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January  2020, 25(1): 1-29. doi: 10.3934/dcdsb.2019170

Singular perturbations and scaling

Christian Lax and  Sebastian Walcher ,

Mathematik A, RWTH Aachen, 52056 Aachen, Germany

* Corresponding author

Received  October 2018 Revised  February 2019 Published  July 2019

Scaling transformations involving a small parameter (degenerate scalings) are frequently used for ordinary differential equations that model chemical reaction networks. They are motivated by quasi-steady state (QSS) of certain chemical species, and ideally lead to slow-fast systems for singular perturbation reductions, in the sense of Tikhonov and Fenichel. In the present paper we discuss properties of such scaling transformations, with regard to their applicability as well as to their determination. Transformations of this type are admissible only when certain consistency conditions are satisfied, and they lead to singular perturbation scenarios only if additional conditions hold, including a further consistency condition on initial values. Given these consistency conditions, two scenarios occur. The first (which we call standard) is well known and corresponds to a classical quasi-steady state (QSS) reduction. Here, scaling may actually be omitted because there exists a singular perturbation reduction for the unscaled system, with a coordinate subspace as critical manifold. For the second (nonstandard) scenario scaling is crucial. Here one may obtain a singular perturbation reduction with the slow manifold having dimension greater than expected from the scaling. For parameter dependent systems we consider the problem to find all possible scalings, and we show that requiring the consistency conditions allows their determination. This lays the groundwork for algorithmic approaches, to be taken up in future work. In the final section we consider some applications. In particular we discuss relevant nonstandard reductions of certain reaction-transport systems.

Keywords: Reaction networks, dimension reduction, invariant sets, critical manifold.
Mathematics Subject Classification: Primary: 92C45, 34E15; Secondary: 80A30, 13P10.
Citation: Christian Lax, Sebastian Walcher. Singular perturbations and scaling. Discrete & Continuous Dynamical Systems - B, 2020, 25 (1) : 1-29. doi: 10.3934/dcdsb.2019170
References:
[1]

E. Feliu and C. Wiuf, Variable elimination in chemical reaction networks with mass-action kinetics, SIAM J. Appl. Math., 72 (2012), 959-981.  doi: 10.1137/110847305.  Google Scholar

[2]

E. Feliu and C. Wiuf, Variable elimination in post-translational modification reaction networks with mass-action kinetics, J. Math. Biol., 66 (2013), 281-310.  doi: 10.1007/s00285-012-0510-4.  Google Scholar

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N. Fenichel, Persistence and smoothness of invariant manifolds for flows, Indiana Univ. Math. J., 21 (1971/1972), 193-226.  doi: 10.1512/iumj.1972.21.21017.  Google Scholar

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

[6]

M. Frank, C. Lax, S. Walcher and O. Wittich, Quasi-steady state reduction for the Michaelis-Menten reaction-diffusion system, J. Math. Chem., 56 (2018), 1759-1781.  doi: 10.1007/s10910-018-0891-8.  Google Scholar

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F. R. Gantmacher, Applications of the Theory of Matrices, Evanusa Interscience Publishers, Inc., New York; Interscience Publishers Ltd., London, 1959.  Google Scholar

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A. Goeke and C. Lax, Quasi-steady state reduction for compartmental systems, Physica D, 327 (2016), 1-12.  doi: 10.1016/j.physd.2016.04.013.  Google Scholar

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A. Goeke, C. Schilli, S. Walcher and E. Zerz, Computing quasi-steady state reductions, J. Math. Chem., 50 (2012), 1495-1513.  doi: 10.1007/s10910-012-9985-x.  Google Scholar

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A. Goeke, Reduktion und Asymptotische Reduktion von Reaktionsgleichungen, Doctoral dissertation, RWTH Aachen, 2013. Google Scholar

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A. Goeke and S. Walcher, A constructive approach to quasi-steady state reduction, J. Math. Chem., 52 (2014), 2596-2626.  doi: 10.1007/s10910-014-0402-5.  Google Scholar

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A. Goeke, S. Walcher and E. Zerz, Determining "small parameters" for quasi-steady state, J. Differential Equations, 259 (2015), 1149-1180.  doi: 10.1016/j.jde.2015.02.038.  Google Scholar

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A. Goeke, S. Walcher and E. Zerz, Classical quasi-steady state reduction – A mathematical characterization, Physica D, 345 (2017), 11-26.  doi: 10.1016/j.physd.2016.12.002.  Google Scholar

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J. Gunawardena, A linear framework for time-scale separation in nonlinear biochemical systems, PLoS ONE, 7 (2012), e36321. Google Scholar

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F. G. Heineken, H. M. Tsuchiya and R. Aris, On the mathematical status of the pseudo-steady state hypothesis of biochemical kinetics, Math. Biosci., 1 (1967), 95-113.  doi: 10.1016/0025-5564(67)90029-6.  Google Scholar

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

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M. Korc and M. Feinberg, Multiple steady states as a source of pattern formation in complex multicell chemical systems, Chem. Eng. Sc., 48 (1993), 4143-4151.  doi: 10.1016/0009-2509(93)80260-W.  Google Scholar

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C. Lax, K. Seliger and S. Walcher, A coordinate-independent version of Hoppensteadt's convergence theorem, Qual.Theory Dyn. Syst., 17 (2018), 7-28.  doi: 10.1007/s12346-017-0235-2.  Google Scholar

[22]

M. Marcondes de Freitas, E. Feliu and C. Wiuf, Intermediates, catalysts, persistence, and boundary steady states, J. Math. Biol., 74 (2017), 887-932.  doi: 10.1007/s00285-016-1046-9.  Google Scholar

[23]

J. D. Murray, Mathematical Biology. I. An Introduction, 3rd edition, Springer-Verlag, New York, 2002.  Google Scholar

[24]

J. Nestruev, Smooth Manifolds and Observables, Springer-Verlag, New York, 2003.  Google Scholar

[25]

V. Noel, D. Grigoriev, S. Vakulenko and O. Radulescu, Tropicalization and tropical equilibrium of chemical reactions, in Tropical and Idempotent Mathematics and Applications (eds. G.L. Litvinov and S.N. Sergeev), Contemporary Math., Amer. Math. Soc., Providence, 616 (2014), 261–275. doi: 10.1090/conm/616/12316.  Google Scholar

[26]

L. Noethen and S. Walcher, Tikhonov's theorem and quasi-steady state, Discrete Contin. Dyn. Syst. Ser. B, 16 (2011), 945-961.  doi: 10.3934/dcdsb.2011.16.945.  Google Scholar

[27]

O. Radulescu, S. Vakulenko and D. Grigoriev, Model reduction of biochemical reactions networks by tropical analysis methods, Math. Model. Nat. Phenom., 10 (2015), 124-138.  doi: 10.1051/mmnp/201510310.  Google Scholar

[28]

M. Saez, C. Wiuf and E. Feliu, Graphical reduction of reaction networks by linear elimination of species, J. Math. Biol., 74 (2017), 195-237.  doi: 10.1007/s00285-016-1028-y.  Google Scholar

[29]

S. S. Samal, D. Grigoriev, H. Fröhlich and O. Radulescu, Analysis of reaction network systems using tropical geometry, in Computer Algebra in Scientific ComputingComputer Algebra in Scientific Computing. 17th International Workshop, CASC 2015 (eds. V.P. Gerdt, W. Koepf, W.M. Seiler and E.V. Vorozhtsov), Lecture Notes in Computer Science, 9301 (2015), Springer-Verlag, Cham, 424–439. Google Scholar

[30]

S. S. Samal, D. Grigoriev, H. Fröhlich, A. Weber and O. Radulescu, A geometric method for model reduction of biochemical networks with polynomial rate functions, Bull. Math. Biol., 77 (2015), 2180-2211.  doi: 10.1007/s11538-015-0118-0.  Google Scholar

[31]

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

[32]

A. Shapiro and F. Horn, On the possibility of sustained oscillations, multiple steady states, and asymmetric steady states in multicell reaction systems, Math. Biosci., 44 (1979), 19-39.  doi: 10.1016/0025-5564(79)90027-0.  Google Scholar

[33]

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

[34]

F. Verhulst, Methods and Applications of Singular Perturbations. Boundary Layers and Multiple Timescale Dynamics, Springer-Verlag, New York, 2005. doi: 10.1007/0-387-28313-7.  Google Scholar

show all references

References:
[1]

E. Feliu and C. Wiuf, Variable elimination in chemical reaction networks with mass-action kinetics, SIAM J. Appl. Math., 72 (2012), 959-981.  doi: 10.1137/110847305.  Google Scholar

[2]

E. Feliu and C. Wiuf, Variable elimination in post-translational modification reaction networks with mass-action kinetics, J. Math. Biol., 66 (2013), 281-310.  doi: 10.1007/s00285-012-0510-4.  Google Scholar

[3]

E. Feliu and C. Wiuf, Simplifying biochemical models with intermediate species, J. Roy. Soc. Interface, 10 (2013), 20130484. doi: 10.1098/rsif.2013.0484.  Google Scholar

[4]

N. Fenichel, Persistence and smoothness of invariant manifolds for flows, Indiana Univ. Math. J., 21 (1971/1972), 193-226.  doi: 10.1512/iumj.1972.21.21017.  Google Scholar

[5]

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

[6]

M. Frank, C. Lax, S. Walcher and O. Wittich, Quasi-steady state reduction for the Michaelis-Menten reaction-diffusion system, J. Math. Chem., 56 (2018), 1759-1781.  doi: 10.1007/s10910-018-0891-8.  Google Scholar

[7]

F. R. Gantmacher, Applications of the Theory of Matrices, Evanusa Interscience Publishers, Inc., New York; Interscience Publishers Ltd., London, 1959.  Google Scholar

[8]

A. Goeke and C. Lax, Quasi-steady state reduction for compartmental systems, Physica D, 327 (2016), 1-12.  doi: 10.1016/j.physd.2016.04.013.  Google Scholar

[9]

A. Goeke, C. Schilli, S. Walcher and E. Zerz, Computing quasi-steady state reductions, J. Math. Chem., 50 (2012), 1495-1513.  doi: 10.1007/s10910-012-9985-x.  Google Scholar

[10]

A. Goeke, Reduktion und Asymptotische Reduktion von Reaktionsgleichungen, Doctoral dissertation, RWTH Aachen, 2013. Google Scholar

[11]

A. Goeke and S. Walcher, A constructive approach to quasi-steady state reduction, J. Math. Chem., 52 (2014), 2596-2626.  doi: 10.1007/s10910-014-0402-5.  Google Scholar

[12]

A. Goeke, S. Walcher and E. Zerz, Determining "small parameters" for quasi-steady state, J. Differential Equations, 259 (2015), 1149-1180.  doi: 10.1016/j.jde.2015.02.038.  Google Scholar

[13]

A. Goeke, S. Walcher and E. Zerz, Classical quasi-steady state reduction – A mathematical characterization, Physica D, 345 (2017), 11-26.  doi: 10.1016/j.physd.2016.12.002.  Google Scholar

[14]

D. A. Goussis, Quasi-steady state and partial equilibrium approximations: Their relation and their validity, Combustion Theory and Modelling, 16 (2012), 869-926.  doi: 10.1080/13647830.2012.680502.  Google Scholar

[15]

J. Gunawardena, A linear framework for time-scale separation in nonlinear biochemical systems, PLoS ONE, 7 (2012), e36321. Google Scholar

[16]

F. G. Heineken, H. M. Tsuchiya and R. Aris, On the mathematical status of the pseudo-steady state hypothesis of biochemical kinetics, Math. Biosci., 1 (1967), 95-113.  doi: 10.1016/0025-5564(67)90029-6.  Google Scholar

[17]

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

[18]

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

[19]

J. Keener and J. Sneyd, Mathematical Physiology I: Cellular Physiology, 2nd edition, Springer-Verlag, New York, 2009. doi: 10.1007/978-0-387-79388-7.  Google Scholar

[20]

M. Korc and M. Feinberg, Multiple steady states as a source of pattern formation in complex multicell chemical systems, Chem. Eng. Sc., 48 (1993), 4143-4151.  doi: 10.1016/0009-2509(93)80260-W.  Google Scholar

[21]

C. Lax, K. Seliger and S. Walcher, A coordinate-independent version of Hoppensteadt's convergence theorem, Qual.Theory Dyn. Syst., 17 (2018), 7-28.  doi: 10.1007/s12346-017-0235-2.  Google Scholar

[22]

M. Marcondes de Freitas, E. Feliu and C. Wiuf, Intermediates, catalysts, persistence, and boundary steady states, J. Math. Biol., 74 (2017), 887-932.  doi: 10.1007/s00285-016-1046-9.  Google Scholar

[23]

J. D. Murray, Mathematical Biology. I. An Introduction, 3rd edition, Springer-Verlag, New York, 2002.  Google Scholar

[24]

J. Nestruev, Smooth Manifolds and Observables, Springer-Verlag, New York, 2003.  Google Scholar

[25]

V. Noel, D. Grigoriev, S. Vakulenko and O. Radulescu, Tropicalization and tropical equilibrium of chemical reactions, in Tropical and Idempotent Mathematics and Applications (eds. G.L. Litvinov and S.N. Sergeev), Contemporary Math., Amer. Math. Soc., Providence, 616 (2014), 261–275. doi: 10.1090/conm/616/12316.  Google Scholar

[26]

L. Noethen and S. Walcher, Tikhonov's theorem and quasi-steady state, Discrete Contin. Dyn. Syst. Ser. B, 16 (2011), 945-961.  doi: 10.3934/dcdsb.2011.16.945.  Google Scholar

[27]

O. Radulescu, S. Vakulenko and D. Grigoriev, Model reduction of biochemical reactions networks by tropical analysis methods, Math. Model. Nat. Phenom., 10 (2015), 124-138.  doi: 10.1051/mmnp/201510310.  Google Scholar

[28]

M. Saez, C. Wiuf and E. Feliu, Graphical reduction of reaction networks by linear elimination of species, J. Math. Biol., 74 (2017), 195-237.  doi: 10.1007/s00285-016-1028-y.  Google Scholar

[29]

S. S. Samal, D. Grigoriev, H. Fröhlich and O. Radulescu, Analysis of reaction network systems using tropical geometry, in Computer Algebra in Scientific ComputingComputer Algebra in Scientific Computing. 17th International Workshop, CASC 2015 (eds. V.P. Gerdt, W. Koepf, W.M. Seiler and E.V. Vorozhtsov), Lecture Notes in Computer Science, 9301 (2015), Springer-Verlag, Cham, 424–439. Google Scholar

[30]

S. S. Samal, D. Grigoriev, H. Fröhlich, A. Weber and O. Radulescu, A geometric method for model reduction of biochemical networks with polynomial rate functions, Bull. Math. Biol., 77 (2015), 2180-2211.  doi: 10.1007/s11538-015-0118-0.  Google Scholar

[31]

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

[32]

A. Shapiro and F. Horn, On the possibility of sustained oscillations, multiple steady states, and asymmetric steady states in multicell reaction systems, Math. Biosci., 44 (1979), 19-39.  doi: 10.1016/0025-5564(79)90027-0.  Google Scholar

[33]

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

[34]

F. Verhulst, Methods and Applications of Singular Perturbations. Boundary Layers and Multiple Timescale Dynamics, Springer-Verlag, New York, 2005. doi: 10.1007/0-387-28313-7.  Google Scholar

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