Skip to main content
SpringerLink
Log in

Quasi-Steady State – Intuition, Perturbation Theory and Algorithmic Algebra

  • Conference paper
  • First Online:
Computer Algebra in Scientific Computing (CASC 2015)

Part of the book series: Lecture Notes in Computer Science ((LNTCS,volume 9301))

Included in the following conference series:

Abstract

This survey of mathematical approaches to quasi-steady state (QSS) phenomena provides an analytical foundation for an algorithmic-algebraic treatment of the associated (parameter-dependent) ordinary differential systems, in particular for reaction networks. Topics include an ad hoc reduction procedure, singular perturbations, and methods to identify suitable parameter regions.

This is a preview of subscription content, log in via an institution to check access.

Access this chapter

Log in via an institution
Chapter
USD 29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD 39.99
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 54.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. Anai, H., Horimoto, K., Kutsia, T.: AB 2007. LNCS, vol. 4545. Springer, Heidlberg (2007)

    Google Scholar 

  2. Boulier, F., Lemaire, F., Sedoglavic, A., Ürgüplü, A.: Towards an Automated Reduction Method for Polynomial ODE Models of Biochemical Reaction Systems. Mathematics in Computer Science 2, 443–464 (2009)

    Google Scholar 

  3. Boulier, F., Lefranc, M., Lemaire, F., Morant, P.E.: Model Reduction of Chemical Reaction Systems using Elimination. Mathematics in Computer Science 5, 289–301 (2011)

    Google Scholar 

  4. Boulier, F., Lemaire, F., Petitot, M., Sedoglavic, A.: Chemical reaction systems, computer algebra and systems biology. In: Gerdt, V.P., Koepf, W., Mayr, E.W., Vorozhtsov, E.V. (eds.) CASC 2011. LNCS, vol. 6885, pp. 73–87. Springer, Heidelberg (2011)

    Google Scholar 

  5. Borghans, J.A.M., de Boer, R.J., Segel, L.A.: Extending the quasi-steady state approximation by changing variables. Bull. Math. Biol. 58, 43–63 (1996)

    Google Scholar 

  6. Briggs, G.E., Haldane, J.B.S.: A note on the kinetics of enzyme actiion. Biochem. J. 19, 338–339 (1925)

    Google Scholar 

  7. Cicogna, G., Gaeta, G., Walcher, S.: Side conditions for ordinary differential equations. J. Lie Theory 25, 125–146 (2015)

    Google Scholar 

  8. Cox, D.A., Little, J., O’Shea, D.: Using algebraic geometry. Graduate Texts in Mathematics, vol. 185, 2nd edn. Springer, New York (2005)

    Google Scholar 

  9. Errami, H., Eiswirth, M., Grigoriev, D., Seiler, W.M., Sturm, T., Weber, A.: Efficient methods to compute hopf bifurcations in chemical reaction networks using reaction coordinates. In: Gerdt, V.P., Koepf, W., Mayr, E.W., Vorozhtsov, E.V. (eds.) CASC 2013. LNCS, vol. 8136, pp. 88–99. Springer, Heidelberg (2013)

    Google Scholar 

  10. Decker, W., Greuel, G.-M., Pfister, G., Schönemann, H.: Singular 3-1-3 – A computer algebra system for polynomial computations (2011). http://www.singular.uni-kl.de

  11. Decker, W., Lossen, Ch.: Computing in algebraic geometry. Algorithms and computation in mathematics, vol. 16. Springer, Berlin (2006)

    Google Scholar 

  12. Duchêne, P., Rouchon, P.: Kinetic scheme reduction via geometric singular perturbation techniques. Chem. Eng. Sci. 12, 4661–4672 (1996)

    Google Scholar 

  13. Fenichel, N.: Geometric singular perturbation theory for ordinary differential equations. J. Differential Equations 31(1), 53–98 (1979)

    Google Scholar 

  14. Gatermann, K., Huber, B.: A family of sparse polynomial systems arising in chemical reaction systems. J. Symbolic Comput. 33, 275–305 (2002)

    Google Scholar 

  15. Goeke, A.: Reduktion und asymptotische Reduktion von Reaktionsgleichungen. Doctoral dissertation, RWTH Aachen (2013)

    Google Scholar 

  16. Goeke, A., Walcher, S.: A constructive approach to quasi-steady state reduction. J. Math. Chem. 52, 2596–2626 (2014)

    Google Scholar 

  17. Goeke, A., Walcher, S., Zerz, E.: Determining “small parameters” for quasi-steady state. J. Diff. Equations 259, 1149–1180 (2015)

    Google Scholar 

  18. Heineken, F.G., Tsuchiya, H.M., Aris, R.: On the mathematical status of the pseudo-steady state hypothesis of biochemical kinetics. Math. Biosci. 1, 95–113 (1967)

    Google Scholar 

  19. Henri, V.: Lois générales de l’action des diastases. Hermann, Paris (1903)

    Google Scholar 

  20. Horimoto, K., Regensburger, G., Rosenkranz, M., Yoshida, H.: AB 2008. LNCS, vol. 5147. Springer, Heidelberg (2008)

    Google Scholar 

  21. Hubert, E., Labahn, G.: Scaling Invariants and Symmetry Reduction of Dynamical Systems. Found. Comput. Math. 13, 479–516 (2013)

    Google Scholar 

  22. Laidler, K.J.: Theory of the transient phase in kinetics, with special reference to enzyme systems. Can. J. Chem. 33, 1614–1624 (1955)

    Google Scholar 

  23. Lam, S.H., Goussis, D.A.: The CSP method for simplifying kinetics. Int. J. Chemical Kinetics 26, 461–486 (1994)

    Google Scholar 

  24. Lee, C.H., Othmer, H.G.: A multi-time-scale analysis of chemical reaction networks: I Deterministic systems. J. Math. Biol. 60, 387–450 (2009)

    Google Scholar 

  25. Michaelis, L., Menten, M.L.: Die Kinetik der Invertinwirkung. Biochem. Z 49, 333–369 (1913)

    Google Scholar 

  26. Niu, W., Wang, D.: Algebraic analysis of bifurcations and limit cycles for biological systems. In: [20], pp. 156–171

    Google Scholar 

  27. Noethen, L., Walcher, S.: Quasi-steady state and nearly invariant sets. SIAM J. Appl. Math. 70(4), 1341–1363 (2009)

    Google Scholar 

  28. Noethen, L., Walcher, S.: Tikhonov’s theorem and quasi-steady state. Discrete Contin. Dyn. Syst. Ser. B 16(3), 945–961 (2011)

    Google Scholar 

  29. Schauer, M., Heinrich, R.: Analysis of the quasi-steady-state approximation for an enzymatic one-substrate reaction. J. Theoret. Biol. 79, 425–442 (1979)

    Google Scholar 

  30. Schauer, M., Heinrich, R.: Quasi-steady-state approximation in the mathematical modeling of biochemical networks. Math. Biosci. 65, 155–170 (1983)

    Google Scholar 

  31. Sedoglavic, A.: Reduction of algebraic parametric systems by rectification of their affine expanded Lie symmetries. In: [1], pp. 277–291

    Google Scholar 

  32. Segel, L.A., Slemrod, M.: The quasi-steady-state assumption: A case study in perturbation. SIAM Review 31, 446–477 (1989)

    Google Scholar 

  33. Shafarevich, I.R.: Basic algebraic geometry. Springer, New York (1977)

    Google Scholar 

  34. Shiu, A., Sturmfels, B.: Siphons in chemical reaction networks. Bull. Math. Biol. 72, 1448–1463 (2010)

    Google Scholar 

  35. Stiefenhofer, M.: Quasi-steady-state approximation for chemical reaction networks. J. Math. Biol. 36, 593–609 (1998)

    Google Scholar 

  36. Tikhonov, A.N.: Systems of differential equations containing a small parameter multiplying the derivative (in Russian). Math. Sb. 31, 575–586 (1952)

    Google Scholar 

  37. Verhulst, F.: Methods and Applications of Singular Perturbations. Boundary Layers and Multiple Timescale Dynamics. Springer, New York (2005)

    Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Lehrstuhl A für Mathematik, RWTH Aachen, 52056, Aachen, Germany

    Alexandra Goeke & Sebastian Walcher

  2. Lehrstuhl D für Mathematik, RWTH Aachen, 52056, Aachen, Germany

    Eva Zerz

Authors
  1. Alexandra Goeke
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Sebastian Walcher
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Eva Zerz
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Alexandra Goeke .

Editor information

Editors and Affiliations

  1. Laboratory of Information Technologies, Joint Institute of Nuclear Research, Dubna, Russia

    Vladimir P. Gerdt

  2. Institute for Mathematics, Universität Kassel, Kassel, Germany

    Wolfram Koepf

  3. Institut für Mathematik, Universität Kassel, Kassel, Hessen, Germany

    Werner M. Seiler

  4. Novosibirsk, Russia

    Evgenii V. Vorozhtsov

Rights and permissions

Reprints and permissions

Copyright information

© 2015 Springer International Publishing Switzerland

About this paper

Cite this paper

Goeke, A., Walcher, S., Zerz, E. (2015). Quasi-Steady State – Intuition, Perturbation Theory and Algorithmic Algebra. In: Gerdt, V., Koepf, W., Seiler, W., Vorozhtsov, E. (eds) Computer Algebra in Scientific Computing. CASC 2015. Lecture Notes in Computer Science(), vol 9301. Springer, Cham. https://doi.org/10.1007/978-3-319-24021-3_10

Download citation

  • DOI: https://doi.org/10.1007/978-3-319-24021-3_10

  • Published:

  • Publisher Name: Springer, Cham

  • Print ISBN: 978-3-319-24020-6

  • Online ISBN: 978-3-319-24021-3

  • eBook Packages: Computer ScienceComputer Science (R0)

Publish with us

Policies and ethics

Search

Navigation