Dihedral angle

A dihedral angle is a measurement of the space between two intersecting surfaces. In geometry, this usually refers to two planes or two half-planes. To measure this angle, you must look at a third plane. This third plane is perpendicular to the line where the first two surfaces meet.

By viewing the intersection from this specific angle, you can see the plane angle formed between the two surfaces. In higher dimensions, this concept expands to represent the angle between two hyperplanes.

In mathematics, we can describe these planes using Cartesian coordinates. If we have two equations representing the planes, the dihedral angle is determined by their coefficients. Another way to find the angle is by using normal vectors. A normal vector is a line that stands perfectly upright from a plane. The angle depends on the dot product of these vectors and the product of their lengths. Mathematicians often use absolute values in these formulas to ensure the result remains consistent. This is because changing the signs of the coefficients does not change the actual plane.

Chemistry uses dihedral angles to describe the shape of molecules, known as molecular conformation. This happens when we look at a chain of four atoms connected by chemical bonds. Any three atoms that do not sit in a straight line define a single half-plane. When you have four atoms in a row, you have two such half-planes. The angle between these two planes is the dihedral angle.

This measurement is vital in polymer physics, where scientists study chains of points and links. In these chains, bond vectors are defined by the positions of consecutive points. Scientists often look at the angle between half-planes created by three consecutive bond vectors.

Because these angles can be measured in different directions, they are often given a specific sign. In chemistry, the clockwise angle is used to define the direction. This allows the angle to exist in a range between 0 and 360 degrees. In polymer physics, the angle is often defined within an interval of -180 to 180 degrees. This is done using a mathematical function called atan2. This specific method ensures the angle does not change even if you reverse the order of the chain.

Chemists use specific names to categorize these molecular shapes based on their angles. Arrangements between 0 and 90 degrees are called syn. Arrangements between 90 and 180 degrees are called anti. Other terms like clinal and periplanar describe even narrower ranges. For example, an angle between 0 and 30 degrees is synperiplanar, also known as the cis-conformation. An angle between 150 and 180 degrees is antiperiplanar, also called the trans-conformation. Another common shape is the gauche conformation, which occurs at angles around 60 degrees.

In 1963, G. N. Ramachandran, C. Ramakrishnan, and V. Sasisekharan developed a way to study these angles in proteins. They created the Ramachandran plot, which helps visualize allowed regions for backbone dihedral angles. Proteins are made of amino acids, and their backbone has three specific angles: omega, phi, and psi.

The omega angle usually stays at 180 degrees because of the planarity of the peptide bond. The phi and psi angles are essential for determining how a protein folds. Most peptide bonds in proteins are in the trans position, though some, like those in proline, can be cis. This distinction affects the distance between atoms, such as the 3.8 Å distance in trans isomers versus 2.9 Å in cis isomers.

Dihedral angles also appear in the study of polyhedra, which are solid 3D shapes with flat faces. Every edge of a polyhedron has a dihedral angle where two faces meet. This is sometimes called the face angle. If the angle is 180 degrees, the faces are parallel, like a flat tiling. If the angle is 0 degrees, the faces overlap, which creates a degenerate polyhedron. Some special shapes, like the five Platonic solids, are isohedral. This means every dihedral angle in the shape has the exact same value.