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Dihedral angle

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Collection: Angle, Euclidean Solid Geometry, Protein Structure, Stereochemistry
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Dihedral angle

Dihedral angle between the two planes R and R'.

In solid geometry a dihedral or torsion angle is the union of a line and two half-planes that have this line as a common edge. The line is the edge of the dihedral angle and the union of the edge and one of the planes is a face.

The plane angle of a dihedral angle is an angle formed by the two rays wich are the intersections of the faces of the dihedral angle and a plane perpendicular to the edge. Any two plane angles of a dihedral angle are congruent.

The measure of a dihedral angle is the measure of one of its plane angles. A dihedral angle is acute, right or obtuse according as its plane angles are acute, right or obtuse.[1]

A dihedral angle can be oriented by choosing which face is the first. Thus the oriented dihedral angle of a face A and a face B is the angle through which face A must be rotated around their edge c to align it with face B. It follows that the oriented angle between B and A is the opposite of the oriented angle between A and B. However, there is no definition which of the two opposite angles is positive, and which one is negative.

Sometimes the dihedral angle is defined not as an angle between two half planes but as an angle between two intersecting planes.

A common notation for describing a dihedral angle is to use an (ordered) 4-tuple of points, (P, Q, R, S) where P, Q and P are in face A, and Q, R and S are in face B, (so Q and R are on the line of intersection \overline{AB} .)

In higher dimension, a dihedral angle represents the angle between two hyperplanes.[2]

Contents

  • Calculating the angle of a dihedral angle 1
    • Angle between two planes 1.1
    • Angle between three vectors 1.2
    • Angle between faces of a polyhedron 1.3
  • Computation 2
  • Methods of computation 3
  • Dihedral angles in polyhedra 4
  • Dihedral angles in stereochemistry 5
    • IUPAC definition 5.1
    • Dihedral angles of four atoms 5.2
      • Improper dihedral angle 5.2.1
  • Dihedral angles in bio-chemistry 6
    • Dihedral angles of biological molecules 6.1
  • See also 7
  • References 8
  • External links 9

Calculating the angle of a dihedral angle

To avoid ambiguity, the angle measure of a dihedral angle (oriented or not) is conventionally taken to be between 0 and a straight angle. Similarly, the measure of an angle of planes around their intersecting line is assumed to be between 0 and a right angle.

Depending on orientation, conventions and uses of the discipline in which the dihedral angle is used the measure of the angle \varphi (in degrees) can also be -\varphi, 180^\circ - \varphi (supplementary angle), 180^\circ + \varphi or 360^\circ - \varphi .

Angle between two planes

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

The dihedral angle \varphi_{AB} between two faces A and B can be calculated as the angle between their two normal vectors \mathbf{n}_{A} and \mathbf{n}_{B}:

\varphi_{AB} = \arccos \left( \frac{ |\mathbf{n}_A \cdot \mathbf{n}_B|}{|\mathbf{n}_A | |\mathbf{n}_B|}\right) = \arcsin \left(\frac{|\mathbf{n}_A \times \mathbf{n}_B|}{|\mathbf{n}_A| |\mathbf{n}_B|}\right)

Where · and × denote respectively the dot product and the cross product, and arccos and arcsin are inverse trigonometric functions. When the two intersecting planes are analyticly described by \Pi_1 : a_1 x + b_1 y + c_1 z + d_1 = 0 and \Pi_2 : a_2 x + b_2 y + c_2 z + d_2 = 0 , then the dihedral angle between them is the angle between their normal directions, given by:

\cos(\varphi_{AB}) = \frac{a_1 a_2 + b_1 b_2 + c_1 c_2}{\sqrt{a_1^2+b_1^2+c_1^2}\sqrt{a_2^2+b_2^2+c_2^2}}.

Angle between three vectors

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.

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

Taking one of each pair as laying along the edge in the same direction, we can also define the dihedral angle of a triple of non-collinear vectors \mathbf{b}_1, \mathbf{b}_2 and \mathbf{b}_3 (red, green and blue, respectively, in the diagram). The vector \mathbf{b}_2 is along the edge. 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, given by:[3]

\varphi = \operatorname{atan2} \left( \left([\mathbf{b}_1 \times \mathbf{b}_2]\times [\mathbf{b}_2 \times \mathbf{b}_3]\right) \cdot \frac{\mathbf{b}_2}{|\mathbf{b}_2|}, [\mathbf{b}_1 \times \mathbf{b}_2] \cdot [\mathbf{b}_2 \times \mathbf{b}_3] \right).

Angle between faces of a polyhedron

A third way to calculate the dihedral angle is as angle between faces of a polyhedron

Given 3 faces around a common vertex P with edges AP , BP and CP then then the dihedral angle between the faces containing APC and BPC is:[4]

\varphi =\arccos \left( \frac{ \cos (\angle APB) - \cos (\angle APC) \cos (\angle BPC)}{ \sin(\angle APC) \sin(\angle BPC)} \right).

Computation

Whichever method is used for defining planes, it is usually not difficult to compute points on the plane and on their intersection. Therefore we describe how to compute the angle of two planes or half planes, starting for two points A and B on the intersection of the planes, and two other points C1 and D1 that are not on this intersection, one in each plane or half plane.

In the case of half planes, the angle \varphi is given by

\varphi=\arccot\left(\frac{1}{\sqrt{AB\cdot AB}}\frac{(AB\times AC)\cdot (AB\times AD)}{AB\cdot (AC\times AD)} \right)\,,

where the words of two letters denote the vectors defined by the corresponding points, · and × denote respectively the dot product and the cross product, and arccot is the principal value of the inverse trigonometric function arccotangent.

In the case of plane, as one want an angle in the interval (0, π), one must take the absolute value of the argument of the cotangent.

In both cases, the formula does not induce a division by zero, if the planes are distinct (or if the half planes are not supported by the same plane).

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. 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 UA = (A2A1) × (A3A1) is orthogonal to plane A and UB = (B2B1) × (B3B1) is orthogonal to plane B. The (unsigned) dihedral angle can therefore be computed with either

\varphi_{AB}=\arccos \left(\frac{U_A \cdot U_B}{|U_A| |U_B|}\right) = \arcsin \left(\frac{|U_A \times U_B|}{|U_A| |U_B|}\right)

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 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 that have a codimension 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.[5] If

\mathbf{A} is a matrix of orthonormal basis vectors for flat A, and
\mathbf{B} is a matrix of orthonormal basis vectors for flat B, and
u \;= the eigenvector with the smallest corresponding eigenvalue of \left(\mathbf{B}^T\mathbf{A}\right)^T\left(\mathbf{B}^T\mathbf{A}\right), and
v \;= the eigenvector with the smallest corresponding eigenvalue of \left(\mathbf{A}^T\mathbf{B}\right)^T\left(\mathbf{A}^T\mathbf{B}\right),

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

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 polyhedra, the two quasiregular solids, and two quasiregular dual solids.

Dihedral angles in stereochemistry

IUPAC definition

In the IUPAC Gold Book gives the definition of a dihedral angle (here named a torsion angle) for the use in stereochemistry:


In a chain of atoms A-B-C-D, the dihedral angle between the plane containing the atoms A,B,C and that containing B,C,D. In a Newman projection the torsion angle is the angle (having an absolute value between 0° and 180°) between bonds to two specified (fiducial) groups, one from the atom nearer (proximal) to the observer and the other from the further (distal) atom. The torsion angle between groups A and D is then considered to be positive if the bond A-B is rotated in a clockwise direction through less than 180° in order that it may eclipse the bond C-D: a negative torsion angle requires rotation in the opposite sense. Stereochemical arrangements corresponding to torsion angles between 0° and ±90° are called syn (s), those corresponding to torsion angles between ±90° and 180° anti (a). Similarly, arrangements corresponding to torsion 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).
syn/anti peri/clinal
The two types of terms can be combined so as to define four ranges of torsion 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). [6]

Dihedral angles of four atoms

Dihedral angle defined by three bond vectors (shown in red, green and blue) connecting four atoms.
Dihedral angle defined by three bond vectors (shown in red, green and blue) connecting four atoms. This perspective is looking at the second bond vector (green) end-on (coming out of the page).

The structure of a molecule can be defined with high precision by the dihedral angles between three successive chemical bond vectors. The dihedral angle \varphi 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, which is equivalent to the Newman projection 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 \varphi 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, the dihedral angle is 180°.

Improper dihedral angle

An "improper" dihedral angle is a similar geometric analysis of four atoms, but typically involves a central atom with three others attached to it rather than the standard arrangement of all four of them bonded sequentially each to the next. One of the vectors is the bond from the central atom to one of its attachments. The other two vectors are pairs of the attachments, and thus together represent the plane of the attachments. Improper dihedral angles are useful for analyzing the planarity of the central atom: as the angle deviates from zero, the central atom moves out of the plane defined by the three attached to it.[7]

Dihedral angles in bio-chemistry

Dihedral angles of biological molecules

The backbone dihedral angles of a protein

The backbone dihedral angles of proteins 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 usually restricts ω 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 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 χ15 for each successive bond along that chain. 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 stability of certain 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 acids (DNA and RNA) and for polysaccharides.

See also

References

  1. ^ James; James (1992). Mathematics dictionary (5 ed.). New York: Van Nostrand Reinhold. p. 122.  
  2. ^ Olshevsky, George, Dihedral angle at Glossary for Hyperspace.
  3. ^ 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.  
  4. ^ "dihedral angle calculator polyhedron". www.had2know.com. Retrieved 25 October 2015. 
  5. ^ Gashler, M.; Martinez, T. (2011). Tangent Space Guided Intelligent Neighbor Finding (PDF). Proceedings of the IEEE International Joint Conference on Neural Networks (IJCNN'11). pp. 2617–2624. 
  6. ^ IUPAC, Compendium of Chemical Terminology, 2nd ed. (the "Gold Book") (1997). Online corrected version:  (2006–) "torsion angle".
  7. ^ CHARMM parmfile.doc definition of "IMPH" energy parameter

External links

  • The Dihedral Angle in Woodworking at Tips.FM
  • Analysis of the 5 Regular Polyhedra gives a step-by-step derivation of these exact values.
  • Weisstein, Eric W., "Dihedral angle", MathWorld.
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