Structural model, least square refinement

Electron density maps, calculated with the program PROTEIN (9), were interpreted using FRODO (10). The model xmderwent multiple cycles of restrained structure-factor least-squares refinement using PROLSQ (11). In addition to protein atoms and water molecules, the final models also include a well-ordered HEPES molecule that cocrystallized with the native protein. At least 60 water molecules were added to each structure during refinement. 

This final section of this chapter is devoted to the interplay between theory and the model-driven experimental techniques, such as diffraction methods. Gas-phase electron diffraction and powder diffraction methods are more heavily reliant on direct input from simulation, for example in providing starting structures for least-squares refinement analyses (Section 2.11.3), whereas in the spectroscopic techniques discussed above the need is more for validation and assignment. 

Traditionally, least-squares methods have been used to refine protein crystal structures. In this method, a set of simultaneous equations is set up whose solutions correspond to a minimum of the R factor with respect to each of the atomic coordinates. Least-squares refinement requires an N x N matrix to be inverted, where N is the number of parameters. It is usually necessary to examine an evolving model visually every few cycles of the refinement to check that the structure looks reasonable. During visual examination it may be necessary to alter a model to give a better fit to the electron density and prevent the refinement falling into an incorrect local minimum. X-ray refinement is time consuming, requires substantial human involvement and is a skill which usually takes several years to acquire. 

A preliminary least-squares refinement with the conventional, spherical-atom model indicated no disorder in the low-temperature structure, unlike what had been observed in a previous room-temperature study [4], which showed disorder in the butylic chain at Cl. The intensities were then analysed with various multipole models [12], using the VALRAY [13] set of programs, modified to allow the treatment of a structure as large as LR-B/081 the original maximum number of atoms and variables have been increased from 50 to 70 and from 349 to 1200, respectively. The final multipole model adopted to analyse the X-ray diffraction data is described here. 

The modeling of electron diffraction by the pattern decomposition method, for which no structural information is required, can be successfully applied for extraction of the diffraction information from the pattern. Several parameters can be refined during the procedure of decomposition, including the tilt angle of the specimen the unit cell parameters peak-shape parameters intensities. The procedure consists of fitting, usually with a least-squares refinement, a calculated model to the whole observed diffraction pattern. 

When the model used for Fcalc is that obtained by least-squares refinement of the observed structure factors, and the phases of Fca,c are assigned to the observations, the map obtained with Eq. (5.9) is referred to as a residual density map. The residual density is a much-used tool in structure analysis. Its features are a measure for the shortcomings of the least-squares minimization, and the functions which constitute the least-squares model for the scattering density. 

As discussed in the previous section, a residual density calculated after least-squares refinement will have minimal features. This is confirmed by experience (Dawson 1964, O Connell et al. 1966, Ruysink and Vos 1974). Least-biased structural parameters are needed if the adequacy of a charge density model is to be investigated. Such parameters can be obtained by neutron diffraction, from high-order X-ray data, or by using the modified scattering models discussed in chapter 3. 

As anticipated, lower temperature increases the number of observations from an X-ray diffraction data collection (at constant radiation dose). This is however just one of the advantages that could improve a structure solution or a refinement. In fact, a reduced thermal motion usually implies a more reliable standard model, given that for smaller atomic displacements the harmonic approximation is more appropriate and less correlation is found between variables within a least squares refinement. This returns higher precision of the parameters calculated from those variables (for example bond distances, bond angles, etc.). 

Full-matrix least-squares refinement of the structure model was carried out with programs orfls (9). Since this program in the form we used refines structure factors Fhki rather than the intensities of the powder lines, it was necessary to decompose the intensity of each line having more than one component into contributions from the individual component reflections. This was done by assuming that the F°mz 2 for the several components of a powder line were in the same ratios as the corresponding FcftfcZ 2 obtained from a structure model—the original trial structure or the previous... 

Once the phase problem is solved, then the positions of the atoms may he relined by successive structure-factor calculations (Eq. 21 and Fourier summations (Eq. 3) or by a nonlinear least-squares procedure in which one minimizes, for example, )T u ( F , - F,il(, )- with weights w lakcn in a manner appropriate to the experiment. Such a least-squares refinement procedure presupposes that a suitable calculalional model is known. 

The model can be improved in another way by least-squares refinement of the atomic coordinates. This method entails adjusting the atomic coordinates to improve the agreement between amplitudes calculated from the current model and the original measured amplitudes in the native data set. In the latter stages of structure determination, the crystallographer alternates between map interpretation and least-squares refinement. 

By isostructural substitution, using the Er3+—Y3+ pair, the structures of chloride and nitrate complexes of Er3+ in DMSO solution have been determined and can be compared with the structures of corresponding complexes in aqueous solution (37). The deconvoluted RDFs for 1 M erbium(III) nitrate and chloride solutions, calculated from difference curves, are shown in Fig. 27. The structures for the complexes were derived from these RDFs and the final parameter values were obtained by least-squares refinements using the intensity difference curves. The bonding of the ligands and a comparison between experimental values and values calculated for the derived models are shown in Fig. 27. 

The x, y, and z for the TPAOH complex were chosen from the structural model and not refined by least squares ... 

The electron density distribution for solvent molecules can be improved if the contribution from bulk water to the X-ray scattering is included in the model. This affects the low-angle j X-ray intensity data which are omitted in early stages of the least-squares refinement of protein crystal structures. If they are included in refinement and properly accounted for, the signal-to-noise ratio in the electron density maps is significantly improved and the interpretation of solvent sites is less ambiguous. 

Other well-established methods of analysis include modeling the experimental data using geometrical solid-body representations of the scattering species. This method allows for the construction of many kinds of geometrical models (e.g., hollow shells, core-shell particles, lamellar structures, etc. and provides for the interactive least-squares refinement of the dimensions of the models to fit the data. This approach has been widely used to analyze SANS data obtained from colloidal and polymeric systems. ... 

The least-squares method is a very powerful tool, provided the model is sufficiently close to the true structure. If the initial model is basically correct, the shifts in parameters indicated by the least-squares refinement will drive the energy of the structure to a global minimum. Unfortunately, if the model is not quite correct, the least-squares refinement will produce a structure that is trapped in a local minimum of energy which is not the true structure (see Figure 10.12). This problem manifests itself by monitoring the R value. Often there is a hint of trouble in that the R value is higher than expected and will not decrease to acceptable values. Several crystal structures have been reported with this type of problem, but they are generally corrected in the subsequent literature. 

This process may be repeated as many times as needed until all atoms in the unit cell are located and the following Fourier map(s) do not improve the model. Equations 2.132 to 2.134 may be combined with a least squares refinement using the observed data, which results in a more accurate model of the crystal structure, including positional and displacement parameters of the individual atoms already included in the model. The success in the solution of the crystal structure is critically dependent on both the accuracy of the initial model (initial set of phase angles) and the accuracy of the experimental structure amplitudes. Needless to say, when the precision of the latter is low, then the initial model should be more detailed and precise. 

The application of the Patterson technique to locate strongly scattering atoms is often called the heavy atom method (which comes from the fact that heavy atoms scatter x-rays better and the Patterson technique is most often applied to analyze x-ray diffraction data). This allows constructing of a partial structure model ( heavy atoms only), which for the most part define phase angles of all reflections (see Eq. 2.107). The heavy atoms-only model can be relatively easily completed using sequential Fourier syntheses (either or both standard, Eq. 2.133, and difference, Eq. 2.135), sometimes enhanced by a least squares refinement of all found atoms. 

Figure 7.39. Relative shifts of individual atoms (left-hand scale) and average shift (A) to standard deviation (cr) ratio during the first 12 cycles of the least squares refinement of the coordinates of all atoms in the model of the crystal structure of the anhydrous monoclinic FeP04. Both the P-0 distances and O-P-0 angles were soft restrained with the weight of 4 during the first five cycles. The weight was set to 10 beginning from cycle No. 6. The first five cycles indicate erratic shifts of P and O atoms. Beginning from the sixth cycle, the shifts of all atoms steadily decrease and approach zero after cycle No. 12.

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