An Efficient Characterization of Complex-Balanced, Detailed-Balanced, and Weakly Reversible Systems

Very often, models in biology, chemistry, physics, and engineering are systems of polynomial or power-law ordinary differential equations, arising from a reaction network. Such dynamical systems can be generated by many different reaction networks. On the other hand, networks with special properties (such as reversibility or weak reversibility) are known or conjectured to give rise to dynamical systems that have special properties: existence of positive steady states, persistence, permanence, and (for well-chosen parameters) complex balancing or detailed balancing. These last two are related to thermodynamic equilibrium, and therefore the positive steady states are unique and stable. We describe a computationally efficient characterization of polynomial or power-law dynamical systems that can be obtained as complex-balanced, detailed-balanced, weakly reversible, and reversible mass-action systems.

[1]  Maya Mincheva,et al.  Graph-theoretic methods for the analysis of chemical and biochemical networks. II. Oscillations in networks with delays , 2007, Journal of mathematical biology.

[2]  Gábor Szederkényi,et al.  Semistability of complex balanced kinetic systems with arbitrary time delays , 2017, Syst. Control. Lett..

[3]  Murad Banaji,et al.  Cycle Structure in SR and DSR Graphs: Implications for Multiple Equilibria and Stable Oscillation in Chemical Reaction Networks , 2010, ACSD/Petri Nets Workshops.

[4]  David F. Anderson,et al.  A Proof of the Global Attractor Conjecture in the Single Linkage Class Case , 2011, SIAM J. Appl. Math..

[5]  M. Feinberg.  Complex balancing in general kinetic systems , 1972 .

[6]  Murad Banaji,et al.  Local and Global Stability of Equilibria for a Class of Chemical Reaction Networks , 2012, SIAM J. Appl. Dyn. Syst..

[7]  G. Szederkényi,et al.  Finding complex balanced and detailed balanced realizations of chemical reaction networks , 2010, 1010.4477.

[8]  Murad Banaji,et al.  Graph-theoretic approaches to injectivity and multiple equilibria in systems of interacting elements , 2009, 0903.1190.

[9]  M. Feinberg,et al.  Chemical mechanism structure and the coincidence of the stoichiometric and kinetic subspaces , 1977 .

[10]  Gheorghe Craciun,et al.  Toric Differential Inclusions and a Proof of the Global Attractor Conjecture , 2015, 1501.02860.

[11]  David F. Anderson,et al.  Product-Form Stationary Distributions for Deficiency Zero Chemical Reaction Networks , 2008, Bulletin of mathematical biology.

[12]  Friedrich Hasenöhrl,et al.  Neuer Beweis zweier Sätze über das Wärmegleichgewicht unter mehratomigen Gasmolekülen , 2012 .

[13]  F. Horn.  Necessary and sufficient conditions for complex balancing in chemical kinetics , 1972 .

[14]  G. Szederkényi,et al.  Optimization-based design of kinetic feedbacks for nonnegative polynomial systems , 2013, 2013 IEEE 9th International Conference on Computational Cybernetics (ICCC).

[15]  Gheorghe Craciun,et al.  Mathematical Analysis of Chemical Reaction Systems , 2018, 1805.10371.

[16]  J. Bauer,et al.  Chemical reaction network theory for in-silico biologists , 2003 .

[17]  Minh-Binh Tran,et al.  A reaction network approach to the convergence to equilibrium of quantum Boltzmann equations for Bose gases , 2016, ESAIM: Control, Optimisation and Calculus of Variations.

[18]  A. Tudorascu,et al.  Chemical reaction-diffusion networks: convergence of the method of lines , 2017, Journal of Mathematical Chemistry.

[19]  Z. Tuza,et al.  Finding weakly reversible realizations of chemical reaction networks using optimization , 2011, 1103.4741.

[20]  Jeffrey D Orth,et al.  What is flux balance analysis? , 2010, Nature Biotechnology.

[21]  Balázs Boros.  Existence of Positive Steady States for Weakly Reversible Mass-Action Systems , 2019, SIAM J. Math. Anal..

[22]  Ezra Miller,et al.  A Geometric Approach to the Global Attractor Conjecture , 2013, SIAM J. Appl. Dyn. Syst..

[23]  Bao Quoc Tang,et al.  Trend to Equilibrium for Reaction-Diffusion Systems Arising from Complex Balanced Chemical Reaction Networks , 2016, SIAM J. Math. Anal..

[24]  Casian Pantea,et al.  On the Persistence and Global Stability of Mass-Action Systems , 2011, SIAM J. Math. Anal..

[25]  E. Voit,et al.  Recasting nonlinear differential equations as S-systems: a canonical nonlinear form , 1987 .

[26]  Martin Feinberg,et al.  Multiple Equilibria in Complex Chemical Reaction Networks: I. the Injectivity Property * , 2006 .

[27]  J. Velázquez,et al.  Dynamical Properties of Models for the Calvin Cycle , 2013, 1303.3621.

[28]  Julien Clinton Sprott,et al.  Coexistence and chaos in complex ecologies , 2005 .

[29]  David Angeli,et al.  A tutorial on Chemical Reaction Networks dynamics , 2009, 2009 European Control Conference (ECC).

[30]  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..

[31]  Benjamin Weitz,et al.  An inverse problem in reaction kinetics , 2011 .

[32]  M. Feinberg,et al.  Understanding bistability in complex enzyme-driven reaction networks. , 2006, Proceedings of the National Academy of Sciences of the United States of America.

[33]  R. Jackson,et al.  General mass action kinetics , 1972 .

[34]  B. Palsson,et al.  Metabolic Flux Balancing: Basic Concepts, Scientific and Practical Use , 1994, Bio/Technology.

[35]  Fedor Nazarov,et al.  Persistence and Permanence of Mass-Action and Power-Law Dynamical Systems , 2010, SIAM J. Appl. Math..

[36]  Gábor Szederkényi,et al.  Kinetic feedback design for polynomial systems , 2016 .