Graphtheoretic methods for the analysis of chemical and biochemical networks. I. Multistability and oscillations in ordinary differential equation models
 Maya Mincheva,
 Marc R. Roussel
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A chemical mechanism is a model of a chemical reaction network consisting of a set of elementary reactions that express how molecules react with each other. In classical massaction kinetics, a mechanism implies a set of ordinary differential equations (ODEs) which govern the time evolution of the concentrations. In this article, ODE models of chemical kinetics that have the potential for multiple positive equilibria or oscillations are studied. We begin by considering some methods of stability analysis based on the digraph of the Jacobian matrix. We then prove two theorems originally given by A. N. Ivanova which correlate the bifurcation structure of a massaction model to the properties of a bipartite graph with nodes representing chemical species and reactions. We provide several examples of the application of these theorems.
 Aguda B.D. and Clarke B.L. (1987). Bistability in chemical reaction networks: theory and application to the peroxidaseoxidase reaction. J. Chem. Phys. 87: 3461–3470 CrossRef
 Angeli D. (2006). New analysis technique for multistability detection. IEE Proc. Syst. Biol. 153: 61–69 CrossRef
 Asner B.A. (1970). On the total nonnegativity of the Hurwitz matrix. SIAM J. Appl. Math. 18: 407–414 CrossRef
 Berge C. (1962). The Theory of Graphs and its Applications. Wiley, New York
 Cinquin O. and Demongeot J. (2002). Roles of positive and negative feedback in biological systems. C. R. Biol. 325: 1085–1095 CrossRef
 Cinquin O. and Demongeot J. (2002). Positive and negative feedback: Striking a balance between necessary antagonists. J. Theor. Biol. 216: 229–241 CrossRef
 Clarke B.L. (1974). Graph theoretic approach to the stability analysis of steady state chemical reaction networks. J. Chem. Phys. 60: 1481–1492 CrossRef
 Clarke B.L. (1974). Stability analysis of a model reaction network using graph theory. J. Chem. Phys. 60: 1493–1501 CrossRef
 Clarke B.L. (1975). Theorems on chemical network stability. J. Chem. Phys. 62: 773–775 CrossRef
 Clarke B.L. (1975). Stability of topologically similar chemical networks. J. Chem. Phys. 62: 3726–3738 CrossRef
 Clarke B.L. (1980). Stability of complex reaction networks. Adv. Chem. Phys. 43: 1–217 CrossRef
 Clarke B.L. and Jiang W. (1993). Method for deriving Hopf and saddlenode bifurcation hypersurfaces and application to a model of the BelousovZhabotinskii system. J. Chem. Phys. 99: 4464–4478 CrossRef
 CornishBowden A. and Hofmeyr J.H.S. (2002). The role of stoichiometric analysis in studies of metabolism: an example. J. Theor. Biol. 216: 179–191 CrossRef
 Craciun G. and Feinberg M. (2005). Multiple equilibria in complex chemical reaction networks: I. The injectivity property. SIAM J. Appl. Math. 65: 1526–1546 CrossRef
 Craciun G. and Feinberg M. (2006). Multiple equilibria in complex chemical reaction networks: II. The speciesreactions graph. SIAM J. Appl. Math. 66: 1321–1338 CrossRef
 Craciun G. and Feinberg M. (2006). Multiple equilibria in complex chemical reaction networks: extensions to entrapped species models. IEE Proc. Syst. Biol. 153: 179–186 CrossRef
 Craciun G., Tang Y. and Feinberg M. (2006). Understanding bistability in complex enzymedriven reaction networks. Proc. Natl. Acad. Sci. USA 103: 8697–8702 CrossRef
 Eiswirth M., Freund A. and Ross J. (1991). Mechanistic classification of chemical oscillators and the role of species. Adv. Chem. Phys. 80: 127–199 CrossRef
 Elowitz M. and Leibler S. (2000). A synthetic oscillatory network of transcriptional regulators. Nature 403: 335–338 CrossRef
 Engel G. and Schneider H. (1976). The Hadamard–Fisher inequality for a class of matrices defined by eigenvalue monotonicity. Linear Multilinear Algebra 4: 155–176
 Ermakov G.L. (2003). A theoretical graph method for search and analysis of critical phenomena in biochemical systems. I. Graphical rules for detecting oscillators. Biochemistry (Moscow) 68: 1109–1120 CrossRef
 Ermakov G.L. (2003). A theoretical graph method for search and analysis of critical phenomena in biochemical systems. II. Kinetic models of biochemical oscillators including two and three substances. Biochemistry (Moscow) 68: 1121–1131 CrossRef
 Ermakov G.L. and Goldstein B.N. (2002). Simplest kinetic schemes for biochemical oscillators. Biochemistry (Moscow) 67: 473–484 CrossRef
 Escher C. (1979). Models of chemical reaction systems with exactly evaluable limit cycle oscillations. Z. Phys. B 35: 351–361 CrossRef
 Escher C. (1979). On chemical reaction systems with exactly evaluable limit cycle oscillations. J. Chem. Phys. 70: 4435–4436 CrossRef
 Feinberg M. (1972). Complex balancing in general kinetic systems. Arch. Ration. Mech. Anal. 49: 187–194 CrossRef
 Feinberg M. (1987). Chemical reaction network structure and the stability of complex isothermal reactors—I. The deficiency zero and deficiency one theorems. Chem. Eng. Sci. 42: 2229–2268 CrossRef
 Feinberg M. (1995). The existence and uniqueness of steady states for a class of chemical reaction networks. Arch. Ration. Mech. Anal. 132: 311–370 CrossRef
 Feinberg M. (1995). Multiple steady states for chemical reaction networks of deficiency one. Arch. Ration. Mech. Anal. 132: 371–406 CrossRef
 Gantmakher F.R. (1959). Applications of the Theory of Matrices. Interscience Publishing, New York
 Gatermann K., Eiswirth M. and Sensse A. (2005). Toric ideals and graph theory to analyze Hopf bifurcations in mass action systems. J. Symb. Comp. 40: 1361–1382 CrossRef
 Goldbeter A. (1995). A model for circadian oscillations in the Drosophila period protein (PER). Proc. R. Soc. Lond. B 261: 319–324 CrossRef
 Goldstein B. (2007). Switching mechanism for branched biochemical fluxes: graphtheoretical analysis. Biophys. Chem. 125: 314–319 CrossRef
 Goldstein B.N., Ermakov G., Centelles J.J., Westerhoff H.V. and Cascante M. (2004). What makes biochemical networks tick? a graphical tool for the identification of oscillophores. Eur. J. Biochem. 271: 3877–3887 CrossRef
 Goldstein B.N. and Ivanova A.N. (1987). Hormonal regulation of 6phosphofructo2kinase/fructose2, 6bisphosphatase: Kinetic models. FEBS Lett. 217: 212–215 CrossRef
 Goldstein B.N. and Maevsky A.A. (2002). Critical switch of the metabolic fluxes by phosphofructo2kinase:fructose2,6bisphosphatase. A kinetic model. FEBS Lett. 532: 295–299 CrossRef
 Goldstein B.N. and Selivanov V.A. (1990). Graphtheoretic approach to metabolic pathways. Biomed. Biochim. Acta 49: 645–650
 Gray P. and Scott S.K. (1984). Autocatalytic reactions in the isothermal continuous stirred tank reactor: oscillations and instabilities in the system A + 2B→ 3B; B→C. Chem. Eng. Sci. 39: 1087–1097 CrossRef
 Guckenheimer J., Myers M. and Sturmfels B. (1997). Computing Hopf bifurcation I. SIAM J. Numer. Anal. 34: 1–21 CrossRef
 Harary F. (1969). Graph Theory. AddisonWesley, Reading
 Holme P., Huss M. and Jeong H. (2003). Subnetwork hierarchies of biochemical pathways. Bioinformatics 19: 532–538 CrossRef
 Horn F. (1972). Necessary and sufficient conditions for complex balancing in chemical kinetics. Arch. Ration. Mech. Anal. 49: 172–186 CrossRef
 Horn F. and Jackson R. (1972). General mass action kinetics. Arch. Ration. Mech. Anal. 47: 81–116 CrossRef
 Hunt K.L.C., Hunt P.M. and Ross J. (1990). Nonlinear dynamics and thermodynamics of chemical reactions far from equilibrium. Annu. Rev. Phys. Chem. 41: 409–439 CrossRef
 Ivanova A.N. (1979). Conditions for the uniqueness of the stationary states of kinetic systems, connected with the structures of their reaction mechanisms. 1. Kinet. Katal. 20: 1019–1023
 Ivanova A.N. and Tarnopolskii B.L. (1979). One approach to the determination of a number of qualitative features in the behavior of kinetic systems and realization of this approach in a computer (critical conditions, autooscillations). Kinet. Katal. 20: 1541–1548
 Jeffries C. (1974). Qualitative stability and digraphs in model ecosystems. Ecology 55: 1415–1419 CrossRef
 Jeffries C., Klee V. and Driessche P. (1977). When is a matrix sign stable?. Can. J. Math. 29: 315–326
 Klonowski W. (1983). Simplifying principles for chemical and enzyme reaction kinetics. Biophys. Chem. 18: 73–87 CrossRef
 Krischer K., Eiswirth M. and Ertl G. (1992). Oscillatory CO oxidation on Pt(110): modeling of temporal selforganization. J. Chem. Phys. 96: 9161–9172 CrossRef
 Kuznetsov Y.A. (1998). Elements of Applied Bifurcation Theory, 2nd edn. Springer, New York
 Lancaster P. and Tismenetsky M. (1985). The Theory of Matrices. Academic, Orlando
 Markevich N., Hoek J. and Kholodenko B. (2004). Signaling switches and bistability arising from multiple phosphorylation in protein kinase cascades. J. Cell Biol. 164: 353–359 CrossRef
 Maybee J., Olesky D., Wiener G. and Driessche P. (1989). Matrices, digraphs and determinants. SIAM J. Matrix Anal. Appl. 10: 500–519 CrossRef
 Maybee J. and Quirk J. (1969). Qualitative problems in matrix theory. SIAM Rev. 11: 30–51 CrossRef
 Mincheva, M., Roussel, M.R.: Graphtheoretic methods for the analysis of chemical and biochemical networks. II. Oscillations in networks with delays. doi:10.1007/s0028500700982
 Perelson A. and Wallwork D. (1977). The arbitrary dynamic behavior of open chemical reaction systems. J. Chem. Phys. 66: 4390–4394 CrossRef
 Rockafellar R.T. (1970). Convex Analysis. Princeton University Press, Princeton
 Sauro H.M. and Kholodenko B.N. (2004). Quantitative analysis of signaling networks. Prog. Biophys. Mol. Biol. 86: 5–43 CrossRef
 Schilling C.H., Schuster S., Palsson B.O. and Heinrich R. (1999). Metabolic pathway analysis: basic concepts and scientific applications in the postgenomic era. Biotechnol. Prog. 15: 296–303 CrossRef
 Schlosser P.M. and Feinberg M. (1994). A theory of multiple steady states in isothermal homogeneous CFSTRs with many reactions. Chem. Eng. Sci. 49: 1749–1767 CrossRef
 Schuster S. and Höfer T. (1991). Determining all extreme semipositive conservation relations in chemical reaction systems: a test criterion for conservativity. J. Chem. Soc. Faraday Trans. 87: 2561–2566 CrossRef
 Sel’kov E.E. (1968). Selfoscillations in glycolysis. 1. A simple model. Eur. J. Biochem. 4: 79–86 CrossRef
 Slepchenko B.M. and Terasaki M. (2003). Cyclin aggregation and robustness of bioswitching. Mol. Biol. Cell 14: 4695–4706 CrossRef
 Soulé C. (2003). Graphic requirements for multistationarity. ComPlexUs 1: 123–133 CrossRef
 Thomas R. and Kaufman M. (2001). Multistationarity, the basis of cell differentiation and memory. I. Structural conditions of multistationarity and other nontrivial behavior. Chaos 11: 170–179 CrossRef
 Tyson J.J. (1975). Classification of instabilities in chemical reaction systems. J. Chem. Phys. 62: 1010–1015 CrossRef
 Tyson J.J., Chen K. and Novak B. (2001). Network dynamics and cell physiology. Nat. Rev. Mol. Cell Biol. 2: 908–916 CrossRef
 Vasilev V., Volpert A. and Hudyaev S. (1973). A method of quasistationary concentrations for the equations of chemical kinetics. Zh. Vychysl. Mat. Fiz. 13: 683–697
 Volpert, A., Hudyaev, S.: Analyses in Classes of Discontinuous Functions and Equations of Mathematical Physics, Chap. 12. Martinus Nijhoff, Dordrecht (1985)
 Volpert, A., Ivanova, A.: Mathematical models in chemical kinetics. In: Mathematical Modeling (Russian), Nauka, Moscow, pp. 57–102 (1987)
 Walker D.A. (1992). Concerning oscillations. Photosynth. Res. 34: 387–395 CrossRef
 Walter W. (1998). Ordinary Differential Equations. Springer Verlag, New York
 Wilhelm T. and Heinrich R. (1995). Smallest chemical reaction system with Hopf bifurcation. J. Math. Chem. 17: 1–14 CrossRef
 Zeigarnik A.V. and Temkin O.N. (1994). A graphtheoretical model of complex reaction mechanisms: bipartite graphs and the stoichiometry of complex reactions. Kinet. Catal. 35: 647–655
 ZevedeiOancea I. and Schuster S. (2005). A theoretical framework for detecting signal transfer routes in signalling networks. Comput. Chem. Eng. 29: 597–617 CrossRef
 Title
 Graphtheoretic methods for the analysis of chemical and biochemical networks. I. Multistability and oscillations in ordinary differential equation models
 Journal

Journal of Mathematical Biology
Volume 55, Issue 1 , pp 6186
 Cover Date
 20070701
 DOI
 10.1007/s0028500700991
 Print ISSN
 03036812
 Online ISSN
 14321416
 Publisher
 SpringerVerlag
 Additional Links
 Topics
 Keywords

 Chemical reactions
 Graph
 Multistability
 Oscillations
 34C23
 Industry Sectors
 Authors

 Maya Mincheva ^{(1)} ^{(2)}
 Marc R. Roussel ^{(1)}
 Author Affiliations

 1. Department of Chemistry and Biochemistry, University of Lethbridge, Lethbridge, AB T1K 3M4, Canada
 2. Department of Mathematics, University of WisconsinMadison, Madison, WI, 537061388, USA