Journal of Mathematical Biology

, Volume 61, Issue 4, pp 581–616

Graph-theoretic characterizations of monotonicity of chemical networks in reaction coordinates

  • David Angeli
  • Patrick De Leenheer
  • Eduardo Sontag

DOI: 10.1007/s00285-009-0309-0

Cite this article as:
Angeli, D., De Leenheer, P. & Sontag, E. J. Math. Biol. (2010) 61: 581. doi:10.1007/s00285-009-0309-0


This paper derives new results for certain classes of chemical reaction networks, linking structural to dynamical properties. In particular, it investigates their monotonicity and convergence under the assumption that the rates of the reactions are monotone functions of the concentrations of their reactants. This is satisfied for, yet not restricted to, the most common choices of the reaction kinetics such as mass action, Michaelis-Menten and Hill kinetics. The key idea is to find an alternative representation under which the resulting system is monotone. As a simple example, the paper shows that a phosphorylation/dephosphorylation process, which is involved in many signaling cascades, has a global stability property. We also provide a global stability result for a more complicated example that describes a regulatory pathway of a prevalent signal transduction module, the MAPK cascade.


Biochemical reaction networks Monotone systems Global convergence Reaction coordinates Persistence Futile cycle EGF pathway model 

Mathematics Subject Classification (2000)

Primary: 92C42 Systems biology, networks Secondary: 80A30 Chemical kinetics [See also 76V05, 92C45, 92E20] 92C40 Biochemistry, molecular biology 

Copyright information

© Springer-Verlag 2009

Authors and Affiliations

  • David Angeli
    • 1
    • 2
  • Patrick De Leenheer
    • 3
  • Eduardo Sontag
    • 4
  1. 1.Dipartimento di Sistemi e InformaticaUniversity of FlorenceFlorenceItaly
  2. 2.Department of Electrical and Electronic EngineeringImperial CollegeLondonUK
  3. 3.Department of MathematicsUniversity of FloridaGainesvilleUSA
  4. 4.Department of MathematicsRutgers UniversityNew BrunswickUSA

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