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  • Open Access

Stabilization of Active Tissue Deformation by a Dynamic Signaling Gradient

Muhamet Ibrahimi1,2 and Matthias Merkel1,*

  • 1Aix Marseille Univ, Université de Toulon, CNRS, CPT (UMR 7332), Turing Centre for Living Systems, Marseille, France
  • 2Laboratory of Artificial and Natural Evolution (LANE), Department of Genetics and Evolution, University of Geneva, 1211 Geneva, Switzerland

  • *Contact author: matthias.merkel@cnrs.fr
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PRX Life 3, 043013 – Published 19 November, 2025

DOI: https://doi.org/10.1103/yq1s-p2zl

Abstract

A key process during animal morphogenesis is oriented tissue deformation, which is often driven by internally generated active stresses. Yet, such active oriented materials are prone to well-known instabilities, raising the question of how oriented tissue deformation can be robust during morphogenesis. Here we study under which conditions active oriented deformation can be stabilized by the concentration pattern of a signaling molecule, which is secreted by a localized source region, diffuses across the tissue, and degrades. Consistent with earlier results, we find that oriented tissue deformation is always unstable in the gradient-contractile case, i.e., when active stresses act to contract the tissue along the direction of the signaling gradient, and we show that this is true even in the limit of large diffusion. However, active deformationcan be stabilized in the gradient-extensile case, i.e., when active stresses act to extend the tissue along the direction of the signaling gradient. Specifically, we show that gradient-extensile systems can be stable when the tissue is already elongated in the direction of the gradient. We moreover point out the existence of a formerly unknown, additional instability of the tissue shape change. This instability results from the interplay of active tissue shear and signal diffusion, and it indicates that some additional feedback mechanism may be required to control the target tissue shape. Taken together, our theoretical results provide quantitative criteria for robust active tissue deformation and explain the lack of gradient-contractile systems in the biological literature, suggesting that the active matter instability acts as an evolutionary selection criterion.

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References (63)

  1. L. Wolpert, C. Tickle, and A. M. Arias, Principles of Development (Oxford University Press, Oxford, UK, 2015).
  2. J. A. Zallen and E. Wieschaus, Patterned gene expression directs bipolar planar polarity in Drosophila, Dev. Cell 6, 343 (2004).
  3. B. Bénazéraf, P. François, R. E. Baker, N. Denans, C. D. Little, and O. Pourquié, A random cell motility gradient downstream of FGF controls elongation of an amniote embryo, Nature (London) 466, 248 (2010).
  4. A. Shindo, Models of convergent extension during morphogenesis, WIREs Dev. Biol. 7, e293 (2018).
  5. K. A. Johansen, D. D. Iwaki, and J. A. Lengyel, Localized JAK/STAT signaling is required for oriented cell rearrangement in a tubular epithelium, Development 130, 135 (2003).
  6. C. M. Karner, R. Chirumamilla, S. Aoki, P. Igarashi, J. B. Wallingford, and T. J. Carroll, Wnt9b signaling regulates planar cell polarity and kidney tubule morphogenesis, Nat. Genet. 41, 793 (2009).
  7. A. Saxena, B. Denholm, S. Bunt, M. Bischoff, K. VijayRaghavan, and H. Skaer, Epidermal growth factor signal- ling controls myosin II planar polarity to orchestrate convergent extension movements during Drosophila tubulogenesis, PLoS Biol. 12, e1002013 (2014).
  8. R. Etournay, M. Popović, M. Merkel, A. Nandi, C. Blasse, B. Aigouy, H. Brandl, G. Myers, G. Salbreux, F. Jülicher, and S. Eaton, Interplay of cell dynamics and epithelial tension during morphogenesis of the Drosophila pupal wing, eLife 4, e07090 (2015).
  9. S. Hopyan, Biophysical regulation of early limb bud morphogenesis, Dev. Biol. 429, 429 (2017).
  10. H. Tao, M. Zhu, K. Lau, O. K. W. Whitley, M. Samani, X. Xiao, X. X. Chen, N. A. Hahn, W. Liu, M. Valencia, M. Wu, X. Wang, K. D. Fenelon, C. C. Pasiliao, D. Hu, J. Wu, S. Spring, J. Ferguson, E. P. Karuna, R. M. Henkelman et al., Oscillatory cortical forces promote three dimensional cell intercalations that shape the murine mandibular arch, Nat. Commun. 10, 1703 (2019).
  11. C. Bertet, L. Sulak, and T. Lecuit, Myosin-dependent junction remodelling controls planar cell intercalation and axis elongation, Nature (London) 429, 667 (2004).
  12. F. Bosveld, I. Bonnet, B. Guirao, S. Tlili, Z. Wang, A. Petitalot, R. Marchand, P.-L. Bardet, P. Marcq, F. Graner, and Y. Bellaiche, Mechanical control of morphogenesis by fat/dachsous/four-jointed planar cell polarity pathway, Science 336, 724 (2012).
  13. C. Collinet, M. Rauzi, P. F. Lenne, and T. Lecuit, Local and tissue-scale forces drive oriented junction growth during tissue extension, Nat. Cell Biol. 17, 1247 (2015).
  14. M. Behrndt, G. Salbreux, P. Campinho, R. Hauschild, F. Oswald, J. Roensch, S. W. Grill, and C. P. Heisenberg, Forces driving epithelial spreading in zebrafish gastrulation, Science 338, 257 (2012).
  15. S. J. Streichan, M. Lefebvre, N. Noll, E. F. Wieschaus, and B. I. Shraiman, Global morphogenetic flow is accurately predicted by the spatial distribution of myosin motors, eLife 7, e27454 (2018).
  16. A. Stokkermans, A. Chakrabarti, K. Subramanian, L. Wang, S. Yin, P. Moghe, P. Steenbergen, G. Mönke, T. Hiiragi, R. Prevedel, L. Mahadevan, and A. Ikmi, Muscular hydraulics drive larva-polyp morphogenesis, Curr. Biol. 32, 4707 (2022).
  17. E. W. Gehrels, B. Chakrabortty, M.-E. Perrin, M. Merkel, and T. Lecuit, Curvature gradient drives polarized tissue flow in the Drosophila embryo, Proc. Natl. Acad. Sci. USA 120, e2214205120 (2023).
  18. N. A. Dye, M. Popovic, K. V. Iyer, J. F. Fuhrmann, R. Piscitello-Gómez, S. Eaton, and F. Julicher, Self-organized patterning of cell morphology via mechanosensitive feedback, eLife 10, e57964 (2021).
  19. M. Serra, G. S. Nájera, M. Chuai, A. M. Plum, S. Santhosh, V. Spandan, C. J. Weijer, and L. Mahadevan, A mechanochemical model recapitulates distinct vertebrate gastrulation modes, Sci. Adv. 9, eadh8152 (2023).
  20. S. Gsell, S. Tlili, M. Merkel, and P.-F. Lenne, Marangoni-like tissue flows enhance symmetry breaking of embryonic organoids, Nat. Phys. 21, 644 (2025).
  21. K. Barrett, S. Anand, V. Thome, L. Kodjabachian, M. Merkel, and P.-F. Lenne, Epithelial-mesenchymal interface guides cell shapes and axis elongation in embryonic explants, Development 152, dev204344 (2025).
  22. N. H. Claussen, F. Brauns, and B. I. Shraiman, A geometric-tension-dynamics model of epithelial convergent extension, Proc. Natl. Acad. Sci. USA 121, e2321928121 (2024).
  23. A. Ioratim-Uba, T. B. Liverpool, and S. Henkes, Mechanochemical active feedback generates convergence extension in epithelial tissue, Phys. Rev. Lett. 131, 238301 (2023).
  24. R. A. Simha and S. Ramaswamy, Hydrodynamic fluctuations and instabilities in ordered suspensions of self-propelled particles, Phys. Rev. Lett. 89, 058101 (2002).
  25. R. Voituriez, J. F. Joanny, and J. Prost, Spontaneous flow transition in active polar gels, Europhys. Lett. 70, 404 (2005).
  26. M. C. Marchetti, J. F. Joanny, S. Ramaswamy, T. B. Liverpool, J. Prost, M. Rao, and R. A. Simha, Hydrodynamics of soft active matter, Rev. Mod. Phys. 85, 1143 (2013).
  27. S. von der Hardt, J. Bakkers, A. Inbal, L. Carvalho, L. Solnica-Krezel, C. P. Heisenberg, and M. Hammerschmidt, The bmp gradient of the zebrafish gastrula guides migrating lateral cells by regulating cell-cell adhesion, Curr. Biol. 17, 475 (2007).
  28. K. W. Rogers and A. F. Schier, Morphogen gradients: From generation to interpretation, Annu. Rev. Cell Dev. Biol. 27, 377 (2011).
  29. P. Müller, K. W. Rogers, S. R. Yu, M. Brand, and A. F. Schier, Morphogen transport, Development 140, 1621 (2013).
  30. G. Tkačik, J. O. Dubuis, M. D. Petkova, and T. Gregor, Positional information, positional error, and readout precision in morphogenesis: A mathematical framework, Genetics 199, 39 (2015).
  31. L. S. Mosby, A. E. Bowen, and Z. Hadjivasiliou, Morphogens in the evolution of size, shape and patterning, Development 151, dev202412 (2024).
  32. H. Ninomiya, R. P. Elinson, and R. Winklbauer, Antero-posterior tissue polarity links mesoderm convergent extension to axial patterning, Nature (London) 430, 364 (2004).
  33. J. Lavalou, Q. Mao, S. Harmansa, S. Kerridge, A. C. Lellouch, J. M. Philippe, S. Audebert, L. Camoin, and T. Lecuit, Formation of polarized contractile interfaces by self-organized Toll-8/Cirl GPCR asymmetry, Dev. Cell 56, 1574 (2021).
  34. Z. Wang, M. C. Marchetti, and F. Brauns, Patterning of morphogenetic anisotropy fields, Proc. Natl. Acad. Sci. USA 120, e2220167120 (2023).
  35. M. Ibrahimi and M. Merkel, Deforming polar active matter in a scalar field gradient, New J. Phys. 25, 013022 (2023).
  36. M. F. Lefebvre, N. H. Claussen, N. P. Mitchell, H. J. Gustafson, and S. J. Streichan, Geometric control of myosin-II orientation during axis elongation, eLife 12, e78787 (2023).
  37. A. M. Plum and M. Serra, Morphogen patterning in dynamic tissues, PRX Life 3, 043009 (2025).
  38. O. Wartlick, P. Mumcu, A. Kicheva, T. Bittig, C. Seum, F. Jülicher, and M. González-Gaitán, Dynamics of Dpp signaling and proliferation control, Science 331, 1154 (2011).
  39. R. Mateus, L. Holtzer, C. Seum, Z. Hadjivasiliou, M. Dubois, F. Jülicher, and M. Gonzalez-Gaitan, BMP signaling gradient scaling in the zebrafish pectoral fin, Cell Rep. 30, 4292 (2020).
  40. M. Romanova-Michaelides, Z. Hadjivasiliou, D. Aguilar-Hidalgo, D. Basagiannis, C. Seum, M. Dubois, F. Jülicher, and M. Gonzalez-Gaitan, Morphogen gradient scaling by recycling of intracellular dpp, Nature (London) 602, 287 (2022).
  41. O. Wartlick, A. Kicheva, and M. González-Gaitán, Morphogen gradient formation, Cold Spring Harbor Perspect. Biol. 1, a001255 (2009).
  42. Y. Wang, X. Wang, T. Wohland, and K. Sampath, Extracellular interactions and ligand degradation shape the nodal morphogen gradient, eLife 5, e13879 (2016).
  43. A. Huang, C. Amourda, S. Zhang, N. S. Tolwinski, and T. E. Saunders, Decoding temporal interpretation of the morphogen bicoid in the early Drosophila embryo, eLife 6, 1 (2017).
  44. A. C. Paré, A. Vichas, C. T. Fincher, Z. Mirman, D. L. Farrell, A. Mainieri, and J. A. Zallen, A positional toll receptor code directs convergent extension in Drosophila, Nature (London) 515, 523 (2014).
  45. M. A. Benton, M. Pechmann, N. Frey, D. Stappert, K. H. Conrads, Y. T. Chen, E. Stamataki, A. Pavlopoulos, and S. Roth, Toll genes have an ancestral role in axis elongation, Curr. Biol. 26, 1609 (2016).
  46. D. Jia, Q. Xu, Q. Xie, W. Mio, and W. M. Deng, Automatic stage identification of Drosophila egg chamber based on DAPI images, Sci. Rep. 6, 18850 (2016).
  47. H. Alégot, P. Pouchin, O. Bardot, and V. Mirouse, Jak-Stat pathway induces Drosophila follicle elongation by a gradient of apical contractility, eLife 7, e32943 (2018).
  48. D. D. Iwaki, K. A. Johansen, J. B. Singer, and J. A. Lengyel, Drumstick, bowl, and lines are required for patterning and cell rearrangement in the Drosophila embryonic hindgut, Dev. Biol. 240, 611 (2001).
  49. M. L. Williams and L. Solnica-Krezel, Nodal and planar cell polarity signaling cooperate to regulate zebrafish convergence and extension gastrulation movements, eLife 9, e54445 (2020).
  50. B. Steventon, F. Duarte, R. Lagadec, S. Mazan, J. F. Nicolas, and E. Hirsinger, Species-specific contribution of volumetric growth and tissue convergence to posterior body elongation in vertebrates, Development (Cambridge) 143, 1732 (2016).
  51. A. Kicheva, P. Pantazis, T. Bollenbach, Y. Kalaidzidis, T. Bittig, F. Jülicher, and M. González-Gaitán, Kinetics of morphogen gradient formation, Science 315, 521 (2007).
  52. A. Tiribocchi, R. Wittkowski, D. Marenduzzo, and M. E. Cates, Active model H: Scalar active matter in a momentum-conserving fluid, Phys. Rev. Lett. 115, 188302 (2015).
  53. T. R. Kirkpatrick and J. K. Bhattacherjee, Driven active matter: Fluctuations and a hydrodynamic instability, Phys. Rev. Fluids 4, 024306 (2019).
  54. B. Gao, Wnt regulation of planar cell polarity (PCP), Curr. Topics Dev. Biol. 101, 263 (2012).
  55. M. Merkel, A. Sagner, F. S. Gruber, R. Etournay, C. Blasse, E. Myers, S. Eaton, and F. Jülicher, The balance of prickle/spiny-legs isoforms controls the amount of coupling between core and fat PCP systems, Curr. Biol. 24, 2111 (2014).
  56. See Supplemental Material at http://link.aps.org/supplemental/10.1103/yq1s-p2zl for movies showing the simulations.
  57. S. S. Lienkamp, K. Liu, C. M. Karner, T. J. Carroll, O. Ronneberger, J. B. Wallingford, and G. Walz, Vertebrate kidney tubules elongate using a planar cell polarity-dependent, rosette-based mechanism of convergent extension, Nat. Genet. 44, 1382 (2012).
  58. O. Ossipova, K. Kim, and S. Y. Soko, Planar polarization of Vangl2 in the vertebrate neural plate is controlled by Wnt and Myosin II signaling, Biology Open 4, 722 (2015).
  59. M. Ibrahimi, Robustness of active anisotropic deformation in developing biological tissues, Ph.D. thesis, Aix-Marseille Université, 2022.
  60. M. Lefebvre, J. Colen, N. Claussen, F. Brauns, M. Raich, N. Mitchell, M. Fruchart, V. Vitelli, and S. J. Streichan, Learning a conserved mechanism for early neuroectoderm morphogenesis, arXiv:2405.18382v1.
  61. M. Ibrahimi and M. Merkel, Scripts for “Stabilization of active tissue deformation by a dynamic signaling gradient“, PRX Life (2025), doi: 10.24433/CO.8370709.v1.
  62. M. Roberts and J. C. Bowman, Dealiased convolutions for pseudospectral simulations, J. Phys.: Conf. Ser. 318, 072037 (2011).
  63. J. C. Bowman and M. Roberts, Efficient dealiased convolutions without padding, SIAM J. Sci. Comput. 33, 386 (2011).

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PRX Life 3, 043013– Published 19 November, 2025
  • Received 21 March 2025
  • Accepted 17 October 2025
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Published by the American Physical Society under the terms of the Creative Commons Attribution 4.0 International license. Further distribution of this work must maintain attribution to the author(s) and the published article's title, journal citation, and DOI.

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