Difference Patterson

By far, the most common procedure for the determination of heavy-atom positions is the difference Patterson method it is often used in combination with the difference Fourier technique to locate sites in second and third derivatives. 

The Patterson function (Patterson, 1934) is a phaseless Fourier summation similar to that in Eq. 4 but employing as coefficients, thus it can be calculated directly from the experimentally measured amplitudes (Fp) without the need to determine the phase angle. In the case of macromolecules, (Fpn —Fp ) are used as coefficients in Eq. 4 to produce a Patterson map (hence the name difference Patterson). Such a map contains peaks of vectors between atoms (interatomic vectors). Thus in the case of a difference Patterson of macromolecules, it is a heavy-atom vector map. For example if a structure has an atom at position (0.25, 0.11, 0.32) and another atom at position (0.10, 0.35, 0.15), there will be a peak in the Paterson map at position (0.25-0.10, 0.11-0.35, 0.32-0.15), meaning a peak at (0.15, —0.24, 0.17). 

As can be seen from Eq. 4, a Fourier synthesis requires phase angles as input, thus it cannot be used to locate heavy-atom positions in a derivative if no phase information exists. However, it can be used to determine such positions in a derivative, if phases are already available from one or more other derivatives. As in the case of a difference Patterson, the Fourier s)mthesis here also employs difference coefficients. They are of the form ... 

Difference Fourier techniques are most useful in locating sites in a multisite derivative, when a Patterson map is too complicated to be interpretable. The phases for such a Fourier must be calculated from the heavy-atom model of other derivatives in which a difference Patterson map was successfully interpreted, and should not be obtained from the derivative being tested, in order not to bias the phases. Also, difference Fourier techniques can be used to test the correctness of an already identified heavy-atom site by removing that site from the phasing model and seeing whether it will appear in... 

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... 

An entire data set must be collected for each of these derivatives. The evaluation of the phases from these data is a complex mathematical process which usually involves the calculation first of a "difference Patterson projection."406 This is derived by Fourier transformation of the differences between the scattering intensities from the native and heavy atom-containing crystals. The Patterson map is used to locate the coordinates of the heavy metal atoms which are then refined and used to compute the phases for the native protein. 

To obtain the Patterson function solely for the heavy atoms in derivative crystals, we construct a difference Pattersonfunction, in which the amplitudes are (AF)2 = (IFHpl — IFpl)2. The difference between the structure-factor amplitudes with and without the heavy atom reflects the contribution of the heavy atom alone. The difference Patterson function is... 

In words, the difference Patterson function is a Fourier series of simple sine and cosine terms. (Remember that the exponential term is shorthand for these trigonometric functions.) Each term in the series is derived from one reflection hkl in both the native and derivative data sets, and the amplitude of each term is (IFHpI — IFpl)2, which is the amplitude contribution of the heavy atom to structure factor FHp. Each term has three frequencies h in the u-direction, k in the v-direction, and l in the w-direction. Phases of the structure factors are not included at this point, they are unknown. 

A GA method has been developed [92, 93] for ab initio phasing of low-resolution X-ray diffraction data from highly symmetric structures. The direct-space parameterization used incorporates information on structural symmetry, and has been applied to study the structures of viruses, with resolution as high as 3 A [93]. A GA has also been introduced [94] to speed up molecular replacement searches by allowing simultaneous searching of the rotational and translational parameters of a test model, while maximizing the correlation coefficient between the observed and calculated diffraction data. An alternative GA for sixdimensional molecular replacement searches has been described [95,96] and GA methods have also been used [97] to search for heavy atom sites in difference Patterson functions. 

Find the locations of the heavy aiom[s) from a difference Patterson map. 

Anomalous scattering can also be used directly if the protein is small and a suitable anomalous scatterer can be used. The three-dimensional structure of the small protein, crambin, was determined by W ayne A. Hendrickson and Martha Teeter by the use of anomalous dispersion measurements. This protein contains 45 amino acid residues and diffracts to 0.88 A resolution. It crystallizes with 72 water and four ethanol molecules per protein molecule. Since there is a sulfur atom in the protein molecule, the use of its anomalous scattering was made. The nearest absorption edge of sulfur lies at 5.02 A, but for Cu Ka radiation, wavelength 1.5418 A, values of A/ and A/" for sulfur are 0.3 and 0.557, respectively. Friedel-related pairs of reflections were measured to 1.5 A resolution, and sulfur atom positions were computed from difference Patterson maps. The structure is now fully refined and a portion of an a helix was shown in Figure 12.27 (Chapter 12). 

Soak complexes of heavy atoms of various kinds into a crystal. Screen diffraction data from heavy-atom derivatives of the protein for differences in intensities from those in the native data set. Calculate difference Patterson maps and locate the positions of the heavy atoms in the unit cell. 

As with the isomorphous replacement technique it is necessary to identify the positions, the x, y, z coordinates of the anomalous scatterers. This can be done by anomalous difference Patterson maps, which are Patterson syntheses that use the anomalous differences Fhki — F—h—k—i as coefficients (Blow and Rossmann, 1961). These maps are interpreted identically to isomorphous difference Patterson maps (see Chapter 9). Rapidly surpassing Patterson approaches, particularly for selenomethionine problems and others where the number of anomalous scatterers tends to be large, are direct methods (see below). These are strictly mathematical methods that have proved to be surprisingly effective in revealing the constellation of anomalous scatterers in a unit cell. 

The Patterson function has been employed since its formulation in 1935 for determining the locations of heavy atoms in crystals of conventional compounds. This alone made possible application of the heavy atom technique (see Chapter 8) for structure determination. For conventional molecules the information for the heavy atom positions is contained entirely within the native diffraction data, unlike macromolecules, where the information is embedded in differences between two independent data sets, or differences between Friedel mates. Aside from the coefficients employed, use of the function is virtually identical in all cases. Perhaps the major difference arises from the fact that diffraction data from macromolecular crystals, and therefore corresponding difference Patterson maps, contain more noise than... 

FIGURE 9.2 A section from a difference Patterson map calculated between a heavy atom derivative and native diffraction data (known as a difference Patterson map). This map is for a mercury derivative of a crystal of bacterial xylanase. The plane of Patterson density shown here corresponds to all values of u and w for which v =. Because the space group of this crystal is P2, this section of the Patterson map is a Harker section containing peaks denoting vectors between 2t symmetry related heavy atoms. 

FIGURE 9.11 The w = j plane of the difference Patterson map for the K2HgI4 heavy atom derivative of the hexagonal crystal form of the protein canavalin. The space group is P6, so w = is a Harker section. The derivative crystal contained two major K2HgI4 substitution sites and one minor substitution site per asymmetric unit. The Patterson peaks corresponding to those sites are marked with crosses. Note that the Patterson peak corresponding to the minor site cannot be discriminated from noise peaks in the Patterson map as is often the case. 

Another problem that frequently arises with multiple isomorphous derivatives is that of handedness. In space group P2i2i2i, Patterson maps for two independent derivatives may be interpreted to yield a set of symmetry related sites for one derivative and, independently, a second set for the other. Because handedness is completely absent in a Patterson map (because it contains a center of symmetry), there is an equal chance that the heavy atom constellation for the first will be right handed, and the constellation for the other will be left handed, and vice versa. This won t do. The two heavy atom sets will not cooperate when used to obtain phase information. There are ways of unraveling this problem too, and once again, it involves difference Patterson maps between the two derivative data sets and cross vectors. This case can also be resolved by calculating phases based on only one derivative and then computing a difference Fourier map (see Chapter 10) for the other. 

Fp(jH) thus AF(jh)P are the Patterson coefficients. Because the differences primarily are due to the heavy atoms, the resulting isomorphous difference Patterson map reveals the location of the heavy atoms. Programs, such as SOLVE (Terwilliger... 

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... 

A heavy atom derivative data set was analysed for the protein glucose isomerase by Farber et al (1988). The difference Fourier map showed that the heavy atom positions for the Laue and monochromatic data agreed. However, the isomorphous difference Patterson calculated from the Laue data was uninterpretable. 

Figure 7.13 (cont.) The Laue maps (i) and (ii) compare very favourably with (iii), the monochromatic map. From Helliwell et al (1989a) with the permission of the American Institute of Physics, (b) (i) Difference Patterson maps based on... 

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