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Emerging morphologies in round bacterial colonies: comparing volumetric versus chemotactic expansion

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Abstract

Biological experiments performed on living bacterial colonies have demonstrated the microbial capability to develop finger-like shapes and highly irregular contours, even starting from an homogeneous inoculum. In this work, we study from the continuum mechanics viewpoint the emergence of such branched morphologies in an initially circular colony expanding on the top of a Petri dish coated with agar. The bacterial colony expansion, based on either a source term, representing volumetric mitotic processes, or a nonconvective mass flux, describing chemotactic expansion, is modeled at the continuum scale. We demonstrate that the front of the colony is always linearly unstable, having similar dispersion curves to the ones characterizing branching instabilities. We also perform finite element simulations, which not only prove the emergence of branching, but also highlight dramatic differences between the two mechanisms of colony expansion in the nonlinear regime. Furthermore, the proposed combination of analytical and numerical analysis allowed studying the influence of different model parameters on the selection of specific patterns. A very good agreement has been found between the resulting simulations and the typical structures observed in biological assays. Finally, this work provides a new interpretation of the emergence of branched patterns in living aggregates, depicted as the results of a complex interplay among chemical, mechanical and size effects.

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References

  • Adler J (1966) Chemotaxis in bacteria. Science 153:708–716

    Article  Google Scholar 

  • Ambrosi D, Athesian GA, Arruda EM, Cowin SC, Dumais J, Goriely A, Holzapfel GA, Humphrey JD, Kemkemer R, Kuhl E, Olberding JE, Taber LA, Garikipati K (2011) Perspectives on growth and remodeling. J Mech Phys Solids 59(4):863–883

    Article  MathSciNet  MATH  Google Scholar 

  • Bassler BL, Losick R (2006) Bacterially speaking. Cell 125:237–246

    Article  Google Scholar 

  • Be’er A, Zhang HP, Florin EL, Payne SM, Ben-Jacob E, Swinney HL (2009) Deadly competition between sibling bacterial colonies. PNAS 106:428–433

    Article  Google Scholar 

  • Ben Amar M (2013) Chemotaxis migration and morphogenesis of living colonies. Eur Phys J E Soft Matter 36(6). doi:10.1140/epje/i2013-13064-5

  • Ben Amar M, Chatelain C, Ciarletta P (2011) Contour instabilities in early tumor growth models. Phys Rev Lett 106:148101

    Article  Google Scholar 

  • Ben-Jacob E (1993) From snowflakes formation to growth of bacterial colonies. Part I: Diffusive patterning in azoic systems. Contemp Phys 34:247–273

    Article  Google Scholar 

  • Ben-Jacob E (1997) From snowflakes formation to growth of bacterial colonies II: Cooperative formation of complex colonial patterns. Contemp Phys 38(3):205–241

    Article  Google Scholar 

  • Ben-Jacob E, Cohen I, Gutnick DL (1998) Cooperative organization of bacterial colonies: from genotype to morphotype. Annu Rev Microbiol 52:779–806

    Article  Google Scholar 

  • Ben-Jacob E, Cohen I, Levine H (2000) Cooperative self-organization of microorganisms. Adv Phys 49:395–554

    Article  Google Scholar 

  • Ben-Jacob E, Levine H (2006) Self-engineering capabilities of bacteria. J R Soc Interface 3:197–214

    Article  Google Scholar 

  • Ben-Jacob E, Schultz D (2010) Bacteria determine fate by playing dice with controlled odds. PNAS 107:13197–13198

    Article  Google Scholar 

  • Bonachela JA, Nadell CD, Xavier JB, Levin SA (2011) Universality in bacterial colonies. J Stat Phys 144(2):303–315

    Article  MATH  Google Scholar 

  • Cerretti F, Perthame B, Schmeiser C, Tang M, Vauchelet N (2011) Waves for a hyperbolic Keller-Segel model and branching instabilities. Math Models Methods Appl Sci 21:825–842

    Article  MathSciNet  MATH  Google Scholar 

  • Ciarletta P, Ambrosi D, Maugin GA (2012) Mass transport in morphogenetic processes: a second gradient theory for volumetric growth and material remodeling. J Mech Phys Solids 60(3):432–450

    Article  MathSciNet  MATH  Google Scholar 

  • Ciarletta P (2012) Free boundary morphogenesis in living matter. Eur Biophys J 41:681–686

    Article  Google Scholar 

  • Cross MC, Hohenberg PC (1993) Pattern formation outside of equilibrium. Rev Mod Phys 65(3):851–1112

    Article  Google Scholar 

  • Croze OA, Ferguson GP, Cates ME, Poon WC (2011) Migration of chemotactic bacteria in soft agar: role of gel concentration. Biophys J 101(3):525–534

    Article  Google Scholar 

  • Dockery J, Klapper I (2001) Finger formation in biofilm layers. SIAM J Appl Math 62(3):853–869

    MathSciNet  MATH  Google Scholar 

  • Eden M (1961) A two-dimensional growth process. Fourth Berkeley Symp. Math Stat Prob 4:223–239

    MathSciNet  MATH  Google Scholar 

  • Farrell FDC, Hallatschek O, Marenduzzo D, Waclaw B (2013) Mechanically driven growth of quasi-two dimensional microbial colonies. Phys Rev Lett 111:168101

    Article  Google Scholar 

  • Flemming HC, Wingender J (2010) The biofilm matrix. Nat Rev 8:623–633

    Google Scholar 

  • Ford RM, Lauffenburger DA (1991) Analysis of chemotactic bacterial distributions in population migration assays using a mathematical model applicable to steep or shallow attractant gradients. Bull Math Bio 53:721–749

    Article  MATH  Google Scholar 

  • Friedl P, Hegerfeldt Y, Tusch M (2004) Collective cell migration in morphogenesis and cancer. Int J Dev Biol 48:441–449

    Article  Google Scholar 

  • Fujikawa H, Matsushita M (1989) Fractal growth of Bacillus subtilis on agar plates. J Phys Soc Jpn 58(11):3875–3878

    Article  Google Scholar 

  • Fujikawa H (1994) Diversity of the growth patterns of Bacillus subtilis colonies on agar plates. FEMS Microbiol Ecol 13:159–168

    Article  Google Scholar 

  • Ganghoffer JF (2010) Mechanical modeling of growth considering domain variation—Part II: Volumetric and surface growth involving Eshelby tensors. J Mech Phys Solids 58:1434–1459

    Article  MATH  Google Scholar 

  • Golding I, Kozlovsky Y, Cohen I, Ben-Jacob E (1998) Studies of bacterial branching growth using reaction-diffusion models for colonial development. Phys A 260:510–554

    Article  Google Scholar 

  • Guyon E, Hulin JP, Petit L, De Gennes PG (2001) Hydrodynamique physique. EDP Sciences, Les Ulis

    Google Scholar 

  • Hamill OP, Martinac B (2001) Molecular basis of mechanotransduction in living cells. Phys Rev 81(2):685–740

    Google Scholar 

  • Humphrey JD (2003) Continuum biomechanics of soft biological tissues. Proc R Soc Lond. A 459:3–46

    Article  MathSciNet  MATH  Google Scholar 

  • Kawasaki K, Mochizuki A, Matsushita M, Umeda T, Shigesada N (1997) Modeling spatio-temporal patterns generated by Bacillus subtilis. J Theor Biol 188:177185

    Article  Google Scholar 

  • Keller EF, Segel LA (1971) Model for chemotaxis. J Theor Biol 30:225–234

    Article  MATH  Google Scholar 

  • Kozlovsky Y, Cohen I, Golding I, Ben-Jacob E (1999) Lubricating bacteria model for branching growth of bacterial colony. Phys Rev E Phys Plasmas Fluids Relat Interdiscip Top 50:7025–7035

    Google Scholar 

  • Langer JS (1980) Instabilities and pattern formation in crystal growth. Rev Mod Phys 52(1):1–30

    Article  Google Scholar 

  • Lecuit T, Lenne PF (2007) Cell surface mechanics and the control of cell shape, tissue patterns and morphogenesis. Nat Rev Mol Cell Biol 8:633–644

    Article  Google Scholar 

  • Manz BN, Groves JT (2010) Spatial organization and signal transduction at intercellular junctions. Nat Rev Mol Cell Biol 11:342–352

    Article  Google Scholar 

  • Mark S, Shlomovitz R, Gov NS, Poujade M, Grasland-Mongrain E, Silberzan P (2010) Physical model of the dynamic instability in an expanding cell culture. Biophys J 98:361–370

    Article  Google Scholar 

  • Marrocco A, Henry H, Holland IB, Plapp M, Séror SJ, Perthame B (2010) Models of self-organizing bacterial communities and comparisons with experimental observations. Math Model Nat Phenom 5(1):148–162

    Article  MathSciNet  MATH  Google Scholar 

  • Matsushita M, Wakita J, Itoh H, Ràfols I, Matsuyama T, Sakaguchi H, Mimura M (1998) Interface growth and pattern formation in bacterial colonies. Phys A 249:517–524

    Article  Google Scholar 

  • Matsushita M, Hiramatsu F, Kobayashi N, Ozawa T, Yamazaki Y, Matsuyama T (2004) Colony formation in bacteria: experiments and modeling. Biofilms 1:305–317

    Article  Google Scholar 

  • Mimura M, Sakaguchi H, Matsushita M (2000) Reaction-diffusion modelling of bacterial colony patterns. Phys A 282:283–303

    Article  Google Scholar 

  • Nikolić DL, Boettiger AN, Bar-Sagi D, Carbeck JD, Shvartsman SY (2006) Role of boundary conditions in an experimental model of epithelial wound healing. Am J Physiol Cell Physiol 291:68–75

    Article  Google Scholar 

  • Nobes CD, Hall A (1999) Rho GTPases control polarity, protrusion, and adhesion during cell movement. J Cell Biol 144:1235–1244

    Article  Google Scholar 

  • O’May C, Tufenkji N (2011) The swarming motility of Pseudomonas aeruginosa is blocked by cranberry proanthocyanidins and other tannin-containing materials. Appl Environ Microbiol 77:3061–3067

    Article  Google Scholar 

  • Parent CA, Devreotes PN (1999) A cells sense of direction. Science 284:765–770

    Article  Google Scholar 

  • Paterson L (1981) Radial fingering in a Hele-Shaw cell. J Fluid Mech 113:513–529

    Article  Google Scholar 

  • Poujade M, Grasland-Mongrain E, Hertzog A, Jouanneau J, Chavrier P, Ladoux B, Buguin A, Silberzan P (2007) Collective migration of an epithelial monolayer in response to a model wound. PNAS 104(41):15988–15993

    Article  Google Scholar 

  • Saffman PG, Taylor G (1958) The penetration of a fluid into a medium or hele-shaw cell containing a more viscous liquid. Proc Soc Lond Ser A 245:312–329

    Article  MathSciNet  MATH  Google Scholar 

  • Seminara A et al (2012) Osmotic spreading of Bacillus subtilis biofilms driven by an extracellular matrix. PNAS 109:1116–1121

    Article  Google Scholar 

  • Shapiro JA (1988) Bacteria as multicellular organisms. Sci Am 258:82–89

    Article  Google Scholar 

  • Taber LA (1995) Biomechanics of growth, remodeling, and morphogenesis. Appl Mech Rev 48:487–545

    Article  Google Scholar 

  • Tindall MJ, Maini PK, Porter SL, Armitage JP (2008) Overview of mathematical approaches used to model bacterial chemotaxis II: bacterial populations. Bull Math Biol 70:1570–1607

    Article  MathSciNet  MATH  Google Scholar 

  • Volfson D, Cookson S, Hasty J, Tsimring LS (2008) Biomechanical ordering of dense cell populations. PNAS 105(40):15346–15351

    Article  Google Scholar 

  • Wakita J, Komatsu K, Nakahara A, Matsuyama T, Matsushita M (1994) Experimental investigation on the validity of population dynamics approach to bacterial colony formation. J Phys Soc Jpn 63:1205–1211

    Article  Google Scholar 

  • Wakita J, Itoh H, Matsuyama T, Matsushita M (1997) Self-affinity for the growing interface of bacterial colonies. J Phys Soc Jpn 66(1):67–72

    Article  Google Scholar 

  • Yu SR, Burkhardt M, Nowak M, Ries J, Petrášek Z, Scholpp S, Schwille P, Brand M (2009) Fgf8 morphogen gradient forms by a source-sink mechanism with freely diffusing molecules. Nature 461(7263):533–537

    Article  Google Scholar 

  • Zhou S, Lo WC, Suhalim JL, Digman MA, Gratton E, Nie Q, Lander AD (2012) Free extracellular diffusion creates the Dpp morphogen gradient of the Drosophila wing disc. Curr Biol 22:668–675

    Article  Google Scholar 

  • Ziebert F, Aranson IS (2013) Effects of adhesion dynamics and substrate compliance on the shape and motility of crawling cells. PLoS One 8:e64511

    Article  Google Scholar 

Download references

Acknowledgments

We are grateful to Davide Ambrosi for helpful discussions.

Author information

Authors and Affiliations

  1. MOX - Politecnico di Milano and Fondazione CEN, Piazza Leonardo da Vinci 32, 20133, Milan, Italy

    Chiara Giverso

  2. MOX - Politecnico di Milano, Piazza Leonardo da Vinci 32, 20133, Milan, Italy

    Marco Verani

  3. CNRS and Institut Jean le Rond d’Alembert, UMR 7190, Université Paris 6, 4 Place Jussieu, Case 162, 75005, Paris, France

    Pasquale Ciarletta

Authors
  1. Chiara Giverso
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  2. Marco Verani
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  3. Pasquale Ciarletta
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Corresponding author

Correspondence to Pasquale Ciarletta.

Additional information

This work was partially supported by the “Start-up Packages and Ph.D. Program” project, co-funded by Regione Lombardia through the “Fondo per lo sviluppo e la coesione 2007–2013—formerly FAS” and by the “Progetto Giovani GNFM 2014”, funded by the National Group of Mathematical Physics (GNFM-INdAM).

Appendix

Appendix

In Sect. 3.2, the linear stability analysis applied to the quasi-stationary problem leads to the definition of the dispersion equation in the compact form (26), as a function of the unperturbed and perturbed pressure fields. Here, we report some details on how Eqs. (25) and (26) have been obtained and the specific expressions for the perturbed pressure and the dispersion equations in the chemotactic growth model (Table 3) and in the bulk growth model (Table 4). The boundary conditions (25) for the perturbed pressure at the interface can be easily obtained, provided that (6) should hold, therefore

$$\begin{aligned} p(R^*+ \varepsilon e^{\lambda t} \cos (k \theta ))=p_0- \sigma _b C(R^*+ \varepsilon e^{\lambda t} \cos (k \theta )).\nonumber \\ \end{aligned}$$
(27)

Computing the curvature of the perturbed interface and considering on both sides only the first-order terms, the following relation holds

$$\begin{aligned}&p^*(R^*)+ \varepsilon e^{\lambda t} \cos (k \theta ) \left( \dfrac{\partial p^*}{\partial r}(R^*)+ p_1(R^*) \right) \nonumber \\&\quad \approx p_0+ \sigma _b \left( \dfrac{1}{R^*} + \varepsilon e^{\lambda t} \cos (k \theta ) \dfrac{1}{R^*} (k^2-1) \right) . \end{aligned}$$
(28)

The derivation of (25) is then straightforward.

In a similar way, the dispersion equation (26) can be retrieved imposing the boundary condition (7) at the perturbed interface and neglecting the terms of order higher than the first, in the series expansion.

The coefficients \(A,\, A_0,\, A_1\) can be found in Table 2, and they are the same for both models. As it is evident from Tables 3 and 4, the dispersion equations link the time-growth mode \(\lambda \) to the wavenumber k in an implicit way, as a function of the four dimensionless parameters \(\beta _i,\, \sigma ,\, R^*\) and \(R_\mathrm{out}\).

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Giverso, C., Verani, M. & Ciarletta, P. Emerging morphologies in round bacterial colonies: comparing volumetric versus chemotactic expansion. Biomech Model Mechanobiol 15, 643–661 (2016). https://doi.org/10.1007/s10237-015-0714-9

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