Abstract
Chemical mechanisms for even simple reaction networks involve many highly reactive and short-lived species (intermediates), present in small concentrations, in addition to the main reactants and products, present in larger concentrations. The chemical mechanism also often contains many rate constants whose values are unknown a priori and must be determined from experimental measurements of the large species concentrations. A classic model reduction method known as the quasi-steady-state assumption (QSSA) is often used to eliminate the highly reactive intermediate species and remove the large rate constants that cannot be determined from concentration measurements of the reactants and products. Mathematical analysis based on the QSSA is ubiquitous in modeling enzymatic reactions. In this chapter, we focus attention on the QSSA, how it is “taught” to students of chemistry, biology, and chemical and biological engineering, and how it is “practiced” when researchers confront realistic and complex examples. We describe the main types of difficulties that appear when trying to apply the standard ideas of the QSSA, and propose a new strategy for overcoming them, based on rescaling the reactive intermediate species. First, we prove mathematically that the program taught to beginning students for applying the 100-year-old approach of classic QSSA model reduction cannot be carried out for many of the relevant kinetics problems, and perhaps even most of them. By using Galois theory, we prove that the required algebraic equations cannot be solved for as few as five bimolecular reactions between five species (with three intermediates). We expect that many practitioners have suspected this situation regarding nonsolvability to exist, but we have seen no statement or proof of this fact, especially when the kinetics are restricted to unimolecular and bimolecular reactions. We describe algorithms that can test any mechanism for solvability. We also show that an alternative to solving the QSSA equations, the Horiuti–Temkin theory, also does not work for many examples. Of course, the reduced model (and the full model, for that matter) can be solved numerically, which is the standard approach in practice. The remaining difficulty, however, is how to obtain the values of the large kinetic parameters appearing in the model. These parameters cannot be estimated from measurements of the large-concentration reactants and products. We show here how the concept of rescaling the reactive intermediate species allows the large kinetic parameters to be removed from the parameter estimation problem. In general, the number of parameters that can be removed from the full model is less than or equal to the number of intermediate species. The outcome is a reduced model with a set of rescaled parameters that is often identifiable from routinely available measurements. New and freely available computational software (parest_dae) for estimating the reduced model’s kinetic parameters and confidence intervals is briefly described.
This is a preview of subscription content, log in via an institution.
Buying options
Tax calculation will be finalised at checkout
Purchases are for personal use only
Learn about institutional subscriptionsNotes
- 1.
Note that these are not quite the true parameter values, because we used the full model rather than the reduced model to generate the data.
References
N.H. Abel, Mémoire sur les équations algébriques, où l’on démontre l’impossibilité de la résolution de l’équation générale du cinquième degré in Oeuvres complètes de Niels Henrik Abel, Édition de Christiana, vol. 1 (Gröndahl and Son, Oslo, 1881), pp. 28–33
A.C. Atkinson, A.N. Donev, Optimum Experimental Designs (Oxford University Press, New York, 1992)
Y. Bard, Nonlinear Parameter Estimation (Academic, New York, 1974)
M. Bodenstein, Eine Theorie der photochemischen Reaktionsgeschwindigkeiten. Z. Phys. Chem. 85, 329–397 (1913)
D.L. Chapman, L.K. Underhill, The interaction of chlorine and hydrogen. The influence of mass. J. Chem. Soc. Trans. 103, 496–508 (1913)
A. Ciliberto, F. Capuani, J.J. Tyson, Modeling networks of coupled enzymatic reactions using the total quasi-steady state approximation. PLoS Comput. Biol. 3(3) e45 (2007)
D. Cox, J. Little, D. O’Shea, Ideals, Varieties, and Algorithms: An Introduction to Computational Algebraic Geometry and Commutative Algebra (Springer, New York, 2006)
G. Craciun, C. Pantea, Identifiability of chemical reaction networks. J. Math. Chem. 44, 244–259 (2007)
A. Goldbeter, D.E. Koshland, An amplified sensitivity arising from covalent modification in biological systems. Proc. Natl. Acad. Sci. U.S.A. 78(11), 6840–6844 (1981)
S. Hanson, S. Schnell, Reactant stationary approximation in enzyme kinetics. J. Phys. Chem. A 112(37), 8654–8658 (2008)
A.V. Hill, The possible effects of the aggregation of the molecules of hemoglobin on its dissociation curves. J. Physiol. 40, 4–7 (1910)
A.C. Hindmarsh, P.N. Brown, K.E. Grant, S.L. Lee, R. Serban, D.E. Shumaker, C.S. Woodward, SUNDIALS: suite of nonlinear and differential/algebraic equation solvers. ACM Trans. Math. Softw. 31(3), 363–396 (2005)
J. Horiuti, T. Nakamura, Stoichiometric number and the theory of steady reaction. Z. Phys. Chem. 11, 358–365 (1957)
M. Isaacs, Algebra: A Graduate Course (Brooks Cole, Pacific Grove, 1993)
A.G. Khovanskii, On solvability and unsolvability of equations in explicit form. Russ. Math. Surv. 59(4), 661–736 (2004)
S.H. Lam, D.A. Goussis, The CSP method for simplifying kinetics. Int. J. Chem. Kinet. 26, 461–486 (1994)
Maple is a division of Waterloo Maple, Inc. (2007)
J.D. Murray, Mathematical Biology: I. An Introduction (Springer, New York, 2002)
J.B. Rawlings, J.G. Ekerdt, Chemical Reactor Analysis and Design Fundamentals (Nob Hill Publishing, Madison, 2004)
S. Schnell, P.K. Maini, A century of enzyme kinetics: reliability of the K M and v max estimates. Comments Theor. Biol. 8, 169–187 (2003)
L.A. Segel, M. Slemrod, The quasi-steady state assumption: a case study in perturbation. SIAM Rev. 31(3), 446–477 (1989)
M.I. Temkin, The kinetics of some industrial heterogeneous catalytic reactions. Adv. Catal. 28, 173–291 (1979)
T. Turanyi, A.S. Tomlin, M.J. Pilling, On the error of the quasi-steady-state approximation. J. Phys. Chem. 97(1), 163–172 (1993)
J.J. Tyson, K.C. Chen, B. Novak, Sniffers, buzzers, toggles and blinkers: dynamics of regulatory and signaling pathways in the cell. Curr. Opin. Cell Biol. 15, 221–231 (2003)
M.R. Wright, An Introduction to Chemical Kinetics (Wiley, Chichester/Hoboken, 2004)
A. Zagaris, H.G. Kaper, T.J. Kaper, Analysis of the computational singular perturbation reduction method for chemical kinetics. J. Nonlinear Sci. 14, 59–91 (2004)
A. Zagaris, H.G. Kaper, T.J. Kaper, Fast and slow dynamics for the computational singular perturbation method. Multiscale Model. Simul. 2(4), 613–638 (2004)
Acknowledgements
We thank Andrei Căldăraru and Lev Borisov for useful suggestions and discussions. The work of CP and GC was partially supported by the DOE BACTER Institute, the National Science Foundation, and NIH grant R01GM86881.
Author information
Authors and Affiliations
Corresponding author
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2014 Springer-Verlag Berlin Heidelberg
About this chapter
Cite this chapter
Pantea, C., Gupta, A., Rawlings, J.B., Craciun, G. (2014). The QSSA in Chemical Kinetics: As Taught and as Practiced. In: Jonoska, N., Saito, M. (eds) Discrete and Topological Models in Molecular Biology. Natural Computing Series. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-40193-0_20
Download citation
DOI: https://doi.org/10.1007/978-3-642-40193-0_20
Published:
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-642-40192-3
Online ISBN: 978-3-642-40193-0
eBook Packages: Computer ScienceComputer Science (R0)