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  • Open Access

Multi-Timescale Adaptation and Emergent Learning in Single Bacterial Cells

Josiah C. Kratz1,2, Huijing Wang3, Fangwei Si3, and Shiladitya Banerjee2,*

  • *Contact author: sbanerjee347@gatech.edu
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PRX Life 4, 023015 – Published 15 May, 2026

DOI: https://doi.org/10.1103/5zbg-8vll

Abstract

How do single-celled organisms adapt and learn to survive in unpredictable environments without the benefit of a nervous system? In this study, we provide experimental and theoretical evidence that single bacterial cells exhibit sophisticated learninglike behavior in changing environments. Using a custom microfluidic platform, we tracked individual E. coli cells in fluctuating nutrient conditions and found that bacteria do not merely react to the present; they integrate their environmental history to tune their future growth. We show that this adaptation process is scale-free, meaning cells possess a memory that stretches across multiple timescales, from minutes to hours. To explain this, we developed a mathematical framework based on fractional-order dynamics, demonstrating that a dynamic, power-law memory governs how cells integrate environmental history, which we validate with additional nutrient perturbation experiments. We demonstrate that this memory emerges naturally from the cell's internal machinery, specifically a heterogeneous population of ribosomes that react at different speeds. This mechanism reveals a strategic tradeoff between cellular adaptation speed and growth rate, which we confirm experimentally in pulsatile nutrient environments. By mapping the reaction-network architecture underlying bacterial physiology onto the logic of artificial recurrent neural networks, we show that even the simplest unicellular organisms implement complex computational strategies. Our findings establish a new foundation for understanding learning in non-neuronal systems, revealing how single cells use their history to navigate a changing environment.

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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.

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References (64)

  1. N. Pavelka, G. Rancati, J. Zhu, W. D. Bradford, A. Saraf, L. Florens, B. W. Sanderson, G. L. Hattem, and R. Li, Aneuploidy confers quantitative proteome changes and phenotypic variation in budding yeast, Nature (London) 468, 321 (2010).
  2. S. Hui, J. M. Silverman, S. S. Chen, D. W. Erickson, M. Basan, J. Wang, T. Hwa, and J. R. Williamson, Quantitative proteomic analysis reveals a simple strategy of global resource allocation in bacteria, Mol. Syst. Biol. 11, 784 (2015).
  3. M. Mori, Z. Zhang, A. Banaei‐Esfahani, J. Lalanne, H. Okano, B. C. Collins, A. Schmidt, O. T. Schubert, D. Lee, G. Li, R. Aebersold, T. Hwa, and C. Ludwig, From coarse to fine: The absolute Escherichia coli proteome under diverse growth conditions, Mol. Syst. Biol. 17, e9536 (2021).
  4. C. Wu, M. Mori, M. Abele, A. Banaei-esfahani, Z. Zhang, H. Okano, R. Aebersold, C. Ludwig, and T. Hwa, Enzyme expression kinetics by Escherichia coli during transition from rich to minimal media depends on proteome reserves, Nat. Microbiol. 8, 347 (2023).
  5. A. Mitchell and Y. Pilpel, A mathematical model for adaptive prediction of environmental changes by microorganisms, Proc. Natl. Acad. Sci. USA 108, 7271 (2011).
  6. A. Schmidt, K. Kochanowski, S. Vedelaar, E. Ahrné, B. Volkmer, L. Callipo, K. Knoops, M. Bauer, R. Aebersold, and M. Heinemann, The quantitative and condition-dependent Escherichia coli proteome, Nat. Biotechnol. 34, 104 (2016).
  7. J. Nguyen, V. Fernandez, S. Pontrelli, U. Sauer, M. Ackermann, and R. Stocker, A distinct growth physiology enhances bacterial growth under rapid nutrient fluctuations, Nat. Commun. 12, 3662 (2021).
  8. S. Banerjee, K. Lo, N. Ojkic, R. Stephens, N. F. Scherer, and A. R. Dinner, Mechanical feedback promotes bacterial adaptation to antibiotics, Nat. Phys. 17, 403 (2021).
  9. D. Muzzey, C. A. Gómez-Uribe, J. T. Mettetal, and A. van Oudenaarden, A systems-level analysis of perfect adaptation in yeast osmoregulation, Cell 138, 160 (2009).
  10. Z. Bódi, Z. Farkas, D. Nevozhay, D. Kalapis, V. Lázár, B. Csörgő, Á. Nyerges, B. Szamecz, G. Fekete, B. Papp, et al., Phenotypic heterogeneity promotes adaptive evolution, PLoS Biol. 15, e2000644 (2017).
  11. A. M. Brauer, H. Shi, P. A. Levin, and K. C. Huang, Physiological and regulatory convergence between osmotic and nutrient stress responses in microbes, Curr. Opin. Cell Biol. 81, 102170 (2023).
  12. R. L. Hastings and S. Boeynaems, Designer condensates: A toolkit for the biomolecular architect, J. Mol. Biol. 433, 166837 (2021).
  13. A. Larkin, C. Kunze, M. Seman, S. Lemieux, A. S. Khalil, K. Ragunathan, A. Larkin, C. Kunze, M. Seman, A. Levashkevich, J. Curran, and D. Morris-evans, Mapping the dynamics of epigenetic adaptation in S. pombe during heterochromatin misregulation, Dev. Cell 59, 2222 (2024).
  14. O. Patange, C. Schwall, M. Jones, C. Villava, D. A. Griffith, A. Phillips, and J. C. Locke, Escherichia coli can survive stress by noisy growth modulation, Nat. Commun. 9, 5333 (2018).
  15. Y. K. Kohanim, D. Levi, G. Jona, B. D. Towbin, A. Bren, and U. Alon, A bacterial growth law out of steady state, Cell Rep. 23, 2891 (2018).
  16. D. Molenaar, R. Van Berlo, D. De Ridder, and B. Teusink, Shifts in growth strategies reflect tradeoffs in cellular economics, Mol. Syst. Biol. 5, 323 (2009).
  17. M. Scott, C. W. Gunderson, E. M. Mateescu, Z. Zhang, and T. Hwa, Interdependence of cell growth and gene expression: Origins and consequences, Science 330, 1099 (2010).
  18. D. W. Erickson, S. J. Schink, V. Patsalo, J. R. Williamson, U. Gerland, and T. Hwa, A global resource allocation strategy governs growth transition kinetics of Escherichia coli, Nature (London) 551, 119 (2017).
  19. P. P. Pandey and S. Jain, Analytic derivation of bacterial growth laws from a simple model of intracellular chemical dynamics, Theory Biosci. 135, 121 (2016).
  20. G. Lambert and E. Kussel, Memory and fitness optimization of bacteria under fluctuating environments, PLoS Genet. 10, e1004556 (2014).
  21. W. H. Lin and E. Kussell, Complex interplay of physiology and selection in the emergence of antibiotic resistance, Curr. Biol. 26, 1486 (2016).
  22. D. H. de Groot, A. J. Tjalma, F. J. Bruggeman, and E. van Nimwegen, Effective bet-hedging through growth rate dependent stability, Proc. Natl. Acad. Sci. USA 120, e2211091120 (2023).
  23. J. T. George, Optimal phenotypic adaptation in fluctuating environments, Biophys. J. 122, 4414 (2023).
  24. M. Mori, S. Schink, D. W. Erickson, U. Gerland, and T. Hwa, Quantifying the benefit of a proteome reserve in fluctuating environments, Nat. Commun. 8, 1225 (2017).
  25. N. Giordano, F. Mairet, J. L. Gouzé, J. Geiselmann, and H. de Jong, Dynamical allocation of cellular resources as an optimal control problem: Novel insights into microbial growth strategies, PLoS Comput. Biol. 12, e1004802 (2016).
  26. J. C. Kratz and S. Banerjee, Gene expression tradeoffs determine bacterial survival and adaptation to antibiotic stress, PRX Life 2, 013010 (2024).
  27. See Supplemental Material at http://link.aps.org/supplemental/10.1103/5zbg-8vll for (i) derivation of minimal model, (ii) derivation of relation between value estimation and Bayesian inference, (iii) analysis of related memory models, and (iv) derivation of power law memory kernel model. In addition, the Supplemental Material contains 10 Supplemental Figures S1–S10, and Ref. [64].
  28. J. C. Kratz and S. Banerjee, Dynamic proteome trade-offs regulate bacterial cell size and growth in fluctuating nutrient environments, Commun. Biol. 6, 486 (2023).
  29. M. Scott, S. Klumpp, E. M. Mateescu, and T. Hwa, Emergence of robust growth laws from optimal regulation of ribosome synthesis, Mol. Syst. Biol. 10, 747 (2014).
  30. B. D. Towbin, Y. Korem, A. Bren, S. Doron, R. Sorek, and U. Alon, Optimality and sub-optimality in a bacterial growth law, Nat. Commun. 8, 14123 (2017).
  31. C. F. Lorenzo and T. T. Hartley, Variable order and distributed order fractional operators, Nonlin. Dyn. 29, 57 (2002).
  32. W. Ding, S. Patnaik, S. Sidharth, and F. Semperlotti, Applications of distributed-order fractional operators: A review, Entropy 23, 110 (2020).
  33. B. N. Lundstrom, M. H. Higgs, W. J. Spain, and A. L. Fairhall, Fractional differentiation by neocortical pyramidal neurons, Nat. Neurosci. 11, 1335 (2008).
  34. M. Khalighi, G. Sommeria-Klein, D. Gonze, K. Faust, and L. Lahti, Quantifying the impact of ecological memory on the dynamics of interacting communities, PLoS Comput. Biol. 18, e1009396 (2022).
  35. A. Bonfanti, J. L. Kaplan, G. Charras, and A. Kabla, Fractional viscoelastic models for power-law materials, Soft Matter 16, 6002 (2020).
  36. C. Wu, R. Balakrishnan, N. Braniff, M. Mori, G. Manzanarez, Z. Zhang, and T. Hwa, Cellular perception of growth rate and the mechanistic origin of bacterial growth law, Proc. Natl. Acad. Sci. USA 119, e2201585119 (2022).
  37. M. Zhu and X. Dai, Stringent response ensures the timely adaptation of bacterial growth to nutrient downshift, Nat. Commun. 14, 467 (2023).
  38. T. M. Schmeing and V. Ramakrishnan, What recent ribosome structures have revealed about the mechanism of translation, Nature (London) 461, 1234 (2009).
  39. X. Dai, M. Zhu, M. Warren, R. Balakrishnan, V. Patsalo, H. Okano, J. R. Williamson, K. Fredrick, Y. P. Wang, and T. Hwa, Reduction of translating ribosomes enables Escherichia coli to maintain elongation rates during slow growth, Nat. Microbiol. 2, 16231 (2016).
  40. X. Dai and M. Zhu, Coupling of ribosome synthesis and translational capacity with cell growth, Trends Biochem. Sci. 45, 681 (2020).
  41. M. Ueta, R. L. Ohniwa, H. Yoshida, Y. Maki, C. Wada, and A. Wada, Role of HPF (Hibernation promoting factor) in translational activity in Escherichia coli, J. Biochem. 143, 425 (2008).
  42. Y. S. Polikanov, G. M. Blaha, and T. A. Steitz, How hibernation factors RMF, HPF, and YfiA turn off protein synthesis, Science 336, 915 (2012).
  43. T. K. Wood and S. Song, Forming and waking dormant cells: The ppGpp ribosome dimerization persister model, Biofilm 2, 100018 (2020).
  44. L. Calabrese, J. Grilli, M. Osella, C. P. Kempes, M. Cosentino Lagomarsino, and L. Ciandrini, Protein degradation sets the fraction of active ribosomes at vanishing growth, PLoS Comput. Biol. 18, e1010059 (2022).
  45. M. Gupta, A. N. Johnson, E. R. Cruz, E. J. Costa, R. L. Guest, S. H. J. Li, E. M. Hart, T. Nguyen, M. Stadlmeier, B. P. Bratton, T. J. Silhavy, N. S. Wingreen, Z. Gitai, and M. Wühr, Global protein turnover quantification in Escherichia coli reveals cytoplasmic recycling under nitrogen limitation, Nat. Commun. 15, 5890 (2024).
  46. R. Schuetz, N. Zamboni, M. Zampieri, M. Heinemann, and U. Sauer, Multidimensional optimality of microbial metabolism, Science 336, 601 (2012).
  47. M. Basan, T. Honda, D. Christodoulou, M. Hörl, Y. F. Chang, E. Leoncini, A. Mukherjee, H. Okano, B. R. Taylor, J. M. Silverman, C. Sanchez, J. R. Williamson, J. Paulsson, T. Hwa, and U. Sauer, A universal trade-off between growth and lag in fluctuating environments, Nature (London) 584, 470 (2020).
  48. D. A. F. Fuentes, P. Manfredi, U. Jenal, and M. Zampieri, Pareto optimality between growth-rate and lag-time couples metabolic noise to phenotypic heterogeneity in Escherichia coli, Nat. Commun. 12, 3204 (2021).
  49. K. Krishnamurthy, T. Can, and D. J. Schwab, Theory of gating in recurrent neural networks, Phys. Rev. X 12, 011011 (2022).
  50. J. Pereira and X. J. Wang, A tradeoff between accuracy and flexibility in a working memory circuit endowed with slow feedback mechanisms, Cereb. Cortex 25, 3586 (2015).
  51. K. Greff, R. K. Srivastava, J. Koutnik, B. R. Steunebrink, and J. Schmidhuber, LSTM: A search space odyssey, IEEE Trans. Neural Networks Learn. Syst. 28, 2222 (2017).
  52. L. Wang, X. Zhang, H. Su, and J. Zhu, A comprehensive survey of continual learning: Theory, method and application, IEEE Trans. Pattern Anal. Mach. Intell. 46, 5362 (2024).
  53. e_coli-learning-model github repository, https://github.com/BanerjeeLab/e_coli-learning-model (2026).
  54. e_coli experimental dataset github repository, https://github.com/Si-Lab-CMU/Power-law-memory-governs-bacterial-adaptation-and-learning-in-fluctuating-environments (2026).
  55. P. Wang, L. Robert, J. Pelletier, W. L. Dang, F. Taddei, A. Wright, and S. Jun, Robust growth of Escherichia coli, Curr. Biol. 20, 1099 (2010).
  56. R. Thiermann, M. Sandler, G. Ahir, J. T. Sauls, J. Schroeder, S. Brown, G. Le Treut, F. Si, D. Li, J. D. Wang, and S. Jun, Tools and methods for high-throughput single-cell imaging with the mother machine, eLife 12, RP88463 (2023).
  57. E. Lyons, M. Freeling, S. Kustu, and W. Inwood, Using genomic sequencing for classical genetics in E. coli K12, PLoS ONE 6, e16717 (2011).
  58. J. T. Sauls, J. W. Schroeder, F. Si, S. D. Brown, G. Le Treut, and J. D. Wang, Mother machine image analysis with MM3, bioRxiv (2019).
  59. R. Droghetti, P. Fuchs, I. Iuliani, V. Firmano, G. Tallarico, L. Calabrese, J. Grilli, B. Sclavi, L. Ciandrini, and M. Cosentino Lagomarsino, Incoherent feedback from coupled amino acids and ribosome pools generates damped oscillations in growing E. coli, Nat. Commun. 16, 3063 (2025).
  60. S. Chakraborty, P. Singh, and A. S. N. Seshasayee, Understanding the genome-wide transcription response to various cAMP levels in bacteria using phenomenological models, mSystems 7, e00900-22 (2022).
  61. R. S. Sutton and A. G. Barto, Reinforcement Learning: An Introduction, 2nd ed. (MIT Press, Cambridge, MA, 2018).
  62. K. Diethelm, N. J. Ford, and A. D. Freed, A predictor-corrector approach for the numerical solution of fractional differential equations, Nonlin. Dyn. 29, 3 (2002).
  63. R. Garrappa, Numerical solution of fractional differential equations: A survey and a software tutorial, Mathematics 6, 16 (2018).
  64. A. R. Masegosa, D. Ramos-López, A. Salmerón, H. Langseth, and T. D. Nielsen, Variational inference over nonstationary data streams for exponential family models, Mathematics 8, 1942 (2020).

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PRX Life 4, 023015– Published 15 May, 2026
  • Received 12 August 2025
  • Accepted 16 April 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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