Tikhonov's theorem and quasi-steady state

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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

Abstract
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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 and 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.
[2]	J. M. Berg, J. L. Tymovzko and L. Stryer, "Biochemistry: International Edition,'' 6th edition, Palgrave Macmillan, New York, 2006.
[3]	Yu. N. Bibikov, "Local Theory of Nonlinear Analytic Ordinary Differential Equations,'' Lecture Notes in Mathematics, 702, Springer, Berlin, 1979.
[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.
[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.
[6]	P. Duchêne and P. Rouchon, Kinetic scheme reduction via geometric singular perturbation techniques, Chem. Engineering Sci., 51 (1996), 4461-4472.
[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.
[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.
[9]	W. Gröbner and H. Knapp (Eds.), "Contributions to the Method of Lie Series,'' Bibliographisches Institut, Mannheim, 1967
[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.
[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.
[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.
[13]	J. Keener and J. Sneyd, "Mathematical Physiology,'' Springer, New York, 1998.
[14]	S. Lang, "Algebra,'' 2nd edition, Addison-Wesley, Reading, 1984.
[15]	L. Michaelis and M. L. Menten, Die Kinetik der Invertinwirkung, Biochem. Z., 49 (1913), 333-369.
[16]	J. D. Murray, "Mathematical Biology,'' 2nd edition Springer, New York, 1993. doi: 10.1007/b98869.
[17]	L. Noethen, "Quasi-Stationarität und fast-invariante Mengen gewöhnlicher Differentialgleichungen,'' Ph.D. thesis, RWTH Aachen, 2008.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[25]	M. Stiefenhofer, Quasi-steady-state approximation for chemical reaction networks, J. Math. Biol., 36 (1998), 593-609. doi: 10.1007/s002850050116.
[26]	A. N. Tikhonov, Systems of differential equations containing a small parameter multiplying the derivative (in Russian), Mat. Sb., 31 (1952), 575-586.
[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.
[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.
[29]	F. Verhulst, "Methods and Applications of Singular Perturbations,'' Springer, Berlin, 2005. doi: 10.1007/0-387-28313-7.

show all references

References:

[1]	P. Atkins and J. de Paula, "Physical Chemistry,'' 8th edition, Oxford Univ. Press, Oxford, 2006.
[2]	J. M. Berg, J. L. Tymovzko and L. Stryer, "Biochemistry: International Edition,'' 6th edition, Palgrave Macmillan, New York, 2006.
[3]	Yu. N. Bibikov, "Local Theory of Nonlinear Analytic Ordinary Differential Equations,'' Lecture Notes in Mathematics, 702, Springer, Berlin, 1979.
[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.
[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.
[6]	P. Duchêne and P. Rouchon, Kinetic scheme reduction via geometric singular perturbation techniques, Chem. Engineering Sci., 51 (1996), 4461-4472.
[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.
[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.
[9]	W. Gröbner and H. Knapp (Eds.), "Contributions to the Method of Lie Series,'' Bibliographisches Institut, Mannheim, 1967
[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.
[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.
[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.
[13]	J. Keener and J. Sneyd, "Mathematical Physiology,'' Springer, New York, 1998.
[14]	S. Lang, "Algebra,'' 2nd edition, Addison-Wesley, Reading, 1984.
[15]	L. Michaelis and M. L. Menten, Die Kinetik der Invertinwirkung, Biochem. Z., 49 (1913), 333-369.
[16]	J. D. Murray, "Mathematical Biology,'' 2nd edition Springer, New York, 1993. doi: 10.1007/b98869.
[17]	L. Noethen, "Quasi-Stationarität und fast-invariante Mengen gewöhnlicher Differentialgleichungen,'' Ph.D. thesis, RWTH Aachen, 2008.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[25]	M. Stiefenhofer, Quasi-steady-state approximation for chemical reaction networks, J. Math. Biol., 36 (1998), 593-609. doi: 10.1007/s002850050116.
[26]	A. N. Tikhonov, Systems of differential equations containing a small parameter multiplying the derivative (in Russian), Mat. Sb., 31 (1952), 575-586.
[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.
[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.
[29]	F. Verhulst, "Methods and Applications of Singular Perturbations,'' Springer, Berlin, 2005. doi: 10.1007/0-387-28313-7.

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