BIOINORGANIC CHEMISTRY A Short Course Second Edition

X-RAY CRYSTALLOGRAPHY 87

relationships between the molecules. Common symmetry operations are two -
or threefold screw (rotation) axes, mirror planes, inversion centers (centers of
symmetry), and rotation followed by inversion. There are 230 different ways
to combine allowed symmetry operations in a crystal leading to 230space
groups.^13 Not all of these are allowed for protein crystals because of amino
acid asymmetry (onlyl - amino acids are found in naturally occurring proteins).
Only those space groups without symmetry (triclinic) or with rotation or screw
axes are allowed. However, mirror lines and inversion centers may occur in
protein structures along an axis.
Seven crystal systems as described in Table 3.1 occur in the 32 point groups
that can be assigned to protein crystals. For crystals with symmetry higher than
triclinic, particles within the cell are repeated as a consequence of symmetry
operations. The number of asymmetric units within the unit cell is related but
not necessarily equal to the number of molecules in a unit cell, depending on
how the molecules are related by symmetry operations. From the symmetry
in the X - ray diffraction pattern and the systematic absence of specifi c refl ec-
tions in the pattern, it is possible to deduce the space group to which the crystal
belongs.
In summary, it is important to determine crystal quality, unit cell dimensions
of the crystal (a larger crystal absorbs X rays more strongly, 0.3 – 0.5 mm is
considered the optimal size), the crystal ’ s space group, and how many protein
molecules are in the unit cell and in one asymmetric unit. Actually, the great
majority of crystals useable for X - ray crystallography are not ideal but contain
lattice defects. This is true for protein crystals, which are also weak scatterers
since the great majority of the component atoms are light atoms, C, N, and O.

TABLE 3.1 The Seven Crystal Systems

Crystal System Conditions Imposed on Cell Geometry

Minimum Point

Group Symmetry

Triclinic None 1
Monoclinic α = γ = 90 ° ( b is the unique axis; for proteins
this is a twofold axis or screw axis)
or
α = β = 90 ° ( c is the unique axis; for proteins
this is a twofold axis or screw axis)

2

Orthorhombic α = β = γ = 90 ° 222
Tetragonal a = b ; α = β = γ = 90 ° 4
Trigonal a = b ; α = β = 90 ° ; a = b ; γ = 120 ° (hexagonal axes)
or
a = b = c ; α = β = γ (rhombohedral axes)

3

Hexagonal a = b ; α = β = 90 ° ; a = b ; γ = 120 ° 6
Cubic a = b = c ; α = β = γ = 90 ° 23

Source : Adapted from reference 13.