Electron density map, from X-ray

Computation of the electron density map from X-ray diffraction data requires knowledge of the intensities and the phase angle of each measured reflection. The lack of phase angle information in recorded diffraction... 

Details of the ferrous coordination environments are available only for deoxyHr. A difference electron density map from X-ray diffraction studies of deoxyHr and metHr suggests only small structural differences between the two forms (19). Although the crystal structure of deoxyHr is of low resolution (3.9 A), there appears to be a decrease in the correlated motions of the two iron atoms suggesting a weakening of the Fe-p.-0 bonds and an increase in the Fe-Fe distance. These changes in core dimensions are confirmed by an EXAFS analysis of deoxyHr (rpe-o 1.98 A, rpe-Fe = 3.57 A) (39) they suggest that the oxo bridge has become protonated in deoxyHr, in line with the diminished Lewis acidity of Fe(II). 

The lack of electronic interference allows for a simple density map. An electron-density map from X-ray diffraction data includes not only the structural disorders but also the electron clouds. In contrast, the nuclear density map from neutron diffraction data does not include the electron clouds. 

The Fourier theorem states that any periodic function may be resolved into cosine and sine terms involving known constants. Since a crystal has a periodically repeating internal structure, this can be represented, in a mathematically useful way, by a three-dimensional Fourier series, to give a three-dimensional Fourier or electron density map. In X-ray diffraction studies the magnitudes of the coefficients may be derived from... 

Fig. 1. Model for the position of sialic acid in the binding pocket of influenza virus hemagglutinin. This model for the best fit has been deduced from the difference electron density maps of X-ray crystallographic studies. Some of the hydrogen bonds proposed in this model are shown by dashed lines. (Taken from ref. [31] with permission from the authors.)...

Once a suitable crystal is obtained and the X-ray diffraction data are collected, the calculation of the electron density map from the data has to overcome a hurdle inherent to X-ray analysis. The X-rays scattered by the electrons in the protein crystal are defined by their amplitudes and phases, but only the amplitude can be calculated from the intensity of the diffraction spot. Different methods have been developed in order to obtain the phase information. Two approaches, commonly applied in protein crystallography, should be mentioned here. In case the structure of a homologous protein or of a major component in a protein complex is already known, the phases can be obtained by molecular replacement. The other possibility requires further experimentation, since crystals and diffraction data of heavy atom derivatives of the native crystals are also needed. Heavy atoms may be introduced by covalent attachment to cystein residues of the protein prior to crystallization, by soaking of heavy metal salts into the crystal, or by incorporation of heavy atoms in amino acids (e.g., Se-methionine) prior to bacterial synthesis of the recombinant protein. Determination of the phases corresponding to the strongly scattering heavy atoms allows successive determination of all phases. This method is called isomorphous replacement. 

In the following sections we shall focus on the structure and properties of the two-dimensional phases formed by the bent-core liquid crystals. In Sect. 2 we describe the structure studies by the X-ray diffraction (XRD) method, optical studies, and the response of different structures to the external electric field. In Sect. 3 we give theoretical models of the director and layer structure in 2D modulated phases and discuss how to reconstruct electron density maps from XRD data. 

Lastly, it should be pointed out that there are often significant differences between H positions determined by X-ray analysis and those determined from neutron data. Neutron diffraction provides true nuclear positions, whereas X-ray diffraction measures the electron density distribution. Thus, X-ray Fourier maps often give H peaks that, because of the perturbing influence of the M-H bonding electrons, appear closer to the M atoms than they really are A thorough analysis of this effect has... 

The essentials of the method can be stated very simply, the scattering of X-rays by a crystal yields an electron density map from which the positions of individual atoms in a molecule can be found. How this is achieved is summarised in what follows detailed coverage can be found in Rhodes (1993) and Drenth (1999). 

Hajdu, J., Machin, P. A., Campbell, J. W., Greenhough, T. J., Clifton, I. J.. Zurek, S., Cover, S., Johnson, L. N. and Elder, M. Millisecond X-ray diffraction and the first electron density maps from Lane photographs of a protein crystal. Nature London) 329, 178-181 (1987). 

Equation 2.133 is most commonly used to calculate the distributions of the electron (nuclear) density in the unit cell, which are also known as Fourier maps, from x-ray (neutron) diffraction data, respectively. The locations of peaks on the Fourier map calculated using x-ray diffraction data represent coordinates of atoms, while the electron density integrated over the range of the peak corresponds to the number of electrons in the atom. The major problem in using Eq. 2.133 is that only the absolute values of the... 

FIGURE 30.11 The phase problem. The experimental data obtained in an X-ray experiment are the intensities of the reflections. By using an inverse Fourier transform, it is possible to calculate electron-density maps from these intensities. However, it is essential for this calculation to know the phase associated with each reflection. Approximate initial phases can be obtained from heavy-atom derivatives, anomalous dispersion or molecular replacement (see text). More accurate phases can be derived from the refined model, once it has been obtained. 

Hajdu. J. Machin. P.A. Campbell, J.W., et al. Millisecond X-ray-diffraction and the 1st electron-density map from Laue photographs of a protein crystal. Nature 1987. 329. 

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. 

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. 

Since the phase angles cannot be measured in X-ray experiments, structure solution usually involves an iterative process, in which starting from a rough estimate of the phases, the structure suggested by the electron density map obtained from Eq. (13-3) and the phase computed by Eq. (13-1) are gradually refined, until the computed structure factor amplitudes from Eq. (13-1) converge to the ones observed experimentally. 

The result is the electron density map of the protein crystal. The final task for the crystallographer is to build the appropriate protein model, i. e., putting amino acid for amino acid into the electron density. Routinely the theoretical amplitudes and phases are calculated from the model and compared to the experimental data in order to check the correctness of model building. The positions of the protein backbone and the amino acid side chains are well defined by X-ray structures at a... 

The intrazeolite cations necessary to balance the negative charge on the framework aluminum atoms are poorly shielded and as a result high electric (electrostatic) fields on the order of 1-10 V/nm are found in their vicinity. The magnitudes of the electric fields can be calculated from measured effects on the vibrational frequencies or intensities of IR bands of small diatomics such as CO or N2.24 They can also be determined from difference electron density maps determined by X-ray diffraction methods.25 These high electric fields can dramatically influence the stabilities of transition states with significant charge separations. 

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