Patterson synthesis crystals

Multiple isomorphous replacement allows the ab initio determination of the phases for a new protein structure. Diffraction data are collected for crystals soaked with different heavy atoms. The scattering from these atoms dominates the diffraction pattern, and a direct calculation of the relative position of the heavy atoms is possible by a direct method known as the Patterson synthesis. If a number of heavy atom derivatives are available, and... 

Many ingenious applications of vector maps have been suggested and used. For instance, pairs of isomorphous crystals are often used for difficult structures, and if the replaceable atoms are not at symmetry centres, it is necessary to find their parameters. If tlie replaceable atoms are heavy enough, they may be located readily as in picryl iodide if not, the vector maps of the two isomorphous crystals may be compared the differences indicate which peaks are due to the replaceable atoms. Alternatively, a Patterson synthesis may be computed in which the differences between structure amplitudes of corresponding... 

Special problems arise when two different heavy atom derivatives, with the heavy atoms in different sites, are used for the purpose of determining phase angles in non-centrosymmetric crystals (see p. 387) it is essential to know the relative positions of the two heavy atoms. Perutz (1956) found that a sort of combination difference Patterson synthesis —a Fourier synthesis in which the coefficients are... 

Vectors in non-centrosymmetric crystals. The ordinary Patterson synthesis of the X-ray data of a non-centrosymmetric crystal gives a centrosymmetric vector distribution and even if the X-ray data obtained under anomalous scattering conditions are used (it will be remembered that the diffraction pattern is non-centrosymmetric under these conditions), the vector distribution obtained is still centro-symmetric because the cosine function has this symmetry. It has been shown by Okaya, Saito, and Pepinsky (1955) that a synthesis of the Patterson type, but using sines instead of cosines,... 

This important development does for non-centrosymmetric crystals what the Patterson synthesis does for centrosymmetric crystals it has... 

The Patterson synthesis (Patterson, 1935), or Patterson map as it is more commonly known, will be discussed in detail in the next chapter. It is important in conjunction with all of the methods above, except perhaps direct methods, but in theory it also offers a means of deducing a molecular structure directly from the intensity data alone. In practice, however, Patterson techniques can be used to solve an entire structure only if the structure contains very few atoms, three or four at most, though sometimes more, up to a dozen or so if the atoms are arranged in a unique motif such as a planar ring structure. Direct deconvolution of the Patterson map to solve even a very small macromolecule is impossible, and it provides no useful approach. Substructures within macromolecular crystals, such as heavy atom constellations (in isomorphous replacement) or constellations of anomalous scattered, however, are amenable to direct Patterson interpretation. These substructures may then be used to solve the phase problem by one of the other techniques described below. 

If the real unit cell contains only a very few atoms, as is often the case with an ionic crystal or salt, the Patterson map, calculated by including its diffraction intensities as coefficients in the Patterson synthesis, may be treated as a puzzle. The object is to contrive a distribution of atoms whose interatomic vectors yield the highest peaks in the Patterson map. This direct approach in fact provided the means by which many of the first small molecules and simple ionic crystals were solved. It is not practical for larger, more complicated structures. 

Heavy atom derivatives of a macromolecular crystal can be prepared (Green, Ingram and Perutz 1954) which for a minimum of two derivatives (and the native crystal) and in the absence of errors, leads to a unique determination of the phase ahkt in equation (2.7) (figure 2.13(a)). This requires the site and occupancy of the heavy atom to be known for the calculation of the vector FH (the heavy atom structure factor). In the absence of any starting phase information the heavy atom is located using an isomorphous difference Patterson synthesis P(u,v,w) where the isomorphous difference is given by... 

D20.4 The phase problem arises with the analysis of data in X-ray diffraction when seeking to perform a Fourier synthesis of the electron density. In order to carry out the sum it is necessary to know the signs of the structure factors however, because diffraction intensities are proportional to the square of the structure factors, the intensities do not provide information on the sign. For non-centrosymmetric crystals, the structure factors may be complex, and the phase a in the expression F/m = F w e is indeterminate. The phase problem may be evaded by the use of a Patterson synthesis or tackled directly by using the so-called direct methods of phase allocation. 

Simple Inorganic Compounds. The class of simple inorganic compounds includes substances such as metals and salts, that contain very few atoms in the asymmetric unit. In such cases, the complete crystal structure can often be derived as soon as the crystal symmetry (space group) has been determined. If several possibilities exist, the correct structure can be arrived at by trial and error. When this approach also fails, a Patterson synthesis (see Eq. 27) can be used to solve the structure. 

The complete interpretation of a Patterson synthesis is generally made much more difficult because the maxima are not resolved, and cannot be explicitly identified. Figure 36 illustrates this situation for the two-dimensional case. Corresponding to the 6 maxima in the crystal space, there are 19 maxima in the vector space these must be placed in the same unit cell volume that accommodates the 6 atoms. What is more, the maxima in the vector space require more volume than atoms in the crystal space, because the indeterminacies of each pair of atoms add at the Patterson maximum. The half-width values of the maxima in the vector space are thus roughly twice those in the crystal space. As a consequence, the peaks overlap strongly. The relative weights of the maxima are indicated in Figure 36. 

Even without having the stmcture factor phases, e.g. from electron microscopy images, it is possible to get some insight into the atomic architecture of a crystal. A simple but powerful method to get this information was introduced hy A.L. Patterson about 70 years ago. Following Patterson the Fourier synthesis is carried out using the squared stmcture factor amplitudes Fha which are equal to the measured intensities for the reflections with index hkl. Moreover, all phase values must be set to zero, which leads to the following (auto-correlation) function ... 

Lipson. H., and Beevers, C. A. An improved numerical method of two-dimensional Fourier synthesis for crystals. Proc. Phys. Soc. 48, 772-780 (1936). Patterson, A. L., and Tunell, G. A method for the summation of the Fourier series used in the X-ray analysis of crystal structures. Amer. Mineralogist 27. 655-679 (1942). 

All methods of deduction of the relative phases for Bragg reflections from a protein crystal depend, at least to some extent, on a Patterson map, commonly designated P(uvw) (46, 47). This map can be used to determine the location of heavy atoms and to compare orientations of structural domains in proteins if there are more than one per asymmetric unit. The Patterson map indicates all the possible relationships (vectors) between atoms in a crystal structure. It is a Fourier synthesis that uses the indices, l, and the square of the structure factor amplitude f(hkl) of each diffracted beam. This map exists in vector space and is described with respect to axes u, v, and w, rather than x,y,z as for electron-density maps. 

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