- Physical Review Journals
All JournalsPhysics Magazine

PRX Life

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

Marr's Three Levels for Embryonic Development: Information, Dynamical Systems, Gene Networks

David B. Brückner*

Gašper Tkačik†

  • *Contact author: d.b.brueckner@unibas.ch
  • †Contact author: gtkacik@ist.ac.at
PDF

PRX Life 4, 017001 – Published 23 January, 2026

DOI: https://doi.org/10.1103/fdcf-dkws

Abstract

Developmental patterning comprises processes that range from purely instructed, where external signals specify cell fates, to fully self-organized, where spatial patterns emerge autonomously through cellular interactions. We propose that both extremes—as well as the continuum of intermediate cases—can be conceptualized as information-processing systems, whose operation can be described using “Marr's three levels of analysis”: the computational problem being solved, the algorithms employed, and their molecular implementation. At the first level, we argue that normative theories, such as information-theoretic optimization principles, provide a formalization of the computational problem. At the second level, we show how simplified information-processing architectures provide a framework for developmental algorithms, which are formalized mathematically using dynamical systems theory. At the third level, the implementation of developmental algorithms is described by mechanistic biophysical and gene regulatory network models.

View figure in article

Physics Subject Headings (PhySH)

Collections

This article appears in the following collection:

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.

Article Text

References (121)

  1. A. Kicheva and J. Briscoe, Control of tissue development by morphogens, Annu. Rev. Cell Dev. Biol. 39, 91 (2023).
  2. S. Dupont and S. A. Wickström, Mechanical regulation of chromatin and transcription, Nat. Rev. Genet. 23, 624 (2022).
  3. C. Collinet and T. Lecuit, Programmed and self-organized flow of information during morphogenesis, Nat. Rev. Mol. Cell Biol. 22, 245 (2021).
  4. T. Gregor, D. W. Tank, E. F. Wieschaus, and W. Bialek, Probing the limits to positional information, Cell 130, 153 (2007).
  5. S. Wennekamp, S. Mesecke, F. Nédélec, and T. Hiiragi, A self-organization framework for symmetry breaking in the mammalian embryo, Nat. Rev. Mol. Cell Biol. 14, 452 (2013).
  6. A. Lamba, M. Zhu, M. Meglicki, S. Czukiewska, L. Balasubramaniam, R. Hadas, N. Weishaupt, E. M. Patel, Y. H. Kavanagh, R. Wang, N. Jing, and M. Zernicka-Goetzshow, Asynchronous mouse embryo polarization leads to heterogeneity in cell fate specification, eLife 13, RP101140 (2024).
  7. N. Barkai and B.-Z. Shilo, Variability and robustness in biomolecular systems, Mol. Cell 28, 755 (2007).
  8. T. Nakagaki, H. Yamada, and Á. Tóth, Maze-solving by an amoeboid organism, Nature (London) 407, 470 (2000).
  9. K. Alim, G. Amselem, F. Peaudecerf, M. P. Brenner, and A. Pringle, Random network peristalsis in Physarum polycephalum organizes fluid flows across an individual, Proc. Natl. Acad. Sci. USA 110, 13306 (2013).
  10. S. E. Sultan, Phenotypic plasticity for plant development, function and life history, Trends Plant Sci. 5, 537 (2000).
  11. P. Vanderhaeghen and F. Polleux, Developmental mechanisms underlying the evolution of human cortical circuits, Nat. Rev. Neurosci. 24, 213 (2023).
  12. G. Balázsi, A. van Oudenaarden, and J. Collins, Cellular decision making and biological noise: From microbes to mammals, Cell 144, 910 (2011).
  13. J. M. Raser and E. K. O'Shea, Noise in gene expression: Origins, consequences, and control, Science 309, 2010 (2005).
  14. G. Tkačik and P. R. ten Wolde, Information processing in biochemical networks, Annu. Rev. Biophys. 54, 249 (2025).
  15. P. Moghe, E. Hannezo, and T. Hiiragi, Optimality as a framework for understanding developmental robustness, Trends Cell Biol. S0962892425002041 (2025).
  16. J. O. Dubuis, G. Tkacik, E. F. Wieschaus, T. Gregor, and W. Bialek, Positional information, in bits, Proc. Natl. Acad. Sci. USA 110, 16301 (2013).
  17. G. Tkačik and T. Gregor, The many bits of positional information, Development 148, dev176065 (2021).
  18. D. B. Brückner and G. Tkačik, Information content and optimization of self-organized developmental systems, Proc. Natl. Acad. Sci. USA 121, e2322326121 (2024).
  19. 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).
  20. L. G. Morelli, K. Uriu, S. Ares, and A. C. Oates, Computational approaches to developmental patterning, Science 336, 187 (2012).
  21. D. Marr and T. Poggio, From understanding computation to understanding neural circuitry, AIM-357, MIT (1976), https://dspace.mit.edu/handle/1721.1/5782.
  22. D. Marr, Vision: A Computational Investigation into the Human Representation and Processing of Visual Information (MIT press, Cambridge, MA, 2010).
  23. L. Wolpert, Positional information and the spatial pattern of cellular differentiation, J. Theor. Biol. 25, 1 (1969).
  24. J. Sharpe, Wolpert's french flag: What's the problem? Development 146, dev185967 (2019).
  25. 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).
  26. A. Pezzotta and J. Briscoe, Optimal control of gene regulatory networks for morphogen-driven tissue patterning, Cell Syst. 14, 940 (2023).
  27. 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).
  28. D. J. Cislo, A. Pavlopoulos, and B. I. Shraiman, A “morphogenetic action” principle for 3D shape formation by the growth of thin sheets, Phys. Rev. X 15, 021068 (2025).
  29. M. Kracikova, G. Akiri, A. George, R. Sachidanandam, and S. A. Aaronson, A threshold mechanism mediates p53 cell fate decision between growth arrest and apoptosis, Cell Death Differ. 20, 576 (2013).
  30. 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).
  31. 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).
  32. T. Erdmann, M. Howard, and P. R. ten Wolde, Role of spatial averaging in the precision of gene expression patterns, Phys. Rev. Lett. 103, 258101 (2009).
  33. 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).
  34. J. Cooke and E. C. Zeeman, A clock and wavefront model for control of the number of repeated structures during animal morphogenesis, J. Theor. Biol. 58, 455 (1976).
  35. U. Alon, Network motifs: Theory and experimental approaches, Nat. Rev. Genet. 8, 450 (2007).
  36. O. Gozlan and D. Sprinzak, Notch signaling in development and homeostasis, Development 150, dev201138 (2023).
  37. A. Bailles, E. W. Gehrels, and T. Lecuit, Mechanochemical principles of spatial and temporal patterns in cells and tissues, Annu. Rev. Cell Dev. Biol. 38, 321 (2022).
  38. U. Alon, An Introduction to Systems Biology: Design Principles of Biological Circuits (Chapman & Hall, 2006).
  39. J. M. Smith, R. Burian, S. Kauffman, P. Alberch, J. Campbell, B. Goodwin, R. Lande, D. Raup, and L. Wolpert, Developmental constraints and evolution: A perspective from the mountain lake conference on development and evolution, Q. Rev. Biol. 60, 265 (1985).
  40. W. Młynarski, M. Hledík, T. R. Sokolowski, and G. Tkačik, Statistical analysis and optimality of neural systems, Neuron 109, p1227 (2021).
  41. B. Zoller, T. Gregor, and G. Tkačik, Eukaryotic gene regulation at equilibrium, or non? Curr. Opin. Syst. Biol. 31, 100435 (2022).
  42. W. Bialek, Biophysics: Searching for Principles (Princeton University Press, Princeton, NJ, 2012).
  43. V. Balasubramanian and P. Sterling, Receptive fields and functional architecture in the retina, J. Physiol. 587, 2753 (2009).
  44. O. Shoval, H. Sheftel, G. Shinar, Y. Hart, O. Ramote, A. Mayo, E. Dekel, K. Kavanagh, and U. Alon, Evolutionary trade-offs, pareto optimality, and the geometry of phenotype space, Science 336, 1157 (2012).
  45. J. S. Edwards, R. U. Ibarra, and B. O. Palsson, In silico predictions of Escherichia coli metabolic capabilities are consistent with experimental data, Nat. Biotechnol. 19, 125 (2001).
  46. D. De Martino, A. Mc Andersson, T. Bergmiller, C. C. Guet, and G. Tkačik, Statistical mechanics for metabolic networks during steady state growth, Nat. Commun. 9, 2988 (2018).
  47. Y. Morishita and Y. Iwasa, Optimal placement of multiple morphogen sources, Phys. Rev. E 77, 041909 (2008).
  48. P. François and E. D. Siggia, Predicting embryonic patterning using mutual entropy fitness and in silico evolution, Development 137, 2385 (2010).
  49. K. S. Iyer, C. Prabhakara, S. Mayor, and M. Rao, Cellular compartmentalisation and receptor promiscuity as a strategy for accurate and robust inference of position during morphogenesis, eLife 12, e79257 (2023).
  50. E. Hannezo and B. D. Simons, Statistical theory of branching morphogenesis, Dev. Growth Differ. 60, 512 (2018).
  51. J. D. Victor, M. M. Conte, and C. F. Chubb, Textures as probes of visual processing, Annu. Rev. Vis. Sci. 3, 275 (2017).
  52. M. Zagorski, Y. Tabata, N. Brandenberg, M. P. Lutolf, G. Tkačik, T. Bollenbach, J. Briscoe, and A. Kicheva, Decoding of position in the developing neural tube from antiparallel morphogen gradients, Science 356, 1379 (2017).
  53. D. Mortimer, J. Feldner, T. Vaughan, I. Vetter, Z. Pujic, W. J. Rosoff, K. Burrage, P. Dayan, L. J. Richards, and G. J. Goodhill, A bayesian model predicts the response of axons to molecular gradients, Proc. Natl. Acad. Sci. USA 106, 10296 (2009).
  54. E. Libby, T. J. Perkins, and P. S. Swain, Noisy information processing through transcriptional regulation, Proc. Natl. Acad. Sci. USA 104, 7151 (2007).
  55. T. J. Kobayashi, Implementation of dynamic bayesian decision making by intracellular kinetics, Phys. Rev. Lett. 104, 228104 (2010).
  56. C. G. Bowsher and P. S. Swain, Environmental sensing, information transfer, and cellular decision-making, Curr. Opin. Biotechnol. 28, 149 (2014).
  57. P. François, V. Hakim, and E. D. Siggia, Deriving structure from evolution: Metazoan segmentation, Mol. Syst. Biol. 3, 154 (2007).
  58. H. Booth and Z. Hadjivasiliou, Gene network organization, mutation, and selection collectively drive developmental pattern evolvability and predictability, PRX Life 3, 033023 (2025).
  59. H. Ronellenfitsch and E. Katifori, Global optimization, local adaptation, and the role of growth in distribution networks, Phys. Rev. Lett. 117, 138301 (2016).
  60. Q. Zhang, S. Bhattacharya, and M. E. Andersen, Ultrasensitive response motifs: Basic amplifiers in molecular signalling networks, Open Biol. 3, 130031 (2013).
  61. C. A. Doherty, R. Diegmiller, M. Kapasiawala, E. R. Gavis, and S. Y. Shvartsman, Coupled oscillators coordinate collective germline growth, Dev. Cell 56, 860 (2021).
  62. C. Waddington, The Strategy of the Genes (Routledge, New York, 1957).
  63. E. H. Davidson, D. R. McClay, and L. Hood, Regulatory gene networks and the properties of the developmental process, Proc. Natl. Acad. Sci. USA 100, 1475 (2003).
  64. G. Tkačik, A. M. Walczak, and W. Bialek, Optimizing information flow in small genetic networks, Phys. Rev. E 80, 031920 (2009).
  65. W. H. de Ronde, F. Tostevin, and P. R. ten Wolde, Feed-forward loops and diamond motifs lead to tunable transmission of information in the frequency domain, Phys. Rev. E 86, 021913 (2012).
  66. 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).
  67. 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).
  68. U. Kadiyala, D. Sprinzak, N. A. Monk, S. E. Taylor, B. Verd, K. F. Sonnen, L. Moon, A. H. Roeder, R. Perez-Carrasco, and P. Formosa-Jordan, From genes to patterns: Five key dynamical systems concepts to decode developmental regulatory mechanisms, Development 152, dev204617 (2025).
  69. T. R. Sokolowski and G. Tkačik, Optimizing information flow in small genetic networks. IV. Spatial coupling, Phys. Rev. E 91, 062710 (2015).
  70. P. Hillenbrand, U. Gerland, and G. Tkačik, Beyond the french flag model: Exploiting spatial and gene regulatory interactions for positional information, PLoS One 11, e0163628 (2016).
  71. 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).
  72. D. N. Arnosti, S. Barolo, M. Levine, and S. Small, The eve stripe 2 enhancer employs multiple modes of transcriptional synergy, Development 122, 205 (1996).
  73. F. Etoc, J. Metzger, A. Ruzo, C. Kirst, A. Yoney, M. Z. Ozair, A. H. Brivanlou, and E. D. Siggia, A balance between secreted inhibitors and edge sensing controls gastruloid self-organization, Dev. Cell 39, 302 (2016).
  74. F. Corson and E. D. Siggia, Gene-free methodology for cell fate dynamics during development, eLife 6, e30743 (2017).
  75. E. Camacho-Aguilar, A. Warmflash, and D. A. Rand, Quantifying cell transitions in C. elegans with data-fitted landscape models, PLoS Comput. Biol. 17, e1009034 (2021).
  76. 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).
  77. I. Hajji, F. Corson, and W. Keil, Quantitative guiding of developmental cell fate transitions using a dynamical landscape model, bioRxiv (2024).
  78. D. J. Cislo, M. J. Delas, J. Briscoe, and E. D. Siggia, Reconstructing Waddington's landscape from data, Proc. Natl. Acad. Sci. USA 122, e2521762122 (2025).
  79. M. Fontaine, M. J. Delas, M. Saez, R. J. Maizels, E. Finnie, J. Briscoe, and D. A. Rand, Dynamic landscape analysis of cell fate decisions: Predictive models of neural development from single-cell data, bioRxiv (2025), https://doi.org/10.1101/2025.05.28.656648.
  80. M. Sáez, J. Briscoe, and D. A. Rand, Dynamical landscapes of cell fate decisions, Interface Focus 12, 20220002 (2022).
  81. E. Gallo, S. De Renzis, J. Sharpe, R. Mayor, and J. Hartmann, Versatile system cores as a conceptual basis for generality in cell and developmental biology, Cell Syst. 15, 790 (2024).
  82. C. Schwayer and D. B. Brückner, Connecting theory and experiment in cell and tissue mechanics, J. Cell Sci. 136, jcs261515 (2023).
  83. D. B. Brückner and E. Hannezo, Tissue active matter: Integrating mechanics and signaling into dynamical models, Cold Spring Harbor Perspect. Biol. 18, a041653 (2024).
  84. G. Von Dassow, E. Meir, E. M. Munro, and G. M. Odell, The segment polarity network is a robust developmental module, Nature (London) 406, 188 (2000).
  85. A. P. Pomreinke, G. H. Soh, K. W. Rogers, J. K. Bergmann, A. J. Bläßle, and P. Müller, Dynamics of BMP signaling and distribution during zebrafish dorsal-ventral patterning, eLife 6, e25861 (2017).
  86. S. Sick, S. Reinker, J. Timmer, and T. Schlake, WNT and DKK determine hair follicle spacing through a reaction-diffusion mechanism, Science 314, 1447 (2006).
  87. R. Sheth, L. Marcon, M. F. Bastida, M. Junco, L. Quintana, R. Dahn, M. Kmita, J. Sharpe, and M. A. Ros, Hox genes regulate digit patterning by controlling the wavelength of a Turing-type mechanism, Science 338, 1476 (2012).
  88. 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).
  89. 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).
  90. 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).
  91. S. Chhabra, L. Liu, R. Goh, X. Kong, and A. Warmflash, Dissecting the dynamics of signaling events in the BMP, WNT, and NODAL cascade during self-organized fate patterning in human gastruloids, PLoS Biol. 17, e3000498 (2019).
  92. S. Lehr, D. B. Brückner, T. G. Minchington, M. Greunz-Schindler, J. Merrin, E. Hannezo, and A. Kicheva, Self-organized pattern formation in the developing mouse neural tube by a temporal relay of BMP signaling, Dev. Cell 60, P567 (2025).
  93. 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 (2025), https://doi.org/10.1101/2025.01.14.632683.
  94. Y. E. Antebi, J. M. Linton, H. Klumpe, B. Bintu, M. Gong, C. Su, R. McCardell, and M. B. Elowitz, Combinatorial signal perception in the BMP pathway, Cell 170, 1184 (2017).
  95. N. Balaskas, A. Ribeiro, J. Panovska, E. Dessaud, N. Sasai, K. 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).
  96. S. Kondo and T. Miura, Reaction-diffusion model as a framework for understanding biological pattern formation, Science 329, 1616 (2010).
  97. T. M. Cover, Elements of Information Theory (John Wiley & Sons, 1999).
  98. A. Tripathi, J. Dunkel, and D. J. Skinner, Collective is different: Information exchange and speed-accuracy trade-offs in self-organized patterning, arXiv:2510.01397.
  99. G. Ermentrout and L. Edelstein-Keshet, Cellular automata approaches to biological modeling, J. Theor. Biol. 160, 97 (1993).
  100. J. B. A. Green and J. Sharpe, Positional information and reaction-diffusion: Two big ideas in developmental biology combine, Development 142, 1203 (2015).
  101. E. Piddini and J.-P. Vincent, Interpretation of the wingless gradient requires signaling-induced self-inhibition, Cell 136, 296 (2009).
  102. S. G. Kuntz and M. B. Eisen, Drosophila embryogenesis scales uniformly across temperature in developmentally diverse species, PLoS Genet. 10, e1004293 (2014).
  103. C. K. Mirth, T. E. Saunders, and C. Amourda, Growing up in a changing world: Environmental regulation of development in insects, Annu. Rev. Entomol. 66, 81 (2021).
  104. E. P. Simoncelli and B. A. Olshausen, Natural image statistics and neural representation, Annu. Rev. Neurosci. 24, 1193 (2001).
  105. A. M. Plum and M. Serra, Morphogen patterning in dynamic tissues, PRX Life 3, 043009 (2025).
  106. N. Noll, M. Mani, I. Heemskerk, S. J. Streichan, and B. I. Shraiman, Active tension network model suggests an exotic mechanical state realized in epithelial tissues, Nat. Phys. 13, 1221 (2017).
  107. F. Brauns, N. H. Claussen, M. F. Lefebvre, E. F. Wieschaus, and B. I. Shraiman, The geometric basis of epithelial convergent extension, eLife 13, RP95521 (2024).
  108. A. M. Plum and M. Serra, Dynamical systems of fate and form in development, Sem. Cell Dev. Biol. 172, 103620 (2025).
  109. G. S. Nájera, A. M. Plum, B. Steventon, C. J. Weijer, and M. Serra, Control of modular tissue flows shaping the embryo in avian gastrulation, bioRxiv (2024).
  110. D. Ben-Zvi, B. Z. Shilo, A. Fainsod, and N. Barkai, Scaling of the BMP activation gradient in Xenopus embryos, Nature (London) 453, 1205 (2008).
  111. T. Gregor, W. Bialek, R. R. De Ruyter Van Steveninck, D. W. Tank, and E. F. Wieschaus, Diffusion and scaling during early embryonic pattern formation, Proc. Natl. Acad. Sci. USA 102, 18403 (2005).
  112. M. Nikolić, V. Antonetti, F. Liu, G. Muhaxheri, M. D. Petkova, M. Scheeler, E. M. Smith, W. Bialek, and T. Gregor, Scale invariance in early embryonic development Proc. Natl. Acad. Sci. USA 121, e2403265121 (2024).
  113. D. Ben-Zvi and N. Barkai, Scaling of morphogen gradients by an expansion-repression integral feedback control, Proc. Natl. Acad. Sci. USA 107, 6924 (2010).
  114. M. Le Verge-Serandour and K. Alim, Physarum polycephalum: Smart network adaptation, Annu. Rev. Condens. Matter Phys. 15, 263 (2024).
  115. S. Arzash, I. Tah, A. J. Liu, and M. L. Manning, Rigidity of epithelial tissues as a double optimization problem, Phys. Rev. Res. 7, 013157 (2025).
  116. D. S. Banerjee, M. J. Falk, M. L. Gardel, A. M. Walczak, T. Mora, and S. Vaikuntanathan, Learning via mechanosensitivity and activity in cytoskeletal networks, PRX Life 4, 013001 (2025).
  117. M. Stern, D. Hexner, J. W. Rocks, and A. J. Liu, Supervised learning in physical networks: From machine learning to learning machines, Phys. Rev. X 11, 021045 (2021).
  118. M. Stern and A. Murugan, Learning without neurons in physical systems, Annu. Rev. Condens. Matter Phys. 14, 417 (2023).
  119. T. Hain, C. Santangelo, and M. L. Manning, Optimizing properties on the critical rigidity manifold of underconstrained central-force networks, Phys. Rev. E 111, 015418 (2025).
  120. M. Hledík, N. Barton, and G. Tkačik, Accumulation and maintenance of information in evolution, Proc. Natl. Acad. Sci. USA 119, e2123152119 (2022).
  121. T. Tlusty and A. Libchaber, Life sets off a cascade of machines, Proc. Natl. Acad. Sci. USA 122, e2418000122 (2025).

Outline

Information

PRX Life 4, 017001– Published 23 January, 2026
  • Received 24 October 2025
  • Accepted 23 December 2025
Crossmark - Check for updates button
Creative Commons Logo

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.

Published by the American Physical Society

Sign In to Your Journals Account

Filter

Filter

Article Lookup

Enter a citation