Error density maps

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... 

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... 

After a simple Fourier inversion of a set of magnetic structure factors MbU, one can retrieve the magnetisation density. A much better result, e.g. the most probable density map, can be obtained using the Maximum Entropy (MaxEnt) method. It takes into account the lack and the uncertainty of the information not all the Bragg reflections are accessible on the instrument, and all the values contained in the error bars are satisfactory and have to be considered. However, as this method extracts all the information contained in the data, it is important to keep in mind that it may show spurious small details associated to a low accuracy and/or a specific lack of information located in (/-space. 

In summary, there are three important generalizations about error estimation in protein crystallography. The first is that the level of information varies enormously as a function primarily of resolution, but also of sequence knowledge and extent of refinement. The second generalization is that no single item of information is completely immune from possible error. If the electron density map is available or indicators such as temperature factors are known from refinement, then it is possible to tell which parameters are most at risk. The third important generalization is that errors occur at a very low absolute rate 95% of the reported information is completely accurate, and it represents a detailed and objective storehouse of knowledge with which all other studies of proteins must be reconciled. 

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. 

Bhat, T. N. and Cohen, C. H. (1984). OMITMAP An electron density map suitable for the examination of errors in a macromolecular model. /. Appl. Cryst. 17, 244-248. 

Though Eq. (5.31) can be evaluated directly, provided the phase errors can be estimated, it can be reduced to a simpler, computationally more convenient equation for the average covariance in a density map (Rees 1976). For the space group Pi, Eq. (5.31) becomes... 

Fig. 214. Sodium benzyl penicillin, (a) Approximate electron density map, b projection, with atomic coordinates appropriate to extended molecular configuration. (6) Error synthesis map. (c) The same electron density map as (a), with atomic coordinates appropriate to a curled configuration of the same molecule.

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. 

Figure 8. Methyl-orientation results for the three methyl groups in N-acetyl-L-leucine. A, x-ray determined p(r) charge density map. B, graph showing relation between the H-C-C-H methyl torsion angles determined from the x-ray results and those determined by using a quantum chemical geometry optimization. The rms error is 8°.

Rees [27] has calculated the effect on the deformation electron density maps of experimental random errors in centrosymmetric crystals ... 

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