Electron density modeling Crystallographic

The comparison with experiment can be made at several levels. The first, and most common, is in the comparison of derived quantities that are not directly measurable, for example, a set of average crystal coordinates or a diffusion constant. A comparison at this level is convenient in that the quantities involved describe directly the structure and dynamics of the system. However, the obtainment of these quantities, from experiment and/or simulation, may require approximation and model-dependent data analysis. For example, to obtain experimentally a set of average crystallographic coordinates, a physical model to interpret an electron density map must be imposed. To avoid these problems the comparison can be made at the level of the measured quantities themselves, such as diffraction intensities or dynamic structure factors. A comparison at this level still involves some approximation. For example, background corrections have to made in the experimental data reduction. However, fewer approximations are necessary for the structure and dynamics of the sample itself, and comparison with experiment is normally more direct. This approach requires a little more work on the part of the computer simulation team, because methods for calculating experimental intensities from simulation configurations must be developed. The comparisons made here are of experimentally measurable quantities. 

The structure was refined by block-diagonal least squares in which carbon and oxygen atoms were modeled with isotropic and then anisotropic thermal parameters. Although many of the hydrogen atom positions were available from difference electron density maps, they were all placed in ideal locations. Final refinement with all hydrogen atoms fixed converged at crystallographic residuals of R=0.061 and R =0.075. 

The X-ray structure of the dibromine complex with toluene (measured at 123 K) is more complicated, and shows multiple crystallographically independent donor/acceptor moieties [68]. Most important, however, is the fact that in all cases the acceptor shows an over-the-rim location that is similar to that in the benzene complex. In both systems, the acceptor is shifted by 1.4 A from the main symmetry axis, the shortest Br C distances of 3.1 A being significantly less than the sum of the van der Waals radii of 3.55 A [20]. Furthermore, the calculated hapticity in the benzene/Br2 complex (x] = 1.52) is midway between the over-atom (rj = 1.0) and over-bond (rj = 2.0) coordination. In the toluene complex, the latter varies from rj = 1.70 to 1.86 (in four non-equivalent coordination modes) and thus lies closer to the over-bond coordination model. Moreover, the over-bond bromine is remarkably shifted toward the ortho- and para-carbons that correspond to the positions of highest electron density (and lead to the transition states for electrophilic aromatic bromination [12]). Such an experimental location of bromine is in good agreement with the results of high level theoretical... 

This was averaged over the total distribution of ionic and dipolar spheres in the solution phase. Parameters in the calculations were chosen to simulate the Hg/DMSO and Ga/DMSO interfaces, since the mean-spherical approximation, used for the charge and dipole distributions in the solution, is not suited to describe hydrogen-bonded solvents. Some parameters still had to be chosen arbitrarily. It was found that the calculated capacitance depended crucially on d, the metal-solution distance. However, the capacitance was always greater for Ga than for Hg, partly because of the different electron densities on the two metals and partly because d depends on the crystallographic radius. The importance of d is specific to these models, because the solution is supposed (perhaps incorrectly see above) to begin at some distance away from the jellium edge. 

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... 

Computational models, which calculate the optimized parameters for structures based on the crystallographic and spectroscopic information. These can be used to determine the likely patterns of electron density and protonation for the complexes. 

The crystallographic data contain all the information about the data that were used for determining the model. The most important information is the resolution. This refers to the minimum d spacing (O Eq. 22.1) and indicates the smallest distance between two atoms that can be resolved, i.e., completely separated based on electron density. The table also contains space group (P2i2i2i) and unit-cell information along with the statistical measurements for the reflection data. 

Let consider now the ESP-distribution along main crystallographic directions, reconstructed by the K-model. The ESP and electron density for LiF as characteristic ones are shown in Fig.7. 

Below we briefly describe the crystallographic software pipelines using AutoRickshaw as an example, with its flexibility and the ability to decide on the path to be taken dependent on the outcome of a previous step. On one hand, AutoRickshaw has features and general steps, which are also shared by many other pipelines. On the other hand, AutoRickshaw is perhaps the first software pipeline which aims not at the delivery of a fully built, refined, and validated model but rather at fast evaluation of the quality of the X-ray data in terms of interpretability of the obtained electron density map. 

Using time-resolved crystallographic experiments, molecular structure is eventually linked to kinetics in an elegant fashion. The experiments are of the pump-probe type. Preferentially, the reaction is initiated by an intense laser flash impinging on the crystal and the structure is probed a time delay. At, later by the x-ray pulse. Time-dependent data sets need to be measured at increasing time delays to probe the entire reaction. A time series of structure factor amplitudes, IF, , is obtained, where the measured amplitudes correspond to a vectorial sum of structure factors of all intermediate states, with time-dependent fractional occupancies of these states as coefficients in the summation. Difference electron densities are typically obtained from the time series of structure factor amplitudes using the difference Fourier approximation (Henderson and Moffatt 1971). Difference maps are correct representations of the electron density distribution. The linear relation to concentration of states is restored in these maps. To calculate difference maps, a data set is also collected in the dark as a reference. Structure factor amplitudes from the dark data set, IFqI, are subtracted from those of the time-dependent data sets, IF,I, to get difference structure factor amplitudes, AF,. Using phases from the known, precise reference model (i.e., the structure in the absence of the photoreaction, which may be determined from... 

Equation (5.15) describes one structure factor in terms of diffractive contributions from all atoms in the unit cell. Equation (5.16) describes one structure factor in terms of diffractive contributions from all volume elements of electron density in the unit cell. These equations suggest that we can calculate all of the structure factors either from an atomic model of the protein or from an electron density function. In short, if we know the structure, we can calculate the diffraction pattern, including the phases of all reflections. This computation, of course, appears to go in just the opposite direction that the crystallographer desires. It turns out, however, that computing structure factors from a model of the unit cell (back-transforming the model) is an essential part of crystallography, for several reasons. 

With each successive map, new molecular features are added as they can be discerned, and errors in the model, such as side-chain conformations that no longer fit the electron density, are corrected. As the structure nears completion, the crystallographer may use 2F0 Fc and FQ— Fc maps simultaneously to track down the most subtle disagreements between the model and the data. 

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