III.C. CRYSTALS, SYMMETRY, AND DIFFRACTION
Biomacromolecules quite often occur naturally or in vitro as organized structures composed of subunits arranged in a symmetrical way. Such structures are readily studied by diffraction methods. The fundamental concepts concerning crystalline matter, symmetry relationships, and diffraction theory form a basic framework for understanding the principles and practice of image processing and interpretation of structural results. The main concepts, summarized here, are presented in more detailed, introductory form in excellent texts such as Eisenberg and Crothers (1979), Glusker and Trueblood (1972), Holmes and Blow (1965), and Wilson (1966). For additional references, see the reading list.
III.C.1. Definitions of Terms (compiled from Glusker & Trueblood, and Eisenberg & Crothers)
Asymmetric Unit: Part of the symmetric object from which the whole is built up by repeats. Thus, it is the smallest unit from which the object can be generated by the symmetry operations of its point group. Fig.III.3.
A crystal is a regular arrangement of atoms, ions, or molecules, and is conceptually built up by the continuing translational repetition of some structural pattern. This pattern, or unit cell, may contain one or more molecules or a complex assembly of molecules. In three dimensions (3D), the unit cell is defined by three edge lengths a,b,c and three interaxial angles a, b, g. The different 3D crystal systems (triclinic, monoclinic, orthorhombic, tetragonal, trigonal, hexagonal and cubic) arise from the seven fundamental unit cell shapes. The simplest, triclinic, is a parallelepiped, with no restrictions on cell lengths or angles. A cubic cell has equal (a=b=c) and orthogonal (a=b=g=90°) edges.
A lattice is a mathematical formalism that defines an infinite array of imaginary points: each point in the lattice is identical to every other point. That is, the view from each point is identical with the view in the same direction from any other point. This condition is not obeyed at the boundary of a finite, but otherwise perfect crystal. The crystal structure and the crystal lattice are NOT equivalent: the structure is an array of objects whereas the lattice is an array of imaginary, infinitely small points. A 2D lattice is defined by two translations, a,b and two axes at an angle a to each other. A 3D lattice is defined by three translations, a,b,c, and three axes at angles a, b, g to each other.
Crystal lattices (2D or 3D) may be primitive, with one lattice point per unit cell, or centered, containing two or four points per cell. There are four 2D lattice systems (Table III.1) which are subdivided into a total of five 2D lattices (Fig. III.1). There are seven 3D crystal systems corresponding to the seven basic spacefilling shapes that unit cells can adopt. These are subdivided into a total of fourteen, socalled Bravais lattices (Table III.2 and Fig. III.2). For example, the cubic crystal system includes three Bravais lattices: primitive (P), body centered (I), and face centered (F).





Fig. III.3. An asymmetric object. (From Eisenberg and Crothers, p. 756) 
The crystal structure is built by placing a motif at every lattice point. The motif is the object that is translated, and may be asymmetric (e.g. a single polypeptide chain: Fig.III.3) or symmetric (i.e. containing two or more symmetrically arranged subunits). The crystal structure, crystal lattice, and motif are all restricted in the symmetries they can display, but biomacromolecular assemblies themselves are not restricted in the sense that they may display additional internal (noncrystallographic) symmetry. From this emerges the corollary that an asymmetric unit of the crystal structure may itself contain a symmetrical arrangement of identical, asymmetric molecules. The following introduction to symmetry will help clarify this.




Fig. III.7. (a) The generation of a 2D "crystal structure" from a lattice and a structural motif. The replacement of each lattice point by an apple leads to a 2D structure. There are many ways in which unit cells may be chosen in a repeating pattern. Various alternative choices are shown, each having the same area despite varying shape (note that the total content of any chosen unit cell is one apple). Infinite repetition in 2D of any one of these choices for unit cell will reproduce the entire pattern. (b) Perspective view of a triclinic lattice. To see the perspective the lattice points are joined by lines that define imaginary edges of a unit cell. One unit cell is shaded; it could have been chosen with a different shape (but the same volume), as in the 2D example in (a). (From Glusker and Trueblood, p. 10) 
Biological objects may display symmetry about a point or along a line. An object is symmetrical if it is indistinguishable from its initial appearance when spatially manipulated. (Again, we ignore the loss of translational symmetry at the boundary of finite, crystalline objects.)
a. Symmetry operators
There are four types of symmetry operations which lead to superimposition of an object on itself: rotation, translation, reflection and inversion. A symmetry element is a geometrical entity such as a point, line, or plane about which a symmetry operation is performed. The symmetry of any object can be described by some combination of these symmetry operations. The symmetry of any aggregate or crystal of a biological molecule is only described by rotation and/or translation operations. This is because, for example, biological protein molecules mainly consist of lamino acids, hence, reflection or inversion symmetries are not allowed.
b. Asymmetric unit
Symmetry operations give rise to other groups of atoms (the asymmetric unit) which are in equivalent, general positions. The number of equivalent general positions related by the symmetry operators equals the number of asymmetric units in the unit cell. The number of asymmetric units may be less than, equal, or greater than the number of molecules in the unit cell. If the number of asymmetric units is equal to or less than the number of molecules in the cell, than the molecule either contains no symmetry or it contains noncrystallographic symmetry (i.e. symmetry that is not contained within the allowed lattice symmetries). If the number of asymmetric units is greater than the number of molecules in the cell, than the molecules must occupy special positions and possess the appropriate symmetry element of the space group (see Sec. III.C.5.e).
c. Point groups (Symmetry about a single point)
1) Schoenflies and HermanMauguin point group notations
A point group is a collection of symmetry operations that define the symmetry about a point. Two systems of notations are used for point groups: i) the S or Schoenflies notation (capital letters; mainly used by spectroscopists) and ii) the HM or HermannMauguin symbol (an explicit list of the symmetry elements, commonly preferred by crystallographers).
2) Types of point group symmetry operators
The four types of symmetry about a point are rotational symmetry, mirror symmetry, inversion symmetry, and improper rotations. Each of these is described below with accompanying illustrations.
Rotational symmetry (n) : the object appears identical if rotated about an axis by a = 360/n = 2p/n degrees. The only allowed nfold axes for crystal lattices are n = 1,2,3,4, and 6 since lattices must be space filling. Figs.III.8,9,12.
Fig.III.8. Two, three, four, five, and sixfold rotational symmetries. (From Bernal, pp.45, 48, 50, 35, and 52) 
Fig. III.9. A fourfold rotation axis, parallel to c and through the origin of a tetragonal unit cell (a=b), moves a point at x, y, z to a point at (y, x, z) by a rotation of 90 about the axis. The sketch on the right shows four equivalent points resulting from successive rotations. (From Glusker and Trueblood, p. 72) 
Mirror (reflection) symmetry (m): each point in the object is converted to an identical point by projecting through a mirror plane and extending an equal distance beyond this plane.Figs.III.10 and III.11
Inversion symmetry (i): each point in the object is converted to an identical point by projecting through a common center and extending an equal distance beyond this center. Objects with i symmetry are said to be centrosymmetric. Fig.III.10.
Fig. III.10. (Left) A mirror symmetry operation. (Right) An inversion symmetry operation. (From Buerger, p. 8) 
Fig. III.11. Additional examples of mirror symmetry operations. (Right) m_{y} and m_{x}. (From Bernal, pp. 26,27, and 34) 
Improper rotations: rotations followed by m or i. These include the rotoinversion (n followed by i) and rotoreflection (n followed by m). The only inversion axes for crystal lattices are , , , , . Fig.III.12 and III.13.
Fig.III.12. Crystallographic rotation (a) and inversionrotation (b) symmetry axes and their action on an asymmetric figure  a tetrahedron. (From Vainshtein, p. 67) 
Fig. III.13. The operation , a twofold rotaryinversion axis parallel to b and through the origin, converts a point at x, y, z to a point at x, y, z. This is the result of, first, a twofold rotation about an axis through the origin and parallel to b (x, y, z to x, y, z) and then an inversion about the origin (x, y, z to x, y, z). This is the same as the effect of a miror plane perpendicular to the b axis. Note that a right hand has been converted to a left hand. (From Glusker and Trueblood, p. 72) 
3) Types of point groups
Cyclic, dihedral, and cubic (tetrahedral, octahedral and icosahedral) point groups define the collection of symmetry operations about a point. Klug (1969) and others (e.g. Wilson, 1966; Glusker and Trueblood, 1972; Eisenberg and Crothers, 1979; Bernal, Hamilton and Ricci, 1972) describe this and other aspects of symmetry in detail.
Fig. III.14. Examples of cyclic point groups. (From left to right and top to bottom) 2, 2mm, 3, 3m, 4, 4mm, 6, and 6mm (From Bernal, pp. 45, and 4753) 
These contain a single nfold axis of rotation, where n can be any positive integer and also may contain one or more planes of reflection (Fig.III.14). Point groups which contain only an nfold axis of rotation are given the symbol n in the HM system and C_{n} in the S system (C stands for cyclic). For example, the doubledisk structure of tobacco mosaic virus (TMV) stacked disk aggregates contains 34 subunits (polypeptide chains) arranged with C_{17} symmetry (S notation) (Fig.III.15). Nonbiological molecules can also have mirror planes of symmetry either parallel (e.g. nm or nmm in the HM notation or C_{nv} in the S notation where v stands for vertical) or perpendicular (e.g. n/m in the HM notation or C_{nh} in the S notation where h stands for horizontal) to the nfold axis of symmetry.
Dihedral point groups have axes of rotation at right angles to each other. These point groups consist of an nfold axis perpendicular to n 2fold axes. Most oligomeric enzymes display dihedral symmetry (Mathews and Bernhard, 1973). For example, the enzyme ribulose bisphosphate carboxylase/oxygenase (RuBisCO) has D_{4 } symmetry (422 in HM notation). The number of asymmetric units in the point group D_{n} is 2n, thus RuBisCO has eight asymmetric units. In this particular enzyme, each asymmetric unit contains two polypeptide chains: a large (~55 kD), catalytic subunit and a small (~15 kD) subunit whose function (regulatory?) is unknown.
The essential characteristic is four 3fold axes arranged as the four body diagonals (lines connecting opposite corners) of a cube. The three cubic point groups that biological molecules can occupy are T (tetrahedral = 23 in HM notation), O (octahedral = 432) and I (icosahedral = 532). The tetrahedral point group contains 12 asymmetric units. Aspartateßdecarboxylase is presumed to display this point group symmetry (Eisenberg and Crothers, 1979). Dihydrolipoyl transsuccinylase contains 24 asymmetric subunits arranged with octahedral (432) symmetry. There are a large number of plant, animal, and bacterial viruses, each containing 60 asymmetric units, which display icosahedral (532) symmetry (Fig.III.16). In most cases these spherical viruses contain a multiple of 60 copies of chemically identical or distinct protein or glycoprotein subunits. In those cases where the asymmetric unit contains more than one "subunit", not all subunits are equivalently arranged. Instead they occupy quasiequivalent positions.


4) Lattice restrictions and noncrystallographic symmetry
The crystal structure, crystal lattice, and motif may only contain one, two, three, four, or sixfold rotational symmetry axes (because the crystal lattice must be space filling). However, the 34 subunits in the disc structure of TMV, for example, are arranged about a 17fold axis of rotation (Fig.III.15). The TMV disc forms true 3D crystals and has been studied by xray crystallography. In the crystal, the disc occupies a general position in the unit cell, and therefore displays noncrystallographic symmetry. Many of the small, spherical viruses are icosahedral (cubic point group) and they contain symmetry elements compatible with allowed lattice symmetries, and crystallize and display crystallographic as well as noncrystallographic symmetry.
d. Translational symmetry (Symmetry along a line)
1) Repetition in one dimension
Fig.III.17. A onedimensional 'crystal' of right feet. (From Bernal, p.27) 
2) Screw axes
Fig. III.18. A twofold screw axis, 2_{1}, parallel to b and through the origin, which combines both a twofold rotatiion (x, y, z to x, y, z) and a translation of b/2 (x, y, z to x, 1/2+y, z). A second screw operation will convert the point x, 1/2+y, z to x, 1+y, z, which is the equivalent of x, y, z i the next unit cell along b. Note that the left hand is never converted to a right hand. (From Glusker and Trueblood, p. 73) 
A screw axis combines translation and rotation operations to produce a structure with helical symmetry (Figs. III.1820). Screw axes are symmetry elements of crystals that are helices with an integral number of asymmetric units per turn of the helix. An n_{m} screw axis combines a rotation of 2p/n radians about an axis, followed by a translation of m/n of the repeat distance (unit cell edge). The screw axes found in crystals include 2_{1}, 3_{1}, 3_{2}, 4_{1}, 4_{2}, 4_{3}, 6_{1}, 6_{2}, 6_{3}, and 6_{5}. A crystal lattice only accommodates an integral number of asymmetric units per turn of the helix, although this need not apply to helices in general.
Fig. III.19. Some crystallographic fourfold screw axes showing two identiy points for each. Note that the effect of 4_{1} on a left hand is the mirror image of the effect of 4_{3} on a right hand. The right hand has been moved slightly to make this relation obvious. (From Glusker and Trueblood, p. 74) 
Fig.III.20. Crystallographic screw axes and their action on an asymmetrioc tetrahedron. (From Vainshtein, p. 69) 
Aggregates such as actin thin filaments, microtubules, and the TMV particle are helical structures which display symmetry along a line. For example, one turn of the basic helix of the TMV rod contains 16.33 protein subunits (Fig.III.21). The true repeat in the structure is, therefore, three turns of the basic helix containing 49 subunits. Most helical biological aggregates do not form 3D crystals suitable for diffraction studies, not because of strict symmetry constraints (they could occupy noncrystallographic positions), but rather as a consequence of their shape and large size.
Fig. III.21. Drawing of part of the helical structure of tobacco mosaic virus. Each shoeshaped protein subunit is bound to three RNA nucleotides. Part of the RNA chain is shown stripped of its protein subunits in a configuration if could not maintain without them. Each turn of 16.3 protein subunits is closely related to the disk structure of Fig.III.15.. (From Eisenberg and Crothers, p. 782) 
e. Plane groups and space groups
Fig. III.22. The rotational symmetry elements (a) 1 (b) 2 (c) 4 (d) 3 and (e) 62. A single unit cell is outlined by a double set of lines. Other unit cells are shown with single lines. The asymmetric unit is shown shaded. (From Blundell and Johnson, p. 87) 





Asymmetric unit motif structure
Glide plane symmetry (Figs. III.27III.28) is produced by a translation followed by a mirror operation (or vice versa). Biological molecules do not, in general, display glide plane symmetries because they do not exist in enantiomorphic pairs. Note, however, that biological molecules (or crystals) when viewed in twodimensions (i.e. in projection) can display mirror symmetry.


f. Examples of symmetrical biological molecules
1) Helical symmetry
Actin filament Bacterial flagella
2) Point group symmetry
MOLECULE/AGGREGATE  S  HM  # asymmetric units  
Asymmetric aggregates: e.g. ribosome (monomer)  1  1  
Fibrous molecules: e.g. fibrinogen  2  2  
Enzymes:  
lactate dehydrogenase  222  4  
catalase  222  4  
aspartate transcarbamylase  32  6  
ribulose bisphosphate carboxylase/oxygenase  422  8  
glutamine synthetase  622  12  
asparatebdecarboxylase  23  12  
dihydrolipoyl transsuccinylase  432  24  
Spherical viruses: e.g. polyoma, polio, rhino, tomato bushy stunt, human wart, etc.  532  60 
3) Plane group symmetry (twodimensional crystals)
Bacterial cell walls (e.g. Bacillus brevis T layer)
4) Space group symmetry (3D crystals)