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  • Open Access

Theoretical Analysis of Resource-Induced Phase Transitions in Estimation Strategies

Takehiro Tottori1,2,3,* and Tetsuya J. Kobayashi3,†

  • *Contact author: takehiro.tottori@riken.jp
  • †Contact author: tetsuya@mail.crmind.net
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Phys. Rev. Lett. 137, 058401 – Published 28 July, 2026

DOI: https://doi.org/10.1103/5ynb-7k4v

Abstract

Organisms adapt to volatile environments by integrating sensory information with internal memory, yet their information processing is constrained by resource limitations. Such limitations can fundamentally alter optimal estimation strategies in biological systems. For example, recent experiments suggest that organisms exhibit phase transitions between memoryless and memory-based estimation strategies depending on energy availability and sensory reliability. However, a theoretical understanding of how resource limitations induce these transitions is still missing. This Letter presents an analytical characterization of the resource-induced phase transitions in optimal estimation strategies. Our results identify the conditions under which resource limitations alter estimation strategies and analytically reveal the mechanism underlying the emergence of discontinuous, nonmonotonic, and scaling behaviors. These results provide a theoretical foundation for understanding how limited resources shape information processing in biological systems.

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References (99)

  1. T. J. Kobayashi, Implementation of dynamic Bayesian decision making by intracellular kinetics, Phys. Rev. Lett. 104, 228104 (2010).
  2. M. Hinczewski and D. Thirumalai, Cellular signaling networks function as generalized Wiener-Kolmogorov filters to suppress noise, Phys. Rev. X 4, 041017 (2014).
  3. N. B. Becker, A. Mugler, and P. R. ten Wolde, Optimal prediction by cellular signaling networks, Phys. Rev. Lett. 115, 258103 (2015).
  4. T. Mora and I. Nemenman, Physical limit to concentration sensing in a changing environment, Phys. Rev. Lett. 123, 198101 (2019).
  5. K. Nakamura and T. J. Kobayashi, Connection between the bacterial chemotactic network and optimal filtering, Phys. Rev. Lett. 126, 128102 (2021).
  6. R. A. Heinonen, L. Biferale, A. Celani, and M. Vergassola, Optimal policies for Bayesian olfactory search in turbulent flows, Phys. Rev. E 107, 055105 (2023).
  7. J. Rode, M. Novak, and B. M. Friedrich, Information theory of chemotactic agents using both spatial and temporal gradient sensing, PRX Life 2, 023012 (2024).
  8. K. Krishnamurthy, M. R. Nassar, S. Sarode, and J. I. Gold, Arousal-related adjustments of perceptual biases optimize perception in dynamic environments, Nat. Hum. Behav. 1, 0107 (2017).
  9. A. T. Piet, A. El Hady, and C. D. Brody, Rats adopt the optimal timescale for evidence integration in a dynamic environment, Nat. Commun. 9, 4265 (2018).
  10. A. Kutschireiter, M. A. Basnak, R. I. Wilson, and J. Drugowitsch, Bayesian inference in ring attractor networks, Proc. Natl. Acad. Sci. U.S.A. 120, e2210622120 (2023).
  11. G. Lan, P. Sartori, S. Neumann, V. Sourjik, and Y. Tu, The energy–speed–accuracy trade-off in sensory adaptation, Nat. Phys. 8, 422 (2012).
  12. C. C. Govern and P. R. ten Wolde, Energy dissipation and noise correlations in biochemical sensing, Phys. Rev. Lett. 113, 258102 (2014).
  13. C. C. Govern and P. R. ten Wolde, Optimal resource allocation in cellular sensing systems, Proc. Natl. Acad. Sci. U.S.A. 111, 17486 (2014).
  14. A. H. Lang, C. K. Fisher, T. Mora, and P. Mehta, Thermodynamics of statistical inference by cells, Phys. Rev. Lett. 113, 148103 (2014).
  15. S. Ito and T. Sagawa, Maxwell’s demon in biochemical signal transduction with feedback loop, Nat. Commun. 6, 7498 (2015).
  16. G. Malaguti and P. R. ten Wolde, Theory for the optimal detection of time-varying signals in cellular sensing systems, eLife 10, e62574 (2021).
  17. A. J. Tjalma, V. Galstyan, J. Goedhart, L. Slim, N. B. Becker, and P. R. ten Wolde, Trade-offs between cost and information in cellular prediction, Proc. Natl. Acad. Sci. U.S.A. 120, e2303078120 (2023).
  18. S. J. Bryant and B. B. Machta, Physical constraints in intracellular signaling: The cost of sending a bit, Phys. Rev. Lett. 131, 068401 (2023).
  19. G. Nicoletti and D. M. Busiello, Tuning transduction from hidden observables to optimize information harvesting, Phys. Rev. Lett. 133, 158401 (2024).
  20. S. B. Laughlin, R. R. de Ruyter van Steveninck, and J. C. Anderson, The metabolic cost of neural information, Nat. Neurosci. 1, 36 (1998).
  21. P. Lennie, The cost of cortical computation, Curr. Biol. 13, 493 (2003).
  22. J. G. Burns, J. Foucaud, and F. Mery, Costs of memory: Lessons from “mini” brains, Proc. R. Soc. Ser. B Biol. Sci. 278, 923 (2010).
  23. F. Lieder and T. L. Griffiths, Resource-rational analysis: Understanding human cognition as the optimal use of limited computational resources, Behav. Brain Sci. 43, e1 (2020).
  24. R. Polanía, D. Burdakov, and T. A. Hare, Rationality, preferences, and emotions with biological constraints: It all starts from our senses, Trends Cognit. Sci. 28, 264 (2024).
  25. G. Tavoni, V. Balasubramanian, and J. I. Gold, What is optimal in optimal inference?, Curr. Opin. Behav. Sci. 29, 117 (2019).
  26. R. Bhui, L. Lai, and S. J. Gershman, Resource-rational decision making, Curr. Opin. Behav. Sci. 41, 15 (2021).
  27. D. Kahneman, Thinking, Fast and Slow (Farrar, Straus and Giroux, New York, 2011).
  28. M. T. Gailliot, R. F. Baumeister, C. N. DeWall, J. K. Maner, E. A. Plant, D. M. Tice, L. E. Brewer, and B. J. Schmeichel, Self-control relies on glucose as a limited energy source: Willpower is more than a metaphor, J. Pers. Soc. Psychol. 92, 325 (2007).
  29. E. Masicampo and R. F. Baumeister, Toward a physiology of dual-process reasoning and judgment: Lemonade, willpower, and expensive rule-based analysis, Psychol. Sci. 19, 255 (2008).
  30. G. Tavoni, T. Doi, C. Pizzica, V. Balasubramanian, and J. I. Gold, Human inference reflects a normative balance of complexity and accuracy, Nat. Hum. Behav. 6, 1153 (2022).
  31. T. Tottori and T. J. Kobayashi, Memory-limited partially observable stochastic control and its mean-field control approach, Entropy 24, 1599 (2022).
  32. T. Tottori and T. J. Kobayashi, Forward-backward sweep method for the system of HJB-FP equations in memory-limited partially observable stochastic control, Entropy 25, 208 (2023).
  33. T. Tottori and T. J. Kobayashi, Decentralized stochastic control with finite-dimensional memories: A memory limitation approach, Entropy 25, 791 (2023).
  34. T. Tottori and T. J. Kobayashi, Resource limitations induce phase transitions in biological information processing, Phys. Rev. Res. 7, L042012 (2025).
  35. 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).
  36. K. P. Körding and D. M. Wolpert, Bayesian integration in sensorimotor learning, Nature (London) 427, 244 (2004).
  37. J. Burge, M. O. Ernst, and M. S. Banks, The statistical determinants of adaptation rate in human reaching, J. Vis. 8, 20 (2008).
  38. A. Prat-Carrabin, R. C. Wilson, J. D. Cohen, and R. Azeredo da Silveira, Human inference in changing environments with temporal structure, Psychol. Rev. 128, 879 (2021).
  39. J. Yong and X. Y. Zhou, Stochastic Controls (Springer, New York, 1999).
  40. M. Nisio, Stochastic Control Theory, Probability Theory and Stochastic Modelling Vol. 72 (Springer Japan, Tokyo, 2015).
  41. A. Bensoussan, Estimation and Control of Dynamical Systems, Interdisciplinary Applied Mathematics Vol. 48 (Springer International Publishing, Cham, 2018).
  42. A. Pezzotta, M. Adorisio, and A. Celani, Chemotaxis emerges as the optimal solution to cooperative search games, Phys. Rev. E 98, 042401 (2018).
  43. A. Pezzotta and J. Briscoe, Optimal control of gene regulatory networks for morphogen-driven tissue patterning, Cell Syst. 14, 940 (2023).
  44. C. Briat and M. Khammash, Continuous-time and sampled-data optimal control of linear stochastic reaction networks, in 2022 IEEE 61st Conference on Decision and Control (CDC) (IEEE, Piscataway, New Jersey, 2022), pp. 3865–3870.
  45. S. A. Horiguchi and T. J. Kobayashi, Cellular gradient flow structure linking single-cell-level rules and population-level dynamics, Phys. Rev. Res. 5, L022052 (2023).
  46. S. K. Schnyder, J. J. Molina, R. Yamamoto, and M. S. Turner, Understanding Nash epidemics, Proc. Natl. Acad. Sci. U.S.A. 122, e2409362122 (2025).
  47. See Supplemental Material at http://link.aps.org/supplemental/10.1103/5ynb-7k4v for detailed derivations and further discussions, which includes Refs. [48–60].
  48. A. Bensoussan, J. Frehse, and S. C. P. Yam, The master equation in mean field theory, J. Math. Pures Appl. 103, 1441 (2015).
  49. A. Bensoussan, J. Frehse, and S. C. P. Yam, On the interpretation of the master equation, Stochastic Processes Appl. 127, 2093 (2017).
  50. R. Carmona and F. Delarue, Probabilistic Theory of Mean Field Games with Applications I, Probability theory and stochastic modelling Vol. 83 (Springer Nature, Cham, 2018).
  51. R. Carmona and F. Delarue, Probabilistic Theory of Mean Field Games with Applications II, Probability Theory and Stochastic Modelling Vol. 84 (Springer International Publishing, Cham, 2018).
  52. M. F. Krummel, F. Bartumeus, and A. Gérard, T cell migration, search strategies and mechanisms, Nat. Rev. Immunol. 16, 193 (2016).
  53. D. R. Sowinski, J. Carroll-Nellenback, R. N. Markwick, J. Piñero, M. Gleiser, A. Kolchinsky, G. Ghoshal, and A. Frank, Semantic information in a model of resource gathering agents, PRX Life 1, 023003 (2023).
  54. K. V. B. Verano, E. Panizon, and A. Celani, Olfactory search with finite-state controllers, Proc. Natl. Acad. Sci. U.S.A. 120, e2304230120 (2023).
  55. A. Celani and E. Panizon, Olfactory search, arXiv:2405.03374.
  56. G. M. Viswanathan, S. V. Buldyrev, S. Havlin, M. G. E. da Luz, E. P. Raposo, and H. E. Stanley, Optimizing the success of random searches, Nature (London) 401, 911 (1999).
  57. O. Bénichou, M. Coppey, M. Moreau, P.-H. Suet, and R. Voituriez, Optimal search strategies for hidden targets, Phys. Rev. Lett. 94, 198101 (2005).
  58. G. M. Viswanathan, E. P. Raposo, and M. G. E. da Luz, Lévy flights and superdiffusion in the context of biological encounters and random searches, Phys. Life Rev. 5, 133 (2008).
  59. O. Bénichou, C. Loverdo, M. Moreau, and R. Voituriez, Intermittent search strategies, Rev. Mod. Phys. 83, 81 (2011).
  60. A. M. Hein, F. Carrara, D. R. Brumley, R. Stocker, and S. A. Levin, Natural search algorithms as a bridge between organisms, evolution, and ecology, Proc. Natl. Acad. Sci. U.S.A. 113, 9413 (2016).
  61. N. Tishby, F. C. Pereira, and W. Bialek, The information bottleneck method, arXiv:physics/0004057.
  62. G. Chechik, A. Globerson, N. Tishby, and Y. Weiss, Information bottleneck for Gaussian variables, J. Mach. Learn. Res. 6, 165 (2005).
  63. F. Creutzig, A. Globerson, and N. Tishby, Past-future information bottleneck in dynamical systems, Phys. Rev. E 79, 041925 (2009).
  64. D. Strouse and D. J. Schwab, The deterministic information bottleneck, Neural Comput. 29, 1611 (2017).
  65. V. Sachdeva, T. Mora, A. M. Walczak, and S. E. Palmer, Optimal prediction with resource constraints using the information bottleneck, PLoS Comput. Biol. 17, e1008743 (2021).
  66. V. Galstyan, A. Tjalma, and P. R. ten Wolde, Intuitive dissection of the Gaussian information bottleneck method with an application to optimal prediction, Phys. Rev. Res. 7, 043261 (2025).
  67. U. Alon, An Introduction to Systems Biology: Design Principles of Biological Circuits, 2nd ed. (Chapman and Hall/CRC, London, 2019).
  68. U. Alon, Network motifs: Theory and experimental approaches, Nat. Rev. Genet. 8, 450 (2007).
  69. S. Mangan, A. Zaslaver, and U. Alon, The coherent feedforward loop serves as a sign-sensitive delay element in transcription networks, J. Mol. Biol. 334, 197 (2003).
  70. S. Mangan and U. Alon, Structure and function of the feed-forward loop network motif, Proc. Natl. Acad. Sci. U.S.A. 100, 11980 (2003).
  71. E. Dekel, S. Mangan, and U. Alon, Environmental selection of the feed-forward loop circuit in gene-regulation networks, Phys. Biol. 2, 81 (2005).
  72. R. Milo, S. Shen-Orr, S. Itzkovitz, N. Kashtan, D. Chklovskii, and U. Alon, Network motifs: Simple building blocks of complex networks, Science 298, 824 (2002).
  73. S. S. Shen-Orr, R. Milo, S. Mangan, and U. Alon, Network motifs in the transcriptional regulation network of Escherichia coli, Nat. Genet. 31, 64 (2002).
  74. R. Milo, S. Itzkovitz, N. Kashtan, R. Levitt, S. Shen-Orr, I. Ayzenshtat, M. Sheffer, and U. Alon, Superfamilies of evolved and designed networks, Science 303, 1538 (2004).
  75. V. Sourjik and N. S. Wingreen, Responding to chemical gradients: Bacterial chemotaxis, Curr. Opin. Cell Biol. 24, 262 (2012).
  76. Y. Tu, Quantitative modeling of bacterial chemotaxis: Signal amplification and accurate adaptation, Annu. Rev. Biophys. 42, 337 (2013).
  77. A. Gosztolai and M. Barahona, Cellular memory enhances bacterial chemotactic navigation in rugged environments, Commun. Phys. 3, 1 (2020).
  78. P. Lertsethtakarn, K. M. Ottemann, and D. R. Hendrixson, Motility and chemotaxis in campylobacter and helicobacter, Annu. Rev. Microbiol. 65, 389 (2011).
  79. X. Liu and K. M. Ottemann, Methylation-independent chemotaxis systems are the norm for gastric-colonizing helicobacter species, J. Bacteriol. 204, e00231 (2022).
  80. M. Kargeti, I. Kalita, S. Hoch, M. Ratnikava, W. Xu, B. Ni, R. L. Dy, R. Colin, and V. Sourjik, Experimental evolution partially restores functionality of bacterial chemotaxis network with reduced number of components, PLoS Genet. 21, e1011784 (2025).
  81. H. H. Mattingly, K. Kamino, B. B. Machta, and T. Emonet, Escherichia coli chemotaxis is information limited, Nat. Phys. 17, 1426 (2021).
  82. H. H. Mattingly, K. Kamino, J. Ong, R. Kottou, T. Emonet, and B. B. Machta, E. coli chemosensing accuracy is not limited by stochastic molecule arrivals, Nat. Phys. 22, 123 (2026).
  83. K. Nakamura and T. J. Kobayashi, Optimal sensing and control of run-and-tumble chemotaxis, Phys. Rev. Res. 4, 013120 (2022).
  84. G. Tkačik, C. G. Callan, and W. Bialek, Information flow and optimization in transcriptional regulation, Proc. Natl. Acad. Sci. U.S.A. 105, 12265 (2008).
  85. J. O. Dubuis, G. Tkačik, E. F. Wieschaus, T. Gregor, and W. Bialek, Positional information, in bits, Proc. Natl. Acad. Sci. U.S.A. 110, 16301 (2013).
  86. 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).
  87. G. Tkačik and T. Gregor, The many bits of positional information, Development 148, dev176065 (2021).
  88. 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. U.S.A. 118, e2109011118 (2021).
  89. M. Bauer and W. Bialek, Information bottleneck in molecular sensing, PRX Life 1, 023005 (2023).
  90. T. Mijatović, A. R. Kok, M. Brüggen, J. W. Zwanikken, and M. Bauer, Weak transcription factor clustering at binding sites can facilitate information transfer from molecular signals, PRX Life 4, 013003 (2026).
  91. V. Krishnan, S. Sinha, and L. Mahadevan, Hamiltonian bridge for scale-dependent optimal control of particles and fields, Newton 2, 100365 (2025).
  92. J. I. Gold and M. N. Shadlen, The neural basis of decision making, Annu. Rev. Neurosci. 30, 535 (2007).
  93. R. Ratcliff and G. McKoon, The diffusion decision model: Theory and data for two-choice decision tasks, Neural Comput. 20, 873 (2008).
  94. R. Ratcliff, A theory of memory retrieval, Psychol. Rev. 85, 59 (1978).
  95. R. Bogacz, E. Brown, J. Moehlis, P. Holmes, and J. D. Cohen, The physics of optimal decision making: A formal analysis of models of performance in two-alternative forced-choice tasks, Psychol. Rev. 113, 700 (2006).
  96. B. W. Brunton, M. M. Botvinick, and C. D. Brody, Rats and humans can optimally accumulate evidence for decision-making, Science 340, 95 (2013).
  97. S. Tajima, J. Drugowitsch, and A. Pouget, Optimal policy for value-based decision-making, Nat. Commun. 7, 12400 (2016).
  98. S. Tajima, J. Drugowitsch, N. Patel, and A. Pouget, Optimal policy for multi-alternative decisions, Nat. Neurosci. 22, 1503 (2019).
  99. I. Neri and S. Pigolotti, Amendable decisions in living systems, bioRxiv (2026), 10.1101/2025.06.03.657747.

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Phys. Rev. Lett. 137, 058401– Published 28 July, 2026
  • Received 18 November 2025
  • Accepted 3 June 2026
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