research papers
Phasing from an envelope
^{a}Cornell HighEnergy Synchrotron Source and Department of Molecular Biology and Genetics, Cornell University, Ithaca, NY 14853, USA
^{*}Correspondence email: qh22@cornell.edu
Solution of the γ and three translational parameters). The program FSEARCH incorporating this method has been generalized to handle molecules from all space groups. The program can also be used in general sixdimensional cases for a molecularreplacement solution given a predetermined envelope from any source, such as electronmicroscopic images or solution scattering, provided that the envelope can be converted to the standard CCP4 map format or expressed in terms of spherical harmonics. It is hoped that this method will greatly facilitate the ab initio of proteins and provide a good foundation for further structure refinement.
is central to crystallographic Conventional molecularreplacement methods are ineffective in the absence of knowledge of the structure of a homologous protein. A recent method utilizing the lowresolution molecular shape determined from solution Xray scattering data has been shown to be successful in locating the molecular shape within the crystallographic for the cases of the trimeric nitrite reductase (AxNiR, 105 kDa) and the dimeric superoxide dismutase (SOD, 32 kDa). This was achieved by performing a direct realspace search for orientation and translation using the orientation of the noncrystallographic axis obtained by performing a selfrotation on the crystallographic data. This effectively reduces the potential sixdimensional search to a fourdimensional one (Eulerian angleKeywords: FSEARCH; phasing from an envelope; molecular replacement.
1. Introduction
Solution Xray scattering data obtained using synchrotronradiation Xrays have proven to be very useful in providing lowresolution structural details of proteins and other macromolecules in solution. Owing to the significant progress that has been made with the development of ab initio phasing methods for lowresolution shape restoration in terms of spherical harmonics (Svergun & Stuhrmann, 1991; Svergun et al., 1996), the spatial parameters of a structure's molecular envelope can be determined in a modelindependent manner which does not, for example, require the use of coordinates for interpretation. This method has been used to analyse scattering data from a nitrogenase protein complex to provide a stable and unique shape restoration at ∼15 Å resolution (Grossmann et al., 1997). The lowresolution structure of nitrite reductase (105 kDa) from Alcaligenes xylosoxidans (AxNiR) had been determined from scattering data (Grossmann & Hasnain, 1997). Although the was previously solved by the molecularreplacement method at 2.8 Å, the molecular shape of nitrite reductase (AxNiR) determined from solution scattering was successfully used as a candidate for ab initio phasing by locating the molecule within the crystallographic (Hao et al., 1999). In order to test the generality of this method, the fairly small dimeric molecule superoxide dismutase (SOD) from bovine erythrocytes (32 kDa) was similarly treated (Ockwell et al., 2000).
2. Xray scattering experiments and molecularshape determination
In the case of a monodisperse system consisting of randomly orientated particles it is possible to obtain structural information of the particles from smallangle scattering data. Smallangle scattering is performed on a dilute suspension of biological macromolecules whose shape is to be determined. Then, from the onedimensional scattering profile, the threedimensional shape of the particles can be characterized in terms of a series of multipole coefficients. Knowledge of these multipole coefficients allows one to generate an approximate model of the shape of the macromolecules being examined (Stuhrmann, 1970; Svergun et al., 1996).
If we assume that the scattering is caused by a globular homogeneous molecule, one can define its molecular envelope by a twodimensional angular function F(θ, φ) describing the molecular boundary such that the ρ(r) is unity inside and vanishes elsewhere. The function F(θ, φ) can conveniently be expanded into a series of spherical harmonics Y_{lm}(θ, φ) according to (Stuhrmann, 1970),
with f_{lm} being complex multipole coefficients and L representing the multipole order. R_{0} is a scale factor [≃ (3V/4π)^{1/3}], where V is the volume of the particle. Furthermore,
where P are associated Legendre functions (with argument ) and l and m are integers with −l ≤ m ≤ l. Consequently, the ratio of quadrupolar term and zeroorder term 5^{1/2}f_{20}/f_{00} is a good indication of the deviation of the molecular shape from sphericity. A computational procedure to evaluate the multipole coefficients from the experimental scattering curve by minimizing a residual R was developed by Svergun & Stuhrmann (1991). Details of the algorithm are presented for example in Svergun et al. (1996).
The range of experimentally available scattering data generally allows the determination of 15–25 variables in the shape description. This imposes an upper limit for the multipole resolution L, since the number of independent parameters in the above series is equal to (L + 1)^{2} − 6 (arbitrary rotations and translations of the molecule do not alter the scattering curve and therefore lead to a reduction of six variables). Consequently, in general unique shape calculations with the multipole order of L = 4 are possible. In addition, molecular symmetry imposes restrictions on the multipole coefficients f_{lm} which can improve the reliability of the shape restoration by reducing the number of parameters to be calculated. The higher the symmetry, the more multipole coefficients can be omitted, which results in an enhanced resolution (i.e. multipole expansions with L = 6 or 7 are achievable). AxNiR contains three chemically identical subunits and is known to be a trimer in solution; assuming the trimer has threefold symmetry (which is shown by the selfrotation function) there are additional constraints on the multipole coefficients. The multipole expansion up to L = 7 for this symmetry group requires only 22 free parameters, of which 19 parameters are found to have values larger than 0.01 (i.e. all coefficients other than m = 3n, where n is an integer, should vanish provided that the Cartesian coordinate system for the trimeric molecule is chosen so that the threefold axis coincides with the z axis). The restored envelope of AxNiR is displayed in Fig. 1. The shape of the protein neatly reproduces the molecular features when compared with the overall details from the (Grossmann et al., 1997). In the case of the dimeric SOD, the multipole expansion up to order L = 6 requires 25 parameters (all coefficients other than m = 2n, where n is an integer, should vanish) to be determined from the scattering profile (Ockwell et al., 2000).
3. FSEARCH program: locating the molecular shape in the crystallographic unit cell
The conventional method for correctly positioning a known search molecule in a crystallographic ). However, attempts to locate the molecular shape determined from solution scattering by performing a Patterson search at different resolutions using AMoRe (Navaza, 1994) were not successful as the density inside the envelope is uniform, hence there is no discrimination between intraenvelope Patterson vectors.
– an important first step in solving macromolecular structures by the molecularreplacement method – is by the use of the crossrotation function (Rossmann & Blow, 1962In the case of a macromolecule lacking F_{obs} and F_{c} appears necessary. For a molecule in possession of the search may be performed in two separate stages. Utilizing a selfrotation function using ALMN in the CCP4 suite (Collaborative Computational Project, Number 4, 1994) with crystallographic data can yield two Eulerian angles, α and β, for the noncrystallographic (NCS) axis of the molecular shape. Once the orientation of this NCS axis is known, the potential sixdimensional search is reduced to four (Eulerian angle γ and three translational parameters), resulting in a significant reduction in calculation time when locating the molecular shape within the crystallographic It is worth noting that for any given orientation of a noncrystallographic axis, it is possible that the macromolecule can take either of two orientations with respect to the axis; hence, a search based on the original and the `flipped' molecule is necessary.
a full sixdimensional search (three rotational and three translational) to find the best match betweenThe program FSEARCH can be used to conduct a simultaneous (one to sixdimensional) search on orientation and translation to find the best match between F_{obs} and F_{c}. It has been generalized to handle molecules from all space groups and, in particular, those in possession of A flow chart of the program FSEARCH is shown in Fig. 2.
The FSEARCH program has been tested with the solution scattering and crystallographic data from the proteins AxNiR and SOD. In both cases, the correct orientation and translation of the molecular mask which had been determined from solution scattering profile was clearly identified by the program. Full details of the solutions have been published elsewhere (Hao et al., 1999; Ockwell et al., 2000) and a summary of the results is shown in Table 1 and Figs. 3 and 4.

4. Concluding remarks
It has been demonstrated that the molecular shape determined from solution scattering can be located in the crystallographic FSEARCH. The program has been generalized to handle molecules from all space groups and adapted to be compatible with the standard CCP4 libraries and file formats. FSEARCH can also be used in general sixdimensional cases for a molecularreplacement solution given a predetermined envelope from any source, such as electronmicroscopic images or solution scattering, provided that the envelope can be converted to the standard CCP4 map format or expressed in terms of spherical harmonics. The knowledge of the orientation of a axis (conveniently determined by a selfrotation function) can reduce the potential sixdimensional search to four (Eulerian angle γ and three translational parameters). The actual CPU time consumed by a fourdimensional search on an SGI Origin 200 server (one 175 MHz R10000 processor) was about 1 h. However, if no such axis exists in the molecule, a sixdimensional search would be necessary and a time frame in the order of 1000 CPU hours would be expected.
using the programIt is anticipated that the lowresolution phases calculated from the correctly positioned molecular shape can be used as a good starting point for phase extension through the use of genetic algorithms, whereby the mask would be used as the arena for ascertaining a macromolecule's internal structure. Some preliminary tests are promising and full results will be reported in due course. Once the resolution of the structure has been improved to ∼5 Å using this method, phase extension to higher resolutions may be achieved by maximum e.g. solvent flattening, histogram matching, NCS averaging). Thus, it is hoped that this method will greatly facilitate the ab initio of proteins and provide a good foundation for further structure refinement.
and densitymodification methods (The FSEARCH program can be obtained by contacting the author of this paper.
Acknowledgements
I am grateful to Drs J. Grossmann, F. Dodd, M. Hough and Professor S. Hasnain for providing the test data and useful discussions. I would like to thank D. Ockwell for improving and testing the FSEARCH program.
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