Atomic coordinates, precision

When specifying atomic coordinates, interatomic distances etc., the corresponding standard deviations should also be given, which serve to express the precision of their experimental determination. The commonly used notation, such as d = 235.1(4) pm states a standard deviation of 4 units for the last digit, i.e. the standard deviation in this case amounts to 0.4 pm. Standard deviation is a term in statistics. When a standard deviation a is linked to some value, the probability of the true value being within the limits 0 of the stated value is 68.3 %. The probability of being within 2cj is 95.4 %, and within 3ct is 99.7 %. The standard deviation gives no reliable information about the trueness of a value, because it only takes into account statistical errors, and not systematic errors. 

The way to start a CFF parametrization is Select a set of PEFs, with associated parameters. Choose a set of molecules, closely related to the problem in hand (for carbohydrates alkanes, cycloalkanes, ethers, alcohols) their structures should be determined and their vibrational spectra assigned to a reasonable precision. Put in their structures by specifying atomic coordinates they need not be accurate. 

It is usually believed that NO inhibits enzymes by reacting with heme or nonheme iron or copper or via the S-nitrosilation or oxidation of sulfhydryl groups, although precise mechanisms are not always evident. By the use of ESR spectroscopy, Ichimori et al. [76] has showed that NO reacts with the sulfur atom coordinated to the xanthine oxidase molybdenum center, converting xanthine oxidase into a desulfo-type enzyme. Similarly, Sommer et al. [79] proposed that nitric oxide and superoxide inhibited calcineurin, one of the major serine and threonine phosphatases, by oxidation of metal ions or thiols. 

When the general arrangement is known it is then necessary to determine precise atomic coordinates. Sometimes the positions of certain atoms are invariant—they are fixed by symmetry considerations—but in complex crystals most of the atoms are in general5 positions not restricted in any way by symmetry. The variable parameters must be determined by successive approximations here the work of calculating structure amplitudes for postulated atomic positions can be much shortened by the use of graphical methods, to be described later in this chapter. It cannot be denied, however, that the complete determination of a complex structure is a task not to be undertaken lightly the time taken must usually be reckoned in months. 

Although many solution mechanistic tools are inappropriate for reaction studies in solids, their absence is more than compensated by the availability of other techniques that are unique to single crystals. Perhaps the most significant is X-ray diffraction, which can establish precise atomic coordinates not only for the starting material, but also for the environment in which reaction occurs. Availability of this kind of information puts discussions of mechanisms and solvent effects on a completely different footing from those for fluid reactions. 

More than the resolution, we would like to know the precision with which atoms in the model have been located. For years, crystallographers used the Luzzati plot (Fig. 8.3) to estimate the precision of atom locations in a refined crystallographic model. At best, this is an estimate of the upper limit of error in atomic coordinates. The numbers to the right of each smooth curve on the Luzzati plot are theoretical estimates of the average uncertainty in the positions of atoms in the refined model (more precisely, the rms errors in atom positions). The average uncertainty has been shown to depend upon R-factors derived from the final model in various resolution ranges. To prepare data for a Luzzati plot, we separate the intensity data into groups of reflections in... 

A precise description of the structure of a crystalline compound necessitates prior knowledge of its space group and atomic coordinates in the asymmetric unit. Illustrative examples of compounds belonging to selected space groups in the seven crystal systems are presented in the following sections. 

The estimated precision in bond lengths obtained by a least-square refinement of a data set measured by X-ray diffraction can be 0.003 A ( 0.3 pm), for a structure with unweighted R-factor less than 3%. If the data set is collected at low temperatures (20 K or 80 K), the decrease in thermal vibration can yield even better bond distances and angles. For H atom coordinates, the precision is one or two orders of magnitude lower, since the electron density around an H atom is relatively low in these cases a neutron diffraction study (which requires very large crystals) can yield better H atom positions. 

Figure 4 A schematic representation of the experimentai approach for time-resoived XAS measurements. XAS provides local structural and electronic information about the nearest coordination environment surrounding the catalytic metal ion within the active site of a metalloprotein in solution. Spectral analysis of the various spectral regions yields complementary electronic and structural information, which allows the determination of the oxidation state of the X-ray absorbing metal atom and precise determination of distances between the absorbing metal atom and the protein atoms that surround it. Time-dependent XAS provides insight into the lifetimes and local atomic structures of metal-protein complexes during enzymatic reactions on millisecond to minute time scales, (a) The drawing describes a conventional stopped-flow machine that is used to rapidly mix the reaction components (e.g., enzyme and substrate) and derive kinetic traces as shown in (b). (b) The enzymatic reaction is studied by pre-steady-state kinetic analysis to dissect out the time frame of individual kinetic phases, (c) The stopped-flow apparatus is equipped with a freeze-quench device. Sample aliquots are collected after mixing and rapidly froze into X-ray sample holders by the freeze-quench device, (d) Frozen samples are subjected to X-ray data collection and analysis.

There is, however, an alternative (but still indirect) way to view these molecules. It involves studies of crystalline solids and the use of the phenomenon of diffraction. The radiation used is either X rays, with a wavelength on the order of 10 cm, or neutrons of similar wavelengths. The result of analyses by these diffraction techniques, described in this volume, is a complete three-dimensional elucidation of the arrangement of atoms in the crystal under study. The information is obtained as atomic positional coordinates and atomic displacement parameters. The coordinates indicate the position of each atom in a repeat unit within the crystal, while the displacement parameters indicate the extent of atomic motion or disorder in the molecule. From atomic coordinates, it is possible to calculate, with high precision, interatomic distances and angles of the atomic components of the crystal and to learn about the shape (conformation) of molecules in the crystalline state. 

Atomic coordinates of atoms in the model of a protein structure can bo refined to more precise values that give a better fit to the experimental data. The so-called real-space refinement procedure optimizes the fit... 

Preliminary three-dimensional atomic coordinates of atoms in crystal structures are usually derived from electron-density maps by fitting atoms to individual peaks in the map. The chemically reasonable arrangement of atoms so obtained is, however, not very precise. The observed structure amplitudes and their relative phase angles, needed to calculate the electron-density map, each contain errors and these may cause a misinterpretation of the computed electron-density map. Even with the best electron-density maps, the precisions of the atomic coordinates of a preliminary structure are likely to be no better than several hundredths of an A. In order to understand the chemistry one needs to know the atomic positions more precisely so that better values of bond lengths and bond angles will be available. The process of obtaining atomic parameters that are more precise than those obtained from an initial model, referred to as refinement of the crystal structure, is an essential part of any crystal structure analysis. 

The relevant atomic coordinates for the calculation of molecular geometry consist of the values of x, y, and z, obtained as precisely as possible, for all atoms. From values of x, y, and z we can derive geometrical information such as bond distances, bond angles, torsion angles, and the mean planes through groups of atoms. In this way the geometry of the molecule or ion is found. 

---