A Graph-Theoretic Condition for Delay Stability of Reaction Systems

Delay mass-action systems provide a model of chemical kinetics when past states influence the current dynamics. In this work, we provide a graph-theoretic condition for delay stability, i.e., linear stability independent of both rate constants and delay parameters. In particular, the result applies when the system has no delay, implying asymptotic stability for the ODE system. The graph-theoretic condition is about cycles in the directed species-reaction graph of the network, which encodes how different species in the system interact.

[1]  Murad Banaji,et al.  Graph-theoretic criteria for injectivity and unique equilibria in general chemical reaction systems , 2008, Adv. Appl. Math..

[2]  Elisenda Feliu,et al.  Identifying parameter regions for multistationarity , 2016, PLoS Comput. Biol..

[3]  M. Serra,et al.  Kinetics and thermodynamics of DNA, RNA, and hybrid duplex formation. , 2013, Biochemistry.

[4]  N. Destainville,et al.  Physics of base-pairing dynamics in DNA , 2015, 1510.05574.

[5]  M. Bodnar,et al.  The nonnegativity of solutions of delay differential equations , 2000, Appl. Math. Lett..

[6]  Martin Feinberg,et al.  Multiple Equilibria in Complex Chemical Reaction Networks: Ii. the Species-reactions Graph , 2022 .

[7]  Murad Banaji,et al.  P-matrices and signed digraphs , 2011, Discret. Math..

[8]  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.

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

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

[11]  János Lejtovicz,et al.  A Computer Program for the Deficiency Zero and Deficiency One Theorems , 1991, Comput. Chem..

[12]  Eduardo D. Sontag,et al.  Graph-Theoretic Analysis of Multistability and Monotonicity for Biochemical Reaction Networks , 2011 .

[13]  M. Fiedler,et al.  On matrices with non-positive off-diagonal elements and positive principal minors , 1962 .

[14]  R. A. Cox.  Biophysical Chemistry Part III: The Behavior of Biological Macromolecules , 1981 .

[15]  M. Roussel.  The Use of Delay Differential Equations in Chemical Kinetics , 1996 .

[16]  Maya Mincheva,et al.  Graph-theoretic methods for the analysis of chemical and biochemical networks. I. Multistability and oscillations in ordinary differential equation models , 2007, Journal of mathematical biology.

[17]  M. Feinberg,et al.  Structural Sources of Robustness in Biochemical Reaction Networks , 2010, Science.

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

[19]  Marcel Abendroth,et al.  Biological delay systems: Linear stability theory , 1990 .

[20]  Irving R. Epstein,et al.  Differential delay equations in chemical kinetics: Some simple linear model systems , 1990 .

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

[22]  Danielle Robbins,et al.  Theoretical foundations for traditional and generalized sensitivity functions for nonlinear delay differential equations. , 2013, Mathematical biosciences and engineering : MBE.

[23]  A. Moore.  4 – Production of Fluoroelastomers , 2006 .

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

[25]  Polly Y. Yu,et al.  Delay stability of reaction systems. , 2020, Mathematical biosciences.

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

[27]  Saqalgghmnvh Rrdrarlrl,et al.  Report of the Chicago Charitable Eye and Ear Infirmary , 1860, The Chicago Medical Journal.

[28]  Srividya Mohan,et al.  Mechanism of RNA double helix-propagation at atomic resolution. , 2009, The journal of physical chemistry. B.

[29]  Josef Hofbauer,et al.  Diagonal dominance and harmless off-diagonal delays , 2000 .

[30]  David Angeli,et al.  Graph-theoretic characterizations of monotonicity of chemical networks in reaction coordinates , 2010, Journal of mathematical biology.

[31]  A. Perelson,et al.  The arbitrary dynamic behavior of open chemical reaction systems , 1977 .

[32]  Charles R. Johnson.  Second, third, and fourth order D-stability , 1974 .

[33]  C. Chern,et al.  Emulsion polymerization mechanisms and kinetics , 2006 .

[34]  David Angeli,et al.  Combinatorial approaches to Hopf bifurcations in systems of interacting elements , 2013, 1301.7076.

[35]  M. Feinberg.  Chemical reaction network structure and the stability of complex isothermal reactors—I. The deficiency zero and deficiency one theorems , 1987 .

[36]  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.

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

[38]  Eberhard O. Voit,et al.  150 Years of the Mass Action Law , 2015, PLoS Comput. Biol..