Fourier electron density map

Several steps were needed to determine the structure of the core particle to higher resolution (Fig. Id). The X-ray phases of the low-resolution models were insufficient to extend the structure to higher resolution, since the resolution of the early models of the NCP was severely limited by disorder in the crystals. The disorder was presumed to derive from both the random sequences of the DNA and from heterogeneity of the histone proteins caused by variability in post-translational modification of the native proteins. One strategy for developing an atomic position model of the NCP was to develop a high-resolution structure of the histone core. This structure could then be used with molecular replacement techniques to determine the histone core within the NCP and subsequently identify the DNA in difference Fourier electron density maps. 

The presence of a heavy atom greatly simplifies the structure determination since the prominent Patterson peaks are those corresponding to vectors between heavy atoms. Once the heavy atoms are located, the other atoms can be positioned from Fourier electron density maps using the phase angles determined by the heavy-atom coordinates. The ease of solving such structures is partly responsible for the rapid growth of metal organic chemistry. 

X-Ray diffraction from single crystals is the most direct and powerful experimental tool available to determine molecular structures and intermolecular interactions at atomic resolution. Monochromatic CuKa radiation of wavelength (X) 1.5418 A is commonly used to collect the X-ray intensities diffracted by the electrons in the crystal. The structure amplitudes, whose squares are the intensities of the reflections, coupled with their appropriate phases, are the basic ingredients to locate atomic positions. Because phases cannot be experimentally recorded, the phase problem has to be resolved by one of the well-known techniques the heavy-atom method, the direct method, anomalous dispersion, and isomorphous replacement.1 Once approximate phases of some strong reflections are obtained, the electron-density maps computed by Fourier summation, which requires both amplitudes and phases, lead to a partial solution of the crystal structure. Phases based on this initial structure can be used to include previously omitted reflections so that in a couple of trials, the entire structure is traced at a high resolution. Difference Fourier maps at this stage are helpful to locate ions and solvent molecules. Subsequent refinement of the crystal structure by well-known least-squares methods ensures reliable atomic coordinates and thermal parameters. 

FIGURE 5.5 Electron-density mapping corresponding to the Fourier transforms (A) for denatured (extruded at 100 °C) and native WPI, an (B) inverse reciprocal spacing of electron-density images of native and denatured WPI (Onwulata et al., 2006). 

For molecules of molecular weight above 20,000 g/mol, X-ray diffraction remains the only experimental approach available to obtain detailed and reliable three-dimensional atomic models. The major steps of the method include the obtention of large and well-ordered crystals, their exposure to X-rays and collection of diffraction data and the phasing of these data to obtain by Fourier analysis a three-dimensional view (or map) of the electron density of the molecule. Finally a three-dimensional atomic model of the protein is fitted like a hand in a glove within this map, using a kit containing all the available biochemical and spectroscopic information (Table 6.2). The reliability of the final atomic model is of course dependent on the qnality of the electron density map. This qnality depends on the number of X-ray data per atom and on the resolution and accnracy of these data, which in turn are highly dependent on the size and quality of the crystals. 

The second approach is to use Fourier methods to calculate the electron density based on the model (using calculated Fs and phases, the vector Fc) and compare this with the electron density based on the observations (with calculated phases, the vector Fo). An electron-density map is calculated based on I To I — I. Pc I- This so-called difference map will give an accurate representation of where the errors are in the model compared with the experimental data. If an atom is located in the model where there is no experimental observation for it, then the difference map will show a negative density peak. Conversely, when there is no atom in the model where there should be, then a positive peak will be present. This map can be used to manually move, remove, or add atoms into the model. 

As discussed in the following chapter, difference electron density maps, representing Ap = pobs — pcak, are based on the Fourier transform of the complex difference structure factors AF, defined as... 

Fig. 26. Electron density map, calculated with Fourier coefficients 2 Fo - [fj and phases calculated from the final model, of the hydrophobic pocket of the Val-14S- Tyr mutant of human carbonic anhydrase II (Alexander etai, 1991). This mutation nearly obliterates the pocket and results in a 10 -fold loss of activity (Fierke et ai, 1991).

The limitations mentioned do not apply if the work can be done on a machine various types of mechanical and electrical analogue machines for Fourier synthesis have been described (see Lipson and Cochran, 1953), culminating in Pepinsky s XRAC (X-ray analogue computer), in which structure amplitudes and phases for a two -dimensional synthesis are put in on an array of dials, and the electron density map appears at once on a cathode ray tube (Pepinsky, 1952). An increasing proportion of Fourier syntheses (and indeed crystallographic calculations of all types) is done on electronic digital computers. 

There is another important point. If a Fourier series is cut off sharply when the terms are still appreciable, false detail will appear in the electron density map. To avoid this, for crystals giving strong reflections at large angles, an artificial temperature factor may be applied to the intensities, to make the F s fade off gradually instead of... 

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... 

Here is the Fourier series that gives the first electron-density map... 

Just as the auto mechanic sometimes has parts left over, electron-density maps occasionally show clear, empty density after all known contents of the crystal have been located. Apparent density can appear as an artifact of missing Fourier terms, but this density disappears when a more complete set of data is obtained. Among the possible explanations for density that is not artifactual are ions like phosphate and sulfate from the mother liquor reagents like mercaptoethanol, dithiothreitol, or detergents used in purification or crystallization or cofactors, inhibitors, allosteric effectors, or other small molecules that survived the protein purification. Later discovery of previously unknown but important ligands has sometimes resulted in subsequent interpretation of empty density. 

Plate 12 Electron-density maps at increasing resolution. Maps were calculated using final phases, and Fourier series were truncated at the resolution limits indicated (a) 6.0 A (b) 4.5 A (c) 3.0 A (d) 1.6 A. (For discussion, see Chapter 7.) (Continues)... 

In structure determination from X-ray diffraction data, it sometimes happens that, on the Fourier maps, parts of the coming out structure are unclear. Fuzzy electron density maps may present problems in determining even the approximate positions of the respective fragments of the structure being analyzed. For example, the layered structure of the inclusion (intercalation) compound formed by Ni(NCS)2 (4-methylpyridine)4 (host) and methylcellosolve (guest) [1], The guest molecules are (Fig. 11.1) located on twofold crystal axes of unit cell symmetry and are orientationally disordered as shown in the picture. 

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