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The Michaelis-Menten-Stueckelberg Theorem
Department of Mathematics, University of Leicester, Leicester, LE1 7RH, UK
Department of Mathematics, Hazara University, Mansehra, 21300, Pakistan
* Author to whom correspondence should be addressed.
Received: 25 January 2011; in revised form: 28 March 2011 / Accepted: 12 May 2011 / Published: 20 May 2011
Abstract: We study chemical reactions with complex mechanisms under two assumptions: (i) intermediates are present in small amounts (this is the quasi-steady-state hypothesis or QSS) and (ii) they are in equilibrium relations with substrates (this is the quasiequilibrium hypothesis or QE). Under these assumptions, we prove the generalized mass action law together with the basic relations between kinetic factors, which are sufficient for the positivity of the entropy production but hold even without microreversibility, when the detailed balance is not applicable. Even though QE and QSS produce useful approximations by themselves, only the combination of these assumptions can render the possibility beyond the “rarefied gas” limit or the “molecular chaos” hypotheses. We do not use any a priori form of the kinetic law for the chemical reactions and describe their equilibria by thermodynamic relations. The transformations of the intermediate compounds can be described by the Markov kinetics because of their low density (low density of elementary events). This combination of assumptions was introduced by Michaelis and Menten in 1913. In 1952, Stueckelberg used the same assumptions for the gas kinetics and produced the remarkable semi-detailed balance relations between collision rates in the Boltzmann equation that are weaker than the detailed balance conditions but are still sufficient for the Boltzmann H-theorem to be valid. Our results are obtained within the Michaelis-Menten-Stueckelbeg conceptual framework.
Keywords: chemical kinetics; Lyapunov function; entropy; quasiequilibrium; detailed balance; complex balance
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Gorban, A.N.; Shahzad, M. The Michaelis-Menten-Stueckelberg Theorem. Entropy 2011, 13, 966-1019.
Gorban AN, Shahzad M. The Michaelis-Menten-Stueckelberg Theorem. Entropy. 2011; 13(5):966-1019.
Gorban, Alexander N.; Shahzad, Muhammad. 2011. "The Michaelis-Menten-Stueckelberg Theorem." Entropy 13, no. 5: 966-1019.