Skip to main content

Thank you for visiting You are using a browser version with limited support for CSS. To obtain the best experience, we recommend you use a more up to date browser (or turn off compatibility mode in Internet Explorer). In the meantime, to ensure continued support, we are displaying the site without styles and JavaScript.

Colocalization for super-resolution microscopy via optimal transport


Super-resolution fluorescence microscopy is a widely used technique in cell biology. Stimulated emission depletion (STED) microscopy enables the recording of multiple-color images with subdiffraction resolution. The enhanced resolution leads to new challenges regarding colocalization analysis of macromolecule distributions. We demonstrate that well-established methods for the analysis of colocalization in diffraction-limited datasets and for coordinate-stochastic nanoscopy are not equally well suited for the analysis of high-resolution STED images. We propose optimal transport colocalization, which measures the minimal transporting cost below a given spatial scale to match two protein intensity distributions. Its validity on simulated data as well as on dual-color STED recordings of yeast and mammalian cells is demonstrated. We also extend the optimal transport colocalization methodology to coordinate-stochastic nanoscopy.

This is a preview of subscription content, access via your institution

Relevant articles

Open Access articles citing this article.

Access options

Rent or buy this article

Prices vary by article type



Prices may be subject to local taxes which are calculated during checkout

Fig. 1: Illustration of confocal and STED images obtained by Gaussian convolution of two point sources that are located at a distance of 45 nm.
Fig. 2: Illustration of optimal transport based on two protein distributions.
Fig. 3: OTC analysis.
Fig. 4: Evaluation of simulated images with sparse structures.
Fig. 5: Comparison of pixel-based methods and OTC for confocal and STED images.
Fig. 6: Proof of concept for OTC analysis on proteins with known proximity.
Fig. 7: The application of a 3D STED PSF enhances colocalization analysis.

Data availability

All data57 used to create the figures in the main text as well as in the supplement can be found in the Zenodo archive at as well as in the GitHub repository. The data for all figures and Extended Data figures are available in Source Data.

Code availability

The code58 is available on GitHub. The specific version of the OTC package and the scripts generating all figures in this paper can be found at To speed up computation we used the solver CPLEX (v12.6.3.0)59. This IBM product is free for academic use. To download the solver sign up for the IBM academic initiative and download the solver afterwards. To use the solver, download the transport package50 from CRAN as a tar.gz file and change the settings in the makevars file before installing the package. To reproduce any results from the paper please just run the respective script. Without the CPLEX solver, the runtime may take much longer or will not terminate on a standard laptop. With the CPLEX solver, the script for Fig. 5 requires less than 10 min runtime on a standard laptop. If you want to use the OTC package with your own data please see the read me on GitHub.


  1. Sahl, S. J., Hell, S. W. & Jakobs, S. Fluorescence nanoscopy in cell biology. Nat. Rev. Mol. Cell Biol. 18, 685–701 (2017).

    Google Scholar 

  2. Sigal, Y. M., Zhou, R. & Zhuang, X. Visualizing and discovering cellular structures with super-resolution microscopy. Science 361, 880–887 (2018).

    Google Scholar 

  3. Betzig, E. et al. Imaging intracellular fluorescent proteins at nanometer resolution. Science 313, 1642–1645 (2006).

    Article  Google Scholar 

  4. Hess, S. T., Girirajan, T. P. K. & Mason, M. D. Ultra-high resolution imaging by fluorescence photoactivation localization microscopy. Biophys. J. 91, 4258–4272 (2006).

    Google Scholar 

  5. Rust, M. J., Bates, M. & Zhuang, X. Sub-diffraction-limit imaging by stochastic optical reconstruction microscopy (STORM). Nat. Methods 3, 793–796 (2006).

    Google Scholar 

  6. Hell, S. W. Far-field optical nanoscopy. Science 316, 1153–1158 (2007).

    Google Scholar 

  7. Klar, T. A., Jakobs, S., Dyba, M., Egner, A. & Hell, S. W. Fluorescence microscopy with diffraction resolution barrier broken by stimulated emission. Proc. Natl Acad. Sci. USA 97, 8206–8210 (2000).

    Google Scholar 

  8. Hofmann, M., Eggeling, C., Jakobs, S. & Hell, S. W. Breaking the diffraction barrier in fluorescence microscopy at low light intensities by using reversibly photoswitchable proteins. Proc. Natl Acad. Sci. USA 102, 17565–17569 (2005).

    Google Scholar 

  9. Demandolx, D. & Davoust, J. Multicolour analysis and local image correlation in confocal microscopy. J Microsc. 185, 21–36 (1997).

    Google Scholar 

  10. Bolte, S. & Cordelières, F. P. A guided tour into subcellular colocalization analysis in light microscopy. J. Microsc. 224, 213–232 (2006).

    MathSciNet  Google Scholar 

  11. Worz, S. et al. 3D geometry-based quantification of colocalizations in multichannel 3D microscopy images of human soft tissue tumors. IEEE Trans. Med. Imaging 29, 1474–1484 (2010).

    Google Scholar 

  12. Zinchuk, V. & Grossenbacher-Zinchuk, O. Quantitative colocalization analysis of fluorescence microscopy images. Curr. Protoc. Cell Biol. 62, 4.19.1–4.19.14 (2014).

    Google Scholar 

  13. Costes, S. V. et al. Automatic and quantitative measurement of protein-protein colocalization in live cells. Biophys. J. 86, 3993–4003 (2004).

    Google Scholar 

  14. Dunn, K. W., Kamocka, M. M. & McDonald, J. H. A practical guide to evaluating colocalization in biological microscopy. Am. J. Physiol. Cell Physiol. 300, C723–C742 (2011).

    Google Scholar 

  15. Barlow, A. L., MacLeod, A., Noppen, S., Sanderson, J. & Guérin, C. J. Colocalization analysis in fluorescence micrographs: verification of a more accurate calculation of Pearson’s correlation coefficient. Microsc. Microanal. 16, 710–724 (2010).

    Google Scholar 

  16. Comeau, J. W. D., Costantino, S. & Wiseman, P. W. A guide to accurate fluorescence microscopy colocalization measurements. Biophys. J. 91, 4611–4622 (2006).

    Google Scholar 

  17. Manders, E. M., Stap, J., Brakenhoff, G. J., van Driel, R. & Aten, J. A. Dynamics of three-dimensional replication patterns during the S-phase, analysed by double labelling of DNA and confocal microscopy. J. Cell Sci. 103, 857–862 (1992).

    Google Scholar 

  18. Manders, E. M. M., Verbeek, F. J. & Aten, J. A. Measurement of co-localization of objects in dual-colour confocal images. J. Microsc. 169, 375–382 (1993).

    Google Scholar 

  19. Rizk, A. et al. Segmentation and quantification of subcellular structures in fluorescence microscopy images using Squassh. Nat. Protoc. 9, 586–596 (2014).

    Google Scholar 

  20. Wang, S., Fan, J., Pocock, G. & Yuan, M. Structured correlation detection with application to colocalization analysis in dual-channel fluorescence microscopic imaging. Statistica Sinica. 31, 333–360 (2021).

    Google Scholar 

  21. Wang, S. et al. Spatially adaptive colocalization analysis in dual-color fluorescence microscopy. Preprint at (2017).

  22. Coltharp, C., Yang, X. & Xiao, J. Quantitative analysis of single-molecule superresolution images. Curr. Opin. Struct. Biol. 28, 112–121 (2014).

    Google Scholar 

  23. Georgieva, M. et al. Nanometer resolved single-molecule colocalization of nuclear factors by two-color super resolution microscopy imaging. Methods 105, 44–55 (2016).

    Google Scholar 

  24. Lehmann, M. et al. Quantitative multicolor super-resolution microscopy reveals tetherin HIV-1 interaction. PLoS Pathog. 7, e1002456 (2011).

    Google Scholar 

  25. Malkusch, S. et al. Coordinate-based colocalization analysis of single-molecule localization microscopy data. Histochem. Cell Biol. 137, 1–10 (2012).

    Google Scholar 

  26. Lagache, T., Sauvonnet, N., Danglot, L. & Olivo-Marin, J.-C. Statistical analysis of molecule colocalization in bioimaging. Cytometry A 87, 568–579 (2015).

    Google Scholar 

  27. Lagache, T. et al. Mapping molecular assemblies with fluorescence microscopy and object-based spatial statistics. Nat. Commun. 9, 698 (2018).

    Google Scholar 

  28. Mukherjee, S., Gonzalez-Gomez, C., Danglot, L., Lagache, T. & Olivo-Marin, J. Generalizing the statistical analysis of objects’ spatial coupling in bioimaging. IEEE Signal Proces. Lett. 27, 1085–1089 (2020).

    Google Scholar 

  29. Ripley, B. D. The second-order analysis of stationary point processes. J. Appl. Probab. 13, 255–266 (1976).

    MathSciNet  MATH  Google Scholar 

  30. Helmuth, J. A., Paul, G. & Sbalzarini, I. F. Beyond co-localization: inferring spatial interactions between sub-cellular structures from microscopy images. BMC Bioinformatics 11, 372 (2010).

    Google Scholar 

  31. Shivanandan, A., Radenovic, A. & Sbalzarini, I. F. MosaicIA: an ImageJ/Fiji plugin for spatial pattern and interaction analysis. BMC Bioinformatics 14, 349 (2013).

    Google Scholar 

  32. Blom, H. et al. Nearest neighbor analysis of dopamine D1 receptors and Na(+)-K(+)-ATPases in dendritic spines dissected by STED microscopy. Microsc. Res. Tech. 75, 220–228 (2012).

    Google Scholar 

  33. Levet, F. et al. A tessellation-based colocalization analysis approach for single-molecule localization microscopy. Nature Commun. 10, 2379 (2019).

    Google Scholar 

  34. Zaritsky, A. et al. Decoupling global biases and local interactions between cell biological variables. eLife 6, e22323 (2017).

    Google Scholar 

  35. Peyré, G. & Cuturi, M. Computational optimal transport. Preprint at (2018).

  36. Klatt, M., Tameling, C. & Munk, A. Empirical regularized optimal transport: statistical theory and applications. SIAM J. Math. Data Sci. 2, 419–443 (2020).

    MathSciNet  Google Scholar 

  37. Monge, G. Mémoire sur la théorie des déblais et des remblais (De l’Imprimerie Royale, 1781).

  38. Kantorovich, L. V. On a problem of Monge. Usp. Mat. Nauk 3, 225–226 (1948).

    Google Scholar 

  39. Villani, C. Optimal Transport: Old and New (Springer Science & Business Media, 2008).

  40. Sommerfeld, M. & Munk, A. Inference for empirical Wasserstein distances on finite spaces. J. R. Stat. Soc. B 80, 219–238 (2018).

    MathSciNet  MATH  Google Scholar 

  41. Wang, S., Arena, E. T., Eliceiri, K. W. & Yuan, M. Automated and robust quantification of colocalization in dual-color fluorescence microscopy: a nonparametric statistical approach. IEEE Trans. Image Process. 27, 622–636 (2018).

    MathSciNet  MATH  Google Scholar 

  42. Göttfert, F. et al. Coaligned dual-channel STED nanoscopy and molecular diffusion analysis at 20 nm resolution. Biophys. J. 105, L01–L02 (2013).

    Google Scholar 

  43. Jans, D.C. et al. STED super-resolution microscopy reveals an array of MINOS clusters along human mitochondria. Proc. Natl Acad. Sci. USA 110, 8936–8941 (2013).

    Google Scholar 

  44. Stoldt, S. et al. Spatial orchestration of mitochondrial translation and OXPHOS complex assembly. Nat. Cell Biol. 20, 528–534 (2018).

    Google Scholar 

  45. Vogel, F., Bornhövd, C., Neupert, W. & Reichert, A. S. Dynamic subcompartmentalization of the mitochondrial inner membrane. J. Cell Biol. 175, 237–247 (2006).

    Google Scholar 

  46. de Chaumont, F. et al. Icy: an open bioimage informatics platform for extended reproducible research. Nat. Methods 9, 690–696 (2012).

    Google Scholar 

  47. Balzarotti, F. et al. Nanometer resolution imaging and tracking of fluorescent molecules with minimal photon fluxes. Science 355, 606–612 (2017).

    Google Scholar 

  48. Schmitzer, B. A sparse multiscale algorithm for dense optimal transport. J. Math. Imaging Vis. 56, 238–259 (2016).

    MathSciNet  MATH  Google Scholar 

  49. R Core Team R: A Language and Environment for Statistical Computing (R Foundation for Statistical Computing, 2018);

  50. Schuhmacher, D. et al. Transport: optimal transport in various forms. R package version 0.9-4 (2017).

  51. Billingsley, P. Convergence of Probability Measures (Wiley, 2013).

  52. Harke, B. et al. Resolution scaling in STED microscopy. Opt. Express 16, 4154–4162 (2008).

    Google Scholar 

  53. Lagache, T. Colocalization Studio in ICY. ICY (2021).

  54. Kehrein, K. et al. Organization of mitochondrial gene expression in two distinct ribosome-containing assemblies. Cell Rep. 10, 843–853 (2015).

    Google Scholar 

  55. Melin, J. et al. Presequence recognition by the Tom40 channel contributes to precursor translocation into the mitochondrial matrix. Mol. Cell. Biol. 34, 3473–3485 (2014).

    Google Scholar 

  56. Wurm, C. A., Neumann, D., Schmidt, R., Egner, A. & Jakobs, S. in Live Cell Imaging: Methods in Molecular Biology (ed. Papkovsky, D.) 185–199 (Humana Press, 2010).

  57. Tameling, C. et al. Simluated and real data. Zenodo (2021).

  58. Tameling, C. & Naas, J. ctameling/OTC: optimal transport colocalization. Zenodo (2021).

  59. IBM ILOG CPLEX Optimization Studio (IBM, 2018);

Download references


We thank P. Rehling from the University Medical Center Göttingen for providing an antiserum specific to Tom40. We are grateful to R. Schmitz-Salue for excellent technical assistance. Further thanks to J. Keller-Findeisen and F. Werner for helpful discussions about the simulation of STED images and to B. Schmitzer on the shielding algorithm. C.T. and J.N. gratefully acknowledge support by the DFG RTN 2088 Project A1. S.J. and A.M. acknowledge support of the DFG Cluster of Excellence MBExC 2067 and DFG-CRC 1456, Project C06. S.J. acknowledges support from the European Research Council (ERCAdG No. 835102).

Author information

Authors and Affiliations



A.M. and C.T. developed statistical methodology and algorithms. Furthermore, they performed computer experiments and art work jointly with J.N. S.J., T.S. and S.S. performed experiments and analyzed data jointly with A.M. and C.T. S.J., A.M. and C.T. wrote the manuscript with contributions from all authors. All authors read and approved the final manuscript.

Corresponding author

Correspondence to Axel Munk.

Ethics declarations

Competing interests

The authors declare no competing interests.

Additional information

Peer review informationNature Computational Science thanks Thomas Huser, Suvadip Mukherjee, and the other, anonymous, reviewer(s) for their contribution to the peer review of this work. Ananya Rastogi was the primary editor on this article and managed its editorial process and peer review in collaboration with the rest of the editorial team.

Publisher’s note Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Supplementary information

Supplementary Information

Supplementary Figs. 1–16, Tables 1–20 and additional simulations.

Reporting Summary

Supplementary Video 1

Simulation of increasing resolution and influence on Manders’ method and Pearson’s method.

Supplementary Video 2

Visualization of optimal transport between images Tom40 and Mprl4 in yeast cells.

Supplementary Video 3

Visualization of optimal transport of images of two different stainings of Tom40 in yeast cells.

Source data

Source Data Fig. 4

Statistical and image source data.

Source Data Fig. 5

Statistical and image source data.

Source Data Fig. 6

Statistical and image source data.

Source Data Fig. 7

Statistical and image source data.

Rights and permissions

Reprints and Permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Tameling, C., Stoldt, S., Stephan, T. et al. Colocalization for super-resolution microscopy via optimal transport. Nat Comput Sci 1, 199–211 (2021).

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI:

This article is cited by


Quick links

Nature Briefing

Sign up for the Nature Briefing newsletter — what matters in science, free to your inbox daily.

Get the most important science stories of the day, free in your inbox. Sign up for Nature Briefing