Bulletin of Mathematical Biology

, Volume 81, Issue 5, pp 1613–1644 | Cite as

Computing Weakly Reversible Deficiency Zero Network Translations Using Elementary Flux Modes

  • Matthew D. JohnstonEmail author
  • Evan Burton


We present a computational method for performing structural translation, which has been studied recently in the context of analyzing the steady states and dynamical behavior of mass-action systems derived from biochemical reaction networks. Our procedure involves solving a binary linear programming problem where the decision variables correspond to interactions between the reactions of the original network. We call the resulting network a reaction-to-reaction graph and formalize how such a construction relates to the original reaction network and the structural translation. We demonstrate the efficacy and efficiency of the algorithm by running it on 508 networks from the European Bioinformatics Institutes’ BioModels database. We also summarize how this work can be incorporated into recently proposed algorithms for establishing mono- and multistationarity in biochemical reaction systems.


Biochemical reactions Deficiency Multistationarity Binary programming 

Mathematics Subject Classification

92C42 90C10 



  1. Alon U (2007) An introduction to systems biology: design principles of biological circuits. Chapman & Hall/CRC, LondonzbMATHGoogle Scholar
  2. Angeli D, de Leenheer P, Sontag E (2007) A Petri net approach to the study of persistence in chemical reaction networks. Math Biosci 210(2):598–618MathSciNetCrossRefzbMATHGoogle Scholar
  3. Clarke BL (1980) Stability of complex reaction networks. Adv Chem Phys 43:1–215Google Scholar
  4. Clarke BL (1988) Stoichiometric network analysis. Cell Biophys 12:237–253CrossRefGoogle Scholar
  5. Conradi C, Feliu E, Mincheva M, Wiuf C (2017) Identifying parameter regions for multistationarity. PLoS Comput Biol 13(10):e1005751CrossRefGoogle Scholar
  6. Conradi C, Shiu A (2015) A global convergence result for processive multisite phosphorylation systems. Bull Math Biol 77(1):126–155MathSciNetCrossRefzbMATHGoogle Scholar
  7. Craciun G, Dickenstein A, Shiu A, Sturmfels B (2009) Toric dynamical systems. J Symb Comput 44(11):1551–1565MathSciNetCrossRefzbMATHGoogle Scholar
  8. Dickenstein A, Pérez Millán M (2011) How far is complex balancing from detailed balancing? Bull Math Biol 73:811–828MathSciNetCrossRefzbMATHGoogle Scholar
  9. Dickenstein A, Pérez Millán M (2018) The structure of MESSI systems. SIAM J Appl Dyn Syst 17(2):1650–1682MathSciNetCrossRefzbMATHGoogle Scholar
  10. Feinberg M (1972) Complex balancing in general kinetic systems. Arch Ration Mech Anal 49:187–194MathSciNetCrossRefGoogle Scholar
  11. Feinberg M (1979) Lectures on chemical reaction networks. Unpublished written versions of lectures given at the Mathematics Research Center, University of Wisconsin.
  12. Feinberg M (1987) Chemical reaction network structure and the stability of complex isothermal reactors : I. The deficiency zero and deficiency one theorems. Chem Eng Sci 42(10):2229–2268CrossRefGoogle Scholar
  13. Feinberg M (1988) Chemical reaction network structure and the stability of complex isothermal reactors: II. Multiple steady states for networks of deficiency one. Chem Eng Sci 43(1):1–25CrossRefGoogle Scholar
  14. Feinberg M (1989) Necessary and sufficient conditions for detailed balancing in mass action systems of arbitrary complexity. Chem Eng Sci 44(9):1819–1827CrossRefGoogle Scholar
  15. Feinberg M (1995a) The existence and uniqueness of steady states for a class of chemical reaction networks. Arch Ration Mech Anal 132:311–370MathSciNetCrossRefzbMATHGoogle Scholar
  16. Feinberg M (1995b) Multiple steady states for chemical reaction networks of deficiency one. Arch Ration Mech Anal 132:371–406MathSciNetCrossRefzbMATHGoogle Scholar
  17. Guldberg CM, Waage P (1864) Studies concerning affinity. Videnskabs-Selskabet i Chistiana, C. M, Forhandlinger, p 35Google Scholar
  18. Hill A (1910) The possible effects of the aggregation of the molecules of haemoglobin on its dissociation curves. J Physiol 40(4):4–7Google Scholar
  19. Horn F (1972) Necessary and sufficient conditions for complex balancing in chemical kinetics. Arch Ration Mech Anal 49:172–186MathSciNetCrossRefGoogle Scholar
  20. Horn F, Jackson R (1972) General mass action kinetics. Arch Ration Mech Anal 47:81–116MathSciNetCrossRefGoogle Scholar
  21. Ingalls BP (2013) Mathematical modeling in systems biology: an introduction. MIT Press, New YorkzbMATHGoogle Scholar
  22. Johnston MD (2014) Translated chemical reaction networks. Bull Math Biol 76(5):1081–1116MathSciNetCrossRefzbMATHGoogle Scholar
  23. Johnston MD (2015) A computational approach to steady state correspondence of regular and generalized mass action systems. Bull Math Biol 77(6):1065–1100MathSciNetCrossRefzbMATHGoogle Scholar
  24. Johnston MD, Müller S, Pantea C (2018) Rational parametrizations of steady state manifolds for a class of mass-action systems. arXiv:1805.09295
  25. Jones JDG, Dangl JL (2006) The plant immune system. Nature 444:323–329CrossRefGoogle Scholar
  26. Lenstra HW (1983) Integer programming with a fixed number of variables. Math Oper Res 8:538–548MathSciNetCrossRefzbMATHGoogle Scholar
  27. Li C, Donizello M, Rodriguez N, Dharuri H, Endler L, Chelliah V, Li L, He E, Henry A, Stefan MI, Snoep JL, Hucka M, Le Lovere N, Laibe C (2010) Biomodels database: an enhance, curated and annotated resource for published quantitative kinetic models. BMC Syst Biol 4:92CrossRefGoogle Scholar
  28. Markevich NI, Hoek JB, Kholodenko BN (2004) Signaling switches and bistability arising from multisite phosphorylation in protein kinase cascades. J Cell Biol 164(3):353–359CrossRefGoogle Scholar
  29. Michaelis L, Menten M (1913) Die kinetik der invertinwirkung. Biochem Z 49:333–369Google Scholar
  30. Millán MP, Dickenstein A, Shiu A, Conradi C (2012) Chemical reaction systems with toric steady states. Bull Math Biol 74(5):1027–1065MathSciNetCrossRefzbMATHGoogle Scholar
  31. Müller S, Regensburger G (2012) Generalized mass action systems: complex balancing equilibria and sign vectors of the stoichiometric and kinetic-order subspaces. SIAM J Appl Math 72(6):1926–1947MathSciNetCrossRefzbMATHGoogle Scholar
  32. Müller S, Regensburger G, (2014) Generalized mass-action systems and positive solutions of polynomial equations with real and symbolic exponents (invited talk). In: Gerdt VP, Koepf W, Seiler WM, Vorozhtsov EV (eds) Computer algebra in scientific computing. CASC. Lecture notes in computer science, 8660. Springer, Berlin, pp 302–323Google Scholar
  33. Müller S, Feliu E, Regensburger G, Conradi C, Shiu A, Dickenstein A (2016) 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
  34. Orth JD, Thiele I, Palsson BO (2010) What is flux balance analysis? Nat Biotechnol 28:245–248CrossRefGoogle Scholar
  35. Pritchard L, Birch PRJ (2014) The zigzag model of plant-microbe interactions: is it time to move on? Mol Plant Pathol 15(9):865–870CrossRefGoogle Scholar
  36. Shinar G, Feinberg M (2010) Structural sources of robustness in biochemical reaction networks. Science 327(5971):1389–1391CrossRefGoogle Scholar
  37. Tonello E (2016) CrnPy: a python library for the analysis of chemical reaction networks.
  38. Tonello E, Johnston MD (2018) Network translation and steady state properties of chemical reaction systems. Bull Math Biol 80(9):2306–2337MathSciNetCrossRefzbMATHGoogle Scholar
  39. Wiback SJ, Palsson BO (2002) Extreme pathway analysis of human red blood cell metabolism. Biophys J 83(2):808–818CrossRefGoogle Scholar
  40. Zanghellini J, Ruckerbauer DE, Hanscho M, Jungreuthmayer C (2013) Elementary flux modes in a nutshell: properties, calculation and applications. Biotechnol J 8(9):1009–1016CrossRefGoogle Scholar

Copyright information

© Society for Mathematical Biology 2019

Authors and Affiliations

  1. 1.Department of MathematicsSan José State UniversitySan JoseUSA

Personalised recommendations