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Bulletin of Mathematical Biology

, Volume 74, Issue 9, pp 2183–2203 | Cite as

A Model of Oscillatory Protein Dynamics in Bacteria

  • Peter Rashkov
  • Bernhard A. Schmitt
  • Lotte Søgaard-Andersen
  • Peter Lenz
  • Stephan Dahlke
Original Article

Abstract

Spatial oscillations of proteins in bacteria have recently attracted much attention. The cellular mechanism underlying these oscillations can be studied at molecular as well as at more macroscopic levels. We construct a minimal mathematical model with two proteins that is able to produce self-sustained regular pole-to-pole oscillations without having to take into account molecular details of the proteins and their interactions. The dynamics of the model is based solely on diffusion across the cell body and protein reactions at the poles, and is independent of stimuli coming from the environment. We solve the associated system of reaction–diffusion equations and perform a parameter scan to demonstrate robustness of the model for two possible sets of the reaction functions.

Keywords

Biological modeling Protein dynamics Dynamical systems Oscillatory behavior 

Notes

Acknowledgements

We would like to thank Thorsten Raasch for work done at the early stages of the project, Steffen Beck for advice on the MATLAB code, and Bruno Eckhardt for helpful discussions. This work has been supported by the Center for Synthetic Microbiology (SYNMIKRO) in Marburg, promoted by the LOEWE Excellence Program of the state of Hessen, Germany.

References

  1. Adams, D. W., & Errington, J. (2009). Bacterial cell division: assembly, maintenance and disassembly of the Z ring. Nat. Rev., Microbiol., 7, 642–653. CrossRefGoogle Scholar
  2. Arnold, V. I. (1980). Ordinary differential equations (1st ed.). Cambridge: MIT Press. Transl. and ed. by R.A. Silverman. zbMATHGoogle Scholar
  3. Bulyha, I., Schmidt, C., Lenz, P., Jakovljevic, V., Höne, A., Maier, B., Hopper, M., & Søgaard-Andersen, L. (2009). Regulation of the type IV pili molecular machine by dynamic localization of two motor proteins. Mol. Microbiol., 74, 691–705. CrossRefGoogle Scholar
  4. Bulyha, I., Hot, E., Huntley, S., & Søgaard-Andersen, L. (2011). GTPases in bacterial cell polarity and signaling. Curr. Opin. Microbiol., 14, 726–733. CrossRefGoogle Scholar
  5. Cytrynbaum, E. N., & Marshall, B. D. L. (2007). A multistranded polymer model explains MinDE dynamics in E. coli cell division. Biophys. J., 93, 1134–1150. CrossRefGoogle Scholar
  6. Edelstein-Keshet, L. (2005). Mathematical models in biology. Philadelphia: SIAM. zbMATHCrossRefGoogle Scholar
  7. Elowitz, M. B., Surette, M. G., Wolf, P.-E., Stock, J. B., & Leibler, S. (1999). Protein mobility in the cytoplasm of Escherichia coli. J. Bacteriol., 181, 197–203. Google Scholar
  8. Fange, D., & Elf, J. (2006). Noise induced min phenotypes in E. coli. PLoS Comput. Biol., 2, e80. CrossRefGoogle Scholar
  9. Gerdes, K., Howard, M., & Szardenings, F. (2010). Pushing and pulling in prokaryotic DNA segregation. Cell, 141, 927–942. CrossRefGoogle Scholar
  10. Gitai, Z., Dye, N. A., Reisenauer, A., Wachi, M., & Shapiro, L. (2005). MreB actin-mediated segregation of a specific region of a bacterial chromosome. Cell, 120, 329–341. CrossRefGoogle Scholar
  11. Guckenheimer, J., & Holmes, P. (1983). Nonlinear oscillations, dynamical systems, and bifurcations of vector fields. Berlin: Springer. zbMATHGoogle Scholar
  12. Hastings, S. P., & Murray, J. D. (1975). The existence of oscillatory solutions in the Field–Noyes model for the Belousov–Zhabotinskii reaction. SIAM J. Appl. Math., 28, 678–688. MathSciNetCrossRefGoogle Scholar
  13. Howard, M., Rutenberg, A. D., & de Vet, S. (2001). Dynamic compartmentalization of bacteria: accurate division in E. coli. Phys. Rev. Lett., 87(21), 278102. CrossRefGoogle Scholar
  14. Huang, K. C., Meir, Y., & Wingreen, N. S. (2003). Dynamic structures in Escherichia coli: spontaneous formation of MinE rings and MinD polar zones. Proc. Natl. Acad. Sci. USA, 100(22), 12724–12728. CrossRefGoogle Scholar
  15. Kruse, K. (2002). A dynamic model for determining the middle of Escherichia coli. Biophys. J., 82, 618–627. CrossRefGoogle Scholar
  16. Laub, M. T., Shapiro, L., & McAdams, H. H. (2007). Systems biology of caulobacter. Annu. Rev. Genet., 41, 429–441. CrossRefGoogle Scholar
  17. Lenz, P., & Søgaard-Andersen, L. (2011). Temporal and spatial oscillations in bacteria. Nat. Rev., Microbiol., 9, 565–577. CrossRefGoogle Scholar
  18. Leonardy, S., Bulyha, I., & Søgaard-Andersen, L. (2008). Reversing cells and oscillating proteins. Mol. BioSyst., 4, 1009–1014. CrossRefGoogle Scholar
  19. Leonardy, S., Miertzschke, M., Bulyha, I., Sperling, E., Wittinghofer, A., & Søgaard-Andersen, L. (2010). Regulation of dynamic polarity switching in bacteria by a Ras-like G-protein and its cognate GAP. EMBO J., 29, 2276–2289. CrossRefGoogle Scholar
  20. Lichtenberg, A. J., & Lieberman, M. A. (1992). Regular and chaotic dynamics (2nd ed.). Berlin: Springer. zbMATHGoogle Scholar
  21. Loose, M., Fischer-Friedrich, E., Ries, J., Kruse, K., & Schwille, P. (2008). Spatial regulators for bacterial cell division self-organize into surface waves in vitro. Science, 320, 789–792. CrossRefGoogle Scholar
  22. Loose, M., Kruse, K., & Schwille, P. (2011). Protein self-organization: lessons from the min system. Annu. Rev. Biophys., 40, 315–336. CrossRefGoogle Scholar
  23. López, D., & Kolter, R. (2010). Functional microdomains in bacterial membranes. Genes Dev., 24, 1893–1902. CrossRefGoogle Scholar
  24. Lutkenhaus, J. (2007). Assembly dynamics of the bacterial MinCDE system and spatial regulation of the Z ring. Annu. Rev. Biochem., 76, 539–562. CrossRefGoogle Scholar
  25. Matsumoto, K., Kusaka, J., Nishibori, A., & Hara, H. (2006). Lipid domains in bacterial membranes. Mol. Microbiol., 61, 1110–1117. CrossRefGoogle Scholar
  26. Meacci, G., Ries, J., Fischer-Friedrich, E., Kahya, N., Schwille, P., & Kruse, K. (2006). Mobility of min-proteins in Escherichia coli measured by fluorescence correlation spectroscopy. Phys. Biol., 3, 255–263. CrossRefGoogle Scholar
  27. Meinhardt, H., & de Boer, P. A. J. (2001). Pattern formation in Escherichia coli: a model for the pole-to-pole oscillations of min proteins and the localization of the division site. Proc. Natl. Acad. Sci. USA, 98, 14202–14207. CrossRefGoogle Scholar
  28. Montero Llopis, P., Jackson, A. F., Sliusarenko, O., Surovtsev, I., Heinritz, J., Emonet, T., & Jacobs-Wagner, C. (2010). Spatial organization of the flow of genetic information in bacteria. Nature, 466, 77–81. CrossRefGoogle Scholar
  29. Nan, B., Mauriello, E. M. F., Wong, A., Sun, I. H., & Zusman, D. R. (2010). A multi-protein complex from Myxococcus xanthus required for bacterial gliding motility. Mol. Microbiol., 76, 1539–1554. CrossRefGoogle Scholar
  30. Nelson, W. J. (2003). Adaptation of core mechanisms to generate cell polarity. Nature, 422, 766–774. CrossRefGoogle Scholar
  31. Nevo-Dinur, K., Nussbaum-Shochat, A., Ben-Yehuda, S., & Amster-Choder, O. (2011). Translation-independent localization of mRNA in E. coli. Science, 331, 1081–1084. CrossRefGoogle Scholar
  32. Prigogine, I., & Nicolis, G. (1967). On symmetry-breaking instabilities in dissipative systems. J. Chem. Phys., 46, 3542–3551. CrossRefGoogle Scholar
  33. Prigogine, I., & Lefever, R. (1968). On symmetry-breaking instabilities in dissipative systems (II). J. Chem. Phys., 46, 1695–1700. CrossRefGoogle Scholar
  34. Ringgaard, S., van Zon, J., Howard, M., & Gerdes, K. (2009). Movement and equipositioning of plasmids by ParA filament disassembly. Proc. Natl. Acad. Sci. USA, 106, 19369–19374. CrossRefGoogle Scholar
  35. Rinzel, J., & Troy, W. C. (1982). Bursting phenomena in a simplified Oregonator flow system model. J. Chem. Phys., 76, 1775–1789. MathSciNetCrossRefGoogle Scholar
  36. Seydel, R. (2010). Practical bifurcation and stability analysis (3rd ed.). New York: Springer. zbMATHCrossRefGoogle Scholar
  37. Shapiro, L., McAdams, H. H., & Losick, R. (2009). Why and how bacteria localize proteins. Science, 326, 1225–1228. CrossRefGoogle Scholar
  38. Teleman, A. A., Graumann, P. L., Lin, D. C.-H., Grossman, A. D., & Losick, R. (1998). Chromosome arrangement within a bacterium. Curr. Biol., 8, 1102–1109. CrossRefGoogle Scholar
  39. Viollier, P. H., Thanbichler, M., McGrath, P. T., West, L., Meewan, M., McAdams, H. H., & Shapiro, L. (2004). Rapid and sequential movement of individual chromosomal loci to specific subcellular locations during bacterial DNA replication. Proc. Natl. Acad. Sci. USA, 101, 9257–9262. CrossRefGoogle Scholar
  40. Verhulst, F. (1990). Nonlinear differential equations and dynamical systems. Berlin: Springer. zbMATHCrossRefGoogle Scholar
  41. Wang, X., Liu, X., Possoz, C., & Sherratt, D. J. (2006). The two Escherichia coli chromosome arms locate to separate cell halves. Genes Dev., 20, 1727–1731. CrossRefGoogle Scholar
  42. Weiner, R., Schmitt, B. A., & Podhaisky, H. (1997). ROWMAP—a ROW-code with Krylov techniques for large stiff ODEs. Appl. Numer. Math., 25(2–3), 303–319. MathSciNetzbMATHCrossRefGoogle Scholar
  43. Zhang, Y., Franco, M., Ducret, A., & Mignot, T. (2010). A bacterial Ras-like small GTP-binding protein and its cognate GAP establish a dynamic spatial polarity axis to control directed motility. PLoS Biol., 8, e1000430. CrossRefGoogle Scholar

Copyright information

© Society for Mathematical Biology 2012

Authors and Affiliations

  • Peter Rashkov
    • 1
  • Bernhard A. Schmitt
    • 1
  • Lotte Søgaard-Andersen
    • 2
  • Peter Lenz
    • 3
  • Stephan Dahlke
    • 1
  1. 1.Department of Mathematics and InformaticsPhilipps-Universität MarburgMarburgGermany
  2. 2.Max Planck Institute for Terrestrial MicrobiologyMarburgGermany
  3. 3.Department of PhysicsPhilipps-Universität MarburgMarburgGermany

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