Calculations of electron-density maps

The electron density in a crystal is periodic from unit cell to unit cell. Therefore it can be represented as a Fourier series (as discussed in Chapter 6). The coefficients of this Fourier series are the amplitudes of the Bragg reflections the periodicities h,k, and /) are the indices of each Bragg reflection. Only the relative phase angles a hkl) are still needed, and once these have been estimated (see Chapter 8), all of the information for calculating the electron density becomes available. 

The formula for the Fourier summation used to calculate an electron-density map is  

Equation 9.1 may also be written, in a more convenient form, as 

The resulting electron-density map gives a representation of that part of the electron density that has not been correctly accounted for in the 

Three general ways in which electron-density maps are used in structure determination will be considered in turn. 

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On Output, a refinement job will produce a res-file (which is a valid ins-file for the next round of refinement) containing the new description of the model, a pdb-file containing the coordinates of the refined model, an Ist-file containing logging information, and an fcf-file containing stmcture factor moduli and phases for the calculation of electron density maps. The fcf-file can be read directly by Xfit (McRee, 1999) and Coot (Emsley and Cowtan, 2004) or converted into other formats with SHELXPRO. 

MATLAB code for the calculation of electron density maps (plane group p2) ... 

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... 

It is impossible to directly measure phases of diffracted X-rays. Since phases determine how the measured diffraction intensities are to be recombined into a three-dimensional electron density, phase information is required to calculate an electron density map of a crystal structure. In this chapter we discuss how prior knowledge of the statistical distribution of the electron density within a crystal can be used to extract phase information. The information can take various forms, for example ... 

Figure 39 Comparison of sections of electron density maps around Arg-42 of bovine pancreatic trypsin inhibitor, projected onto the xy plane.240. (Left) Map calculated by using the experimental 2.5-A phases. (Right) Map obtained by using the calculated phases in a late stage of refinement.

R. Huber considered the reaction centre to be a dull photosyntheric protein which cannot do anything. If it would be a receptor, I would be personally interested, he said. D. Oesterhelt generously considered the project as one of the young people. J. Deisenhofer calculated the electron density maps, and we frequendy sat together to try to interpret and to incorporate the new sequence information which we were gathering in D. Oesterhelt s department, into the model. R. Huber suddenly changed his mind when he interpreted the electron density map of phycocyanin (another dull... 

The values given are the crystal radii of Shannon, calculated using electron density maps and internuclear distances from X-ray data. Some of the trends that can be seen in these radii are the following ... 

Finally, the determination of phases of structure invariants (groups of phases that have combined values independent of the choice of origin) by t/)-scans of multiple reflections is a recent technique. A comparison of the usefulness in a general laboratory of the various methods is given in Table 8.3. The result of all these methods consist of the relative phase angles of the reflections, h,k,l, which is all the information needed in order to calculate an electron density map. 

The electron density in a crystal, p (xyz), is a continuous function, and it can be evaluated at any point x,y,z in the unit cell by use of the Fourier series in Equations 9.1 and 9.2. It is convenient (because of the amount of computing that would otherwise be required) to confine the calculation of electron density to points on a regularly spaced three-dimensional grid, as shown in Figure 9.3, rather than try to express the entire continuous three-dimensional electron-density function. The electron-density map resulting from such a calculation consists of numbers, one at each of a series of grid points. In order to reproduce the electron density properly, these grid points should sample the unit cell at intervals of approximately one third of the resolution of the diffraction data. They are therefore typically 0.3 A apart in three dimensions for the crystal structures of small molecules where the resolution is 0.8 A. 

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. 

Preliminary three-dimensional atomic coordinates of atoms in crystal structures are usually derived from electron-density maps by fitting atoms to individual peaks in the map. The chemically reasonable arrangement of atoms so obtained is, however, not very precise. The observed structure amplitudes and their relative phase angles, needed to calculate the electron-density map, each contain errors and these may cause a misinterpretation of the computed electron-density map. Even with the best electron-density maps, the precisions of the atomic coordinates of a preliminary structure are likely to be no better than several hundredths of an A. In order to understand the chemistry one needs to know the atomic positions more precisely so that better values of bond lengths and bond angles will be available. The process of obtaining atomic parameters that are more precise than those obtained from an initial model, referred to as refinement of the crystal structure, is an essential part of any crystal structure analysis. 

A Patterson map, different for each space group, is a unique puzzle that must be solved to gain a foothold on the phase problem. It is by finding the absolute atomic coordinates of a heavy atom, for both small molecule and macromolecular crystals, that initial estimates (later to be improved upon) can be obtained for the phases of the structure factors needed to calculate an electron density map. 

Attempts to calculate the site of metabolism based on molecular orbital theory have met with some success. Currently the state-of-the-art method is to calculate an electron density map of molecules using quantum mechanics and then calculate a steric factor using knowledge of the CYP active site.116 Taken together these two parameters have successfully predicted the sites of oxidation for limited sets of molecules.116 This approach may be dramatically enhanced as the crystal structures of more mammalian CYP enzymes become available.126-128... 

PLATE V Calculated difference electron density maps in Si02. Differences are relative to atomic wavefunc-tions. Red colours indicate depletion of electron density blue colours indicate enhancement. 

The direct experimental result of a crystallographic analysis is an electron-density map, and not the atomic model everybody looks at If errors occur in crystal structures, they most often occur at the level of the (subjective) interpretation of the electron-density maps by the crystallographer. A severe problem, especially at low resolution (lower than 3.0 A), is the so-called model bias. To calculate an electron-density map, one needs amplitudes and phases. The amplitudes are determined experimentally, but the phases cannot be measured directly. In later stages of refinement, they are calculated from the model, which means that if the model contains errors, the phases will contain the same errors. Since phases make up at least 50% of the information which is used to calculate the electron-density maps, wrong features may still have reasonable electron density because of these phase errors. 

Calculation of an electron density map. Several heavy-atom derivatives are prepared and isomorphous replacement has been used to estimate phases for aU Fp(h, k, 1). These phases are used to calculate an electron density map of the crystal and aU data to a certain resolution. 

Patterson methods are specially oriented to solving structures containing heavy atoms, where it becomes easy to derive the projected positions of the heavy atoms, and their coordinates represent a good initial model for solving the structure. The calculated phases, are a good approximation of the true phase for calculating an electron density map by means of eqn [11], using the observed amplitudes, Po hkI), and the calculated phases to find new atomic positions and proceed by successive Fourier cycles vmtil the structure is completed. 

In case of the C0I2 phase, it was not possible to calculate an electron density map. This is due to the insufficient resolution of the X-ray dilfiaction detector which does not allow a sufficient separation of the individual reflexes. Nonetheless, the C0I2 phase is expected to look similar to the Coli phase except for the shape of the columns which should be cireular instead of oval, as its lattice parameters a and b have the same values (cf. Fig. 5.18d). 

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