International
Tables for Crystallography Volume B Reciprocal space Edited by U. Shmueli © International Union of Crystallography 2006 
International Tables for Crystallography (2006). Vol. B, ch. 2.3, pp. 248250
Section 2.3.5. Noncrystallographic symmetry ^{a}Department of Biological Sciences, Purdue University, West Lafayette, Indiana 47907, USA, and ^{b}CABM & Rutgers University, 679 Hoes Lane, Piscataway, New Jersey 088545638, USA 
The interpretation of Pattersons can be helped by using various types of chemical or physical information. An obvious example is the knowledge that one or two heavy atoms per crystallographic asymmetric unit are present. Another example is the exploitation of a rigid chemical framework in a portion of a molecule (Nordman & Nakatsu, 1963; Burnett & Rossmann, 1971). One extremely useful constraint on the interpretation of Pattersons is noncrystallographic symmetry. Indeed, the structural solution of large biological assemblies such as viruses is only possible because of the natural occurrence of this phenomenon. The term `molecular replacement' is used for methods that utilize noncrystallographic symmetry in the solution of structures [for earlier reviews see Rossmann (1972) and Argos & Rossmann (1980)]. These methods, which are only partially dependent on Patterson concepts, are discussed in Sections 2.3.6 –2.3.8.
Crystallographic symmetry applies to the whole of the threedimensional crystal lattice. Hence, the symmetry must be expressed both in the lattice and in the repeating pattern within the lattice. In contrast, noncrystallographic symmetry is valid only within a limited volume about the noncrystallographic symmetry element. For instance, the noncrystallographic twofold axes in the plane of the paper of Fig. 2.3.5.1 are true only in the immediate vicinity of each local dyad. In contrast, the crystallographic twofold axes perpendicular to the plane of the paper (Fig. 2.3.5.1) apply to every point within the lattice. Two types of noncrystallographic symmetry can be recognized: proper and improper rotations. A proper symmetry element is independent of the sense of rotation, as, for example, a fivefold axis in an icosahedral virus; a rotation either left or right by onefifth of a revolution will leave all parts of a given icosahedral shell (but not the whole crystal) in equivalent positions. Proper noncrystallographic symmetry can also be recognized by the existence of a closed point group within a defined volume of the lattice. Improper rotation axes are found when two molecules are arbitrarily oriented relative to each other in the same asymmetric unit or when they occur in two entirely different crystal lattices. For instance, in Fig. 2.3.5.2, the object can be rotated by +θ about the axis at P to orient it identically with . However, the two objects will not be coincident after a rotation of by −θ or of by +θ. The envelope around each noncrystallographic object must be known in order to define an improper rotation. In contrast, only the volume about the closed point group need be defined for proper noncrystallographic operations. Hence, the boundaries of the repeating unit need not correspond to chemically covalently linked units in the presence of proper rotations.
Translational components of noncrystallographic rotation elements are said to be `precise' in a direction parallel to the axis and `imprecise' perpendicular to the axis (Rossmann et al., 1964). The position, but not direction, of a rotation axis is arbitrary. However, a convenient choice is one that leaves the translation perpendicular to the axis at zero after rotation (Fig. 2.3.5.3).
Noncrystallographic symmetry has been used as a tool in structural analysis primarily in the study of biological molecules. This is due to the propensity of proteins to form aggregates with closed point groups, as, for instance, viruses with 532 symmetry. At best, only part of such a point group can be incorporated into the crystal lattice. Since biological materials cannot contain inversion elements, all studies of noncrystallographic symmetries have been limited to rotational axes. Reflection planes and inversion centres could also be considered in the application of molecular replacement to nonbiological materials.
In this chapter, the relationship will be used to describe noncrystallographic symmetry, where x and x′ are position vectors, expressed as fractional coordinates, with respect to the crystallographic origin, [C] is a rotation matrix, and d is a translation vector. Crystallographic symmetry will be described as where [T] and t are the crystallographic rotation matrix and translation vector, respectively. The noncrystallographic asymmetric unit will be defined as having n copies within the crystallographic asymmetric unit, and the unit cell will be defined as having m crystallographic asymmetric units. Hence, there are noncrystallographic asymmetric units within the unit cell. Clearly, the n noncrystallographic asymmetric units cannot completely fill the volume of one crystallographic asymmetric unit. The remaining space must be assumed to be empty or to be occupied by solvent molecules which disobey the noncrystallographic symmetry.
If noncrystallographic symmetry is present, an atom at a general position within the relevant volume will imply the presence of others within the same crystallographic asymmetric unit. If the noncrystallographic symmetry is known, then the positions of equivalent atoms may be generated from a single atomic position. The additional vector interactions which arise from crystallographically and noncrystallographically equivalent atoms in a crystal may be predicted and exploited in an interpretation of the Patterson function.
An object in real space which has a closed point group may incorporate some of its symmetry in the crystallographic symmetry. If there are l such objects in the cell, then there will be equivalent positions within each object. The `selfvectors' formed between these positions within the object will be independent of the position of the objects. This distinction is important in that the selfvectors arising from atoms interacting with other atoms within a single particle may be correctly predicted without the knowledge of the particle centre position. In fact, this distinction may be exploited in a twostage procedure in which an atom may be first located relative to the particle centre by use of the selfvectors and subsequently the particle may be positioned relative to crystallographic symmetry elements by use of the `crossvectors' (Table 2.3.5.1).

The interpretation of a heavyatom difference Patterson for the holoenzyme of lobster glyceraldehyde3phosphate dehydrogenase (GAPDH) provides an illustration of how the known noncrystallographic symmetry can aid the solution (Rossmann et al., 1972; Buehner et al., 1974). The GAPDH enzyme crystallized in a cell (a = 149.0, b = 139.1, c = 80.7 Å) containing one tetramer per asymmetric unit. A rotationfunction analysis had indicated the presence of three mutually perpendicular molecular twofold axes which suggested that the tetramer had 222 symmetry, and a locked rotation function determined the precise orientation of the tetramer relative to the crystal axes (see Table 2.3.5.2). Packing considerations led to assignment of a tentative particle centre near .

An isomorphous difference Patterson was calculated for the K_{2}HgI_{4} derivative of GAPDH using data to a resolution of 6.8 Å. From an analysis of the three Harker sections, a tentative first heavyatom position was assigned (atom at x, y, z). At this juncture, the known noncrystallographic symmetry was used to obtain a full interpretation. From Table 2.3.5.2 we see that molecular axis 2 will generate a second heavy atom with coordinates roughly (if the molecular centre was assumed to be at ). Starting from the tentative coordinates of site , the site related by molecular axis 1 was detected at about the predicted position and the second site generated acceptable crossvectors with the earlier determined site . Further examination enabled the completion of the set of four noncrystallographically related heavyatom sites, such that all predicted Patterson vectors were acceptable and all four sites placed the molecular centre in the same position. Following refinement of these four sites, the corresponding SIR phases were used to find an additional set of four sites in this compound as well as in a number of other derivatives. The multiple isomorphous replacement phases, in conjunction with realspace electrondensity averaging of the noncrystallographically related units, were then sufficient to solve the GAPDH structure.
When investigators studied larger macromolecular aggregates such as the icosahedral viruses, which have 532 point symmetry, systematic methods were developed for utilizing the noncrystallographic symmetry to aid in locating heavyatom sites in isomorphous heavyatom derivatives. Argos & Rossmann (1974, 1976) introduced an exhaustive Patterson search procedure for a single heavyatom site within the noncrystallographic asymmetric unit which has been successfully applied to the interpretation of both virus [satellite tobacco necrosis virus (STNV) (Lentz et al., 1976), southern bean mosaic virus (Rayment et al., 1978), alfalfa mosaic virus (Fukuyama et al., 1983), cowpea mosaic virus (Stauffacher et al., 1987)] and enzyme [catalase (Murthy et al., 1981)] heavyatom difference Pattersons. A heavy atom is placed in turn at all plausible positions within the volume of the noncrystallographic asymmetric unit and the corresponding vector set is constructed from the resulting constellation of heavy atoms. Argos & Rossmann (1976) found a spherical polar coordinate search grid to be convenient for spherical viruses. After all vectors for the current search position are predicted, the vectors are allocated to the nearest grid point and the list is sorted to eliminate recurring ones. The criterion used by Argos & Rossmann for selecting a solution is that the sum of the lookup Patterson density values achieves a high value for a correct heavyatom position. The sum is corrected for the carpet of crossvectors by the second term in the sum.
An additional criterion, which has been found useful for discriminating correct solutions, is a unit vector density criterion where is the number of vectors expected to contribute to the Patterson density value (Arnold et al., 1987). This criterion can be especially valuable for detecting correct solutions at special search positions, such as an icosahedral fivefold axis, where the number of vector lookup positions may be drastically reduced owing to the higher symmetry. An alternative, but equivalent, method for locating heavyatom positions from isomorphous difference data is discussed in Section 2.3.3.5.
Even for a single heavyatom site at a general position in the simplest icosahedral or virus, there are 60 equivalent heavy atoms in one virus particle. The number of unique vectors corresponding to this selfparticle vector set will depend on the crystal symmetry but may be as many as for a virus particle at a general crystallographic position. Such was the case for the STNV crystals which were in space group C2 containing four virus particles at general positions. The method of Argos & Rossmann was applied successfully to a solution of the K_{2}HgI_{4} derivative of STNV using a 10 Å resolution difference Patterson. Application of the noncrystallographic symmetry vector search procedure to a K_{2}Au(CN)_{2} derivative of human rhinovirus 14 (HRV14) crystals (space group ) has succeeded in establishing both the relative positions of heavy atoms within one particle and the positions of the virus particles relative to the crystal symmetry elements (Arnold et al., 1987). The particle position was established by incorporating interparticle vectors in the search and varying the particle position along the crystallographic threefold axis until the best fit for the predicted vector set was achieved.
References
Argos, P. & Rossmann, M. G. (1974). Determining heavyatom positions using noncrystallographic symmetry. Acta Cryst. A30, 672–677.Argos, P. & Rossmann, M. G. (1976). A method to determine heavyatom positions for virus structures. Acta Cryst. B32, 2975–2979.
Argos, P. & Rossmann, M. G. (1980). Molecular replacement methods. In Theory and practice of direct methods in crystallography, edited by M. F. C. Ladd & R. A. Palmer, pp. 361–417. New York: Plenum.
Arnold, E., Vriend, G., Luo, M., Griffith, J. P., Kamer, G., Erickson, J. W., Johnson, J. E. & Rossmann, M. G. (1987). The structure determination of a common cold virus, human rhinovirus 14. Acta Cryst. A43, 346–361.
Buehner, M., Ford, G. C., Moras, D., Olsen, K. W. & Rossmann, M. G. (1974). Structure determination of crystalline lobster Dglyceraldehyde3phosphate dehydrogenase. J. Mol. Biol. 82, 563–585.
Burnett, R. M. & Rossmann, M. G. (1971). The determination of the crystal structure of trans2,4dihydroxy2,4dimethylcyclohexanetrans1acetic acid γlactone, C_{10}H_{16}O_{3}, using rotation and translation functions in reciprocal space. Acta Cryst. B27, 1378–1387.
Fukuyama, K., AbdelMeguid, S. S., Johnson, J. E. & Rossmann, M. G. (1983). Structure of a T = 1 aggregate of alfalfa mosaic virus coat protein seen at 4.5 Å resolution. J. Mol. Biol. 167, 873–894.
Lentz, P. J. Jr, Strandberg, B., Unge, T., Vaara, I., Borell, A., Fridborg, K. & Petef, G. (1976). The determination of the heavyatom substitution sites in the satellite tobacco necrosis virus. Acta Cryst. B32, 2979–2983.
Murthy, M. R. N., Reid, T. J. III, Sicignano, A., Tanaka, N. & Rossmann, M. G. (1981). Structure of beef liver catalase. J. Mol. Biol. 152, 465–499.
Nordman, C. E. & Nakatsu, K. (1963). Interpretation of the Patterson function of crystals containing a known molecular fragment. The structure of an Alstonia alkaloid. J. Am. Chem. Soc. 85, 353–354.
Rayment, I., Johnson, J. E., Suck, D., Akimoto, T. & Rossmann, M. G. (1978). An 11 Å resolution electron density map of southern bean mosaic virus. Acta Cryst. B34, 567–578.
Rossmann, M. G. (1972). The molecular replacement method. New York: Gordon & Breach.
Rossmann, M. G., Blow, D. M., Harding, M. M. & Coller, E. (1964). The relative positions of independent molecules within the same asymmetric unit. Acta Cryst. 17, 338–342.
Rossmann, M. G., Ford, G. C., Watson, H. C. & Banaszak, L. J. (1972). Molecular symmetry of glyceraldehyde3phosphate dehydrogenase. J. Mol. Biol. 64, 237–249.
Stauffacher, C. V., Usha, R., Harrington, M., Schmidt, T., Hosur, M. V. & Johnson, J. E. (1987). The structure of cowpea mosaic virus at 3.5 Å resolution. In Crystallography in molecular biology, edited by D. Moras, J. Drenth, B. Strandberg, D. Suck & K. Wilson, pp. 293–308. New York, London: Plenum.