foambryo-2d

foambryo-2d is a python package based on Polyscope designed to infer surface tensions in 2D epithelial-cell-clusters pip install foambryo-2d It is the lightweight 2D adaptation of the 3d force-inference software foambryo, available soon.

We rely on delaunay-watershed, a tool we designed to reconstruct efficiently multimaterial meshes from instance segmentations. From these multimaterial meshes, we can efficiently and robustly extract junctional angles and invert Young-Dupré laws, to find back the physical parameters involved in the mechanical equilibrium: surface tensions $\gamma_{ij}$.

Example

Load an instance segmentation, reconstruct its multimaterial mesh, infer and visualize the forces

pip install foambryo-2d

from dw2d import geometry_reconstruction_2d
from foambryo2d import infer_tension,plot_tension_inference

## Load the labels
import skimage.io as io
Segmentation = io.imread("Segmentation.tif")

## Reconstruct a multimaterial mesh from the labels
DW = geometry_reconstruction_2d(label, min_dist=3)
Mesh = DW.return_dcel()

## Infer and view the forces
_,dict_tensions,_ = infer_tension(Mesh,mean_tension=1,check_topo_changes=False,mode='YD')
plot_tension_inference(Mesh,dict_tensions)

Installation

pip install foambryo2d

API and Documentation

1 - Creating a Mesh object

The first step is to load your multimaterial mesh into a DCEL_Data object via the builder DCEL_Data(Verts, Edges_multimaterial):

• Verts is an V x 2 numpy array of vertex positions
• Edges_multimaterial is an F x 4 numpy array of face and material indices, where at each row the 2 first indices refers to a vertex and the 2 last refer to a given material, 0 being the exterior media

2 - Infer the forces

Then the second step is to use this Mesh object to infer the tensions and pressions

• infer_tension(Mesh,mean_tension=1,mode='YD'): We infer tensions ratios by inverting junctional equilibrium relations
  • Mesh is a DCEL_Data object
  • mean_tension has to be defined as we only infer ratio between tensions
  • check-topo-changes checks if a mechanical equilibrium can be attained or if the resulting configuration will lead to a topology change
  • mode is the formula used to infer the tensions. It has to be choosen among: YD, Eq, Projection_YD, cotan, inv_cotan, Lamy, inv_Lamy, Lamy_Log, Variational

3 - Visualize

• `plot_tension_inference(Mesh,dict_tensions=None,original_image = None,plot_original_image = False)
  • Mesh is a DCEL_Data object
  • dict_tensions is the dictionnary obtained with infer_tension, and is computed automatically if unspecified.
  • original_image: the original image is plotted if provided

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Credits, contact, citations

If you use this tool, please cite the associated preprint. Do not hesitate to contact Hervé Turlier for practical questions and applications. We hope that foambryo2d could help biologists and physicists to shed light on the mechanical aspects of early development.

@article {Yamamoto2023.03.07.531437,
	author = {Yamamoto, Kazunori and Ichbiah, Sacha and Perez, Matthieu and Borrego-Pinto, Joana and Delbary, Fabrice and Goehring, Nate and Turlier, Herv{\'e} and Charras, Guillaume},
	title = {Spatiotemporal mapping of the contractile and adhesive forces sculpting early C. elegans embryos},
	elocation-id = {2023.03.07.531437},
	year = {2025},
	doi = {10.1101/2023.03.07.531437},
	publisher = {Cold Spring Harbor Laboratory},
	abstract = {Embryo shape is determined by individual cell mechanics, intercellular interaction strength, and geometrical constraints. Models based on surface tensions at cell interfaces can predict 3D static cellular arrangements within aggregates. However, predicting the dynamics of such arrangements is challenging due to difficulties in measuring temporal changes in tensions. Here, we characterise the spatiotemporal changes in cellular tensions shaping the early nematode embryo using AFM, live microscopy, and tension inference. Using excoriated embryos, we validate a hybrid inference pipeline that calibrates relative inferred tensions temporally using cortical myosin enrichment and absolute tensions using AFM measurements. Applied to embryos within their native shell, we infer a spatiotemporal map of absolute tensions, revealing that ABa, ABp, and EMS compaction is driven by increased tension at free surfaces, while P2{\textquoteright}s initial exclusion is due to high tension at intercellular contacts. We uncover a direct and non-affine contribution of cadherins to cell-cell contact tension, comparable to cadherins{\textquoteright} indirect contribution via actomyosin regulation.HighlightsP lineage cells have lower cortical tensions than AB lineage cellsEnrichment of Myosin-II at the cell cortex is a good predictor of cell-medium tension but is not sufficient to determine tension at cell-cell contacts.Myosin-informed tension inference allows determination of the spatiotemporal evolution of all surface tensions within the embryo.ABa, ABp, and EMS compact due to high tensions at their cell-medium interfaces compared to their cell-cell interfaces, while P2 is initially excluded due to high cell-cell contact tensions.Cadherins contribute directly in a non-linear way by reducing cell-cell contact tension by nearly 50\%.Open Access For the purpose of Open Access, the author has applied a CC BY public copyright license to any Author Accepted Manuscript version arising from this submission.Competing Interest StatementThe authors have declared no competing interest.Grant-in-Aid for JSPS Fellows, 16J09469Uehara Memorial Foundation, https://ror.org/00gc20a07European Research Council, CoG-647186Cancer Research UK, FC001086Medical Research Council, FC001086Wellcome Trust, FC001086European Research Council, Grant agreement No. 949267NIH Office of Research Infrastructure Programs, P40 OD010440},
	issn = {2692-8205},
	URL = {https://www.biorxiv.org/content/early/2025/07/01/2023.03.07.531437},
	eprint = {https://www.biorxiv.org/content/early/2025/07/01/2023.03.07.531437.full.pdf},
	journal = {bioRxiv}
}

License

This work is licensed under a Creative Commons Attribution-NonCommercial-ShareAlike 4.0 International License.