Bulk-solvent and overall scaling revisited: faster calculations, improved results. Corrigendum.

The article by Afonine et al. [ Acta Cryst. (2013). D69, 625–634] is corrected.

Abstract

Equations in Sections 2.3 and 2.4 of the article by Afonine et al. [Acta Cryst. (2013). D69, 625–634] are corrected.

In the article by Afonine et al. (2013 ▸) some improper notations and errors in several equations in Sections 2.3 and 2.4 have been corrected. We note that the Computational Crystallography Toolbox (Grosse-Kunstleve et al., 2002 ▸) has been using the correct version of these equations since 2013. Updated versions of Section 2.3 and equations (42), (43) and (45) are given below.

2.3. Bulk-solvent parameters and overall isotropic scaling

Assuming the resolution-dependent scale factors k _mask(s) and k _isotropic(s) to be constants k _mask and k _isotropic in each thin resolution shell, the determination of their values is reduced to minimizing the residual

where the sum is calculated over all reflections s in the given resolution shell, and k _overall and k _anisotropic(s) are calculated previously and fixed. This minimization problem is generally highly over-determined because the number of reflections per shell is usually much larger than two.

Introducing w _s = |F _mask(s)|², + , u _s = |F _calc(s)|², and and substituting them into (22) leads to the minimization of

with respect to K and k _mask. This leads to a system of two equations:

Developing these equations with respect to k _mask,

and introducing new notations for the coefficients, we obtain

Multiplying the second equation by Y ₂ and substituting KY ₂ from the first equation into the new second equation, we obtain a cubic equation with fixed coefficients

The senior coefficient in equation (27) satisfies the Cauchy–Schwarz inequality:

Therefore, equation (27) can be rewritten as

and solved using a standard procedure.

The corresponding values of K are obtained by substituting the roots of equation (29) into the first equation in equation (26),

If no positive root exists, k _mask is assigned a zero value, which implies the absence of a bulk-solvent contribution. If several roots with k _mask ≥ 0 exist then the one that gives the smallest value of LS(K, k _mask) is selected.

If desired, one can fit the right-hand side of expression (10) to the array of k _mask values by minimizing the residual

for all k _mask > 0. This can be achieved analytically as described in Appendix A. Similarly, one can fit k _overallexp(−B _overall  s ²/4) to the array of K values.

Equations (42), (43) and (45) in Section 2.4 of Afonine et al. (2013 ▸) are also updated as follows