International Workshop on Computer Algebra in Scientific Computing

Computer Algebra in Scientific Computing pp 135-151 | Cite as

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

Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 9301)

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.

MSC (2010)

92C45 34E15 80A30 13P10 

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References

  1. 1.
    Anai, H., Horimoto, K., Kutsia, T.: AB 2007. LNCS, vol. 4545. Springer, Heidlberg (2007)Google Scholar
  2. 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. 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. 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. 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. 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. 7.
    Cicogna, G., Gaeta, G., Walcher, S.: Side conditions for ordinary differential equations. J. Lie Theory 25, 125–146 (2015)Google Scholar
  8. 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. 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. 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. 11.
    Decker, W., Lossen, Ch.: Computing in algebraic geometry. Algorithms and computation in mathematics, vol. 16. Springer, Berlin (2006)Google Scholar
  12. 12.
    Duchêne, P., Rouchon, P.: Kinetic scheme reduction via geometric singular perturbation techniques. Chem. Eng. Sci. 12, 4661–4672 (1996)Google Scholar
  13. 13.
    Fenichel, N.: Geometric singular perturbation theory for ordinary differential equations. J. Differential Equations 31(1), 53–98 (1979)Google Scholar
  14. 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. 15.
    Goeke, A.: Reduktion und asymptotische Reduktion von Reaktionsgleichungen. Doctoral dissertation, RWTH Aachen (2013)Google Scholar
  16. 16.
    Goeke, A., Walcher, S.: A constructive approach to quasi-steady state reduction. J. Math. Chem. 52, 2596–2626 (2014)Google Scholar
  17. 17.
    Goeke, A., Walcher, S., Zerz, E.: Determining “small parameters” for quasi-steady state. J. Diff. Equations 259, 1149–1180 (2015)Google Scholar
  18. 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. 19.
    Henri, V.: Lois générales de l’action des diastases. Hermann, Paris (1903)Google Scholar
  20. 20.
    Horimoto, K., Regensburger, G., Rosenkranz, M., Yoshida, H.: AB 2008. LNCS, vol. 5147. Springer, Heidelberg (2008)Google Scholar
  21. 21.
    Hubert, E., Labahn, G.: Scaling Invariants and Symmetry Reduction of Dynamical Systems. Found. Comput. Math. 13, 479–516 (2013)Google Scholar
  22. 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. 23.
    Lam, S.H., Goussis, D.A.: The CSP method for simplifying kinetics. Int. J. Chemical Kinetics 26, 461–486 (1994)Google Scholar
  24. 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. 25.
    Michaelis, L., Menten, M.L.: Die Kinetik der Invertinwirkung. Biochem. Z 49, 333–369 (1913)Google Scholar
  26. 26.
    Niu, W., Wang, D.: Algebraic analysis of bifurcations and limit cycles for biological systems. In: [20], pp. 156–171Google Scholar
  27. 27.
    Noethen, L., Walcher, S.: Quasi-steady state and nearly invariant sets. SIAM J. Appl. Math. 70(4), 1341–1363 (2009)Google Scholar
  28. 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. 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. 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. 31.
    Sedoglavic, A.: Reduction of algebraic parametric systems by rectification of their affine expanded Lie symmetries. In: [1], pp. 277–291Google Scholar
  32. 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. 33.
    Shafarevich, I.R.: Basic algebraic geometry. Springer, New York (1977)Google Scholar
  34. 34.
    Shiu, A., Sturmfels, B.: Siphons in chemical reaction networks. Bull. Math. Biol. 72, 1448–1463 (2010)Google Scholar
  35. 35.
    Stiefenhofer, M.: Quasi-steady-state approximation for chemical reaction networks. J. Math. Biol. 36, 593–609 (1998)Google Scholar
  36. 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. 37.
    Verhulst, F.: Methods and Applications of Singular Perturbations. Boundary Layers and Multiple Timescale Dynamics. Springer, New York (2005)Google Scholar

Copyright information

© Springer International Publishing Switzerland 2015

Authors and Affiliations

  1. 1.Lehrstuhl A für MathematikRWTH AachenAachenGermany
  2. 2.Lehrstuhl D für MathematikRWTH AachenAachenGermany

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