Isotropic refinement

For an isotropic refinement, four parameters per atom (three for the coordinates and one for the B factor) thus have to be optimized during the refinement process. If anisotropic B factors and occupancies are used, the number of parameters increases almost three fold and therefore overfltting can become an important problem. One way to test for overfitting is the above mentioned cross-validation using Bfree. 

Fig. 10.1 Schematic views of different parameterizations. Isotropic refinement involves 3 parameters for the coordinates and 1 isotropic B-value. In anisotropic refinement, 6 parameters are used to model the probability function for the location of an atom. When an atom needs to be modeled using more than one site, the number of parameters increases accordingly. However, to define the occupancy of two sites only one extra parameter is needed, as the occupancy p2 of the second site can be calculated from the occupancy of the first site,pi, by/ 2 = 1.0—pi.

When ADP s are introduced into the refinement, the value of / free should be monitored closely. If the drop in / free is less than 1-1.5%, it is better to revert to an isotropic refinement. As an intermediate solution, one can make only the heavy atoms in the stmcture anisotropic (e.g. sulfurs and/or metals) by applying the ANIS instruction only to the respective atoms (see example in Chapter 11 of the SHELXL manual). 

The structure was refined with the program RESTRAIN (35) using all data between 12.0 and 1.72 A. In the final cycles of refinement all the protein and solvent atoms were allowed to refine isotropically, but... 

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 structure was refined with block diagonal least squares. In cases of pseudo-symmetry, least squares refinement is usually troublesome due to the high correlations between atoms related by false symmetry operations. Because of the poor quality of the data, only those reflections not suffering from the effects of decomposition were used in the refinement. With all non-hydrogen atoms refined with isotropic thermal parameters and hydrogen atoms included at fixed positions, the final R and R values were 0.142 and 0.190, respectively. Refinement with anisotropic thermal parameters resulted in slightly more attractive R values, but the much lower data to parameter ratio did not justify it. 

An isotropic extinction parameter, of type I and Lorentzian distribution (in the formalism of Becker and Coppens [16]), was also refined. The motions of the non-H atoms were described by anisotropic parameters, while those of the H atoms by isotropic B s. All these displacement parameters were included among the refinable quantities of the model, for a total of 1161 variables in a single least-squares matrix. 

With data averaged in point group m, the first refinements were carried out to estimate the atomic coordinates and anisotropic thermal motion parameters IP s. We have started with the atomic coordinates and equivalent isotropic thermal parameters of Joswig et al. [14] determined by neutron diffraction at room temperature. The high order X-ray data (0.9 < s < 1.28A-1) were used in this case in order not to alter these parameters by the valence electron density contributing to low order structure factors. Hydrogen atoms of the water molecules were refined isotropically with all data and the distance O-H were kept fixed at 0.95 A until the end of the multipolar refinement. The inspection of the residual Fourier maps has revealed anharmonic thermal motion features around the Ca2+ cation. Therefore, the coefficients up to order 6 of the Gram-Charlier expansion [15] were refined for the calcium cation in the scolecite. 

Crystal data and parameters of the data collection (at -173°, 50 < 20 < 450) are shown in Table I. A data set collected on a parallelopiped of dimensions 0.09 x 0.18 x 0.55 mm yielded the molecular structure with little difficulty using direct methods and Fourier techniques. Full matrix refinement using isotropic thermal parameters converged to R = 0.I7. Attempts to use anisotropic thermal parameters, both with and without an absorption correction, yielded non-positive-definite thermal parameters for over half of the atoms and the residual remained at ca. 0.15. 

The isotropic thermal parameter listed for those atoms refined anisotropically is the isotropic equivalent. 

Refinement takes place by adjusting the model to find closer agreement between the calculated and observed structure factors. For proteins the refinements can yield R-factors in the range of 10-20%. An example taken from reference 10 is instructive. In a refinement of a papain crystal at 1.65-A resolution, 25,000 independent X-ray reflections were measured. Parameters to be refined were the positional parameters (x, y, and z) and one isotropic temperature factor parameter... 

Reflection intensity in the SAED negatives was measured with a microdensitometer. The refinement of the structure analysis was performed by the least square method over the intensity data (25 reflections) thus obtained. A PPX single-crystal is a mosaic crystal which gives an "N-pattem". Therefore we used the 1/d hko as the Lorentz correction factor [28], where d hko is the (hkO) spacing of the crystal. In this case, the reliability factor R was 31%, and the isotropic temperature factor B was 0.076nm. The molecular conformation of the P-form took after that of the P-form since R was minimized with this conformation benzene rings are perpendicular to the trans-zigzag plane of -CH2-CH2-. 

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