Resolution and precision of atomic positions

In microscopy, the phrase resolution of 2 A implies that we can resolve objects that are 2 A apart. If this phrase had the same meaning for a crystallo- 

In X-ray crystallography, 2-A model means that analysis included reflections out to a distance in the reciprocal lattice of 1/(2 A) from the center of the diffraction pattern. This means that the model takes into account diffraction from sets of equivalent, parallel planes spaced as closely as 2 A in the unit cell. (Presumably, data farther out than the stated resolution was unobtainable or was too weak to be reliable.) Although the final 2-A map, viewed as an empty contour surface, may indeed not allow us to discern adjacent atoms, structural constraints on the model greatly increase the precision of atom positions. The main constraint is that we know we can fit the map with groups of atoms — amino-acid residues — having known connectivities, bond lengths, bond angles, and stereochemistry. 

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 

Publications, especially older ones, of refined structures may include a Luzzati plot, allowing the reader to assess very roughly the average uncertainty of atom positions in the model. Alternatively, they may simply report the uncertainty as determined by the method of Luzzati. For highly refined models, rms errors as low as 0.15 A are sometimes attained. In Fig. 8.5 a, Section III.C, the jagged curve represents the data for the refined model of the protein ALBP. The position of the curve on the Luzzati plot indicates that rms error for this model is about 0.34 A, about one-fifth the length of a carbon-carbon bond. 

In crystallography, unlike microscopy, the term resolution simply refers to the amount of data ultimately phased and used in the structure determination. In contrast, the precision of atom positions depends in part upon the resolution limits of the data, but also depends critically upon the quality of the data, as reflected by the R-factor. Good data can yield atom positions that are precise to within one-fifth to one-tenth of the stated resolution. 

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