On the quasi-steady-state approximation in an open Michaelis–Menten reaction mechanism

Justin Eilertsen; Marc R. Roussel; Santiago Schnell; Sebastian Walcher; Justin Eilertsen; Marc R. Roussel; Santiago Schnell; Sebastian Walcher

doi:10.3934/math.2021398

AIMS Mathematics

2021, Volume 6, Issue 7: 6781-6814. doi: 10.3934/math.2021398

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On the quasi-steady-state approximation in an open Michaelis–Menten reaction mechanism

1.
Department of Molecular & Integrative Physiology, University of Michigan Medical School, Ann Arbor, Michigan 49109, USA
2.
Alberta RNA Research and Training Institute, Department of Chemistry and Biochemistry, University of Lethbridge, Lethbridge, Alberta, Canada, T1K 3M4
3.
Department of Computational Medicine & Bioinformatics, University of Michigan Medical School, Ann Arbor, Michigan 49109, USA
4.
Mathematik A, RWTH Aachen, D-52056 Aachen, Germany

Received: 30 December 2020 Accepted: 09 April 2021 Published: 21 April 2021
MSC : 92C45, 34N05, 34C45

The conditions for the validity of the standard quasi-steady-state approximation in the Michaelis–Menten mechanism in a closed reaction vessel have been well studied, but much less so the conditions for the validity of this approximation for the system with substrate inflow. We analyze quasi-steady-state scenarios for the open system attributable to singular perturbations, as well as less restrictive conditions. For both settings we obtain distinguished invariant manifolds and time scale estimates, and we highlight the special role of singular perturbation parameters in higher order approximations of slow manifolds. We close the paper with a discussion of distinguished invariant manifolds in the global phase portrait.
- singular perturbation,
- slow manifold,
- quasi-steady state,
- Michaelis–Menten mechanism,
- critical manifold,
- Gronwall lemma,
- Poincaré sphere
Citation: Justin Eilertsen, Marc R. Roussel, Santiago Schnell, Sebastian Walcher. On the quasi-steady-state approximation in an open Michaelis–Menten reaction mechanism[J]. AIMS Mathematics, 2021, 6(7): 6781-6814. doi: 10.3934/math.2021398

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Abstract

The conditions for the validity of the standard quasi-steady-state approximation in the Michaelis–Menten mechanism in a closed reaction vessel have been well studied, but much less so the conditions for the validity of this approximation for the system with substrate inflow. We analyze quasi-steady-state scenarios for the open system attributable to singular perturbations, as well as less restrictive conditions. For both settings we obtain distinguished invariant manifolds and time scale estimates, and we highlight the special role of singular perturbation parameters in higher order approximations of slow manifolds. We close the paper with a discussion of distinguished invariant manifolds in the global phase portrait.

References

[1]	A. V. P. Bobadilla, B. J. Bartmanski, R. Grima, H. G. Othmer, The status of the QSSA approximation in stochastic simulations of reaction networks, in 2018 MATRIX Annals, Springer International Publishing, 2020,137–147.
[2]	M. Bodenstein, Eine Theorie der photochemischen Reaktionsgeschwindigkeiten, Z. Phys. Chem., 85 (1913), 329–397.
[3]	J. R. Bowen, A. Acrivos, A. K. Oppenheim, Singular perturbation refinement to quasi-steady state approximation in chemical kinetics, Chem. Eng. Sci., 18 (1963), 177–187. doi: 10.1016/0009-2509(63)85003-4
[4]	G. E. Briggs, J. B. S. Haldane, A note on the kinetics of enzyme action, Biochem. J., 19 (1925), 338–339. doi: 10.1042/bj0190338
[5]	M. S. Calder, D. Siegel, Properties of the Michaelis-Menten mechanism in phase space, J. Math. Anal. Appl., 339 (2008), 1044–1064. doi: 10.1016/j.jmaa.2007.06.078
[6]	D. L. Chapman, L. K. Underhill, LV.–-The interaction of chlorine and hydrogen. The influence of mass, J. Chem. Soc., Trans., 103 (1913), 496–508. doi: 10.1039/CT9130300496
[7]	J. Eilertsen, S. Schnell, The quasi-steady-state approximations revisited: Timescales, small parameters, singularities, and normal forms in enzyme kinetics, Math. Biosci., 325 (2020), 108339. doi: 10.1016/j.mbs.2020.108339
[8]	N. Fenichel, Persistence and smoothness of invariant manifolds for flows, Indiana U. Math. J., 21 (1971), 193–226. doi: 10.1512/iumj.1972.21.21017
[9]	N. Fenichel, Geometric singular perturbation theory for ordinary differential equations, J. Differ. Equations, 31 (1979), 53–98. doi: 10.1016/0022-0396(79)90152-9
[10]	E. H. Flach, S. Schnell, Use and abuse of the quasi-steady-state approximation, IEEE Proc. Syst. Biol., 153 (2006), 187–191. doi: 10.1049/ip-syb:20050104
[11]	S. J. Fraser, The steady state and equilibrium approximations: A geometrical picture, J. Chem. Phys., 88 (1988), 4732–4738. doi: 10.1063/1.454686
[12]	S. J. Fraser, M. R. Roussel, Phase-plane geometries in enzyme kinetics, Can. J. Chem., 72 (1994), 800–812. doi: 10.1139/v94-107
[13]	A. Goeke, C. Schilli, S. Walcher, E. Zerz, Computing quasi-steady state reductions, J. Math. Chem., 50 (2012), 1495–1513. doi: 10.1007/s10910-012-9985-x
[14]	A. Goeke, S. Walcher, Quasi-steady state: Searching for and utilizing small parameters, in Recent Trends in Dynamical Systems (eds. A. Johann, H.-P. Kruse, F. Rupp and S. Schmitz), Springer Basel, Basel, 2013,153–178.
[15]	A. Goeke, S. Walcher, A constructive approach to quasi-steady state reductions, J. Math. Chem., 52 (2014), 2596–2626. doi: 10.1007/s10910-014-0402-5
[16]	A. Goeke, S. Walcher, E. Zerz, Determining "small parameters" for quasi-steady state, J. Differ. Equations, 259 (2015), 1149–1180. doi: 10.1016/j.jde.2015.02.038
[17]	A. Goeke, S. Walcher, E. Zerz, Classical quasi-steady state reduction – A mathematical characterization, Phys. D, 345 (2017), 11–26. doi: 10.1016/j.physd.2016.12.002
[18]	D. Gonze, W. Abou-Jaoudé, D. A. Ouattara, J. Halloy, How molecular should your molecular model be? On the level of molecular detail required to simulate biological networks in systems and synthetic biology, Meth. Enzymol., 487 (2011), 171–215. doi: 10.1016/B978-0-12-381270-4.00007-X
[19]	A. N. Gorban, Model reduction in chemical dynamics: Slow invariant manifolds, singular perturbations, thermodynamic estimates, and analysis of reaction graph, Curr. Opin. Chem. Eng., 21 (2018), 48–59. doi: 10.1016/j.coche.2018.02.009
[20]	A. N. Gorban, I. V. Karlin, Method of invariant manifold for chemical kinetics, Chem. Eng. Sci., 58 (2003), 4751–4768. doi: 10.1016/j.ces.2002.12.001
[21]	A. N. Gorban, I. V. Karlin, A. Yu. Zinovyev, Constructive methods of invariant manifolds for kinetic problems, Phys. Rep., 396 (2004), 197–403. doi: 10.1016/j.physrep.2004.03.006
[22]	F. G. Heineken, H. M. Tsuchiya, R. Aris, On the mathematical status of the pseudo-steady hypothesis of biochemical kinetics, Math. Biosci., 1 (1967), 95–113. doi: 10.1016/0025-5564(67)90029-6
[23]	V. Henri, Théorie générale de l'action de quelques diastases, C. R. Acad. Sci., 135 (1902), 916–919.
[24]	J. H. Hubbard, B. H. West, Differential Equations: A Dynamical Systems Approach, vol. 5 of Texts in Applied Mathematics, Springer, New York, 1991.
[25]	H. G. Kaper, T. J. Kaper, Asymptotic analysis of two reduction methods for systems of chemical reactions, Phys. D, 165 (2002), 66–93. doi: 10.1016/S0167-2789(02)00386-X
[26]	C. Kuehn, Multiple Time Scale Dynamics, vol. 191 of Applied Mathematical Sciences, Springer, 2015.
[27]	S. H. Lam, Using CSP to understand complex chemical kinetics, Combust. Sci. Technol., 89 (1993), 375–404. doi: 10.1080/00102209308924120
[28]	U. Maas, S. Pope, Simplifying chemical kinetics: Intrinsic low-dimensional manifolds in composition space, Combust. Flame, 88 (1992), 239–264. doi: 10.1016/0010-2180(92)90034-M
[29]	L. Michaelis, M. L. Menten, Die Kinetik der Invertinwirkung, Biochem. Z., 49 (1913), 333–369.
[30]	D. L. Nelson, M. M. Cox, Lehninger Principles of Biochemistry, 5th edition, Freeman, New York, 2008.
[31]	A. H. Nguyen, S. J. Fraser, Geometrical picture of reaction in enzyme kinetics, J. Chem. Phys., 91 (1989), 186–193. doi: 10.1063/1.457504
[32]	L. Noethen, S. Walcher, Quasi-steady state and nearly invariant sets, SIAM J. Appl. Math., 70 (2009), 1341–1363. doi: 10.1137/090758180
[33]	L. Noethen, S. Walcher, Tikhonov's theorem and quasi-steady state, Discrete Contin. Dyn. Syst. Ser. B, 16 (2011), 945–961.
[34]	L. Perko, Differential equations and dynamical systems, 3rd edition, no. 7 in Texts in Applied Mathematics, Springer, New York, 2001.
[35]	A. J. Roberts, The utility of an invariant manifold description of the evolution of a dynamical system, SIAM J. Math. Anal., 20 (1989), 1447–1458. doi: 10.1137/0520094
[36]	M. R. Roussel, S. J. Fraser, Geometry of the steady-state approximation: Perturbation and accelerated convergence methods, J. Chem. Phys., 93 (1990), 1072–1081. doi: 10.1063/1.459171
[37]	M. R. Roussel, Forced-convergence iterative schemes for the approximation of invariant manifolds, J. Math. Chem., 21 (1997), 385–393. doi: 10.1023/A:1019151225744
[38]	M. R. Roussel, Heineken, Tsushiya and Aris on the mathematical status of the pseudo-steady state hypothesis: A classic from volume 1 of Mathematical Biosciences, Math. Biosci., 318 (2019), 108274. doi: 10.1016/j.mbs.2019.108274
[39]	M. R. Roussel, S. J. Fraser, On the geometry of transient relaxation, J. Chem. Phys., 94 (1991), 7106–7113. doi: 10.1063/1.460194
[40]	M. Schauer, 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
[41]	L. A. Segel, M. Slemrod, The quasi-steady-state assumption: A case study in perturbation, SIAM Rev., 31 (1989), 446–477. doi: 10.1137/1031091
[42]	L. A. Segel, On the validity of the steady state assumption of enzyme kinetics, Bull. Math. Biol., 50 (1988), 579–593. doi: 10.1016/S0092-8240(88)80057-0
[43]	H. L. Smith, P. Waltman, The Theory of the Chemostat. Dynamics of Microbial Competition, no. 13 in Cambridge Studies in Mathematical Biology, Cambridge University Press, Cambridge, 1995.
[44]	I. Stoleriu, F. A. Davidson, J. L. Liu, Quasi-steady state assumptions for non-isolated enzyme-catalysed reactions, J. Math. Biol., 48 (2004), 82–104. doi: 10.1007/s00285-003-0225-7
[45]	P. Thomas, A. V. Straube, R. Grima, Limitations of the stochastic quasi-steady-state approximation in open biochemical reaction networks, J. Chem. Phys., 135 (2011), 181103. doi: 10.1063/1.3661156
[46]	A. Tikhonov, Systems of differential equations containing small parameters in their derivatives, Mat. Sb. (N.S.), 31 (1952), 575–586.
[47]	S. Walcher, On transformations into normal form, J. Math. Anal. Appl., 180 (1993), 617–632. doi: 10.1006/jmaa.1993.1420
[48]	S. Walcher, On the Poincaré problem, J. Differ. Equations, 166 (2000), 51–78. doi: 10.1006/jdeq.2000.3801
[49]	M. Wechselberger, Geometric Singular Perturbation Theory Beyond the Standard Form, no. 6 in Frontiers in Applied dynamical systems: Tutorials and Reviews, Springer, 2020.
[50]	A. Wurtz, Sur la papaïne. Nouvelle contribution à l'histoire des ferments solubles, C. R. Acad. Sci., 91 (1880), 787–791.

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