Sign Conditions for Injectivity of Generalized Polynomial Maps with Applications to Chemical Reaction Networks and Real Algebraic Geometry
We give necessary and sufficient conditions in terms of sign vectors for the injectivity of families of polynomial maps with arbitrary real exponents defined on the positive orthant. Our work relates and extends existing injectivity conditions expressed in terms of Jacobian matrices and determinants. In the context of chemical reaction networks with power-law kinetics, our results can be used to preclude as well as to guarantee multiple positive steady states. In the context of real algebraic geometry, our work recognizes a prior result of Craciun, Garcia-Puente, and Sottile, together with work of two of the authors, as the first partial multivariate generalization of the classical Descartes’ rule, which bounds the number of positive real roots of a univariate real polynomial in terms of the number of sign variations of its coefficients.
KeywordsSign vector Restricted injectivity Power-law kinetics Descartes’ rule of signs Oriented matroid
Mathematics Subject Classification13P15 12D10 70K42 37C10 80A30 52C40
This project began during the Dagstuhl Seminar on “Symbolic methods for chemical reaction networks” held in November 2012 at Schloss Dagstuhl, Germany. The authors also benefited from discussions during the AIM workshop on “Mathematical problems arising from biochemical reaction networks” held in March 2013, in Palo Alto. EF was supported by a postdoctoral grant “Beatriu de Pinós” from the Generalitat de Catalunya and the Spanish research project MTM2012-38122-C03-01. CC was supported by BMBF grant Virtual Liver (FKZ 0315744) and the research focus dynamical systems of the state Saxony-Anhalt. AS was supported by the NSF (DMS-1004380 and DMS-1312473). AD was partially supported by UBACYT 20020130100207BA, CONICET PIP 11220110100580, and ANPCyT PICT-2013-1110, Argentina. The authors thank an anonymous referee for helpful comments.
- 2.R. M. Anderson and R. M. May, Infectious diseases of humans: Dynamics and control, Oxford University Press, Oxford, 1991.Google Scholar
- 3.D. L. Applegate, W. Cook, S. Dash, and D. G. Espinoza, Exact solutions to linear programming problems, Oper. Res. Lett. 35 (2007), 693–699.Google Scholar
- 4.D. L. Applegate, W. Cook, S. Dash, and D. G. Espinoza, QSopt_ex (2009). Available online at http://www.math.uwaterloo.ca/~bico//qsopt/ex/
- 5.E. Babson, L. Finschi, and K. Fukuda, Cocircuit graphs and efficient orientation reconstruction in oriented matroids, European J. Combin. 22 (2001), 587–600.Google Scholar
- 6.M. Banaji and G. Craciun, Graph-theoretic approaches to injectivity and multiple equilibria in systems of interacting elements, Commun. Math. Sci. 7 (2009), 867–900.Google Scholar
- 7.M. Banaji and G. Craciun, Graph-theoretic criteria for injectivity and unique equilibria in general chemical reaction systems, Adv. Appl. Math. 44 (2010), 168–184.Google Scholar
- 8.M. Banaji, P. Donnell, and S. Baigent, \(P\) matrix properties, injectivity, and stability in chemical reaction systems, SIAM J. Appl. Math. 67 (2007), 1523–1547.Google Scholar
- 9.M. Banaji and C. Pantea, Some results on injectivity and multistationarity in chemical reaction networks, Available online at arXiv:1309.6771, 2013.
- 10.S. Basu, R. Pollack, and M. F. Roy, Algorithms in real algebraic geometry, second ed., Algorithms and Computation in Mathematics, vol. 10, Springer-Verlag, Berlin, 2006, Updated online version available at http://perso.univ-rennes1.fr/marie-francoise.roy/bpr-ed2-posted2.html.
- 11.F. Bihan and A. Dickenstein, Descartes’ rule of signs for polynomial systems supported on circuits, Preprint, 2014.Google Scholar
- 13.A. Björner, M. Las Vergnas, B. Sturmfels, N. White, and G. M. Ziegler, Oriented matroids, second ed., Encyclopedia Math. Appl., vol. 46, Cambridge University Press, Cambridge, 1999.Google Scholar
- 14.S. Boyd, S. J. Kim, L. Vandenberghe, and A. Hassibi, A tutorial on geometric programming, Optim. Eng. 8 (2007), 67–127.Google Scholar
- 16.S. Chaiken, Oriented matroid pairs, theory and an electric application, Matroid theory (Seattle, WA, 1995), Contemp. Math., vol. 197, Amer. Math. Soc., Providence, RI, 1996, pp. 313–331.Google Scholar
- 17.C. Conradi and D. Flockerzi, Multistationarity in mass action networks with applications to ERK activation, J. Math. Biol. 65 (2012), 107–156.Google Scholar
- 18.C. Conradi and D. Flockerzi, Switching in mass action networks based on linear inequalities, SIAM J. Appl. Dyn. Syst. 11 (2012), 110–134.Google Scholar
- 19.C. Conradi, D. Flockerzi, and J. Raisch, Multistationarity in the activation of a MAPK: parametrizing the relevant region in parameter space, Math. Biosci. 211 (2008), 105–131.Google Scholar
- 20.G. Craciun and M. Feinberg, Multiple equilibria in complex chemical reaction networks. I. The injectivity property, SIAM J. Appl. Math. 65 (2005), 1526–1546.Google Scholar
- 21.G. Craciun and M. Feinberg, Multiple equilibria in complex chemical reaction networks: extensions to entrapped species models, Systems Biology, IEE Proceedings 153 (2006), 179–186.Google Scholar
- 22.G. Craciun and M. Feinberg, Multiple equilibria in complex chemical reaction networks. II. The species-reaction graph, SIAM J. Appl. Math. 66 (2006), 1321–1338.Google Scholar
- 23.G. Craciun and M. Feinberg, Multiple equilibria in complex chemical reaction networks: semiopen mass action systems, SIAM J. Appl. Math. 70 (2010), 1859–1877.Google Scholar
- 24.G. Craciun, L. Garcia-Puente, and F. Sottile, Some geometrical aspects of control points for toric patches, Mathematical Methods for Curves and Surfaces (Heidelberg) (M. Dæhlen, M. S. Floater, T. Lyche, J. L. Merrien, K. Morken, and L. L. Schumaker, eds.), Lecture Notes in Computer Science, vol. 5862, Springer, 2010, pp. 111–135.Google Scholar
- 25.G. Craciun, J. W. Helton, and R. J. Williams, Homotopy methods for counting reaction network equilibria, Math. Biosci. 216 (2008), 140–149.Google Scholar
- 26.M. Feinberg, Complex balancing in general kinetic systems, Arch. Rational Mech. Anal. 49 (1972/73), 187–194.Google Scholar
- 27.M. Feinberg, Lectures on chemical reaction networks, Available online at http://www.crnt.osu.edu/LecturesOnReactionNetworks, (1980).
- 28.M. Feinberg, Chemical reaction network structure and the stability of complex isothermal reactors–I. The deficiency zero and deficiency one theorems, Chem. Eng. Sci. 42 (1987), 2229–2268.Google Scholar
- 29.M. Feinberg, Chemical reaction network structure and the stability of complex isothermal reactors–II. Multiple steady states for networks of deficiency one, Chem. Eng. Sci. 43 (1988), 1–25.Google Scholar
- 30.M. Feinberg, The existence and uniqueness of steady states for a class of chemical reaction networks, Arch. Rational Mech. Anal. 132 (1995), 311–370.Google Scholar
- 31.M. Feinberg, Multiple steady states for chemical reaction networks of deficiency one, Arch. Rational Mech. Anal. 132 (1995), 371–406.Google Scholar
- 35.I. M. Gelfand, M. M. Kapranov, and A. V. Zelevinsky, Discriminants, resultants, and multidimensional determinants, reprint of the 1994 ed., Boston, MA: Birkhäuser, 2008.Google Scholar
- 36.A. M. Gleixner, D. E. Steffy, and K. Wolter, Improving the accuracy of linear programming solvers with iterative refinement, Proceedings of the 37th International Symposium on Symbolic and Algebraic Computation (ISSAC ’12) (New York, NY, USA), ACM, 2012, pp. 187–194.Google Scholar
- 38.C. M. Guldberg and P. Waage, Studies Concerning Affinity, C. M. Forhandlinger: Videnskabs-Selskabet i Christiana 35 (1864).Google Scholar
- 39.J. Gunawardena, Chemical reaction network theory for in-silico biologists, Available online at http://vcp.med.harvard.edu/papers/crnt.pdf, 2003.
- 40.J. W. Helton, V. Katsnelson, and I. Klep, Sign patterns for chemical reaction networks, J. Math. Chem. 47 (2010), 403–429.Google Scholar
- 41.J. W. Helton, I. Klep, and R. Gomez, Determinant expansions of signed matrices and of certain Jacobians, SIAM J. Matrix Anal. Appl. 31 (2009), 732–754.Google Scholar
- 44.F. Horn and R. Jackson, General mass action kinetics, Arch. Ration. Mech. Anal. 47 (1972), 81–116.Google Scholar
- 45.I. Itenberg and M. F. Roy, Multivariate Descartes’ rule, Beiträge zur Algebra und Geometrie 37 (1996), 337–346.Google Scholar
- 46.B. Joshi and A. Shiu, Simplifying the Jacobian criterion for precluding multistationarity in chemical reaction networks, SIAM J. Appl. Math. 72 (2012), 857–876.Google Scholar
- 47.M. Joswig and T. Theobald, Polyhedral and algebraic methods in computational geometry, Universitext, Springer, London, 2013.Google Scholar
- 48.A. G. Khovanskiĭ, Fewnomials, Translations of Mathematical Monographs, vol. 88, American Mathematical Society, Providence, RI, 1991, Translated from the Russian by Smilka Zdravkovska.Google Scholar
- 49.V. Klee, R. Ladner, and R. Manber, Signsolvability revisited, Linear Algebra Appl. 59 (1984), 131–157.Google Scholar
- 50.F. Kubler and K. Schmedders, Tackling multiplicity of equilibria with Gröbner bases, Oper. Res. 58 (2010), 1037–1050.Google Scholar
- 51.O. L. Mangasarian, Nonlinear programming, Classics in Applied Mathematics, vol. 10, Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA, 1994, Corrected reprint of the 1969 original.Google Scholar
- 52.M. Mincheva and G. Craciun, Multigraph conditions for multistability, oscillations and pattern formation in biochemical reaction networks, Proceedings of the IEEE 96 (2008), 1281–1291.Google Scholar
- 54.S. Müller and G. Regensburger, Generalized mass-action systems and positive solutions of polynomial equations with real and symbolic exponents, Computer Algebra in Scientific Computing (V. P. Gerdt, W. Koepf, W. M. Seiler, and E. V. Vorozhtsov, eds.), Lecture Notes in Computer Science, vol. 8660, Springer International Publishing, 2014, pp. 302–323.Google Scholar
- 55.J. D. Murray, Mathematical biology: I. An introduction, third ed., Interdisciplinary Applied Mathematics, vol. 17, Springer-Verlag, New York, 2002.Google Scholar
- 56.C. Pantea, H. Koeppl, and G. Craciun, Global injectivity and multiple equilibria in uni- and bi-molecular reaction networks, Discrete Contin. Dyn. Syst. Ser. B 17 (2012), 2153–2170.Google Scholar
- 58.J. Rambau, TOPCOM: Triangulations of point configurations and oriented matroids, Mathematical software (Beijing, 2002), World Sci. Publ., River Edge, NJ, 2002, pp. 330–340.Google Scholar
- 59.J. Richter-Gebert and G. M. Ziegler, Oriented matroids, Handbook of discrete and computational geometry, CRC, Boca Raton, FL, 1997, pp. 111–132.Google Scholar
- 60.M. Safey El Din, RAGLib, Available online at http://www-polsys.lip6.fr/~safey/RAGLib/, 2013.
- 61.I. W. Sandberg and A. N. Willson, Existence and uniqueness of solutions for the equations of nonlinear DC networks, SIAM J. Appl. Math. 22 (1972), 173–186.Google Scholar
- 63.M. A. Savageau and E. O. Voit, Recasting nonlinear differential equations as S-systems: a canonical nonlinear form, Math. Biosci. 87 (1987), 83–115.Google Scholar
- 64.A. Schrijver, Theory of linear and integer programming, Wiley-Interscience Series in Discrete Mathematics, John Wiley & Sons Ltd., Chichester, 1986, A Wiley-Interscience Publication.Google Scholar
- 65.G. Shinar and M. Feinberg, Concordant chemical reaction networks, Math. Biosci. 240 (2012), 92–113.Google Scholar
- 66.G. Shinar and M. Feinberg, Concordant chemical reaction networks and the species-reaction graph, Math. Biosci. 241 (2013), 1–23.Google Scholar
- 67.A. J. Sommese and C. W. Wampler, II, The numerical solution of systems of polynomials arising in engineering and science, World Scientific Publishing Co. Pte. Ltd., Hackensack, NJ, 2005.Google Scholar
- 68.F. Sottile and C. Zhu, Injectivity of 2D toric Bézier patches, Proceedings of 12th International Conference on Computer-Aided Design and Computer Graphics (Jinan, China) (R. Martin, H. Suzuki, and C. Tu, eds.), IEEE CPS, 2011, pp. 235–238.Google Scholar
- 69.F. Sottile, Real solutions to equations from geometry, University Lecture Series, vol. 57, American Mathematical Society, Providence, RI, 2011.Google Scholar
- 70.W. A. Stein et al., Sage Mathematics Software, Available online at http://www.sagemath.org, 2013.
- 71.D. J. Struik (ed.), A source book in mathematics, 1200-1800, Source Books in the History of the Sciences. Cambridge, Mass.: Harvard University Press, XIV, 427 p., 1969.Google Scholar
- 72.M. Uhr, Structural analysis of inference problems arising in systems biology, Ph.D. thesis, ETH Zurich, 2012.Google Scholar