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

Singular perturbations and scaling

Mathematik A, RWTH Aachen, 52056 Aachen, Germany

* Corresponding author

Received  October 2018 Revised  February 2019 Published  January 2020 Early access  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.

Citation: Christian Lax, Sebastian Walcher. Singular perturbations and scaling. Discrete and 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.

[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.

[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.

[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.

[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.

[6]

M. FrankC. LaxS. 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.

[7]

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

[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.

[9]

A. GoekeC. SchilliS. Walcher and E. Zerz, Computing quasi-steady state reductions, J. Math. Chem., 50 (2012), 1495-1513.  doi: 10.1007/s10910-012-9985-x.

[10]

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

[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.

[12]

A. GoekeS. 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.

[13]

A. GoekeS. 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.

[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.

[15]

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

[16]

F. G. HeinekenH. 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.

[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.

[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.

[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.

[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.

[21]

C. LaxK. 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.

[22]

M. Marcondes de FreitasE. 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.

[23]

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

[24]

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

[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.

[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.

[27]

O. RadulescuS. 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.

[28]

M. SaezC. 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.

[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.

[30]

S. S. SamalD. GrigorievH. FröhlichA. 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.

[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.

[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.

[33]

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

[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.

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.

[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.

[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.

[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.

[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.

[6]

M. FrankC. LaxS. 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.

[7]

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

[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.

[9]

A. GoekeC. SchilliS. Walcher and E. Zerz, Computing quasi-steady state reductions, J. Math. Chem., 50 (2012), 1495-1513.  doi: 10.1007/s10910-012-9985-x.

[10]

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

[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.

[12]

A. GoekeS. 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.

[13]

A. GoekeS. 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.

[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.

[15]

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

[16]

F. G. HeinekenH. 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.

[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.

[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.

[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.

[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.

[21]

C. LaxK. 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.

[22]

M. Marcondes de FreitasE. 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.

[23]

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

[24]

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

[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.

[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.

[27]

O. RadulescuS. 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.

[28]

M. SaezC. 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.

[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.

[30]

S. S. SamalD. GrigorievH. FröhlichA. 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.

[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.

[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.

[33]

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

[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.

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