Copyright © International Union of Crystallography 2008
AMoRe: classical and modern Correspondence e-mail: stefano.trapani/at/cbs.cnrs.fr ConferenceMolecular replacement Received February 16, 2007; Accepted September 11, 2007. This is an open-access article distributed under the terms described at http://journals.iucr.org/services/termsofuse.html. | |||||||||||||||||||||||
Abstract An account is given of the latest developments of the AMoRe package: new rotational search algorithms, exploitation of noncrystallographic symmetry, generation and use of ensemble models and interactive graphical molecular replacement. Keywords: AMoRe, molecular replacement | |||||||||||||||||||||||
In this paper, we give an account of the latest developments of the AMoRe package. The newly introduced features follow the general guidelines that determined the success of the AMoRe molecular-replacement (MR) strategy (Navaza, 1994 ):
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The rotation function , as defined by Rossmann & Blow (1962 ), measures the overlap between one Patterson function (the target object) and a rotated version of another (the search object) as a function of the applied rotation R. The detection of rotation-function peaks aims at determining the possible orientations of a MR probe (cross-rotation function) or the NCS rotational components (self-rotation function). Methods for an optimal and rapid sampling of the rotation function have long been an object of study (for a review, see Navaza, 2001 ). Here, we describe how FFT acceleration, first introduced by Crowther (1972 ) to sample two-dimensional sections of the rotation domain, has been extended in AMoRe to three angular variables (Trapani & Navaza, 2006 ). Also, it is shown how distortion-free sections (Burdina, 1971 ; Lattman, 1972 ) can be economically sampled by FFT (Trapani et al., 2007 ). A natural way of representing the rotation function is to expand it in terms of the complete set of Wigner functions, i.e. the elements of the rotation-group irreducible-representation matrices (Wigner, 1959 ), 2.1. Three-dimensional FFT sampling of the rotation function (4) requires the evaluation of the reduced Wigner functions up to the degree max for each sampled β value. This can be carried out by means of several recursion formulas. Alternatively, one can use the Fourier representation of , According to Navaza (1993 ), calculation of the coefficients requires the evaluation of spherical harmonic functions on a nonregular grid corresponding to the reciprocal-lattice directions. Since spherical harmonics correspond to special restrictions of the Wigner functions, (5) can be further exploited to obtain by FFT, rather than by recursion, a set of spherical harmonic values that are very finely sampled. In addition to rapidity, this approach avoids numerical stability issues found in most recursive algorithms. Accurate results up to at least = 1000 and β ≥ 10−4 rad can be obtained. 2.2. Metric based FFT sampling of the rotation function The Wigner expansion in (1) and the Fourier expansion in (6) are of quite general validity for functions defined on the domain of rotations, although they were obtained for the specific case of the Patterson-overlap rotation function. The summation limit max, which should be infinity in theory, is set to a convenient finite number in all practical cases and is generally associated with the angular resolution of the rotation function. Indeed, according to (6), max determines the maximum oscillation frequency in α, β and γ. Also, according to standard FFT requirements, max determines the minimum number (2max + 1) of equispaced samples along α, β and γ.The choice of a suitable set of samples for the rotation function is a nontrivial issue if the actual distance between sampling points is to be taken into account. Since the metric of the rotation group, independent of its parametrization, cannot be reduced to a Euclidean metric, FFT sampling based on (6) will result in an unevenly distributed set of points in the rotation domain. The problem can be partially solved in two-dimensional β-sections, where the rotation metric becomes equivalent to a Euclidean metric (Burdina, 1971 ; Lattman, 1972 ). The rotation length element ds may be expressed using Euler angles as It seems physically reasonable to assume that if a sample spacing Δ (as defined above) permits the recovery of one β-section from its samples, then the same sample spacing should also be applicable to any other β-section. Δ would then represent the angular resolution of the rotation function. Under this hypothesis, however, the number of Fourier coefficients of a β-section should vary according to sinβ, while, after (3) and (4), the number of S m,m′ coefficients is dictated uniquely by the value of max independently of β. We have shown numerically (Trapani et al., 2007 ) that this apparent contradiction is resolved by an intrinsic feature of the reduced Wigner functions which renders the S m,m′ coefficients vanishingly small when their indices (m, m′) do not satisfy the condition If in (3) we limit the summation to those indices that satisfy inequality (10), then one can sample the rotation function on economic grids with sinβ fewer points than in the classical Crowther’s development, while still computing it by FFT techniques, and recover distortion-free sections, which facilitates peak-searching procedures. In Fig. 1 we show two plots of the same section (β = 137.8°) of the IBDV VP2 self-rotation function computed by FFT using the classical sampling (96 100 points) and the metric based sampling (64 736 points). As expected, both plots display the same features. | |||||||||||||||||||||||
When several copies of the same molecule are present in the asymmetric unit, each molecule can in principle be superimposed on another of the same type by a rigid-body movement, although the structural correspondence between the two molecules may not be exact owing to the different crystalline environments. This movement is not an element of the crystal symmetry space group; it defines a noncrystallographic symmetry operation (see Rossmann, 1990 ; Blow, 2001 and references therein). The rotational component of the NCS operations can be detected by analysis of the self-rotation function, while no straightforward method exists for determination of the NCS translational components. An exception occurs when there is pure translational NCS, which should result in very strong peaks in the Patterson map. The knowledge of the NCS operations can be exploited to help the MR search when the standard procedures fail.
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The number of protein structures available from the PDB has grown to a point where many of the known protein folds are currently over-represented. As a consequence, it is often possible to find families of homologues that are potentially exploitable as probes for a given MR problem. In these cases, the logical choice of a model must clearly favour a very closely related molecule, if one is available. On the other hand, it may be necessary to test several trial models when only medium/low-similarity homologues are available. Accumulated experience shows that even if structural homology is certain, models are likely to fail if similarity is not high enough. When the MR search using each member of a whole family of structures fails, it may be worthwhile to combine information from all available models in order to take into account structural variability within the family and thus improve the effectiveness of the model. Hybrid models can be built up on the basis of structure and sequence alignments. Their use has indeed led to positive results in some difficult MR cases. The outcome, however, depends highly on the quality of the alignment employed for model construction (Schwarzenbacher et al., 2004 ). It should also be noticed that structural differences among homologues may arise not only from sequence diversity, but also from molecular flexibility, which can range from small side-chain torsional changes to large domain movements. As an alternative to hybrid models, one can treat a whole ensemble of superposed homologous structures as an MR probe. In this way, regions of structural variability/flexibility are implicitly weighted within the model itself. This type of model closely resembles NMR-based models, whose usability as MR probes has previously been examined (Chen, 2001 ). In a recent study (the results of which are briefly summarized in Table 1 ), we used single structures as well as ensembles of homologues to solve two difficult MR cases: the antibody Fab Q11 B13 crystal structure (unpublished data) and the Escherichia coli gene product YECD crystal structure (PDB code 1j2r; Abergel et al., 2003 ). We observed that the ensembles enhanced the effectiveness of the single-structure probes. More interestingly, whole sets of individually unfruitful structures could be correctly placed when used as ensembles. Notice that for the Fab structure rather large ensembles were used (Fig. 3 ). Also, many of the ensemble members corresponded to the same molecules in different crystalline environments. In order to superpose molecular structures and thus generate ensembles, it is common practice to use algorithms which optimize a certain set of interatomic distances. In the work described above, we used a different approach based on the maximization of the electron-density correlation (EDC). By expressing the EDC in terms of the molecular Fourier transforms, the superposition problem can be straightforwardly reduced to an MR-like problem. Exploiting the existing AMoRe procedures, an automatic EDC-based model-superposition utility, SUPER, has been developed and is now available as part of the software package. A somewhat related technique has been implemented in MOLREP (Vagin & Isupov, 2001 ). Although both approaches aim to maximize the overlap between two electron densities considered as rigid bodies, in MOLREP the putative translations are first determined by means of the spherically averaged phased translation function and the orientations are then looked for by means of a phased rotation function, whereas in SUPER we first use the standard fast rotation function to determine the putative orientations and then compute the phased translation function. The EDC maximization presents some advantages with respect to distance-based superposition methods.
According to our experience, ensemble-based MR searches have the potential to combine and exploit the richness of structural information in the PDB in a relatively easy though effective way. EDC-based ensembles of structures should be considered by developers of databases for automatic structure-solution pipelines as a valuable alternative to homology-based representative models. | |||||||||||||||||||||||
A molecular-graphics interface has been developed to assist the AMoRe user in the interpretation and interactive manipulation of the MR search results. The program permits the following.
A foreseen use of the program, in addition to facilitating crystal-packing analysis, is for the specific cases in which one wants to position small-size components of molecular complexes, especially when there is some prior knowledge of the regions of interaction between the components. | |||||||||||||||||||||||
Acknowledgments We acknowledge Alberto Podjarny for kindly providing the Fab Q11 B13 crystal structure factors. | |||||||||||||||||||||||
References
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