Tihonov theory and center manifolds for inhibitory mechanisms in enzyme kinetics

Abstract

In this paper we study the chemical reaction of inhibition, determine the appropriate parameter 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.

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

“…characterized by two blocks. The Jacobian allows the separation between the eigenvalues of the first reaction and the eigenvalues of the second reaction, so that we can apply Palsson's technique to find the perturbation parameter , as one can see considering the results obtained in [26]. An easy computation shows that…”

“…The mechanism we are studying in this paper does not belong to this class of reactions. Nevertheless Palsson's technique was adapted to the fully competitive inhibition, too [26].…”

“…In [26] the authors study the chemical reaction of inhibition and determine the appropriate parameter for the application of Tikhonov'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 tend asymptotically to the center manifold of the system. In other words, the above quoted paper gives another example of connection among Tikhonov's Theorem, center manifold and tQSSA, after the fundamental article [20].…”

“…In this paper, we determine the asymptotic expansions beyond the tQSSA, up to the 1-st order in the parameter (introduced in [26], adopting Palsson's technique) for the inner and outer solutions and the corresponding uniform expansions.…”

“…The paper is organized in the following way: in Section 2 we recall the most important mathematical background concerning Palsson's theory. In Section 3 we recall the mathematical model of the enzymatic inhibition reaction and its adimensionalization, as done in [26], arriving at a suitable perturbation parameter which will be used in Section 4, in a study case, to determine the 0-th and the 1-st order asymptotic expansions for the inner and outer solutions and the corresponding uniform expansions, obtained by means of singular perturbation techniques. Numerical results are shown, for different values of .…”