A study case for the analysis of asymptotic expansions beyond the tQSSA for inhibitory mechanisms in enzyme kinetics.

Bersani, A. M.; Borri, A.; Milanesi, A.; Tomassetti, G.; Vellucci, P.

doi:10.1515/caim-2019-0019

Abstract

In this paper we study the model of the chemical reaction of fully competitive inhibition and determine the appropriate parameter ∊ (related to the chemical constants of the model), for the application of singular perturbation techniques. We determine the inner and the outer solutions up to the first perturbation order and the uniform expansions. Some numerical results are discussed.

References

1. J. Berg, J. Tymoczko, and L. Stryer, Biochemistry. W. H. Freeman New York, 2002.
Search in Google Scholar Back to article
2. J. Murray, Mathematical Biology: An introduction. Springer-Verlag New York, 2002.
Search in Google Scholar Back to article
3. J. Borghans, R. de Boer, and L. Segel, Extending the quasi-steady state approximation by changing variables, Bull.Math.Biol., vol. 58, pp. 43–63, 1996.10.1007/BF02458281
Search in Google Scholar Back to article
4. A. M. Bersani, E. Bersani, G. Dell’Acqua, and M. G. Pedersen, New trends and perspectives in nonlinear intracellular dynamics: one century from michaelis–menten paper, Continuum Mechanics and Thermodynamics, vol. 27, no. 4, pp. 659–684, 2015.10.1007/s00161-014-0367-4
Search in Google Scholar Back to article
5. A. Cornish-Bowden, One hundred years of michaelis-menten kinetics, Perspectives in Science, vol. 4, pp. 3 – 9, 2015.10.1016/j.pisc.2014.12.002
Search in Google Scholar Back to article
6. F. G. Heineken, H. M. Tsuchiya, and R. Aris, On the mathematical status of the pseudo-steady state hypothesis of biochemical kinetics, Math. Biosc., vol. 1, pp. 95–11, 1967.10.1016/0025-5564(67)90029-6
Search in Google Scholar Back to article
7. L. A. Segel and M. Slemrod, The quasi steady-state assumption: a case study in pertubation., Siam Rev., vol. 31, pp. 446–477, 1989.10.1137/1031091
Search in Google Scholar Back to article
8. G. Dell’Acqua and A. M. Bersani, A perturbation solution of michaelis–menten kinetics in a “total” framework, Journal of Mathematical Chemistry, vol. 50, no. 5, pp. 1136–1148, 2012.
Search in Google Scholar Back to article
9. J. Carr, Applications of Center Manifold Theory. Springer-Verlag New York, Heidelberg, Berlin, 1981.
Search in Google Scholar Back to article
10. S. Wiggins, Normally Hyperbolic Invariant Manifolds in Dynamical Systems, vol. 105. Springer-Verlag New York, 1994.10.1007/978-1-4612-4312-0
Search in Google Scholar Back to article
11. S. Wiggins, Introduction to applied nonlinear dynamical systems and chaos, vol. 2. Springer-Verlag New York, 2003.
Search in Google Scholar Back to article
12. F. C. Hoppensteadt, Singular perturbations on the infinite interval, Transactions of the American Mathematical Society, vol. 123, no. 2, pp. 521–535, 1966.10.1090/S0002-9947-1966-0194693-9
Search in Google Scholar Back to article
13. A. Tikhonov, On the dependence of the solutions of differential equations on a small parameter (in russian), Mat. Sb. (N.S.), vol. 22, no. 2, pp. 193 – 204, 1948.
Search in Google Scholar Back to article
14. A. Tikhonov, On a system of differential equations containing parameters (in russian), Mat. Sb. (N.S.), vol. 27, pp. 147–156, 1950.
Search in Google Scholar Back to article
15. A. Tikhonov, Systems of differential equations containing small parameters in the derivatives (in russian), Mat. Sb. (N.S.), vol. 31, no. 3, pp. 575–586, 1952.
Search in Google Scholar Back to article
16. A. B. Vasil’eva, Asymptotic behaviour of solutions to certain problems involving non-linear differential equations containing a small parameter multiplying the highest derivatives (in russian), Russian Mathematical Surveys, vol. 18, no. 3, pp. 13–84, 1963.10.1070/RM1963v018n03ABEH001137
Search in Google Scholar Back to article
17. W. Wasov, Asymptotic Expansions for Ordinary Di erential Equations. Wiley-InterScience, 1965.
Search in Google Scholar Back to article
18. I. Dvořák and J.Šiška, Analysis of metabolic systems with complex slow and fast dynamics, Bulletin of Mathematical Biology, vol. 51, no. 2, pp. 255–274, 1989.10.1007/BF02458446
Search in Google Scholar Back to article
19. A. Kumar and K. Josić, Reduced models of networks of coupled enzymatic reactions, Journal of Theoretical Biology, vol. 278, no. 1, pp. 87 – 106, 2011.10.1016/j.jtbi.2011.02.02521377474
Search in Google Scholar Back to article
20. N. Fenichel, Geometric singular perturbation theory for ordinary differential equations, Journal of Di erential Equations, vol. 31, no. 1, pp. 53–98, 1979.10.1016/0022-0396(79)90152-9
Search in Google Scholar Back to article
21. A. J. Roberts, Model emergent dynamics in complex systems. SIAM, 2015.10.1137/1.9781611973563
Search in Google Scholar Back to article
22. B. O. Palsson and E. N. Lightfoot, Mathematical modelling of dynamics and control in metabolic networks. i. on michaelis-menten kinetics, Journal of Theoretical Biology, vol. 111, no. 2, pp. 273 – 302, 1984.10.1016/S0022-5193(84)80211-8
Search in Google Scholar Back to article
23. B. O. Palsson, R. Jamier, and E. N. Lightfoot, Mathematical modelling of dynamics and control in metabolic networks. ii. simple dimeric enzymes, Journal of Theoretical Biology, vol. 111, no. 2, pp. 303 – 321, 1984.10.1016/S0022-5193(84)80212-X
Search in Google Scholar Back to article
24. B. O. Palsson, H. Palsson, and E. N. Lightfoot, Mathematical modelling of dynamics and control in metabolic networks. iii. linear reaction sequences, Journal of Theoretical Biology, vol. 113, no. 2, pp. 231 – 259, 1985.10.1016/S0022-5193(85)80226-5
Search in Google Scholar Back to article
25. B. O. Palsson, On the dynamics of the irreversible michaelis-menten reaction mechanism, Chemical Engineering Science, vol. 42, no. 3, pp. 447 – 458, 1987.10.1016/0009-2509(87)80007-6
Search in Google Scholar Back to article
26. A. Bersani, A. Borri, A. Milanesi, and P. Vellucci, Tihonov theory and center manifolds for inhibitory mechanisms in enzyme kinetics, Communications in Applied and Industrial Mathematics, vol. 8, no. 1, pp. 81–102, 2017.10.1515/caim-2017-0005
Search in Google Scholar Back to article
27. S. Schnell and C. Mendoza, Time-dependent closed form solutions for fully competitive enzyme reactions, Bulletin of Mathematical Biology, vol. 62, no. 2, pp. 321–336, 2000.10.1006/bulm.1999.0156
Search in Google Scholar Back to article
28. S. Schnell and C. Mendoza, Closed form solution for time-dependent enzyme kinetics, Journal of Theoretical Biology, vol. 187, no. 2, pp. 207 – 212, 1997.10.1006/jtbi.1997.0425
Search in Google Scholar Back to article
29. M. G. Pedersen, A. M. Bersani, and E. Bersani, The total quasi-steady-state approximation for fully competitive enzyme reactions, Bulletin of Mathematical Biology, vol. 69, no. 1, pp. 433–457, 2006.10.1007/s11538-006-9136-2
Search in Google Scholar Back to article
30. M. G. Pedersen, A. M. Bersani, and E. Bersani, Quasi steady-state approximations in complex intra-cellular signal transduction networks – a word of caution, Journal of Mathematical Chemistry, vol. 43, no. 4, pp. 1318–1344, 2008.
Search in Google Scholar Back to article
31. S. Schnell and P. Maini, Enzyme kinetics far from the standard quasi-steady-state and equilibrium approximations, Mathematical and Computer Modelling, vol. 35, no. 1-2, pp. 137–144, 2002.10.1016/S0895-7177(01)00156-X
Search in Google Scholar Back to article
32. C.-C. Lin and L. A. Segel, Mathematics applied to deterministic problems in the natural sciences, vol. 1. Siam, 1988.
Search in Google Scholar Back to article
33. M. G. Pedersen, A. M. Bersani, E. Bersani, and G. Cortese, The total quasi-steady-state approximation for complex enzyme reactions, Mathematics and Computers in Simulation, vol. 79, no. 4, pp. 1010 – 1019, 2008.
Search in Google Scholar Back to article
34. A. Tzafriri and E. Edelman, The total quasi-steady-state approximation is valid for reversible enzyme kinetics, Journal of Theoretical Biology, vol. 226, no. 3, pp. 303 – 313, 2004.10.1016/j.jtbi.2003.09.00614643644
Search in Google Scholar Back to article
35. M. Sabouri-Ghomi, A. Ciliberto, S. Kar, B. Novak, and J. J. Tyson, Antagonism and bistability in protein interaction networks, Journal of Theoretical Biology, vol. 250, no. 1, pp. 209 – 218, 2008.10.1016/j.jtbi.2007.09.00117950756
Search in Google Scholar Back to article
36. S. Rubinow and J. L. Lebowitz, Time-dependent michaelis-menten kinetics for an enzyme-substrate-inhibitor system, Journal of the American Chemical Society, vol. 92, no. 13, pp. 3888–3893, 1970.
Search in Google Scholar Back to article
37. A. Bersani, E. Bersani, and L. Mastroeni, Deterministic and stochastic models of enzymatic networks-applications to pharmaceutical research, Computers & Mathematics with Applications, vol. 55, no. 5, pp. 879 – 888, 2008. Modeling and Computational Methods in Genomic Sciences.10.1016/j.camwa.2006.12.092
Search in Google Scholar Back to article
38. A. Ciliberto, F. Capuani, and J. J. Tyson, Modeling networks of coupled enzymatic reactions using the total quasi-steady state approximation, PLOS Computational Biology, vol. 3, pp. 1–10, 03 2007.10.1371/journal.pcbi.0030045182870517367203
Search in Google Scholar Back to article
39. M. G. Pedersen and A. M. Bersani, Introducing total substrates simplifies theoretical analysis at non-negligible enzyme concentrations: pseudo first-order kinetics and the loss of zero-order ultrasensitivity, Journal of Mathematical Biology, vol. 60, no. 2, pp. 267–283, 2010.10.1007/s00285-009-0267-619333602
Search in Google Scholar Back to article
40. C. Kwang-Hyun, S. Sung-Young, K. Hyun-Woo, O. Wolkenhauer, B. McFerran, and W. Kolch, Mathematical modeling of the influence of rkip on the erk signaling pathway, in Computational Methods in Systems Biology (C. Priami, ed.), pp. 127–141, Berlin, Heidelberg: Springer Berlin Heidelberg, 2003.10.1007/3-540-36481-1_11
Search in Google Scholar Back to article
41. S. MacNamara and K. Burrage, Krylov and steady-state techniques for the solution of the chemical master equation for the mitogen-activated protein kinase cascade, Numerical Algorithms, vol. 51, no. 3, pp. 281–307, 2009.10.1007/s11075-008-9239-y
Search in Google Scholar Back to article
42. G. Dell’Acqua and A. M. Bersani, Quasi-steady state approximations and multistability in the double phosphorylation-dephosphorylation cycle, in Biomedical Engineering Systems and Technologies (A. Fred, J. Filipe, and H. Gamboa, eds.), pp. 155–172, Berlin, Heidelberg: Springer Berlin Heidelberg, 2013.10.1007/978-3-642-29752-6_12
Search in Google Scholar Back to article
43. G. Dell’Acqua and A. M. Bersani, Bistability and the complex depletion paradox in the double phosphorylation-dephosphorylation cycle., in BIOINFORMATICS, pp. 55–65, 2011.
Search in Google Scholar Back to article
44. A. M. Bersani, G. Dell’Acqua, and G. Tomassetti, On stationary states in the double phosphorylation-dephosphorylation cycle, AIP Conference Proceedings, vol. 1389, no. 1, pp. 1208–1211, 2011.
Search in Google Scholar Back to article
45. A. Bersani, A. Borri, A. Milanesi, G. Tomassetti, and P. Vellucci, Asymptotic analysis of the double phosphorylation mechanism, in a tqssa framework. in preparation.
Search in Google Scholar Back to article
46. A. Bersani, A. Borri, A. Milanesi, G. Tomassetti, and P. Vellucci, Uniform asymptotic expansions beyond the tqssa for the goldbeter-koshland switch. in preparation.
Search in Google Scholar Back to article
47. A. R. Tzafriri and E. R. Edelman, Quasi-steady-state kinetics at enzyme and substrate concentrations in excess of the michaelis-menten constant, Journal of Theoretical Biology, vol. 245, no. 4, pp. 737 – 748, 2007.10.1016/j.jtbi.2006.12.00517234216
Search in Google Scholar Back to article
48. H. P. Fischer, Mathematical modeling of complex biological systems: from parts lists to understanding systems behavior, Alcohol Research & Health, vol. 31, no. 1, p. 49, 2008.
Search in Google Scholar Back to article

A study case for the analysis of asymptotic expansions beyond the tQSSA for inhibitory mechanisms in enzyme kinetics.

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