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October  2011, 16(3): 945-961. doi: 10.3934/dcdsb.2011.16.945

Tikhonov's theorem and quasi-steady state

Lena Noethen 1, and  Sebastian Walcher 1,

1. 

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

Received  July 2010 Revised  March 2011 Published  June 2011

There exists a systematic approach to asymptotic properties for quasi-steady state phenomena via the classical theory of Tikhonov and Fenichel. This observation allows, on the one hand, to settle convergence issues, which are far from trivial in asymptotic expansions. On the other hand, even if one takes convergence for granted, the approach yields a natural way to compute a reduced system on the slow manifold, with a reduced equation that is frequently simpler than the one obtained by the ad hoc approach. In particular, the reduced system is always rational. The paper includes a discussion of necessary and sufficient conditions for applicability of Tikhonov's and Fenichel's theorems, computational issues and a direct determination of the reduced system. The results are applied to several relevant examples.
Keywords: enzyme kinetics., Singular perturbations, Michaelis-Menten reaction.
Mathematics Subject Classification: Primary: 34E15, 92C45; Secondary: 80A3.
Citation: Lena Noethen, Sebastian Walcher. Tikhonov's theorem and quasi-steady state. Discrete & Continuous Dynamical Systems - B, 2011, 16 (3) : 945-961. doi: 10.3934/dcdsb.2011.16.945
References:
[1]

P. Atkins and J. de Paula, "Physical Chemistry,'' 8th edition, Oxford Univ. Press, Oxford, 2006. Google Scholar

[2]

J. M. Berg, J. L. Tymovzko and L. Stryer, "Biochemistry: International Edition,'' 6th edition, Palgrave Macmillan, New York, 2006. Google Scholar

[3]

Yu. N. Bibikov, "Local Theory of Nonlinear Analytic Ordinary Differential Equations,'' Lecture Notes in Mathematics, 702, Springer, Berlin, 1979.  Google Scholar

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J. A. M. Borghans, R. J. de Boer and L. A. Segel, Extending the quasi-steady state approximation by changing variables, Bull. Math. Biol., 58 (1996), 43-63. doi: 10.1007/BF02458281.  Google Scholar

[5]

S. Cunha Orfao, G. Jank, K. Mottaghy, S. Walcher and E. Zerz, Qualitative properties and stabilizability of a model for blood thrombin formation, J. Math. Anal. Appl., 346 (2008), 218-226. doi: 10.1016/j.jmaa.2008.05.060.  Google Scholar

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P. Duchêne and P. Rouchon, Kinetic scheme reduction via geometric singular perturbation techniques, Chem. Engineering Sci., 51 (1996), 4461-4472. Google Scholar

[7]

N. Fenichel, Geometric singular perturbation theory for ordinary differential equations, J. Diff. Eqs., 31 (1979), 53-98. doi: 10.1016/0022-0396(79)90152-9.  Google Scholar

[8]

S. J. Fraser and M. R. Roussel, Phase plane geometries in enzyme kinetics, Can. J. Chem., 72 (1993), 800-812. doi: 10.1139/v94-107.  Google Scholar

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W. Gröbner and H. Knapp (Eds.), "Contributions to the Method of Lie Series,'' Bibliographisches Institut, Mannheim, 1967  Google Scholar

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F. C. Hoppensteadt, Singular perturbations on the infinite interval, Trans. Amer. Math. Soc., 123 (1966), 521-535. doi: 10.1090/S0002-9947-1966-0194693-9.  Google Scholar

[11]

C. K. R. T. Jones, Geometric singular perturbation theory, in "Dynamical Systems, Montecatini Terme, 1994,'' (ed. R. Johnson), Lecture Notes in Mathematics, 1609, Springer, Berlin, (1995), 44 - 118.  Google Scholar

[12]

H. G. Kaper and T. J. Kaper, Asymptotic analysis of two reduction methods for systems of chemical reactions, Physica D, 165 (2002), 66-93. doi: 10.1016/S0167-2789(02)00386-X.  Google Scholar

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J. Keener and J. Sneyd, "Mathematical Physiology,'' Springer, New York, 1998.  Google Scholar

[14]

S. Lang, "Algebra,'' 2nd edition, Addison-Wesley, Reading, 1984.  Google Scholar

[15]

L. Michaelis and M. L. Menten, Die Kinetik der Invertinwirkung, Biochem. Z., 49 (1913), 333-369. Google Scholar

[16]

J. D. Murray, "Mathematical Biology,'' 2nd edition Springer, New York, 1993. doi: 10.1007/b98869.  Google Scholar

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L. Noethen, "Quasi-Stationarität und fast-invariante Mengen gewöhnlicher Differentialgleichungen,'' Ph.D. thesis, RWTH Aachen, 2008. Google Scholar

[18]

L. Noethen and S. Walcher, Quasi-steady state in the Michaelis-Menten system, Nonlin. Analysis Real World Appl., 8 (2007), 1512-1535. doi: 10.1016/j.nonrwa.2006.08.004.  Google Scholar

[19]

L. Noethen and S. Walcher, Quasi-steady state and nearly invariant sets, SIAM J. Appl. Math., 70 (2009), 1341-1363. doi: 10.1137/090758180.  Google Scholar

[20]

M. Schauer and R. Heinrich, Analysis of the quasi-steady-state approximation for an enzymatic one-substrate reaction, J. Theor. Biol., 79 (1979), 425-442. doi: 10.1016/0022-5193(79)90235-2.  Google Scholar

[21]

M. Schauer and R. Heinrich, Quasi-steady-state approximation in the mathematical modeling of biochemical reaction networks, Math. Biosci., 65 (1983), 155-170. doi: 10.1016/0025-5564(83)90058-5.  Google Scholar

[22]

S. Schnell and P. K. Maini, Enzyme kinetics at high enzyme concentration, Bull. Math. Biol., 62 (2000), 483-499. doi: 10.1006/bulm.1999.0163.  Google Scholar

[23]

L. A. Segel and M. Slemrod, The quasi-steady-state assumption: A case study in perturbation, SIAM Review, 31 (1989), 446-477. doi: 10.1137/1031091.  Google Scholar

[24]

M. Seshadri and G. Fritzsch, Analytical solutions of a simple enzyme kinetic problem by a perturbative procedure, Biophys. Struct. Mech., 6 (1980), 111-123. doi: 10.1007/BF00535748.  Google Scholar

[25]

M. Stiefenhofer, Quasi-steady-state approximation for chemical reaction networks, J. Math. Biol., 36 (1998), 593-609. doi: 10.1007/s002850050116.  Google Scholar

[26]

A. N. Tikhonov, Systems of differential equations containing a small parameter multiplying the derivative (in Russian), Mat. Sb., 31 (1952), 575-586.  Google Scholar

[27]

A. R. Tzafriri and E. R. Edelman, The total quasi-steady-state approximation is valid for reversible enzyme kinetics, J. Theoret. Biol., 226 (2004), 303-313. doi: 10.1016/j.jtbi.2003.09.006.  Google Scholar

[28]

A. B. Vasil'eva, Asymptotic behavior of solutions to certain problems involving nonlinear differential equations containing a small parameter multiplying the highest derivatives, Russ. Math. Surveys, 18 (1963), 13-84. doi: 10.1070/RM1963v018n03ABEH001137.  Google Scholar

[29]

F. Verhulst, "Methods and Applications of Singular Perturbations,'' Springer, Berlin, 2005. doi: 10.1007/0-387-28313-7.  Google Scholar

show all references

References:
[1]

P. Atkins and J. de Paula, "Physical Chemistry,'' 8th edition, Oxford Univ. Press, Oxford, 2006. Google Scholar

[2]

J. M. Berg, J. L. Tymovzko and L. Stryer, "Biochemistry: International Edition,'' 6th edition, Palgrave Macmillan, New York, 2006. Google Scholar

[3]

Yu. N. Bibikov, "Local Theory of Nonlinear Analytic Ordinary Differential Equations,'' Lecture Notes in Mathematics, 702, Springer, Berlin, 1979.  Google Scholar

[4]

J. A. M. Borghans, R. J. de Boer and L. A. Segel, Extending the quasi-steady state approximation by changing variables, Bull. Math. Biol., 58 (1996), 43-63. doi: 10.1007/BF02458281.  Google Scholar

[5]

S. Cunha Orfao, G. Jank, K. Mottaghy, S. Walcher and E. Zerz, Qualitative properties and stabilizability of a model for blood thrombin formation, J. Math. Anal. Appl., 346 (2008), 218-226. doi: 10.1016/j.jmaa.2008.05.060.  Google Scholar

[6]

P. Duchêne and P. Rouchon, Kinetic scheme reduction via geometric singular perturbation techniques, Chem. Engineering Sci., 51 (1996), 4461-4472. Google Scholar

[7]

N. Fenichel, Geometric singular perturbation theory for ordinary differential equations, J. Diff. Eqs., 31 (1979), 53-98. doi: 10.1016/0022-0396(79)90152-9.  Google Scholar

[8]

S. J. Fraser and M. R. Roussel, Phase plane geometries in enzyme kinetics, Can. J. Chem., 72 (1993), 800-812. doi: 10.1139/v94-107.  Google Scholar

[9]

W. Gröbner and H. Knapp (Eds.), "Contributions to the Method of Lie Series,'' Bibliographisches Institut, Mannheim, 1967  Google Scholar

[10]

F. C. Hoppensteadt, Singular perturbations on the infinite interval, Trans. Amer. Math. Soc., 123 (1966), 521-535. doi: 10.1090/S0002-9947-1966-0194693-9.  Google Scholar

[11]

C. K. R. T. Jones, Geometric singular perturbation theory, in "Dynamical Systems, Montecatini Terme, 1994,'' (ed. R. Johnson), Lecture Notes in Mathematics, 1609, Springer, Berlin, (1995), 44 - 118.  Google Scholar

[12]

H. G. Kaper and T. J. Kaper, Asymptotic analysis of two reduction methods for systems of chemical reactions, Physica D, 165 (2002), 66-93. doi: 10.1016/S0167-2789(02)00386-X.  Google Scholar

[13]

J. Keener and J. Sneyd, "Mathematical Physiology,'' Springer, New York, 1998.  Google Scholar

[14]

S. Lang, "Algebra,'' 2nd edition, Addison-Wesley, Reading, 1984.  Google Scholar

[15]

L. Michaelis and M. L. Menten, Die Kinetik der Invertinwirkung, Biochem. Z., 49 (1913), 333-369. Google Scholar

[16]

J. D. Murray, "Mathematical Biology,'' 2nd edition Springer, New York, 1993. doi: 10.1007/b98869.  Google Scholar

[17]

L. Noethen, "Quasi-Stationarität und fast-invariante Mengen gewöhnlicher Differentialgleichungen,'' Ph.D. thesis, RWTH Aachen, 2008. Google Scholar

[18]

L. Noethen and S. Walcher, Quasi-steady state in the Michaelis-Menten system, Nonlin. Analysis Real World Appl., 8 (2007), 1512-1535. doi: 10.1016/j.nonrwa.2006.08.004.  Google Scholar

[19]

L. Noethen and S. Walcher, Quasi-steady state and nearly invariant sets, SIAM J. Appl. Math., 70 (2009), 1341-1363. doi: 10.1137/090758180.  Google Scholar

[20]

M. Schauer and R. Heinrich, Analysis of the quasi-steady-state approximation for an enzymatic one-substrate reaction, J. Theor. Biol., 79 (1979), 425-442. doi: 10.1016/0022-5193(79)90235-2.  Google Scholar

[21]

M. Schauer and R. Heinrich, Quasi-steady-state approximation in the mathematical modeling of biochemical reaction networks, Math. Biosci., 65 (1983), 155-170. doi: 10.1016/0025-5564(83)90058-5.  Google Scholar

[22]

S. Schnell and P. K. Maini, Enzyme kinetics at high enzyme concentration, Bull. Math. Biol., 62 (2000), 483-499. doi: 10.1006/bulm.1999.0163.  Google Scholar

[23]

L. A. Segel and M. Slemrod, The quasi-steady-state assumption: A case study in perturbation, SIAM Review, 31 (1989), 446-477. doi: 10.1137/1031091.  Google Scholar

[24]

M. Seshadri and G. Fritzsch, Analytical solutions of a simple enzyme kinetic problem by a perturbative procedure, Biophys. Struct. Mech., 6 (1980), 111-123. doi: 10.1007/BF00535748.  Google Scholar

[25]

M. Stiefenhofer, Quasi-steady-state approximation for chemical reaction networks, J. Math. Biol., 36 (1998), 593-609. doi: 10.1007/s002850050116.  Google Scholar

[26]

A. N. Tikhonov, Systems of differential equations containing a small parameter multiplying the derivative (in Russian), Mat. Sb., 31 (1952), 575-586.  Google Scholar

[27]

A. R. Tzafriri and E. R. Edelman, The total quasi-steady-state approximation is valid for reversible enzyme kinetics, J. Theoret. Biol., 226 (2004), 303-313. doi: 10.1016/j.jtbi.2003.09.006.  Google Scholar

[28]

A. B. Vasil'eva, Asymptotic behavior of solutions to certain problems involving nonlinear differential equations containing a small parameter multiplying the highest derivatives, Russ. Math. Surveys, 18 (1963), 13-84. doi: 10.1070/RM1963v018n03ABEH001137.  Google Scholar

[29]

F. Verhulst, "Methods and Applications of Singular Perturbations,'' Springer, Berlin, 2005. doi: 10.1007/0-387-28313-7.  Google Scholar

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