International
Tables for
Crystallography
Volume F
Crystallography of biological macromolecules
Edited by M. G. Rossmann and E. Arnold

International Tables for Crystallography (2006). Vol. F, ch. 13.4, pp. 280-281   | 1 | 2 |

Section 13.4.3. Phase determination using NCS

M. G. Rossmanna* and E. Arnoldb

aDepartment of Biological Sciences, Purdue University, West Lafayette, IN 47907-1392, USA, and  bBiomolecular Crystallography Laboratory, CABM & Rutgers University, 679 Hoes Lane, Piscataway, NJ 08854-5638, USA
Correspondence e-mail:  mgr@indiana.bio.purdue.edu

13.4.3. Phase determination using NCS

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The molecular replacement method [cf. Rossmann & Blow (1962[link]); Rossmann (1972,[link] 1990[link]); Argos & Rossmann (1980[link]); Rossmann & Arnold (2001[link])] is dependent upon the presence of NCS, whether it relates objects within one crystal lattice or between crystal lattices. The NCS rotational relationship in real space is exactly mimicked in reciprocal space. Local symmetry in real space has the equivalent effect of rotating a reciprocal lattice onto itself or another (with origins coincident), such that the integral reciprocal-lattice points of one reciprocal space coincide with non-integral reciprocal-lattice positions in the other. As the reciprocal lattice samples the Fourier transform of a molecule only at finite and integral reciprocal-lattice points, the effect of an NCS operation is to permit sampling of the molecular transform at intermediate non-integral reciprocal-lattice positions. If such sampling occurs frequently enough, it will constitute a plot of the continuous transform of the molecule and, hence, amount to a structure determination.

Whenever a molecule exists more than once either in the same unit cell or in different unit cells, then error in the molecular electron-density distribution due to error in phasing can be reduced by averaging the various molecular copies. The number of such copies, N, is referred to as the noncrystallographic redundancy. As the NCS is, by definition, only local (often pertaining to a particular molecular centre), there are holes and gaps between the averaged density, which presumably are solvent space between molecules. Thus, the electron density can be improved both by averaging electron density and by setting the density between molecules to a low, constant value (`solvent flattening'). Phases calculated by Fourier back-transforming the improved density should be more accurate than the original phases. Hence, the observed structure amplitudes (suitably weighted) can be associated with the improved phases, and a new and improved map can be calculated. This, in turn, can again be averaged until convergence has been reached and the phases no longer change. In addition, the back-transformed map can be used to compute phases just beyond the extremity of the resolution of the terms used in the original map. The resultant amplitudes will not be zero because the map had been modified by averaging and solvent flattening. Thus, phases can be gradually extended and improved, starting from a very low resolution approximation to the molecular structure. This procedure was first implemented in reciprocal space (Rossmann & Blow, 1963[link]; Main, 1967[link]; Crowther, 1969[link]) and then, more recently, in real space (Bricogne, 1974,[link] 1976[link]; Johnson, 1978[link]; Jones, 1992[link]; Rossmann et al., 1992[link]). More recently still, there has been an attempt to reproduce the very successful real-space procedure in reciprocal space (Tong & Rossmann, 1995[link]).

Early examples of such a procedure for phase improvement are the structure determinations of deoxyhaemoglobin (Muirhead et al., 1967[link]), α-chymotrypsin (Matthews et al., 1967[link]), lobster glyceraldehyde-3-phosphate dehydrogenase (Buehner et al., 1974[link]), hexokinase (Fletterick & Steitz, 1976[link]), tobacco mosaic virus disk protein (Champness et al., 1976[link]; Bloomer et al., 1978[link]), the influenza virus haemagglutinin spike (Wilson et al., 1981[link]), tomato bushy stunt virus (Harrison et al., 1978[link]) and southern bean mosaic virus (Abad-Zapatero et al., 1980[link]). Early examples of phase extension, using real-space electron-density averaging, were the study of glyceraldehyde-3-phosphate dehydrogenase (Argos et al., 1975[link]), satellite tobacco necrosis virus (Nordman, 1980[link]), haemocyanin (Gaykema et al., 1984[link]), human rhinovirus 14 (Rossmann et al., 1985[link]) and poliovirus (Hogle et al., 1985[link]). Since then, this method has been used in numerous virus structure determinations, with the phase extension being initiated from ever lower resolution.

A once-popular computer program for real-space averaging was written by Gerard Bricogne (1976[link]). Another program has been described by Johnson (1978[link]). Both programs were based on a double-sorting procedure. Bricogne (1976[link]) had suggested that, with interpolation between grid points using linear polynomials, it was necessary to sample electron density at grid intervals finer than one-sixth of the resolution limit of the Fourier terms that were used in calculating the map. With the availability of more computer memory, it was possible to store much of the electron density, thus avoiding time-consuming sorting operations (Hogle et al., 1985[link]; Luo et al., 1989[link]). Simultaneously, the storage requirements could be drastically reduced by using interpolation with quadratic polynomials. While the latter required a little extra computation time, this was far less than what would have been needed for sorting. Furthermore, it was found that Bricogne's estimate for the fineness of the map storage grid was too pessimistic, even for linear interpolation, which works well to about 1/2.5 of the resolution limit of the map.

In addition to changes in strategy brought about by computers with much larger memories, experience has been gained in program requirements for real-space averaging for phase determination (Dodson et al., 1992[link]). Here we give a general procedure for electron-density averaging.

References

Abad-Zapatero, C., Abdel-Meguid, S. S., Johnson, J. E., Leslie, A. G. W., Rayment, I., Rossmann, M. G., Suck, D. & Tsukihara, T. (1980). Structure of southern bean mosaic virus at 2.8 Å resolution. Nature (London), 286, 33–39.Google Scholar
Argos, P., Ford, G. C. & Rossmann, M. G. (1975). An application of the molecular replacement technique in direct space to a known protein structure. Acta Cryst. A31, 499–506.Google Scholar
Argos, P. & Rossmann, M. G. (1980). Molecular replacement method. In Theory and practice of direct methods in crystallography, edited by M. F. C. Ladd & R. A. Palmer, pp. 361–417. New York: Plenum.Google Scholar
Bloomer, A. C., Champness, J. N., Bricogne, G., Staden, R. & Klug, A. (1978). Protein disk of tobacco mosaic virus at 2.8 Å resolution showing the interactions within and between subunits. Nature (London), 276, 362–368.Google Scholar
Bricogne, G. (1974). Geometric sources of redundancy in intensity data and their use for phase determination. Acta Cryst. A30, 395–405.Google Scholar
Bricogne, G. (1976). Methods and programs for direct-space exploitation of geometric redundancies. Acta Cryst. A32, 832–847.Google Scholar
Buehner, M., Ford, G. C., Moras, D., Olsen, K. W. & Rossmann, M. G. (1974). Structure determination of crystalline lobster D-glyceraldehyde-3-phosphate dehydrogenase. J. Mol. Biol. 82, 563–585.Google Scholar
Champness, J. N., Bloomer, A. C., Bricogne, G., Butler, P. J. G. & Klug, A. (1976). The structure of the protein disk of tobacco mosaic virus at 5 Å resolution. Nature (London), 259, 20–24.Google Scholar
Crowther, R. A. (1969). The use of non-crystallographic symmetry for phase determination. Acta Cryst. B25, 2571–2580.Google Scholar
Dodson, E. J., Gover, S. & Wolf, W. (1992). Editors. Proceedings of the CCP4 study weekend. Molecular replacement. Warrington: Daresbury Laboratory.Google Scholar
Fletterick, R. J. & Steitz, T. A. (1976). The combination of independent phase information obtained from separate protein structure determinations of yeast hexokinase. Acta Cryst. A32, 125–132.Google Scholar
Gaykema, W. P. J., Hol, W. G. J., Vereijken, J. M., Soeter, N. M., Bak, H. J. & Beintema, J. J. (1984). 3.2 Å structure of the copper-containing, oxygen-carrying protein Panulirus interruptus haemocyanin. Nature (London), 309, 23–29.Google Scholar
Harrison, S. C., Olson, A. J., Schutt, C. E., Winkler, F. K. & Bricogne, G. (1978). Tomato bushy stunt virus at 2.9 Å resolution. Nature (London), 276, 368–373.Google Scholar
Hogle, J. M., Chow, M. & Filman, D. J. (1985). Three-dimensional structure of poliovirus at 2.9 Å resolution. Science, 229, 1358–1365.Google Scholar
Johnson, J. E. (1978). Appendix II. Averaging of electron density maps. Acta Cryst. B34, 576–577.Google Scholar
Jones, T. A. (1992). a, yaap, asap, @#*? A set of averaging programs. In Proceedings of the CCP4 study weekend. Molecular replacement, edited by E. Dodson, S. Gover & W. Wolf, pp. 91–105. Warrington: Daresbury Laboratory.Google Scholar
Luo, M., Vriend, G., Kamer, G. & Rossmann, M. G. (1989). Structure determination of Mengo virus. Acta Cryst. B45, 85–92.Google Scholar
Main, P. (1967). Phase determination using non-crystallographic symmetry. Acta Cryst. 23, 50–54.Google Scholar
Matthews, B. W., Sigler, P. B., Henderson, R. & Blow, D. M. (1967). Three-dimensional structure of tosyl-α-chymotrypsin. Nature (London), 214, 652–656.Google Scholar
Muirhead, H., Cox, J. M., Mazzarella, L. & Perutz, M. F. (1967). Structure and function of haemoglobin. III. A three-dimensional Fourier synthesis of human deoxyhaemoglobin at 5.5 Å resolution. J. Mol. Biol. 28, 117–156.Google Scholar
Nordman, C. E. (1980). Procedures for detection and idealization of non-crystallographic symmetry with application to phase refinement of the satellite tobacco necrosis virus structure. Acta Cryst. A36, 747–754.Google Scholar
Rossmann, M. G. (1972). Editor. The molecular replacement method. New York: Gordon & Breach.Google Scholar
Rossmann, M. G. (1990). The molecular replacement method. Acta Cryst. A46, 73–82.Google Scholar
Rossmann, M. G. & Arnold, E. (2001). Patterson and molecular-replacement techniques. In International tables for crystallography, Vol. B. Reciprocal space, edited by U. Shmueli, ch. 2.3. Dordrecht: Kluwer Academic Publishers.Google Scholar
Rossmann, M. G., Arnold, E., Erickson, J. W., Frankenberger, E. A., Griffith, J. P., Hecht, H. J., Johnson, J. E., Kamer, G., Luo, M., Mosser, A. G., Rueckert, R. R., Sherry, B. & Vriend, G. (1985). Structure of a human common cold virus and functional relationship to other picornaviruses. Nature (London), 317, 145–153.Google Scholar
Rossmann, M. G. & Blow, D. M. (1962). The detection of sub-units within the crystallographic asymmetric unit. Acta Cryst. 15, 24–31.Google Scholar
Rossmann, M. G. & Blow, D. M. (1963). Determination of phases by the conditions of non-crystallographic symmetry. Acta Cryst. 16, 39–45.Google Scholar
Rossmann, M. G., McKenna, R., Tong, L., Xia, D., Dai, J.-B., Wu, H., Choi, H.-K. & Lynch, R. E. (1992). Molecular replacement real-space averaging. J. Appl. Cryst. 25, 166–180.Google Scholar
Tong, L. & Rossmann, M. G. (1995). Reciprocal-space molecular-replacement averaging. Acta Cryst. D51, 347–353.Google Scholar
Wilson, I. A., Skehel, J. J. & Wiley, D. C. (1981). Structure of the haemagglutinin membrane glycoprotein of influenza virus at 3 Å resolution. Nature (London), 289, 366–373.Google Scholar








































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