Vector maps

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. 

Global velocity distribution behind flame front. Upward propagation in 5.15% methane/air mixture, (a) vector map, (b) and (c) scalar maps of axial and radial velocity components, respectively. Spots are caused by condensation of water vapor on the glass walls. 

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). 

Fig. 1. The expression vector pTDI. (a) pTDI vector map. (b) DMA sequence of the pTDI vector around the multiple cloning sites.

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... 

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,... 

Fig. 224. Picryl iodide. Vector map, 110 projection. (Huse and Powell, 1940.)...

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... 

Similar in principle to this is the use of two different X-ray wavelengths on the same crystal, one near to and the other far from the absorption edge of a marker atom, to give different diffracting powers this is the "ideal isomorphism method advocated by Lipson and Cochran (195 ) and Pepinsky (1956). A comparison of the vector maps for the two wavelengths, or a vector map based on the differences of structure amplitudes for corresponding reflections for the two wavelengths, indicates which peaks are due to the marker atoms. 

Unravelling vector maps. Many attempts have been made to devise systematic methods of deducing the actual atomic arrangement... 

Figure 9 (a) Three-dimensional view of 3D solid concentration with a horizontal gas jet in the fluidized bed (b) 3D voxel-volume-averaged solid phase velocity vector map in the Y-Z plane of the fluidized bed (c) 3D voxel-volume-averaged solid phase vector map (Wang et al., 2008) (see Plate 12 in Color Plate Section at the end of this book). 

Figure 18 Tomographic images of local oil fraction distributions in oscillatory baffled reactor (top left), with estimated velocity profiles (top right) from crosscorrelation signals between adjacent electrode rings (bottom right) and vector map (bottom left) (Vilar et al, 2008 Vilar, 2008) (see Plate 15 in Color Plate Section at the end of this book).

Figure 9.2. Shuttle vector map retrieved from Riken Gene Bank. The map of shuttle vector pYAC 3/4/5 shows major restriction sites.

Wilkins MR, Sanchez JC, Williams KL, Hochstrasser DF. Current challenges and future applications for protein maps and post-translational vector maps in proteome projects. Electrophoresis 1996 17(5) 830—838. 

MATHEMATICAL METHODS IN PHYSICS AND ENGINEERING, John W. Denman. Algebraically based approach to vectors, mapping, diffraction, other topics in applied math. Also generalized functions, analytic function theory, more. Exercises. 448pp. 5X 85. 65649-7 Pa. 8.95... 

The computational labor associated with two-dimensional Fourier syntheses is not too formidable, and two-dimensional Fourier maps can be constructed without machine help. The labor associated with two-dimensional Patterson sysntheses is even less, and a two-dimensional vector map can often be obtained from measured intensities in a few hours. For Fourier and Patterson syntheses in three-dimensions, however, machine help is virtually indispensable. Before application of automatic computers to x-ray diffraction, the main obstacle standing in the way of a structure determination was generally the computational effort involved. In the 1950 8, the use of computers became commonplace, and the main obstacle became the conversion of measured intensities to amplitudes (the so-called phase problem ). There is still no general way of attacking this problem that is applicable in all situations, but enough methods have been developed so that by use of one, or a combination of them, all but very complicated structures may, with time and ingenuity, be determined. 

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