Voronoi diagram-based compensation of preferred particle orientations in cryo-EM 3D reconstructions

Igor M. Orlov; Bruno P. Klaholz; Igor M. Orlov; Bruno P. Klaholz

doi:10.3934/biophy.2026007

AIMS Biophysics

2026, Volume 13, Issue 1: 111-130. doi: 10.3934/biophy.2026007

Previous Article Next Article

Research article Special Issues

Voronoi diagram-based compensation of preferred particle orientations in cryo-EM 3D reconstructions

Igor M. Orlov ^1,2,3,4,
Bruno P. Klaholz ^{1,2,3,4
,
,}

1.
Centre for Integrative Biology (CBI), Department of Integrated Structural Biology, IGBMC, CNRS, Inserm, Université de Strasbourg, 1 rue Laurent Fries, 67404 Illkirch, France
2.
Institute of Genetics and of Molecular and Cellular Biology (IGBMC), 1 rue Laurent Fries, Illkirch, France
3.
Centre National de la Recherche Scientifique (CNRS), UMR 7104, Illkirch, France
4.
Institut National de la Santé et de la Recherche Médicale (Inserm), U964, Illkirch, France
5.
Université de Strasbourg, Illkirch, France

Received: 03 December 2025 Revised: 05 March 2026 Accepted: 10 March 2026 Published: 20 March 2026

To reveal functional properties of biological complexes using single-particle cryo electron microscopy (cryo-EM), it is important to obtain maps without distortions. 3D reconstructions can be obtained from back-projection of the individual 2D projections of the particles, a concept that assumes equally distributed orientations. However, most biological samples exhibit preferential views due to preferred particle orientations, e.g., at the air–water interface or on a carbon surface of a cryo-EM grid. Here, we present a method based on Voronoi diagrams that takes into account the Euler angle distribution of particles through individual image weighting, thus reducing 3D reconstruction artefacts emanating from preferential particle orientations. Voronoi diagrams are built from the Euler angle values plotted on a sphere in order to graphically display particle orientations, and the assignment of the projection weights is then calculated, being proportional to the individual Voronoi polygon areas. We implemented this real-space approach into a 3D reconstruction routine that allows performing automated weighting. The method was validated using a back-projection algorithm on synthetic model data and on real experimental data with different point group symmetries and in the presence or absence of preferential views, revealing a particular benefit for asymmetric objects.
- single-particle cryo electron microscopy,
- cryo-EM,
- 3D reconstruction,
- Voronoi diagram,
- back-projection,
- preferred particle orientations,
- preferential views,
- Euler angle distribution,
- weighting
Citation: Igor M. Orlov, Bruno P. Klaholz. Voronoi diagram-based compensation of preferred particle orientations in cryo-EM 3D reconstructions[J]. AIMS Biophysics, 2026, 13(1): 111-130. doi: 10.3934/biophy.2026007

Related Papers:

Abstract

To reveal functional properties of biological complexes using single-particle cryo electron microscopy (cryo-EM), it is important to obtain maps without distortions. 3D reconstructions can be obtained from back-projection of the individual 2D projections of the particles, a concept that assumes equally distributed orientations. However, most biological samples exhibit preferential views due to preferred particle orientations, e.g., at the air–water interface or on a carbon surface of a cryo-EM grid. Here, we present a method based on Voronoi diagrams that takes into account the Euler angle distribution of particles through individual image weighting, thus reducing 3D reconstruction artefacts emanating from preferential particle orientations. Voronoi diagrams are built from the Euler angle values plotted on a sphere in order to graphically display particle orientations, and the assignment of the projection weights is then calculated, being proportional to the individual Voronoi polygon areas. We implemented this real-space approach into a 3D reconstruction routine that allows performing automated weighting. The method was validated using a back-projection algorithm on synthetic model data and on real experimental data with different point group symmetries and in the presence or absence of preferential views, revealing a particular benefit for asymmetric objects.

Acknowledgments

We thank our lab members for discussions and suggestions, Alexandre Urzhumtsev for suggestions and Jonathan Michalon for IT support. This work was supported by CNRS (Centre national de la recherche scientifique), Association pour la Recherche sur le Cancer (ARC), Institut National du Cancer (INCa), the Fondation pour la Recherche Médicale (FRM), Ligue nationale contre le cancer (Ligue) and Agence National pour la Recherche (ANR). The electron microscope facility at CBI/IGBMC was supported by the Region Grand Est, FEDER, the French Infrastructure for Integrated Structural Biology (FRISBI) ANR-10-INSB-05-01/France 2030 program, by Instruct-ULTRA as part of the European Union's Horizon 2020 grant ID 731005, by EquipEx⁺ France-Cryo-EM (ANR-21-ESRE-0046) and by Instruct-ERIC.

Conflict of interest

The authors declare no conflict of interest.

Author contributions

I.M.O. wrote software code, performed image processing and tests with model and real data. I.M.O. & B.P.K. wrote the manuscript.

References

[1]	Orlov I, Myasnikov AG, Andronov L, et al. (2017) The integrative role of cryo electron microscopy in molecular and cellular structural biology. Biol Cell 109: 81-93. https://doi.org/10.1111/boc.201600042
[2]	White HE, Ignatiou A, Clare DK, et al. (2017) Structural study of heterogeneous biological samples by cryoelectron microscopy and image processing. BioMed Res Int 2017: 1032432. https://doi.org/10.1155/2017/1032432
[3]	Costa TRD, Ignatiou A, Orlova EV (2017) Structural analysis of protein complexes by cryo electron microscopy. Methods Mol Biol 1615: 377-413. https://doi.org/10.1007/978-1-4939-7033-9_28
[4]	Earl LA, Falconieri V, Milne JL, et al. (2017) Cryo-EM: beyond the microscope. Curr Opin Struct Biol 46: 71-78. https://doi.org/10.1016/j.sbi.2017.06.002
[5]	von Loeffelholz O, Natchiar SK, Djabeur N, et al. (2017) Focused classification and refinement in high-resolution cryo-EM structural analysis of ribosome complexes. Curr Opin Struct Biol 46: 140-148. https://doi.org/10.1016/j.sbi.2017.07.007
[6]	Chiu W, Downing KH (2017) Editorial overview: Cryo electron microscopy: Exciting advances in cryoEM herald a new era in structural biology. Curr Opin Struct Biol 46: iv-viii. https://doi.org/10.1016/j.sbi.2017.07.006
[7]	Kaelber JT, Hryc CF, Chiu W (2017) Electron cryomicroscopy of viruses at near-atomic resolutions. Annu Rev Virol 4: 287-308. https://doi.org/10.1146/annurev-virology-101416-041921
[8]	De Rosier DJ, Klug A (1968) Reconstruction of three dimensional structures from electron micrographs. Nature 217: 130-134. https://doi.org/10.1038/217130a0
[9]	Klug A (1983) From macromolecules to biological assemblies (Nobel Lecture). Biosci Rep 3: 395-430. https://doi.org/10.1002/anie.198305653
[10]	Radon J (1917) Über die Bestimmung von Funktionen durch ihre Integralwerte langs gewisser Manningfaltigkeiten. (On the determination of functions from their integrals along certain manifolds). Aka Wiss 69: 262-277.
[11]	Herman GT (1980) Image Reconstruction from Projections: the Fundamentals of Computerized Tomography. New York: Academic Press.
[12]	Harauz G, van Heel M (1986) Exact filters for general geometry three dimensional reconstruction. Optik 73: 146-156.
[13]	Russ JC, Neal FB (1995) The Image Processing Handbook. Boca Raton: CRC press. https://doi.org/10.1201/b18983
[14]	Frank J (2006) Three-Dimensional Electron Microscopy of Macromolecular Assemblies: Visualization of Biological Molecules in Their Native State. New York: Oxford Academic. https://doi.org/10.1093/acprof:oso/9780195182187.001.0001
[15]	Bai XC, Fernandez IS, McMullan G, et al. (2013) Ribosome structures to near-atomic resolution from thirty thousand cryo-EM particles. eLife 2: e00461. https://doi.org/10.7554/eLife.00461
[16]	Amunts A, Brown A, Toots J, et al. (2015) Ribosome. The structure of the human mitochondrial ribosome. Science 348: 95-98. https://doi.org/10.1126/science.aaa1193
[17]	Khatter H, Myasnikov AG, Natchiar SK, et al. (2015) Structure of the human 80S ribosome. Nature 520: 640-645. https://doi.org/10.1038/nature14427
[18]	Abeyrathne PD, Koh CS, Grant T, et al. (2016) Ensemble cryo-EM uncovers inchworm-like translocation of a viral IRES through the ribosome. eLife 5: e14874. https://doi.org/10.7554/eLife.14874
[19]	Merk A, Bartesaghi A, Banerjee S, et al. (2016) Breaking cryo-EM resolution barriers to facilitate drug discovery. Cell 165: 1698-1707. https://doi.org/10.1016/j.cell.2016.05.040
[20]	Natchiar SK, Myasnikov AG, Kratzat H, et al. (2017) Visualization of chemical modifications in the human 80S ribosome structure. Nature 551: 472-477. https://doi.org/10.1038/nature24482
[21]	Dai X, Li Z, Lai M, et al. (2017) In situ structures of the genome and genome-delivery apparatus in a single-stranded RNA virus. Nature 541: 112-116. https://doi.org/10.1038/nature20589
[22]	Kühlbrandt W (2014) The resolution revolution. Science 343: 1443-1444. https://doi.org/10.1126/science.1251652
[23]	Glaeser RM, Han BG (2017) Opinion: hazards faced by macromolecules when confined to thin aqueous films. Biophys Rep 3: 1-7. https://doi.org/10.1007/s41048-016-0026-3
[24]	Rice WJ, Cheng A, Noble AJ, et al. (2018) Routine determination of ice thickness for cryo-EM grids. J Struct Biol 204: 38-44. https://doi.org/10.1016/j.jsb.2018.06.007
[25]	Penczek PA, Renka R, Schomberg H (2004) Gridding-based direct Fourier inversion of the three-dimensional ray transform. J Opt Soc Am A 21: 499-509. https://doi.org/10.1364/JOSAA.21.000499
[26]	Kühlbrandt W (2014) Cryo-EM enters a new era. eLife 3: e03678. https://doi.org/10.7554/eLife.03678
[27]	Orlov IM, Morgan DG, Cheng RH (2006) Efficient implementation of a filtered back-projection algorithm using a voxel-by-voxel approach. J Struct Biol 154: 287-296. https://doi.org/10.1016/j.jsb.2006.03.007
[28]	Jenner L, Romby P, Rees B, et al. (2005) Translational operator of mRNA on the ribosome: how repressor proteins exclude ribosome binding. Science 308: 120-123. https://doi.org/10.1126/science.1105639
[29]	van Heel M, Harauz G, Orlova EV, et al. (1996) A new generation of the IMAGIC image processing system. J Struct Biol 116: 17-24. https://doi.org/10.1006/jsbi.1996.0004
[30]	Kennaway CK, Benesch JLP, Gohlke U, et al. (2005) Dodecameric structure of the small heat shock protein Acr1 from Mycobacterium tuberculosis. J Biol Chem 280: 33419-33425. https://doi.org/10.1074/jbc.M504263200
[31]	Simonetti A, Marzi S, Myasnikov AG, et al. (2008) Structure of the 30S translation initiation complex. Nature 455: 416-420. https://doi.org/10.1038/nature07192
[32]	Orlov I, Rochel N, Moras D, et al. (2012) Structure of the full human RXR/VDR nuclear receptor heterodimer complex with its DR3 target DNA. EMBO J 31: 291-300. https://doi.org/10.1038/emboj.2011.445
[33]	Orlov I, Hemmer C, Ackerer L, et al. (2020) Structural basis of nanobody recognition of grapevine fanleaf virus and of virus resistance loss. Proc Natl Acad Sci USA 117: 10848-10855. https://doi.org/10.1073/pnas.1913681117
[34]	Hryc CF, Chen DH, Afonine PV, et al. (2017) Accurate model annotation of a near-atomic resolution cryo-EM map. Proc Natl Acad Sci USA 114: 3103-3108. https://doi.org/10.1073/pnas.1621152114
[35]	Wang J, Moore PB (2017) On the interpretation of electron microscopic maps of biological macromolecules. Protein Sci 26: 122-129. https://doi.org/10.1002/pro.3060
[36]	Wang J, Natchiar SK, Moore PB, et al. (2021) Identification of Mg²⁺ ions next to nucleotides in cryo-EM maps using electrostatic potential maps. Acta Crystallogr Sect Struct Biol 77: 534-539. https://doi.org/10.1107/S2059798321001893
[37]	Bick T, Dominiak PM, Wendler P (2024) Exploiting the full potential of cryo-EM maps. BBA Adv 5: 100113. https://doi.org/10.1016/j.bbadva.2024.100113
[38]	Ray P, Klaholz BP, Finn RD, et al. (2003) Determination of Escherichia coli RNA polymerase structure by single particle cryoelectron microscopy. Method Enzymol 370: 24-42. https://doi.org/10.1016/S0076-6879(03)70003-2
[39]	van Heel M, Gowen B, Matadeen R, et al. (2000) Single-particle electron cryo-microscopy: towards atomic resolution. Q Rev Biophys 33: 307-369. https://doi.org/10.1017/s0033583500003644
[40]	Voronoi G (1907) Nouvelles applications des paramètres continus à la théorie des formes quadratiques. J Reine Angew Math 133: 97-178. https://doi.org/10.1515/crll.1908.133.97
[41]	Aurenhammer F, Klein R (2000) Handbook of Computational Geometry. Amsterdam, Netherlands: North-Holland 201-290.
[42]	Dirichlet GL (1850) Über die Reduktion der positiven quadratischen Formen mit drei unbestimmten ganzen Zahlen. J Reine Angew Math 40: 209-227. https://doi.org/10.1515/crll.1850.40.209
[43]	Emsley P, Lohkamp B, Scott WG, et al. (2010) Features and development of Coot. Acta Crystallogr D Biol Crystallogr 66: 486-501. https://doi.org/10.1107/S0907444910007493
[44]	Pettersen EF, Goddard TD, Huang CC, et al. (2004) UCSF Chimera--a visualization system for exploratory research and analysis. J Comput Chem 25: 1605-1612. https://doi.org/10.1002/jcc.20084
[45]	DeLano WL The PyMOL Molecular Graphics System (DeLano Scientific) (2002). Available from: http://www.pymol.org
[46]	Andronov L, Orlov I, Lutz Y, et al. (2016) ClusterViSu, a method for clustering of protein complexes by Voronoi tessellation in super-resolution microscopy. Sci Rep 6: 24084. https://doi.org/10.1038/srep24084
[47]	Andronov L, Michalon J, Ouararhni K, et al. (2018) 3DClusterViSu: 3D clustering analysis of super-resolution microscopy data by 3D Voronoi tessellations. Bioinformatics 34: 3004-3012. https://doi.org/10.1093/bioinformatics/bty200
[48]	Delaunay B (1928) Sur la sphère vide. Proc Int Math Congr : 695-700.
[49]	Delaunay B (1934) Sur la sphère vide. Izv Akad Nauk SSSR Otd Mat Estestv Nauk : 793-800.
[50]	Okabe A, Boots B, Sugihara K, et al. (2000) Spatial Tessellations: Concepts and Applications of Voronoi Diagrams. Chichester: Wiley.
[51]	van Heel M (1987) Angular reconstitution: a posteriori assignment of projection directions for 3D reconstruction. Ultramicroscopy 21: 111-123. https://doi.org/10.1016/0304-3991(87)90078-7
[52]	Scheres SHW (2012) RELION: implementation of a Bayesian approach to cryo-EM structure determination. J Struct Biol 180: 519-530. https://doi.org/10.1016/j.jsb.2012.09.006
[53]	Punjani A, Rubinstein JL, Fleet DJ, et al. (2017) cryoSPARC: algorithms for rapid unsupervised cryo-EM structure determination. Nat Methods 14: 290-296. https://doi.org/10.1038/nmeth.4169
[54]	Grant T, Rohou A, Grigorieff N (2018) cisTEM, user-friendly software for single-particle image processing. eLife : 7e35383. https://doi.org/10.7554/eLife.35383
[55]	Urzhumtseva L, Barchet C, Klaholz BP, et al. (2024) Program VUE: analysing distributions of cryo-EM projections using uniform spherical grids. J Appl Crystallogr 57: 865-876. https://doi.org/10.1107/S1600576724002383
[56]	van Heel M, Keegstra W, Schutter WG, et al. (1982) Arthropod hemocyanin studied by image analysis. Life Chem Rep Suppl 1: 5.
[57]	Saxton WO, Baumeister W (1982) The correlation averaging of a regularly arranged bacterial cell envelope protein. J Microsc 127: 127-138. https://doi.org/10.1111/j.1365-2818.1982.tb00405.x
[58]	Barchet C, von Loeffelholz O, Bahena-Ceron R, et al. (2026) Explicit correction of severely non-uniform distributions of cryo-EM views. Acta Crystallogr Sect Struct Biol 82: 65-78. https://doi.org/10.1107/S2059798326000306
[59]	Klaholz BP, Myasnikov AG, van Heel M (2004) Visualization of release factor 3 on the ribosome during termination of protein synthesis. Nature 427: 862-865. https://doi.org/10.1038/nature02332
[60]	White HE, Saibil HR, Ignatiou A, et al. (2004) Recognition and separation of single particles with size variation by statistical analysis of their images. J Mol Biol 336: 453-460. https://doi.org/10.1016/j.jmb.2003.12.015
[61]	Penczek PA, Frank J, Spahn CMT (2006) A method of focused classification, based on the bootstrap 3D variance analysis, and its application to EF-G-dependent translocation. J Struct Biol 154: 184-194. https://doi.org/10.1016/j.jsb.2005.12.013
[62]	Scheres SHW (2010) Classification of structural heterogeneity by maximum-likelihood methods. Methods Enzymol 482: 295-320. https://doi.org/10.1016/S0076-6879(10)82012-9
[63]	Klaholz BP (2015) Structure sorting of multiple macromolecular states in heterogeneous cryo-EM samples by 3D multivariate statistical analysis. Open J Stat 5: 820-836. https://doi.org/10.4236/ojs.2015.57081
[64]	Barchet C, Fréchin L, Holvec S, et al. (2023) Focused classifications and refinements in high-resolution single particle cryo-EM analysis. J Struct Biol 215: 108015. https://doi.org/10.1016/j.jsb.2023.108015
[65]	Holvec S, Barchet C, Lechner A, et al. (2024) The structure of the human 80S ribosome at 1.9 Å resolution reveals the molecular role of chemical modifications and ions in RNA. Nat Struct Mol Biol 31: 1251-1264. https://doi.org/10.1038/s41594-024-01274-x
[66]	Fréchin L, Holvec S, von Loeffelholz O, et al. (2023) High-resolution cryo-EM performance comparison of two latest-generation cryo electron microscopes on the human ribosome. J Struct Biol 215: 107905. https://doi.org/10.1016/j.jsb.2022.107905
[67]	Urzhumtsev AG (2026) Quantifying distributions of cryo-EM projections. Acta Crystallogr Sect Struct Biol 82: 100-112. https://doi.org/10.1107/S2059798325011611
[68]	Zhu D, Cao W, Li J, et al. (2024) Correction of preferred orientation-induced distortion in cryo-electron microscopy maps. Sci Adv 10: eadn0092. https://doi.org/10.1126/sciadv.adn0092
[69]	Liu YT, Fan H, Hu JJ, et al. (2025) Overcoming the preferred-orientation problem in cryo-EM with self-supervised deep learning. Nat Methods 22: 113-123. https://doi.org/10.1038/s41592-024-02505-1
[70]	Yan Y, Fan S, Yuan F, et al. (2026) A comprehensive foundation model for cryo-EM image processing. Nat Methods 23: 88-95. https://doi.org/10.1038/s41592-025-02916-8
[71]	Zhang H, Zheng D, Wu Q, et al. (2025) CryoPROS: Correcting misalignment caused by preferred orientation using AI-generated auxiliary particles. Nat Commun 16: 4565. https://doi.org/10.1038/s41467-025-59797-w
[72]	Sorzano COS, Semchonok D, Lin SC, et al. (2021) Algorithmic robustness to preferred orientations in single particle analysis by CryoEM. J Struct Biol 213: 107695. https://doi.org/10.1016/j.jsb.2020.107695
[73]	Myasnikov AG, Afonina ZA, Klaholz BP (2013) Single particle and molecular assembly analysis of polyribosomes by single- and double-tilt cryo electron tomography. Ultramicroscopy 126: 33-39. https://doi.org/10.1016/j.ultramic.2012.12.009

Reader Comments

Your name:*

Email:*
© 2026 the Author(s), licensee AIMS Press. This is an open access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/4.0)