Abstract
Dynamical systems arising from chemical reaction networks with mass action kinetics are the subject of chemical reaction network theory (CRNT). In particular, this theory provides statements about uniqueness, existence, and stability of positive steady states for all rate constants and initial conditions. In terms of the corresponding polynomial equations, the results guarantee uniqueness and existence of positive solutions for all positive parameters.
We address a recent extension of CRNT, called generalized mass-action systems, where reaction rates are allowed to be power-laws in the concentrations. In particular, the (real) kinetic orders can differ from the (integer) stoichiometric coefficients. As with mass-action kinetics, complex balancing equilibria are determined by the graph Laplacian of the underlying network and can be characterized by binomial equations and parametrized by monomials. In algebraic terms, we focus on a constructive characterization of positive solutions of polynomial equations with real and symbolic exponents.
Uniqueness and existence for all rate constants and initial conditions additionally depend on sign vectors of the stoichiometric and kinetic-order subspaces. This leads to a generalization of Birch’s theorem, which is robust with respect to certain perturbations in the exponents. In this context, we discuss the occurrence of multiple complex balancing equilibria.
We illustrate our results by a running example and provide a MAPLE worksheet with implementations of all algorithmic methods.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Preview
Unable to display preview. Download preview PDF.
References
Adrovic, D., Verschelde, J.: A polyhedral method to compute all affine solution sets of sparse polynomial systems (2013), http://arxiv.org/abs/1310.4128, arXiv:1310.4128 [cs.SC]
Bachem, A., Kern, W.: Linear programming duality. Springer, Berlin (1992)
Basu, S., Pollack, R., Roy, M.F.: Algorithms in real algebraic geometry, 2nd edn. Springer, Berlin (2006)
Ben-Israel, A., Greville, T.N.E.: Generalized inverses, 2nd edn. Springer, New York (2003)
Birch, M.W.: Maximum likelihood in three-way contingency tables. J. Roy. Statist. Soc. Ser. B 25, 220–233 (1963)
Björner, A., Las Vergnas, M., Sturmfels, B., White, N., Ziegler, G.M.: Oriented matroids, 2nd edn. Cambridge University Press, Cambridge (1999)
Boulier, F., Lemaire, F., Petitot, M., Sedoglavic, A.: Chemical reaction systems, computer algebra and systems biology. In: Gerdt, V.P., Koepf, W., Mayr, E.W., Vorozhtsov, E.V. (eds.) CASC 2011. LNCS, vol. 6885, pp. 73–87. Springer, Heidelberg (2011)
Brualdi, R.A., Ryser, H.J.: Combinatorial matrix theory. Cambridge University Press, Cambridge (1991)
Conradi, C., Flockerzi, D., Raisch, J.: Multistationarity in the activation of a MAPK: parametrizing the relevant region in parameter space. Math. Biosci. 211, 105–131 (2008)
Corless, R.M., Jeffrey, D.J.: The turing factorization of a rectangular matrix. SIGSAM Bull. 31, 20–30 (1997)
Corless, R.M., Jeffrey, D.J.: Linear Algebra in Maple. In: CRC Handbook of Linear Algebra, 2nd edn. Chapman and Hall/CRC (2013)
Craciun, G., Dickenstein, A., Shiu, A., Sturmfels, B.: Toric dynamical systems. J. Symbolic Comput. 44, 1551–1565 (2009)
Dickenstein, A.: A world of binomials. In: Foundations of Computational Mathematics, Hong Kong, pp. 42–67. Cambridge Univ. Press, Cambridge (2009)
Errami, H., Seiler, W.M., Eiswirth, M., Weber, A.: Computing Hopf bifurcations in chemical reaction networks using reaction coordinates. In: Gerdt, V.P., Koepf, W., Mayr, E.W., Vorozhtsov, E.V. (eds.) CASC 2012. LNCS, vol. 7442, pp. 84–97. Springer, Heidelberg (2012)
Feinberg, M.: Complex balancing in general kinetic systems. Arch. Rational Mech. Anal. 49, 187–194 (1972)
Feinberg, M.: Lectures on chemical reaction networks (1979), http://crnt.engineering.osu.edu/LecturesOnReactionNetworks
Feinberg, M.: 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 (1987)
Feinberg, M.: Chemical reaction network structure and the stability of complex isothermal reactors–II. Multiple steady states for networks of deficiency one. Chem. Eng. Sci. 43, 1–25 (1988)
Feinberg, M.: The existence and uniqueness of steady states for a class of chemical reaction networks. Arch. Rational Mech. Anal. 132, 311–370 (1995)
Feinberg, M.: Multiple steady states for chemical reaction networks of deficiency one. Arch. Rational Mech. Anal. 132, 371–406 (1995)
Feinberg, M., Horn, F.J.M.: Chemical mechanism structure and the coincidence of the stoichiometric and kinetic subspaces. Arch. Rational Mech. Anal. 66, 83–97 (1977)
Fienberg, S.E.: Introduction to Birch (1963) Maximum likelihood in three-way contingency tables. In: Kotz, S., Johnson, N.L. (eds.) Breakthroughs in statistics, vol. II, pp. 453–461. Springer, New York (1992)
Fulton, W.: Introduction to toric varieties. Princeton University Press, Princeton (1993)
Gatermann, K., Wolfrum, M.: Bernstein’s second theorem and Viro’s method for sparse polynomial systems in chemistry. Adv. in Appl. Math. 34, 252–294 (2005)
Gatermann, K.: Counting stable solutions of sparse polynomial systems in chemistry. In: Symbolic Computation: Solving Equations in Algebra, Geometry, and Engineering, pp. 53–69. Amer. Math. Soc., Providence (2001)
Gatermann, K., Eiswirth, M., Sensse, A.: Toric ideals and graph theory to analyze Hopf bifurcations in mass action systems. J. Symbolic Comput. 40, 1361–1382 (2005)
Gatermann, K., Huber, B.: A family of sparse polynomial systems arising in chemical reaction systems. J. Symbolic Comput. 33, 275–305 (2002)
Gopalkrishnan, M., Miller, E., Shiu, A.: A Geometric Approach to the Global Attractor Conjecture. SIAM J. Appl. Dyn. Syst. 13, 758–797 (2014)
Grigoriev, D., Weber, A.: Complexity of solving systems with few independent monomials and applications to mass-action kinetics. In: Gerdt, V.P., Koepf, W., Mayr, E.W., Vorozhtsov, E.V. (eds.) CASC 2012. LNCS, vol. 7442, pp. 143–154. Springer, Heidelberg (2012)
Gunawardena, J.: Chemical reaction network theory for in-silico biologists (2003), http://vcp.med.harvard.edu/papers/crnt.pdf
Gunawardena, J.: A linear framework for time-scale separation in nonlinear biochemical systems. PLoS ONE 7, e36321 (2012)
Horn, F.: Necessary and sufficient conditions for complex balancing in chemical kinetics. Arch. Rational Mech. Anal. 49, 172–186 (1972)
Horn, F., Jackson, R.: General mass action kinetics. Arch. Rational Mech. Anal. 47, 81–116 (1972)
Johnston, M.D.: Translated Chemical Reaction Networks. Bull. Math. Biol. 76(5), 1081–1116 (2014)
Jungnickel, D.: Graphs, networks and algorithms, 4th edn. Springer, Heidelberg (2013)
Lemaire, F., Ürgüplü, A.: MABSys: Modeling and analysis of biological systems. In: Horimoto, K., Nakatsui, M., Popov, N. (eds.) ANB 2010. LNCS, vol. 6479, pp. 57–75. Springer, Heidelberg (2012)
Mirzaev, I., Gunawardena, J.: Laplacian dynamics on general graphs. Bull. Math. Biol. 75, 2118–2149 (2013)
Mnëv, N.E.: The universality theorems on the classification problem of configuration varieties and convex polytopes varieties. In: Topology and geometry—Rohlin Seminar. Lecture Notes in Math., vol. 1346, pp. 527–543. Springer, Berlin (1988)
Müller, S., Feliu, E., Regensburger, G., Conradi, C., Shiu, A., Dickenstein, A.: Sign conditions for injectivity of generalized polynomial maps with applications to chemical reaction networks and real algebraic geometry (2013) (submitted), http://arxiv.org/abs/1311.5493, arXiv:1311.5493 [math.AG]
Müller, S., Regensburger, G.: Generalized mass action systems: Complex balancing equilibria and sign vectors of the stoichiometric and kinetic-order subspaces. SIAM J. Appl. Math. 72, 1926–1947 (2012)
Pachter, L., Sturmfels, B.: Statistics. In: Algebraic statistics for computational biology, pp. 3–42. Cambridge Univ. Press, New York (2005)
Pérez Millán, M., Dickenstein, A., Shiu, A., Conradi, C.: Chemical reaction systems with toric steady states. Bull. Math. Biol. 74, 1027–1065 (2012)
Rambau, J.: TOPCOM: triangulations of point configurations and oriented matroids. In: Mathematical Software (Beijing 2002), pp. 330–340. World Sci. Publ, River Edge (2002)
Richter-Gebert, J., Ziegler, G.M.: Oriented matroids. In: Handbook of Discrete and Computational Geometry, pp. 111–132. CRC, Boca Raton (1997)
Samal, S.S., Errami, H., Weber, A.: PoCaB: A software infrastructure to explore algebraic methods for bio-chemical reaction networks. In: Gerdt, V.P., Koepf, W., Mayr, E.W., Vorozhtsov, E.V. (eds.) CASC 2012. LNCS, vol. 7442, pp. 294–307. Springer, Heidelberg (2012)
Savageau, M.A.: Biochemical systems analysis: II. The steady state solutions for an n-pool system using a power-law approximation. J. Theor. Biol. 25, 370–379 (1969)
Thomson, M., Gunawardena, J.: The rational parameterisation theorem for multisite post-translational modification systems. J. Theoret. Biol. 261, 626–636 (2009)
Voit, E.O.: Biochemical systems theory: A review. In: ISRN Biomath. 2013, 897658 (2013)
Zeilberger, D.: A combinatorial approach to matrix algebra. Discrete Math. 56, 61–72 (1985)
Ziegler, G.M.: Lectures on polytopes. Springer, New York (1995)
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2014 Springer International Publishing Switzerland
About this paper
Cite this paper
Müller, S., Regensburger, G. (2014). Generalized Mass-Action Systems and Positive Solutions of Polynomial Equations with Real and Symbolic Exponents (Invited Talk). In: Gerdt, V.P., Koepf, W., Seiler, W.M., Vorozhtsov, E.V. (eds) Computer Algebra in Scientific Computing. CASC 2014. Lecture Notes in Computer Science, vol 8660. Springer, Cham. https://doi.org/10.1007/978-3-319-10515-4_22
Download citation
DOI: https://doi.org/10.1007/978-3-319-10515-4_22
Publisher Name: Springer, Cham
Print ISBN: 978-3-319-10514-7
Online ISBN: 978-3-319-10515-4
eBook Packages: Computer ScienceComputer Science (R0)