## addenda and errata

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

^{a}Lawrence Berkeley National Laboratory, One Cyclotron Road, MS64R0121, Berkeley, CA 94720, USA, ^{b}Department of Bioengineering, University of California Berkeley, Berkeley, CA 94720, USA, ^{c}IGBMC, CNRS–INSERM–UdS, 1 Rue Laurent Fries, BP 10142, 67404 Illkirch, France, and ^{d}Université Nancy: Département de Physique – Nancy 1, BP 239, Faculté des Sciences et des Technologies, 54506 Vandoeuvre-lès-Nancy, France^{*}Correspondence e-mail: pafonine@lbl.gov

Equations in Sections 2.3 and 2.4 of the article by Afonine *et al.* [*Acta Cryst.* (2013). D**69**, 625–634] are corrected.

Keywords: *Phenix*; anisotropy; bulk solvent; scaling.

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*_{overall }exp(−*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

### References

Afonine, P. V., Grosse-Kunstleve, R. W., Adams, P. D. & Urzhumtsev, A. (2013). *Acta Cryst.* D**69**, 625–634. Web of Science CrossRef CAS IUCr Journals Google Scholar

Grosse-Kunstleve, R. W., Sauter, N. K., Moriarty, N. W. & Adams, P. D. (2002). *J. Appl. Cryst.* **35**, 126–136. Web of Science CrossRef CAS IUCr Journals Google Scholar

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