×

zbMATH — the first resource for mathematics

Toric ideals and graph theory to analyze Hopf bifurcations in mass action systems. (English) Zbl 1120.13033
Summary: A family of polynomial differential systems describing the behavior of a chemical reaction network with generalized mass action kinetics is investigated. The coefficients and monomials are given by graphs. The aim of this investigation is to clarify the algebraic-discrete aspects of a Hopf bifurcation in these special differential equations. We apply concepts from toric geometry and convex geometry. As usual in stoichiometric network analysis we consider the solution set as a convex polyhedral cone and we intersect it with the deformed toric variety of the monomials. Using Gröbner bases the polynomial entries of the Jacobian are expressed in different coordinate systems. Then the Hurwitz criterion is applied in order to determine parameter regions where a Hopf bifurcation occurs. Examples from chemistry illustrate the theoretical results.

MSC:
13P10 Gröbner bases; other bases for ideals and modules (e.g., Janet and border bases)
68W30 Symbolic computation and algebraic computation
14M25 Toric varieties, Newton polyhedra, Okounkov bodies
05C50 Graphs and linear algebra (matrices, eigenvalues, etc.)
34C23 Bifurcation theory for ordinary differential equations
37G15 Bifurcations of limit cycles and periodic orbits in dynamical systems
80A30 Chemical kinetics in thermodynamics and heat transfer
92E20 Classical flows, reactions, etc. in chemistry
PDF BibTeX XML Cite
Full Text: DOI
References:
[1] Capani, A.; Niesi, G.; Robbiano, L., Cocoa, A system for doing computations in commutative algebra, (2000), Available via anonymous ftp from
[2] Clarke, B.L., Stability of complex reaction networks, Adv. chem. phys., 43, 1-213, (1980)
[3] Cox, D.; Little, J.; O’Shea, D., Ideals, varieties, and algorithms, an introduction to computational algebraic geometry and commutative algebra, () · Zbl 0861.13012
[4] Dupont, G.; Goldbeter, A., Allosteric regulation, cooperativity, and biochemical oscillations, Biophys. chem., 37, 341-353, (1990)
[5] Eiswirth, M., Instability and oscillations in chemistry, Suuri kagaku, 372, 59-64, (1994)
[6] Eiswirth, M.; Bürger, J.; Strasser, P.; Ertl, G., Oscillating langmuir – hinshelwood mechanisms, J. phys. chem., 100, 19118-19123, (1996)
[7] Eiswirth, M.; Freund, A.; Ross, J., Mechanistic classification of chemical oscillators and the role of species, Adv. chem. phys., 127-197, (1991)
[8] Feinberg, M., The existence and uniqueness of steady states for a class of chemical reaction networks, Arch. ration. mech. anal., 132, 311-370, (1995) · Zbl 0853.92024
[9] Gantmacher, F.R., Matrizentheorie, (1986), Springer Berlin
[10] Gatermann, K., Counting stable solutions of sparse polynomial systems in chemistry, (), 53-69 · Zbl 1009.68198
[11] Gatermann, K.; Huber, B., A family of sparse polynomial systems arising in chemical reaction systems, J. symbolic comput., 33, 275-305, (2002) · Zbl 0994.92040
[12] Gatermann, K.; Sensse, A., Storch 1.0, stability of reactions in chemistry. Maple 7, (2002), Available at
[13] Gatermann, K., Wolfrum, M., 2003. Bernstein’s second theorem and Viro’s method for sparse polynomial systems in chemistry. Adv. Appl. Math. (in press). Available at http://www.zib.de/gatermann/publi.html · Zbl 1075.65074
[14] Gawrilov, E., Joswig, M., 1997. polymake, version 2.0. http://www.math.tu-berlin.de/diskregeom/polymake, 1997-2003. With contributions by T. Schröder and N. Witte
[15] Greuel, G.-M., Pfister, G., Schönemann, H., 2001. Singular 2.0. A Computer Algebra System for Polynomial Computations, Centre for Computer Algebra, University of Kaiserslautern. http://www.singular.uni-kl.de.
[16] Guckenheimer, J.; Holmes, P., Nonlinear oscillations, dynamical systems, and bifurcations of vector fields, () · Zbl 0515.34001
[17] Hanusse, P., Etudes des systemes dissipatifs chimiques a deux et trois especes intermediaires, C. R. acad. sci. Paris, 277, C-263, (1973)
[18] Heinrich, R.; Schuster, S., The regulation of cellular systems, (1996), Chapman & Hall New York · Zbl 0895.92013
[19] Holme, P.; Huss, M.; Jeong, H., Subnetwork hierarchies of biochemical pathways, Bioinformatics, 19, 532-538, (2003)
[20] Horn, F.; Jackson, R., General mass action kinetics, Arch. ration mech. anal., 41, 81-116, (1972)
[21] Huber, B.; Thomas, R., Tigers, (1999), Available at
[22] Huber, B.; Thomas, R., Computing Gröbner fans of toric ideals, Exp. math., 9, 321-331, (2000) · Zbl 0978.13019
[23] Ivanova, A.N., Conditions for uniqueness of the stationary states of kinetic systems, connected with the structures of their reaction mechanisms, Kinet. katal., 20, 4, 1019-1023, (1979)
[24] Jeffries, C.; Klee, V.; van den Driessche, P., When is a matrix sign-stable?, Canad. J. math., 29, 315-326, (1977) · Zbl 0383.15005
[25] Klamt, S., 2002. FluxAnalyzer. An interactive program in Matlab. Max-Planck Institut Magdeburg. Available on request
[26] Klamt, S.; Stelling, J.; Ginkel, M.; Gilles, E.D., Fluxanalyzer: exploring structure, pathways and flux distributions in metabolic networks on interactive flux maps, Bioinformatics, 19, 2, 261-269, (2003)
[27] Klee, V.; van den Driessche, P., Linear algorithms for testing the sign-stability of a matrix and for finding z-maximum matchings in acyclic graphs, Numer. math., 28, 273-285, (1977) · Zbl 0348.65032
[28] Murray, J.D., Mathematical biology, (2002), Springer New York
[29] Prajna, S.; Papachristodoulou, A.; Parrilo, P.A., SOSTOOLS: sum of squares optimization toolbox for MATLAB, (2002), Available from
[30] Schuster, S., Elementary flux modes in biochemical reaction systems: algebraic properties, validated calculation procedure and example from nucleotide metabolism, J. math. biol., 45, 153-181, (2002) · Zbl 1012.92016
[31] Sensse, A., 2002. Algebraic methods for the analysis of Hopf bifurcations in biochemical networks. Diploma thesis. Humboldt Universität Berlin
[32] Sensse, A.; Gatermann, K.; Eiswirth, M., Analytic solution for the electrocatalytic oxidation of formic acid, J. electroanal. chem., 577, 35-46, (2005)
[33] Sturmfels, B., Gröbner bases and convex polytopes, () · Zbl 0856.13020
[34] Theis, C., Singular library toric.lib, (2000)
[35] Vol’pert, A.I.; Hudjaev, S.I., Equations of chemical kinetics, (), 607-667, (Chapter 12)
[36] Bican Xia, 2003. Private communication
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.