Multistationarity in the Space of Total Concentrations for Systems that Admit a Monomial Parametrization

Abstract

We apply tools from real algebraic geometry to the problem of multistationarity of chemical reaction networks. A particular focus is on the case of reaction networks whose steady states admit a monomial parametrization. For such systems, we show that in the space of total concentrations multistationarity is scale invariant: If there is multistationarity for some value of the total concentrations, then there is multistationarity on the entire ray containing this value (possibly for different rate constants)—and vice versa. Moreover, for these networks it is possible to decide about multistationarity independent of the rate constants by formulating semi-algebraic conditions that involve only concentration variables. These conditions can easily be extended to include total concentrations. Hence, quantifier elimination may give new insights into multistationarity regions in the space of total concentrations. To demonstrate this, we show that for the distributive phosphorylation of a protein at two binding sites multistationarity is only possible if the total concentration of the substrate is larger than either the total concentration of the kinase or the total concentration of the phosphatase. This result is enabled by the chamber decomposition of the space of total concentrations from polyhedral geometry. Together with the corresponding sufficiency result of Bihan et al., this yields a characterization of multistationarity up to lower-dimensional regions.

This is a preview of subscription content, log in to check access.

Fig. 1
Fig. 2
Fig. 3

References

  1. Banaji M, Craciun G (2009) Graph-theoretic approaches to injectivity and multiple equilibria in systems of interacting elements. Commun Math Sci 7(4):867–900

    MathSciNet  MATH  Article  Google Scholar 

  2. Banaji M, Craciun G (2010) Graph-theoretic criteria for injectivity and unique equilibria in general chemical reaction systems. Adv Appl Math 44(2):168–184

    MathSciNet  MATH  Article  Google Scholar 

  3. Bates DJ, Hauenstein JD, Sommese AJ, Wampler CW (2006) Bertini: Software for Numerical Algebraic Geometry. https://bertini.nd.edu with permanent https://doi.org/10.7274/R0H41PB5

  4. Becker E, Neuhaus R (1993) Computation of real radicals of polynomial ideals. Computational algebraic geometry. Springer, Berlin, pp 1–20

    Google Scholar 

  5. Bihan F, Dickenstein A, Giaroli M (2018) Lower bounds for positive roots and regions of multistationarity in chemical reaction networks. preprint, arXiv:1807.05157

  6. Bradford R, Davenport J, England M, Errami H, Gerdt VP, Grigoriev D, Hoyt C, Kosta M, Radulescu O, Sturm T, Weber A (2017) A case study on the parametric occurrence of multiple steady states. In: Proceedings of the 42nd international symposium on symbolic and algebraic computation (ISSAC ’17), ACM, pp 45–52

  7. Brake D, Niemberg M (2016) Paramotopy. http://paramotopy.com

  8. Brown CW (2003) QEPCAD B: a program for computing with semi-algebraic sets using CADs. ACM SIGSAM Bull 37(4):97–108

    MATH  Article  Google Scholar 

  9. Brown CW, Strzeboński A (2010) Black-box/white-box simplification and applications to quantifier elimination. In: Proceedings of the 2010 international symposium on symbolic and algebraic computation. ACM, pp 69–76

  10. Conradi C, Flockerzi D (2012) Multistationarity in mass action networks with applications to ERK activation. J Math Biol 65(1):107–156

    MathSciNet  MATH  Article  Google Scholar 

  11. Conradi C, Mincheva M (2014) Catalytic constants enable the emergence of bistability in dual phosphorylation. J R Soc Interface 11(95):20140158

    Article  Google Scholar 

  12. Conradi C, Shiu A (2018) Dynamics of posttranslational modification systems: recent progress and future directions. Biophys J 114(3):507–515

    Article  Google Scholar 

  13. Conradi C, Pantea C (2019) Chapter 9–multistationarity in biochemical networks: results, analysis, and examples. In: Robeva R, Macauley M (eds) Algebraic and combinatorial computational biology. Academic Press, Cambridge, pp 279–317

    Google Scholar 

  14. Conradi C, Saez-Rodriguez J, Gilles E-D, Raisch J (2005) Using chemical reaction network theory to discard a kinetic mechanism hypothesis, systems biology. IEE Proc (now IET Syst Biol) 152(4):243–248

    Google Scholar 

  15. Conradi C, Flockerzi D, Raisch J (2008) Multistationarity in the activation of a MAPK: parametrizing the relevant region in parameter space. Math Biosci 211(1):105–131

    MathSciNet  MATH  Article  Google Scholar 

  16. Conradi C, Feliu E, Mincheva M, Wiuf C (2017) Identifying parameter regions for multistationarity. PLOS Comput Biol 13(10):1–25

    Article  Google Scholar 

  17. Coste M (2002) An introduction to semialgebraic geometry. RAAG Netw Sch 145:30

    Google Scholar 

  18. Cox DA, Little JB, O’Shea D (1996) Ideals, varieties, and algorithms, 2nd edn. Springer, New York

    Google Scholar 

  19. Craciun G, Dickenstein A, Shiu A, Sturmfels B (2009) Toric dynamical systems. J Symb Comput 44(11):1551–1565 Ordner: Gatermann

    MathSciNet  MATH  Article  Google Scholar 

  20. Craciun G, Pantea C, Rempala GA (2009) Algebraic methods for inferring biochemical networks: a maximum likelihood approach. Comput Biol Chem 33(5):361–367

    MathSciNet  MATH  Article  Google Scholar 

  21. De Loera JA, Kim ED, Onn S, Santos F (2009) Graphs of transportation polytopes. J Comb Theory Ser A 116(8):1306–1325

    MathSciNet  MATH  Article  Google Scholar 

  22. Dickenstein A (2016) Biochemical reaction networks: an invitation for algebraic geometers, vol 656. Mathematical congress of the Americas. American Mathematical Society, Providence, pp 65–83

    Google Scholar 

  23. Dickenstein A, Péréz-Millán M, Shiu A, Tang X (2019) Multistationarity in structured reaction networks. Bull Math Biol 81(5):1527–1581

    MathSciNet  MATH  Article  Google Scholar 

  24. Dolzmann A, Sturm T (1997) REDLOG: computer algebra meets computer logic. SIGSAM Bull 31(2):2–9

    Article  Google Scholar 

  25. Eisenbud D, Sturmfels B (1996) Binomial ideals. Duke Math J 84(1):1–45

    MathSciNet  MATH  Article  Google Scholar 

  26. Ellison PR (1998) The Advanced Deficiency Algorithm and its applications to mechanism discrimination, Ph.D. thesis, The University of Rochester

  27. Ellison P, Feinberg M (2000) How catalytic mechanisms reveal themselves in multiple steady-state data: I. Basic principles. J Mol Catal A Chem 154(1–2):155–167

    Article  Google Scholar 

  28. Ellison P, Feinberg M, Yueb M-H, Saltsburg H (2000) How catalytic mechanisms reveal themselves in multiple steady-state data: II. An ethylene hydrogenation example. J Mol Catal A Chem 154(1–2):169–184

    Article  Google Scholar 

  29. Érdi P, Tóth J (1989) Mathematical models of chemical reactions: theory and applications of deterministic and stochastic models. Manchester University Press, Manchester

    Google Scholar 

  30. Feinberg M (1995a) The existence and uniqueness of steady states for a class of chemical reaction networks. Arch Ration Mech Anal 132(4):311–370

    MathSciNet  MATH  Article  Google Scholar 

  31. Feinberg M (1995b) Multiple steady states for chemical reaction networks of deficiency one. Arch Ration Mech Anal 132(4):371–406

    MathSciNet  MATH  Article  Google Scholar 

  32. Feliu E, Wiuf C (2012) Preclusion of switch behavior in networks with mass-action kinetics. Appl Math Comput 219(4):1449–1467

    MathSciNet  MATH  Google Scholar 

  33. Flockerzi D, Holstein K, Conradi C (2014) N-site phosphorylation systems with 2n–1 steady states. Bull Math Biol 76:1–25

    MathSciNet  MATH  Article  Google Scholar 

  34. Gawrilow E, Joswig M (2000) polymake: a framework for analyzing convex polytopes, polytopes–combinatorics and computation, vol 29. Birkhäuser, Basel, pp 43–47

    Google Scholar 

  35. Gleixner A, Bastubbe M, Eifler L, Gally T, Gamrath G, Gottwald RL, Hendel G, Hojny C, Koch T, Lübbecke ME, Maher SJ, Miltenberger M, Müller B, Pfetsch ME, Puchert C, Rehfeldt D, Schlösser F, Schubert C, Serrano F, Shinano Y, Viernickel JM, Walter M, Wegscheider F, Witt JT, Witzig J (2018) The SCIP optimization suite 6.0, Technical report, Optimization Online

  36. Gross E, Harrington HA, Rosen Z, Sturmfels B (2016) Algebraic systems biology: a case study for the wnt pathway. Bull Math Biol 78(1):21–51

    MathSciNet  MATH  Article  Google Scholar 

  37. Holstein K, Flockerzi D, Conradi C (2013) Multistationarity in sequential distributed multisite phosphorylation networks. Bull Math Biol 75(11):2028–2058

    MathSciNet  MATH  Article  Google Scholar 

  38. Kahle T, Miller E (2014) Decompositions of commutative monoid congruences and binomial ideals. Algebra Number Theory 8(6):1297–1364

    MathSciNet  MATH  Article  Google Scholar 

  39. Lazard D, Rouillier F (2007) Solving parametric polynomial systems. J Symb Comput 42(6):636–667

    MathSciNet  MATH  Article  Google Scholar 

  40. Maplesoft (2017) a division of Waterloo Maple Inc., Waterloo, Ontario, Maple

  41. Markevich NI, Hoek JB, Kholodenko BN (2004) Signaling switches and bistability arising from multisite phosphorylation in protein kinase cascades. J Cell Biology 164(3):353–359

    Article  Google Scholar 

  42. 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):69–97

    MathSciNet  MATH  Article  Google Scholar 

  43. Neuhaus R (1998) Computation of real radicals of polynomial ideals–II. J Pure Appl Algebra 124(1–3):261–280

    MathSciNet  MATH  Article  Google Scholar 

  44. Péréz-Millán M, Dickenstein A (2018) The structure of MESSI biological systems. SIAM J Appl Dyn Syst 17(2):1650–1682

    MathSciNet  MATH  Article  Google Scholar 

  45. Péréz-Millán M, Dickenstein A, Shiu A, Conradi C (2012) Chemical reaction systems with toric steady states. Bull Math Biol 74(5):1027–1065

    MathSciNet  MATH  Article  Google Scholar 

  46. Rambau J (2002) TOPCOM: triangulations of point configurations and oriented matroids. In: Arjeh MC, Xiao-Shan G, Nobuki T (eds) Mathematical software–ICMS 2002. World Scientific, Singapore, pp 330–340

    Google Scholar 

  47. Rockafellar RT (1970) Convex analysis. Princeton University Press, Princeton

    Google Scholar 

  48. Sadeghimanesh AH, Feliu E (2019a) Gröbner bases of reaction networks with intermediate species. Adv Appl Math 107:74–101

    MathSciNet  MATH  Article  Google Scholar 

  49. Sadeghimanesh AH, Feliu E (2019b) The multistationarity structure of networks with intermediates and a binomial core network. Bull Math Biol 81:2428–2462

    MathSciNet  MATH  Article  Google Scholar 

  50. Schlosser PM, Feinberg M (1994) A theory of multiple steady states in isothermal homogeneous CFSTRs with many reactions. Chem Eng Sci 49(11):1749–1767

    Article  Google Scholar 

  51. Shinar G, Feinberg M (2012) Concordant chemical reaction networks. Math Biosci 240(2):92–113

    MathSciNet  MATH  Article  Google Scholar 

  52. Shinar G, Feinberg M (2013) Concordant chemical reaction networks and the species-reaction graph. Math Biosci 241(1):1–23

    MathSciNet  MATH  Article  Google Scholar 

  53. Shiu A (2010) Algebraic methods for biochemical reaction network theory, Ph.D. thesis, University of California, Berkeley

  54. Shiu A, Sturmfels B (2010) Siphons in chemical reaction networks. Bull Math Biol 72(6):1448–1463 Ordner: Gatermann

    MathSciNet  MATH  Article  Google Scholar 

  55. Wang L, Sontag E (2008) On the number of steady states in a multiple futile cycle. J Math Biol 57:29–52

    MathSciNet  MATH  Article  Google Scholar 

  56. 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(4):1685–1721

    MathSciNet  MATH  Article  Google Scholar 

  57. Wolfram Research, Inc., Mathematica, Version 11.2, Champaign, IL (2017)

  58. Ziegler Günter M (2012) Lectures on polytopes, GTM, vol 152. Springer, Berlin

    Google Scholar 

Download references

Acknowledgements

This project is funded by the Deutsche Forschungsgemeinschaft, 284057449. Alexandru Iosif and Thomas Kahle are also partially supported by the DFG-RTG “MathCore,” 314838170. We thank the anonymous reviewers for their valuable comments. One reviewer helped to improve the paper by providing a simpler proof of Lemma 3.10, clarifying the statements of Theorems 3.15 and 3.18, and pointing us to Frédéric et al. (2018, Theorem 4.1) which yields Corollary 4.13.

Author information

Affiliations

Authors

Corresponding author

Correspondence to Carsten Conradi.

Additional information

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Rights and permissions

Reprints and Permissions

About this article

Verify currency and authenticity via CrossMark

Cite this article

Conradi, C., Iosif, A. & Kahle, T. Multistationarity in the Space of Total Concentrations for Systems that Admit a Monomial Parametrization. Bull Math Biol 81, 4174–4209 (2019). https://doi.org/10.1007/s11538-019-00639-4

Download citation

Keywords

  • Polynomial systems in biology
  • Chemical reaction networks
  • Steady states
  • Multistationarity

Mathematics Subject Classification

  • Primary: 13P15
  • 37N25
  • Secondary: 92C42
  • 13P25
  • 13P10