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  • Open Access

Ignition Criteria for Trigger Waves in Cell Signaling

Brandon H. Schlomann1,*, William S. DeWitt2, Yuanzhao Zhang3, and Kasturi Shah4

  • *Contact author: bschloma@berkeley.edu
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PRX Life 4, 023003 – Published 6 April, 2026

DOI: https://doi.org/10.1103/z7hq-1gxg

Abstract

To rapidly coordinate collective behaviors in tissues, cells communicate with one another through traveling fronts of signaling activity called trigger waves. The stimulus threshold for wave propagation is a key biological parameter that determines when the system produces a global response to a local stimulus. However, it is unclear how this stimulus threshold is determined by the properties of the underlying signaling network. To address this gap, we studied a model of trigger waves with a realistic Hill-type autoactivation function. We obtained an analytic expression for the wave ignition threshold in one dimension (1D) and numerical solutions in 2D and 3D. In the limit of high sensitivity, we found that the trigger wave threshold depends on both the effective dissociation constant, KD, and the effective Hill coefficient, n, of the positive feedback circuit, with the dominant contribution scaling as KDn/(n−1) in all dimensions. This result leads to a simple method for predicting the trigger wave ignition threshold from bulk activation data and is potentially of use for developing synthetic trigger wave circuits with desired sensitivities.

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References (45)

  1. B. Alberts, R. Heald, A. Johnson, D. Morgan, M. Raff, K. Roberts, and P. Walter, Molecular Biology of the Cell (Seventh Edition) (W. W. Norton, New York, 2022).
  2. A. Einstein, Über die von der molekularkinetischen Theorie der Wärme geforderte Bewegung von in ruhenden Flüssigkeiten suspendierten Teilchen, Ann. Phys. 322, 549 (1905).
  3. R. Phillips, J. Kondev, J. Theriot, and H. Garcia, Physical Biology of the Cell (Garland Science, New York, 2012).
  4. L. Gelens, G. A. Anderson, and J. E. Ferrell Jr, Spatial trigger waves: Positive feedback gets you a long way, Mol. Biol. Cell 25, 3486 (2014).
  5. K Hara, Cinematographic observation of “surface contraction waves” (SCW) during the early cleavage of axolotl eggs, W. Roux' Arch Entwicklungsmech. Org. 167, 183 (1971).
  6. J. B. Chang and J. E. Ferrell, Jr., Mitotic trigger waves and the spatial coordination of the Xenopus cell cycle, Nature (Lond.) 500, 603 (2013).
  7. M. Vergassola, V. E. Deneke, and S. Di Talia, Mitotic waves in the early embryogenesis of Drosophila: Bistability traded for speed, Proc. Natl. Acad. Sci. USA 115, E2165 (2018).
  8. A. H. Cornell-Bell, S. M. Finkbeiner, M. S. Cooper, and S. J. Smith, Glutamate induces calcium waves in cultured astrocytes: long-range glial signaling, Science, 247, 470 (1990).
  9. Y. Fan, C. Chai, P. Li, X. Zou, J. E. Ferrell, Jr., and B. Wang, Ultrafast distant wound response is essential for whole-body regeneration, Cell 186, 3606 (2023).
  10. H. K. C. Co, C.-C. Wu, Y.-C. Lee, and Sheng-hong Chen, Emergence of large-scale cell death through ferroptotic trigger waves, Nature (London) 631, 654 (2024).
  11. P. Yde, M. Hogh Jensen, and A. Trusina, Analyzing inflammatory response as excitable media, Phys. Rev. E 84, 051913 (2011).
  12. P. Yde, B. Mengel, M. H. Jensen, S. Krishna, and A. Trusina, Modeling the NF−κB mediated inflammatory response predicts cytokine waves in tissue, BMC Syst. Biol. 5, 115 (2011).
  13. R Luther, Räumliche Fortpflanzung chemischer Reaktionen, Z. Elektrochem. Angew. Phys. Chem. 12, 596 (1906).
  14. C. Trentesaux, T. Yamada, O. D. Klein, and W. A. Lim, Harnessing synthetic biology to engineer organoids and tissues, Cell Stem Cell 30, 10 (2023).
  15. O. A. Mornev, On the Conditions of Excitation of One-dimensional Autowave Media: Autowave Processes in Systems with Diffusion (Institute of Applied Physics of the USSR Academy of Sciences, Gorky, 1981), pp. 92–98.
  16. H. P. McKean and V. Moll, A threshold for a caricature of the nerve equation, Bull. Am. Math. Soc. 12, 255 (1985).
  17. G. Flores, The stable manifold of the standing wave of the Nagumo equation, J. Diff. Equ. 80, 306 (1989).
  18. J. C. Neu, R. S. Preissig, and W. Krassowska, Initiation of propagation in a one-dimensional excitable medium, Physica D 102, 285 (1997).
  19. I. Idris and V. N. Biktashev, Analytical approach to initiation of propagating fronts, Phys. Rev. Lett. 101, 244101 (2008).
  20. B. Bezekci and V. N. Biktashev, Fast-slow asymptotic for semi-analytical ignition criteria in FitzHugh-Nagumo system, Chaos 27, 093916 (2017).
  21. B. Bezekci and V. N. Biktashev, Strength-Duration relationship in an excitable medium, Commun. Nonlin. Sci. Numer. Simul. 80, 104954 (2020).
  22. J. Zhao, M. L. Perkins, M. Norstad, and H. G. Garcia, A bistable autoregulatory module in the developing embryo commits cells to binary expression fates, Curr. Biol. 33, 2851 (2023).
  23. H. Tran, J. Desponds, C. A. P. Romero, M. Coppey, C. Fradin, N. Dostatni, and A. M. Walczak, Precision in a rush: Trade-offs between reproducibility and steepness of the hunchback expression pattern, PLoS Comput. Biol. 14, e1006513 (2018).
  24. G. Birzu, O. Hallatschek, and K. S. Korolev, Fluctuations uncover a distinct class of traveling waves, Proc. Natl. Acad. Sci. USA 115, E3645 (2018).
  25. W. van Saarloos, Front propagation into unstable states, Phys. Rep. 386, 29 (2003).
  26. M. O. Lavrentovich and D. R. Nelson, Asymmetric mutualism in two- and three-dimensional range eExpansions, Phys. Rev. Lett. 112, 138102 (2014).
  27. M. Gralka and O. Hallatschek, Environmental heterogeneity can tip the population genetics of range expansions, eLife 8, e44359 (2019).
  28. S. R. Gandhi, E. A. Yurtsev, K. S. Korolev, and J. Gore, Range expansions transition from pulled to pushed waves as growth becomes more cooperative in an experimental microbial population, Proc. Natl. Acad. Sci. USA 113, 6922 (2016).
  29. V. Volpert and S. Petrovskii, Reaction-diffusion waves in biology: New trends, recent developments, Phys. Life Rev. 52, 1 (2025).
  30. A. M. Turing, The chemical basis of morphogenesis, Philos. Trans. R. Soc. Lond. B 237, 37 (1952).
  31. R. A. Fisher, The wave of advance of advantageous genes, Ann. Eugenics 7, 355 (1937).
  32. A. Kolmogorov, I. Petrovskii, and N. Piskunov, A study of the diffusion equation with increase in the amount of substance, and its application to a biological problem, Mosc. Univ. Math. Bull. 1, 1 (1937).
  33. M. Son, T. Frank, T. Holst-Hansen, A. G. Wang, M. Junkin, S. S. Kashaf, A. Trusina, and S. Tay, Spatiotemporal NF−κB dynamics encodes the position, amplitude, and duration of local immune inputs, Sci. Adv. 8, eabn6240 (2022).
  34. C. Sokolik, Y. Liu, D. Bauer, J. McPherson, M. Broeker, G. Heimberg, L. S. Qi, D. A. Sivak, and Matt Thomson, Transcription factor competition allows embryonic stem cells to distinguish authentic sSignals from noise, Cell Syst. 1, 117 (2015).
  35. R. Swan, Factorization of polynomials over finite fields, Pacific J. Math. 12, 1099 (1962).
  36. V. Arnold, Ordinary Differential Equations (MIT Press, Cambridge, MA, 1973).
  37. L. J. M. Bruurs, M. Müller, J. G. Schipper, H. H. Rabouw, S. Boersma, F. J. M. van Kuppeveld, and M. E. Tanenbaum, Antiviral responses are shaped by heterogeneity in viral replication dynamics, Nat. Microbiol. 8, 2115 (2023).
  38. R. A. Gottschalk, A. J. Martins, B. R. Angermann, B. Dutta, C. E. Ng, S. Uderhardt, J. S. Tsang, I. D. C. Fraser, M. Meier-Schellersheim, and R. N. Germain, Distinct NF−κB and MAPK activation thresholds uncouple steady-state microbe sensing from anti-pathogen inflammatory responses, Cell Syst. 2, 378 (2016).
  39. R. Singh, S. Harsh, A. Bajpai, S. Pal, R. K. Pandey, and P. Sinha, A Serpin27A-dependent Toll signaling underlies host genetics of cancer resistance in Drosophila, bioRxiv [Preprint] (2021), doi:10.1101/2021.03.13.435228.
  40. G. Zahn and A. Greischel, Pharmacokinetics of tumor necrosis factor alpha after intravenous administration in rats. Dose dependence and influence of tumor necrosis factor beta, Arzneimittel-Forsch. 39, 1180 (1989).
  41. T. Moritz, N. Niederle, J. Baumann, D. May, E. Kurschel, R. Osieka, J. Kempeni, E. Schlick, and C. G. Schmidt, Phase I study of recombinant human tumor necrosis factor alpha in advanced malignant disease, Cancer Immunol. Immunother. CII 29, 144 (1989).
  42. A. Adelaja, B. Taylor, K. M. Sheu, Y. Liu, S. Luecke, and A. Hoffmann, Six distinct NFκB signaling codons convey discrete information to distinguish stimuli and enable appropriate macrophage responses, Immunity 54, 916 (2021).
  43. W. S. DeWitt, immunowave (2026), https://github.com/dewitt-lab/immunowave
  44. R. Descartes and I. Maclean, A, Discourse on the Method, Oxford World's Classics (Oxford University Press, Oxford, 2006).
  45. P. Kidger, On neural differential equations, Ph.D. thesis, University of Oxford, 2021.

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PRX Life 4, 023003– Published 6 April, 2026
  • Received 22 August 2025
  • Accepted 24 February 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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