Read (1986, 1990) proposed another graphical method to obtain the average positional error (Ar). In this approach, aA is plotted as function of S in the following form:


where oa is defined by Eq. (6.36). The summation goes over all N atoms of the whole structure and P atoms of the partially known structure. The argument of the logarithm on the right-hand side of Eq. (7.46) should be l if the structure refinement has been finished. However, this is never the case because of the disordered structure of the solvent atoms in the crystal. These atoms contribute considerably to low-resolution reflections which should, therefore, be ignored at the final stage of refinement. oa plots can, for example, be calculated with program SIGMAA (Read, 1986) or within CNS (Brünger et al., 1998).

7.3 Verification and Accuracy of Structure Determination | 179 The Diffraction-Component Precision Index

Cruickshank (1999) introduced a quick and rough guide for the diffraction-data-only error component for atoms with isotropic B-factor equal to the average B-factor, Bavg, of the biomacromolecular structure. This is named the diffraction component precision index (DPI), and is given by:

where N is the number of fully occupied atomic sites, p is the difference between the number of observations nobs and the number of parameters nparam, and C is the fractional completeness of the diffraction data to dmin, the minimal lattice plane distance.

For low-resolution structures, the number of parameters may exceed the number of diffraction data. In Eq. (7.47), p is then negative, so that a(x) is imaginary, but this problem may be circumvented empirically by replacing p with nobs and R with Rfree. The counterpart of the DPI (Eq. 7.47) is then ff(x, Bavg) = 1.0(Ni/n.obs)1/2C-1/3Rfreedmin . (7.48)

Here, nobs is the number of the reflections included in the refinement, not the number in the Rfree set.

Often, an estimate of a position error (Ar), rather than a coordinate error (Ax), is required. In the isotropic approximation we obtain a(x, Bavg) = 31/2ct(x, Bavg) . (7.49)

Consequently, the DPI formulae for the position errors are a(x, Bavg) = 31/2(Ni/p)1/2C-1/3Rdmin (7.50)

with R and a(x, Bavg) = 31/2(Ni/»obs)1/2C-1/3Rfreedmin (7.51) with Rfree.

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