Multistationarity and Bistability for Fewnomial Chemical Reaction Networks

  • Elisenda Feliu
  • Martin HelmerEmail author


Bistability and multistationarity are properties of reaction networks linked to switch-like responses and connected to cell memory and cell decision making. Determining whether and when a network exhibits bistability is a hard and open mathematical problem. One successful strategy consists of analyzing small networks and deducing that some of the properties are preserved upon passage to the full network. Motivated by this, we study chemical reaction networks with few chemical complexes. Under mass action kinetics, the steady states of these networks are described by fewnomial systems, that is polynomial systems having few distinct monomials. Such systems of polynomials are often studied in real algebraic geometry by the use of Gale dual systems. Using this Gale duality, we give precise conditions in terms of the reaction rate constants for the number and stability of the steady states of families of reaction networks with one non-flow reaction.


Chemical reaction networks Multistationarity and bistability Fewnomial systems Gale duality Real algebraic geometry Steady states of dynamical systems 



This work was partially funded by the Independent Research Fund of Denmark.


  1. Basu S, Pollack R, Coste-Roy M-F (2007) Algorithms in real algebraic geometry, vol 10. Springer, BerlinzbMATHGoogle Scholar
  2. Bates D, Bihan F, Sottile F (2007) Bounds on the number of real solutions to polynomial equations. Int Math Res Not 2007(9):rnm114MathSciNetzbMATHGoogle Scholar
  3. Bihan F, Dickenstein A (2016) Descartes rule of signs for polynomial systems supported on circuits. Int Math Res Not 2017(22):6867–6893MathSciNetGoogle Scholar
  4. Bihan F, Sottile F (2007) New fewnomial upper bounds from gale dual polynomial systems. Mosc Math J 7(3):387–407MathSciNetzbMATHGoogle Scholar
  5. Bihan F, Sottile F (2008) Gale duality for complete intersections (dualité de gale pour des intersections complètes). Ann Inst Fourier 58(3):877–891MathSciNetCrossRefzbMATHGoogle Scholar
  6. Craciun G, Feinberg M (2006) Multiple equilibria in complex chemical reaction networks: extensions to entrapped species models. Syst Biol (Stevenage) 153:179–186CrossRefGoogle Scholar
  7. Conradi C, Flockerzi D (2012) Switching in mass action networks based on linear inequalities. SIAM J Appl Dyn Syst 11(1):110–134MathSciNetCrossRefzbMATHGoogle Scholar
  8. Conradi C, Feliu E, Mincheva M, Wiuf C (2017) Identifying parameter regions for multistationarity. PLoS Comput Biol 13:e1005751CrossRefGoogle Scholar
  9. Donnell P, Banaji M, Marginean A, Pantea C (2014) Control: an open source framework for the analysis of chemical reaction networks. Bioinformatics 30(11):1633–1634CrossRefGoogle Scholar
  10. Dimitrov DK, Rafaeli FR (2009) Descartes rule of signs for orthogonal polynomials. East J Approx 15(2):233–262MathSciNetzbMATHGoogle Scholar
  11. Ellison P, Feinberg M, Ji H, Knight D (2012) Chemical reaction network toolbox, version 2.2. Accessed 2018
  12. Feinberg M (1980) Lectures on chemical reaction networks. Accessed 2018
  13. Ferrell JE (2012) Bistability, bifurcations, and Waddington’s epigenetic landscape. Curr Biol 22(11):R458–466CrossRefGoogle Scholar
  14. Feliu E, Wiuf C (2013a) A computational method to preclude multistationarity in networks of interacting species. Bioinformatics 29:2327–2334CrossRefGoogle Scholar
  15. Feliu E, Wiuf C (2013b) Simplifying biochemical models with intermediate species. J R Soc Interface 10:20130484CrossRefGoogle Scholar
  16. Gunawardena J (2003) Chemical reaction network theory for in-silico biologists. Accessed 2018
  17. Joshi B (2013) Complete characterization by multistationarity of fully open networks with one non-flow reaction. Appl Math Comput 219(12):6931–6945MathSciNetzbMATHGoogle Scholar
  18. Joshi B, Shiu A (2013) Atoms of multistationarity in chemical reaction networks. J Math Chem 51(1):153–178MathSciNetzbMATHGoogle Scholar
  19. 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
  20. Joshi B, Shiu A (2017) Which small reaction networks are multistationary? SIAM J Appl Dyn Syst 16(2):802–833MathSciNetCrossRefzbMATHGoogle Scholar
  21. Laurent M, Kellershohn N (1999) Multistability: a major means of differentiation and evolution in biological systems. Trends Biochem Sci 24(11):418–422CrossRefGoogle Scholar
  22. Millán MP, Dickenstein A, Shiu A, Conradi C (2012) Chemical reaction systems with toric steady states. Bull Math Biol 74:1027–1065MathSciNetCrossRefzbMATHGoogle Scholar
  23. Perko L (2001) Differential equations and dynamical systems. Texts in applied mathematics, vol 7, 3rd edn. Springer, New YorkCrossRefzbMATHGoogle Scholar
  24. Pólya G, Szegö G (1997) Problems and theorems in analysis II: theory of functions. Zeros. Polynomials. Determinants. Number theory. Geometry. Springer, BerlinzbMATHGoogle Scholar
  25. Shiu A, de Wolff T (2018) Nondegenerate multistationarity in small reaction networks. Discrete Contin Dyn Syst Ser B. To appearGoogle Scholar
  26. Sottile F (2011) Real solutions to equations from geometry, vol 57. American Mathematical Society, ProvidencezbMATHGoogle Scholar
  27. Xiong W, Ferrell JE Jr (2003) A positive-feedback-based bistable ’memory module’ that governs a cell fate decision. Nature 426(6965):460–465CrossRefGoogle Scholar

Copyright information

© Society for Mathematical Biology 2018

Authors and Affiliations

  1. 1.Department of Mathematical SciencesUniversity of CopenhagenCopenhagenDenmark

Personalised recommendations