The Patterson Function

The Patterson function is the convolution of the electron density p(r) with itself inverted with respect to the origin  

Since P(r ) is sampled only at the nodes of the reciprocal lattice (where r = h), we have  

The above features can make it difficult to derive the atomic positions from analysis of the Patterson map. The most traditional approach involves the use of the so-called Marker sections they contain the interatomic vectors between an atom and its symmetry equivalents. For instance, let us consider the space group P2j, with equivalent positions  

The Harker vector (2x, 0.5, 2z) lies in the Marker section (u, 0.5, w) and may be used to derive the position of the atom indeed x = m/2, y undefined (the origin in P2i floats freely along the z-axis), z = w/2. 

The use of the Harker sections is made easier when a few heavy atoms are present in the unit cell their Harker maxima can be recognized and used to locate the heavy atoms. If they have a sufficiently high atomic number, they can be used as a good initial model to which one can apply the so-called Method of Fourier Recycling, to obtain the light atom positions and then to recover the complete structure. 

As we saw earlier, the use of the electron density function in stmctural studies is not compatible with the fact that we are only measuring the square modulus of the stmcture factor, whereas it would actually be necessary to know the complex value of this factor. Patterson [PAT 34, PAT 35] thought of a method to obtain stmctural 

Let us assume, as we did before, that the electron density in a point with coordinates (x,y,z) is denoted by p(x,y,z) and given by relation [4.19]. The values of this function p(x,y,z) ate very small, except in certain points which correspond to the positions of the atoms. Now consider the function p(x + u, y + v, z + w), where u, V and w ate constant parameters for any values of x, y and z. The values of the product p(x,y,z)Xp(x + u, y + v, z + w) ate small, except when the triplet (u,v,w) is comprised of the components of a vector that connects the positions of two atoms in 

Let us now replace the two distributions with their expressions given in Relation 

Generally, the projections of the Patterson function are used, rather than its three-dimensional representation. 

If we denote by a(x,y) the projection onto the plane (x,y) of the electron density p(x,y,z), the corresponding Patterson function will be written  

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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. 

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 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). 

Patterson and Pattersons Fifty years of the Patterson function... 

Patterson and Pattersons fifty years of the Patterson function J. P. Glusker, B. K. Patterson, and M. Rossi, editors... 

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. ... 

In a similar way, it can be shown that if a crystal has a plane of symmetry perpendicular to its b axis, the Patterson function has maxima along the 6 axis of the cell (the line 0, y, 0, in Fig. 229) which indicate the distance of atoms from the plane of symmetry. For a glide plane perpendicular to b, with a translation c/2, the distance of atoms from this plane are indicated by maxima along the line 0, y,... 

There are other circumstances in which some of the atomic coordinates in a crystal can be discovered by evaluation of the Patterson function over a particular plane or along a particular line. For instance, it may be known, from a consideration of the space-group and the equivalent positions in the unit cell, that there is one particular atom at the origin of the cell and others somewhere on the plane y = L The... 

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. 

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. 

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... 

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... 

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... 

Patterson and Symmetry Superposition Methods. An older bootstrap method, based on searches of the Patterson function and variants thereof (vector superposition and symmetry superposition functions), should present significant advantages for noncentric structures. Much recent progress has been made in such alternative algorithms, which should be used when direct methods fail. 

The phase problem can be solved, that is, phases of the scattered waves determined, either by Patterson function or by direct methods. The Patterson function P is a self-convolution of the electron density p, and its magnitude at a point u, v, w can be obtained by multiplying p (x, y, z) hy p (x + u, y + V, z + w) and summing these products for every point of the unit cell. In practice, it is calculated as... 

Patterson methods have also been successfiilly used for structure solution from powder diffraction data. By taking advantage of the Patterson function Fi, usefiil information about the crystal structure can be deduced. Compared to Direct methods, Patterson techniques are more suitable for powder diffraction data with lower resolution, and peak overlap causing significant difficulties. The Patterson function can be calculated by using the equation... 

Here Mj is the Madelung constant based on I as unit distance, n is the number of molecules in the unit cell, zy is the charge number of atom j, V is the volume of the unit cell and h is the magnitude of the vector (hi, h2, ha) in reciprocal space or the reciprocal of the spacing of the planes (hihjha). The coordinates of atom j are a i/, X2, Xay. The sums over j are taken over all the atoms in the imit cell. F(h) is the Fourier transform of the Patterson function and (h) is the Fourier transform of the charge distribution /(r). F h) is given by... 

The Patterson function is a map that indicates all the possible relationships (vectors) between atoms in a crystal structure. It was introduced by A. Lindo Patterson " in 1934, inspired by earlier work on radial distribution functions in liquids and powders. In crystals the directionality as well as the lengths of vectors between atoms (atomic distances) can be deduced. By contrast, in liquids and powders the geometric information that can be obtained is limited to interatomic distances, because in these the molecules are randomly oriented. While the use of the Patterson function revolutionized the determination of crystal structures of small molecules in the 1930s to 1950s, direct methods are now the most widely used methods for obtaining structures of small organic molecules. The Patterson function, however, continues to play an essential part in the determination of crystal structures of inorganic compounds and macromolecules. It is also very useful when the structure of a small molecule proves difficult to solve by direct methods. 

This Equation 8.12 has the same form as the equation for electron density (Equation 6.3, Chapter 6), but note that there is no phase angle in the expression The coefficients of the Patterson function are the observed intensities, after some geometric corrections involved with the data collection process are made. Because the Patterson function uses... 

The symmetry of the Patterson function is the same as the Laue symmetry of the crystal. The Patterson function for space groups that have symmetry operations with translational components (screw axes and glide planes) has an added property that is very useful for the determination of the coordinates of heavy atoms. Specific peaks, first described b David Harker, are associated with the vectors between atoms related by these symmetry operators. These peaks are found along lines or sections (Figure 8.17). For example, in the space group P2i2i2i there are atoms at... 

The Patterson function of a crystal consists of two parts self vectors, which are a set of vectors within the particular molecule, and cross vectors, which are a set of vectors from atoms in one molecule in the crystal to atoms in another molecule. Suppose that a vector map is calculated for a molecule or part of a molecule whose structure is known. Then the set of self vectors in this vector map will be identical to those in the Patterson map computed for the crystal, but will probably be rotated by some set of angles. If the two Patterson functions are superimposed, there would be no particular agreement except when the two sets of selfvectors have the same orientation. If one of these Patterson maps is rotated on the second, keeping the locations of the origins of the two... 

FIGURE 8.21 (cont d). (c) The rotation function for crystalline insulin. Peaks (highlighted by arrows) correctly indicate the direction of the local twofold symmetry axes. These twofold axes were also indicated in the Patterson function in (b). The actual crystal structure, in the same orientation as in (a) to (c), is shown in (d), with the unit-cell axes a and b, and the local twofold axes indicated by arrows. (From Ref, 65. Courtesy the authors and Academic Press.)... 

Rotation function The rotation function gives a measure of the of the degree of correspondence between a vector set calculated for a known structure and the Patterson function of that crystal, as a function of the rotation of one with respect to the other about the origin. Peaks in the rotation function define a likely orientation of the known fragment of the structure. All vectors are calculated for a known portion of a crystal structure, and the resulting set of vectors is rotated about its origin until it best matches the orientation of vectors in the Patterson map. 

Glusker, J. P., Patterson, B. K., and Rossi, M. (Eds.) Patterson and Pattersons. Fifty Years of the Patterson Function. International Union of Crystallography Crystallographic Symposia. Oxford University Press Oxford (1987). 

Nordman, C. E., and Nakatsu, K. Interpretation of the Patterson function of crystals containing an unknown molecular fragment. The structure of an Alstonia alkaloid. J. Amer. Chem. Soc. 85, 353-354 (1963). 

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