# Toward Real Real-Space Refinement of Atomic Models

^{1}

^{2}

^{3}

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Results

#### 2.1. Gaussian Atomic Model

#### 2.2. Schemes of Reciprocal-Space Refinement

#### 2.3. Schemes of Real-Space Refinement

#### 2.4. Map as an Analytic Function

#### 2.5. Comparison of Schemes of Real-Space Refinement

## 3. Discussion

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## References

- Cardone, G.; Heymann, J.B.; Steven, A.C. One number does not fit all: Mapping local variations in resolution in cryo-EM reconstructions. J. Struct. Biol.
**2013**, 184, 226–236. [Google Scholar] [CrossRef] [Green Version] - Urzhumtsev, A.G.; Lunin, V.Y. Introduction to crystallographic refinement of macromolecular atomic models. Crystallogr. Rev.
**2019**, 25, 164–262. [Google Scholar] [CrossRef] - Brown, A.; Long, F.; Nicholls, R.A.; Toots, I.; Emsley, P.; Murshudov, G. Tools for molecular model building and refinement into electron cryo-microscopy reconstructions. Acta Crystallogr.
**2015**, D71, 136–153. [Google Scholar] - Lunin, V.Y.; Afonine, P.V.; Urzhumtsev, A.G. Likelihood-based refinement. I. Irremovable model errors. Acta Crystallogr.
**2002**, A58, 270–282. [Google Scholar] [CrossRef] [PubMed] - Kostrewa, D. Bulk Solvent Correction: Practical Application and Effects in Reciprocal and Real Space. Jt. CCP4 ESF-EACBM Newsl. Protein Crystallogr.
**1995**, 34, 9–22. [Google Scholar] - Afonine, P.V.; Adams, P.D.; Sobolev, O.V.; Urzhumtsev, A. A mosaic bulk-solvent model improves density maps and the fit between model and data. bioRxiv. 2021. Available online: https://www.biorxiv.org/content/10.1101/2021.12.09.471976v1.full (accessed on 6 October 2022).
- Afonine, P.V.; Urzhumtsev, A.; Adams, P.D. On the analysis of residual density distribution on an absolute scale. Comput. Cryst. Newsl.
**2012**, 3, 43–46. [Google Scholar] - Palmer, C.M.; Aylett, C.H.S. Real space in cryo-EM: The future is local. Acta Crystallogr.
**2022**, D78, 136–143. [Google Scholar] [CrossRef] [PubMed] - Lunin, V.Y.; Urzhumtsev, A. Program construction for macromolecule atomic model refinement based on the fast Fourier transform and fast differentiation algorithms. Acta Crystallogr.
**1985**, A41, 327–333. [Google Scholar] [CrossRef] - Urzhumtsev, A.G.; Lunin, V.Y. Fast differentiation algorithm and efficient calculation of the exact matrix of the second derivatives. Acta Crystallogr.
**2001**, A57, 451–460. [Google Scholar] [CrossRef] [Green Version] - Diamond, R. A real-space procedure for proteins. Acta Crystallogr.
**1971**, A27, 436–451. [Google Scholar] [CrossRef] - Abagyan, R.A.; Totrov, M.M.; Kuznetsov, D.A. Icm—A new method for protein modeling and design—Applications to docking and structure prediction from the distorted native conformation. J. Comput. Chem.
**1994**, 15, 488–506. [Google Scholar] [CrossRef] - Rice, L.M.; Brünger, A.T. Torsion angle dynamics: Reduced variable conformational sampling enhances crystallographic structure refinement. Proteins Struct. Funct. Genet.
**1994**, 19, 277–290. [Google Scholar] [CrossRef] [PubMed] - Afonine, P.V.; Grosse-Kunstleve, R.W.; Urzhumtsev, A.G.; Adams, P.D. Automatic multiple-zone rigid-body refinement with a large convergence radius. J. Appl. Crystallogr.
**2009**, 42, 607–615. [Google Scholar] [CrossRef] - Merritt, E.A. To B or not to B: A question of resolution? Acta Crystallogr.
**2012**, D68, 468–477. [Google Scholar] [CrossRef] [PubMed] [Green Version] - Cruickshank, D.W.J. The analysis of the anisotropic thermal motion of molecules in crystals. Acta Crystallogr.
**1956**, 9, 754–756. [Google Scholar] [CrossRef] - Schomaker, V.; Trueblood, K.N. On the rigid-body motion of molecules in crystals. Acta Crystallogr.
**1968**, B24, 63–76. [Google Scholar] [CrossRef] - Doyle, P.A.; Turner, P.S. Relativistic Hartree-Fock X-ray and electron scattering factors. Acta Crystallogr.
**1968**, A24, 390–397. [Google Scholar] [CrossRef] - Agarwal, R.C. A new least-squares refinement technique based on the fast Fourier transform algorithm. Acta Crystallogr.
**1978**, A41, 327–333. [Google Scholar] [CrossRef] - Waasmaier, D.; Kirfel, A. New analytical scattering-factor functions for free atoms and ions. Acta Crystallogr.
**1985**, A34, 791–809. [Google Scholar] [CrossRef] - Peng, L.-M. Electron atomic scattering factors and scattering potentials of crystals. Micron
**1999**, 30, 625–648. [Google Scholar] [CrossRef] - Grosse-Kunstleve, R.W.; Sauter, N.K.; Adams, P.D. CCTBX news. Newsl. IUCr Comm. Crystallogr. Comput.
**2004**, 3, 22–31. [Google Scholar] - Brown, P.J.; Fox, A.G.; Maslen, E.N.; O’Keefe, M.A.; Willis, B.T.M. Intensity of diffracted intensities. In International Tables for X-ray Crystallography; Prince, E., Ed.; Springer: Dordrecht, The Netherlands, 2006; Volume C, pp. 554–595. [Google Scholar]
- Marques, M.A.; Purdy, M.D.; Yeager, M. CryoEM maps are full of potential. Curr. Opin. Struct. Biol.
**2019**, 58, 214–223. [Google Scholar] [CrossRef] - Carugo, O. B-factor accuracy in protein crystal structures. Acta Crystallogr.
**2022**, D78, 69–74. [Google Scholar] [CrossRef] [PubMed] - Masmaliyeva, R.C.; Murshudov, G.N. Analysis and validation of macromolecular B values. Acta Crystallogr.
**2019**, D75, 505–518. [Google Scholar] [CrossRef] [PubMed] - Sayre, D. The Calculation of Structure Factors by Fourier Summation. Acta Crystallogr.
**1951**, 4, 327–333. [Google Scholar] [CrossRef] - Ten Eyck, L.F. Efficient structure-factor calculation for large molecules by the fast Fourier transform. Acta Crystallogr.
**1977**, A33, 486–492. [Google Scholar] [CrossRef] - Cooley, J.W.; Tukey, J.W. An algorithm for machine calculation of complex Fourier series. Math. Comput.
**1965**, 19, 297–301. [Google Scholar] [CrossRef] - Navaza, J. On the computation of structure factors by FFT techniques. Acta Crystallogr.
**2002**, A58, 568–573. [Google Scholar] [CrossRef] [PubMed] [Green Version] - Afonine, P.V.; Urzhumtsev, A. On a fast and accurate calculation of structure factors at a subatomic resolution. Acta Crystallogr.
**2004**, A60, 19–32. [Google Scholar] [CrossRef] [PubMed] - Rossmann, M.G. Fitting atomic models into electron-microscopy maps. Acta Crystallogr.
**2000**, D56, 1341–1349. [Google Scholar] [CrossRef] [PubMed] [Green Version] - Rossmann, M.G.; Bernal, R.; Pletnev, S.V. Combining electron microscopic with x-ray crystallographic structures. J. Struct. Biol.
**2001**, 136, 190–200. [Google Scholar] [CrossRef] [Green Version] - Emsley, P.; Cowtan, K. Coot: Model-building tools for molecular graphics. Acta Crystallogr.
**2004**, D60, 2126–2132. [Google Scholar] [CrossRef] [PubMed] [Green Version] - Afonine, P.V.; Poon, B.K.; Read, R.J.; Sobolev, O.V.; Terwilliger, T.C.; Urzhumtsev, A.G.; Adams, P.D. Real-space refinement in PHENIX for cryo-EM and crystallography. Acta Crystallogr.
**2018**, D74, 531–544. [Google Scholar] - Lunin, V.Y.; Urzhumtsev, A. Improvement of protein phases by coarse model modification. Acta Crystallogr.
**1984**, A40, 269–277. [Google Scholar] [CrossRef] - Mooij, W.T.M.; Hartshorn, M.J.; Tickle, I.J.; Sharff, A.J.; Verdonk, M.L.; Jhoti, H. Automated protein-ligand crystallography for structure-based drug design. ChemMedChem
**2006**, 1, 827–838. [Google Scholar] [CrossRef] [PubMed] - DiMaio, F.; Song, Y.; Li, X.; Brunner, M.J.; Xu, C.; Conticello, V.; Egelman, E.; Marlovits, T.; Cheng, Y.; Baker, D. Atomic-accuracy models from 4.5-Å cryo-electron microscopy data with density-guided iterative local refinement. Nat. Methods
**2015**, 12, 361–365. [Google Scholar] [CrossRef] [PubMed] [Green Version] - Chapman, M.S. Restrained real-space macromolecular atomic refinement using a new resolution-dependent electron-density function. Acta Crystallogr.
**1995**, A51, 69–80. [Google Scholar] [CrossRef] - Chapman, M.S.; Trzynka, A.; Chapman, B.K. Atomic modeling of cryo-electron microscopy reconstructions—Joint refinement of model and imaging parameters. J. Struct. Biol.
**2013**, 182, 10–21. [Google Scholar] [CrossRef] [PubMed] [Green Version] - Afonine, P.V.; Lunin, V.Y.; Muzet, N.; Urzhumtsev, A. On the possibility of the observation of valence electron density for individual bonds in proteins in conventional difference maps. Acta Crystalogr.
**2004**, D60, 260–274. [Google Scholar] [CrossRef] [Green Version] - Urzhumtsev, A.; Urzhumtseva, L.; Lunin, V.Y. Direct calculation of cryo EM and crystallographic model maps for real-space refinement. bioRxiv
**2022**. Available online: https://doi.org/10.1101/2022.07.17.500345 (accessed on 6 October 2022). [CrossRef] - Urzhumtsev, A.G.; Lunin, V.Y. Analytic representation of inhomogeneous-resolution maps of three-dimensional scalar fields. bioRxiv
**2022**. Available online: https://doi.org/10.1101/2022.03.28.486044 (accessed on 6 October 2022). [CrossRef] - Simonetti, A.; Marzi, S.; Fabbretti, A.; Myasnikov, A.G.; Hazemann, I.; Jenner, L.; Urzhumtsev, A.; Gualerzi, C.O.; Klaholz, B.P. Crystal structure of the protein core of translation initiation factor IF2 in apo, GTP and GDP forms. Acta Crystallogr.
**2013**, D69, 925–933. [Google Scholar] - Schrödinger, L.; DeLano, W.L. Pymol. Version 2.5.2. 2020. Available online: http://www.pymol.org (accessed on 6 October 2022).
- Ten Eyck, L.F. Crystallographic fast Fourier transforms. Acta Crystallogr.
**1973**, A33, 183–191. [Google Scholar] [CrossRef]

**Figure 1.**Levels of macromolecular parameterization in MX and cryo-EM. By ‘density distribution’, we consider various kinds of scalar functions in space, such as an electron or nuclear scattering density distribution in crystallography or scattering electrostatic potential in cryo-EM, etc. The term ‘density map’ stands for maps of any of these distributions. Atomic parameters are usually the coordinates of the centers of atoms and their displacement parameters, ADP. Common parameters may be dihedral angles [11,12,13], rigid-body parameters [14], common ADP values for all atoms of the residue [15] or TLS parameters [16,17], or something else, describing common features of an atomic group. Black arrows show the step-by-step hierarchic recalculation of the model parameters; the red and blue arrows illustrate alternative direct calculations of structure factors and maps from model parameters.

**Figure 2.**Error in the Fourier transform of a density of a Gaussian virtual atom. Error $\u2206\left(X\right)$ is given as function (5) of the dimensionless truncation radius, $X={r}_{FT}/\sqrt{B}$, and for different values of the parameter $t=s\sqrt{B}$. $\u2206\left(0\right)$ is equal to the exact value $\left|F\left(\mathit{s};\mathit{B}\right)\right|$ for the respective $s\sqrt{B}$.

**Figure 3.**Map of an inhomogeneous resolution calculated in a single run. Map resolution varies from 2 Å around the molecular center (red sphere) to 5 Å at the periphery. Color arrows indicate the regions of a high resolution and small ADP (blue), high resolution and large ADP (magenta), low resolution and small ADP (grey), and low resolution and large ADP (red). Figure has been prepared using Pymol [45].

**Figure 4.**CPU time for the three-step map calculation for different grid steps expressed as a part of the resolution D. (

**a**) CPU time, in seconds, to calculate a density distribution for the test protein model using different truncation distance ${r}_{FT}$. (

**b**) CPU time to calculate FFT on a grid as defined in (a).

**Figure 5.**CPU time for the single-step map calculation, as a function of the truncation distance and the grid step, expressed as a part of the resolution D. (

**a**) CPU time, in seconds, to calculate a density map for the test protein model using the simplified decomposition (10) of atomic images at a fixed resolution. (

**b**) The same using the variable-resolution terms (8).

**Figure 6.**CPU time, in seconds, for different values of parameters used to calculate the model map. Multicolor columns represent the multi-step calculation with the green part for ${T}_{density}$, beige for ${T}_{FFT\_SF}$, and red for ${T}_{FFT\_map}$. Index ‘D’ indicates the grid step for the density, as a part of the resolution, D/3 or D/4; index ‘r’ value is equal to the truncation distance times ten. Blue and variable-blue columns stand for ${T}_{direct}$ for the fixed-resolution and variable-resolution decompositions, respectively, as indicated by ‘F’ and ‘V’ letters. The grid step of the resulted model map is equal to (

**a**) D/2; (

**b**) D/3; and (

**c**) D/4.

Publisher’s Note: MDPI stays neutral with regard to jurisdictional claims in published maps and institutional affiliations. |

© 2022 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (https://creativecommons.org/licenses/by/4.0/).

## Share and Cite

**MDPI and ACS Style**

Urzhumtsev, A.G.; Lunin, V.Y.
Toward Real Real-Space Refinement of Atomic Models. *Int. J. Mol. Sci.* **2022**, *23*, 12101.
https://doi.org/10.3390/ijms232012101

**AMA Style**

Urzhumtsev AG, Lunin VY.
Toward Real Real-Space Refinement of Atomic Models. *International Journal of Molecular Sciences*. 2022; 23(20):12101.
https://doi.org/10.3390/ijms232012101

**Chicago/Turabian Style**

Urzhumtsev, Alexandre G., and Vladimir Y. Lunin.
2022. "Toward Real Real-Space Refinement of Atomic Models" *International Journal of Molecular Sciences* 23, no. 20: 12101.
https://doi.org/10.3390/ijms232012101