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Computational model of fractal interface formation in bacterial biofilms

Caelan Brooks1,2,3,4,*, Meiyi Yao5,*, Jake T. McCool2,3,6, Alan Gillman6,†, Gürol M. Süel6, Andrew Mugler1, and Joseph W. Larkin2,3,7,‡

  • *These authors contributed equally to this work.
  • †Present Address: Evident Scientific, Waltham, Massachusetts, USA.
  • ‡Contact author: jwlarkin@bu.edu
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Phys. Rev. E 112, 064408 – Published 8 December, 2025

DOI: https://doi.org/10.1103/2zm9-r3qs

Abstract

Bacteria benefit from cellular heterogeneity: cells differentiate into diverse gene expression states. As colonies grow, cellular phenotypes arrange into spatial patterns. To uncover the functional role of these emergent patterns, we must understand how they arise from cellular growth and mechanical interactions. Here we present a simple, agent-based model to predict patterns of motile and extracellular matrix-producing cells in biofilms of Bacillus subtilis. By incorporating phenotypic inheritance, mechanical interactions, and peripheral motile cell dispersal, our model predicts the emergence of a pattern: matrix cells surround a fractal-like interior motile population. We find that, while some properties of the motile-matrix interface depend on initial conditions, the motile distribution at large radii depends solely on the model's growth mechanism. The phenotypic interface exhibits a fractal dimension that increases as biofilms grow but reaches a maximum as the peripheral layer of matrix cells exceeds the capacity of the inner cells to push it out of the way. By varying parameters, we find correlations between the interface fractal dimension and expansion of motile cells. We validate findings using experiments on B. subtilis biofilms in microfluidics. Our model demonstrates the emergence of colony-level phenotypes from single cell-level interactions and cells modifying their own environment.

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References (43)

  1. J.-W. Veening, W. K. Smits, and O. P. Kuipers, Bistability, epigenetics, and bet-hedging in bacteria, Annu. Rev. Microbiol. 62, 193 (2008).
  2. N. Nikolic, F. Schreiber, A. Dal Co, D. J. Kiviet, T. Bergmiller, S. Littmann, Marcel M. M. Kuypers, and M. Ackermann, Cell-to-cell variation and specialization in sugar metabolism in clonal bacterial populations, PLoS Genet. 13, e1007122 (2017).
  3. H. Maamar, A. Raj, and D. Dubnau, Noise in gene expression determines cell fate in Bacillus subtilis, Science 317, 526 (2007).
  4. D. Lopez, H. Vlamakis, and R. Kolter, Generation of multiple cell types in Bacillus subtilis, FEMS Microbiol. Rev. 33, 152 (2009).
  5. H. Vlamakis, Y. Chai, P. Beauregard, R. Losick, and R. Kolter, Sticking together: Building a biofilm the Bacillus subtilis way, Nat. Rev. Microbiol. 11, 157 (2013).
  6. D. B. Kearns and R. Losick, Cell population heterogeneity during growth of Bacillus subtilis, Genes Dev. 19, 3083 (2005).
  7. C. R. Armbruster, C. K. Lee, J. Parker-Gilham, J. de Anda, A. Xia, K. Zhao, K. Murakami, B. S. Tseng, L. R. Hoffman, F. Jin, C. S. Harwood, G. C. Wong, and M. R. Parsek, Heterogeneity in surface sensing suggests a division of labor in Pseudomonas Aeruginosa populations, eLife 8, e45084 (2021).
  8. N. Steinberg, A. Keren-Paz, Q. Hou, S. Doron, K. Yanuka-Golub, T. Olender, R. Hadar, G. Rosenberg, R. Jain, J. Cámara-Almirón, D. Romero, S. van Teeffelen, and I. Kolodkin-Gal, The extracellular matrix protein tasa is a developmental cue that maintains a motile subpopulation within Bacillus subtilis biofilms, Sci. Signal. 13, eaaw8905 (2020).
  9. P. S. Stewart and M. J. Franklin, Physiological heterogeneity in biofilms, Nat. Rev. Microbiol. 6, 199 (2008).
  10. H. Vlamakis, C. Aguilar, R. Losick, and R. Kolter, Control of cell fate by the formation of an architecturally complex bacterial community, Genes Dev. 22, 945 (2008).
  11. G. O'Toole, H. B. Kaplan, and R. Kolter, Biofilm formation as microbial development, Annu. Rev. Microbiol. 54, 49 (2000).
  12. P. Stoodley, K. Sauer, D. G. Davies, and J. W. Costerton, Biofilms as complex differentiated communities, Annu. Rev. Microbiol. 56, 187 (2002).
  13. S. M. Rosenberg, Life, death, differentiation, and the multicellularity of bacteria, PLoS Genet. 5, e1000418 (2009).
  14. H.-C. Flemming and J. Wingender, The biofilm matrix, Nat. Rev. Microbiol. 8, 623 (2010).
  15. D. B. Kearns, F. Chu, S. S. Branda, R. Kolter, and R. Losick, A master regulator for biofilm formation by Bacillus subtilis, Mol. Microbiol. 55, 739 (2005).
  16. K. P. Lemon, A. M. Earl, H. C. Vlamakis, C. Aguilar, and R. Kolter, Biofilm development with an emphasis on Bacillus subtilis, in Bacterial Biofilms, Current Topics in Microbiology and Immunology, Vol. 322, edited by T. Romeo (Springer Berlin Heidelberg, Berlin, Heidelberg, 2008), pp. 1–16.
  17. A. D. Grossman, Genetic networks controlling the initiation of sporulation and the development of genetic competence in Bacillus subtilis, Annu. Rev. Genet. 29, 477 (1995).
  18. S. Arnaouteli, N. C. Bamford, N. R. Stanley-Wall, and Á. T. Kovács, Bacillus subtilis biofilm formation and social interactions, Nat. Rev. Microbiol. 19, 600 (2021).
  19. S. Srinivasan, I. D. Vladescu, S. A. Koehler, X. Wang, M. Mani, and S. M. Rubinstein, Matrix production and sporulation in Bacillus subtilis biofilms localize to propagating wave fronts, Biophys. J. 114, 1490 (2018).
  20. Y. Chai, F. Chu, R. Kolter, and R. Losick, Bistability and biofilm formation in Bacillus subtilis, Mol. Microbiol. 67, 254 (2008).
  21. T. M. Norman, N. D. Lord, J. Paulsson, and R. Losick, Memory and modularity in cell-fate decision making, Nature (London) 503, 481 (2013).
  22. M. Asally, M. Kittisopikul, P. Rué, Y. Du, Z. Hu, T. Çağatay, A. B. Robinson, H. Lu, J. Garcia-Ojalvo, and G. M. Süel, Localized cell death focuses mechanical forces during 3d patterning in a biofilm, Proc. Natl. Acad. Sci. USA 109, 18891 (2012).
  23. Y. Chai, T. Norman, R. Kolter, and R. Losick, An epigenetic switch governing daughter cell separation in Bacillus subtilis, Genes Dev. 24, 754 (2010).
  24. A. Persat, C. D. Nadell, M. K. Kim, F. Ingremeau, A. Siryaporn, K. Drescher, N. S. Wingreen, B. L. Bassler, Z. Gitai, and H. A. Stone, The mechanical world of bacteria, Cell 161, 988 (2015).
  25. D. Volfson, S. Cookson, J. Hasty, and L. S. Tsimring, Biomechanical ordering of dense cell populations, Proc. Natl. Acad. Sci. USA 105, 15346 (2008).
  26. W. P. J. Smith, Y. Davit, J. M. Osborne, W. Kim, K. R. Foster, and J. M. Pitt-Francis, Cell morphology drives spatial patterning in microbial communities, Proc. Natl. Acad. Sci. USA 114, E280 (2017).
  27. T. J. Rudge, F. Federici, P. J. Steiner, A. Kan, and J. Haseloff, Cell polarity-driven instability generates self-organized, fractal patterning of cell layers, ACS Synth. Biol. 2, 705 (2013).
  28. J. J. Winkle, O. A. Igoshin, M. R. Bennett, K. Josić, and W. Ott, Modeling mechanical interactions in growing populations of rod-shaped bacteria, Phys. Biol. 14, 055001 (2017).
  29. A. V. Narla, D. B. Borenstein, and N. S. Wingreen, A biophysical limit for quorum sensing in biofilms, Proc. Natl. Acad. Sci. USA 118, e2022818118 (2021).
  30. A. Martínez-Calvo, C. Trenado-Yuste, H. Lee, J. Gore, N. S. Wingreen, and S. S. Datta, Interfacial morphodynamics of proliferating microbial communities, Phys. Rev. X 15, 011016 (2025).
  31. C. J. Comerci, A. L. Gillman, L. Galera-Laporta, E. Gutierrez, A. Groisman, J. W. Larkin, J. Garcia-Ojalvo, and G. M. Süel, Localized electrical stimulation triggers cell-type-specific proliferation in biofilms, Cell Syst. 13, 488 (2022).
  32. C. Picioreanu, J.-U. Kreft, and M. C. M. van Loosdrecht, Particle-based multidimensional multispecies biofilm model, Appl. Environ. Microbiol. 70, 3024 (2004).
  33. K. Drescher, J. Dunkel, C. D. Nadell, S. van Teeffelen, I. Grnja, N. S. Wingreen, H. A. Stone, and B. L. Bassler, Architectural transitions in Vibrio cholerae biofilms at single-cell resolution, Proc. Natl. Acad. Sci. USA 113, E2066 (2016).
  34. B. J. Cooley, T. W. Thatcher, S. M. Hashmi, G. L'her, H. H. Le, D. A. Hurwitz, D. Provenzano, A. Touhami, and V. D. Gordon, The extracellular polysaccharide Pel makes the attachment of P. Aeruginosa to surfaces symmetric and short-ranged, Soft Matter 9, 3871 (2013).
  35. S. S. Branda, F. Chu, D. B. Kearns, R. Losick, and R. Kolter, A major protein component of the Bacillus subtilis biofilm matrix, Mol. Microbiol. 59, 1229 (2006).
  36. P. Stoodley, S. Wilson, Luanne Hall-Stoodley, J. D. Boyle, H. M. Lappin-Scott, and J. W. Costerton, Growth and detachment of cell clusters from mature mixed-species biofilms, Appl. Environ. Microbiol. 67, 5608 (2001).
  37. See Supplemental Material at http://link.aps.org/supplemental/10.1103/2zm9-r3qs for collapse radii at varied shoving capacity, replicates of radial motile distributions, decrease in fractal dimension as islands break off the motile interior, effect of preferred growth direction, and average distance between cells of different phenotypes.
  38. T. Vicsek, M. Cserző, and V. K. Horváth, Self-affine growth of bacterial colonies, Phys. A (Amsterdam, Neth.) 167, 315 (1990).
  39. H. Koorehdavoudi, G. W. Paul Bogdan, R. Marculescu, J. Zhuang, R. W. Carlsen, and M. Sitti, Multi-fractal characterization of bacterial swimming dynamics: A case study on real and simulated Serratia marcescens, Proc. R. Soc. A 473, 20170154 (2017).
  40. J. J. Gagnepain and C. Roques-Carmes, Fractal approach to two-dimensional and three-dimensional surface roughness, Wear 109, 119 (1986).
  41. Github repository (2025), https://github.com/caelan-brooks/frac-interface-model.
  42. J. Schindelin, I. Arganda-Carreras, E. Frise, V. Kaynig, M. Longair, T. Pietzsch, S. Preibisch, C. Rueden, S. Saalfeld, B. Schmid, J.-Y. Tinevez, D. J. White, V. Hartenstein, K. Eliceiri, P. Tomancak, and A. Cardona, Fiji: An open-source platform for biological-image analysis, Nat. Methods 9, 676 (2012).
  43. W. Zhang, A. Seminara, M. Suaris, M. P. Brenner, D. A. Weitz, and T. E. Angelini, Nutrient depletion in Bacillus subtilis biofilms triggers matrix production, New J. Phys. 16, 015028 (2014).

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Phys. Rev. E 112, 064408– Published 8 December, 2025
  • Received 16 December 2024
  • Accepted 9 September 2025
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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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