ELECTRON DENSITY, REFINEMENT, AND DIFFERENCE FOURIER MAPS

ELECTRON DENSITY, REFINEMENT, AND DIFFERENCE FOURIER MAPS 

Prior to the advent of computer graphics systems, electron density maps were, in actuality, produced as described above. Each section was printed on paper as a field of numbers, each 

Introduction to Macromolecular Crystallography, Second Edition By Alexander McPherson Copyright 2009 John Wiley Sons, Inc. 

FIGURE 10.1 An electron density map section from a yeast phenylalanine tRNA crystal showing contour lines drawn about areas of high electron density. In the map can be seen what appear to be broken lengths of continuous chains of density. When electron density sections above and below are included, the chains assume greater continuity. 

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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. 

For trypanosomal TIM we experimented with three different cocktails of 32 compounds (Table 4). Molecules were chosen in such a way that they would be compatible, soluble, cheap, and as varied as possible. Each compound was present at a concentration of 1 m M The final cocktail solutions were clear and devoid of precipitate. Since this was a pilot experiment both subcocktails were checked at each stage of the dichotomic strategy. Only the soak with cocktail 1 revealed electron density that could not be accounted for by water molecules, hereafter called peak X. The soaks with cocktails 2 and 3 led to featureless difference Fourier maps. The quality of the data and refinement can be inspected from Table 5, while Figure 9 illustrates the dichotomic search to identify peak X. An oxidized molecule of DTT, identified in the high-resolution structure of the native TIM crystals [24], served as an internal reference to judge the quality of the data and the noise level in the final difference Fourier maps. 

In the X-ray analysis of a protein crystal structure, solvent molecules appear as spheres of electron density in difference Fourier maps calculated at the end of a refinement. In a strict sense, the electron density map exhibits preferred.s/tes of hydration which are occupied by freely interchanging solvent molecules. This electron density is well defined for the tightly bound solvent molecules and can be as spurious as just above background for ill-defined molecules which exhibit large temperature factors and/or only partly occupied atomic positions. Since these two parameters are correlated in least-squares refinement, this gives rise to methodological problems. 

The average remaining electron density in ZnAPO-34 sample was close to zero, indicating that zinc atoms were incorporated into the framework. The difference Fourier map of as-synthesised MnZnAPO-34 showed some positive diffuse electron density remaining in the cavity, that prevented the determination of the position of a manganese atom, which is probably situated at several positions in the cavity 7 and is most likely responsible for the observed electron density. Also in the case of calcined sample, where the cavities were free of template, manganese could not be located from the positive electron density in the cavities due to low crystallinity of the sample which caused low resolution of the Fourier maps. Low crystallinity was also the reason for non-stable refinement of calcined samples and that is only unit cell parameters are given for comparison (Table 1). Unit cells of calcined samples are smaller compared to as-synthesised ones, as expected. 

A difference Fourier map, calculated at this point, reveals an additional small electron density maximum in the tetrahedral cavity next to the partially occupied V2. Thus, it is reasonable to assume that the V2 site splits into two independent partially occupied positions with the coordinates, which distribute V atoms in a random fashion in two adjacent tetrahedral positions rather than being simply vanadium-deficient. We label these two sites as V2a (corresponding to the former V2) and V2b (corresponding to the Fourier peak). Refinement of this model slightly improves the fit. Subsequently, additional profile parameters (F, F , and sample displacement) were included in the refinement, followed by a typical procedure of refining the porosity in the Suortti approximation with fixed atomic coordinates and Ui o, and then fixing the porosity parameters for the remainder of the refinement. 

Simulated annealing refinement is usually unable to correct very large errors in the atomic model or to correct for missing parts of the structure. The atomic model needs to be corrected by inspection of a difference Fourier map. In order to improve the quality and resolution of the difference map, the observed phases are often replaced or combined with calculated phases, as soon as an initial atomic model has been built. These combined electron density maps are then used to improve and to refine the atomic model. The inclusion of calculated phase information brings with it the danger of biasing the refinement process towards the current atomic model. This model bias can obscure the detection of errors in atomic models if sufficient experimental phase information is unavailable. In fact during the past decade several cases of incorrect or partly incorrect atomic models have been reported where model bias may have played a role [67]. 

The difference Fourier technique is also very useful in refinement of protein structures. An analysis of the errors and their treatment is discussed by Henderson and Moffat, who find that a difference Fourier map is able to detect much smaller features of electron density than those revealed by a normal Fourier map with the same phases. 

The difference electron density distribution also finds an interesting application when the fully refined model of the crystal structure is used to compute a Fourier transformation. Although it may seem that such a Fourier map should result in zero electron density throughout the unit cell (since the differences in Eq. 2.135 are expected to approach zero), this is true only if electron shells of atoms in the crystal structure were not deformed. In reality, atoms do interact and form chemical bonds with their neighbors. This causes a redistribution of the electron density when compared to isolated atoms, for which atomic scattering functions are known. 

Refinement also may be accomplished in real space by calculating difference Fourier syntheses using as coefficients A F = F i,s — Fcaic and phases caic derived from the trial structure. In general, if an atom has been incorrectly placed near its true location, a negative peak will appear at its assigned position and a positive peak will appear at its proper location. That is, in the difference Fourier we have subtracted electron density from where the atom isn t, and not subtracted density from where it is. The atom is then shifted by altering x, y, z, improved atomic parameters are included in a new round of structure factor calculations, and another difference Fourier computed. The process is repeated until the map is devoid of significant features. 

When the application of Eq. (11) to a least squares analysis of x-ray structure factors has been completed, it is usual to calculate a Fourier synthesis of the difference between observed and calculated structure factors. The map is constructed by computation of Eq. (9), but now IFhid I is replaced by Fhki - F/f /, where the phase of the calculated structure factor is assumed in the observed structure factor. In this case the series termination error is virtually too small to be observed. If the experimental errors are small and atomic parameters are accurate, the residual density map is a molecular bond density convoluted onto the motion of the nuclear frame. A molecular bond density is the difference between the true electron density and that of the isolated Hartree-Fock atoms placed at the mean nuclear positions. An extensive study of such residual density maps was reported in 1966.7 From published crystallographic data of that period, the authors showed that peaking of electron density in the aromatic C-C bonds of five organic molecular crystals was systematic. The random error in the electron density maps was reduced by averaging over chemically equivalent bonds. The atomic parameters from the model Eq. (11), however, will refine by least squares to minimize residual densities in the unit cell. 

The three crystal structures were solved by a combination of direct methods (MULTAN80) and fourier techniques. The amide hydrogen atoms were found directly from electron density difference maps at an intermediate stage of the refinement the remaining hydrogens were introduced in calculated positions. The hydrogen atoms were... 

The procedure is called hybrid electronic maps (or the over-fit ) method. Figure 5.23, and can be obtained, for example, by non-refining (i.e., non-minimizing) completely the D factor of (5.60), so generating a structure such as Eq. (5.61), yet unrefined, therefore being called as the structure by omission, this is compared (by superimposing) with the density difference (or synthesis of the Fourier maps differences)... 

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