Journal of Mathematical Biology

, Volume 70, Issue 7, pp 1669–1684 | Cite as

Implicit dose-response curves

  • Mercedes Pérez Millán
  • Alicia Dickenstein


We develop tools from computational algebraic geometry for the study of steady state features of autonomous polynomial dynamical systems via elimination of variables. In particular, we obtain nontrivial bounds for the steady state concentration of a given species in biochemical reaction networks with mass-action kinetics. This species is understood as the output of the network and we thus bound the maximal response of the system. The improved bounds give smaller starting boxes to launch numerical methods. We apply our results to the sequential enzymatic network studied in Markevich et al. (J Cell Biol 164(3):353–359, 2004) to find nontrivial upper bounds for the different substrate concentrations at steady state. Our approach does not require any simulation, analytical expression to describe the output in terms of the input, or the absence of multistationarity. Instead, we show how to extract information from effectively computable implicit dose-response curves, with the use of resultants and discriminants. We moreover illustrate in the application to an enzymatic network, the relation between the exact implicit dose-response curve we obtain symbolically and the standard hysteresis diagram provided by a numerical ode solver. The setting and tools we propose could yield many other results adapted to any autonomous polynomial dynamical system, beyond those where it is possible to get explicit expressions.


Chemical reaction networks Steady states Bounds  Resultants Maximal response 

Mathematics Subject Classification

92C45 13P15 



This work was partially supported by UBACYT 20020100100242, CONICET PIP 11220110100580 and ANPCyT 2008-0902, Argentina.

Supplementary material

285_2014_809_MOESM1_ESM.txt (3 kb)
Supplementary material 1 (txt 3 KB)


  1. Cox D, Little J, O’Shea D (2007) Ideals, varieties and algorithms. Undergraduate texts in Mathematics, 3rd edn. Springer, New YorkCrossRefGoogle Scholar
  2. Decker W, Greuel G-M, Pfister G, Schönemann H (2012) Singular 3-1-6: a computer algebra system for polynomial computations.
  3. Feinberg M (1979) Lectures on chemical reaction networks. Ohio State University, Columbus.
  4. Feinberg M, Horn F (1977) Chemical mechanism structure and the coincidence of the stoichiometric and kinetic subspaces. Arch Ration Mech Anal 66(1):83–97CrossRefzbMATHMathSciNetGoogle Scholar
  5. 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
  6. Feliu E, Knudsen M, Andersen L, Wiuf C (2012) An algebraic approach to signaling cascades with \(n\) layers. Bull Math Biol 74(1):45–72CrossRefzbMATHMathSciNetGoogle Scholar
  7. Flockerzi D, Holstein K, Conradi C (2013) N-site phosphorylation systems with 2N–1 steady states. Available at arXiv:1312.4774
  8. Gelf’and I, Kapranov M, Zelevinsky A (1994) Discriminants, resultants and multidimensional determinants. Birkhäuser, BostonCrossRefGoogle Scholar
  9. Horn F, Jackson R (1972) General mass action kinetics. Arch Ration Mech Anal 47(2):81–116CrossRefMathSciNetGoogle Scholar
  10. Karp R, Pérez Millán M, Dasgupta T, Dickenstein A, Gunawardena J (2012) Complex-linear invariants of biochemical networks. J Theor Biol 311:130–138CrossRefGoogle Scholar
  11. Maple 17 (2013) Maplesoft: a division of Waterloo Maple Inc., Waterloo, OntarioGoogle Scholar
  12. Markevich N, Hoek J, Kholodenko B (2004) Signaling switches and bistability arising from multisite phosphorylation in protein kinase cascades. J Cell Biol 164(3):353–359CrossRefGoogle Scholar
  13. MATLAB (2014) version 8.3.0. Natick, Massachusetts: The MathWorks IncGoogle Scholar
  14. Pérez Millán M, Dickenstein A, Shiu A, Conradi C (2012) Chemical reaction systems with toric steady states. Bull Math Biol 74(5):1027–1065CrossRefzbMATHMathSciNetGoogle Scholar
  15. Shafarevich I (1994) Basic algebraic geometry 1. Varieties in projective space, 2nd edn. Springer, BerlinCrossRefzbMATHGoogle Scholar
  16. Tobis EA (2005) Libraries for counting real roots, Reports on computer algebra, ZCA, University of Kaiserslautern, p 34Google Scholar
  17. Vol’pert AI, Hudjaev SI (1985) Analysis in classes of discontinuous functions and equations of mathematical physics, vol 8. Mechanics: analysis, Martinus Nijhoff Publishers, DordrechtGoogle Scholar
  18. Wang L, Sontag E (2008) On the number of steady states in a multiple futile cycle. J Math Biol 57(1):29–52CrossRefzbMATHMathSciNetGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2014

Authors and Affiliations

  1. 1.Dto. de Matemática, FCENUniversidad de Buenos Aires, Ciudad UniversitariaBuenos AiresArgentina
  2. 2.Dto. de Ciencias Exactas, CBCUniversidad de Buenos AiresBuenos AiresArgentina
  3. 3.IMAS, CONICETCiudad UniversitariaBuenos AiresArgentina

Personalised recommendations