Coordinates Patterson space

If we know the structure, the relative x, y, z coordinates in space of one member of a homologous pair, or series, then we can compute its continuous molecular transform, or diffraction pattern, as well as its Patterson function. If we additionally have an observed diffraction pattern recorded from crystals of another, homologous molecule, then we can compare the computed transform of the known molecule with the observed transform from the unknown crystalline homologue. This can be accomplished even though the former transform is continuous and the latter discreet. The computed transform will initially, of course, have an arbitrary and undefined orientation with respect to that observed, but the spatial relationship can ultimately be resolved using the same rotation function search procedure described above. There are some other complexities that must also be addressed in this approach, but they are usually not insurmountable as is witnessed by the remarkable number of successful applications currently swelling the literature. 

The Patterson function is yet another example of a convolution function (see Chapter 3), and it maps vector relationships in real space into a second coordinate system, which is Patterson space. It will be instructive here to examine the Patterson function s relationship to a real atom distribution by asking how it may be physically generated if the distribution of atoms in real space is known. Examples are illustrated in Figures 9.3 and 9.4. The Patterson function of a known structure is formed in the following way. 

A Patterson coordinate system is defined based on unit vectors u, v, and w, which are parallel with the axes of the real unit cell of the crystal. Each point (u, v, and w) in Patterson space defines the end point of a vector u having a unique direction and length from the origin of Patterson space to that point, that is, u = (u,v, w). Every point, or vector, in Patterson space (u, v, w) will have associated with it a value P(u). 

FIGURE 9.9 Two RNase A molecules are related by a twofold axis along y in a crystal. Assume that the points indicated correspond to sites of heavy atoms. The vector between any pair of symmetry-equivalent heavy atom sites is U, 0, W in Patterson space and is Xi — x, yi — yi, Zi — Zi in real space. Thus V = x2 — xt, V = yi — yq = 0, and IT = -2 — z,. U, V, and VT can be obtained by inspection of the Patterson map calculated from the diffraction intensities of the crystal and x and z, two coordinates of the heavy atoms, be determined unambiguously. 

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. 

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... 

With a knowledge of the space-group symmetry, strong vectors can be analyzed to give the fractional atomic coordinates of those atoms in the structure that have the highest atomic numbers. If there is only one such heavy atom in the asymmetric unit, the interpretation of the Patterson map is simplified because the map is dominated by heavy-atom-hea ... 

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... 

Complete the solution of the crystal structure and perform Rietveld refinement of the model of tungsten oxide peroxide hydrate, W02(02)(H20), which crystallizes in the space group symmetry P2]/n with a = 12.07, b = 3.865, c = 7.36 A, p = 102.9". The location of W has been found from a Patterson map and it has the coordinates x = 0.680, y = 0.066, z = 0.364. Note that W usually exhibits octahedral or square-pyramidal coordination (with the peroxide group, 0-0, counted as one ligand). The experimental powder diffraction pattern is found on the CD in the file Ch7Pr08 CuKa.raw. 

Remembering that Fflkl is simply the measured intensity / <, a scaler quantity, then the expression can be recognized as simply the electron density equation (see Chapter 5) with squared coefficients and all phases 4>hkt set equal to zero. The normalization constant is here l/V2 because of the squared coefficient, where V is the volume of the unit cell. The units it implies for the function, something per volume squared, immediately indicates that P(u, v, w) is not electron density but some other spatial function. Because the equation yields something other than electron density, existing in some unique space, we cannot denote it by p(x, y, z) in jc, y, z (real space) we must designate it by P(u, v, w) in some alternative coordinate space whose variables are u, v, w. Otherwise, P(u, v, w) is the equation for aperiodic function in u, v, w space. The Patterson function, or Patterson wave... 

Second, algebraic differences between the equivalent positions for the space group are formed. For each pair of equivalent positions, one coordinate difference will turn out to be a constant, namely 0, 5, 3, 5, depending on the symmetry operator. These define the Harker sections for that space group, which are the planes having one coordinate u,v, or w constant, and that will contain peaks corresponding to vectors between symmetry equivalent atoms. In focusing attention only on Harker sections, the Patterson coordinates u,v,w... 

Consider the example in Figure 9.10 of a 2i axis along z in the unit cell of a crystal. The corresponding Harker peaks will be at 2x, 2y, If, in examining the w = section of the Patterson map, we find a propitious peak at U,V,, then we can properly conclude that the real space coordinates of the heavy atom are at x = U/2, y = V/2. Again, we are able to deduce actual, real space positions of the heavy atoms in the unit cell, x and y, from coordinates of peaks on the appropriate Harker section of the Patterson map, U and V. 

A Patterson map, different for each space group, is a unique puzzle that must be solved to gain a foothold on the phase problem. It is by finding the absolute atomic coordinates of a heavy atom, for both small molecule and macromolecular crystals, that initial estimates (later to be improved upon) can be obtained for the phases of the structure factors needed to calculate an electron density map. 

This function is similar to the electron density function given earlier. Here, P(uvw) is the value of the Patterson function at Patterson coordinates u, v, w these are the traditional coordinate symbols (instead of x, y, z) used for squared ( F jt,p) space. All other symbols have their usual meaning. The Patterson function is a Fourier summation using the intensities as coefficients and setting all equal to 0. The resulting contoured map will have peaks corresponding to vector differences between all atoms in the structure. A vector between an atom and itself is a zero vector therefore, the Patterson functions always have a very large peak at u,v,w = 0, 0, 0. 

Apart from the high peak at the origin (the sum of the vectors from each atom to itself), there should be one high peak in the Patterson map at v = A, and the position of this peak will give values forx and z of the heavy atom in the unit cell (with the screw axes at x = 0, z = 0). They coordinate of one atom is arbitrary in this particular space group. This is evident in the general positions of the space group (listed above) since if there is an atom aty there is also one at Vi + y, but no infor-... 

Patterson, E.S., Watts-Perotti, J.C., and Woods, D.D. 1999. Voice loops as coordination aids in space shuttle mission control. Computer Supported Cooperative Work 8, 353-71. 

Watts, J.C., Woods, D.D., and Patterson, E.S. 1996. Functionally Distributed Coordination During Anomaly Response in Space Shuttle Mission Control. Proceedings of Human Interaction with Complex Systems. IEEE Computer Society Press, Los Alamitos, CA. 

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