International
Tables for Crystallography Volume F Crystallography of biological macromolecules Edited by M. G. Rossmann and E. Arnold © International Union of Crystallography 2006 
International Tables for Crystallography (2006). Vol. F, ch. 18.5, pp. 412414
Section 18.5.8. Luzzati plots^{a}Chemistry Department, UMIST, Manchester M60 1QD, England 
Luzzati (1952) provided a theory for estimating, at any stage of a refinement, the average positional shifts which would be needed in an idealized refinement to reach . He did not provide a theory for estimating positional errors at the end of a normal refinement.
Luzzati gave families of curves for R versus for varying average positional errors for both centrosymmetric and noncentrosymmetric structures. The curves do not depend on the number N of atoms in the cell. They all rise from at to the Wilson (1950) values 0.828 and 0.586 for random structures at high . Table 18.5.8.1 gives as a function of for threedimensional noncentrosymmetric structures.

In a footnote (p. 807), Luzzati suggested that at the end of a normal refinement (with R nonzero due to experimental and model errors, etc.), the curves would indicate an upper limit for . He noted that typical smallmolecule 's of 0.01–0.02 Å, if used as in the plots, would give much smaller R's than are found at the end of a refinement.
As examples, the Luzzati plots for the two structures of TGFβ2 are shown in Fig. 18.5.8.1. Daopin et al. (1994) inferred average 's around 0.21 Å for 1TGI and 0.23 Å for 1TGF.

Luzzati plots showing the refined R factor as a function of resolution for 1TGI (solid squares) and 1TGF (open squares) (Daopin et al., 1994). 
Of the three Luzzati assumptions summarized above, the most attractive is the third, which does not require the atoms to be identical nor the position errors to be small. For proteins, there are very obvious difficulties with assumption (2). Errors do depend very strongly on Z and B. In the highangle data shells, atoms with large B's contribute neither to nor to , and so have no effect on R in these shells. In their important paper on protein accuracy, Chambers & Stroud (1979) said `the [Luzzati] estimate derived from reflections in this range applies mainly to [the] best determined atoms.'
Thus a Luzzati plot seems to allow a cautious upperlimit statement about the precision of the best parts of a structure, but it gives little indication for the poor parts.
One reason for the past popularity of Luzzati plots has been that the R values for the middle and outer shells of a structure often roughly follow a Luzzati curve. Evidently, the effective average for the structure must be decreasing as increases, since atoms of high B are ceasing to contribute, whereas the proportionate experimental errors must be increasing. This also suggests that the upper limit for for the lowB atoms could be estimated from the lowest Luzzati theoretical curve touched by the experimental R plot. Thus in Fig. 18.5.8.1 the upper limits for the lowB atoms could be taken as 0.18 and 0.21 Å, rather than the 0.21 and 0.23 Å chosen by Daopin et al.
From the introduction of by Brünger (1992) and the discussion of by Tickle et al. (1998b), it can be seen that Luzzati plots should be based on a residual more akin to than R in order to avoid bias from the fitting of data.
The mean positional error of atoms can also be estimated from the plots of Read (1986, 1990). This method arose from Read's analysis of improved Fourier coefficients for maps using phases from partial structures with errors. It is preferable in several respects to the Luzzati method, but like the Luzzati method it assumes that the coordinate distribution is the same for all atoms. Luzzati and/or Read estimates of are available for some of the structures in Tables 18.5.7.2 and 18.5.7.3. Often, the two estimates are not greatly different.
Luzzati plots are fundamentally different from other statistical estimates of error. The Luzzati theory applies to an idealized incomplete refinement and estimates the average shifts needed to reach . In the leastsquares method, the equations for shifts are quite different from the equations for estimating variances in a converged refinement. However, Luzzatistyle plots of R versus can be reinterpreted to give statistically based estimates of .
During Cruickshank's (1960) derivation of the approximate equation (18.5.6.2) for in diagonal least squares, he reached an intermediate equation He then assumed R to be independent of and took R outside the summation to reach (18.5.6.2) above.
Luzzati (1952) calculated the acentric residual R as a function of , the average radial error of the atomic positions. His analysis shows that R is a linear function of s and for a substantial range of , with The theoretical Luzzati plots of R are nearly linear for smalltomedium (see Fig. 18.5.8.1). If we substitute this R in the leastsquares estimate (18.5.8.1) and use the threedimensionalGaussian relation , some manipulation (Cruickshank, 1999) along the lines of Section 18.5.6 eventually yields a statistically based formula, where is the value of R at some value of on the selected Luzzati curve. Equation (18.5.8.3) provides a means of making a very rough statistical estimate of error for an atom with (the average B for fully occupied sites) from a plot of R versus .
Protein structures always show a great range of B values. The Luzzati theory effectively assumes that all atoms have the same B. Nonetheless, the Luzzati method applied to highangle data shells does provide an upper limit for for the atoms with low B. It is an upper limit since experimental errors and model imperfections are not allowed for in the theory.
Lowresolution structures can be determined validly by using restraints, even though the number of diffraction observations is less than the number of atomic coordinates. The Luzzati method, based preferably on , can be applied to the atoms of low B in such structures. As the number of observations increases, and the resolution improves, the Luzzati increasingly overestimates the true of the lowB atoms.
In the use of Luzzati plots, the method of refinement, and its degree of convergence, is irrelevant. A Luzzati plot is a statement for the lowB atoms about the maximum errors associated with a given structure, whether converged or not.
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