Patterson synthesis

In 1934, Patterson suggested that instead of using the structure factors themselves, F h,k,l), we may use the square modulus of the structure factors, F(/j,k,/)p, which are the intensities of the diffraction. Equation (20.6) is now modified to the form 

In an x-ray diffraction experiment, if heavy atoms are introduced as part of the molecule, the scattering from these heavy atoms tends to dominate over die scattering from the light atoms. The summation in Patterson function may be taken only over the heavy-atom positions. The light atoms are considered to be arranged in the form of a random walk around the heavy atoms. Frequently, some light atoms 

The direct method is a mathematical process in which diffrent phase and amplitudes of the molecular transform are related to the contribution of individual atoms. This method is based on statistical procedures used to extract the phase information. The missing phase information is believed to be present in the statistical intensity distribution. One of the most well-known statistical proposals was offered by Karle and Hauptmann (1986). These authors assumed that the Fourier transform of the intensity represents the probability of finding interatomic distances in a molecule and that the transform is nonnegative. 

The intensity data collected from an experiment can be transformed into normalized structure factor magnitudes by 

The symbol s means sign of and the vector kr gives the restriction that Fk and iFh k must have large values. If the sign is plus, then the phase is a if the sign is minus, then the phase is n. These two values, ct and n, are the only values possible for centrosymmetric 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... 

In contrast to Fourier synthesis, which yields with electron diffraction data high electrostatic potential at the positions of the atoms, the maps obtained from Patterson synthesis show peaks at the tips of vectors. The length of each vector (drawn from the origin of the Patterson map) corresponds always to the distances between pairs of atoms and the direction each vector points... 

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... 

As I described earlier, this entails extracting the relatively simple diffraction signature of the heavy atom from the far more complicated diffraction pattern of the heavy-atom derivative, and then solving a simpler "structure," that of one heavy atom (or a few) in the unit cell of the protein. The most powerful tool in determining the heavy-atom coordinates is a Fourier series called the Pattersonfunction P(u,v,w), a variation on the Fourier series used to compute p(x,y,z) from structure factors. The coordinates (u,v,w) locate a point in a Patterson map, in the same way that coordinates (x,y,z) locate a point in an electron-density map. The Patterson function or Patterson synthesis is a Fourier series without phases. The amplitude of each term is the square of one structure factor, which is proportional to the measured reflection intensity. Thus we can construct this series from intensity measurements, even though we have no phase information. Here is the Patterson function in general form... 

Since the intensities of the standards were observed to diminish (finally to 85% of their original values) in a regular and nearly isotropic manner, the data were scaled linearly between each pair of standards. Associated with this decrease we also noted a decrease in the parameters b and y (which were in the end reduced by 0.02 A and 0.21 from their initial values). Broadening of the scans of file standards from 0.10 to 0.35 was also observed. The positions of the heavy atoms were determined from a three-dimensional Patterson synthesis. These positions were subjected to least-squares refinement as xenon atoms, after which it was possible to separate the antimony atoms by exploiting temperature factor differences. The positions were then further refined. A difference Fourier revealed positions for 12 of the 14 fluorine atoms. Least-.squares refinement of these positions was followed by another difference Fourier which revealed the positions of the final two fluorine atoms. Refinement of all these positions, with anisotropic temperature factors, resulted in a conventional/ factor of 0.06. Wei ting sch es were as previously described. ... 

Structure Reflnement. At the outset, it was assumed that the structure would be centric, so the two acentric data sets were averaged to give a single centric data set. A three-dimensional Patterson synthesis r ealed the positions of two of the three heavy atoms, but it also indicated that the non-centiic group (Pca2j, No. 

The structure was solved by heavy-atom methods at the U.C. Berkeley CHEXRaY facility using full-matrix least-squares refinement procedures detailed elsewhere. Systematically absent reflections were eliminated from the data set, and those remaining were corrected for absorption by means of the calculated absorption coefficient. A three-dimensional Patterson synthesis gave peaks that were consistent with Xe atoms in Wyckoff position 4c and Ge atoms in 4a in space group Pnmb (see Pnma, No. 62). Three cycles of... 

The Patterson synthesis indicated that the space group C2cm (No. 40) was unlikely since this requires fourfold sets of As and S atoms to be atZ = /4 or 0, V2 and appropriate vectors... 

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. 

The mathematical function that must be interpreted in order to deduce the heavy atom coordinates, a puzzle really, is called a Patterson function or Patterson synthesis (Patterson, 1935). It has a form similar to the equation for electron density except that all phases are effectively zero. It yields, also in a similar manner, a three-dimensional density distribution. The peaks in this map, however, do not correspond to electron density centers but mark the interatomic vectors relating those centers. 

The Patterson synthesis was not derived by a mathematician and then put into use by an experimenter. On the contrary, it was calculated by a perceptive experimenter, who was initially uncertain of its meaning, who examined its features, and deduced its properties. Only some time later, and after considerable discussion among interested investigators was its actual physical basis deduced and its meaning mathematically established (see Glusker et al., 1987). It seems reasonable therefore to follow that same path and simply state how it is calculated, what it means in terms of atom distributions, and then later show how it may be used in structure determination. 

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... 

All the studies described below use the wavelength normalisation method of Campbell et al (1986). The structure of an organo-metallic compound was solved with Laue data whereby a Patterson synthesis yielded iron and rhodium positions (Harding et al (1988) and figure 7.12). The unit cell parameters came from monochromatic photographs. It was then possible to locate the remaining atoms in the subsequent difference Fourier syntheses. The final R-factor was quoted as 0.14 (from... 

The positions of the anomalous scatterers can be established using the °FA(h) values entered as coefficients in a Patterson synthesis or direct methods program. Alternatively, the values of AF(h) - AF(h) can be... 

Figure 9.13 (a) A Harker section in the Patterson synthesis calculated using optimised zIano terms as coefficients for Fe cytochrome C4. The agreement between the observed peaks and the predicted positions is self-evident. All the expected Harker peaks occur in the map and are significantly above background noise. 

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. 

---