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Intermediate physical interactions induce spatiotemporal dynamics in Turing patterns

Cathelijne ter Burg and David Zwicker*

  • *Contact author: david.zwicker@ds.mpg.de
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Phys. Rev. Research 8, 033156 – Published 10 August, 2026

DOI: https://doi.org/10.1103/mnpt-c563

Abstract

Turing patterns are a central paradigm for describing spatial patterns in nature. The corresponding theory of reaction-diffusion dynamics combines ideal diffusion with nonlinear reactions, resulting in patterns when species diffuse at different rates and reactions are sufficiently nonlinear. However, real systems are more complex and particularly involve physical interactions between constituents. While such interactions can promote patterns, we here show that they can also induce persistent spatiotemporal patterns. These patterns exhibit well-defined length and timescales, which result from cycles of droplet coarsening and fission. The dynamical patterns combine properties of traditional Turing patterns and chemically active droplets, which emerge for strong physical interactions. Our analysis thus reveals three qualitatively different regimes that emerge when two components interact physically and undergo nonlinear reactions.

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References (44)

  1. A. M. Turing, The chemical basis of morphogenesis, Philos. Trans. R. Soc. London Ser. B, Biol. Sci. 237, 37 (1952).
  2. S. T. Vittadello, T. Leyshon, D. Schnoerr, and M. P. H. Stumpf, Turing pattern design principles and their robustness, Philos. Trans. R. Soc. A 379, 20200272 (2021).
  3. S. Kondo and T. Miura, Reaction-diffusion model as a framework for understanding biological pattern formation, Science 329, 1616 (2010).
  4. F. Yuki, K. Hiroyasu, B. Kamran, and K. Aharon, Nanoscale Turing patterns in a bismuth monolayer, Nat. Phys. 17, 1031 (2021).
  5. P. Recho, A. Hallou, and E. Hannezo, Theory of mechanochemical patterning in biphasic biological tissues, Proc. Natl. Acad. Sci. USA 116, 5344 (2019).
  6. L. Goehring, Pattern formation in the geosciences, Philos. Trans. R. Soc. A 371, 20120352 (2013).
  7. L. Menou, C. Luo, and D. Zwicker, Physical interactions in non-ideal fluids promote Turing patterns, J. R. Soc. Interface 20, 20230244 (2023).
  8. S. Banani, H. Lee, A. Hyman, and M. K. Rosen, Biomolecular condensates: Organizers of cellular biochemistry, Nat. Rev. Mol. Cell Biol. 18, 285 (2017).
  9. T. Y.-C. Tsai, R. M. Garner, and S. G. Megason, Adhesion-based self-organization in tissue patterning, Annu. Rev. Cell Dev. Biol. 38, 349 (2022).
  10. Q. Liu, A. Doelman, V. Rottschäfer, M. de Jager, P. Herman, M. Rietkerk, and J. van de Koppel, Phase separation explains a new class of self-organized spatial patterns in ecological systems, Proc. Natl. Acad. Sci. USA 110, 11905 (2013).
  11. G. L. Dignon, R. B. Best, and J. Mittal, Biomolecular phase separation: From molecular driving forces to macroscopic properties, Annu. Rev. Phys. Chem. 71, 53 (2020).
  12. D. Zwicker, O. W. Paulin, and C. ter Burg, Physics of droplet regulation in biological cells, Rep. Prog. Phys. 88, 116601 (2025).
  13. S. C. Glotzer, D. Stauffer, and N. Jan, Monte Carlo simulations of phase separation in chemically reactive binary mixtures, Phys. Rev. Lett. 72, 4109 (1994).
  14. J. J. Christensen, K. Elder, and H. C. Fogedby, Phase segregation dynamics of a chemically reactive binary mixture, Phys. Rev. E 54, R2212 (1996).
  15. D. Zwicker, A. A. Hyman, and F. Julicher, Suppression of Ostwald ripening in active emulsions, Phys. Rev. E 92, 012317 (2015).
  16. D. Carati and R. Lefever, Chemical freezing of phase separation in immiscible binary mixtures, Phys. Rev. E 56, 3127 (1997).
  17. D. Zwicker, The intertwined physics of active chemical reactions and phase separation, Curr. Opin. Colloid Interface Sci. 61, 101606 (2022).
  18. C. ter Burg and D. Zwicker, Physical interactions enable energy-efficient Turing patterns, Phys. Rev. Res. 7, L042070 (2025).
  19. D. Osmanović and E. Franco, Complex dynamics in reaction-phase separation systems, Phys. Rev. E 111, 025404 (2025).
  20. E. J. Kramer, P. Green, and C. J. Palmstrøm, Interdiffusion and marker movements in concentrated polymer-polymer diffusion couples, Polymer 25, 473 (1984).
  21. S. Safran, Statistical Thermodynamics of Surfaces, Interfaces and Membranes (CRC Press, New York, 2018).
  22. M. Rubinstein and R. Colby, Polymer Physics (Oxford University Press, Oxford, 2003), Vol. 23.
  23. J. W. Cahn and J. E. Hilliard, Free energy of a nonuniform system. I. Interfacial free energy, J. Chem. Phys. 28, 258 (1958).
  24. P. I. Flory, Thermodynamics of high polymer solutions, J. Chem. Phys. 10, 51 (1942).
  25. M. L. Huggins, Solutions of long chain compounds, J. Chem. Phys. 9, 440 (1941).
  26. A. Gierer and H. Meinhardt, A theory of biological pattern formation, Biol. Cybern. 12, 30 (1972).
  27. J. Kirschbaum and D. Zwicker, Controlling biomolecular condensates via chemical reactions, J. R. Soc. Interface 18, 20210255 (2021).
  28. P. W. Voorhees, The theory of Ostwald ripening, J. Stat. Phys. 38, 231 (1985).
  29. P. W. Voorhees, Ostwald ripening of two-phase mixtures, Annu. Rev. Mater. Sci. 22, 197 (1992).
  30. D. Zwicker, R. Seyboldt, C. A. Weber, A. A. Hyman, and F. Jülicher, Growth and division of active droplets provides a model for protocells, Nat. Phys. 13, 408 (2017).
  31. A. M. Bergmann, J. Bauermann, G. Bartolucci, C. Donau, M. Stasi, A.-L. Holtmannspötter, F. Jülicher, C. A. Weber, and J. Boekhoven, Liquid spherical shells are a non-equilibrium steady state of active droplets, Nat. Commun. 14, 6552 (2023).
  32. W. Verstraeten, M. P. Tran, A. Taskina, M. Hamberger, E. W. Green, M. Platten, C. Helbig, and K. Göpfrich, Genetic encoding and mutagenesis of RNA droplet phenotypes, ChemRxiv, https://doiorg/10.26434/chemrxiv-2025-5ffnn-v2.
  33. K. J. Painter and T. Hillen, Spatio-temporal chaos in a chemotaxis model, Physica D 240, 363 (2011).
  34. B. Chakrabarti, M. Das, C. Dasgupta, S. Ramaswamy, and A. K. Sood, Spatiotemporal rheochaos in nematic hydrodynamics, Phys. Rev. Lett. 92, 055501 (2004).
  35. M. Das, B. Chakrabarti, C. Dasgupta, S. Ramaswamy, and A. K. Sood, Routes to spatiotemporal chaos in the rheology of nematogenic fluids, Phys. Rev. E 71, 021707 (2005).
  36. D. Chakraborty, C. Dasgupta, and A. K. Sood, Banded spatiotemporal chaos in sheared nematogenic fluids, Phys. Rev. E 82, 065301(R) (2010).
  37. M. Yan, E. Frey, M. Müller, and S. Klumpp, Stochastic bubble dynamics in phase-separated scalar active matter, arXiv:2501.11442.
  38. C. Luo and D. Zwicker, Influence of physical interactions on spatiotemporal patterns, Phys. Rev. E 108, 034206 (2023).
  39. C. Luo, Y. Qiang, and D. Zwicker, Beyond pairwise: Higher-order physical interactions affect phase separation in multicomponent liquids, Phys. Rev. Res. 6, 033002 (2024).
  40. N. Ziethen, J. Kirschbaum, and D. Zwicker, Nucleation of chemically active droplets, Phys. Rev. Lett. 130, 248201 (2023).
  41. L. Demarchi, A. Goychuk, I. Maryshev, and E. Frey, Enzyme-enriched condensates show self-propulsion, positioning, and coexistence, Phys. Rev. Lett. 130, 128401 (2023).
  42. Y. Qiang, C. Luo, and D. Zwicker, Self-propulsion via nontransitive phase coexistence in chemically active mixtures, Phys. Rev. Lett. 135, 268301 (2025).
  43. C. ter Burg and D. Zwicker, Code for the paper “Intermediate physical interactions induce spatiotemporal dynamics in Turing patterns”, Zenodo, 2026, https://doi.org/10.5281/zenodo.21060374.
  44. D. Zwicker, py-pde: A Python package for solving partial differential equations, J. Open Source Software 5, 2158 (2020).

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Phys. Rev. Research 8, 033156– Published 10 August, 2026
  • Received 27 January 2026
  • Accepted 24 June 2026
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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. Open access publication funded by Max Planck Society.

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