Local dissipative Hamiltonian description of reversible reaction networks

In this letter we show that closed reversible chemical reaction networks with independent elementary reactions admit a global pseudo-Hamiltonian structure which is at least locally dissipative around any equilibrium point. The structure matrix of the Hamiltonian description reflects the graph topology of the reaction network and it is a smooth function of the concentrations of the chemical species in the positive orthant. The physical interpretation of the description is briefly explained and two illustrative examples are presented for global and local dissipative Hamiltonian description, respectively.

[1]  H. Ch. Öttinger,et al.  Beyond Equilibrium Thermodynamics , 2005 .

[2]  Jacquelien M.A. Scherpen,et al.  On Distributed Port-Hamiltonian Process Systems , 2004 .

[3]  A. Schaft,et al.  An intrinsic Hamiltonian formulation of the dynamics of LC-circuits , 1995 .

[4]  Alexander N Gorban,et al.  Invariant grids for reaction kinetics , 2003 .

[5]  I. Prigogine,et al.  Formative Processes. (Book Reviews: Self-Organization in Nonequilibrium Systems. From Dissipative Structures to Order through Fluctuations) , 1977 .

[6]  Arjan van der Schaft,et al.  Stabilization of port-controlled Hamiltonian systems via energy balancing , 1999 .

[7]  Eduardo D. Sontag.  Structure and stability of certain chemical networks and applications to the kinetic proofreading model of T-cell receptor signal transduction , 2001, IEEE Trans. Autom. Control..

[8]  Nicolas Robidoux,et al.  UNIFIED APPROACH TO HAMILTONIAN SYSTEMS, POISSON SYSTEMS, GRADIENT SYSTEMS, AND SYSTEMS WITH LYAPUNOV FUNCTIONS OR FIRST INTEGRALS , 1998 .

[9]  Shuzhi Sam Ge,et al.  Approximate dissipative Hamiltonian realization and construction of local Lyapunov functions , 2007, Syst. Control. Lett..

[10]  Alessandro Astolfi,et al.  On feedback equivalence to port controlled Hamiltonian systems , 2005, Syst. Control. Lett..

[11]  Antonio A. Alonso,et al.  Stabilization of distributed systems using irreversible thermodynamics , 2001, Autom..

[12]  Daizhan Cheng,et al.  Pseudo-Hamiltonian realization and its application , 2002, Commun. Inf. Syst..

[13]  Jacquelien M.A. Scherpen,et al.  Relating lagrangian and hamiltonian formalisms of LC circuits , 2003 .

[14]  P. Rysselberghe,et al.  Thermodynamic theory of affinity : a book of principles , 1937 .

[15]  Gábor Szederkényi,et al.  Dynamic analysis and control of biochemical reaction networks , 2008, Math. Comput. Simul..

[16]  Antonio A. Alonso,et al.  Process systems and passivity via the Clausius-Planck inequality , 1997 .