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.
1. J. Berg J. Tymoczko and L. Stryer Biochemistry. W. H. Freeman New York 2002.
2. J. Murray Mathematical Biology: An introduction. Springer-Verlag New York 2002.
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.
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.
5. A. Cornish-Bowden One hundred years of michaelis-menten kinetics Perspectives in Science vol. 4 pp. 3 – 9 2015.
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.
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.
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.
9. J. Carr Applications of Center Manifold Theory. Springer-Verlag New York Heidelberg Berlin 1981.
10. S. Wiggins Normally Hyperbolic Invariant Manifolds in Dynamical Systems vol. 105. Springer-Verlag New York 1994.
11. S. Wiggins Introduction to applied nonlinear dynamical systems and chaos vol. 2. Springer-Verlag New York 2003.
12. F. C. Hoppensteadt Singular perturbations on the infinite interval Transactions of the American Mathematical Society vol. 123 no. 2 pp. 521–535 1966.
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.
14. A. Tikhonov On a system of differential equations containing parameters (in russian) Mat. Sb. (N.S.) vol. 27 pp. 147–156 1950.
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.
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.
17. W. Wasov Asymptotic Expansions for Ordinary Di erential Equations. Wiley-InterScience 1965.
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.
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.
20. N. Fenichel Geometric singular perturbation theory for ordinary differential equations Journal of Di erential Equations vol. 31 no. 1 pp. 53–98 1979.
21. A. J. Roberts Model emergent dynamics in complex systems. SIAM 2015.
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.
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.
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.
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.
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.
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.
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.
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.
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.
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.
32. C.-C. Lin and L. A. Segel Mathematics applied to deterministic problems in the natural sciences vol. 1. Siam 1988.
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.
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.
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.
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.
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.
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.
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.
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.
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.
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.
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.
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.
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.
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.
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.
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.