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  • Open Access

Collective is Different: Information Exchange and Speed-Accuracy Trade-Offs in Self-Organized Patterning

Ashutosh Tripathi1,2, Jörn Dunkel1, and Dominic J. Skinner2,*

  • *Contact author: dskinner@flatironinstitute.org
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PRX Life 4, 013015 – Published 4 February, 2026

DOI: https://doi.org/10.1103/lfnt-8qbt

Abstract

During development, highly ordered structures emerge as cells collectively coordinate with each other. While recent advances have clarified how individual cells process and respond to external signals, understanding collective cellular decision making remains a major challenge. Here, we introduce a minimal, analytically tractable model of cell patterning via local cell-cell communication. Using this framework, we identify a trade-off between the speed and accuracy of collective pattern formation and, by adapting techniques from stochastic chemical kinetics, quantify how information flows between cells during patterning. Our analysis reveals counterintuitive features of collective patterning: globally optimized solutions do not necessarily maximize intercellular information transfer and individual cells may appear suboptimal in isolation. Moreover, the model predicts that instantaneous information shared between cells can be nonmonotonic in time as patterning occurs. An analysis of recent experimental data from lateral inhibition in Drosophila pupal abdomen finds a qualitatively similar effect.

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PRX Life Collection on adaptation and learning

PRX Life presents a special collection on ‘Biological systems that learn’, curated by the organizers of the National Institute for Theory and Mathematics in Biology (NITMB) workshop together with early-career researchers. The papers in this collection illustrate how living systems acquire functions, showing it is useful to describe them as tunable matter with many individually adjustable degrees of freedom characterizing microscopic properties. Drawing inspiration from AI and non-biological systems that can acquire desired functions, the featured research highlights the timeliness of the topic and its potential to drive unifying frameworks for emergent collective function in living tunable matter.

Every article in this collection is subject to a rigorous peer review process, upholding the high standards applied to all papers in the journal. The PRX Life editorial team manages peer review and makes all editorial decisions.

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References (89)

  1. D. B. Brückner and G. Tkačik, Information content and optimization of self-organized developmental systems, Proc. Natl. Acad. Sci. USA 121, e2322326121 (2024).
  2. F. Schweisguth and F. Corson, Self-organization in pattern formation, Dev. Cell 49, 659 (2019).
  3. A. Kicheva and J. Briscoe, Control of tissue development by morphogens, Annu. Rev. Cell Dev. Biol. 39, 91 (2023).
  4. M. D. Petkova, G. Tkačik, W. Bialek, E. F. Wieschaus, and T. Gregor, Optimal decoding of cellular identities in a genetic network, Cell 176, 844 (2019).
  5. I. Hajji, F. Corson, and W. Keil, Quantitative guiding of developmental cell fate transitions using a dynamical landscape model, bioRxiv, doi:10.1101/2024.11.16.623539.
  6. W. Wang, K. Ni, D. Poe, and J. Xing, Transiently increased coordination in gene regulation during cell phenotypic transitions, PRX Life 2, 043009 (2024).
  7. T. A. Leathers and C. D. Rogers, Time to go: Neural crest cell epithelial-to-mesenchymal transition, Development 149, dev200712 (2022).
  8. S. J. Bray and A. Bigas, Modes of notch signalling in development and disease, Nat. Rev. Mol. Cell Biol. 26 522 (2025).
  9. M.-S. Phan, J.-m. Kim, C. Picciotto, L. Couturier, N. Veits, K. Mazouni, and F. Schweisguth, Symmetry breaking and fate divergence during lateral inhibition in drosophila, Development 151, dev203165 (2024).
  10. F. Corson, L. Couturier, H. Rouault, K. Mazouni, and F. Schweisguth, Self-organized notch dynamics generate stereotyped sensory organ patterns in Drosophila, Science 356, eaai7407 (2017).
  11. F. Bocci, J. N. Onuchic, and M. K. Jolly, Understanding the principles of pattern formation driven by notch signaling by integrating experiments and theoretical models, Front. Physiol. 11, 929 (2020).
  12. E. D. Siggia and M. Vergassola, Decisions on the fly in cellular sensory systems, Proc. Natl. Acad. Sci. USA 110, E3704 (2013).
  13. T. J. Kobayashi, Implementation of dynamic bayesian decision making by intracellular kinetics, Phys. Rev. Lett. 104, 228104 (2010).
  14. K. Nakamura and T. J. Kobayashi, Connection between the bacterial chemotactic network and optimal filtering, Phys. Rev. Lett. 126, 128102 (2021).
  15. T. Tottori and T. J. Kobayashi, Theory for optimal estimation and control under resource limitations and its applications to biological information processing and decision-making, Phys. Rev. Res. 7, 043048 (2025).
  16. F. Corson and E. D. Siggia, Geometry, epistasis, and developmental patterning, Proc. Natl. Acad. Sci. USA 109, 5568 (2012).
  17. P. W. Sternberg, Vulval development, WormBook, ed. The C. elegans Research Community, WormBook, 2005, doi:10.1895/wormbook.1.6.1.
  18. A. Roure, P. Lemaire, and S. Darras, An otx/nodal regulatory signature for posterior neural development in ascidians, PLoS Genet. 10, e1004548 (2014).
  19. R. Bettoni, G. Dupont, A. M. Walczak, and S. de Buyl, Optimizing information transmission in neural induction constrains cell surface contacts of ascidian embryos, PRX Life 3, 033004 (2025).
  20. M. Kato and T. J. Kobayashi, Optimality theory of stigmergic collective information processing by chemotactic cells, PRX Life 3, 043005 (2025).
  21. N. Ohta and Y. Satou, Multiple signaling pathways coordinate to induce a threshold response in a chordate embryo, PLoS Genet. 9, e1003818 (2013).
  22. T. Tottori and T. J. Kobayashi, Memory-limited partially observable stochastic control and its mean-field control approach, Entropy 24, 1599 (2022).
  23. T. Rashid, M. Samvelyan, C. S. De Witt, G. Farquhar, J. Foerster, and S. Whiteson, Monotonic value function factorisation for deep multi-agent reinforcement learning, J. Mach. Learn. Res. 21, 1 (2020) .
  24. M. Samvelyan, T. Rashid, C. S. de Witt, G. Farquhar, N. Nardelli, T. G. J. Rudner, C.-M. Hung, P. H. S. Torr, J. Foerster, and S. Whiteson, The starcraft multi-agent challenge, arXiv:1902.04043.
  25. E. H. Durfee, Distributed problem solving and planning, in Multi-Agent Systems and Applications (Springer, Berlin, 2001), pp. 118–149.
  26. J. N. Gray, Notes on data base operating systems, in Operating Systems: An Advanced Course (Springer, Berlin, 1978), pp. 393–481.
  27. S. Strogatz, S. Walker, J. M. Yeomans, C. Tarnita, E. Arcaute, M. De Domenico, O. Artime, and K.-I. Goh, Fifty years of ‘more is different', Nat. Rev. Phys. 4, 508 (2022).
  28. P. W. Anderson, More is different, Science 177, 393 (1972).
  29. J. R. Collier, N. A. Monk, P. K. Maini, and J. H. Lewis, Pattern formation by lateral inhibition with feedback: A mathematical model of delta-notch intercellular signalling, J. Theor. Biol. 183, 429 (1996).
  30. K. Friston, M. Levin, B. Sengupta, and G. Pezzulo, Knowing one's place: A free-energy approach to pattern regulation, J. R. Soc. Interface 12, 20141383 (2015).
  31. R. Kuintzle, L. A. Santat, and M. B. Elowitz, Diversity in notch ligand–receptor signaling interactions, eLife 12, RP91422 (2024).
  32. A.-L. Moor and C. Zechner, Dynamic information transfer in stochastic biochemical networks, Phys. Rev. Res. 5, 013032 (2023).
  33. H. K. Hartline, H. G. Wagner, and F. Ratliff, Inhibition in the eye of limulus, J. Gen. Physiol. 39, 651 (1956).
  34. O. Barad, D. Rosin, E. Hornstein, and N. Barkai, Error minimization in lateral inhibition circuits, Sci. Signaling 3, ra51 (2010).
  35. D. Riddle, T. Blumenthal, B. Meyer, and J. Priess, C. Elegans II (Cold Spring Harbor Laboratory Press, Cold Spring Harbor, NY, 1997).
  36. P. François and E. D. Siggia, A case study of evolutionary computation of biochemical adaptation, Phys. Biol. 5, 026009 (2008).
  37. B. Zoller, A. Bénichou, T. Gregor, and G. Tkačik, Invariant non-equilibrium dynamics of transcriptional regulation optimize information flow, arXiv:2507.12395.
  38. A. Hilfinger, T. M. Norman, G. Vinnicombe, and J. Paulsson, Constraints on fluctuations in sparsely characterized biological systems, Phys. Rev. Lett. 116, 058101 (2016).
  39. B. Kell, R. Ripsman, and A. Hilfinger, Noise properties of adaptation-conferring biochemical control modules, Proc. Natl. Acad. Sci. USA 120, e2302016120 (2023).
  40. J. Yan, A. Hilfinger, G. Vinnicombe, and J. Paulsson, Kinetic uncertainty relations for the control of stochastic reaction networks, Phys. Rev. Lett. 123, 108101 (2019).
  41. D. Zhang, Y. Cao, Q. Ouyang, and Y. Tu, The energy cost and optimal design for synchronization of coupled molecular oscillators, Nat. Phys. 16, 95 (2020).
  42. D. Zhang, Y. Cao, Q. Ouyang, and Y. Tu, Altruistic resource-sharing mechanism for synchronization: The energy-speed-accuracy trade-off, Phys. Rev. Lett. 135, 037401 (2025).
  43. See Supplemental Material at http://link.aps.org/supplemental/10.1103/lfnt-8qbt for additional algorithmic and numerical details, Figs. S1–S10, and Tables S1–S2, which includes Refs.  [82, 83, 84, 85, 86, 87, 88, 89].
  44. Z.-q. Luo, W.-k. Ma, A. M.-c. So, Y. Ye, and S. Zhang, Semidefinite relaxation of quadratic optimization problems, IEEE Signal Process. Mag. 27, 20 (2010).
  45. J. A. Owen, T. R. Gingrich, and J. M. Horowitz, Universal thermodynamic bounds on nonequilibrium response with biochemical applications, Phys. Rev. X 10, 011066 (2020).
  46. M. Razo-Mejia, S. Marzen, G. Chure, R. Taubman, M. Morrison, and R. Phillips, First-principles prediction of the information processing capacity of a simple genetic circuit, Phys. Rev. E 102, 022404 (2020).
  47. J. O. Dubuis, G. Tkačik, E. F. Wieschaus, T. Gregor, and W. Bialek, Positional information, in bits, Proc. Natl. Acad. Sci. USA 110, 16301 (2013).
  48. G. Tkačik, A. M. Walczak, and W. Bialek, Optimizing information flow in small genetic networks, Phys. Rev. E 80, 031920 (2009).
  49. A. M. Walczak, G. Tkačik, and W. Bialek, Optimizing information flow in small genetic networks. II. Feed-forward interactions, Phys. Rev. E 81, 041905 (2010).
  50. G. Tkačik, A. M. Walczak, and W. Bialek, Optimizing information flow in small genetic networks. III. A self-interacting gene, Phys. Rev. E 85, 041903 (2012).
  51. L. Hahn, A. M. Walczak, and T. Mora, Dynamical information synergy in biochemical signaling networks, Phys. Rev. Lett. 131, 128401 (2023).
  52. G. Tkačik and P. R. ten Wolde, Information processing in biochemical networks, Annu. Rev. Biophys. 54, 249 (2025).
  53. Y. Tang, A. Adelaja, F. X. F. Ye, E. Deeds, R. Wollman, and A. Hoffmann, Quantifying information accumulation encoded in the dynamics of biochemical signaling, Nat. Commun. 12, 1272 (2021).
  54. O. Witteveen, S. J. Rosen, R. S. Lach, M. Z. Wilson, and M. Bauer, Optimizing information transmission in the canonical wnt pathway, arXiv:2506.22633.
  55. M. Kramar, L. Hahn, A. M. Walczak, T. Mora, and M. Coppey, Single cells can resolve graded stimuli, PRX Life 3, 043016 (2025).
  56. R. Cheong, A. Rhee, C. J. Wang, I. Nemenman, and A. Levchenko, Information transduction capacity of noisy biochemical signaling networks, Science 334, 354 (2011).
  57. M. Bauer, M. D. Petkova, T. Gregor, E. F. Wieschaus, and W. Bialek, Trading bits in the readout from a genetic network, Proc. Natl. Acad. Sci. USA 118, e2109011118 (2021).
  58. T. R. Sokolowski, T. Gregor, W. Bialek, and G. Tkačik, Deriving a genetic regulatory network from an optimization principle, Proc. Natl. Acad. Sci. USA 122, e2402925121 (2025).
  59. A.-L. Moor, A. Tjalma, M. Reinhardt, P. Rein ten Wolde, and C. Zechner, State- versus reaction-based information processing in biochemical networks, arXiv:2505.13373.
  60. F. Tostevin and P. Rein ten Wolde, Mutual information between input and output trajectories of biochemical networks, Phys. Rev. Lett. 102, 218101 (2009).
  61. A. Das and P. Rein ten Wolde, Exact computation of transfer entropy with path weight sampling, Phys. Rev. Lett. 135, 107404 (2025).
  62. M. Reinhardt, G. Tkačik, and P. Rein ten Wolde, Path weight sampling: Exact Monte Carlo computation of the mutual information between stochastic trajectories, Phys. Rev. X 13, 041017 (2023).
  63. L. Duso and C. Zechner, Selected-node stochastic simulation algorithm, J. Chem. Phys. 148, 164108 (2018).
  64. J. R. Norris, Continuous-time Markov chains II, in Markov Chains, Cambridge Series in Statistical and Probabilistic Mathematics (Cambridge University Press, Cambridge, UK, 1997), pp. 108–127.
  65. S. Watanabe, Information theoretical analysis of multivariate correlation, IBM J. Res. Dev. 4, 66 (1960).
  66. H. H. Mattingly, K. Kamino, B. B. Machta, and T. Emonet, Escherichia coli chemotaxis is information limited, Nat. Phys. 17, 1426 (2021).
  67. H. B. Brandl, J. A. Klarevas-Irby, D. Zuñiga, C. Hansen Wheat, C. Christensen, F. Omengo, C. Nzomo, W. Cherono, B. Nyaguthii, and D. R. Farine, The physiological cost of leadership in collective movements, Curr. Biol. 35, 4003 (2025).
  68. A. Kraskov, H. Stögbauer, and P. Grassberger, Estimating mutual information, Phys. Rev. E 69, 066138 (2004).
  69. C. Schwayer, S. Barbiero, D. B. Brückner, C. Baader, N. A. Repina, O. E. Diaz, L. C. Meylan, V. Kalck, S. Suppinger, Q. Yang, J. Schnabl, U. Kilik, J. G. Camp, B. Stockinger, M. Bühler, M. B. Stadler, E. Hannezo, and P. Liberali, Cell heterogeneity and fate bistability drive tissue patterning during intestinal regeneration, bioRxiv, doi:10.1101/2025.01.14.632683.
  70. S. Okuda, T. Miura, Y. Inoue, T. Adachi, and M. Eiraku, Combining turing and 3D vertex models reproduces autonomous multicellular morphogenesis with undulation, tubulation, and branching, Sci. Rep. 8, 2386 (2018).
  71. C. M. Glen, M. L. Kemp, and E. O. Voit, Agent-based modeling of morphogenetic systems: Advantages and challenges, PLoS Comput. Biol. 15, e1006577 (2019).
  72. S. Bajpai, R. Chelakkot, R. Prabhakar, and M. M. Inamdar, Role of delta-notch signalling molecules on cell–cell adhesion in determining heterogeneous chemical and cell morphological patterning, Soft Matter 18, 3505 (2022).
  73. T. Dullweber and A. Erzberger, Mechanochemical feedback loops in contact-dependent fate patterning, Curr. Opin. Syst. Biol. 32-33, 100445 (2023).
  74. M. Smart and A. Zilman, Emergent properties of collective gene-expression patterns in multicellular systems, Cell Rep. Phys. Sci. 4, 101247 (2023).
  75. M. Faldor and A. Cully, CAX: Cellular automata accelerated in JAX, in International Conference on Learning Representations (OpenReview.net, Singapore, 2025).
  76. A. D. Richardson, T. Antal, R. A. Blythe, and L. J. Schumacher, Learning spatio-temporal patterns with neural cellular automata, PLOS Comput. Biol. 20, e1011589 (2024).
  77. M. Reinhardt, G. Tkačik, and P. Rein ten Wolde, Ml-pws: Estimating the mutual information between experimental time series using neural networks, arXiv:2508.16509.
  78. N. C. Lammers, Y. J. Kim, J. Zhao, and H. G. Garcia, A matter of time: Using dynamics and theory to uncover mechanisms of transcriptional bursting, Curr. Opin. Cell Biol. 67, 147 (2020).
  79. 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).
  80. C. Marshall, Specificity of receptor tyrosine kinase signaling: Transient versus sustained extracellular signal-regulated kinase activation, Cell 80, 179 (1995).
  81. A. Tripathi, J. Dunkel, and D. Skinner, Collective is different: Information exchange and speed-accuracy trade-offs in self-organized patterning [data set], Zenodo (2026), doi:10.5281/zenodo.18189941.
  82. J. Bragantini, I. Theodoro, X. Zhao, T. A. P. M. Huijben, E. Hirata-Miyasaki, S. VijayKumar, A. Balasubramanian, T. Lao, R. Agrawal, S. Xiao, J. Lammerding, S. Mehta, A. X. Falcão, A. Jacobo, M. Lange, and L. A. Royer, Ultrack: Pushing the limits of cell tracking across biological scales, Nat. Methods 22, 2423 (2025).
  83. B. Efron and C. Stein, The jackknife estimate of variance, Ann. Statist. 9, 586 (1981).
  84. I. Loshchilov and F. Hutter, SGDR: Stochastic gradient descent with warm restarts, in International Conference on Learning Representations (OpenReview.net, Toulon, France. 2017).
  85. S. Mohamed, M. Rosca, M. Figurnov, and A. Mnih, Monte Carlo gradient estimation in machine learning, arXiv:1906.10652.
  86. J. Nocedal and S. Wright, Numerical optimization, Springer Series in Operations Research and Financial Engineering (Springer, New York, 2006).
  87. S. J. Reddi, S. Kale, and S. Kumar, On the convergence of Adam and beyond, International Conference on Learning Representations (OpenReview.net, Vancouver, BC, Canada, 2018).
  88. R. E. Spinney, J. T. Lizier, and M. Prokopenko, Transfer entropy in physical systems and the arrow of time, Phys. Rev. E 94, 022135 (2016).
  89. R. E. Spinney, M. Prokopenko, and J. T. Lizier, Transfer entropy in continuous time, with applications to jump and neural spiking processes, Phys. Rev. E 95, 032319 (2017).

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PRX Life 4, 013015– Published 4 February, 2026
  • Received 1 September 2025
  • Accepted 5 January 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.

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