Advertisement

Bulletin of Mathematical Biology

, Volume 81, Issue 5, pp 1527–1581 | Cite as

Multistationarity in Structured Reaction Networks

  • Alicia Dickenstein
  • Mercedes Pérez Millán
  • Anne Shiu
  • Xiaoxian TangEmail author
Article
  • 90 Downloads

Abstract

Many dynamical systems arising in biology and other areas exhibit multistationarity (two or more positive steady states with the same conserved quantities). Although deciding multistationarity for a polynomial dynamical system is an effective question in real algebraic geometry, it is in general difficult to determine whether a given network can give rise to a multistationary system, and if so, to identify witnesses to multistationarity, that is, specific parameter values for which the system exhibits multiple steady states. Here we investigate both problems. First, we build on work of Conradi, Feliu, Mincheva, and Wiuf, who showed that for certain reaction networks whose steady states admit a positive parametrization, multistationarity is characterized by whether a certain “critical function” changes sign. Here, we allow for more general parametrizations, which make it much easier to determine the existence of a sign change. This is particularly simple when the steady-state equations are linearly equivalent to binomials; we give necessary conditions for this to happen, which hold for many networks studied in the literature. We also give a sufficient condition for multistationarity of networks whose steady-state equations can be replaced by equivalent triangular-form equations. Finally, we present methods for finding witnesses to multistationarity, which we show work well for certain structured reaction networks, including those common to biological signaling pathways. Our work relies on results from degree theory, on the existence of explicit rational parametrizations of the steady states, and on the specialization of Gröbner bases.

Keywords

Reaction network Mass-action kinetics Multistationarity Parametrization Binomial ideal Brouwer degree Gröbner basis 

Notes

Acknowledgements

The authors thank Frank Sottile for helpful discussions, Alan Rendall for pointing us to the Calvin Cycle model, and Carsten Conradi for helpful discussions on the ERK network. The authors also thank three conscientious referees whose comments helped improve our work. AD and MPM were partially supported by UBACYT 20020170100048BA, CONICET PIP 11220150100473 and 11220150100483, and ANPCyT PICT 2016-0398, Argentina. AS partially supported by the NSF (DMS-1513364 and DMS-1752672) and the Simons Foundation (#521874). XT was partially supported by the NSF (DMS-1752672).

References

  1. Banaji M, Pantea C (2016) Some results on injectivity and multistationarity in chemical reaction networks. SIAM J Appl Dyn Syst 15(2):807–869MathSciNetCrossRefzbMATHGoogle Scholar
  2. Becker E, Marinari MG, Mora T, Traverso C (1994) The shape of the Shape Lemma. In: Proceedings of ISSAC ’94. ACM, New York, pp 129–133Google Scholar
  3. Bihan F, Dickenstein A, Giaroli M (2018) Lower bounds for positive roots and regions of multistationarity in chemical reaction networks. Preprint arXiv:1807.05157
  4. Conradi C, Feliu E, Mincheva M, Wiuf C (2017) Identifying parameter regions for multistationarity. PLoS Comput Biol 13(10):e1005751CrossRefGoogle Scholar
  5. Conradi C, Shiu A (2018) Dynamics of post-translational modification systems: recent progress and future challenges. Biophys J 114(3):507–515CrossRefGoogle Scholar
  6. Cox D, Little J, O’Shea D (2005) Using algebraic geometry, vol 185. Springer, BerlinzbMATHGoogle Scholar
  7. Cox D, Little J, O’Shea D (2007) Ideals, varieties, and algorithms: an introduction to computational algebraic geometry and commutative algebra. Springer, BerlinCrossRefzbMATHGoogle Scholar
  8. Craciun G, Feinberg M (2005) Multiple equilibria in complex chemical reaction networks. I. The injectivity property. SIAM J Appl Math 65(5):1526–1546MathSciNetCrossRefzbMATHGoogle Scholar
  9. Craciun G, Feinberg M (2010) Multiple equilibria in complex chemical reaction networks: semiopen mass action systems. SIAM J Appl Math 70(6):1859–1877MathSciNetCrossRefzbMATHGoogle Scholar
  10. Craciun G, Helton JW, Williams RJ (2008) Homotopy methods for counting reaction network equilibria. Math Biosci 216(2):140–149MathSciNetCrossRefzbMATHGoogle Scholar
  11. Dickenstein A (2016) Biochemical reaction networks: An invitation for algebraic geometers. In: Mathematical Congress of the Americas, vol 656. American Mathematical Soc, pp 65–83Google Scholar
  12. Enciso G (2014) Fixed points and convergence in monotone systems under positive or negative feedback. Int J Control 87(2):301–311MathSciNetCrossRefzbMATHGoogle Scholar
  13. Feliu E (2014) Injectivity, multiple zeros and multistationarity in reaction networks. Proc R Soc A 471(2173):20140530Google Scholar
  14. Feliu E, Wiuf C (2012) Enzyme-sharing as a cause of multi-stationarity in signalling systems. J R Soc Interface 9(71):1224–1232CrossRefGoogle Scholar
  15. Feliu E, Wiuf C (2013) Simplifying biochemical models with intermediate species. J R Soc Interface 10:20130484CrossRefGoogle Scholar
  16. Feliu E, Wiuf C (2013) Variable elimination in post-translational modification reaction networks with mass-action kinetics. J Math Biol 66(1–2):281–310MathSciNetCrossRefzbMATHGoogle Scholar
  17. Félix B, Shiu A, Woodstock Z (2016) Analyzing multistationarity in chemical reaction networks using the determinant optimization method. Appl Math Comput 287–288:60–73MathSciNetzbMATHGoogle Scholar
  18. Gelfand I, Kapranov M, Zelevinsky A (1994) Discriminants, resultants and multidimensional determinants. Birkhäuser, BostonCrossRefzbMATHGoogle Scholar
  19. Giaroli M, Bihan F, Dickenstein A (2018) Regions of multistationarity in cascades of Goldbeter-Koshland loops. Preprint arXiv:1807.08400
  20. Giusti M, Heintz J, Morais JE, Morgenstern J, Pardo LM (1998) Straight-line programs in geometric elimination theory. J Pure Appl Algebra 124(1):101–146MathSciNetCrossRefzbMATHGoogle Scholar
  21. Giusti M, Lecerf G, Salvy B (2001) A Gröbner free alternative for polynomial system solving. J Complex 17:154–211CrossRefzbMATHGoogle Scholar
  22. Gnacadja G (2011) Reachability, persistence, and constructive chemical reaction networks (part iii): a mathematical formalism for binary enzymatic networks and application to persistence. J Math Chem 49(10):2158–2176MathSciNetCrossRefzbMATHGoogle Scholar
  23. Grimbs S, Arnold A, Koseska A, Kurths J, Selbig J, Nikoloski Z (2011) Spatiotemporal dynamics of the Calvin cycle: multistationarity and symmetry breaking instabilities. BioSystems 103:212–223CrossRefGoogle Scholar
  24. Holstein K, Flockerzi D, Conradi C (2013) Multistationarity in sequential distributed multisite phosphorylation networks. Bull Math Biol 75(11):2028–2058MathSciNetCrossRefzbMATHGoogle Scholar
  25. Johnston MD (2014) Translated chemical reaction networks. Bull Math Biol 76(6):1081–1116MathSciNetCrossRefzbMATHGoogle Scholar
  26. Johnston M, Müller S, Pantea C (2018) A deficiency-based approach to parametrizing positive equilibria of biochemical reaction systems. Preprint arXiv:1805.09295
  27. Joshi B (2013) Complete characterization by multistationarity of fully open networks with one non-flow reaction. Appl Math Comput 219:6931–6945MathSciNetzbMATHGoogle Scholar
  28. Joshi B, Shiu A (2015) A survey of methods for deciding whether a reaction network is multistationary. Math Model Nat Phenom 10(5):47–67MathSciNetCrossRefzbMATHGoogle Scholar
  29. Joshi B, Shiu A (2017) Which small reaction networks are multistationary? SIAM J Appl Dyn Syst 16(2):802–833MathSciNetCrossRefzbMATHGoogle Scholar
  30. Kapur D, Sun Y, Wang D (2010) A new algorithm for computing comprehensive Gröbner systems. In: ISSAC’10 Proceedings of the 35th international symposium on symbolic and algebraic computation, pp 29–36Google Scholar
  31. Mirzaev I, Gunawardena J (2013) Laplacian dynamics on general graphs. Bull Math Biol 75(11):2118–49MathSciNetCrossRefzbMATHGoogle Scholar
  32. Müller S, Feliu E, Regensburger G, Conradi C, Shiu A, Dickenstein A (2016) Sign conditions for injectivity of generalized polynomial maps with applications to chemical reaction networks and real algebraic geometry. Found Comput Math 16(1):69–97MathSciNetCrossRefzbMATHGoogle Scholar
  33. Müller S, Hofbauer J, Regensburger G (2018) On the bijectivity of families of exponential/generalized polynomial maps. Preprint arXiv:1804.01851
  34. Maple 17 (2013) Maplesoft, a division of Waterloo Maple Inc., Waterloo, OntarioGoogle Scholar
  35. Millán MP, Dickenstein A (2018) The structure of MESSI biological systems. SIAM J Appl Dyn Syst 17(2):1650–1682MathSciNetCrossRefzbMATHGoogle Scholar
  36. Millán MP, Dickenstein A, Shiu A, Conradi C (2012) Chemical reaction systems with toric steady states. Bull Math Biol 74(5):1027–1065MathSciNetCrossRefzbMATHGoogle Scholar
  37. Rubinstein BY, Mattingly HH, Berezhkovskii AM, Shvartsman SY (2016) Long-term dynamics of multisite phosphorylation. Mol Biol Cell 27(14):2331–2340CrossRefGoogle Scholar
  38. Sadeghimanesh A, Feliu E (2018) The multistationarity structure of networks with intermediates and a binomial core network. Preprint arXiv:1808.07548
  39. Shinar G, Feinberg M (2012) Concordant chemical reaction networks. Math Biosci 240(2):92–113MathSciNetCrossRefzbMATHGoogle Scholar
  40. Shiu A (2008) The smallest multistationary mass-preserving chemical reaction network. Lect Notes Comput Sci 5147:172–184CrossRefzbMATHGoogle Scholar
  41. Shiu A, de Wolff T (2018) Nondegenerate multistationarity in small reaction networks. Preprint arXiv:1802.00306
  42. Shiu A, Sturmfels B (2010) Siphons in chemical reaction networks. Bull Math Biol 72(6):1448–1463MathSciNetCrossRefzbMATHGoogle Scholar
  43. Thomson M, Gunawardena J (2009) The rational parameterisation theorem for multisite post-translational modification systems. J Theor Biol 261(4):626–636MathSciNetCrossRefzbMATHGoogle Scholar
  44. Tutte WT (1948) The dissection of equilateral triangles into equilateral triangles. Math Proc Camb 44(4):463–482MathSciNetCrossRefzbMATHGoogle Scholar
  45. Wang L, Sontag ED (2008) On the number of steady states in a multiple futile cycle. J Math Biol 57(1):29–52MathSciNetCrossRefzbMATHGoogle Scholar
  46. Wiuf C, Feliu E (2013) Power-law kinetics and determinant criteria for the preclusion of multistationarity in networks of interacting species. SIAM J Appl Dyn Syst 12:1685–1721MathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© Society for Mathematical Biology 2019

Authors and Affiliations

  • Alicia Dickenstein
    • 1
  • Mercedes Pérez Millán
    • 1
  • Anne Shiu
    • 2
  • Xiaoxian Tang
    • 2
    Email author
  1. 1.Departamento de Matemática, FCENUniversidad de Buenos Aires e IMAS (UBA–CONICET)Buenos AiresArgentina
  2. 2.Department of MathematicsTexas A&M UniversityCollege StationUSA

Personalised recommendations