Reuse & Permissions

It is not necessary to obtain permission to reuse this article or its components as it is available under the terms of the Creative Commons Attribution 4.0 International license. This license permits unrestricted use, distribution, and reproduction in any medium, provided attribution to the author(s) and the published article's title, journal citation, and DOI are maintained. Please note that some figures may have been included with permission from other third parties. It is your responsibility to obtain the proper permission from the rights holder directly for these figures.

Export citation

Export citation

Choose format for download:

Download Citation
  • Open Access

Morphogen Patterning in Dynamic Tissues

Alex M. Plum and Mattia Serra*

  • *Contact author: mserra@ucsd.edu

PRX Life 3, 043009 – Published 6 November, 2025

DOI: https://doi.org/10.1103/h74q-3dgj

Abstract

Embryogenesis integrates morphogenesis—coordinated cell movements—with morphogen patterning and cell differentiation. While largely studied independently, morphogenesis and patterning often unfold simultaneously in early embryos. Yet how cell movements affect morphogen transport and cells' exposure over time remains unclear, as most pattern formation models assume static tissues. Here we develop a theoretical framework for morphogen patterning in dynamic tissues, recasting advection-reaction-diffusion equations in the cells' moving reference frames. This framework (i) elucidates how morphogenesis mediates morphogen transport and compartmentalization: cell-cell diffusive transport is enhanced at multicellular flow attractors, while repellers act as barriers, affecting cell fate induction and bifurcations. (ii) It formalizes cell-cell signaling ranges in dynamic tissues, deconfounding morphogenetic movements to identify which cells could communicate via morphogens. (iii) It provides two new nondimensional numbers to assess when and where morphogenesis affects morphogen transport. We demonstrate this framework by analyzing classical patterning models with common morphogenetic motifs as well as experimental tissue flows. Our work rationalizes dynamic tissue patterning in development, constraining candidate patterning mechanisms and parameters using accessible cell motion data.

View figure in article

Physics Subject Headings (PhySH)

Article Text

Supplemental Material

References (105)

  1. L. Wolpert, C. Tickle, and A. M. Arias, Principles of Development (Oxford University Press, New York, 2015).
  2. A. M. Turing, The chemical basis of morphogenesis, Philos. Trans. R. Soc. Lond. Ser. B 237, 37 (1952).
  3. F. Crick, Diffusion in embryogenesis, Nature (London) 225, 420 (1970).
  4. E. A. Zamir, B. J. Rongish, and C. D. Little, The ECM moves during primitive streak formation—Computation of ECM versus cellular motion, PLoS Biol. 6, e247 (2008).
  5. T. Fulton, B. Verd, and B. Steventon, The unappreciated generative role of cell movements in pattern formation, R. Soc. Open Sci. 9, 211293 (2022).
  6. L. Busby and B. Steventon, Tissue tectonics and the multi-scale regulation of developmental timing, Interface Focus 11, 20200057 (2021).
  7. J. V. Veenvliet, P.-F. Lenne, D. A. Turner, I. Nachman, and V. Trivedi, Sculpting with stem cells: How models of embryo development take shape, Development 148, dev192914, (2021).
  8. R. D. J. G. Ho, K. Kishi, M. Majka, A. Kicheva, and M. Zagorski, Dynamics of morphogen source formation in a growing tissue, PLoS Comput. Biol. 20, e1012508 (2024).
  9. M. Ibrahimi and M. Merkel, Stabilization of active tissue deformation by a dynamic morphogen gradient, arXiv:2412.15774.
  10. T. Rozario and D. W. DeSimone, The extracellular matrix in development and morphogenesis: A dynamic view, Dev. Biol. 341, 126 (2010).
  11. A. Kumar, J. K. Placone, and A. J. Engler, Understanding the extracellular forces that determine cell fate and maintenance, Development 144, 4261 (2017).
  12. G. Halder, S. Dupont, and S. Piccolo, Transduction of mechanical and cytoskeletal cues by YAP and TAZ, Nat. Rev. Mol. Cell Biol. 13, 591 (2012).
  13. P. Caldarelli, A. Chamolly, A. Villedieu, O. Alegria-Prévot, C. Phan, J. Gros, and F. Corson, Self-organized tissue mechanics underlie embryonic regulation, Nature (London) 633, 887 (2024).
  14. C. Collinet and T. Lecuit, Programmed and self-organized flow of information during morphogenesis, Nat. Rev. Mol. Cell Biol. 22, 245 (2021).
  15. A. G. Fletcher, M. Osterfield, R. E. Baker, and S. Y. Shvartsman, Vertex models of epithelial morphogenesis, Biophys. J. 106, 2291 (2014).
  16. 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).
  17. S. Harmansa, A. Erlich, C. Eloy, G. Zurlo, and T. Lecuit, Growth anisotropy of the extracellular matrix shapes a developing organ, Nat. Commun. 14, 1220 (2023).
  18. J. F. Fuhrmann, A. Krishna, J. Paijmans, C. Duclut, G. Cwikla, S. Eaton, M. Popović, F. Jülicher, C. D. Modes, and N. A. Dye, Active shape programming drives drosophila wing eversion, Sci. Adv. 10, eadp0860 (2024).
  19. A. Gierer and H. Meinhardt, A theory of biological pattern formation, Kybernetik 12, 30 (1972).
  20. S. Kondo and R. Asai, A reaction–diffusion wave on the skin of the marine angelfish pomacanthus, Nature (London) 376, 765 (1995).
  21. P. Müller, K. W. Rogers, B. M. Jordan, J. S. Lee, D. Robson, S. Ramanathan, and A. F. Schier, Differential diffusivity of nodal and lefty underlies a reaction-diffusion patterning system, Science 336, 721 (2012).
  22. A. D. Economou, A. Ohazama, T. Porntaveetus, P. T. Sharpe, S. Kondo, M. A. Basson, A. Gritli-Linde, M. T. Cobourne, and J. B. A. Green, Periodic stripe formation by a turing mechanism operating at growth zones in the mammalian palate, Nat. Genet. 44, 348 (2012).
  23. J. Raspopovic, L. Marcon, L. Russo, and J. Sharpe, Digit patterning is controlled by a Bmp-Sox9-Wnt Turing network modulated by morphogen gradients, Science 345, 566 (2014).
  24. E. Grall, C. Feregrino, S. Fischer, A. D. Courten, F. Sacher, T. W. Hiscock, and P. Tschopp, Self-organized BMP signaling dynamics underlie the development and evolution of digit segmentation patterns in birds and mammals, Proc. Natl. Acad. Sci. USA 121, e2304470121 (2024).
  25. L. Wolpert, Positional information and the spatial pattern of cellular differentiation, J. Theor. Biol. 25, 1 (1969).
  26. J. B. A. Green, and J. Sharpe, Positional information and reaction-diffusion: Two big ideas in developmental biology combine, Development 142, 1203 (2015).
  27. P. Gross, K. V. Kumar, N. W. Goehring, J. S. Bois, C. Hoege, F. Jülicher, and S. W. Grill, Guiding self-organized pattern formation in cell polarity establishment, Nat. Phys. 15, 293 (2019).
  28. S. Meindlhumer, F. Brauns, J. R. Finžgar, J. Kerssemakers, C. Dekker, and E. Frey, Directing min protein patterns with advective bulk flow, Nat. Commun. 14, 450 (2023).
  29. E. J. Crampin, E. A. Gaffney, and P. K. Maini, Reaction and diffusion on growing domains: Scenarios for robust pattern formation, Bull. Math. Biol. 61, 1093 (1999).
  30. L. Li, X. Wang, J. Chai, X. Wang, A. Buganza-Tepole, and D. M. Umulis, Determining the role of advection in patterning by bone morphogenetic proteins through neural network model-based acceleration of a 3d finite element model of the zebrafish embryo, Front. Syst. Biol. 2, 983372 (2022).
  31. D. Summerbell, J. H. Lewis, and L. Wolpert, Positional information in chick limb morphogenesis, Nature (London) 244, 492 (1973).
  32. L. Wolpert, Positional information and pattern formation, Philos. Trans. Roy. Soc. Lond. B 295, 441 (1981).
  33. L. Wolpert, The progress zone model for specifying positional information, Int. J. Dev. Biol. 46, 869 (2002).
  34. P. Fried and D. Iber, Dynamic scaling of morphogen gradients on growing domains, Nat. Commun. 5, 5077 (2014).
  35. T. Fulton, V. Trivedi, A. Attardi, K. Anlas, C. Dingare, A. M. Arias, and B. Steventon, Axis specification in zebrafish is robust to cell mixing and reveals a regulation of pattern formation by morphogenesis, Curr. Biol. 30, 2984 (2020).
  36. T. Fulton, K. Spiess, L. Thomson, Y. Wang, B. Clark, S. Hwang, B. Paige, B. Verd, and B. Steventon, Cell rearrangement generates pattern emergence as a function of temporal morphogen exposure, bioRxiv (2022), https://doi.org/10.1101/2021.02.05.429898.
  37. K. Spiess, S. E. Taylor, T. Fulton, K. Toh, D. Saunders, S. Hwang, Y. Wang, B. Paige, B. Steventon, and B. Verd, Approximated gene expression trajectories for gene regulatory network inference on cell tracks, Iscience 27, 110840 (2024).
  38. E. Dessaud, L. L. Yang, K. Hill, B. Cox, F. Ulloa, A. Ribeiro, A. Mynett, B. G. Novitch, and J. Briscoe, Interpretation of the sonic hedgehog morphogen gradient by a temporal adaptation mechanism, Nature (London) 450, 717 (2007).
  39. N. Balaskas, A. Ribeiro, J. Panovska, E. Dessaud, N. Sasai, K. M. Page, J. Briscoe, and V. Ribes, Gene regulatory logic for reading the sonic hedgehog signaling gradient in the vertebrate neural tube, Cell 148, 273 (2012).
  40. J. Dubrulle, M. J. McGrew, and O. Pourquié, FGF signaling controls somite boundary position and regulates segmentation clock control of spatiotemporal Hox gene activation, Cell 106, 219 (2001).
  41. A. Kicheva and J. Briscoe, Control of tissue development by morphogens, Annu. Rev. Cell Dev. Biol. 39, 91 (2023).
  42. E. G. Hagos, and S. T. Dougan, Time-dependent patterning of the mesoderm and endoderm by nodal signals in zebrafish, BMC Dev. Biol. 7, 22 (2007).
  43. E. Camacho-Aguilar, S. T. Yoon, M. A. Ortiz-Salazar, S. Du, M. C. Guerra, and A. Warmflash, Combinatorial interpretation of BMP and WNT controls the decision between primitive streak and extraembryonic fates, Cell Syst. 15, 445 (2024).
  44. S. Teague, G. Primavera, B. Chen, Z.-Y. Liu, L. Yao, E. Freeburne, H. Khan, K. Jo, C. Johnson, and I. Heemskerk, Time-integrated BMP signaling determines fate in a stem cell model for early human development, Nat. Commun. 15, 1471 (2024).
  45. R. Loganathan, B. J. Rongish, C. M. Smith, M. B. Filla, A. Czirok, B. Bénazéraf, and C. D. Little, Extracellular matrix motion and early morphogenesis, Development 143, 2056 (2016).
  46. See Supplemental Material at http://link.aps.org/supplemental/10.1103/h74q-3dgj for additional details, Figs. S1–S5, Algorithm 1, and Movies 1–6.
  47. X. Z. Tang and A. H. Boozer, Finite time Lyapunov exponent and advection-diffusion equation, Physica D 95, 283 (1996).
  48. J.-L. Thiffeault, Advection–diffusion in Lagrangian coordinates, Phys. Lett. A 309, 415 (2003).
  49. G. Haller, D. Karrasch, and F. Kogelbauer, Material barriers to diffusive and stochastic transport, Proc. Natl. Acad. Sci. USA 115, 9074 (2018).
  50. G. Schlissel, M. Meziane, D. Narducci, A. S. Hansen, and P. Li, Diffusion barriers imposed by tissue topology shape hedgehog morphogen gradients, Proc. Natl. Acad. Sci. USA 121, e2400677121 (2024).
  51. M. Serra, S. Streichan, M. Chuai, C. J. Weijer, and L. Mahadevan, Dynamic morphoskeletons in development, Proc. Natl. Acad. Sci. USA 117, 11444 (2020).
  52. S. Mowlavi, M. Serra, E. Maiorino, and L. Mahadevan, Detecting Lagrangian coherent structures from sparse and noisy trajectory data, J. Fluid Mech. 948, A4 (2022).
  53. M. Chuai, G. S. Nájera, M. Serra, L. Mahadevan, and C. J. Weijer, Reconstruction of distinct vertebrate gastrulation modes via modulation of key cell behaviors in the chick embryo, Sci. Adv. 9, eabn5429 (2023).
  54. M. Lange, A. Granados, S. VijayKumar, J. Bragantini, S. Ancheta, Y.-J. Kim, S. Santhosh, M. Borja, H. Kobayashi, E. McGeever et al., A multimodal zebrafish developmental atlas reveals the state-transition dynamics of late-vertebrate pluripotent axial progenitors, Cell 187, 6742 (2024).
  55. C. Sinigaglia, F. Braghin, and M. Serra, Optimal control of short-time attractors in active nematics, Phys. Rev. Lett. 132, 218302 (2024).
  56. C. Guillot, Y. Djeffal, M. Serra, and O. Pourquie, Control of epiblast cell fate by mechanical cues, bioRxiv (2024), https://doi.org/10.1101/2024.06.24.600402.
  57. M. Serra, L. Lemma, L. Giomi, Z. Dogic, and L. Mahadevan, Defect-mediated dynamics of coherent structures in active nematics, Nat. Phys. 19, 1355 (2023).
  58. G. S. Nájera, A. M. Plum, B. Steventon, C. J. Weijer, and M. Serra, Control of tissue flows and embryo geometry in avian gastrulation, Nature Communications 16, 5174 (2025).
  59. T. Ma, V. Grudtsyna, R. V. Bölsterli, and A. Doostmohammadi, Cellular flow architecture exposes the hidden mechanics of biological matter, arXiv:2508.17974.
  60. G. Haller, Lagrangian coherent structures, Annu. Rev. Fluid Mech. 47, 137 (2015).
  61. S. Izumoto, D. M. Escala, J. Heyman, T. Le Borgne, and A. De Wit, Control of chemical waves by fluid stretching and compression, Phys. Rev. Lett. 133, 218001 (2024).
  62. T. Gregor, E. F. Wieschaus, A. P. McGregor, W. Bialek, and D. W. Tank, Stability and nuclear dynamics of the bicoid morphogen gradient, Cell 130, 141 (2007).
  63. O. Voiculescu, L. Bodenstein, I.-J. Lau, and C. D. Stern, Local cell interactions and self-amplifying individual cell ingression drive amniote gastrulation, eLife 3, e01817 (2014).
  64. P. Müller, K. W. Rogers, S. R. Yu, M. Brand, and A. F. Schier, Morphogen transport, Development 140, 1621 (2013).
  65. L. A. Davidson, B. D. Davidson, R. Keller, and D. W. Desimone, Live imaging of cell protrusive activity, and extracellular matrix assembly and remodeling during morphogenesis in the frog, enopus laevis, Dev. Dyn. 237, 2684 (2008).
  66. M. Nebyla, M. Přibyl, and I. Schreiber, Effects of convective transport on chemical signal propagation in epithelia, Biophys. J. 102, 990 (2012).
  67. Abdel Fattah AR, F. Sgualdino, S. Poovathingal, K. Davie, P. Baatsen, K. Vints, N. Gunko, and A. Ranga, Extracellular matrix flow guides in-vitro epithelial morphogenesis, bioRxiv (2022), https://doi.org/10.1101/2022.10.21.513217.
  68. R. T. Foister and TGM V. D. Ven, Diffusion of Brownian particles in shear flows, J. Fluid Mech. 96, 105 (1980).
  69. E. B. Vadas, R. G. Cox, H. L. Goldsmith, and S. G. Mason, The microrheology of colloidal dispersions: II. Brownian diffusion of doublets of spheres, J. Colloid Interface Sci. 57, 308 (1976).
  70. K. M. Reilly, and D. A. Melton, Short-range signaling by candidate morphogens of the TGFβ family and evidence for a relay mechanism of induction, Cell 86, 743 (1996).
  71. L. Liu, A. Nemashkalo, L. Rezende, J. Y. Jung, S. Chhabra, M. C. Guerra, I. Heemskerk, and A. Warmflash, Nodal is a short-range morphogen with activity that spreads through a relay mechanism in human gastruloids, Nat. Commun. 13, 497 (2022).
  72. J. E. M. Dickmann, J. C. Rink, and F. Jülicher, Long-range morphogen gradient formation by cell-to-cell signal propagation, Phys. Biol. 19, 066001 (2022).
  73. C. R. Shalizi, K. L. Shalizi, and R. Haslinger, Quantifying self-organization with optimal predictors, Phys. Rev. Lett. 93, 118701 (2004).
  74. S. Aaronson, S. M. Carroll, and L. Ouellette, Quantifying the rise and fall of complexity in closed systems: The coffee automaton, arXiv:1405.6903.
  75. A. A. Teleman, M. Strigini, and S. M. Cohen, Shaping morphogen gradients, Cell 105, 559 (2001).
  76. A. Kicheva, P. Pantazis, T. Bollenbach, Y. Kalaidzidis, T. Bittig, F. Julicher, and M. Gonzalez-Gaitan, Kinetics of morphogen gradient formation, Science 315, 521 (2007).
  77. E. Rozbicki, M. Chuai, A. I. Karjalainen, F. Song, H. M. Sang, R. Martin, H.-J. Knölker, M. P. MacDonald, and C. J. Weijer, Myosin-II-mediated cell shape changes and cell intercalation contribute to primitive streak formation, Nat. Cell Biol. 17, 397 (2015).
  78. M. Saadaoui, D. Rocancourt, J. Roussel, F. Corson, and J. Gros, A tensile ring drives tissue flows to shape the gastrulating amniote embryo, Science 367, 453 (2020).
  79. G. S. Nájera and C. J. Weijer, Cellular processes driving gastrulation in the avian embryo, Mech. Dev. 163, 103624 (2020).
  80. C. F. Arias, M. A. Herrero, C. D. Stern, and F. Bertocchini, A molecular mechanism of symmetry breaking in the early chick embryo, Sci. Rep. 7, 15776 (2017).
  81. H. C. Lee, C. Hastings, N. M. M. Oliveira, R. Pérez-Carrasco, K. M. Page, L. Wolpert, and C. D. Stern, Neighbourhood watch'model: Embryonic epiblast cells assess positional information in relation to their neighbours, Development 149, dev200295 (2022).
  82. H. C. Lee, N. M. M. Oliveira, C. Hastings, P. Baillie-Benson, A. A. Moverley, H.-C. Lu, Y. Zheng, E. L. Wilby, T. T. Weil, K. M. Page et al., Regulation of long-range BMP gradients and embryonic polarity by propagation of local calcium-firing activity, Nat. Commun. 15, 1463 (2024).
  83. A. Nemashkalo, A. Ruzo, I, Heemskerk, and A. Warmflash, Morphogen and community effects determine cell fates in response to BMP4 signaling in human embryonic stem cells, Development 144, 3042 (2017).
  84. L. Guignard, U.-M. Fiúza, B. Leggio, J. Laussu, E. Faure, G. Michelin, K. Biasuz, L. Hufnagel, G. Malandain, C. Godin et al., Contact area–dependent cell communication and the morphological invariance of ascidian embryogenesis, Science 369, eaar5663 (2020).
  85. R. H. Chisholm, B. D. Hughes, and K. A. Landman, Building a morphogen gradient without diffusion in a growing tissue, PLoS One 5, e12857 (2010).
  86. I. Averbukh, D. Ben-Zvi, S. Mishra, and N. Barkai, Scaling morphogen gradients during tissue growth by a cell division rule, Development 141, 2150 (2014).
  87. G. S. Nájera, Analysis and Modulation of cell behaviours driving avian gastrulation, PhD. thesis, University of Dundee, 2021.
  88. W Thompson d'Arcy, On Growth and Form (Cambridge University Press, Cambridge, UK, 1917).
  89. A. D. Lander, Morpheus unbound: Reimagining the morphogen gradient, Cell 128, 245 (2007).
  90. H. Acloque, O. H. Ocana, D. Abad, C. D. Stern, and M. A. Nieto, Snail2 and Zeb2 repress P-cadherin to define embryonic territories in the chick embryo, Development 144, 649 (2017).
  91. A. Raffaelli and C. D. Stern, Signaling events regulating embryonic polarity and formation of the primitive streak in the chick embryo, Curr. Top. Dev. Biol. 136, 85 (2020).
  92. H. C. Lee, C. Hastings, and C. D. Stern, The extra-embryonic area opaca plays a role in positioning the primitive streak of the early chick embryo, Development 149, dev200303 (2022).
  93. H. C. Lee, N. M. M. Oliveira, and C. D. Stern, Exploring the roles of FGF/MAPK and cVG1/GDF on mesendoderm induction and convergent extension during chick primitive streak formation, Dev. Genes Evol. 232, 115 (2022).
  94. D. Psychoyos and C. D. Stern, Fates and migratory routes of primitive streak cells in the chick embryo, Development 122, 1523 (1996).
  95. J. Michael and W. Slack, From Egg to Embryo: Regional Specification in Early Development (Cambridge University Press, Cambridge, UK, 1991).
  96. Y. Hatada and C. D. Stern, A fate map of the epiblast of the early chick embryo, Development 120, 2879 (1994).
  97. G. Mundhe, V. Dunsing-Eichenauer, J.-M. Philippe, E. D. Silva, C. Collinet, and T. Lecuit, Spatial patterning of contractility by a mechanogen gradient underlies Drosophila gastrulation, bioRxiv (2025), https://doi.org/10.1101/2025.04.11.648359.
  98. S. Javaherian, N. Anesiadis, R. Mahadevan, and A. P. McGuigan, Design principles for generating robust gene expression patterns in dynamic engineered tissues, Integr. Biol. 5, 578 (2013).
  99. K. Mizuno, T. Hirashima, and S. Toda, Robust tissue pattern formation by coupling morphogen signal and cell adhesion, EMBO Rep. 25, 4803 (2024).
  100. F. Corson and E. D. Siggia, Geometry, epistasis, and developmental patterning, Proc. Natl. Acad. Sci. USA 109, 5568 (2012).
  101. D. A. Rand, A. Raju, M. Sáez, F. Corson, and E. D. Siggia, Geometry of gene regulatory dynamics, Proc. Natl. Acad. Sci. USA 118, e2109729118, (2021).
  102. M. Sáez, R. Blassberg, E. Camacho-Aguilar, E. D. Siggia, D. A. Rand, and J. Briscoe, Statistically derived geometrical landscapes capture principles of decision-making dynamics during cell fate transitions, Cell Syst. 13, 12 (2022).
  103. R. Thom, Structural Stability and Morphogenesis (CRC Press, Boca Raton, FL, 2018).
  104. A. Raju and E. D. Siggia, A geometrical perspective on development, Dev. Growth Differ. 65, 245 (2023).
  105. A. M. Plum and M. Serra, Dynamical systems of fate and form in development, Semin. Cell Dev. Biol. 172, 103620 (2025).

Outline

Information

Sign In to Your Journals Account

Filter

Filter

Article Lookup

Enter a citation