Abstract
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.
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References
D. Angeli, P. De Leenheer, E.D. Sontag, “On the structural monotonicity of chemical reaction networks,” Proc. IEEE Conf. Decision and Control, San Diego, Dec. 2006, IEEE Publications, (2006), to appear.
D. Angeli, E.D. Sontag, “Monotone control systems,” IEEE Trans. Autom. Control 48 (2003), pages 1684–1698.
D. Angeli, J.E. Ferrell, Jr., E.D. Sontag, “Detection of multi-stability, bifurcations, and hysteresis in a large class of biological positive-feedback systems,” Proceedings of the National Academy of Sciences USA 101 (2004), pages 1822–1827.
D. Angeli, E.D. Sontag, “A global convergence result for strongly monotone systems with positive translation invariance,” Nonlinear Analysis Series B: Real World Applications, to appear.
J-P. Aubin, A. Cellina, Differential Inclusions: Set-Valued Maps and Viability Theory, Springer-Verlag, 1984.
N.P. Bhatia, G.P. Szegö, Stability Theory of Dynamical Systems, Springer-Verlag, Berlin, 1970.
L.A. Bunimovich, S.G. Dani, R.L. Dobrushin, et al., Dynamical Systems, Ergodic Theory and Applications, Springer-Verlag, 2000.
G. Butler, P. Waltman, “Persistence in dynamical systems,” J. Differential Equations 63 (1986), pages 255–263.
G. Butler, H.I. Freedman, P. Waltman, “Uniformly persistent systems,” Proc. Am. Math. Soc. 96 (1986), pages 425–430.
M. Chaves, E.D. Sontag, R.J. Dinerstein, “Steady-states of receptor-ligand dynamics: A theoretical framework,” J. Theoretical Biology 227 (2004), pages 413–428.
B.L. Clarke, “Stability of complex reaction networks,” Adv. Chem. Phys. 43 (1980), pages 1–216.
F. Commoner, “Deadlocks in Petri Nets,” Tech. Report, Applied Data Research Inc. Wakefield, Massachussetts (1972)
C. Conradi, J. Saez-Rodriguez, E.-D. Gilles, J. Raisch “Using chemical reaction network theory to discard a kinetic mechanism hypothesis,” in Proc. FOSBE 2005 (Foundations of Systems Biology in Engineering), Santa Barbara, Aug. 2005. pages 325–328.
P. De Leenheer, D. Angeli, E.D. Sontag, “Monotone chemical reaction networks,” J. Mathematical Chemistry (2006), to appear.
M. Feinberg, F.J.M. Horn, “Dynamics of open chemical systems and algebraic structure of underlying reaction network,” Chemical Engineering Science 29 (1974), pages 775–787.
M. Feinberg, “Chemical reaction network structure and the stabiliy of complex isothermal reactors-I. The deficiency zero and deficiency one theorems,” Review Article 25, Chemical Engr. Sci. 42(1987), pp. 2229–2268.
M. Feinberg, “The existence and uniqueness of steady states for a class of chemical reaction networks,” Archive for Rational Mechanics and Analysis 132 (1995), pp. 311–370.
M. Feinberg, “Lectures on chemical reaction networks,” Lectures at the Mathematics Research Center, University of Wisconsin, 1979. http://www.che.eng.ohio-state.edu/~feinberg/
T.C. Gard, “Persistence in food webs with general interactions,” Math. Biosci. 51 (1980), pages 165–174.
H. Genrich, R. Küffner, K. Voss, “Executable Petri net models for the analysis of metabolic pathways,” Int. J. on Software Tools for Technology Transfer (STTT) 3 (2001), pages 394–404.
M.W. Hirsch, H.L. Smith, X. Zhao, “Chain transitivity, attractivity, and strong repellors for semidynamical systems,” Journal of Dynamics and Differential Equations 13 (2001), pages 107–131.
M.H.T. Hack, “Analysis of production schemata by Petri-Nets,” Master Thesis, MIT (1972)
M. Hirsch, H.L. Smith, in Handbook of Differential Equations, Ordinary Differential Equations (second volume) (A. Canada, P. Drabek, and A. Fonda, eds.), Elsevier, 2005.
J. Hofbauer, J. W.-H. So, “Uniform persistence and repellors for maps,” Proceedings of the American Mathematical Society 107 (1989), pages 1137–1142.
R. Hofestädt, “A Petri net application to model metabolic processes,” Syst. Anal. Mod. Simul. 16 (1994), pages 113–122.
F.J.M. Horn, R. Jackson, “General mass action kinetics,” Arch. Rational Mech. Anal. 49 (1972), pp. 81–116.
F.J.M. Horn, “The dynamics of open reaction systems,” in Mathematical aspects of chemical and biochemical problems and quantum chemistry (Proc. SIAM-AMS Sympos. Appl. Math., New York, 1974), pp. 125–137. SIAM-AMS Proceedings, Vol. VIII, Amer. Math. Soc., Providence, 1974.
C.-Y.F. Huang, Ferrell, J.E., “Ultrasensitivity in the mitogen-activated protein kinase cascade,” Proc. Natl. Acad. Sci. USA 93 (1996), pages 10078–10083.
J._G. Kemeny, A.W. Knapp, J.L. Snell and J.G. Kemeny, Denumerable Markov Chains, Graduate Texts in Mathematics, Springer-Verlag, 1976.
R. Küffner, R. Zimmer, T. Lengauer, “Pathway analysis in metabolic databases via differential metabolic display (DMD),” Bioinformatics 16 (2000), pages 825–836.
A.R. Asthagiri and D.A. Lauffenburger, “A computational study of feedback effects on signal dynamics in a mitogen-activated protein kinase (MAPK) pathway model,” Biotechnol. Prog. 17 (2001), pages 227–239.
N.I. Markevich, J.B. Hoek, B.N. Kholodenko, “Signaling switches and bistability arising from multisite phosphorilation in protein kinase cascades” Journal of Cell Biology, Vol. 164, N.3, pp. 353–359, 2004
J.S. Oliveira, C.G. Bailey, J.B. Jones-Oliveira, Dixon, D.A., Gull, D.W., Chandler, M.L.A., “A computational model for the identification of biochemical pathways in the Krebs cycle,” J. Comput. Biol. 10 (2003), pages 57–82.
M. Peleg, M., I. Yeh, R. Altman, “Modeling biological processes using workflow and Petri net models,” Bioinformatics 18 (2002), pages 825–837.
J.L. Peterson, Petri Net Theory and the Modeling of Systems Prentice Hall, Lebanon, Indiana 1981.
C.A. Petri, Kommunikation mit Automaten Ph.D. Thesis, University of Bonn, 1962.
V.N. Reddy, M.L. Mavrovouniotis, M.N. Liebman, “Petri net representations in metabolic pathways.,” Proc. Int. Conf. Intell. Syst. Mol. Biol. 1 (1993), pages 328–336.
G. Rozenberg, W. Reisig, Lectures on Petri Nets Basic Models: Basic Models, Lecture Notes in Computer Science 1491, Springer-Verlag, 1998.
H.L. Smith, Monotone dynamical systems: An introduction to the theory of competitive and cooperative systems, Mathematical Surveys and Monographs, vol. 41 (AMS, Providence, RI, 1995).
E.D. Sontag, “Structure and stability of certain chemical networks and applications to the kinetic proofreading model of T-cell receptor signal transduction,” IEEE Trans. Autom. Control 46 (2001), pages 1028–1047. (Errata in IEEE Trans. Autom. Control 47 (2002): 705.)
E.D. Sontag, Mathematical Control Theory: Deterministic Finite Dimensional Systems, Second Edition Springer, New York 1998.
H.R. Thieme, “Uniform persistence and permanence for non-autonomous semiflows in population biology,” Math. Biosci. 166 (2000), pages 173–201.
C. Widmann, G. Spencer, M.B. Jarpe, G.L. Johnson, G.L., “Mitogen-activated protein kinase: Conservation of a three-kinase module from yeast to human,” Physiol. Rev. 79 (1999), pages 143–180.
I. Zevedei-Oancea, S. Schuster, “Topological analysis of metabolic networks based on Petri net theory,” In Silico Biol. 3 (2003), paper 0029.
M. Zhou, Modeling, Simulation, and Control of Flexible Manufacturing Systems: A Petri Net Approach World Scientific Publishing, Hong Kong, 1999.
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Angeli, D., De Leenheer, P., Sontag, E. (2007). A Petri Net Approach to Persistence Analysis in Chemical Reaction Networks. In: Queinnec, I., Tarbouriech, S., Garcia, G., Niculescu, SI. (eds) Biology and Control Theory: Current Challenges. Lecture Notes in Control and Information Sciences, vol 357. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-540-71988-5_9
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