Skip to main content

Thank you for visiting nature.com. 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.

  • Article
  • Published:

TubULAR: tracking in toto deformations of dynamic tissues via constrained maps

Nature Methods volume 20, pages 1980–1988 (2023)Cite this article

Subjects

Abstract

A common motif in biology is the arrangement of cells into tubes, which further transform into complex shapes. Traditionally, analysis of dynamic tissues has relied on inspecting static snapshots, live imaging of cross-sections or tracking isolated cells in three dimensions. However, capturing the interplay between in-plane and out-of-plane behaviors requires following the full surface as it deforms and integrating cell-scale motions into collective, tissue-scale deformations. Here, we present an analysis framework that builds in toto maps of tissue deformations by following tissue parcels in a static material frame of reference. Our approach then relates in-plane and out-of-plane behaviors and decomposes complex deformation maps into elementary contributions. The tube-like surface Lagrangian analysis resource (TubULAR) provides an open-source implementation accessible either as a standalone toolkit or as an extension of the ImSAnE package used in the developmental biology community. We demonstrate our approach by analyzing shape change in the embryonic Drosophila midgut and beating zebrafish heart. The method naturally generalizes to in vitro and synthetic systems and provides ready access to the mechanical mechanisms relating genetic patterning to organ shape change.

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

Access options

Buy this article

  • Purchase on Springer Link
  • Instant access to full article PDF
Buy now

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

Fig. 1: TubULAR is a toolkit for tracking dynamic surfaces such as visceral organs.
Fig. 2: Coordinate parameterization follows 3D deformations of the evolving surface, enabling quantification of cell and tissue-scale dynamics, shown here for the fly midgut as it folds into compartments.
Fig. 3: Dynamic maps to the material frame yield covariant measures of tissue velocity and deformation in three dimensions.
Fig. 4: Tracing the in-plane and out-of-plane dynamics of a beating heart reveals the phased coupling between dilatational and transverse deformations.
Fig. 5: Decomposition of tissue motion in the beating heart reveals two principal components with non-reciprocal dynamics.

Similar content being viewed by others

Data availability

Data generated in this work are available at https://doi.org/10.6084/m9.figshare.c.6178351. Source data are provided with this paper.

Code availability

Software used in this study is available at https://github.com/npmitchell/tubular, with full documentation and tutorials at https://npmitchell.github.io/tubular/. Integration with ImSAnE is provided at https://github.com/npmitchell/imsane.

References

  1. Diaz-de-la-Loza, M.-D.-C. et al. Apical and basal matrix remodeling control epithelial morphogenesis. Dev. Cell 46, 23–39.e5 (2018).

  2. Bakkers, J., Verhoeven, M. C. & Abdelilah-Seyfried, S. Shaping the zebrafish heart: from left-right axis specification to epithelial tissue morphogenesis. Dev. Biol. 330, 213–220 (2009).

    Article  CAS  PubMed  Google Scholar 

  3. Iruela-Arispe, M. L. & Beitel, G. J. Tubulogenesis. Development 140, 2851–2855 (2013).

    Article  CAS  PubMed  PubMed Central  Google Scholar 

  4. Mitchell, N. P. et al. Visceral organ morphogenesis via calcium-patterned muscle constrictions. eLife 11, e77355 (2022).

  5. Shyer, A. E. et al. Villification: how the gut gets its villi. Science 342, 212–218 (2013).

  6. Beitel, G. J. & Krasnow, M. A. Genetic control of epithelial tube size in the Drosophila tracheal system. Development 127, 3271–3282 (2000).

    Article  CAS  PubMed  Google Scholar 

  7. Goodwin, K. et al. Smooth muscle differentiation shapes domain branches during mouse lung development. Development 146, dev181172 (2019).

  8. Gómez, H. F., Dumond, M. S., Hodel, L., Vetter, R. & Iber, D. 3D cell neighbour dynamics in growing pseudostratified epithelia. eLife 10, e68135 (2021).

    Article  PubMed  PubMed Central  Google Scholar 

  9. Klein, Y., Efrati, E. & Sharon, E. Shaping of elastic sheets by prescription of non-Euclidean metrics. Science 315, 1116–1120 (2007).

    Article  CAS  PubMed  Google Scholar 

  10. van Rees, W. M., Vouga, E. & Mahadevan, L. Growth patterns for shape-shifting elastic bilayers. Proc. Natl Acad. Sci. USA 114, 11597–11602 (2017).

    Article  PubMed  PubMed Central  Google Scholar 

  11. Maroudas-Sacks, Y. et al. Topological defects in the nematic order of actin fibres as organization centres of hydra morphogenesis. Nat. Phys. 17, 251–259 (2021).

  12. Stokkermans, A. et al. Muscular hydraulics drive larva-polyp morphogenesis. Curr. Biol. 32, 4707–4718.e8 (2022).

  13. Saadaoui, M., Rocancourt, D., Roussel, J., Corson, F. & Gros, J. A tensile ring drives tissue flows to shape the gastrulating amniote embryo. Science 367, 453–358 (2020).

    Article  CAS  PubMed  Google Scholar 

  14. Lefebvre, M. F., Claussen, N. H., Mitchell, N. P., Gustafson, H. J. & Streichan, S. J. Geometric control of Myosin II orientation during axis elongation. eLife 12, e78787 (2023).

    Article  PubMed  PubMed Central  Google Scholar 

  15. Mietke, A., Jülicher, F. & Sbalzarini, I. F. Self-organized shape dynamics of active surfaces. Proc. Natl Acad. Sci. USA 116, 29–34 (2019).

    Article  CAS  PubMed  Google Scholar 

  16. Forouhar, A. S. et al. The embryonic vertebrate heart tube is a dynamic suction pump. Science 312, 751–753 (2006).

  17. Savin, T. et al. On the growth and form of the gut. Nature 476, 57–62 (2011).

  18. Barbier de Reuille, P. et al. Morphographx: a platform for quantifying morphogenesis in 4F. eLife 4, e05864 (2015).

  19. Etournay, R. et al. Interplay of cell dynamics and epithelial tension during morphogenesis of the Drosophila pupal wing. eLife 4, e07090 (2015).

  20. Le Garrec, J. F. et al. A predictive model of asymmetric morphogenesis from 3D reconstructions of mouse heart looping dynamics. eLife 6, e28951 (2017).

  21. Streichan, S. J., Lefebvre, M. F., Noll, N., Wieschaus, E. F. & Shraiman, B. I. Global morphogenetic flow is accurately predicted by the spatial distribution of myosin motors. eLife 7, e27454 (2018).

  22. Wolff, C. et al. Multi-view light-sheet imaging and tracking with the MaMuT software reveals the cell lineage of a direct developing arthropod limb. eLife 7, e34410 (2018).

  23. Kierzkowski, D. et al. A growth-based framework for leaf shape development and diversity. Cell 177, 1405–1418.e17 (2019).

  24. Dalmasso, G. et al. 4d reconstruction of murine developmental trajectories using spherical harmonics. Dev. Cell 57, 2140–2150.e5 (2022).

  25. Schmid, B. et al. High-speed panoramic light-sheet microscopy reveals global endodermal cell dynamics. Nat. Commun. 4, 2207 (2013).

  26. Heemskerk, I. & Streichan, S. J. Tissue cartography: compressing bio-image data by dimensional reduction. Nat. Methods 12, 1139–1142 (2015).

    Article  CAS  PubMed  PubMed Central  Google Scholar 

  27. Herbert, S. et al. LocalZProjector and DeProj: a toolbox for local 2D projection and accurate morphometrics of large 3D microscopy images. BMC Biol. 19, 136 (2021).

  28. Mitchell, N. P. et al. Morphodynamic atlas for Drosophila development. Preprint at bioRxiv https://doi.org/10.1101/2022.05.26.493584 (2022).

  29. Chen, D.-Y., Lipari, K., Dehghan, Y., Streichan, S. & Bilder, D. Symmetry breaking in an edgeless epithelium by fat2-regulated microtubule polarity. Cell Reports 15, 1125–1133 (2016).

    Article  CAS  PubMed  Google Scholar 

  30. Zhou, Z., Alégot, H. & Irvine, K. D. Oriented cell divisions are not required for Drosophila wing shape. Curr. Biol. 29, 856–864.e3 (2019).

    Article  PubMed  PubMed Central  Google Scholar 

  31. Gallagher, K. D., Mani, M. & Carthew, R. W. Emergence of a geometric pattern of cell fates from tissue-scale mechanics in the Drosophila eye. eLife 11, e72806 (2022).

  32. Madhu, R. et al. Characterizing the cellular architecture of dynamically remodeling vascular tissue using 3-D image analysis and virtual reconstruction. Mol. Biol. Cell 31, 1714–1725 (2020).

  33. Candeo, A. et al. Virtual unfolding of light sheet fluorescence microscopy dataset for quantitative analysis of the mouse intestine. J. Biomed. Optics 21, 56001 (2016).

  34. Immerglück, K., Lawrence, P. A. & Bienz, M. Induction across germ layers in Drosophila mediated by a genetic cascade. Cell 62, 261–268 (1990).

    Article  PubMed  Google Scholar 

  35. Bienz, M. & Tremml, G. Domain of ultrabithorax expression in drosophila visceral mesoderm from autoregulation and exclusion. Nature 333, 576–578 (1988).

    Article  CAS  PubMed  Google Scholar 

  36. Reuter, R. & Scott, M. P. Expression and function of the homoeotic genes antennapedia and sex combs reduced in the embryonic midgut of drosophila. Development 109, 289–303 (1990).

    Article  CAS  PubMed  Google Scholar 

  37. Bakkers, J. Zebrafish as a model to study cardiac development and human cardiac disease. Cardiovasc. Res. 91, 279–88 (2011).

    Article  CAS  PubMed  PubMed Central  Google Scholar 

  38. Bhat, S., Ohn, J. & Liebling, M. Motion-based structure separation for label-free high-speed 3-D cardiac microscopy. IEEE Trans. Image Process. 21, 3638–3647 (2012).

    Article  PubMed  Google Scholar 

  39. Stainier, D., Lee, R. & Fishman, M. Cardiovascular development in the zebrafish. I. Myocardial fate map and heart tube formation. Development 119, 31–40 (1993).

    Article  CAS  PubMed  Google Scholar 

  40. Chan, K. G., Streichan, S. J., Trinh, L. A. & Liebling, M. Simultaneous temporal superresolution and denoising for cardiac fluorescence microscopy. IEEE Trans. Comput. Imag. 2, 348–358 (2016).

    Article  Google Scholar 

  41. Pawley, J. Handbook of Biological Confocal Microscopy Vol. 236 (Springer, 2006).

  42. Keller, P. J., Schmidt, A. D., Wittbrodt, J. & Stelzer, E. H. Reconstruction of zebrafish early embryonic development by scanned light sheet microscopy. Science 322, 1065–1069 (2008).

    Article  CAS  PubMed  Google Scholar 

  43. Krzic, U., Gunther, S., Saunders, T. E., Streichan, S. J. & Hufnagel, L. Multiview light-sheet microscope for rapid in toto imaging. Nat. Methods 9, 730–733 (2012).

    Article  CAS  PubMed  Google Scholar 

  44. Chen, B.-C. et al. Lattice light-sheet microscopy: imaging molecules to embryos at high spatiotemporal resolution. Science 346, 1257998 (2014).

  45. Campos-Ortega, J. & Hartenstein, V. The Embryonic Development of Drosophila melanogaster (Springer, 1997).

  46. Klapper, R. et al. The formation of syncytia within the visceral musculature of the drosophila midgut is dependent on DUF, SNS and MBC. Mech. Dev. 110, 85–96 (2002).

  47. Wolfstetter, G. et al. Fusion of circular and longitudinal muscles in Drosophila is independent of the endoderm but further visceral muscle differentiation requires a close contact between mesoderm and endoderm. Mech. Dev. 126, 721–736 (2009).

  48. Chan, T. F. & Vese, L. A. Active contours without edges. IEEE Trans. Image Process. 10, 266–277 (2001).

    Article  CAS  PubMed  Google Scholar 

  49. Berg, S. et al. ilastik: interactive machine learning for (bio)image analysis. Nat. Methods 16, 1226–1232 (2019).

  50. Landau, L. D. & Lifshitz, E. M. Fluid Mechanics, Volume 6, Course of Theoretical Physics 2nd edn (Butterworth-Heinemann, 1987).

  51. Blanchard, G. B. et al. Tissue tectonics: morphogenetic strain rates, cell shape change and intercalation. Nat. Methods 6, 458–464 (2009).

  52. Crane, K., de Goes, F., Desbrun, M. & Schröder, P. Digital geometry processing with discrete exterior calculus. In Proc. CM SIGGRAPH 2013 courses, SIGGRAPH ’13, Article 7, 1-126 (Association for Computing Machinery, 2013).

  53. Desbrun, M., Hirani, A. N., Leok, M. & Marsden, J. E. Discrete exterior calculus. Preprint at https://doi.org/10.48550/arXiv.math/0508341 (2005).

  54. Romeo, N., Hastewell, A., Mietke, A. & Dunkel, J. Learning developmental mode dynamics from single-cell trajectories. eLife 10, e68679 (2021).

  55. Lévy, B. & Zhang, H. R. Spectral mesh processing. In Proc. ACM SIGGRAPH 2010 Courses, SIGGRAPH ’10, Article 8, 1–312 (Association for Computing Machinery, 2010).

  56. Jolliffe, I. Principal Component Analysis (Springer Verlag, 2002).

  57. Huang, C.-J., Tu, C.-T., Hsiao, C.-D., Hsieh, F.-J. & Tsai, H.-J. Germ-line transmission of a myocardium-specific GFP transgene reveals critical regulatory elements in the cardiac myosin light chain 2 promoter of zebrafish. Dev. Dynamics 228, 30–40 (2003).

    Article  CAS  Google Scholar 

  58. Metzger, R. J., Klein, O. D., Martin, G. R. & Krasnow, M. A. The branching programme of mouse lung development. Nature 453, 745–750 (2008).

    Article  CAS  PubMed  PubMed Central  Google Scholar 

  59. Goodwin, K. & Nelson, C. M. Branching morphogenesis. Development 147, dev184499 (2020).

    Article  CAS  PubMed  Google Scholar 

  60. Karzbrun, E. et al. Human neural tube morphogenesis in vitro by geometric constraints. Nature 599, 268–272 (2021).

  61. Tayar, A. M. et al. Controlling liquid-liquid phase behaviour with an active fluid. Nat. Materials 22, 1401–1408 (2023).

  62. Lemma, B., Mitchell, N. P., Subramanian, R., Needleman, D. J. & Dogic, Z. Active microphase separation in mixtures of microtubules and tip-accumulating molecular motors. Phys. Rev. X 12, 031006 (2022).

    CAS  PubMed  PubMed Central  Google Scholar 

  63. Martin-Bermudo, M. D., Dunin-Borkowski, O. M., Brown, N. H. Specificity of ps integrin function during embryogenesis resides in the alpha subunit extracellular domain. EMBO J. 16, 4184–4193 (1997).

  64. Sens, K. L. et al. An invasive podosome-like structure promotes fusion pore formation during myoblast fusion. J. Cell Biol. 191, 1013–1027 (2010).

  65. Welte, M. A., Gross, S. P., Postner, M., Block, S. M. & Wieschaus, E. F. Developmental regulation of vesicle transport in Drosophila embryos: forces and kinetics. Cell 92, 547–557 (1998).

    Article  CAS  PubMed  Google Scholar 

  66. Osher, S. J. & Fedkiw, R. Level Set Methods and Dynamic Implicit Surfaces Vol. 153 (Springer, 2003).

  67. Marquez-Neila, P., Baumela, L. & Alvarez, L. A morphological approach to curvature-based evolution of curves and surfaces. IEEE Trans. Pattern Anal. Mach. Intell. 36, 2–17 (2014).

    Article  PubMed  Google Scholar 

  68. Lorensen, W. E. & Cline, H. E. Marching cubes: a high resolution 3D surface construction algorithm. In Proc. 14th Annual Conference on Computer Graphics and Interactive Techniques - SIGGRAPH ’87 Vol. 21, 163–169 (ACM Press, 1987).

  69. Thielicke, W. & Sonntag, R. Particle image velocimetry for MATLAB: accuracy and enhanced algorithms in PIVlab. J. Open Res. Softw. 9, 1–14 (2021).

  70. Struik, D. J. Lectures on Classical Differential Geometry (Courier Corporation, 1961).

  71. Arroyo, M. & Desimone, A. Relaxation dynamics of fluid membranes. Phys. Rev. E 79, 031915 (2009).

  72. Efrati, E., Sharon, E. & Kupferman, R. Elastic theory of unconstrained non-Euclidean plates. J. Mech. Phys. Solids 57, 762–775 (2009).

    Article  Google Scholar 

Download references

Acknowledgements

S. Streichan provided insights, mentorship, expertise and the laboratory and computational resources to develop and execute this work, with primary support for this work from the National Science Foundation (NSF) grant no. PHY-2047140. B. Shraiman provided additional insights and mentorship. We thank S. Streichan and M. Liebling for the light-sheet dataset of the beating zebrafish heart and A. Tayar for the dataset of the deforming DNA droplet in a microtubule gel (Supplementary Information Section I). We also thank S. Shankar and F. Brauns for useful discussions. Research reported in this publication was supported by NIH NICHD award no. K99HD110675. N.P.M. acknowledges support from the Helen Hay Whitney Foundation. D.J.C. acknowledges support from the NSF grant no. PHY-1707973. The work was also supported in part by the NSF grants PHY-1748958 and PHY-2309135 to the Kavli Institute for Theoretical Physics.

Author information

Author notes

  1. Dillon J. Cislo

    Present address: Center for Studies in Physics and Biology, The Rockefeller University, New York, NY, USA

  2. These authors contributed equally: Noah P. Mitchell, Dillon J. Cislo.

Authors and Affiliations

  1. Kavli Institute for Theoretical Physics, University of California Santa Barbara, Santa Barbara, CA, USA

    Noah P. Mitchell

  2. Department of Physics, University of California Santa Barbara, Santa Barbara, CA, USA

    Noah P. Mitchell & Dillon J. Cislo

Authors
  1. Noah P. Mitchell
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Dillon J. Cislo
    View author publications

    You can also search for this author in PubMed Google Scholar

Contributions

N.P.M and D.J.C. contributed equally to all aspects of this work.

Corresponding authors

Correspondence to Noah P. Mitchell or Dillon J. Cislo.

Ethics declarations

Competing interests

The authors declare no competing interests.

Peer review

Peer review information

Nature Methods thanks Idse Heemskerk, Timothy Saunders and the other, anonymous, reviewer(s) for their contribution to the peer review of this work. Peer reviewer reports are available. Primary Handling Editor: Madhura Mukhopadhyay, in collaboration with the Nature Methods team.

Additional information

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

Extended data

Extended Data Fig. 1 Global parameterization of tube-like surfaces with material coordinates proceeds by a sequence of mapping steps.

The 3D surface is first mapped via f to the plane, either through Ricci flow (slower but results in a more exactly conformal map) or through minimization of a Dirichlet energy (faster but less precisely conformal, see Supplementary Information Section VIIa). In either case, the material is periodic in the circumferential v dimension and finite in extent along the longitudinal u direction. The resulting coordinate system is then adjusted. First, we apply Z: u → s, where s is a distance along the longitude of the tissue defined by Eq. (1), which we find aids in parameterization for tubes with varying radii (Supplementary Information Section VIIb). If the timepoint under question is the reference timepoint t0, this defines the material coordinates. Otherwise, if t > t0, we then apply Φ: v → ϕ, where ϕ is given by Eq. (1), and then apply J to stabilize the resulting coordinates based on material motion measured through particle image velocimetry (phase correlation analysis) relative to the previous timepoint.

Extended Data Fig. 2 Overlaid pullback images spanning morphogenesis demonstrate the stability of the pullback parameterization against 3D motion of the tissue.

Using planar maps of the folding Drosophila midgut, we perform refined Lagrangian parameterization of the surface. The resulting timepoints at 0, 35, and 70 minutes after constriction onset are overlaid in cyan, magenta, and yellow, respectively. Much of the tissue appears as black and white, indicating that tissue placement in the pullback frame is stationary.

Extended Data Fig. 3 TubULAR aids in measuring the contribution of cell intercalations to tissue-scale convergent extension.

a, In the absence of cell divisions, in-plane tissue-scale convergent extension occurs due to the changing shape of cells as well as the occurrence of oriented cell intercalations (‘T1 events’). b, During constrictions in the fly midgut, no cell divisions take place, but cells change shape and also intercalate in the endodermal layer. Scale bars are 10 μm. c, We can then compare the cumulative effect of each contribution (blue and yellow) to the total tissue-scale convergent extension (orange, constricting along ϕ and extending along s). In the midgut endoderm, we directly measure tissue shear from the deviatoric component of the integrated strain computed from Lagrangian pathlines in 3D. This shear strain is almost entirely oriented along the longitudinal axis s. In order to compare directly this quantity to the cell shape change, we imprint the segmentation of 1260 cells at t = 0 on the tissue surface, follow the outlines of these cells along 3D tissue pathlines obtained from full stabilization J∘Φ∘Z∘f with small Gaussian smoothing applied to the optical flow stabilization J to avoid self-intersections in pathlines, and compute the cell shape anisotropy \((1-\frac{a}{b})\cos 2\theta\) in the tangent plane of the tissue for each cell. Here, a and b are the semimajor and semiminor axes of the ellipse capturing each cell’s moment of inertia tensor, and θ is the cell’s angle with respect to the material frame’s longitudinal axis. We excluded advected polygons that acquire partial self-intersections from advection, so that we excluded 3, 5, and 8 out of a total 1260 advected cell shapes at the latest timepoints 63, 73, and 83 minutes after constriction onset, respectively. We compared these advected segmentation shapes with the true segmentation. After passing pullback images through a skeletonization procedure detailed in ref. 4, we manually selected n= 961, 925, 1028, 1262, 883, 833, 912, 1184, 659, 783, 663, and 964 cells for accuracy with broad organ coverage at each timepoint. Blue and yellow curves represent the net contribution of cell shape change and the net contribution of intercalations averaged across the organ, and the red curve represents the mean tissue shape change. Error bars denote standard error on the mean. Further technical details are found in TubULAR’s generateCellSegmentationPathlines3D method.

Source data

Extended Data Fig. 4 TubULAR measures the kinematic coupling between in-plane and out-of-plane motion, computing the rate of local area change across the organ, shown here for the developing midgut.

a, The underlying out-of-plane deformation, defined as the normal motion vn times twice the mean curvature H, shows negative values at each constriction, where the mean curvature becomes negative. b, DEC computation of the divergence of the in-plane velocity ∇ ⋅ v∥ shows patterns of sinks in the constrictions and sources in the chambers’ lobes, in synchrony with the out-of-plane deformation. c, As a result of the match between in-plane and out-of-plane dynamics, the areal growth rate remains relatively quiescent. d, Regions of tissue which experience positive divergence (the lobes of each gut chamber) tend to experience modest areal growth, while regions with negative tissue divergence experience slight areal compression. This is the relatively quiescent signature of the tissue’s compressibility. Here, data is averaged across three biological repeats, with shaded band denoting standard deviation and tick marks denoting standard error on the mean. e, An example kymograph from a single embryo’s developing midgut showing the small but persistent areal strain rate. The tissue expands in the lobes of each chamber (red in the kymograph) and contracts near each constriction (dashed lines and red arrows). The kymograph is aligned such that each vertical line follows a ring of tissue as it deforms in 3D. In other words, measurements are made in the Lagrangian frame of reference. The anterior-posterior position (horizontal axis) is parameterized in the material frame at the onset of the first constriction.

Source data

Extended Data Fig. 5 TubULAR reveals the tangential deformations of a beating embryonic zebrafish heart.

a, A rendering of cardiomyocyte fluorescence via TexturePatch shows cyclic deformations with a beat period, T. See also Fig. 4a. b, TubULAR maps the tangential component of the 3D velocity field onto the pullback plane. Color denotes the tangential velocity direction along the long axis (purple or orange) or along the circumferential axis (green or red). The opacity of the rendered image reflects the magnitude of the tangential velocity, as do the length of overlaid black arrows.

Source data

Supplementary information

Supplementary Information

Supplementary Figs. 1–19 and Discussion Sections I–XII.

Reporting Summary

Peer Review File

Source data

Source Data Fig. 1

Numerical source data and mesh triangulations.

Source Data Fig. 2

Mesh triangulations.

Source Data Fig. 3

Numerical source data and mesh triangulations.

Source Data Fig. 4

Numerical source data.

Source Data Fig. 5

Numerical source data and mode decomposition images.

Source Data Extended Data Fig. 3

Statistical source data and segmentation images.

Source Data Extended Data Fig. 4

Numerical source data and statistical source data.

Source Data Extended Data Fig. 5

Numerical source data.

Rights and permissions

Springer Nature or its licensor (e.g. a society or other partner) holds exclusive rights to this article under a publishing agreement with the author(s) or other rightsholder(s); author self-archiving of the accepted manuscript version of this article is solely governed by the terms of such publishing agreement and applicable law.

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Mitchell, N.P., Cislo, D.J. TubULAR: tracking in toto deformations of dynamic tissues via constrained maps. Nat Methods 20, 1980–1988 (2023). https://doi.org/10.1038/s41592-023-02081-w

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1038/s41592-023-02081-w

This article is cited by

Search

Advanced search

Quick links

Nature Briefing AI and Robotics

Sign up for the Nature Briefing: AI and Robotics newsletter — what matters in AI and robotics research, free to your inbox weekly.

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