.T06406
^IUPAC,Compendium of Chemical Terminology, 5th ed. (the "Gold Book") (2025). Online version: (2006–) "Dihedral angle".doi:10.1351/goldbook.D01730
^Anslyn, Eric; Dennis Dougherty (2006).Modern Physical Organic Chemistry. University Science. p. 95.ISBN 978-1891389313.
^Ramachandran, G. N.; Ramakrishnan, C.; Sasisekharan, V. (1963). "Stereochemistry of polypeptide chain configurations".Journal of Molecular Biology.7:95–9.doi:10.1016/S0022-2836(63)80023-6.PMID 13990617.
^Richardson, J. S. (1981). "The Anatomy and Taxonomy of Protein Structure".Anatomy and Taxonomy of Protein Structures. Advances in Protein Chemistry. Vol. 34. pp. 167–339.doi:10.1016/S0065-3233(08)60520-3.ISBN 9780120342341.PMID 7020376.
^Singh J, Hanson J, Heffernan R, Paliwal K, Yang Y, Zhou Y (August 2018). "Detecting Proline and Non-Proline Cis Isomers in Protein Structures from Sequences Using Deep Residual Ensemble Learning".Journal of Chemical Information and Modeling.58 (9):2033–2042.doi:10.1021/acs.jcim.8b00442.PMID 30118602.S2CID 52031431.
^"Side Chain Conformation".
^Dunbrack, RL Jr.; Karplus, M (20 March 1993). "Backbone-dependent rotamer library for proteins. Application to side-chain prediction".Journal of Molecular Biology.230 (2):543–74.doi:10.1006/jmbi.1993.1170.PMID 8464064.
^Dunbrack, RL Jr; Karplus, M (May 1994). "Conformational analysis of the backbone-dependent rotamer preferences of protein sidechains".Nature Structural Biology.1 (5):334–40.doi:10.1038/nsb0594-334.PMID 7664040.S2CID 9157373.
^"dihedral angle calculator polyhedron".www.had2know.com. Archived fromthe original on 25 November 2015. Retrieved25 October 2015.
^"Formula Derivations from Polyhedra". Retrieved4 December 2024.