Vector Patterson

FIGURE 8.22. A set of translation vectors between two benzene molecules, (a) The two molecules with atomic numbering used, and (b) their cross vectors with the atoms involved in each vector so marked. Compare these with the self vectors in Figure 8.19. The origin of the vector (Patterson) map is labelled as such. The highest peak in the... 

The integrand in this expression will have a large value at a point r if p(r) and p(r+s) are both large, and P s) will be large if this condition is satisfied systematically over all space. It is therefore a self- or autocorrelation fiinction of p(r). If p(r) is periodic, as m a crystal, F(s) will also be periodic, with a large peak when s is a vector of the lattice and also will have a peak when s is a vector between any two atomic positions. The fiinction F(s) is known as the Patterson function, after A L Patterson [14], who introduced its application to the problem of crystal structure detemiination. 

How do we find phase differences between diffracted spots from intensity changes following heavy-metal substitution We first use the intensity differences to deduce the positions of the heavy atoms in the crystal unit cell. Fourier summations of these intensity differences give maps of the vectors between the heavy atoms, the so-called Patterson maps (Figure 18.9). From these vector maps it is relatively easy to deduce the atomic arrangement of the heavy atoms, so long as there are not too many of them. From the positions of the heavy metals in the unit cell, one can calculate the amplitudes and phases of their contribution to the diffracted beams of the protein crystals containing heavy metals. 

Derivation of the Structure.—The observed intensities reported by Ludi et al. for the silver salt have been converted to / -values by dividing by the multiplicity of the form or pair of forms and the Lorentz and polarization factors (Table 1). With these / -values we have calculated the section z = 0 of the Patterson function. Maxima are found at the positions y2 0, 0 1/2, and 1/21/2. These maxima represent the silver-silver vectors, and require that silver atoms lie at or near the positions l/2 0 2,0 y2 z, V2 V2 z. The section z = l/2 of the Patterson function also shows pronounced maxima at l/2 0,0 y2, and y2 x/2, with no maximum in the neighborhood of y6 ys. These maxima are to be attributed to the silver-cobalt vectors, and they require that the cobalt atom lie at the position 0 0 0, if z for the silver atoms is assigned the value /. Thus the Patterson section for z = /2 eliminates the structure proposed by Ludi et al. 

The structure factor for the 104-atom complex with almost perfect icosahedral symmetry determines the intensities of the diffraction maxima, in correspondence with the inverse relationship between intensity in reciprocal space and the atom-pair vectors in real space that was discovered fifty years ago by Patterson.27 The icosahedral nature of the clusters in the cubic crystal explains the appearance of the Fibonacci numbers and the golden ratio. 

The model protein is used to search the crystal space until an approximate location is found. This is, in a simplistic way, analogous to the child s game of blocks of differing shapes and matching holes. Classical molecular replacement does this in two steps. The first step is a rotation search. Simplistically, the orientation of a molecule can be described by the vectors between the points in the molecule this is known as a Patterson function or map. The vector lengths and directions will be unique to a given orientation, and will be independent of physical location. The rotation search tries to match the vectors of the search model to the vectors of the unknown protein. Once the proper orientation is determined, the second step, the translational search, can be carried out. The translation search moves the properly oriented model through all the 3-D space until it finds the proper hole to fit in. 

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... 

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). 

The second step consists of calculating a convolution of interatomic vectors between s)unmetry-related molecules of the correctly oriented model placed at different origins, with the experimental Patterson fimction. The reciprocal version of the... 

In a methodology they developed called holographic QSAR (44), Hurst and Patterson have used integer-valued vectors to characterize the frequency of occurrence of molecular fragments. However, they do not use the vectors in their native form but rather fold them into a smaller vector by hashing. 

The crux of the method is that the relative positions of the heavy atoms in the two different crystals must be known. When nothing detailed is known of the molecular structure, it is not easy to obtain this information. Perutz (1956) devised methods based on Fourier syntheses of the Patterson type referred to in a later section, which give interatomic vector maps the combined data for the two heavy-atom derivatives, in special correlation functions, give the relative positions... 

Interatomic vectors. Although, in the absence of knowledge of the signs of the Fourier terms, it is not possible to deduce directly the actual positions of the atoms in the cell, it is theoretically possible to deduce interatomic vectors, that is, the lengths and directions of lines joining atomic centres. Patterson (1934,1935 a) showed that aFourier synthesis employing values of F2 (which are of course all positive) yields this information. The Patterson function... 

Calculations of the Patterson function may be carried out in exactly the same way as those of electron densities. Bragg s optical method may also be used indeed, in general it may be applied more readily to the formation of vector maps, since (the signs of the jF2 coefficients being all positive) the question of phase adjustment does not arise. The optical method has been shown to give a correct vector map for the 6 projection of haemoglobin. ... 

The usefulness of the F2 synthesis is subject to the inherent limitation of a vector diagram vectors are all erected from a single point. The vector diagram, when obtained, must be interpreted in terms of actual atomic coordinates. (For the relations between peak positions on vector maps and the equivalent points in the 17 plane-groups, see Patterson, 1935 6.) For simple structures this presents little difficulty,... 

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... 

Patterson function will show maxima on this plane at positions which give immediately the actual coordinates of these atoms. Similar considerations were used in the determination of the structure of potassium sulphamate NH2SO3K (Brown and Cox, 1940) it was known that the y coordinates of the potassium ions are 0 and while those of the sulphur atoms are and consequently, the Patterson function on the plane y = l shows maxima at positions corresponding to K-S vectors. Atomic positions are not given directly, but can be derived from the positions of Patterson peaks by a consideration of the equivalent positions in the space-group. 

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,... 

There are two approaches to the solution of the phase problem that have remained in favor. The first is based on the tremendously important discovery or Patterson in the 1930s ihal the Fourier summation of Eq. 3. with (he experimentally known quantities F2 (htl> replacing F(hkl) leads nol to a map of scattering density, but to a map of all interatomic vectors. The second approach involves the use of so-called direct methods developed principally by Karie and Hauptman of the U.S. Naval Research Laboratory and which led to the award of the 1985 Nobel Prize in Chemistry. Building upon earlier proposals that (he relative intensities of the spots in a diffraction pattern contain information about a crystal phase. Hauptman and Karie developed a mathematical means of extracting the information. A fundamental proposition of (heir direct method is that if thrice intense spots in the pattern have positions whose coordinates add up to zero, their relative phases will cancel out. Compulations done with many triads of spots yield probable phases for a significant number of diffracted waves and further mathematical analysis leads lo a likely solution for the structure of the molecule as a whole. 

The Patterson function has been the most useful and generally applicable approach to the solution of the phase problem, and over the years a number of ingenious methods of unraveling the Patterson function have been proposed. Many of these methods involve multiple superpositions of ports of the map. or "image-seeking with known vectors. Such processes are ideally suited lo machine compulation. Whereas the great increase in the power of x-ray methods of structure determination in the past few years has come simply front our ability lo compute a three-dimensional Patterson function, it is reasonable lo expect that, as machine methods of unraveling the Patterson function are developed, this power will increase many fold. 

Because the Patterson function contains no phases, it can be computed from any raw set of crystallographic data, but what does it tell us A contour map of p(x,y,z) displays areas of high density (peaks) at the locations of atoms. In contrast, a Patterson map, which is a contour map of P(u,v,w), displays peaks at locations corresponding to vectors between atoms. (This is a strange idea at first, but the following example will make it clearer.) Of course, there are more vectors between atoms than there are atoms, so a Patterson map is more complicated than an electron-density map. But if the structure is simple, like that of one or a few heavy atoms in the unit cell, the Patterson map may be simple enough to allow us to locate the atom(s). You can see now that the... 

I will show, in a two-dimensional example, how to construct the Patterson map from a simple crystal structure and then how to use a calculated Patterson map to deduce a structure (Fig. 6.10). The simple molecular structure in Fig. 6.10a contains three atoms (dark circles) in each unit cell. To construct the Patterson map, first draw all possible vectors between atoms in one unit cell, including vectors between the same pair of atoms but in opposite directions. (For example, treat 1 —> 2 and 2 — 1 as distinct vectors.) Two of the six vectors (1 — 3 and 3 —> 2) are shown in the figure. Then draw empty unit cells around an origin (Fig. 6.10b), and redraw all vectors with their tails at the origin. The head of each vector is the location of a peak in the Patterson map, sometimes called a Patterson "atom" (light circles). The coordinates (u,v,w)... 

Figure 6.10 Construction and interpretation of a Patterson map. (a) Structure of unit cell containing three atoms. Two of the six interatomic vectors are shown, (h) Patterson map is constructed by moving all interatomic vectors to the origin. Patterson "atoms" (peaks in the contour map) occur at the head of each vector. (c) Complete Patterson map, containing all peaks from (b) in all unit cells. Peak at origin results from self-vectors. Image of original structure is present (origin and two darkened peaks) amid other peaks, (d) Trial solution of map (c).lf origin and Patterson atoms a and b were the image of the real unit cell, the interatomic vector a - b would produce a peak in the small box. Absence of the peak disproves this trial solution.

Unit-cell symmetry can also simplify the search for peaks in a three-dimensional Patterson map. For instance, in a unit cell with a 2X axis (twofold screw) on edge c, recall (equivalent positions, Chapter 4, Section II.H) that each atom at (x,y,z) has an identical counterpart atom at (-x,-y,V2 + z). The vectors connecting such symmetry-related atoms will all lie at (u,v,w) = (2x,2y,V2) in the Patterson map (just subtract one set of coordinates from the other), which means they all lie in the plane that cuts the Patterson unit cell at w = l/2. Such planes, which contain the Patterson vectors for symmetry-related atoms, are called Harker sections or Harker planes. If heavy atoms bind to the protein at... 

---