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Two continuum models for the spreading of myxobacteria swarms

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Abstract

We analyze the phenomenon of spreading of a Myxococcus xanthus bacterial colony on plates coated with nutrient. The bacteria spread by gliding on the surface. In the first few hours, cell growth is irrelevant to colony spread. In this case, bacteria spread through peninsular protrusions from the edge of the initial colony. We analyze the diffusion through the narrowing reticulum of cells on the surface mathematically and derive formulae for the spreading rates. On the time scale of tens of hours, effective diffusion of the bacteria, combined with cell division and growth, causes a constant linear increase in the colony's radius. Mathematical analysis and numerical solution of reaction-diffusion equations describing the bacterial and nutrient dynamics demonstrate that, in this regime, the spreading rate is proportional to the square root of both the effective diffusion coefficient and the nutrient concentration. The model predictions agree with the data on spreading rate dependence on the type of gliding motility.

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References

  • Alber, M.S., Kiskowski, M.A., Jiang, Y., 2004. Two-stage aggregate formation via streams in myxobacteria. Phys. Rev. Lett. 93, 068102.

    Article  PubMed  Google Scholar 

  • Bees, M.A., Andresen, P., Mosekilde, E., Givskov, M., 2002. Quantitative effects of medium hardness and nutrient availability on the swarming motility of Serratia liquefaciens. Bull. Math. Biol. 64, 565–587.

    Article  PubMed  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., Schochet, O., Tenenbaum, A., Cohen, I., Czirok, A., Vicsek, T., 1994. Generic modelling of cooperative growth patterns in bacterial colonies. Nature 368, 46–49.

    Article  PubMed  Google Scholar 

  • Berg, H., Budrene, E., 1991. Complex patterns formed by motile cells in E. coli. Nature 349, 630–633.

    Article  PubMed  Google Scholar 

  • Berg, H.C., 1993. Random walks in biology. Princeton University Press, Princeton, NJ.

    Google Scholar 

  • Billingham, J., Needham, D.J., 1991a. The development of traveling waves in quadratic and cubic autocatalysis with unequal diffusion rates. I. Permanent form traveling waves. Phil. Trans. R. Soc.: Phys. Sci. Eng. 334, 1–24.

    Google Scholar 

  • Billingham, J., Needham, D.J., 1991b. The development of traveling waves in quadratic and cubic autocatalysis with unequal diffusion rates. II. An initial value problem with an immobilized or nearly immobilized autocatalysis. Phil. Trans. R. Soc.: Phys. Sci. Eng. 336, 497–539.

    Google Scholar 

  • Borner, U., Deutsch, A., Reichenbach, H., Bar, M., 2002. Rippling patterns in aggregates of myxobacteria arise from cell–cell collisions. Phys. Rev. Lett. 89, 078101.

    Article  PubMed  Google Scholar 

  • Bray, D., 2002. Cell Movements. Garland, New York.

    Google Scholar 

  • Burchard, R., 1974. Growth of surface colonies of the gliding bacterium Myxococcus xanthus. Arch. Microbiol. 96, 247–254.

    Article  PubMed  Google Scholar 

  • Deutsch, A., 1995. Towards analyzing complex swarming patterns in biological systems with the help of lattice–gas cellular automata. J. Biol. Syst. 3, 947–955.

    Article  Google Scholar 

  • Dockery, J.D., Keener, J.P., 2001. A mathematical model for quorum sensing in Pseudomonas aeruginosa. Bull. Math. Biol. 63, 95–116.

    Article  PubMed  Google Scholar 

  • Dworkin, M., Eide, D., 1983. Myxococcus xanthus does not respond chemotactically to moderate concentration gradients. J. Bacteriol. 154, 437–442.

    PubMed  Google Scholar 

  • Edelstein-Keshet, L., 1988. Mathematical Models in Biology. Random House, New York.

    MATH  Google Scholar 

  • Esipov, S., Shapiro, D., 1998. Kinetic model of Proteus mirabilis swarm colony development. J. Math. Biol. 36, 249–268.

    Article  MathSciNet  MATH  Google Scholar 

  • Fife, P.C., 1979. Mathematical Aspects of Reacting and Diffusing Systems. Springer, Berlin.

    MATH  Google Scholar 

  • Garcia, A.L., 2000. Numerical Methods for Physics. Prentice-Hall, Englewood Cliffs, NJ.

    Google Scholar 

  • Gray, B.F., Kirwan, N.A., 1974. Growth rates of yeast colonies on solid media. Biophys. Chem. 1, 204–213.

    Article  PubMed  Google Scholar 

  • Igoshin, O., Mogilner, A., Welsch, R., Kaiser, D., Oster, G., 2001. Pattern formation and traveling waves in Myxobacteria: Theory and modeling. Proc. Natl. Acad. Sci. U.S.A. 98, 14913–14918.

    Article  PubMed  Google Scholar 

  • Igoshin, O., Welch, R., Kaiser, D., Oster, G., 2004. Waves and aggregation patterns in Myxobacteria. Proc. Natl. Acad. Sci. U.S.A. 101, 4256–4261.

    Article  PubMed  Google Scholar 

  • Jelsbak, L., Sogaard-Andersen, L., 2000. Pattern formation: Fruiting body morphogenesis in Myxococcus xanthus. Curr. Opin. Microbiol. 3, 637–642.

    Article  PubMed  Google Scholar 

  • Kaiser, D., Crosby, C., 1983. Cell movement and its coordination in swarms of Myxococcus xanthus. Cell Motil. 3, 227–245.

    Article  Google Scholar 

  • Kawasaki, K., Mochizuchi, A., Matsushita, M., Umeda, T., Shigesada, N., 1997. Modeling spatio-temporal patterns generated by Bacillus subtilis. J. Theor. Biol. 188, 177–185.

    Article  PubMed  Google Scholar 

  • Koch, A.L., 1999. Diffusion through agar blocks of finite dimensions: A theoretical analysis of three systems of practical significance in microbiology. Microbiology 145, 643–654.

    Article  PubMed  Google Scholar 

  • Komoto, A., Hanaki, K., Maenosono, S., Wakano, J.Y., Yamaguchi, Y., Yamamoto, K., 2003. Growth dynamics of Bacillus circulans colony. J. Theor. Biol. 225, 91–97.

    Article  PubMed  MathSciNet  Google Scholar 

  • Lutscher, F., Stevens, A., 2002. Emerging patterns in a hyperbolic model for locally interacting cell systems. J. Nonlinear Sci. 12, 619–640.

    Article  MATH  MathSciNet  Google Scholar 

  • Merz, A.J., So, M., Sheetz, M.P., 2000. Pilus retraction powers bacterial twitching motility. Nature 407, 98–102.

    Article  PubMed  Google Scholar 

  • Murray, J.D., 1993. Mathematical Biology. Springer, Berlin.

    MATH  Google Scholar 

  • Nagai, T., Ikeda, T., 1991. Traveling waves in a chemotactic model. J. Math. Biol. 30, 169–184.

    Article  PubMed  MATH  MathSciNet  Google Scholar 

  • Newman, W.I., 1980. Some exact solutions to a nonlinear diffusion problem in population genetics and combustion. J. Theor. Biol. 85, 325–334.

    Article  PubMed  Google Scholar 

  • Othmer, H.G., Stevens, A., 1997. Aggregation, blowup and collapse: The ABC's of taxis in reinforced random walks. SIAM J. Appl. Math. 57, 1044–1081.

    Article  MATH  MathSciNet  Google Scholar 

  • Pfistner, B., 1989. A one-dimensional model for the swarming behavior of Myxobacteria. In: Alt, W., Hoffmann, G. (Eds.), Biological Motion. Springer, Berlin, pp. 556–563.

  • Rauprich, O., Matsushita, M., Weijer, C.J., Siegert, F., Esipov, S.E., Shapiro, J.A., 1996. Periodic phenomena in Proteus mirabilis swarm colony development. J. Bacteriol. 178, 6525–6538.

    PubMed  Google Scholar 

  • Rosenberg, E., Keller, K.H., Dworkin, M., 1977. Cell density-dependent growth of Myxococcus xanthus on casein. J. Bacteriol. 129, 770–777.

    PubMed  Google Scholar 

  • Santoianu, R.A., Maini, P.K., Garduno, F.S., Armitage, J.P., 2001. Traveling waves in a nonlinear degenerate diffusion model for bacterial pattern formation. Disc. Cont. Dyn. Syst. B 1, 339–362.

    Article  Google Scholar 

  • Shapiro, J.A., 1998. Thinking about bacterial populations as multicellular organisms. Annu. Rev. Microbiol. 52, 81–104.

    Article  PubMed  Google Scholar 

  • Shi, W., Kohler, T., Zusman, D.R., 1993. Chemotaxis plays a role in the social behavior of Myxococcus xanthus. Mol. Microbiol. 9, 601–611.

    Article  PubMed  Google Scholar 

  • Shi, W., Ngok, F.K., Zusman, D.R., 1996. Cell density regulates cellular reversal frequency in Myxococcus xanthus. Proc. Natl. Acad. Sci. U.S.A. 93, 4142–4146.

    Article  PubMed  Google Scholar 

  • Shimkets, L.J., Kaiser, D., 1982. Induction of coordinated movement of Myxococcus xanthus cells. J. Bacteriol. 152, 451–461.

    PubMed  Google Scholar 

  • Spormann, A.M., 1999. Gliding motility in bacteria: Insights from studies of Myxococcus xanthus. Microbiol. Mol. Biol. Rev. 63, 621–641.

    PubMed  Google Scholar 

  • Spormann, A.M., Kaiser, A.D., 1995. Gliding movements in Myxococcus xanthus. J. Bacteriol. 177, 5846–5852.

    PubMed  Google Scholar 

  • Spormann, A.M., Kaiser, A.D., 1999. Gliding mutants of Myxococcus xanthus with high reversal frequencies and small displacements. J. Bacteriol. 181, 2593–2601.

    PubMed  Google Scholar 

  • Stevens, A., 1995. Trail following and aggregation of myxobacteria. J. Biol. Syst. 3, 1059–1068.

    Article  Google Scholar 

  • Ward, M.J., Mok, K.C., Zusman, D.R., 1998. Myxococcus xanthus displays Frz-dependent chemokinetic behavior during vegetative swarming. J. Bacteriol. 180, 440–443.

    PubMed  Google Scholar 

  • Wolgemuth, C., Hoiczyk, E., Kaiser, D., Oster, G., 2002. How myxobacteria glide. Curr. Biol. 12, 369–377.

    Article  PubMed  Google Scholar 

  • Woodward, D.E., Tyson, R., Myerscough, M.R., Murray, J.D., Budrene, E.O., Berg, H.C., 1995. Spatio-temporal patterns generated by Salmonella typhimurium. Biophys. J. 68, 2181–2189.

    Article  PubMed  Google Scholar 

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Authors and Affiliations

  1. Department of Mathematics, University of California, Davis, CA, 95616, USA

    Angela Gallegos & Alex Mogilner

  2. Department of Mathematics, University of Utah, Salt Lake City, Utah, 84112, USA

    Barbara Mazzag

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  1. Angela Gallegos
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  2. Barbara Mazzag
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  3. Alex Mogilner
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Correspondence to Alex Mogilner.

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Gallegos, A., Mazzag, B. & Mogilner, A. Two continuum models for the spreading of myxobacteria swarms. Bull. Math. Biol. 68, 837–861 (2006). https://doi.org/10.1007/s11538-005-9031-2

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