In this paper we study the chemical reaction of inhibition, determine the appropriate parameter ε for the application of Tihonov's Theorem, compute explicitly the equations of the center manifold of the system and find sufficient conditions to guarantee that in the phase space the curves which relate the behavior of the complexes to the substrates by means of the tQSSA are asymptotically equivalent to the center manifold of the system. Some numerical results are discussed.
1. J. Murray, Mathematical Biology: An introduction. Springer-Verlag New York, 2002.
2. 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.
3. A. Bersani, E. Bersani, G. Dell'Acqua, and M. Pedersen, New trends and perspectives in nonlinear intracellular dynamics: one century from michaelis-menten paper, CMAT, vol. 27, pp. 659-684, 2015.
4. F. G. Heineken, T. M., and A. R., On the mathematical status of the pseudo-steady state hypothesis of biochemical kinetics, Math. Biosc., vol. 1, pp. 95-11, 1967.
5. L. A. Segel and M. Slemrod, The quasi steady-state assumption: a case study in pertubation., Siam Rev., vol. 31, pp. 446-477, 1989.
6. 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.
7. J. Carr, Applications of Center Manifold Theory. Springer-Verlag New York, Heidelberg, Berlin, 1981.
8. S. Wiggins, Normally Hyperbolic Invariant Manifolds in Dynamical Sys- tems, vol. 105. Springer-Verlag New York, 1994.
9. S. Wiggins, Introduction to applied nonlinear dynamical systems and chaos, vol. 2. Springer-Verlag New York, 2003.
10. A. Tikhonov, On the dependence of the solutions of differential equations on a small parameter, Mat. Sb. (N.S.), vol. 22, no. 2, pp. 193 - 204, 1948.
11. A. Tikhonov, On a system of differential equations containing parameters, Mat. Sb. (N.S.), vol. 27, pp. 147-156, 1950.
12. A. Tikhonov, Systems of differential equations containing small parameters in the derivatives, Mat. Sb. (N.S.), vol. 31, no. 3, pp. 575-586, 1952.
13. W. Wasov, Asymptotic Expansions for Ordinary Differential Equations. Wiley-InterScience, 1965.
14. 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.
15. 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.
16. A. Bersani, E. Bersani, A. Borri, and P. Vellucci, “Dynamical aspects of the total QSSA in enzyme kinematics.” https://arxiv.org/abs/1702.05351. Submitted to Journal of Mathematical Biology.
17. 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.
18. 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.
19. 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.
20. B. O. Palsson, On the dynamics of the irreversible michaelis-menten reaction mechanism, Chemical Engineering Science, vol. 42, no. 3, pp. 447- 458, 1987.
21. A. J. Roberts, Model emergent dynamics in complex systems. SIAM, 2015.