Sign Conditions for Injectivity of Generalized Polynomial Maps with Applications to Chemical Reaction Networks and Real Algebraic Geometry

Foundations of Computational Mathematics

Volumn 16, Issue 1, 2016, Pages 69-97

  Müller, Stefan ^a   Feliu, Elisenda ^b   Regensburger, Georg ^a   Conradi, Carsten ^c   Shiu, Anne ^d   Dickenstein, Alicia ^e  

^a JOHANN RADON INSTITUTE FOR COMPUTATIONAL AND APPLIED MATHEMATICS   (Austria)

^b UNIVERSITY OF COPENHAGEN   (Denmark)

^c MAX PLANCK INSTITUTE FOR DYNAMICS OF COMPLEX TECHNICAL SYSTEMS   (Germany)

^d TEXAS A AND M UNIVERSITY   (United States)

^e UNIVERSIDAD DE BUENOS AIRES   (Argentina)

Author keywords

Descartes rule of signs; Oriented matroid; Power law kinetics; Restricted injectivity; Sign vector

Indexed keywords

ALGEBRA; CHEMICAL REACTIONS; GEOMETRY; JACOBIAN MATRICES; REACTION KINETICS;

DESCARTES; INJECTIVITY; ORIENTED MATROID; POWER-LAW KINETICS; SIGN VECTORS;

POLYNOMIALS;

EID: 84955638834     PISSN: 16153375     EISSN: 16153383     Source Type: Journal    
DOI: 10.1007/s10208-014-9239-3     Document Type: Article

References (74)

1. E. Althaus and D. Dumitriu, Certifying feasibility and objective value of linear programs, Oper. Res. Lett. 40 (2012), 292–297.
2. R. M. Anderson and R. M. May, Infectious diseases of humans: Dynamics and control, Oxford University Press, Oxford, 1991.
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.
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.
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.
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.
8. M. Banaji, P. Donnell, and S. Baigent, $$P$$Pmatrix properties, injectivity, and stability in chemical reaction systems, SIAM J. Appl. Math. 67 (2007), 1523–1547.
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.
12. M. W. Birch, Maximum likelihood in three-way contingency tables, J. Roy. Stat. Soc. B Met. 25 (1963), 220–233.
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.
14. S. Boyd, S. J. Kim, L. Vandenberghe, and A. Hassibi, A tutorial on geometric programming, Optim. Eng. 8 (2007), 67–127.
15. R. Brualdi and B. Shader, Matrices of sign-solvable linear systems, Cambridge University Press, 1995.
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.
17. C. Conradi and D. Flockerzi, Multistationarity in mass action networks with applications to ERK activation, J. Math. Biol. 65 (2012), 107–156.
18. C. Conradi and D. Flockerzi, Switching in mass action networks based on linear inequalities, SIAM J. Appl. Dyn. Syst. 11 (2012), 110–134.
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.
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.
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.
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.
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.
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.
25. G. Craciun, J. W. Helton, and R. J. Williams, Homotopy methods for counting reaction network equilibria, Math. Biosci. 216 (2008), 140–149.
26. M. Feinberg, Complex balancing in general kinetic systems, Arch. Rational Mech. Anal. 49 (1972/73), 187–194.
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.
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.
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.
31. M. Feinberg, Multiple steady states for chemical reaction networks of deficiency one, Arch. Rational Mech. Anal. 132 (1995), 371–406.
32. E. Feliu and C. Wiuf, Preclusion of switch behavior in reaction networks with mass-action kinetics, Appl. Math. Comput. 219 (2012), 1449–1467.
33. E. Feliu and C. Wiuf, A computational method to preclude multistationarity in networks of interacting species, Bioinformatics 29 (2013), 2327–2334.
34. D. Gale and H. Nikaidô, The Jacobian matrix and global univalence of mappings, Math. Ann. 159 (1965), 81–93.
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.
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.
37. G. Gnacadja, A Jacobian criterion for the simultaneous injectivity on positive variables of linearly parameterized polynomials maps, Linear Algebra Appl. 437 (2012), 612–622.
38. C. M. Guldberg and P. Waage, Studies Concerning Affinity, C. M. Forhandlinger: Videnskabs-Selskabet i Christiana 35 (1864).
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.
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.
42. K. Holstein, D. Flockerzi, and C. Conradi, Multistationarity in sequential distributed multisite phosphorylation networks, Bull. Math. Biol. 75 (2013), 2028–2058.
43. F. Horn, Necessary and sufficient conditions for complex balancing in chemical kinetics, Arch. Rational Mech. Anal. 49 (1972), 172–186.
44. F. Horn and R. Jackson, General mass action kinetics, Arch. Ration. Mech. Anal. 47 (1972), 81–116.
45. I. Itenberg and M. F. Roy, Multivariate Descartes’ rule, Beiträge zur Algebra und Geometrie 37 (1996), 337–346.
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.
47. M. Joswig and T. Theobald, Polyhedral and algebraic methods in computational geometry, Universitext, Springer, London, 2013.
48. A. G. Khovanskiĭ, Fewnomials, Translations of Mathematical Monographs, vol. 88, American Mathematical Society, Providence, RI, 1991, Translated from the Russian by Smilka Zdravkovska.
49. V. Klee, R. Ladner, and R. Manber, Signsolvability revisited, Linear Algebra Appl. 59 (1984), 131–157.
50. F. Kubler and K. Schmedders, Tackling multiplicity of equilibria with Gröbner bases, Oper. Res. 58 (2010), 1037–1050.
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.
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.
53. S. Müller and G. Regensburger, Generalized mass action systems: Complex balancing equilibria and sign vectors of the stoichiometric and kinetic-order subspaces, SIAM J. Appl. Math. 72 (2012), 1926–1947.
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.
55. J. D. Murray, Mathematical biology: I. An introduction, third ed., Interdisciplinary Applied Mathematics, vol. 17, Springer-Verlag, New York, 2002.
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.
57. M. Pérez Millán, A. Dickenstein, A. Shiu, and C. Conradi, Chemical reaction systems with toric steady states, Bull. Math. Biol. 74 (2012), 1027–1065.
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.
59. J. Richter-Gebert and G. M. Ziegler, Oriented matroids, Handbook of discrete and computational geometry, CRC, Boca Raton, FL, 1997, pp. 111–132.
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.
62. M. A. Savageau, Biochemical systems analysis. I. Some mathematical properties of the rate law for the component enzymatic reactions, J. Theor. Biol. 25 (1969), 365–369.
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.
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.
65. G. Shinar and M. Feinberg, Concordant chemical reaction networks, Math. Biosci. 240 (2012), 92–113.
66. G. Shinar and M. Feinberg, Concordant chemical reaction networks and the species-reaction graph, Math. Biosci. 241 (2013), 1–23.
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.
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.
69. F. Sottile, Real solutions to equations from geometry, University Lecture Series, vol. 57, American Mathematical Society, Providence, RI, 2011.
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.
72. M. Uhr, Structural analysis of inference problems arising in systems biology, Ph.D. thesis, ETH Zurich, 2012.
73. C. Wiuf and E. Feliu, Power-law kinetics and determinant criteria for the preclusion of multistationarity in networks of interacting species, SIAM J. Appl. Dyn. Syst. 12 (2013), 1685–1721.
74. G. M. Ziegler, Lectures on polytopes, Springer-Verlag, New York, 1995.