- Physical Review Journals
All JournalsPhysics Magazine

Physical Review E

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

Coarsening dynamics of chemotactic aggregates

Henrik Weyer1,*, David Muramatsu1,*, and Erwin Frey1,2,†

  • *These authors contributed equally to this work.
  • †Contact author: frey@lmu.de
PDF

Phys. Rev. E 112, 054406 – Published 10 November, 2025

DOI: https://doi.org/10.1103/c9px-mdhs

Abstract

Autochemotaxis, the directed movement of cells along gradients in chemicals they secrete, is central to the formation of complex spatiotemporal patterns in biological systems. Since the introduction of the Keller-Segel model, numerous variants have been analyzed, revealing phenomena such as coarsening of aggregates, stable aggregate sizes, and spatiotemporally chaotic dynamics. Here we consider general mass-conserving Keller-Segel models, that is, models without cell growth and death, and analyze the generic long-time dynamics of the chemotactic aggregates. Building on and extending our previous work, which demonstrated that chemotactic aggregation can be understood through a generalized Maxwell construction balancing density fluxes and reactive turnover, we use singular perturbation theory to derive the rates of mass competition between well-separated aggregates. We analyze how this mass-competition process drives coarsening in both diffusion- and reaction-limited regimes, with the diffusion-limited rate aligning with our previous quasi-steady-state analyses. Our results generalize earlier mathematical findings, demonstrating that coarsening is driven by self-amplifying mass transport and aggregate coalescence. Additionally, we provide a linear stability analysis of the lateral instability, predicting it through a nullcline-slope criterion that parallels the curvature criterion in spinodal decomposition. Overall, our findings suggest that chemotactic aggregates behave similarly to phase-separating droplets, providing a robust framework for understanding the coarse-grained dynamics of autochemotactic cell populations and a quantitative basis for comparing chemotactic coarsening to canonical nonequilibrium phase separation.

View figure in article

Physics Subject Headings (PhySH)

See Also

Chemotaxis-Induced Phase Separation

Henrik Weyer, David Muramatsu, and Erwin Frey
Phys. Rev. Lett. 135, 208402 (2025)

Article Text

References (64)

  1. Edited by H. C. Berg, E. Coli in Motion, Biological and Medical Physics, Biomedical Engineering (Springer New York, New York, 2004).
  2. Edited by J. D. Murray, Mathematical Biology: II: Spatial Models and Biomedical Applications, Interdisciplinary Applied Mathematics, Vol. 18 (Springer New York, New York, 2003).
  3. E. T. Roussos, J. S. Condeelis, and A. Patsialou, Chemotaxis in cancer, Nat. Rev. Cancer 11, 573 (2011).
  4. X. Trepat, Z. Chen, and K. Jacobson, Cell migration, in Comprehensive Physiology, 1st ed., edited by Y. S. Prakash (Wiley, New York, 2012), pp. 2369–2392.
  5. C. Gerard and B. J. Rollins, Chemokines and disease, Nat. Immunol. 2, 108 (2001).
  6. S. J. Ebbens and J. R. Howse, In pursuit of propulsion at the nanoscale, Soft Matter 6, 726 (2010).
  7. B. Ramm, A. Goychuk, A. Khmelinskaia, P. Blumhardt, H. Eto, K. A. Ganzinger, E. Frey, and P. Schwille, A diffusiophoretic mechanism for ATP-driven transport without motor proteins, Nat. Phys. 17, 850 (2021).
  8. B. Liebchen and H. Löwen, Synthetic chemotaxis and collective behavior in active matter, Acc. Chem. Res. 51, 2982 (2018).
  9. H. Stark, Artificial chemotaxis of self-phoretic active colloids: Collective behavior, Acc. Chem. Res. 51, 2681 (2018).
  10. R. Golestanian, Phoretic active matter, in Active Matter and Nonequilibrium Statistical Physics, 1st ed., edited by J. Tailleur, G. Gompper, M. C. Marchetti, J. M. Yeomans, and C. Salomon (Oxford University Press, Oxford, 2022), pp. 230–293.
  11. M. W. Cotton, R. Golestanian, and J. Agudo-Canalejo, Catalysis-induced phase separation and autoregulation of enzymatic activity, Phys. Rev. Lett. 129, 158101 (2022).
  12. L. Demarchi, A. Goychuk, I. Maryshev, and E. Frey, Enzyme-enriched condensates show self-propulsion, positioning, and coexistence, Phys. Rev. Lett. 130, 128401 (2023).
  13. A. Goychuk, L. Demarchi, I. Maryshev, and E. Frey, Self-consistent sharp interface theory of active condensate dynamics, Phys. Rev. Res. 6, 033082 (2024).
  14. E. O. Budrene and H. C. Berg, Dynamics of formation of symmetrical patterns by chemotactic bacteria, Nature (Lond.) 376, 49 (1995).
  15. J. Cremer, T. Honda, Y. Tang, J. Wong-Ng, M. Vergassola, and T. Hwa, Chemotaxis as a navigation strategy to boost range expansion, Nature (London) 575, 658 (2019).
  16. E. F. Keller and L. A. Segel, Initiation of slime mold aggregation viewed as an instability, J. Theor. Biol. 26, 399 (1970).
  17. T. Hillen and K. J. Painter, A user's guide to PDE models for chemotaxis, J. Math. Biol. 58, 183 (2009).
  18. K. J. Painter and T. Hillen, Spatio-temporal chaos in a chemotaxis model, Physica D 240, 363 (2011).
  19. A. Ziepke, I. Maryshev, I. S. Aranson, and E. Frey, Multi-scale organization in communicating active matter, Nat. Commun. 13, 6727 (2022).
  20. G. Wolansky, Multi-components chemotactic system in the absence of conflicts, Eur. J. Appl. Math. 13, 641 (2002).
  21. W. Liu, J. Cremer, D. Li, T. Hwa, and C. Liu, An evolutionarily stable strategy to colonize spatially extended habitats, Nature (Lond.) 575, 664 (2019).
  22. D. Muramatsu, H. Weyer, F. M. Gartner, and E. Frey, Basic interactions responsible for thymus function explain convoluted medulla shape, PNAS 122, e2415288122 (2025).
  23. M. C. Cross and P. C. Hohenberg, Pattern formation outside of equilibrium, Rev. Mod. Phys. 65, 851 (1993).
  24. J. W. Cahn and J. E. Hilliard, Free energy of a nonuniform system. I. Interfacial free energy, J. Chem. Phys. 28, 258 (1958).
  25. L. Rapp and W. Zimmermann, Universal aspects of collective behavior in chemotactic systems, Phys. Rev. E 100, 032609 (2019).
  26. M. J. Ward, Asymptotic methods for reaction-diffusion systems: Past and present, Bull. Math. Biol. 68, 1151 (2006).
  27. J. Wei and M. Winter, Mathematical Aspects of Pattern Formation in Biological Systems, Applied Mathematical Sciences, Vol. 189 (Springer London, London, 2014).
  28. Y. Dolak and C. Schmeiser, The Keller–Segel model with logistic sensitivity function and small diffusivity, SIAM J. Appl. Math. 66, 286 (2005).
  29. A. B. Potapov and T. Hillen, Metastability in chemotaxis models, J. Dynam. Differ. Eq. 17, 293 (2005).
  30. K. Kang, T. Kolokolnikov, and M. J. Ward, The stability and dynamics of a spike in the 1D Keller–Segel model, IMA J. Appl. Math. 72, 140 (2007).
  31. T. Kolokolnikov, J. Wei, and A. Alcolado, Basic mechanisms driving complex spike dynamics in a chemotaxis model with logistic growth, SIAM J. Appl. Math. 74, 1375 (2014).
  32. F. Kong, M. J. Ward, and J. Wei, Existence, stability and slow dynamics of spikes in a 1D minimal Keller–Segel model with logistic growth, J. Nonlinear Sci. 34, 51 (2024).
  33. E. Kavanagh, Interface motion in the ostwald ripening and chemotaxis systems, Master's thesis, University of British Columbia, 2014.
  34. M. Mimura and T. Tsujikawa, Aggregating pattern dynamics in a chemotaxis model including growth, Physica A 230, 499 (1996).
  35. M. Meyer, L. Schimansky-Geier, and P. Romanczuk, Active Brownian agents with concentration-dependent chemotactic sensitivity, Phys. Rev. E 89, 022711 (2014).
  36. J. O'Byrne and J. Tailleur, Lamellar to micellar phases and beyond: When tactic active systems admit free energy functionals, Phys. Rev. Lett. 125, 208003 (2020).
  37. A. Dinelli, J. O'Byrne, and J. Tailleur, Fluctuating hydrodynamics of active particles interacting via taxis and quorum sensing: Static and dynamics, J. Phys. A: Math. Theor. 57, 395002 (2024).
  38. D. Horstmann, Lyapunov functions and Lp-estimates for a class of reaction-diffusion systems, Colloq. Math. 87, 113 (2001).
  39. H. Weyer, D. Muramatsu, and E. Frey, companion paper, Chemotaxis-induced phase separation, Phys. Rev. Lett. 135, 208402 (2025).
  40. H. Weyer, F. Brauns, and E. Frey, Coarsening and wavelength selection far from equilibrium: A unifying framework based on singular perturbation theory, Phys. Rev. E 108, 064202 (2023).
  41. F. Brauns, J. Halatek, and E. Frey, Phase-space geometry of mass-conserving reaction-diffusion dynamics, Phys. Rev. X 10, 041036 (2020).
  42. E. F. Keller and L. A. Segel, Model for chemotaxis, J. Theor. Biol. 30, 225 (1971).
  43. R. Schaaf, Stationary solutions of chemotaxis systems, Trans. Am. Math. Soc. 292, 531 (1985).
  44. F. Brauns, J. Halatek, and E. Frey, Diffusive coupling of two well-mixed compartments elucidates elementary principles of protein-based pattern formation, Phys. Rev. Res. 3, 013258 (2021).
  45. H. Weyer, T. A. Roth, and E. Frey, Deciphering the interface laws of Turing mixtures and foams, arXiv:2409.20070 [Nat. Phys. (to be published)].
  46. S. Childress and J. Percus, Nonlinear aspects of chemotaxis, Math. Biosci. 56, 217 (1981).
  47. M. Otsuji, S. Ishihara, C. Co, K. Kaibuchi, A. Mochizuki, and S. Kuroda, A mass conserved reaction–diffusion system captures properties of cell polarity, PLoS Comput. Biol. 3, e108 (2007).
  48. J. Halatek and E. Frey, Rethinking pattern formation in reaction–diffusion systems, Nat. Phys. 14, 507 (2018).
  49. P. C. Hohenberg and B. I. Halperin, Theory of dynamic critical phenomena, Rev. Mod. Phys. 49, 435 (1977).
  50. A. J. Bray, Theory of phase-ordering kinetics, Adv. Phys. 51, 481 (2002).
  51. T. Hillen and K. Painter, Global existence for a parabolic chemotaxis model with prevention of overcrowding, Adv. Appl. Math. 26, 280 (2001).
  52. K. Painter and T. Hillen, Volume-filling and quorum-sensing in models for chemosensitive movement, Can. Appl. Math. Quart. 10, 501 (2002).
  53. M. Doi, Soft Matter Physics, 1st ed. (Oxford University Press, Oxford, 2013).
  54. L. M. Pismen, Patterns and Interfaces in Dissipative Dynamics, Springer Series in Synergetics (Springer-Verlag, Berlin, 2006).
  55. T. Kolokolnikov, T. Erneux, and J. Wei, Mesa-type patterns in the one-dimensional Brusselator and their stability, Physica D 214, 63 (2006).
  56. I. M. Lifshitz and V. V. Slyozov, The kinetics of precipitation from supersaturated solid solutions, J. Phys. Chem. Solids 19, 35 (1961).
  57. C. Wagner, Theorie der Alterung von Niederschlägen durch Umlösen (Ostwald-Reifung), Zeitschr. Elektrochem. 65, 581 (1961).
  58. J. Langer, Theory of spinodal decomposition in alloys, Ann. Phys. 65, 53 (1971).
  59. K. B. Glasner and T. P. Witelski, Coarsening dynamics of dewetting films, Phys. Rev. E 67, 016302 (2003).
  60. J. Rubinstein and P. Sternberg, Nonlocal reaction—Diffusion equations and nucleation, IMA J. Appl. Math. 48, 249 (1992).
  61. A. I. Curatolo, N. Zhou, Y. Zhao, C. Liu, A. Daerr, J. Tailleur, and J. Huang, Cooperative pattern formation in multi-component bacterial systems through reciprocal motility regulation, Nat. Phys. 16, 1152 (2020).
  62. M. E. Cates and J. Tailleur, Motility-induced phase separation, Annu. Rev. Condens. Matter Phys. 6, 219 (2015).
  63. J. Tailleur and M. E. Cates, Statistical mechanics of interacting run-and-tumble bacteria, Phys. Rev. Lett. 100, 218103 (2008).
  64. X. Fu, L.-H. Tang, C. Liu, J.-D. Huang, T. Hwa, and P. Lenz, Stripe formation in bacterial systems with density-suppressed motility, Phys. Rev. Lett. 108, 198102 (2012).

Outline

Information

Phys. Rev. E 112, 054406– Published 10 November, 2025
  • Received 9 September 2024
  • Revised 26 May 2025
  • Accepted 10 July 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