Abstract
We present a method to determine the reduction of a (polynomial or rational) ordinary differential equation that models a chemically reacting system, under the assumption that this system admits quasi-steady state (QSS) behavior for certain variables or reactions. We interpret QSS mathematically as a singular perturbation setting to which the classical theorems of Tikhonov and Fenichel apply. Based on a special decomposition of the fast part of the equation, we obtain an explicit formula for a reduced system, defined on the slow manifold (which is a subset of an algebraic variety). Moreover we determine appropriate initial values for the reduced system, which correspond to first integrals of the fast subsystem. These first integrals may not be obtainable in closed form, but locally Taylor expansions are available. We give several examples and applications, and we discuss in detail the separation of a system into fast and slow reactions. It turns out that methods and results from (algorithmic) commutative algebra and algebraic geometry are useful tools for QSS reduction.
Similar content being viewed by others
References
R.C. Aiken (ed.), Stiff Computation (Oxford University Press, New York, 1985)
Y.N. Bibikov, Local Theory of Nonlinear Analytic Ordinary Differential Equations. Lecture Notes in Mathematics 702 (Springer, Berlin, 1979)
F. Boulier, F. Lemaire, M. Moreno Maza, Reduction of chemical reaction systems using algebraic elimination (2011) (Preprint)
D. Bothe, Instantaneous limits of reversible chemical reactions in presence of macroscopic convection. J. Differ. Equ. 193, 27–48 (2003)
M.A. Burke, P.K. Maini, J.D. Murray, On the kinetics of suicide substrates. Biophys. Chem. 37, 81–90 (1990)
D.A. Cox, J. Little, D. O’Shea, Ideals, Varieties, and Algorithms. Undergraduate Texts in Mathematics (Springer, New York, 2007)
W. Decker, G.-M. Greuel, G. Pfister, H. Schönemann, Singular 3-1-3—A Computer Algebra System for Polynomial Computations. http://www.singular.uni-kl.de (2011)
W. Decker, Ch. Lossen, Computing in Algebraic Geometry. Algorithms and Computation in Mathematics 16 (Springer, Berlin, 2006)
M. Feinberg, The existence and uniqueness of steady states for a class of chemical reaction networks. Arch. Ration. Mech. Anal. 132, 311–370 (1995)
N. Fenichel, Geometric singular perturbation theory for ordinary differential equations. J. Differ. Equ. 31(1), 53–98 (1979)
R.J. Field, R.M. Noyes, Oscillations in chemical systems, IV. Limit cycle bahavior in a real chemical reaction. J. Chem. Phys. 60, 1877–1884 (1974)
A. Goeke, Reduktion und asymptotische Reduktion von Reaktionsgleichungen. Doctoral dissertation, RWTH Aachen (2013)
A. Goeke, C. Schilli, S. Walcher, E. Zerz, A note on the kinetics of suicide substrates. J. Math. Chem. 50, 1373–1377 (2012)
A. Goeke, C. Schilli, S. Walcher, E. Zerz, Computing quasi-steady state reductions. J. Math. Chem. 50, 1495–1513 (2012)
A. Goeke, S. Walcher, Quasi-steady state: Searching for and utilizing small parameters, in Recent Trends in Dynamical Systems, Proceedings of a Conference in Honor of Jürgen Scheurle, pp. 153–178. Springer Proceedings in Mathematics & Statistics 35 (Springer, New York, 2013)
D.A. Goussis, Quasi steady state and partial equilibrium approximations: their relation and their validity. Combust. Theory Model. 16(5), 869–926 (2012)
F.G. Heineken, H.M. Tsuchiya, R. Aris, On the mathematical status of the pseudo-steady state hypothesis of biochemical kinetics. Math. Biosci. 1, 95–113 (1967)
F.C. Hoppensteadt, Singular perturbations on the infinite interval. Trans. Am. Math. Soc. 123, 521–535 (1966)
F. Horn, R. Jackson, General mass action kinetics. Arch. Ration. Mech. Anal. 47, 81–116 (1972)
J.E. Humphreys, Linear Algebraic Groups (Springer, New York, 1981)
H.G. Kaper, T.J. Kaper, Asymptotic analysis of two reduction methods for systems of chemical reactions. Phys. D 165, 66–93 (2002)
S.H. Lam, D.A. Goussis, The CSP method for simplifying kinetics. Int. J. Chem. Kinet. 26, 461–486 (1994)
C.H. Lee, H.G. Othmer, A multi-time-scale analysis of chemical reaction networks: I. Deterministic systems. J. Math. Biol. 60, 387–450 (2009)
J.W. Milnor, Topology from the Differentiable Viewpoint (Princeton University Press, Princeton, 1997)
J.D. Murray, Mathematical Biology. I. An Introduction, 3rd edn. (Springer, New York, 2002)
L. Noethen, S. Walcher, Tikhonov’s theorem and quasi-steady state. Discret. Contin. Dyn. Syst. Ser. B 16(3), 945–961 (2011)
I. Prigogine, R. Lefever, Symmetry breaking instabilities in dissipative structures. J. Chem. Phys. 48, 1695–1700 (1968)
M. Schauer, R. Heinrich, Quasi-steady-state approximation in the mathematical modeling of biochemical networks. Math. Biosci. 65, 155–170 (1983)
L.A. Segel, M. Slemrod, The quasi-steady-state assumption: a case study in perturbation. SIAM Rev. 31, 446–477 (1989)
I.R. Shafarevich, Basic Algebraic Geometry (Springer, New York, 1977)
M. Stiefenhofer, Quasi-steady-state approximation for chemical reaction networks. J. Math. Biol. 36, 593–609 (1998)
N. Tatsunami, M. Yago, M. Hosoe, Kinetics of suicide substrates. Steady state treatments and computer-aided exact solutions. Biochem. Biophys. Acta 662, 226–235 (1981)
A.N. Tikhonov, Systems of differential equations containing a small parameter multiplying the derivative. Math. Sb. 31, 575–586 (1952). (in Russian)
F. Verhulst, Methods and Applications of Singular Perturbations. Boundary Layers and Multiple Timescale Dynamics (Springer, New York, 2005)
Acknowledgments
The first named author was supported by the DFG Research Training Group “Experimental and constructive algebra”. The present paper is based on her doctoral dissertations [16].
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Goeke, A., Walcher, S. A constructive approach to quasi-steady state reductions. J Math Chem 52, 2596–2626 (2014). https://doi.org/10.1007/s10910-014-0402-5
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10910-014-0402-5