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  • Open Access

Quantification of Information Flow by Dual Reporter System and Its Application to Bacterial Chemotaxis

Kento Nakamura1,*,†, Hajime Fukuoka2, Akihiko Ishijima2, and Tetsuya J. Kobayashi3,†,‡,§,∥

  • 1RIKEN Center for Brain Science, 2-1 Hirosawa, Wako, Saitama 351-0198, Japan
  • 2Graduate School of Frontier Biosciences, Osaka University, 1-3 Yamadaoka, Suita, Osaka 565-0871, Japan
  • 3Institute of Industrial Science, The University of Tokyo, 4-6-1, Komaba, Meguro-ku, Tokyo 153-8505 Japan

  • *Contact author: kento.nakamura@riken.jp, kento.nakamura.bio@gmail.com
  • †Also at Theoretical Sciences Visiting Program (TSVP), Okinawa Institute of Science and Technology Graduate University, Onna 904-0495, Japan.
  • ‡Contact author: tetsuya@mail.crmind.net
  • §Also at Universal Biology Institute, The University of Tokyo, 7-3-1, Hongo, Bunkyo-ku, Tokyo 113-8654, Japan.
  • ∥Also at Department of Mathematical Informatics, the Graduate School of Information Science and Technology, The University of Tokyo, 7-3-1, Hongo, Bunkyo-ku, Tokyo 113-8654, Japan.
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Phys. Rev. Lett. 136, 108403 – Published 12 March, 2026

DOI: https://doi.org/10.1103/jrph-wj94

Abstract

Mutual information is a theoretically grounded metric for quantifying cellular signaling pathways. However, its measurement demands characterization of both input and output distributions, limiting practical applications. Here, we present alternative method that alleviates this requirement using dual reporter systems. By extending extrinsic-intrinsic noise analysis, we derive a mutual information estimator that eliminates the need to measure input distribution. We demonstrate our method by analyzing the bacterial chemotactic pathway, regarding multiple flagellar motors as natural dual reporters. We show the biological relevance of the measured information flow by comparing it with theoretical bounds on sensory information. This framework opens new directions for quantifying information flow in cellular signaling pathways.

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References (85)

  1. C. V. Rao, D. M. Wolf, and A. P. Arkin, Nature (London) 420, 231 (2002).
  2. C. G. Bowsher and P. S. Swain, Curr. Opin. Biotechnol. 28, 149 (2014).
  3. P. François and A. Zilman, Curr. Opin. Syst. Biol. 18, 111 (2019).
  4. B. W. Andrews and P. A. Iglesias, PLoS Comput. Biol. 3, e153 (2007).
  5. E. Libby, T. J. Perkins, and P. S. Swain, Proc. Natl. Acad. Sci. U.S.A. 104, 7151 (2007).
  6. G. Tkacik, A. M. Walczak, and W. Bialek, Phys. Rev. E 80, 031920 (2009).
  7. F. Tostevin and P. R. Ten Wolde, Phys. Rev. Lett. 102, 218101 (2009).
  8. M. D. Petkova, G. Tkačik, W. Bialek, E. F. Wieschaus, and T. Gregor, Cell 176, 844 (2019).
  9. T. Mora and I. Nemenman, Phys. Rev. Lett. 123, 198101 (2019).
  10. G. Malaguti and P. R. Ten Wolde, eLife 10, e62574 (2021).
  11. H. Mattingly, K. Kamino, B. Machta, and T. Emonet, Nat. Phys. 17, 1426 (2021).
  12. J. Rode, M. Novak, and B. M. Friedrich, PRX Life 2, 023012 (2024).
  13. T. Tottori and T. J. Kobayashi, Phys. Rev. Res. 7, L042012 (2025).
  14. T. Tottori and T. J. Kobayashi, Phys. Rev. Res. 7, 043048 (2025).
  15. G. Tkačik and P. R. t. Wolde, Annu. Rev. Biophys. 54, 249 (2025).
  16. T. J. Kobayashi, Phys. Rev. Lett. 104, 228104 (2010).
  17. C. Zechner, G. Seelig, M. Rullan, and M. Khammash, Proc. Natl. Acad. Sci. U.S.A. 113, 4729 (2016).
  18. K. Nakamura and T. J. Kobayashi, Phys. Rev. Lett. 126, 128102 (2021).
  19. K. Nakamura and T. J. Kobayashi, Phys. Rev. Res. 4, 013120 (2022).
  20. C. E. Shannon, Bell Syst. Tech. J. 27, 379 (1948).
  21. G. Tkačik and W. Bialek, Annu. Rev. Condens. Matter Phys. 7, 89 (2016).
  22. S. Uda, Biophys. Rev. Lett. 12, 377 (2020).
  23. Y. Tang and A. Hoffmann, Rep. Prog. Phys. 85, 086602 (2022).
  24. P. A. Iglesias, IEEE Trans. Mol. Biol. Multiscale Commun. 2, 31 (2016).
  25. J. M. R. Parrondo, J. M. Horowitz, and T. Sagawa, Nat. Phys. 11, 131 (2015).
  26. T. Gregor, E. F. Wieschaus, A. P. McGregor, W. Bialek, and D. W. Tank, Cell 130, 141 (2007).
  27. G. Tkacik, C. G. Callan, Jr., and W. Bialek, Proc. Natl. Acad. Sci. U.S.A. 105, 12265 (2008).
  28. R. C. Yu, C. G. Pesce, A. Colman-Lerner, L. Lok, D. Pincus, E. Serra, M. Holl, K. Benjamin, A. Gordon, and R. Brent, Nature (London) 456, 755 (2008).
  29. A. A. Granados, J. M. J. Pietsch, S. A. Cepeda-Humerez, I. L. Farquhar, G. Tkačik, and P. S. Swain, Proc. Natl. Acad. Sci. U.S.A. 115, 6088 (2018).
  30. R. Cheong, A. Rhee, C. J. Wang, I. Nemenman, and A. Levchenko, Science 334, 354 (2011).
  31. S. Uda, T. H. Saito, T. Kudo, T. Kokaji, T. Tsuchiya, H. Kubota, Y. Komori, Y.-I. Ozaki, and S. Kuroda, Science 341, 558 (2013).
  32. M. Voliotis, R. M. Perrett, C. McWilliams, C. A. McArdle, and C. G. Bowsher, Proc. Natl. Acad. Sci. U.S.A. 111, E326 (2014).
  33. J. Selimkhanov, B. Taylor, J. Yao, A. Pilko, J. Albeck, A. Hoffmann, L. Tsimring, and R. Wollman, Science 346, 1370 (2014).
  34. T. Jetka, K. Nienałtowski, T. Winarski, S. Błoński, and M. Komorowski, PLoS Comput. Biol. 15, e1007132 (2019).
  35. Y. Tang, A. Adelaja, F. X.-F. Ye, E. Deeds, R. Wollman, and A. Hoffmann, Nat. Commun. 12, 1272 (2021).
  36. S. R. Achar, F. X. P. Bourassa, T. J. Rademaker, A. Lee, T. Kondo, E. Salazar-Cavazos, J. S. Davies, N. Taylor, P. François, and G. Altan-Bonnet, Science 376, 880 (2022).
  37. A.-L. Moor and C. Zechner, Phys. Rev. Res. 5, 013032 (2023).
  38. M. Reinhardt, G. c. v. Tkačik, and P. R. ten Wolde, Phys. Rev. X 13, 041017 (2023).
  39. M. Gehri, N. Engelmann, and H. Koeppl, in Proceedings of the 2024 IEEE International Symposium on Information Theory (ISIT) (IEEE, New York, 2024), pp. 1931–1936.
  40. P. S. Swain, M. B. Elowitz, and E. D. Siggia, Proc. Natl. Acad. Sci. U.S.A. 99, 12795 (2002).
  41. M. B. Elowitz, A. J. Levine, E. D. Siggia, and P. S. Swain, Science 297, 1183 (2002).
  42. A. Hilfinger and J. Paulsson, Proc. Natl. Acad. Sci. U.S.A. 108, 12167 (2011).
  43. C. G. Bowsher and P. S. Swain, Proc. Natl. Acad. Sci. U.S.A. 109, E1320 (2012).
  44. H. C. Berg, E. coli in Motion (Springer, New York, 2004).
  45. Y. Tu, Annu. Rev. Biophys. 42, 337 (2013).
  46. H. H. Mattingly and Y. Tu, Nat. Phys. 22, 131 (2026).
  47. T. M. Cover and J. A. Thomas, Elements of Information Theory (Wiley Series in Telecommunications and Signal Processing) (Wiley-Interscience, Hoboken, NJ, USA, 2006).
  48. Y. Uchida, T. Hamamoto, Y.-S. Che, H. Takahashi, J. S. Parkinson, A. Ishijima, and H. Fukuoka, J. Bacteriol. 204, e00278 (2022).
  49. See Supplemental Material at http://link.aps.org/supplemental/10.1103/jrph-wj94 for detailed derivations of the estimators and discussion of applicability, which includes Refs. [50–66].
  50. Ö. D. Özçete, A. Banerjee, and P. S. Kaeser, Mol. Psychiatry 29, 3680 (2024).
  51. J. O. Dubuis, G. Tkačik, E. F. Wieschaus, T. Gregor, and W. Bialek, Proc. Natl. Acad. Sci. U.S.A. 110, 16301 (2013).
  52. R. Shwartz-Ziv and N. Tishby, arXiv:1703.00810.
  53. Z. Goldfeld, E. van den Berg, K. Greenewald, I. Melnyk, N. Nguyen, B. Kingsbury, and Y. Polyanskiy, in Proceedings of the International Conference on Machine Learning, edited by K. Chaudhuri and R. Salakhutdinov, (International Machine Learning Society (IMLS), Long Beach, CA, USA, 2019), Vol. 97, pp. 2299–2308, http://proceedings.mlr.press/v97/goldfeld19a/goldfeld19a.pdf.
  54. J. Fernando and G. Guitchounts, arXiv:2502.12131.
  55. F. Tostevin and P. R. Ten Wolde, Phys. Rev. E 81, 061917 (2010).
  56. A. Kutschireiter, S. C. Surace, and J.-P. Pfister, J. Math. Psychol. 94, 102307 (2020).
  57. T. J. Kobayashi, Phys. Rev. Lett. 106, 228101 (2011).
  58. C. Gardiner, Stochastic Methods (Springer, Berlin, 2009), Vol. 4.
  59. E. Mayer-Wolf and M. Zakai, in Filtering and Control of Random Processes, edited by H. Korezlioglu, G. Mazziotto, and J. Szpirglas (Springer, Berlin, Heidelberg, 1984), pp. 164–171.
  60. S. K. Mitter and N. J. Newton, J. Stat. Phys. 118, 145 (2005).
  61. J. E. Gough and N. H. Amini, arXiv:1710.05553.
  62. J. Yong and X. Y. Zhou, Stochastic Controls: Hamiltonian Systems and HJB Equations (Springer Science & Business Media, New York, 1999), Vol. 43.
  63. A. Segall, IEEE Trans. Autom. Control 22, 179 (1977).
  64. A. Bensoussan, Stochastic Control of Partially Observable Systems (Cambridge University Press, Cambridge, England, 1992).
  65. A. C. Davison and D. V. Hinkley, Bootstrap Methods and Their Application (Cambridge University Press, Cambridge, England, 1997).
  66. M. R. Kosorok, Introduction to Empirical Processes and Semiparametric Inference (Springer, New York, 2008).
  67. D. N. Politis and J. P. Romano, J. Am. Stat. Assoc. 89, 1303 (1994).
  68. S. Terasawa, H. Fukuoka, Y. Inoue, T. Sagawa, H. Takahashi, and A. Ishijima, Biophys. J. 100, 2193 (2011).
  69. H. Mao, P. S. Cremer, and M. D. Manson, Proc. Natl. Acad. Sci. U.S.A. 100, 5449 (2003).
  70. D. L. Englert, M. D. Manson, and A. Jayaraman, Appl. Environ. Microbiol. 75, 4557 (2009).
  71. L. Jiang, Q. Ouyang, and Y. Tu, PLoS Comput. Biol. 6, e1000735 (2010).
  72. M. Flores, T. S. Shimizu, P. R. ten Wolde, and F. Tostevin, Phys. Rev. Lett. 109, 148101 (2012).
  73. Y. S. Dufour, X. Fu, L. Hernandez-Nunez, and T. Emonet, PLoS Comput. Biol. 10, e1003694 (2014).
  74. T. Deguchi, M. K. Iwanski, E.-M. Schentarra, C. Heidebrecht, L. Schmidt, J. Heck, T. Weihs, S. Schnorrenberg, P. Hoess, S. Liu et al., Science 379, 1010 (2023).
  75. D. Watanabe, M. Hiroshima, M. Yasui, and M. Ueda, Nat. Commun. 15, 8975 (2024).
  76. N. Umeki, Y. Kabashima, and Y. Sako, eLife 14, e104432 (2025).
  77. M. Sun and J. Zhang, Nucleic Acids Res. 48, 533 (2020).
  78. H. Kazama and R. I. Wilson, Nat. Neurosci. 12, 1136 (2009).
  79. R. L. Goris, J. A. Movshon, and E. P. Simoncelli, Nat. Neurosci. 17, 858 (2014).
  80. A. R. Palmer and C. Strobeck, Annu. Rev. Ecol. Syst. 17, 391 (1986).
  81. C. P. Klingenberg, Front. Ecol. Evol. 7, 56 (2019).
  82. L. Ham, M. Jackson, and M. P. Stumpf, eLife 10, e69324 (2021).
  83. H. H. Mattingly, K. Kamino, J. Ong, R. Kottou, T. Emonet, and B. B. Machta, Nat. Phys. 22, 123 (2026).
  84. https://github.com/rotala17/nakamura2026-information-flow.
  85. 10.6084/m9.figshare.31304680.

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Phys. Rev. Lett. 136, 108403– Published 12 March, 2026
  • Received 7 July 2025
  • Accepted 6 February 2026
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