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THE PATTERSON DECONVOLUTION METHOD

 

Reference:        Burla, M.C., Caliandro, R., Carrozzini,B., Cascarano, G.L., De Caro, L., Giacovazzo, C. & Polidori, G. (2004). J. Appl. Cryst. 37, 258-264.

Contact:           Rocco.Caliandro@ic.cnr.it

 

THE DECONVOLUTION METHOD

The structure factor moduli contains information on the interatomic vectors of the crystal structure under study. This information can be visualized by the Patterson map, obtained as Fourier transform of the squared moduli. The Patterson map cannot be directly interpreted, because of the strong overlap of its peaks for protein structures. Nevertheless it can be used as a starting point for a procedure aiming at determining the atomic positions of the structure, which is called Patterson deconvolution method.

ThePatterson map of an ideal structure of N atoms can be interpreted as superposition of N images of the structure, each shifted with respect to the other by an interatomic vector [1]. If the structure is not centrosymmetric, N further inverse images have to be considered. An example is given in Fig.1 for the ideal structure of 3 atoms shown on the left. The Patterson peaks are represented (on the right) by points and the 3 corresponding images are highlighted by dashed lines (the actual images are 6 if the inverse images are also considered). The extraction of a single image of the structure from the set of overlapped images contained in the Patterson map can be achieved by means of the deconvolution method as described in the following.

Let us suppose to shift the Patterson map reported on the right of Fig.1 by one of the interatomic vectors of the structure on the left (the vector 1-2, for example) and to superimpose it to the original one. The set of the peaks in overlap will single out a unique image of the structure (together with its inverse). The remaining ambiguity can be definitively eliminated by a second superposition of the Patterson map shifted by another interatomic vector (for example 1-3).

If P(r) is the Patterson map and P(r-r12) is that shifted by the interatomi vector r12, the superposition can be mathematically done by means of the so-called image-seeking functions, the simplest of which is the minimum function: S(r)=min[P(r), P(r-r12)]. By taking the minimum between the heights of the map P(r) and that P(r-r12) for each pixel point, a new map S(r) is obtained, called minimum superposition function, which contains only the peaks in common between the two maps. The interatomic vector r12 is the pivot of the decomposition. In real cases, with N>3 and in presence of false Patterson peaks, is preferable to operate a single decomposition, taking as pivot one of the highest peaks of the Patterson map. The map S(r), once cleaned by filtering operations which allow to eliminate the residual ambiguity, will contain exclusively peaks in correspondence of the atomic position of the structure [2].

 

DECONVOLUTION IN PRESENCE OF SYMMETRY

If the space group is not P1, the position of the deconvolved image in the unit cell is not arbitrary, hence the above described method is not applicable as it is. In this case the Harker section can be used, namely the sections of the Patterson map containing the interatomic vectors among an atom position and its equivalent[3]. As an example, in the space group P21 the equivalent positions are (x, y, z) and (-x, y+0.5, -z), so that the interatomic vector will have coordinates (2x, 0.5, 2z) and will fall in the Harker section (u, 0.5, w). Let us suppose that rp=(up, 0.5, wp) is one of the highest peaks in the Harker section, and it actually correspond to the interatomic vector between the position of a given atom and its equivalent, then the position of the considered atomi will be (up/2, y qualunque, wp/2). Therefore, from the Harker section is possible to obtain information about the atomic positions, following a procedure analogous to that called heavy atom method. By using the same arguments, from the Harker section is possible to obtain a map, called symmetry minimum function and denoted SMF(r), which contains some peaks in correspondence to the atomic positions. In fact, if we define the implication function  (x,y,z)à(2x,0.5,2z), which is defined by the symmetry operation which geners the Harker section, than the SMF map is obtained by the relation SMF(x,y,z)=P(2x,0.5,2z). Note that SMF(r) is constant along y, because of the floating origin in P21. If rp is the position of one of the SMF(r) peaks, then the map S(r)=min[SMF(r), P(r-rp)] will contain the peaks corresponding to the image of the structure correctly positioned, together with those corresponding to other images still superimposed. The spurious peaks can be eliminated by a special filtering of the map.

 

GENERAL FORMULATION

In general terms, an implication transformation I(r) is a function for the atom position r defined over the appropriate Harker section for the sth symmetry operator Cs, with value:

where (r - Csr) is the Harker vector, and ns is the multiplicity, equal to the number of symmetry operators which generate the given Harker vector. A single implication transformation can only provide information about two of the three atomic parameters x, y and z. Spurious peaks lying on the Harker section and the lack of resolution of the Patterson peaks also make a single Is of limited use. When the space-group symmetry has more than two primitive operators, all the implication transformations may be used together:

where the minimum operator Min indicates that the lowest value among the  functions Is has been chosen.

Even if more accurate than equation (1), the map defined by equation (2) will not obey the crystal symmetry; it will provide maxima at the atomic positions r relative to any acceptable origin for that space group, and for their enantiomorphs. It is possible to clean the SMF map by calculating the minimum superposition function S(r) as

where r1, r2, ..., rn are trial atoms selected from the peaks in the SMF(r) map, and F may be either the sum or the product, or the minimum function [4].

 

THE PROCEDURE

The procedure of crystal structure resolution through the Patterson deconvolution method can be summarized as follows:

 

1)    the observed intensities are rescaled and normalized structure factor moduli are calculated;

2)    the Patterson map is calculated by using the normalized moduli: it results to be sharper than that obtained by using the original moduli;

3)     the SMF(r) map is built by taking into account all the independent Harker sections of the space group (if it is P1, SMF(r)=P(r)) and a list of its peaks, ordered by their height, is done and stored;

4)    the position rp of the highest peak is used to pilot the decomposition and the minimum superposition function is calculated accordino to the equation S(r)=min[SMF(r), P(r-rp)];

5)    the S(r) map undergoes an iterative procedure of electron density modification. The starting map is modified by means of specific filters, aiming at enhance the signal over background (low intensities are zeroed, high intensities are powered, etc.) and at eliminating the remnant of the Patterson symmetry. The modified map is then Fourier-inverted and the new phase values, together with the observed moduli, are used to build a new map by Fourier transform, which will be the starting point for a next cycle;

6)    due to the presence in the Patterson map of false peaks or peaks misplaced with respect to their correct positions, the deconvolution procedure might not be able to successfully deconvolve the Patterson by using a single pivot peak rp. Therefore a multisolution approach is used, by taking into account the highest peaks of the SMF(r) map. Their actual number depends on the size of the structure to be solved, on its heavy atom content and on data resolution;

7)    the solution obtained (one for each pivot peak) are assessed by means of a proper figure of merit and the more promising ones are further processed by the direct space refinement procedure.

 

 

Figura 4. Left: ideal structure of three atoms. Right: corresponding Patterson peaks. The dashed lines identify three images of the structure on the left. Three other inverse images can be also recognised.

 

 

References

1.       M.J. Buerger, Vector space ad its application in crystal structure investigation, 1959, Wiley, New York.

2.       M.C. Burla, R. Caliandro, B. Carrozzini, G.L. Cascarano, L. De Caro, C. Giacovazzo, G. Polidori, J. Appl. Cryst., 2004, 37, 258.

3.       D. Harker,, J. Chem. Phys., 1936, 4, 381.

4.       Pavelčík, F., kuchta, L. & Sivy, J. (1992). Acta Cryst. A48, 791-796.

 

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