Graph-theoretic methods for the analysis of chemical and biochemical networks. I. Multistability and oscillations in ordinary differential equation models Authors Maya Mincheva Department of Chemistry and Biochemistry University of Lethbridge Department of Mathematics University of Wisconsin-Madison Marc R. Roussel Department of Chemistry and Biochemistry University of Lethbridge Article

First Online: 31 May 2007 Received: 24 December 2005 Revised: 15 March 2007 DOI :
10.1007/s00285-007-0099-1

Cite this article as: Mincheva, M. & Roussel, M.R. J. Math. Biol. (2007) 55: 61. doi:10.1007/s00285-007-0099-1
Abstract 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 mass-action 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 mass-action 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.

Keywords Chemical reactions Graph Multistability Oscillations This work was funded by the Natural Sciences and Engineering Research Council of Canada.

References 1.

Aguda B.D. and Clarke B.L. (1987). Bistability in chemical reaction networks: theory and application to the peroxidase-oxidase reaction.

J. Chem. Phys. 87: 3461–3470

CrossRef 2.

Angeli D. (2006). New analysis technique for multistability detection.

IEE Proc. Syst. Biol. 153: 61–69

CrossRef 3.

Asner B.A. (1970). On the total nonnegativity of the Hurwitz matrix.

SIAM J. Appl. Math. 18: 407–414

MATH CrossRef MathSciNet 4.

Berge C. (1962). The Theory of Graphs and its Applications. Wiley, New York

MATH 5.

Cinquin O. and Demongeot J. (2002). Roles of positive and negative feedback in biological systems.

C. R. Biol. 325: 1085–1095

CrossRef 6.

Cinquin O. and Demongeot J. (2002). Positive and negative feedback: Striking a balance between necessary antagonists.

J. Theor. Biol. 216: 229–241

CrossRef MathSciNet 7.

Clarke B.L. (1974). Graph theoretic approach to the stability analysis of steady state chemical reaction networks.

J. Chem. Phys. 60: 1481–1492

CrossRef MathSciNet 8.

Clarke B.L. (1974). Stability analysis of a model reaction network using graph theory.

J. Chem. Phys. 60: 1493–1501

CrossRef MathSciNet 9.

Clarke B.L. (1975). Theorems on chemical network stability.

J. Chem. Phys. 62: 773–775

CrossRef MathSciNet 10.

Clarke B.L. (1975). Stability of topologically similar chemical networks.

J. Chem. Phys. 62: 3726–3738

CrossRef 11.

Clarke B.L. (1980). Stability of complex reaction networks.

Adv. Chem. Phys. 43: 1–217

CrossRef 12.

Clarke B.L. and Jiang W. (1993). Method for deriving Hopf and saddle-node bifurcation hypersurfaces and application to a model of the Belousov-Zhabotinskii system.

J. Chem. Phys. 99: 4464–4478

CrossRef 13.

Cornish-Bowden 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 MathSciNet 14.

Craciun G. and Feinberg M. (2005). Multiple equilibria in complex chemical reaction networks: I. The injectivity property.

SIAM J. Appl. Math. 65: 1526–1546

MATH CrossRef MathSciNet 15.

Craciun G. and Feinberg M. (2006). Multiple equilibria in complex chemical reaction networks: II. The species-reactions graph.

SIAM J. Appl. Math. 66: 1321–1338

MATH CrossRef MathSciNet 16.

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 17.

Craciun G., Tang Y. and Feinberg M. (2006). Understanding bistability in complex enzyme-driven reaction networks.

Proc. Natl. Acad. Sci. USA 103: 8697–8702

CrossRef 18.

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 19.

Elowitz M. and Leibler S. (2000). A synthetic oscillatory network of transcriptional regulators.

Nature 403: 335–338

CrossRef 20.

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

MathSciNet 21.

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 22.

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 23.

Ermakov G.L. and Goldstein B.N. (2002). Simplest kinetic schemes for biochemical oscillators.

Biochemistry (Moscow) 67: 473–484

CrossRef 24.

Escher C. (1979). Models of chemical reaction systems with exactly evaluable limit cycle oscillations.

Z. Phys. B 35: 351–361

CrossRef MathSciNet 25.

Escher C. (1979). On chemical reaction systems with exactly evaluable limit cycle oscillations.

J. Chem. Phys. 70: 4435–4436

CrossRef 26.

Feinberg M. (1972). Complex balancing in general kinetic systems.

Arch. Ration. Mech. Anal. 49: 187–194

CrossRef MathSciNet 27.

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 28.

Feinberg M. (1995). The existence and uniqueness of steady states for a class of chemical reaction networks.

Arch. Ration. Mech. Anal. 132: 311–370

MATH CrossRef MathSciNet 29.

Feinberg M. (1995). Multiple steady states for chemical reaction networks of deficiency one.

Arch. Ration. Mech. Anal. 132: 371–406

MATH CrossRef MathSciNet 30.

Gantmakher F.R. (1959). Applications of the Theory of Matrices. Interscience Publishing, New York

31.

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 MathSciNet 32.

Goldbeter A. (1995). A model for circadian oscillations in the Drosophila period protein (PER).

Proc. R. Soc. Lond. B 261: 319–324

CrossRef 33.

Goldstein B. (2007). Switching mechanism for branched biochemical fluxes: graph-theoretical analysis.

Biophys. Chem. 125: 314–319

CrossRef 34.

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 35.

Goldstein B.N. and Ivanova A.N. (1987). Hormonal regulation of 6-phosphofructo-2-kinase/fructose-2, 6-bisphosphatase: Kinetic models.

FEBS Lett. 217: 212–215

CrossRef 36.

Goldstein B.N. and Maevsky A.A. (2002). Critical switch of the metabolic fluxes by phosphofructo-2-kinase:fructose-2,6-bisphosphatase. A kinetic model.

FEBS Lett. 532: 295–299

CrossRef 37.

Goldstein B.N. and Selivanov V.A. (1990). Graph-theoretic approach to metabolic pathways. Biomed. Biochim. Acta 49: 645–650

38.

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 39.

Guckenheimer J., Myers M. and Sturmfels B. (1997). Computing Hopf bifurcation I.

SIAM J. Numer. Anal. 34: 1–21

MATH CrossRef MathSciNet 40.

Harary F. (1969). Graph Theory. Addison-Wesley, Reading

41.

Holme P., Huss M. and Jeong H. (2003). Subnetwork hierarchies of biochemical pathways.

Bioinformatics 19: 532–538

CrossRef 42.

Horn F. (1972). Necessary and sufficient conditions for complex balancing in chemical kinetics.

Arch. Ration. Mech. Anal. 49: 172–186

CrossRef MathSciNet 43.

Horn F. and Jackson R. (1972). General mass action kinetics.

Arch. Ration. Mech. Anal. 47: 81–116

CrossRef MathSciNet 44.

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 45.

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

46.

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

47.

Jeffries C. (1974). Qualitative stability and digraphs in model ecosystems.

Ecology 55: 1415–1419

CrossRef 48.

Jeffries C., Klee V. and Driessche P. (1977). When is a matrix sign stable?.

Can. J. Math. 29: 315–326

MATH 49.

Klonowski W. (1983). Simplifying principles for chemical and enzyme reaction kinetics.

Biophys. Chem. 18: 73–87

CrossRef 50.

Krischer K., Eiswirth M. and Ertl G. (1992). Oscillatory CO oxidation on Pt(110): modeling of temporal self-organization.

J. Chem. Phys. 96: 9161–9172

CrossRef 51.

Kuznetsov Y.A. (1998). Elements of Applied Bifurcation Theory, 2nd edn. Springer, New York

52.

Lancaster P. and Tismenetsky M. (1985). The Theory of Matrices. Academic, Orlando

MATH 53.

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 54.

Maybee J., Olesky D., Wiener G. and Driessche P. (1989). Matrices, digraphs and determinants.

SIAM J. Matrix Anal. Appl. 10: 500–519

MATH CrossRef MathSciNet 55.

Maybee J. and Quirk J. (1969). Qualitative problems in matrix theory.

SIAM Rev. 11: 30–51

MATH CrossRef MathSciNet 56.

Mincheva, M., Roussel, M.R.: Graph-theoretic methods for the analysis of chemical and biochemical networks. II. Oscillations in networks with delays. doi:10.1007/s00285-007-0098-2

57.

Perelson A. and Wallwork D. (1977). The arbitrary dynamic behavior of open chemical reaction systems.

J. Chem. Phys. 66: 4390–4394

CrossRef 58.

Rockafellar R.T. (1970). Convex Analysis. Princeton University Press, Princeton

MATH 59.

Sauro H.M. and Kholodenko B.N. (2004). Quantitative analysis of signaling networks.

Prog. Biophys. Mol. Biol. 86: 5–43

CrossRef 60.

Schilling C.H., Schuster S., Palsson B.O. and Heinrich R. (1999). Metabolic pathway analysis: basic concepts and scientific applications in the post-genomic era.

Biotechnol. Prog. 15: 296–303

CrossRef 61.

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 62.

Schuster S. and Höfer T. (1991). Determining all extreme semi-positive conservation relations in chemical reaction systems: a test criterion for conservativity.

J. Chem. Soc. Faraday Trans. 87: 2561–2566

CrossRef 63.

Sel’kov E.E. (1968). Self-oscillations in glycolysis. 1. A simple model.

Eur. J. Biochem. 4: 79–86

CrossRef 64.

Slepchenko B.M. and Terasaki M. (2003). Cyclin aggregation and robustness of bio-switching.

Mol. Biol. Cell 14: 4695–4706

CrossRef 65.

Soulé C. (2003). Graphic requirements for multistationarity.

ComPlexUs 1: 123–133

CrossRef 66.

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 MathSciNet 67.

Tyson J.J. (1975). Classification of instabilities in chemical reaction systems.

J. Chem. Phys. 62: 1010–1015

CrossRef 68.

Tyson J.J., Chen K. and Novak B. (2001). Network dynamics and cell physiology.

Nat. Rev. Mol. Cell Biol. 2: 908–916

CrossRef 69.

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

70.

Volpert, A., Hudyaev, S.: Analyses in Classes of Discontinuous Functions and Equations of Mathematical Physics, Chap. 12. Martinus Nijhoff, Dordrecht (1985)

71.

Volpert, A., Ivanova, A.: Mathematical models in chemical kinetics. In: Mathematical Modeling (Russian), Nauka, Moscow, pp. 57–102 (1987)

72.

Walker D.A. (1992). Concerning oscillations.

Photosynth. Res. 34: 387–395

CrossRef 73.

Walter W. (1998). Ordinary Differential Equations. Springer Verlag, New York

MATH 74.

Wilhelm T. and Heinrich R. (1995). Smallest chemical reaction system with Hopf bifurcation.

J. Math. Chem. 17: 1–14

MATH CrossRef MathSciNet 75.

Zeigarnik A.V. and Temkin O.N. (1994). A graph-theoretical model of complex reaction mechanisms: bipartite graphs and the stoichiometry of complex reactions. Kinet. Catal. 35: 647–655

76.

Zevedei-Oancea I. and Schuster S. (2005). A theoretical framework for detecting signal transfer routes in signalling networks.

Comput. Chem. Eng. 29: 597–617

CrossRef