Resolution of electron density maps

FIGURE 10.2 A small area of a raw electron density map of a protein, directly from the Fourier calculation as it comes off the computer. The location of each number on the plane corresponds to a specific x, y, z fractional coordinate in the unit cell. In general, one of the three coordinates will be constant for the entire plane, and rows and columns will correspond to the other two coordinates. The value of the number at each position is p(x, y, z), the electron density at that point. Contours are incrementally drawn around areas having p(x, y, z) greater than certain values. This yields a topological map of the electron density on each plane of the unit cell. 

For conventional molecules with relatively small unit cells and low thermal parameters, all the theoretically possible data may be collected and included in the Fourier synthesis. The value X/2 is then the practical as well as the theoretical resolution limit. For macromolecular crystals, X/2 is never the practical limit, simply because the consistency of structural detail from molecule to molecule, and unit cell to unit cell throughout the crystal is not adequate. Thus, beyond a certain Bragg spacing, usually considerably short of the theoretical limit of 0.77 A for CuK radiation, the intensities decline and ultimately become unobservable. In 

ELECTRON DENSITY, REFINEMENT, AND DIFFERENCE FOURIER MAPS 

FIGURE 10.4 Several consecutive sections of the electron density of the hexon capsid protein from adenovirus are stacked and displayed on a light box. Here the continuity of the polypeptide begins to emerge. Some recognizable features such as the two a helices marked by arrows are also evident to 

It should be emphasized that the stated resolution of a particular structure determination does not necessarily reflect the clarity of the features present in the electron density map, or the accuracy of the model based on the map. That is, many 3.0 A resolution electron density maps are far more interpretable in terms of protein structure, and the amino acid side groups more recognizable, than are some maps that include higher resolution terms. This is because resolution says nothing about the accuracy of the structure factors used to compute the map. In particular, it gives no indication of the quality of the phase determination, the mean phase error, which is the essential factor that influences contrast between the electron density of the protein and the background noise level in the Fourier map. 

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It is clear that there is no one correct receptor model. However, since the early modelling work based on bacteriorhodopsin, receptor models have become increasingly more accurate regarding their ability to explain experimental observations. Further improvements in the resolution of electron density maps will only add to the accuracy of these models, allowing them to become a predictive tool also. 

Recently, a systematic experimental study has clearly proved the effect on a gaseous medium of both ASE and picosecond pedestal prior to the arrival of the ultrashort intense laser pulse [26]. The study has been based on sequences of electron density maps obtained from optical interferograms with femtosecond resolution and has been supported by numerical simulation of the ionization of the medium. 

Tsuruta, H., et al. (1998). Imaging RNA and dynamic protein segments with low-resolution virus crystallography experimental design, data processing and imph-cations of electron density maps. /. Mol. Biol. 284, 1439-1452. 

A recent structural determination of the bovine enzyme at 1.7 A resolution suggests that the Cu(l)/Cu(ll) transition, along with their related coordination changes, is modulated by a protein loop that lines the active site cavity [15]. Again, possible photoreduction resulting from exposure to X-rays has been considered and (possibly) ruled out by examination of electron density maps and temperature factors corresponding to atoms at the active site. 

Svergun and coworkers have developed the software GNOM to obtain p r) from an indirect Fourier transform of the scattering profile. Nevertheless, in recent times, the use of p r) displaced by algorithms provide a low-resolution 3D electron density map of the molecule from the ID SAXS profile. 

The advancement of algorithms that permit ab initio reconstruction of low-resolution 3D electron density maps... 

InterpretatioD of Electron-density Maps and Refinement of Protein Structures.—Now that so many proteins have been determined at high resolution, improving the fit of the protein model to the electron density and improving the electron density itself have become central problems. 

Figure 12.3 Two-dimensional crystals of the protein bacteriorhodopsin were used to pioneer three-dimensional high-resolution structure determination from electron micrographs. An electron density map to 7 A resolution (a) was obtained and interpreted in terms of seven transmembrane helices (b).

Deisenhofer, J., et al. X-ray structure analysis of a membrane protein complex. Electron density map at 3 A resolution and a model of the chromophores of the photosynthetic reaction center from Rhodopseudomonas viridis. f. Mol. Biol. 180 385-398, 1984. 

The amplitudes and the phases of the diffraction data from the protein crystals are used to calculate an electron-density map of the repeating unit of the crystal. This map then has to be interpreted as a polypeptide chain with a particular amino acid sequence. The interpretation of the electron-density map is complicated by several limitations of the data. First of all, the map itself contains errors, mainly due to errors in the phase angles. In addition, the quality of the map depends on the resolution of the diffraction data, which in turn depends on how well-ordered the crystals are. This directly influences the image that can be produced. The resolution is measured in A... 

Figure 18.11 Electron-density maps at different resolution show more detail at higher resolution, (a) At low resolution (5.0 A) individual groups of atoms are not resolved, and only the rodlike feature of an

From a map at low resolution (5 A or higher) one can obtain the shape of the molecule and sometimes identify a-helical regions as rods of electron density. At medium resolution (around 3 A) it is usually possible to trace the path of the polypeptide chain and to fit a known amino acid sequence into the map. At this resolution it should be possible to distinguish the density of an alanine side chain from that of a leucine, whereas at 4 A resolution there is little side chain detail. Gross features of functionally important aspects of a structure usually can be deduced at 3 A resolution, including the identification of active-site residues. At 2 A resolution details are sufficiently well resolved in the map to decide between a leucine and an isoleucine side chain, and at 1 A resolution one sees atoms as discrete balls of density. However, the structures of only a few small proteins have been determined to such high resolution. 

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. 

Once an electron density map has become available, atoms may be fitted into the map by means of computer graphics to give an initial structural model of the protein. The quality of the electron density map and structural model may be improved through iterative structural refinement but will ultimately be limited by the resolution of the diffraction data. At low resolution, electron density maps have very few detailed features (Fig. 6), and tracing the protein chain can be rather difficult without some knowledge of the protein structure. At better than 3.0 A resolution, amino acid side chains can be recognized with the help of protein sequence information, while at better than 2.5 A resolution solvent molecules can be observed and added to the structural model with some confidence. As the resolution improves to better than 2.0 A resolution, fitting of individual atoms may be possible, and most of the... 

Figure 3.9 Stereoview of 2 F0bs - Fcaic electron density maps at 2-A resolution for (A) Pox and (B) PN. (C) Stereoview of the Fobs - Fobs (reduced minus oxidized) electron density maps showing differences in the vicinity of the P cluster. (Reprinted with permission from Figure 3 of Peters, J. W. Stowell, M. H. B. Soltis, S. M. Finnegan, M. G. Johnson, M. K. Rees, D. C. Biochemistry, 1997, 36, 1181-1187. Copyright 1997, American Chemical Society.)...

Hill, E., Tsernoglou, D., Webb, L., Banaszak, L. Polypeptide conformation of cytoplasmic malate dehydrogenase from an electron density map at 3.0 A resolution. J. Mol. Biol. 72, 577-591 (1972). 

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