research papers
Implementation of AMoRe
in^{a}CNRSGIF, LGV, 91198 GifsurYvette, France
^{*}Correspondence email: jorge.navaza@gv.cnrsgif.fr
An account is given of the AMoRe. The overall strategy of the method is presented and the main functions used in the package are described. The most important features of AMoRe are the quality of the fast rotation and translation functions and the facility of multiple inputs to translation and rigidbody functions, which allow for a fast multiple exploration of crystal configurations with a high level of automation.
method as implemented in the packageKeywords: AMoRe; molecular replacement.
1. Introduction
The idea of AMoRe, the comparison essentially involves the in terms of amplitudes. This criterion was chosen in the light of the results available one decade ago, results that now may be considered as corresponding to easy or moderately difficult MR problems. At that time, an exhaustive positional search involving in general six variables per independent model using that simple but robust criterion could not be envisaged. Nowadays, a full sixdimensional search would also be too lengthy, although feasible. This explains, perhaps, the fact that the original ideas of Rossmann and Blow, i.e. the splitting of the search into two consecutive threedimensional ones, are still found in filigree in most MR packages.
is to build a tentative using known molecular models similar to the actual molecules that constitute the crystal in order to start model building or The problem is to determine the positions of the models within the crystal cell. This is ultimately performed by comparing observed and calculated structure factors for selected positions of the independent molecules within the cell. InThe main programs in AMoRe aim at selecting a certain number of positions, obtained through the exhaustive exploration of threedimensional domains with fast functions, and computing the correlation coefficients associated with these positions. The idea is to assess many crystal configurations, as it is the contrast in the values of the criterion that gives one confidence in the solution. The fast functions, rotation functions and translation functions are either improved versions of already proposed ones or new ones. Accurate and fast algorithms are used throughout the package in order to save computing time. In particular, molecular scattering factors replace coordinates, which are used only once in the whole procedure.
The main stream in AMoRe is the set of values of the variables that specify the positions of the independent models within the crystal, from which structure factors and inputs to the fast functions are calculated. We will first define these variables and their relationship to the calculated structure factors. We will then describe the strategy for the selection of configurations.
2. Positional variables and crystal configurations
The position of the molecular model within the crystal is determined by the rotation R and the translation T that move the model from a reference initial position, specified by the atomic vectors {r^{o}}, to the current position, specified by the atomic vectors {r},
The translation T is usually given in fractional coordinates (x, y, z) in the crystal cell. The rotation R is parameterized with the Euler angles (φ, θ, ψ) associated with an orthonormal frame (X, Y, Z). Several conventions exist for the names of angles and definitions of the axes involved in this parameterization. We will follow the convention by which (φ, θ, ψ) denotes a rotation of ψ about the Z axis, followed by a rotation of θ about the Y axis and finally a rotation of φ about the Z axis,
The angles take values within the parallelepiped {0 ≤ φ < 360; 0 ≤ θ ≤ 180; 0 ≤ ψ < 360°}. For θ = 0 or 180°, only the combinations φ + ψ or φ − ψ are independent, respectively.
The initial position of the model is usually chosen with its center of mass placed at the origin and its principal axes of inertia parallel to the orthonormal frame, as this leads to an efficient sampling of configurations. A good choice for the orthonormal frame is Z parallel to the highest crystal symmetry axis (nort = 0 in AMoRe). This choice restricts the orientational search to {0 ≤ φ < 36/n}, where n is the order of the rotational symmetry around Z.
Therefore, given the models' initial positions, the crystal unitcell parameters, the spacegroup symmetry and the orientation of the orthonormal frame, a crystal configuration is uniquely determined by giving the positions of the independent molecular models within the
expressed in terms of the positional variables,The labels m′, …, m identify the molecules and the molecular models. Note that some of these models may coincide.
3. Structurefactor calculation
The calculated structure factors are conveniently written in terms of the individual molecular scattering factors f_{m}(s), i.e. the Fourier transform of the electron density corresponding to the isolated molecule in its initial position. These molecular scattering factors are computed with the TABLING program, which translates the model coordinates so that the center of mass is at the origin and rotates the coordinates so that the model's principal axes of inertia are parallel to the model box. An electron density is then constructed and eventually transformed by Fast Fourier techniques. One feature of AMoRe is that the model may well be an electron density or an electronmicroscopy reconstruction, as only the Fourier coefficients are used.
If R_{m} and T_{m} denote the rotation and translation that define the molecule's current position, M_{g} and t_{g} the spacegroup transformation matrix and translation vector of the gth and H the coordinates of a crystal reciprocal vector, the contribution of molecule m to the calculated factor is
D and O_{m} are orthogonalizing and deorthogonalizing matrices. In fact, DR_{m}O_{m} is simply the rotation matrix R_{m} expressed in a mixed basis: it applies (from left to right) to reciprocal coordinates (Miller indices) in the crystal and produces reciprocal coordinates in the model box. If there are M independent molecules we have to add M terms like this. Assuming that the individual molecular scattering factors f_{m}(s) have been set to a common scale, we have
4. Correlation coefficient
As stated in the introduction, the agreement criterion to assess crystal configurations is the (linear)
between observed and calculated amplitudes,where denotes a `centered' variable, e.g.
and means average over reflections. CC_{F} takes values in the interval (1, −1).
5. Strategy
The overall strategy of MR as implemented in AMoRe is easily understood if we consider the between intensities
as the target function for screening. The calculated total intensity is given by
where the overline means `complex conjugate'. The positional variables entering into this expression are successively determined by using different approximations to and, accordingly, CC_{I}. The protocol consists of three main steps.
The actual protocol in AMoRe differs from the one above mainly in the rotational search. The ROTING program, based on the fast rotation function proposed by Crowther, is used to determine the possible orientations of the models (Crowther, 1972). Also, as previously stated, the crystal configurations are assessed with CC_{F} instead of CC_{I}. The translations of the oriented models (onebody and nbody searches) are determined with the TRAING program. Several translation functions have been incorporated, among which the one described in the above protocol, i.e. CC_{I} as a function of T_{m}. The of the positional variables is performed with the fast rigidbody program FITING (Castellano et al., 1992). These fast functions will be described in the following section.
A situation where this protocol fails is often one in which a sixdimensional search fails too. As a rule, this corresponds to a poor quality of the search model or a small size of the search fragment with respect to the
content.The fast structurefactor calculation algorithm (4), the performance of ROTING and the facility of multiple inputs to TRAING and FITING allow for a fast multiple exploration. A link between the input/output of the above programs allows for automation. In fact, three levels of automation may be distinguished.

6. Description of the fast search programs
6.1. The ROTING program
It is possible to determine the rotations R that superimpose a search molecule upon the homologous ones within the target crystal by calculating the overlap within a conveniently chosen region Ω of volume v of the observed (the target function P_{t}) and a rotated version of the corresponding to the isolated search molecule (the search function P_{s}),
(Rossmann & Blow, 1962). should display a local maximum for the sought rotations. Note that when we rotate the search function P_{s} by R, its argument contains R^{−1}.
It may be useful to compare rotation functions obtained under different conditions. For this, some kind of normalization is needed. In fact, is cast into the form of a
by dividing (12) by the norms of the truncated Patterson functions,The reciprocalspace formulation of (12) is obtained by replacing the Patterson functions by their Fourier summations
Taking into account that I(−h) = I(h), we obtain
is the Fourier transform of the function that takes the value 1 within Ω and 0 outside. In principle, the domain of integration could have any shape. However, in order to take full advantage of the properties of the rotation group, Ω is usually chosen as a spherical domain of radius b. Letting s = h − kR^{−1} for short, we have
Although simple, the resulting expression for the rotation function has the disadvantage of containing entangled h, k and R contributions, which renders its computation time consuming if the whole domain of rotations has to be explored. The difficulty may be overcome by expanding the exponentials entering into (15) in spherical harmonics, Y_{l,m}. Taking advantage of their transformation under rotations and using recurrence relationships between spherical Bessel functions j_{l}, we obtain
where are the matrices of the irreducible representations of the rotation group. The awkwardness of (17) is apparent rather than real.

6.2. Computing the fast rotation function
The calculations are organized as follows.

is used in AMoRe just to select a certain number of peaks. The output of ROTING contains, besides the values of , those of the correlation coefficients (CC_{F} and CC_{I} as in P1) for each of the selected orientations. CC_{F} is more efficient, in general.
6.3. The locked rotation function
The rotational NCS, determined with the help of the selfrotation function, may be used to enhance the signaltonoise ratio of crossrotation functions (Rossmann et al., 1972; Tong & Rossmann, 1990). If S_{n}, n = 1, …, N denotes the set of NCS rotations, including the identity, and R is a correct orientation of the cross rotation, then S_{n}R must also correspond to a correct orientation. Here, we are assuming that the rotational NCS forms a group. Otherwise, either S_{n}R or , but not both, corresponds to another correct orientation. Therefore, a function may be defined, the locked cross rotation, whose values are the average of the values of at orientations related by the NCS,
By redefining the target function, it can be computed as an ordinary cross rotation. Indeed, _{L} may be written in a form similar to (12),
with the target
substituted by the average over the NCS of the rotated target functions. The computation of (25) is particularly simple in the case of the fast rotation function. The substitutionwhere we replaced the sum over by a sum over S_{n}, because of the rearrangement theorem of group theory, gives the required target coefficients.
6.4. The TRAING program
The possible translations of an oriented model are selected in AMoRe by means of fast translation functions computed with the TRAING program. The output of this program contains, besides the values of the fast translation function, those of CC_{F}, CC_{I} and the R factor for each of the selected translations. Several fast translation functions may be calculated. If we write the Fourier coefficient of the oriented model, rotated by a given R_{m} and placed at T, as
(see equation 3) and the corresponding intensity as
(see equation 10), then the options are (same notation as in equations 6 and 7)
6.5. The FITING program
Although FITING is not a search program, we include it here as it is one of the main molecularreplacement programs. It performs rigidbody by a fast technique first proposed by Huber & Schneider (1985). The quadratic misfit
is minimized with respect to the positional variables {R_{m}, T_{m}}, the overall scale factor λ and the overall temperature factor B.
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