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.
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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.
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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.
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N.P.M and D.J.C. contributed equally to all aspects of this work.
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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.
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.
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.
Supplementary information
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Supplementary Figs. 1–19 and Discussion Sections I–XII.
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Numerical source data and mesh triangulations.
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Numerical source data and mesh triangulations.
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Numerical source data.
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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.
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Numerical source data.
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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
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DOI: https://doi.org/10.1038/s41592-023-02081-w