Improving electron-density maps

For macromolecules it may be difficult to determine the overall conformation of the protein from the electron-density map, but some interesting methods are in use for aiding in the interpretation of this map. 

Density modification is a procedure that is used to improve protein structures. If phase information is poor, it still is possible to calculate an electron-density map, modify it in some way by use of chemical or crystallographic information, and then to calculate the Fourier transform of this modified map (to give new structure factors and phases), and recompute the electron-density map with what are hopefully improved phases.The new electron density map should be easier to interpret than the first. 

The second type of electron-density modification is noncrystallo-graphic symmetry averaging. If there is more than one copy of the molecule in the asymmetric unit, and the relationship between them is known (for example, from the rotation function), constraints can be placed on the electron density map to make the two molecules look approximately the same. This information can then be used to improve the electron density, and hence the relative phases of the structure, and 

FIGURE 9.14. Protein electron-density map showing several sections. The overall shape of the molecule can be seen in this set of sections of the electron-density map, 

Atomic coordinates of atoms in the model of a protein structure can bo refined to more precise values that give a better fit to the experimental data. The so-called real-space refinement procedure optimizes the fit 

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Density modification A computational method for improvement of phases. The electron density is modified and a new set of phases is then determined by Fourier transformation, so that an improved electron-density map can be calculated. 

To elucidate a crystal structure, a crystallographer collects a set of observed diffraction intensities for a crystal diffracting in an X-ray beam. The intensities alone, however, are insufficient to construct a model the crystallographer needs to predict a set of unknowns called the reflection phases. These are initially estimated by various techniques to give a starting model. The starting model is then refined, and the refined model is used to recalculate the phases, allowing the calculation of an improved electron density map, which can be reinterpreted. The process is cyclical, but hopefully leads to an accurate interpretation of the reflection intensities. 

Jones TA, Zou JY, Cowan SW, Kjeldegaard M. Improved methods for building protein models in electron-density maps and the location of errors in these models. Acta Cryst 1991 A47 110-9... 

The difference electron density map following the last cycle of least squares refinement did not show evidence for a simple disorder model to explain the anomalously high B for the hydroxyl oxygen. Attempts to refine residual peaks with partial oxygen occupancies did not significantly improve the agreement index. 

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... 

Model building is an interpretation of the currently available electron density. Refinement is the adjustment of the built model to fit better to the experimental data. A crucial point here is that a density map computed from the refined model is generally better than the map obtained from the same model before the refinement. This then allows for an even better model to be built. Thus, refinement is needed to improve the outcome of model building by generating a better electron density map and model building is needed to provide a model in the first place and to provide stereochemical restraints for the subsequent refinement to proceed smoothly. This viewpoint merges these two steps into one model optimization process. 

I will discuss the iterative improvement of phases and electron-density maps in Chapter 7. For now just take note that obtaining the final structure entails both calculating p(x,y,z) from structure factors and calculating structure factors from some preliminary form of p(x,y,z). Note further that when we compute structure factors from a known or assumed model, the results include the phases. In other words, the computed results give all the information needed for a "full-color" diffraction pattern, like that shown in Plate 3d, whereas experimentally obtained diffraction patterns lack the phases and are merely black and white, like Plate 3e. 

The most demanding element of macromolecular crystallography (except, perhaps, for dealing with macromolecules that resist crystallization) is the so-called phase problem, that of determining the phase angle ahkl for each reflection. In the remainder of this chapter, I will discuss some of the common methods for overcoming this obstacle. These include the heavy-atom method (also called isomorphous replacement), anomalous scattering (also called anomalous dispersion), and molecular replacement. Each of these techniques yield only estimates of phases, which must be improved before an interpretable electron-density map can be obtained. In addition, these techniques usually yield estimates for a limited number of the phases, so phase determination must be extended to include as many reflections as possible. In Chapter 7,1 will discuss methods of phase improvement and phase extension, which ultimately result in accurate phases and an interpretable electron-density map. 

Having located the heavy atom(s) in the unit cell, the crystallographer can compute the structure factors FH for the heavy atoms alone, using Eq. (5.15). This calculation yields both the amplitudes and the phases of structure factors Fh, giving the vector quantities needed to solve Eq. (6.9) for the phases ahkl of protein structure factors Fp. This completes the information needed to compute a first electron-density map, using Eq. (6.7). This map requires improvement because these first phase estimates contain substantial errors. I will discuss improvement of phases and maps in Chapter 7. 

Displaying an electron-density map and adjusting the models to improve its fit to the map (see Plate 21 and the cover of this book). SPV can display maps of several types (CCP4, X-PLOR, DN6). I am aware of no programs currently available for computation of maps from structure factors on personal computers, but I am sure this will soon change. 

Attempts to improve molecular wavefunctions so as to be able to calculate properties more accurately continue to be made, particularly via the constrained variational procedure. Two-particle hypervirial constraints were considered by Bjoma within the SCF formation,282 and he presented a perturbational approach to their solution.233 Using Scherr s wavefunction, and constraining p to satisfy the molecular virial theorem, a calculation on N2 led to rapid convergence.234-235 The constrained SCF orbitals are believed to be a closer approximation to the true tfi nearer the nucleus than further out. A later paper discussed the electron-density maps in comparison to the SCF derived maps, which confirm the conclusion that the wavefunction near the nucleus is improved.236... 

The electron density distribution for solvent molecules can be improved if the contribution from bulk water to the X-ray scattering is included in the model. This affects the low-angle j X-ray intensity data which are omitted in early stages of the least-squares refinement of protein crystal structures. If they are included in refinement and properly accounted for, the signal-to-noise ratio in the electron density maps is significantly improved and the interpretation of solvent sites is less ambiguous. 

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