research papers
Spherically averaged phased translation function and its application to the search for molecules and fragments in electrondensity maps
^{a}Department of Chemistry, University of York, Heslington, York YO1 5DD, England, and ^{b}School of Chemistry, University of Exeter, Stocker Road, Exeter EX4 4QD, England
^{*}Correspondence email: alexei@ysbl.york.ac.uk
The molecularreplacement method has been extended to locate molecules and their fragments in an electrondensity map. The approach is based on a new spherically averaged phased translation function. The position of the centre of mass of a search model is found prior to determination of its orientation. The orientation is subsequently found by a phased rotation function. The technique also allows superposition of distantly related macromolecules. The method has been implemented in a computer program MOLREP and successfully tested using experimental data sets.
1. Introduction
The molecularreplacement method (MR; Rossmann, 1972) is one of the principal techniques for determining the crystal structures of macromolecules. The location of a search model in the of the crystal of interest is usually divided into two threedimensional searches. Firstly, the orientation of the model is found using a crossrotation function (RF). Points in Eulerian space with high RF value indicate the probable orientations of a search molecule. The positional search is then carried out by applying the translation function (TF) to the search model orientated according to its highest RF values.
In spite of significant improvements in MR algorithms in recent years (Turkenburg & Dodson, 1996; Carter & Sweet, 1997), many cases remain unsolved for a variety of reasons. For some of these cases some prior phase information either from MIR/MAD or from a prepositioned may be available. Two approaches are currently used to locate molecules in an electrondensity map. The phased translation function (PTF; Colman & Fehlhammer, 1976; Bentley, 1997) can be used to position the model in an unknown provided that the orientation of the search molecule is known. The disadvantage of this approach is that the conventional RF used to find the orientation of the model does not take into account prior phase information. Another approach is an exhaustive sixdimensional search of all possible orientations and positions of the search molecule in the electrondensity map calculated either in (ESSENS program; Kleywegt & Jones, 1997) or in (FFFEAR program; Cowtan, 1998). This sixdimensional search is computationally demanding even when a coarse grid is used.
We have extended the molecularreplacement method to locate macromolecules and their fragments in the electron density. The central point of the new approach is to position the centre of mass of the search model in the electron density prior to the determination of the model orientation. This is performed using a spherically averaged phased translation function (SAPTF). A local phased rotation function (PRF) is subsequently calculated to determine the orientation of the search molecule. A phased translation function (PTF) is used to check and refine the solution. This new approach also can be used for the superposition of distantly related macromolecules. The method was implemented in the program MOLREP and tested on a number of cases using experimental Xray data sets.
2. Description of the method
Suppose that some prior phase information (experimental electron density) was available for a crystal of a biological macromolecule either from a MIR/MAD experiment or from a prepositioned αhelix). The suggested approach is to find the position of the centre of mass of the search model in the prior to determination of its orientation. This strategy is based upon the following idea. The electron density calculated for a search model can be spherically averaged within a certain sphere around its centre of mass to give its radial distribution. The experimental electron density can also be spherically averaged within a sphere of the same radius at each point of the An overlap function between the two, i.e. the electron density for the model spherically averaged around its centre of mass and the experimental electron density spherically averaged within the same radius around a given point () in is referred to as spherically averaged phased translation function (SAPTF),
and a search model is available which is either a homologous molecule or a molecular fragment (such as anwhere ρ_{obs} is the observed electron density, ρ_{m} is the electron density of the model, represents spherical averaging of the function to give its radial distribution, is the vector in and is the origin of the new coordinate system.
The SAPTF therefore gives a maximum overlap for a correct position of the model. By expanding SAPTF into spherical harmonics it is possible to represent it as a Fourier series and to calculate it using fast Fourier transforms (Appendix A).
Once a putative centre of mass for a search model in the crystal has been located by SAPTF, its orientation at this position can be determined using a phased rotation function (PRF), which is defined as an overlap function between a rotated model electron density and an experimental electron density calculated within the same sphere. Its value has a maximum at a correct orientation for the model. This function is based on the fast rotationfunction algorithm of Crowther (1972), but has the following modifications.

In step 2, for each highly scoring point of SAPTF a local PRF is calculated to find the orientation of the search molecule.
In step 3, the phased translation function is calculated for the orientations of the model found in step 2 to check and refine the position for the centre of mass that was found in step 1.
The above approach allows the separation of the rotation and translation searches. Since fast Fourier transforms are used to calculate the SAPTF and the PTF functions, this molecularreplacement search against electron density is almost as fast as classical
The most timeconsuming part is calculating the PRF.There is no need to have both the model and experimental electron densities averaged. An alternative approach to the one described above would be to spherically average only the model electron density. (1) can then be rewritten as
This gives the possibility of performing the PTF search with the spherically averaged model electron density as a search model. This would require averaging of the electron density in real space and has not yet been implemented.
The SAPTF approach can also be used to superimpose distantly related macromolecules by fitting together the electron densities generated from the two sets of coordinates. The smaller of the two molecules is then taken to be the model and positioned against electron density generated from the larger molecule after placing it in a suitably large
This method of fitting of distantly related molecules has the following advantages.

3. Applications
The SAPTFbased algorithm has been incorporated into the program MOLREP (Vagin & Teplyakov, 1997), a fully automated molecularreplacement program which utilizes an original fullsymmetry TF combined with a packing function (Vagin, 1989).
The search for molecules and their fragments in electron density was tested on a number of cases and gave satisfactory results. Four of them are described here. The first test case illustrates a search for molecular fragments, namely αhelices, in the SIR density. The second test case is the positioning of a search model with no apparent sequence similarity to a crystallized protein into MIR electron density. The third test case shows the application of the SAPTFbased approach to superimpose two protein molecules which are not related in sequence or function but have similar folds. Test case 4 describes the superposition of nonprotein macromolecules.
3.1. Test 1. Positioning of a molecular fragment into isomorphous density
Hevamine from Hevea brasiliensis (PDB code 2hvm ; Terwisscha van Scheltinga et al., 1994) crystallizes in the orthorhombic P2_{1}2_{1}2_{1}, with unitcell parameters a = 52.3, b = 57.7, c = 82.1 Å. An isomorphous derivative was used for phasing to give SIR phases to 3 Å with an overall FOM of 0.45. The search model was a tenresidue αhelix. A SAPTF function (Fig. 1) was calculated with a radius of integration of 9 Å. Most of the SAPTF maximums are at the known helix positions and six helices of seven were correctly positioned by the procedure. This test case illustrates the necessity for including a relatively large number of possible SAPTF and PRF solutions in the calculations. The suggested algorithm seems to be more successful in searching for αhelices than for βstrands. We attribute this to the relative compactness of αhelices in comparison with elongated and flexible βstrands, which give poor contrast in SAPTF calculations.
3.2. Test 2. The positioning of a distantly related model into isomorphous density
Esterase 713 from an Alcaligenes species (PDB code 1qlw ; Bourne et al., 2000) is a dimer of two identical subunits of 318 residues each folded into a single domain with an α/β hydrolase fold. There is no detectable sequence similarity between esterase 713 and any other enzyme of known structure. The protein crystallized in the orthorhombic P2_{1}2_{1}2_{1}, with unitcell parameters a = 58.6, b = 116.8, c = 132.0 Å. The contains an esterase dimer. MIR phases were derived from three heavyatom derivatives to 3 Å with an overall FOM of 0.395. The search model was part of a monomer of acetylcholineesterase from Torpedo californica (PDB code 2ace ; Sussman et al., 1991) truncated to leave only the α/β hydrolase fold (230 residues of 527). The esterase 713 monomer and the search model superimpose with an r.m.s. deviation of 2.07 Å between their C^{α} positions over 159 residues. Conventional MR search with the model was conducted for a range of resolution limits and integration radii by both MOLREP (Vagin & Teplyakov, 1997) and AMoRe (Navaza, 1994). Both programs failed to find either the correct orientations for the search model or the translation solution for the correctly orientated model. However, a SAPTFbased search with MOLREP using MIR phases at 10–3 Å found the molecularreplacement solution for both subunits (Figs. 2a and 2b). The two correct solutions had the highest peak values for both SAPTF and PRF. The correct solutions displayed even better contrast when densitymodified phases after NCS averaging were used.
3.3. Test 3. The suggested approach can also be used to superimpose distantly related molecules
In this case, a SAPTFbased search is conducted to determine the maximal overlap of electron densities calculated from atomic models rather than the best fit between the atomic positions as in most model superposition algorithms. Thermococcus litoralis pyrrolidone carboxyl peptidase (PDB code 1a2z ; Singleton et al., 1999) and Escherichia coli purine nucleotide phosphorylase (PDB code 1a69 ; Koellner et al., 1998) have similar folds but are not related by sequence or function. These two enzymes were superimposed by MOLREP (Fig. 3). From the threedimensional alignment only it could be seen that the C^{α} backbones of the two models superimpose with an r.m.s. deviation of 1.9 Å over 119 residues.
3.4. Test 4. Superposition of nonprotein macromolecules
The advantage of the suggested approach for the alignment of macromolecules is that the search models can differ in size and conformation and can be extended to other nonprotein molecules. Fig. 4 shows a superposition of the yeast Phe (PDB code 1tn2 ; Brown et al., 1983) with the Ser2 from T. thermophilus from its complex with seryltRNA synthetase (PDB code 1ser ; Belrhali et al., 1994). Although the latter model is incomplete and there are conformational differences between the models, MOLREP correctly superimposed the common core.
4. Distribution
The program MOLREP is written in standard Fortran 77 and can be run under UNIX, Linux and Windows. The program MOLREP is available free from AAV, anonymous ftp account ftp.ysbl.york.ac.uk , https://www.ysbl.york.ac.uk/~alexei/ or from CCP4. Inquiries about the program should be addressed to AAV at alexei@ysbl.york.ac.uk.
APPENDIX A
Spherically averaged phased translation function (SAPTF)
If we have an electron density ρ(), we can expand it within a spherical volume r ≤ a,
where = r (r, υ, φ) in polar coordinates, j_{l} is the spherical Bessel function of order l, λ_{ln} zeroes the Bessel function such that j_{l}_{} (λ_{ln}a) = 0 (n = 1, 2, …), a is the radius of the sphere (in the origin of coordinate system), Y^{m}_{l} are the spherical harmonics and
For the new coordinate system with the origin at point ,
The Fourier series corresponding to a crystal is
and for a new origin
where F is the (R, Y, Φ) is the vector in the polar coordinate system in and is the new origin of the coordinate system. Substituting (4) in (3),
where
and
Series with coefficients a_{00n}, n = 1, 2, …, represent the spherically averaged function
where represents spherical averaging of the function and
For a model with its centre of gravity at the origin of the coordinate system,
The spherically averaged phased translation function (SAPTF) is defined as
The integral above is not 0 when n = n′,
The last expression is a Fourier series with coefficients = c_{00n}(R)j_{0}(2πRa)b_{00n} and can be calculated using the fast Fourier transform.
Acknowledgements
We thank Garib Murshudov, Eleanor Dodson and Alexei Teplyakov for useful discussions and Ewald Schroder for helpful comments on the manuscript. AAV is supported by the Collaborative Computational Project, Number 4. MNI is supported by a postdoctoral fellowship from the BBSRC Chemical and Pharmaceutical Directorate.
References
Belrhali, H., Yaremchuk, A., Tukalo, M., Larsen, K., BerthetColominas, C., Leberman, R., Beijer, B., Sproat, B., AlsNielsen, J., Grubel, G., Legrand, J. F., Lehmann, M. & Cusack, S. (1994). Science, 263, 1432–1436. CrossRef CAS PubMed Web of Science Google Scholar
Bentley, G. A. (1997). Methods Enzymol. 276, 611–619. CrossRef Web of Science Google Scholar
Bourne, P. C., Isupov, M. N. & Littlechild, J. A. (2000). Structure, 8, 143–151. Web of Science CrossRef PubMed CAS Google Scholar
Brown, R. S., Hingerty, B. E., Dewan, J. C. & Klug, A. (1983). Nature (London), 303, 543–546. CrossRef CAS PubMed Web of Science Google Scholar
Carter, C. W. & Sweet, R. M. (1997). Editors. Macromolecular Crystallography Part A. Methods Enzymology, Vol. 276. San Diego: Academic Press. Google Scholar
Colman, P. M. & Fehlhammer, H. (1976). J. Mol. Biol. 100, 278–282. CrossRef PubMed CAS Web of Science Google Scholar
Cowtan, K. (1998). Acta Cryst. D54, 750–756. Web of Science CrossRef CAS IUCr Journals Google Scholar
Crowther, R. A. (1972). The Molecular Replacement Method, edited by M. G. Rossmann, pp. 173–178. New York: Gordon & Breach. Google Scholar
Kleywegt, G. J. & Jones, T. A. (1997). Acta Cryst. D53, 179–185. CrossRef CAS Web of Science IUCr Journals Google Scholar
Koellner, G., Luic, M., Shugar, D., Saenger, W. & Bzowska, A. (1998). J. Mol. Biol. 280, 153–166. Web of Science CrossRef CAS PubMed Google Scholar
Kraulis, P. J. (1991). J. Appl. Cryst. 24, 946–950. CrossRef Web of Science IUCr Journals Google Scholar
Navaza, J. (1994). Acta Cryst. A50, 157–163. CrossRef CAS Web of Science IUCr Journals Google Scholar
Rossmann, M. G. (1972). Editor. The Molecular Replacement Method. New York: Gordon & Breach. Google Scholar
Singleton, M. R., Isupov, M. N. & Littlechild J. A. (1999). Structure Fold Des. 7, 237–244. Web of Science CrossRef PubMed CAS Google Scholar
Sussman, J. L., Harel, M., Frolow, F., Oefner, C., Goldman, A., Toker, L. & Silman, I. (1991). Science, 253, 872–879. CrossRef PubMed CAS Web of Science Google Scholar
Terwisscha van Scheltinga, A. C., Kalk, K. H., Beintema, J. J. & Dijkstra, B. W. (1994). Structure, 2, 1181–1189. CrossRef CAS PubMed Google Scholar
Turkenburg, J. P. & Dodson, E. J. (1996). Curr. Opin. Struct. Biol. 6, 604–610. CrossRef CAS PubMed Web of Science Google Scholar
Vagin, A. A. (1989). CCP4 Newsl. Protein Crystallogr. 29, 117–121. Google Scholar
Vagin, A. & Teplyakov, A. (1997). J. Appl. Cryst. 30, 1022–1025. Web of Science CrossRef CAS IUCr Journals Google Scholar
© International Union of Crystallography. Prior permission is not required to reproduce short quotations, tables and figures from this article, provided the original authors and source are cited. For more information, click here.