-
Previous Article
Quasi-periodic solutions for a class of beam equation system
- DCDS-B Home
- This Issue
- Next Article
Singular perturbations and scaling
Mathematik A, RWTH Aachen, 52056 Aachen, Germany |
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.
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. 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. |
[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. 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. |
[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. |
[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. |
[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. |
[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. 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. |
[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. 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. |
[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. |
[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. 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. |
[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. |
[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. |
[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. 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. |
[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. 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. |
[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. |
[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. |
[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. |
[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. 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. |
[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. 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. |
[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. |
[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. 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. |
[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. |
[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. |
[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. |
[1] |
Guillaume Cantin, M. A. Aziz-Alaoui. Dimension estimate of attractors for complex networks of reaction-diffusion systems applied to an ecological model. Communications on Pure & Applied Analysis, , () : -. doi: 10.3934/cpaa.2020283 |
[2] |
Lisa Hernandez Lucas. Properties of sets of Subspaces with Constant Intersection Dimension. Advances in Mathematics of Communications, 2021, 15 (1) : 191-206. doi: 10.3934/amc.2020052 |
[3] |
D. R. Michiel Renger, Johannes Zimmer. Orthogonality of fluxes in general nonlinear reaction networks. Discrete & Continuous Dynamical Systems - S, 2021, 14 (1) : 205-217. doi: 10.3934/dcdss.2020346 |
[4] |
Annegret Glitzky, Matthias Liero, Grigor Nika. Dimension reduction of thermistor models for large-area organic light-emitting diodes. Discrete & Continuous Dynamical Systems - S, 2020 doi: 10.3934/dcdss.2020460 |
[5] |
Xiaoxian Tang, Jie Wang. Bistability of sequestration networks. Discrete & Continuous Dynamical Systems - B, 2021, 26 (3) : 1337-1357. doi: 10.3934/dcdsb.2020165 |
[6] |
Javier Fernández, Cora Tori, Marcela Zuccalli. Lagrangian reduction of nonholonomic discrete mechanical systems by stages. Journal of Geometric Mechanics, 2020, 12 (4) : 607-639. doi: 10.3934/jgm.2020029 |
[7] |
Jesús A. Álvarez López, Ramón Barral Lijó, John Hunton, Hiraku Nozawa, John R. Parker. Chaotic Delone sets. Discrete & Continuous Dynamical Systems - A, 2021 doi: 10.3934/dcds.2021016 |
[8] |
Riccarda Rossi, Ulisse Stefanelli, Marita Thomas. Rate-independent evolution of sets. Discrete & Continuous Dynamical Systems - S, 2021, 14 (1) : 89-119. doi: 10.3934/dcdss.2020304 |
[9] |
Shuang Chen, Jinqiao Duan, Ji Li. Effective reduction of a three-dimensional circadian oscillator model. Discrete & Continuous Dynamical Systems - B, 2020 doi: 10.3934/dcdsb.2020349 |
[10] |
Awais Younus, Zoubia Dastgeer, Nudrat Ishaq, Abdul Ghaffar, Kottakkaran Sooppy Nisar, Devendra Kumar. On the observability of conformable linear time-invariant control systems. Discrete & Continuous Dynamical Systems - S, 2020 doi: 10.3934/dcdss.2020444 |
[11] |
Fanni M. Sélley. A self-consistent dynamical system with multiple absolutely continuous invariant measures. Journal of Computational Dynamics, 2021, 8 (1) : 9-32. doi: 10.3934/jcd.2021002 |
[12] |
Paul A. Glendinning, David J. W. Simpson. A constructive approach to robust chaos using invariant manifolds and expanding cones. Discrete & Continuous Dynamical Systems - A, 2020 doi: 10.3934/dcds.2020409 |
[13] |
Bixiang Wang. Mean-square random invariant manifolds for stochastic differential equations. Discrete & Continuous Dynamical Systems - A, 2021, 41 (3) : 1449-1468. doi: 10.3934/dcds.2020324 |
[14] |
Qingfang Wang, Hua Yang. Solutions of nonlocal problem with critical exponent. Communications on Pure & Applied Analysis, 2020, 19 (12) : 5591-5608. doi: 10.3934/cpaa.2020253 |
[15] |
Bernold Fiedler. Global Hopf bifurcation in networks with fast feedback cycles. Discrete & Continuous Dynamical Systems - S, 2021, 14 (1) : 177-203. doi: 10.3934/dcdss.2020344 |
[16] |
Lars Grüne. Computing Lyapunov functions using deep neural networks. Journal of Computational Dynamics, 2020 doi: 10.3934/jcd.2021006 |
[17] |
Pedro Aceves-Sanchez, Benjamin Aymard, Diane Peurichard, Pol Kennel, Anne Lorsignol, Franck Plouraboué, Louis Casteilla, Pierre Degond. A new model for the emergence of blood capillary networks. Networks & Heterogeneous Media, 2020 doi: 10.3934/nhm.2021001 |
[18] |
Leslaw Skrzypek, Yuncheng You. Feedback synchronization of FHN cellular neural networks. Discrete & Continuous Dynamical Systems - B, 2020 doi: 10.3934/dcdsb.2021001 |
[19] |
Hua Qiu, Zheng-An Yao. The regularized Boussinesq equations with partial dissipations in dimension two. Electronic Research Archive, 2020, 28 (4) : 1375-1393. doi: 10.3934/era.2020073 |
[20] |
Wolfgang Riedl, Robert Baier, Matthias Gerdts. Optimization-based subdivision algorithm for reachable sets. Journal of Computational Dynamics, 2021, 8 (1) : 99-130. doi: 10.3934/jcd.2021005 |
2019 Impact Factor: 1.27
Tools
Article outline
[Back to Top]