A Petri Net Approach to Persistence Analysis in Chemical Reaction Networks

A positive dynamical system is said to be persistent if every solution that starts in the interior of the positive orthant does not approach the boundary of this orthant. For chemical reaction networks and other models in biology, persistence represents a non-extinction property: if every species is present at the start of the reaction, then no species will tend to be eliminated in the course of the reaction. This paper provides checkable necessary as well as sufficient conditions for persistence of chemical species in reaction networks, and the applicability of these conditions is illustrated on some examples of relatively high dimension which arise in molecular biology. More specific results are also provided for reactions endowed with mass-action kinetics. Overall, the results exploit concepts and tools from Petri net theory as well as ergodic and recurrence theory.

[1]  C. Petri.  Kommunikation mit Automaten , 1962 .

[2]  G. P. Szegö,et al.  Stability theory of dynamical systems , 1970 .

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

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

[5]  Michel Hack,et al.  ANALYSIS OF PRODUCTION SCHEMATA BY PETRI NETS , 1972 .

[6]  M. Feinberg,et al.  Dynamics of open chemical systems and the algebraic structure of the underlying reaction network , 1974 .

[7]  T. Gard,et al.  Persistence in food chains with general interactions , 1980 .

[8]  James Lyle Peterson,et al.  Petri net theory and the modeling of systems , 1981 .

[9]  K. Brown,et al.  Graduate Texts in Mathematics , 1982 .

[10]  J. Aubin,et al.  Differential inclusions set-valued maps and viability theory , 1984 .

[11]  Paul Waltman,et al.  Uniformly persistent systems , 1986 .

[12]  Paul Waltman,et al.  Persistence in dynamical systems , 1986 .

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

[14]  Josef Hofbauer,et al.  Uniform persistence and repellors for maps , 1989 .

[15]  Eduardo D. Sontag,et al.  Mathematical Control Theory: Deterministic Finite Dimensional Systems , 1990 .

[16]  Michael L. Mavrovouniotis,et al.  Petri Net Representations in Metabolic Pathways , 1993, ISMB.

[17]  Ralf Hofestädt,et al.  A petri net application to model metabolic processes , 1994 .

[18]  M. Feinberg.  The existence and uniqueness of steady states for a class of chemical reaction networks , 1995 .

[19]  Hal L. Smith,et al.  Monotone Dynamical Systems: An Introduction To The Theory Of Competitive And Cooperative Systems (Mathematical Surveys And Monographs) By Hal L. Smith , 1995 .

[20]  Wolfgang Reisig,et al.  Lectures on Petri Nets I: Basic Models , 1996, Lecture Notes in Computer Science.

[21]  Chi-Ying F. Huang,et al.  Ultrasensitivity in the mitogen-activated protein kinase cascade. , 1996, Proceedings of the National Academy of Sciences of the United States of America.

[22]  Thomas Lengauer,et al.  Pathway analysis in metabolic databases via differetial metabolic display (DMD) , 2000, German Conference on Bioinformatics.

[23]  MengChu Zhou,et al.  Modeling, Simulation, and Control of Flexible Manufacturing Systems - A Petri Net Approach , 1999, Series in Intelligent Control and Intelligent Automation.

[24]  C. Widmann,et al.  Mitogen-activated protein kinase: conservation of a three-kinase module from yeast to human. , 1999, Physiological reviews.

[25]  Leonid A. Bunimovich,et al.  Dynamical Systems, Ergodic Theory and Applications , 2000 .

[26]  H R Thieme,et al.  Uniform persistence and permanence for non-autonomous semiflows in population biology. , 2000, Mathematical biosciences.

[27]  Xiao-Qiang Zhao,et al.  Chain Transitivity, Attractivity, and Strong Repellors for Semidynamical Systems , 2001 .

[28]  R. Altman,et al.  MODELING BIOLOGICAL PROCESSES USING WORKFLOW AND PETRI NET MODELS , 2001 .

[29]  D. Lauffenburger,et al.  A Computational Study of Feedback Effects on Signal Dynamics in a Mitogen‐Activated Protein Kinase (MAPK) Pathway Model , 2001, Biotechnology progress.

[30]  Hartmann J. Genrich,et al.  Executable Petri net models for the analysis of metabolic pathways , 2001, International Journal on Software Tools for Technology Transfer.

[31]  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..

[32]  Russ B. Altman,et al.  Modelling biological processes using workflow and Petri Net models , 2002, Bioinform..

[33]  David Angeli,et al.  Monotone control systems , 2003, IEEE Trans. Autom. Control..

[34]  Stefan Schuster,et al.  Topological analysis of metabolic networks based on Petri net theory , 2003, Silico Biol..

[35]  Janet B. Jones-Oliveira,et al.  A Computational Model for the Identification of Biochemical Pathways in the Krebs Cycle , 2003, J. Comput. Biol..

[36]  Eduardo Sontag,et al.  Untangling the wires: A strategy to trace functional interactions in signaling and gene networks , 2002, Proceedings of the National Academy of Sciences of the United States of America.

[37]  Eduardo Sontag,et al.  Steady-states of receptor-ligand dynamics: a theoretical framework. , 2004, Journal of theoretical biology.

[38]  B. Kholodenko,et al.  Signaling switches and bistability arising from multisite phosphorylation in protein kinase cascades , 2004, The Journal of cell biology.

[39]  E D Gilles,et al.  Using chemical reaction network theory to discard a kinetic mechanism hypothesis. , 2005, Systems biology.

[40]  David Angeli,et al.  On the structural monotonicity of chemical reaction networks , 2006, Proceedings of the 45th IEEE Conference on Decision and Control.

[41]  B. L. Clarke.  Stability of Complex Reaction Networks , 2007 .

[42]  Eduardo Sontag,et al.  Monotone Chemical Reaction Networks , 2007 .

[43]  David Angeli,et al.  A Global Convergence Result for Strongly Monotone Systems with Positive Translation Invariance , 2008 .

[44]  David Angeli,et al.  Translation-invariant monotone systems, and a global convergence result for enzymatic futile cycles ☆ , 2008 .