Boltzmann maps for networks of chemical reactions and the multi-stability problem

Pages: 501 - 526, Volume 4, Issue 3, September 2009

doi:10.3934/nhm.2009.4.501       Abstract        Full Text (307.6K)       Related Articles

Andrea Picco - Institute for Cancer Research and Treatment (IRCC), Str Prov 142, Km 3.95, 10060 Candiolo (Torino), Italy (email)
Lamberto Rondoni - Dipartimento di Matematica, Politecnico di Torino, Corso Duca degli Abruzzi 24, 10129 Torino, Italy (email)

Abstract: Boltzmann Maps are a class of discrete dynamical systems that may be used in the study of complex chemical reaction processes. In this paper they are generalized to open systems allowing the description of non-stoichiometrically balanced reactions with unequal reaction rates. We show that they can be widely used to describe the relevant dynamics, leading to interesting insights on the multi-stability problem in networks of chemical reactions. Necessary conditions for multistability are thus identified. Our findings indicate that the dynamics produced by laws like the mass action law, can hardly produce multistable phenomena. In particular, we prove that they cannot do it in a wide range of chemical reactions.

Keywords:  Biochemistry, molecular biology, Biophysics, Stochastic methods (Fokker-Planck, Langevin, etc.), General biology and biomathematics.
Mathematics Subject Classification:  Primary: 92C40, 92C05; Secondary: 82C31, 92B05.

Received: March 2008;      Revised: April 2009;      Published: July 2009.