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Pattern formation in bacterial colonies with density-dependent diffusion

Published online by Cambridge University Press:  28 January 2018

JULIEN SMITH-ROBERGE,
DAVID IRON and
THEODORE KOLOKOLNIKOV
JULIEN SMITH-ROBERGE
Affiliation:
Dalhousie University, Dept. Math&Stats, PO BOX 15000, Halifax, NS, Canada emails: jsmirob@gmail.com, tkolokol@gmail.com, iron@mathstat.dal.ca
DAVID IRON
Affiliation:
Dalhousie University, Dept. Math&Stats, PO BOX 15000, Halifax, NS, Canada emails: jsmirob@gmail.com, tkolokol@gmail.com, iron@mathstat.dal.ca
THEODORE KOLOKOLNIKOV
Affiliation:
Dalhousie University, Dept. Math&Stats, PO BOX 15000, Halifax, NS, Canada emails: jsmirob@gmail.com, tkolokol@gmail.com, iron@mathstat.dal.ca
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Abstract

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Recent experiments have shown that patterns can emerge in bacterial colonies genetically modified to have a drop in diffusion when population densities (detected via a quorum sensing mechanism) are sufficiently large. We examine one PDE model of this system, and construct its non-constant stationary solutions in the form of an interface for the bacterial density. We derive the equations for the interface motion and demonstrate analytically that such interface solution is stable when the diffusion rate of bacteria is large and the diffusion rate of signalling molecules, Dh, is small. We further demonstrate that increasing Dh induces a Hopf bifurcation, resulting in a loss of stability, which can lead to either complex spatio-temporal dynamics or extinction of pattern altogether. These results are confirmed by numerical simulations.

Keywords

Bacterial aggregationpattern formationstate-dependent diffusion

Information

Type
Papers
Information
European Journal of Applied Mathematics , Volume 30 , Issue 1 , February 2019, pp. 196 - 218
Copyright
Copyright © Cambridge University Press 2018 

References

[1] Ben-Jacob, E., Cohen, I. & Levine, H. (2000) Cooperative self-organization of microorganisms. Adv. Phys. 49 (4), 395–554.Google Scholar
[2] Horstmann, D. (2004) From 1970 until present: The Keller–Segel model in chemotaxis and its consequences. II, Jahresber. Deutsch. Math.-Verein. 106, 51–69.Google Scholar
[3] Sherratt, J. A. (2000) Wavefront propagation in a competition equation with a new motility term modelling contact inhibition between cell populations. In: Proceedings of the Royal Society of London A: Mathematical, Physical and Engineering Sciences, Vol. 456, The Royal Society, pp. 2365–2386.Google Scholar
[4] del Castillo-Negrete, D., Carreras, B. & Lynch, V. (2002) Front propagation and segregation in a reaction–diffusion model with cross-diffusion. Phys. D: Nonlinear Phenom. 168, 45–60.Google Scholar
[5] Ruiz-Baier, R. & Tian, C. (2013) Mathematical analysis and numerical simulation of pattern formation under cross-diffusion. Nonlinear Anal.: Real World Appl. 14 (1), 601–612.Google Scholar
[6] Tian, C., Lin, Z. & Pedersen, M. (2010) Instability induced by cross-diffusion in reaction–diffusion systems. Nonlinear Anal.: Real World Appl. 11 (2), 1036–1045.Google Scholar
[7] Gambino, G., Lombardo, M. C. & Sammartino, M. (2012) Turing instability and traveling fronts for a nonlinear reaction–diffusion system with cross-diffusion. Math. Comp. Simul. 82 (6), 1112–1132.Google Scholar
[8] Müller, J. & van Saarloos, W. (2002) Morphological instability and dynamics of fronts in bacterial growth models with nonlinear diffusion. Phys. Rev. E 65 (6), 061111.Google Scholar
[9] Rosenau, P. (2002) Reaction and concentration dependent diffusion model. Phys. Rev. Lett. 88 (19), 194501.Google Scholar
[10] Gilding, B. & Kersner, R. (2005) A fisher/kpp-type equation with density-dependent diffusion and convection: Travelling-wave solutions. J. Phys. A: Math. Gen. 38 (15), 3367.Google Scholar
[11] Kawasaki, K., Mochizuki, A., Matsushita, M., Umeda, T. & Shigesada, N. (1997) Modeling spatio-temporal patterns generated by Bacillus subtilis. J. Theor. Biol. 188 (2), 177–185.Google Scholar
[12] Miller, M. B., & Bassler, B. L. (2001) Quorum sensing in bacteria. Annu. Rev. Microbiol. 55 (1), 165–199.Google Scholar
[13] Liu, C., Fu, X., Liu, L., Ren, X., Chau, C. K. L., Li, S., Xiang, L., Zeng, H., Chen, G., Tang, L.-H., Lenz, P., Cui, X., Huang, W., Hwa, T. & Huang, J.-D. (2011) Sequential establishment of stripe patterns in an expanding cell population. Science 334 (6053), 238–241.Google Scholar
[14] Farrell, F., Marchetti, M., Marenduzzo, D. & Tailleur, J. (2012) Pattern formation in self-propelled particles with density-dependent motility. Phys. Rev. Lett. 108 (24), 248101.Google Scholar
[15] Tailleur, J. & Cates, M. (2008) Statistical mechanics of interacting run-and-tumble bacteria. Phys. Rev. Lett. 100 (21), 218103.Google Scholar
[16] Cates, M., Marenduzzo, D., Pagonabarraga, I. & Tailleur, J. (2010) Arrested phase separation in reproducing bacteria creates a generic route to pattern formation. Proc. Natl. Acad. Sci. USA 107 (26), 11715–11720.Google Scholar
[17] Fu, X., Tang, L.-H., Liu, C., Huang, J.-D., Hwa, T. & Lenz, P. (2012) Stripe formation in bacterial systems with density-suppressed motility. Phys. Rev. Lett. 108 (19), 198102.Google Scholar
[18] Wang, Z. & Hillen, T. (2007) Classical solutions and pattern formation for a volume filling chemotaxis model. Chaos: An Interdiscip. J. Nonlinear Sci. 17 (3), 037108–037108.Google Scholar
[19] Hillen, T. & Painter, K. (2001) Global existence for a parabolic chemotaxis model with prevention of overcrowding. Adv. Appl. Math. 26 (4), 280–301.Google Scholar
[20] Painter, K. J. & Hillen, T. (2002) Volume-filling and quorum-sensing in models for chemosensitive movement. Can. Appl. Math. Quart. 10 (4), 501–543.Google Scholar
[21] Kolokolnikov, T., Wei, J. & Alcolado, A. (2014) Basic mechanisms driving complex spike dynamics in a chemotaxis model with logistic growth. SIAM J. Appl. Math. 74 (5), 1375–1396.Google Scholar
[22] Hillen, T., Zielinski, J. & Painter, K. J. Merging-emerging systems can describe spatio-temporal patterning in a chemotaxis model. Discrete Contin. Dyn. Syst.-Ser. B 18 (10).Google Scholar
[23] Iron, D., Ward, M. & Wei, J. (2001) The stability of spike solutions to the one-dimensional Gierer–Meinhardt model. Phys. D: Nonlinear Phenom. 150 (1), 25–62.Google Scholar
[24] Doelman, A., Gardner, R. & Kaper, T. 1998 Stability analysis of singular patterns in the 1d Gray-Scott model: A matched asymptotics approach. Phys. D: Nonlinear Phenom. 122 (1), 1–36.Google Scholar
[25] Erneux, T. & Reiss, E. L. (1983) Brussellator isolas. SIAM J. Appl. Math. 43 (6), 1240–1246.Google Scholar
[26] Kolokolnikov, T., Erneux, T. & Wei, J. (2006) Mesa-type patterns in the one-dimensional brusselator and their stability. Phys. D: Nonlinear Phenom. 214 (1), 63–77.Google Scholar
[27] Oster, G. F. & Murray, J. D. (1989) Pattern formation models and developmental constraints. J. Exp. Zool. 251 (2), 186–202.Google Scholar
[28] Maini, P., Myerscough, M., Winters, K. & Murray, J. (1991) Bifurcating spatially heterogeneous solutions in a chemotaxis model for biological pattern generation. Bull. Math. Biol. 53 (5), 701–719.Google Scholar
[29] van Kampen, N. G. (1992) Stochastic Processes in Physics and Chemistry. Vol. 1, Elsevier.Google Scholar
[30] Bressloff, P. C. (2014) Stochastic Processes in Cell Biology. Vol. 41, Springer, New York.Google Scholar