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

Afonine, P.V.; Grosse-Kunstleve, R.W.; Adams, P.D.; Urzhumtsev, A.

doi:10.1107/S2059798323004825

addenda and errata

STRUCTURAL
BIOLOGY

ISSN: 2059-7983

Volume 79| Part 7| July 2023| Pages 666-667

https://doi.org/10.1107/S2059798323004825

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

P. V. Afonine,^a ^* R. W. Grosse-Kunstleve,^a P. D. Adams ^a,^b and A. Urzhumtsev ^c,^d

^aLawrence Berkeley National Laboratory, One Cyclotron Road, MS64R0121, Berkeley, CA 94720, USA, ^bDepartment of Bioengineering, University of California Berkeley, Berkeley, CA 94720, USA, ^cIGBMC, CNRS–INSERM–UdS, 1 Rue Laurent Fries, BP 10142, 67404 Illkirch, France, and ^dUniversité 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

Edited by R. J. Read, University of Cambridge, United Kingdom (Received 23 May 2023; accepted 2 June 2023; online 20 June 2023)

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

Keywords: Phenix; anisotropy; bulk solvent; scaling.

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

$[\eqalignno {\textstyle \sum \limits_{\bf s} \{|{\bf F}_{\rm calc}({\bf s}) & + k_{\rm mask}{\bf F}_{\rm mask}({\bf s})|^2 \cr &\ \quad -\ [k_{\rm overall}\,k_{\rm anisotropic}({\bf s})k_{\rm isotropic}]^{-2}F_{\rm obs}^2({\bf s})\}^2, &(22)}]$

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, $[{v_{\bf s}} = \textstyle{1 \over 2}[{\bf F}_{\rm calc}({\bf s}) {\bf F}_{\rm mask}^*({\bf s})]$ + $[{\bf F}_{\rm calc}^*({\bf s}){\bf F}_{\rm mask}({\bf s})]]$ , u_s = |F_calc(s)|², $[I_{\bf s} = [k_{\rm overall}\,k_{\rm anisotropic}({\bf s})]^{-2} F_{\rm obs}^2({\bf s})]$ and $[K = k_{\rm isotropic}^{-2}]$ and substituting them into (22) leads to the minimization of

$[{\rm LS}(K,k_{\rm mask}) = \textstyle \sum \limits_{\bf s} [(k_{\rm mask}^2{w_{\bf s} + 2k_{\rm mask}v_{\bf s} + u_{\bf s}) - KI_{\bf s}]^2} \eqno(23)]$

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

$[\cases {\displaystyle{{\partial} \over {\partial K}} {\rm LS}(K,k_{\rm mask}) = - 2\textstyle \sum \limits_{\bf s} [(k_{\rm mask}^2w_{\bf s} + 2k_{\rm mask}v_{\bf s} + u_{\bf s}) - KI_{\bf s}] I_{\bf s} \cr \quad\quad\quad\quad\quad\quad\quad = 0, \cr \displaystyle{{\partial} \over {\partial k_{\rm mask}}} {\rm LS}(K,k_{\rm mask}) = 4\textstyle \sum \limits_{\bf s}[(k_{\rm mask}^2w_{\bf s} + 2k_{\rm mask}v_{\bf s} + u_{\bf s}) - KI_{\bf s}]\cr \quad\quad\quad\quad\quad\quad\quad\quad\quad \times\ (k_{\rm mask}w_{\bf s} + v_{\bf s}) \cr \quad\quad\quad\quad\quad\quad\quad\quad= 0.} \eqno(24)]$

Developing these equations with respect to k_mask,

$[\cases {k_{\rm mask}^2\textstyle \sum \limits_{\bf s} w_{\bf s}I_{\bf s} + 2k_{\rm mask}\textstyle \sum \limits_{\bf s} v_{\bf s}I_{\bf s} + \textstyle \sum \limits_{\bf s} u_{\bf s}I_{\bf s} - K\textstyle \sum \limits_{\bf s} I_{\bf s}^2 = 0, \cr k_{\rm mask}^3\textstyle \sum \limits_{\bf s} w_{\bf s}^2 + 3k_{\rm mask}^2\textstyle \sum \limits_{\bf s} w_{\bf s}v_{\bf s} + k_{\rm mask}\textstyle \sum \limits_{\bf s}(2v_{\bf s}^2 + u_{\bf s}w_{\bf s} - KI_{\bf s}w_{\bf s}) \cr \quad\quad+ \textstyle \sum \limits_{\bf s} u_{\bf s}v_{\bf s} - K\textstyle \sum \limits_{\bf s} I_{\bf s}v_{\bf s} = 0,} \eqno(25)]$

and introducing new notations for the coefficients, we obtain

$[\cases {k_{\rm mask}^2C_2 + k_{\rm mask}B_2 + A_2 - KY_2 = 0, \cr k_{\rm mask}^3D_3 + k_{\rm mask}^2C_3 + k_{\rm mask}(B_3 - KC_2) + A_3 - KY_3 = 0.} \eqno(26)]$

Multiplying the second equation by Y₂ and substituting KY₂ from the first equation into the new second equation, we obtain a cubic equation with fixed coefficients

$[\eqalignno {&k_{\rm mask}^3(D_3Y_2 - C_2^2) + k_{\rm mask}^2(C_3Y_2 - C_2B_2 - C_2Y_3)\cr &\quad+ k_{\rm mask}(B_3Y_2 - C_2A_2 - Y_3B_2) + (A_3Y_2 - Y_3A_2) = 0. &(27)}]$

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

$[D_3Y_2 - C_2^2 = \textstyle \sum \limits_{\bf s} w_{\bf s}^2\textstyle \sum \limits_{\bf s} I_{\bf s}^2 - \textstyle \sum \limits_{\bf s} w_{\bf s}I_{\bf s}\textstyle \sum \limits_{\bf s} w_{\bf s}I_{\bf s} \,\gt \,0. \eqno(28)]$

Therefore, equation (27) can be rewritten as

$[k_{\rm mask}^3 + ak_{\rm mask}^2 + bk_{\rm mask} + c = 0 \eqno(29)]$

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),

$[K = (k_{\rm mask}^2C_2 + k_{\rm mask}B_2 + A_2)/Y_2. \eqno(30)]$

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

$[\textstyle \sum \limits_{\bf s} [k_{\rm mask} - k_{\rm sol}\exp(-B_{\rm sol}\,{s^2}/4)]^2 \eqno(31)]$

for all k_mask > 0. This can be achieved analytically as described in Appendix A. Similarly, one can fit k_overallexp(−B_overall s²/4) 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

$[{\bf b} = \left[\textstyle \sum \limits_{\bf s} I({\bf s})I_1({\bf s}_1), \ldots, \textstyle \sum \limits_{\bf s} I({\bf s})I_N({\bf s}_N),1 \right]^t, \eqno(42)]$

$[{\rm LS}(K,k_{\rm mask}) = \textstyle \sum \limits_{\bf s} \left \{\left[\textstyle \sum \limits_{j=1}^N \alpha_j| {\bf F}_{\rm calc}({\bf s}_j) + k_{\rm mask}{\bf F}_{\rm mask}({\bf s}_j)|^2 \right] - KI_{\bf s} \right\}^2, \eqno(43)]$

$[{\rm LS}(K,k_{\rm mask}) = \textstyle \sum \limits_{\bf s}\left[(k_{\rm mask}^2w_{\bf s} + 2k_{\rm mask}v_{\bf s} + u_{\bf s}) - KI_{\bf s}\right]^2. \eqno(45)]$

References

Afonine, P. V., Grosse-Kunstleve, R. W., Adams, P. D. & Urzhumtsev, A. (2013). Acta Cryst. D69, 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

This is an open-access article distributed under the terms of the Creative Commons Attribution (CC-BY) Licence, which permits unrestricted use, distribution, and reproduction in any medium, provided the original authors and source are cited.

STRUCTURAL
BIOLOGY

ISSN: 2059-7983

Volume 79| Part 7| July 2023| Pages 666-667

https://doi.org/10.1107/S2059798323004825

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addenda and errata\(\def\hfill{\hskip 5em}\def\hfil{\hskip 3em}\def\eqno#1{\hfil {#1}}\)

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

2.3. Bulk-solvent parameters and overall isotropic scaling

References

addenda and errata