Applying a Rigorous Quasi-Steady State Approximation Method for Proving the Absence of Oscillations in Models of Genetic Circuits

  • François Boulier
  • Marc Lefranc
  • François Lemaire
  • Pierre-Emmanuel Morant
Part of the Lecture Notes in Computer Science book series (LNCS, volume 5147)

Abstract

In this paper, we apply a rigorous quasi-steady state approximation method on a family of models describing a gene regulated by a polymer of its own protein. We study the absence of oscillations for this family of models and prove that Poincaré-Andronov-Hopf bifurcations arise if and only if the number of polymerizations is greater than 8. A result presented in a former paper at Algebraic Biology 2007 is thereby generalized. The rigorous method is illustrated over the basic enzymatic reaction.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    Boulier, F., Lefranc, M., Lemaire, F., Morant, P.E., Ürgüplü, A.: On proving the absence of oscillations in models of genetic circuits. In: Anai, H., Horimoto, K., Kutsia, T. (eds.) Ab 2007. LNCS, vol. 4545, pp. 66–80. Springer, Heidelberg (2007), http://hal.archives-ouvertes.fr/hal-00139667 CrossRefGoogle Scholar
  2. 2.
    Fall, C.P., Marland, E.S., Wagner, J.M., Tyson, J.J.: Computational Cell Biology. Interdisciplinary Applied Mathematics, vol. 20. Springer, Heidelberg (2002)MATHGoogle Scholar
  3. 3.
    Goodwin, B.C.: Temporal Organization in Cells. Academic Press, London (1963)Google Scholar
  4. 4.
    Goodwin, B.C.: Advances in Enzyme Regulation, vol. 3, p. 425. Pergamon Press, Oxford (1965)Google Scholar
  5. 5.
    Griffith, J.S.: Mathematics of Cellular Control Processes. I. Negative Feedback to One Gene. Journal of Theoretical Biology 20, 202–208 (1968)Google Scholar
  6. 6.
    Doedel, E.: AUTO software for continuation and bifurcation problems in ODEs (1996), http://indy.cs.concordia.ca/auto
  7. 7.
    Ermentrout, B.: Simulating, Analyzing, and Animating Dynamical Systems: A Guide to XPPAUT for Researchers and Students. Software, Environments, and Tools, vol. 14. SIAM, Philadelphia (2002)MATHGoogle Scholar
  8. 8.
    Conrad, E.D., Tyson, J.J.: Modeling Molecular Interaction Networks with Nonlinear Differential Equations. In: Szallasi, Z., Stelling, J., Periwal, V. (eds.) System Modeling in Cell Biology: From Concepts to Nuts and Bolts, pp. 97–124. The MIT Press, Cambridge (2006)Google Scholar
  9. 9.
    Van Breusegem, V., Bastin, G.: Reduced order dynamical modelling of reaction systems: a singular perturbation approach. In: Proceedings of the 30th IEEE Conference on Decision and Control, Brighton, England, pp. 1049–1054 (December 1991)Google Scholar
  10. 10.
    Okino, M.S., Mavrovouniotis, M.L.: Simplification of Mathematical Models of Chemical Reaction Systems. Chemical Reviews 98(2), 391–408 (1998)CrossRefGoogle Scholar
  11. 11.
    Vora, N., Daoutidis, P.: Nonlinear model reduction of chemical reaction systems. AIChE Journal 47(10), 2320–2332 (2001)CrossRefGoogle Scholar
  12. 12.
    Bennet, M.R., Volfson, D., Tsimring, L., Hasty, J.: Transient Dynamics of Genetic Regulatory Networks. Biophysical Journal 92, 3501–3512 (2007)CrossRefGoogle Scholar
  13. 13.
    Boulier, F., Lefranc, M., Lemaire, F., Morant, P.E.: Model Reduction of Chemical Reaction Systems using Elimination. In: The international conference MACIS 2007 (2007), http://hal.archives-ouvertes.fr/hal-00184558
  14. 14.
    Ritt, J.F.: Differential Algebra. Dover Publications Inc, New York (1950), http://www.ams.org/online_bks/coll33
  15. 15.
    Kolchin, E.R.: Differential Algebra and Algebraic Groups. Academic Press, New York (1973)MATHGoogle Scholar
  16. 16.
    Boulier, F., Lazard, D., Ollivier, F., Petitot, M.: Representation for the radical of a finitely generated differential ideal. In: ISSAC 1995: Proceedings of the 1995 international symposium on Symbolic and algebraic computation, pp. 158–166. ACM Press, New York (1995), http://hal.archives-ouvertes.fr/hal-00138020 CrossRefGoogle Scholar
  17. 17.
    Wang, D.: Elimination Practice: Software Tools and Applications. Imperial College Press, London (2003)Google Scholar
  18. 18.
    Boulier, F.: Differential Elimination and Biological Modelling. Radon Series on Computational and Applied Mathematics (Gröbner Bases in Symbolic Analysis) 2, 111–139 (2007), http://hal.archives-ouvertes.fr/hal-00139364 MathSciNetGoogle Scholar
  19. 19.
    Lemaire, F., Moreno Maza, M., Xie, Y.: The RegularChains library in MAPLE 10. In: Kotsireas, I.S. (ed.) The MAPLE conference, pp. 355–368 (2005)Google Scholar
  20. 20.
    Horn, F., Jackson, R.: General mass action kinetics. Archive for Rational Mechanics and Analysis 47, 81–116 (1972)CrossRefMathSciNetGoogle Scholar
  21. 21.
    Sedoglavic, A., Ürgüplü, A.: Expanded Lie Point Symmetry (MAPLE package) (2007), http://www.lifl.fr/~sedoglav/Software
  22. 22.
    Hale, J.K., Koçak, H.: Dynamics and Bifurcations. Texts in Applied Mathematics, vol. 3. Springer, New York (1991)MATHGoogle Scholar
  23. 23.
    Boulier, F., Lazard, D., Ollivier, F., Petitot, M.: Computing representations for radicals of finitely generated differential ideals. Technical report, Université Lille I, LIFL, 59655, Villeneuve d’Ascq, France (1997); Ref. IT306. December 1998 version published in the HDR memoir of Michel Petitot, http://hal.archives-ouvertes.fr/hal-00139061
  24. 24.
    de Jong, H., Ropers, D.: Qualitative Approaches to the Analysis of Genetic Regulatory Networks. In: Szallasi, Z., Stelling, J., Periwal, V. (eds.) System Modeling in Cell Biology: From Concepts to Nuts and Bolts, pp. 125–147. The MIT Press, Cambridge (2006)Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2008

Authors and Affiliations

  • François Boulier
    • 1
  • Marc Lefranc
    • 2
  • François Lemaire
    • 1
  • Pierre-Emmanuel Morant
    • 2
  1. 1.LIFLUniversity Lille IVilleneuve d’AscqFrance
  2. 2.PHLAMUniversity Lille IVilleneuve d’AscqFrance

Personalised recommendations