Structural Biochemistry/Symmetry Operations

=Symmetry Operations=

There are five symmetry elements

Axis of Rotation, Cn
“An objection has n-fold rotational symmetry.” Order of rotation (n) = A rotation by 360 deg/n around some axis of rotation that renders the object unchanged. Example: A pentagon is said to have a 5-fold symmetry, meaning that if a pentagon is rotated by 72 degrees, the pentagon remains unchanged. A benzene ring is said to have a 6-fold symmetry, meaning that if a benzene ring is rotated by 60 degrees, the benzene ring remains unchanged. Also, an object or a molecule can have more than one rotational axes. Example: A benzene ring not only has a C6 rotation, it also has a C2 and a C3 rotational axes. However, the largest number of n is the principle rotational axes.

Planes of Reflection, sigma
An object is said to have mirror planes when the object remains unchanged when reflected through the plane. For convention, the mirror plane perpendicular to the primary rotation axis is denoted “sigma h.” The mirror plane parallel to the primary rotation axis is denoted “sigma v.”

Improper rotation axes, Sn
An improper rotation is a rotation by 360/(2n) degree followed by a reflection through a mirror plane perpendicular to that axis. In Figure II, the molecule has a C3 primary axis. First rotate the molecule by 60 degrees. Then reflect the molecule through the reflection plane. The final result returns the same molecule. Then, the molecule is said to have a S3 improper rotation.

The inversion center, i
Inversion center is a reflection through the center of the molecule or object.

In Figure III, the triangle on the upper right corner is reflected at the origin. The triangle will end up at the same position as the triangle at the lower left corner. As the result, the figure is said to have an inversion center.

( Cn + sigma ) = Sn

The identity, E
Identity is a rotation by 360 degrees about any arbitrary axis. The resulting object or molecule has the same configuration. All the elements and objects all have the identity.