The Multistationarity Structure of Networks with Intermediates and a Binomial Core Network

  • AmirHosein Sadeghimanesh
  • Elisenda FeliuEmail author


This work addresses whether a reaction network, taken with mass-action kinetics, is multistationary, that is, admits more than one positive steady state in some stoichiometric compatibility class. We build on previous work on the effect that removing or adding intermediates has on multistationarity, and also on methods to detect multistationarity for networks with a binomial steady-state ideal. In particular, we provide a new determinant criterion to decide whether a network is multistationary, which applies when the network obtained by removing intermediates has a binomial steady-state ideal. We apply this method to easily characterize which subsets of complexes are responsible for multistationarity; this is what we call the multistationarity structure of the network. We use our approach to compute the multistationarity structure of the n-site sequential distributive phosphorylation cycle for arbitrary n.


Binomial ideal Phosphorylation cycle Multistationarity Model reduction Determinant criterion Toric ideal 



This work has been supported by the Independent Research Fund of Denmark. We thank Alicia Dickenstein, Martin Helmer and Angélica Torres for comments on a preliminary version of this manuscript.


  1. Basu S, Pollack R, Coste-Roy MF (2007) Algorithms in real algebraic geometry, vol 10. Springer, BerlinzbMATHGoogle Scholar
  2. Bihan F, Dickenstein A, Giaroli M (2019) Lower bounds for positive roots and regions of multistationarity in chemical reaction networks. arXiv:1807.05157
  3. Bradford R, Davenport JH, England M, Errami H, Gerdt V, Grigoriev D, Hoyt C, Košta M, Radulescu O, Sturm T, Weber A (2017) A case study on the parametric occurrence of multiple steady states. In: Proceedings of the international symposium on symbolic and algebraic computation, ISSAC, pp 45–52. Association for Computing MachineryGoogle Scholar
  4. Dickenstein A, Pérez Millán M, Shiu A, Tang X (2019) Multistationarity in structured reaction networks. Bull Math Biol 81:1527–1581MathSciNetCrossRefGoogle Scholar
  5. Eisenbud D, Sturmfels B (1996) Binomial ideals. Duke Math J 84(1):1–45MathSciNetCrossRefzbMATHGoogle Scholar
  6. England M, Bradford R, Davenport JH (2015) Improving the use of equational constraints in cylindrical algebraic decomposition. In: Proceedings of the international symposium on symbolic and algebraic computation, ISSAC, pp 165–172. Association for Computing MachineryGoogle Scholar
  7. Feinberg M (1980) Lectures on chemical reaction networks.
  8. Feinberg M (1995) The existence and uniqueness of steady states for a class of chemical reaction networks. Arch Ration Mech Anal 132(4):311–370MathSciNetCrossRefzbMATHGoogle Scholar
  9. 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
  10. Feliu E, Wiuf C (2013) Simplifying biochemical models with intermediate species. J R Soc Interface 10(87):20130484CrossRefGoogle Scholar
  11. Gerhard J, Jeffrey D, Moroz G (2010) A package for solving parametric polynomial systems. ACM Commun Comput Algebra 43(3/4):61–72zbMATHGoogle Scholar
  12. Gunawardena J (2003) Chemical reaction network theory for in-silico biologists.
  13. Lazard D, Rouillier F (2007) Solving parametric polynomial systems. J Symb Comput 42(6):636–667MathSciNetCrossRefzbMATHGoogle Scholar
  14. 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
  15. Pérez Millán M, Dickenstein A (2018) The structure of MESSI biological systems. SIAM J Appl Dyn Syst 17(2):1650–1682MathSciNetCrossRefzbMATHGoogle Scholar
  16. Pérez Millán M, Dickenstein A, Shiu A, Conradi C (2012) Chemical reaction systems with toric steady states. Bull Math Biol 74(5):1027–1065MathSciNetCrossRefzbMATHGoogle Scholar
  17. Sadeghimanesh AH, Feliu E (2019) Gröbner bases of reaction networks with intermediate species. Adv Appl Math 107(2):74–101MathSciNetCrossRefGoogle Scholar
  18. Sáez M, Wiuf C, Feliu E (2017) Graphical reduction of reaction networks by linear elimination of species. J Math Biol 74(1):195–237MathSciNetCrossRefzbMATHGoogle Scholar
  19. Schilling CH, Letscher D, Palsson BØ (2000) Theory for the systemic definition of metabolic pathways and their use in interpreting metabolic function from a pathway-oriented perspective. J Theor Biol 203(3):229–248CrossRefGoogle Scholar
  20. Wang L, Sontag ED (2008) On the number of steady states in a multiple futile cycle. J Math Biol 57(1):29–52MathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© Society for Mathematical Biology 2019

Authors and Affiliations

  1. 1.Department of Mathematical SciencesUniversity of CopenhagenCopenhagenDenmark

Personalised recommendations