research papers
AMoRe: classical and modern
^{a}IBS, Institut de Biologie Structurale JeanPierre Ebel, 41 Rue Jules Horowitz, F38027 Grenoble; CNRS, Université Joseph Fourier; CEA, France
^{*}Correspondence email: stefano.trapani@cbs.cnrs.fr
An account is given of the latest developments of the AMoRe package: new rotational search algorithms, exploitation of generation and use of ensemble models and interactive graphical molecular replacement.
Keywords: AMoRe; molecular replacement.
1. Introduction
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 molecularreplacement (MR) strategy (Navaza, 1994):

2. Fast rotational sampling procedures
The rotation function , as defined by Rossmann & Blow (1962), measures the overlap between one (the target object) and a rotated version of another (the search object) as a function of the applied rotation R. The detection of rotationfunction peaks aims at determining the possible orientations of a MR probe (crossrotation function) or the NCS rotational components (selfrotation 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 twodimensional sections of the rotation domain, has been extended in AMoRe to three angular variables (Trapani & Navaza, 2006). Also, it is shown how distortionfree 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 rotationgroup irreduciblerepresentation matrices (Wigner, 1959),
This expression is easily reduced to a Fourier expansion in twodimensional sections of the rotation domain if Euler angles (α, β, γ) are used to parametrize rotations. In fact, from the simple harmonic dependence of on the α and γ angles,
it follows that
where the Fourier coefficients of a βsection are given by
(4) was first obtained by Crowther (1972) through spherical harmonic expansion of the source and target Patterson functions restricted to a spherical domain. Efficient algorithms for the evaluation of the Wigner expansion coefficients were subsequently elaborated by Navaza (1993).
2.1. Threedimensional FFT sampling of the rotation function
(4) requires the evaluation of the reduced Wigner functions up to the degree ℓ_{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 ,
which permits a full threedimensional Fourier representation of the rotation function (Trapani & Navaza, 2006),
where the Fourier coefficients W_{m,u,m′} are given by
Threedimensional FFT calculation of the rotation function (6) has been implemented in AMoRe, aiming at permitting accurate and more rapid computations at high values of ℓ_{max} for large particles such as viruses. Test calculations on the icosahedral IBDV VP2 subviral particle (ℓ_{max} = 178) showed that the new code performs on average 1.5 times faster than the previous twodimensional FFTbased program with no loss of accuracy. When using precalculated coefficients, as implemented in AMoRe, speed improvements of up to sixfold were observed.
According to Navaza (1993), calculation of the coefficients requires the evaluation of spherical harmonic functions on a nonregular grid corresponding to the reciprocallattice 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 ℓ = 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 Pattersonoverlap rotation function. The summation limit ℓ_{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), ℓ_{max} determines the maximum oscillation frequency in α, β and γ. Also, according to standard FFT requirements, ℓ_{max} determines the minimum number (2ℓ_{max} + 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 twodimensional βsections, 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 follows that in an undistorted Cartesian representation of a βsection the α and γ coordinates must define an oblique twodimensional whose sides of length 2π form an angle equal to β (Fig. 1a). By means of an appropriate coordinate transformation, a rectangular centred can be obtained with side lengths (Fig. 1b):
A uniform sampling on these orthogonal axes will then correspond to a true constant distance Δ between rotations. Notice that the size (area) of a βsection is proportional to sinβ and that the section reduces to a onedimensional segment if β = 0 or π.
It seems physically reasonable to assume that if a sample spacing Δ (as defined above) permits the recovery of one βsection from its samples, then the same sample spacing should also be applicable to any other βsection. Δ would then represent the angular resolution of the rotation function. Under this hypothesis, however, the number of Fourier coefficients of a βsection should vary according to sinβ, while, after (3) and (4), the number of S_{m,m′} coefficients is dictated uniquely by the value of ℓ_{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
with
(10) defines circular regions of radius l_{max} in the twodimensional reciprocal sections (Fig. 2). The longest reciprocalvector lengths to include in calculations are therefore limited by the Wignerexpansion truncation limit ℓ_{max}. Accordingly, the resolution in is given by Δ = π/ℓ_{max}.
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 distortionfree sections, which facilitates peaksearching procedures. In Fig. 1 we show two plots of the same section (β = 137.8°) of the IBDV VP2 selfrotation 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.
3. Exploiting NCS
When several copies of the same molecule are present in the ; Blow, 2001 and references therein). The rotational component of the NCS operations can be detected by analysis of the selfrotation 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.
each molecule can in principle be superimposed on another of the same type by a rigidbody 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 it defines a operation (see Rossmann, 1990The knowledge of the NCS operations can be exploited to help the MR search when the standard procedures fail.
4. Ensemble models: taking structural variability into account
The number of protein structures available from the PDB has grown to a point where many of the known protein folds are currently overrepresented. 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/lowsimilarity 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 sidechain 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 NMRbased 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 (unpublished data) and the Escherichia coli gene product YECD (PDB code 1j2r ; Abergel et al., 2003). We observed that the ensembles enhanced the effectiveness of the singlestructure 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 electrondensity correlation (EDC). By expressing the EDC in terms of the molecular Fourier transforms, the superposition problem can be straightforwardly reduced to an MRlike problem. Exploiting the existing AMoRe procedures, an automatic EDCbased modelsuperposition 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 distancebased

According to our experience, ensemblebased MR searches have the potential to combine and exploit the richness of structural information in the PDB in a relatively easy though effective way. EDCbased ensembles of structures should be considered by developers of databases for automatic structuresolution pipelines as a valuable alternative to homologybased representative models.
5. Graphical interactive molecular replacement
A moleculargraphics 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 crystalpacking analysis, is for the specific cases in which one wants to position smallsize components of molecular complexes, especially when there is some prior knowledge of the regions of interaction between the components.
Acknowledgements
We acknowledge Alberto Podjarny for kindly providing the Fab Q11 B13
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