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. Bravais lattice: One of the 14 possible arrays of points repeated periodically in 3D space in such a way that the arrangement of points about any one of the points in the array is identical in every respect to that about any other point in the array. Fig.III.2. Center of Symmetry (or Center of Inversion): A point through which an inversion operation is performed, converting an object into its mirror image. Fig.III.10. Crystal: A solid having a regularly repeating internal arrangement of its atoms. Figs.III.5,6,7. Crystal lattice: Crystals are composed of groups of atoms repeated at regular intervals in three dimensions with the same orientation. For certain purposes it is sufficient to regard each such group of atoms as replaced by a representative point; the collection of points so formed is the space lattice or lattice of the crystal. Each crystal lattice is a Bravais lattice. Figs.III.1,2,4,7. Crystal Structure: The mutual arrangement of atoms, molecules or ions that are packed together in a lattice to form a crystal. Figs.III.5,6,7. Crystal System: The seven crystal systems, best classified in terms of their symmetry, correspond to the seven fundamental shapes for unit cells consistent with the 14 Bravais lattices. Fig.III.2 and Table III.2. Glide Plane: A symmetry element for which the symmetry operation is reflection across the plane combined with translation in a direction parallel to the plane. Figs.III.27 and III.28. Inversion: A symmetry operation in which each point of an object is converted to an equivalent point by projecting through a common center (called center of inversion or center of symmetry) and extending an equal distance beyond this center. If the center of symmetry is at the origin of the coordinates, every point x,y,z becomes -x,-y,-z. It converts an object or a structure into one of opposite "handedness", related to the first as is any object and its mirror image. Figs.III.10 and III.12. Lattice: Rule for translation. Fig.III.1 and Tables III.1 and III.2. Mirror Plane: A symmetry element for which the corresponding symmetry operation resembles reflection in a mirror coincident with the plane. It converts an object or a structure into one of opposite "handedness", related to the first as is any object and its mirror image. Figs.III.10,11. Motif: Object that is translated. Figs.III.3,4,7. Plane Group: Symmetry of a two-dimensional structure. There are 17 plane group symmetries possible (only 5 for biological structures). Figs.III.6,22-26. Point Group: The collection of symmetry operations that describe the symmetry of an object about a point. Figs.III.8-16. Reciprocal Lattice: The lattice with axes a*, b*, c* related to the crystal lattice or direct lattice (with axes a,b,c) in such a way that a* is perpendicular to b and c; b* is perpendicular to a and c; and c* is perpendicular to a and b. Rows of points in the direct lattice are normal to nets of the reciprocal lattice, and vice versa. For a given crystal, the direct lattice and reciprocal lattice have the same symmetry. The repeat distance between points in a particular row of the reciprocal lattice is inversely proportional to the interplanar spacing between the nets of the crystal lattice that are normal to this row of points. Rotation Axis: An axis of symmetry for an object, such as a crystal. When the object is rotated by (360/n)° about an n-fold rotation axis, the new orientation is indistinguishable from the original one. Figs.III.9,12. Rotary-Inversion Axis: An axis for which the corresponding symmetry operation is a rotation by (360/n)° combined with inversion through a center of symmetry lying on the axis. Figs.III.12,13. Screw Axis: An axis (designated nm) for which the corresponding symmetry operation is a rotation about the axis by (360/n)° followed by a translation parallel to the axis by m/n of the unit cell length in that direction. Figs.III.18-20. Space group: A group or array of operations consistent with an infinitely extended regularly repeating pattern. It is the symmetry of a 3D structure. There are 230 space group symmetries possible (only 65 for biological structures). Symmetry: An object is symmetric if some spatial manipulation of it results in an indistinguishable object. A symmetric object can be superimposed on itself by some operation. Symmetry Element: Geometrical entity (such as a line, a point or a plane) about which a symmetry operation is carried out. Symmetry Operation: The operation that leads to superimposition of an object on itself (i.e. results in moving the object to a position in which its appearance is indistinguishable from its initial appearance). Symmetry operations include rotation, inversion, reflection and translation. Translation: A motion in which all points of an object move in the same direction, that is, along the same or parallel lines. Fig.III.17. Unit Cell: The fundamental portion of a crystal structure that is repeated infinitely by translation in three dimensions. It is characterized by three vectors a,b,c, not in one plane, which form the edges of a parallelepiped. Figs.III.1,2,7.

III.C.2. Crystals

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.

III.C.3. Lattices

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 space-filling shapes that unit cells can adopt. These are subdivided into a total of fourteen, so-called 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).

Table III.1. Plane lattices. (From Sherwood, p.70).
Fig. III.1. The five two-dimensional lattices. (From Eisenberg and Crothers, p. 786)
Table III.2. The fourteen Bravais lattices. (From Sherwood, p.73).
Fig. III.2. The 14 three-dimensional Bravais lattices. (From Eisenberg and Crothers, p. 790)

III.C.4. Crystal Structure

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 symmet-rically 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 (non-crystallographic) 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.4. Lattices and motifs. Each of the diagrams of FIg.III.4 may be reproduced by associating the motifs (a), (b), or (c) with the rectangular array of lattice points. (From Sherwood, p. 61)
Fig. III.5. Three two-dimensional 'crystals'. Each pattern is different, but they have the same underlying rectangular structure. (From Sherwood, p. 60)
Fig. III.6. The five two-dimensional plane groups for biological structures. The asymmetric unit is a triangle in every case; the motif is some group of triangles (1,2,3,4, or 6). (From Eisenberg and Crothers, p. 786)

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)

III.C.5. Symmetry

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 l-amino 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 non-crystallographic 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 Herman-Mauguin 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 H-M or Hermann-Mauguin 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 n-fold 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 six-fold rotational symmetries. (From Bernal, pp.45, 48, 50, 35, and 52)

Fig. III.9. A four-fold 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) my and mx. (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 inversion-rotation (b) symmetry axes and their action on an asymmetric figure - a tetrahedron. (From Vainshtein, p. 67)

Fig. III.13. The operation , a two-fold rotary-inversion 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 two-fold 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 47-53)

Cyclic point groups

These contain a single n-fold 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 n-fold axis of rotation are given the symbol n in the H-M system and Cn in the S system (C stands for cyclic). For example, the double-disk structure of tobacco mosaic virus (TMV) stacked disk aggregates contains 34 subunits (polypeptide chains) arranged with C17 symmetry (S notation) (Fig.III.15). Non-biological molecules can also have mirror planes of symmetry either parallel (e.g. nm or nmm in the H-M notation or Cnv in the S notation where v stands for vertical) or perpendicular (e.g. n/m in the H-M notation or Cnh in the S notation where h stands for horizontal) to the n-fold axis of symmetry.

Dihedral point groups

Dihedral point groups have axes of rotation at right angles to each other. These point groups consist of an n-fold axis perpendicular to n 2-fold axes. Most oligomeric enzymes display dihedral symmetry (Mathews and Bernhard, 1973). For example, the enzyme ribulose bisphosphate carboxylase/oxygenase (RuBisCO) has D4 symmetry (422 in H-M notation). The number of asymmetric units in the point group Dn 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.

Cubic point groups

The essential characteristic is four 3-fold 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 H-M notation), O (octahedral = 432) and I (icosahedral = 532). The tetrahedral point group contains 12 asymmetric units. Aspartate-ß-decarboxylase 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 quasi-equivalent positions.

Fig.III.15. The TMV stacked disk structure with C17 symmetry. (From Eisenberg and Crothers, p.757)
Fig. III.16. Schematic drawing of an icosahedral virus (532 symmetry) consisting of 60 hands that all have identical environments. (From Eisenberg and Crothers, p. 767)

4) Lattice restrictions and non-crystallographic symmetry

The crystal structure, crystal lattice, and motif may only contain one-, two-, three-, four-, or six-fold 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 17-fold axis of rotation (Fig.III.15). The TMV disc forms true 3D crystals and has been studied by x-ray crystallography. In the crystal, the disc occupies a general position in the unit cell, and therefore displays non-crystallographic 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 non-crystallographic symmetry.

d. Translational symmetry (Symmetry along a line)

1) Repetition in one dimension

Fig.III.17. A one-dimensional 'crystal' of right feet. (From Bernal, p.27)
Translation is the symmetry operation of shifting an object a given distance, say t, in a given direction, say the x direction, as illustrated in Fig.III.17 (one-dimensional crystal of right feet). The group of feet can be superimposed on itself if it is shifted in the x direction by the translation t. In this one-dimensional crystal, each foot occupies one unit cell and the distance t is the unit cell edge of the crystal.

2) Screw axes

Fig. III.18. A two-fold screw axis, 21, parallel to b and through the origin, which combines both a two-fold 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.18-20). Screw axes are symmetry elements of crystals that are helices with an integral number of asymmetric units per turn of the helix. An nm 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 21, 31, 32, 41, 42, 43, 61, 62, 63, and 65. 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 four-fold screw axes showing two identiy points for each. Note that the effect of 41 on a left hand is the mirror image of the effect of 43 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 non-crystallographic 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 shoe-shaped 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

The symmetry of a structure is described by a plane group if it is 2D or by a space group if it is 3D. All possible crystal symmetries are generated by combining all types of lattice symmetries with all types of motif symmetries. If the internal structure of the crystal is considered, additional symmetry exists due to the presence of screw axis and glide plane symmetries. This leads to 17 possible plane groups in two-dimensions and 230 space groups in three-dimensions. Thus, there are only 17 ways in which to generate a two-dimensional regular pattern from a motif associated with a two-dimensional lattice and 230 ways in which to generate a 3D regular pattern from a motif associated with a 3D lattice. Volume I of the International Tables for X-ray Crystallography (1969) provides descriptions and formulae for all the 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)

Only five plane groups and65 space groups are compatible with the enantiomorphic biological structures. Fig.III.6 and III.22 shows some examples of 2D plane groups. The initial stage of most X-ray crystallographic structure analyses involves the space group determination. When this is known, the number of asymmetric units in the unit cell is also known, thus it is often possible to learn the packing and number of molecules within the unit cell, and prove if the molecules are symmetric. Symmetry operations in the unit cell give rise to systematic absences in the diffraction patterns which often proves useful for determining the correct space group. The number of molecules per unit cell or per asymmetric unit is usually deduced from estimates of the molecular weight and measurements of the crystal density and unit cell volume. In many instances, image processing of electron micrographs of periodic biological specimens provides an objective means for determining or verifying these types of structural information.

Any periodic structure can be generated by placing a motif at every point of a lattice. The lattice is a rule for translation and the motif is the object that is translated. Notice that the motif does not always have to be an asymmetric object such as a right foot. In biological crystals, the motif often has the symmetry of one of the point groups or one of the screw axes. In such cases the periodic structure contains translational symmetry plus rotational (or reflection or inversion in some, mostly small biological molecules) symmetry. The periodic structure can be thought of as being built up in two steps. First a motif is generated from the asymmetric unit by the symmetry operations of the point group. Second, the structure is generated from the motif by the translational symmetry operations of the lattice:

Fig. III.23. Plane group symmetry P1. (From Bernal, p. 58)
Fig. III.24. Plane group symmetry P2. (From Bernal, p. 59)
Fig. III.25. Plane group symmetry Pg. (From Bernal, p. 61)
Fig. III.26. Plane group symmetry Pm. (From Bernal, p. 60)

Asymmetric unit motif structure

Glide plane symmetry (Figs. III.27-III.28) is produced by a translation followed by a mir-ror operation (or vice versa). Biological molecules do not, in general, display glide plane symme-tries because they do not exist in enantiomorphic pairs. Note, however, that biological molecules (or crystals) when viewed in two-dimensions (i.e. in projection) can display mirror symmetry.

Fig. III.27. A glide symmetry operation. (From Buerger, p. 8)
Fig. III.28. A b-glide plane normal to c and through the origin involves a translation of b/2 and areflection in a plane normal to c. It converts a point at x, y, z to one at x, y+1/2, -z. Note that left hands ARE converted to right hands, and vice versa.. (From Glusker and Trueblood, p. 74)

f. Examples of symmetrical biological molecules

1) Helical symmetry

Actin filament Bacterial flagella Chromatin fibers Neurotubules Bacterial pili Sickle cell hemoglobin fibers Tobacco mosaic virus Enzyme aggregates (e.g. catalase tubes) T4 bacteriophage sheath (extended or contracted configuration)

2) Point group symmetry

MOLECULE/AGGREGATE S H-M # asymmetric units
Asymmetric aggregates: e.g. ribosome (monomer)
1 1
Fibrous molecules: e.g. fibrinogen
2 2
lactate dehydrogenase
222 4
222 4
aspartate transcarbamylase
32 6
ribulose bisphosphate carboxylase/oxygenase
422 8
glutamine synthetase
622 12
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 (two-dimensional crystals)

Bacterial cell walls (e.g. Bacillus brevis T layer) Bladder luminal membrane Gap junctions Purple membrane

4) Space group symmetry (3D crystals) Various intracellular inclusions Various in vitro grown crystals suitable for x-ray crystallography