Crystallographic structure refinement methods

Transferred electron density fragments obtained by AFDF method can provide excellent approximations. One such approach, formulated in terms of transferability of fragment density matrices within the AFDF framework is a tool that has been suggested as an approach to macromolecular quantum chemistry [114, 115, 130, 142-146] and to a new density fitting algorithm in the crystallographic structure refinement process [161]. 

Crystallographic structure refinement is generally understood to be the last step in the determination of a crystal structure by diffraction methods. The usual procedure of a crystal structure analysis includes collection of X-ray or neutron diffraction intensities, data reduction yielding structure factor amplitudes, the solution of the crystallographic phase problem yielding approximate structural parameters and finally refinement of these parameters to obtain a best fit of the observed structure factor amplitudes with... 

The number of reflection intensities measured in a crystallographic experiment is large, and commonly exceeds the number of parameters to be determined. It was first realized by Hughes (1941) that such an overdetermination is ideally suited for the application of the least-squares methods of Gauss (see, e.g., Whittaker and Robinson 1967), in which an error function S, defined as the sum of the squares of discrepancies between observation and calculation, is minimized by adjustment of the parameters of the observational equations. As least-squares methods are computationally convenient, they have largely replaced Fourier techniques in crystal structure refinement. 

Data analysis procedures have developed substantially over the last few years. In particular, use of least square refinement methods have been developed. Recent progress with theoretical development for the treatment of multiple scattering has resulted in Ugand group refinement such as an imidazole. We can expect further development in this area which ought to lead us to restrained least square refinement procedures for EXAFS data analysis. This type of restrained refinement is commonly used for macromolecular crystallographic structure determination where a similar problem of imderdeterminancy exists... 

Distance least squares (DLS), a method developed by Meier and Vill-iger (1) for generating model structures (DLS models) of prescribed symmetry and optimum interatomic distances, can supply atomic coordinates which closely approach the values obtained by extensive structure refinement. DLS makes use of the available information on interatomic distances, bond angles, and other geometric features. It is primarily based on the fact that the number of crystallographically non-equivalent interatomic distances exceeds the number of coordinates in framework-type structures. A general DLS program is available (8) which allows any combination of prescribed parameters (interatomic distances, ratios of distances, unit cell constants etc). In addition, subsidiary conditions (as discussed in Refs. 1 and 8) can also be prescribed. 

Two types of information are obtained from any molecular mechanics study, the minimum value of the strain energy and the structure associated with that minimum. Agreement between the energy-minimized and experimental (crystallographic) structures has often been used as the primary check on the validity of the force field and to refine the force field further, but often little predictive use has been made of the structures obtained. As force fields become more reliable, the potential value of structure predictions increases. More importantly, when no unequivocal determination of a structure is available by experimental methods then structure prediction may be the only means of obtaining a three-dimensional model of the molecule. This is often the case, for instance, in metal-macromolecule adducts, and structures obtained by molecular mechanics can be a genuine aid in the visualization of these interactions. In this chapter we consider the ways in which structure prediction by molecular mechanics calcluations has been used, and point to future directions. 

Toraya s WPPD approach is quite similar to the Rietveld method it requires knowledge of the chemical composition of the individual phases (mass absorption coefficients of phases of the sample), and their unit cell parameters from indexing. The benefit of this method is that it does not require the structural model required by the Rietveld method. Furthermore, if the quality of the crystallographic structure is poor and contains disordered pharmaceutical or poorly refined solvent molecules, quantification by the WPPD approach will be unbiased by an inadequate structural model, in contrast to the Rietveld method. If an appropriate internal standard of known quantity is introduced to the sample, the method can be applied to determine the amorphous phase composition as well as the crystalline components.9 The Rietveld method uses structural-based parameters such as atomic coordinates and atomic site occupancies are required for the calculation of the structure factor, in addition to the parameters refined by the WPPD method of Toraya. The additional complexity of the Rietveld method affords a greater amount of information to be extracted from the data set, due to the increased number of refinable parameters. Furthermore, the method is commonly referred to as a standardless method, since the structural model serves the role of a standard crystalline phase. It is generally best to minimize the effect of preferred orientation through sample preparation. In certain instances models of its influence on the powder pattern can be used to improve the refinement.12... 

Three main tendencies have been underlined in recent studies of structure and action mechanism ofbacterial photosynthetic reaction centers. The crystallographic structure of the reaction centers from Rps. viridis and Rb. spheroids was initially determined to be 2.8 and 3 A resolutions (Michel and Deisenhofer et al., 1985 Allen et al., 1986). Resolution and refinement of these structures have been subsequently extended to 2.2, 2.3 and 2.6 A. (Rees et al., 1989 Stowell et al., 1997, Fyfe and Johns, 2000 Rutherford and Faller, 2001). Investigations of the electronic structure of donor and acceptor centers in the ground and exited states by modern physical methods with a combination ofpico-and femtosecond kinetic techniques have become more precise and elaborate. Extensive experimental and theoretical investigations on the role of orbital overlap and protein dynamics in the processes of electron and proton transfer have been done. All the above-mentioned research directions are accompanied by extensive use of methods of sit-directed mutagenesis and substitution of native pigments for artificial compounds of different redox potential. 

An additional major problem with natural membrane systems is that the composition of the one-dimensional unit cell may be unknown because of the presence of proteins of unknown structure and composition. Nevertheless, it is possible to model the electron density profile of a multilamellar system to arrive at a crude model for the distribution of lipids and proteins. The models improve considerably if the lipid composition and lipid stmctures are known. In any case, the basic crystallographic structural method is used The model is refined on the basis of comparisons between the calculated and the experimentally determined intensities through an iterative process. 

The linear size dependence of the AFDF family of methods is an advantage that can be exploited in the rapid generation of macromolecular electron density representations which are of better quality than those obtained by locally spherical distributions in the usual structure refinement process. The proposed approach, a quantum crystallographic application of the AFDF method (QCR-AFDF) is outlined below. 

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