Bulletin of Mathematical Biology

, Volume 74, Issue 5, pp 1027–1065 | Cite as

Chemical Reaction Systems with Toric Steady States

  • Mercedes Pérez Millán
  • Alicia Dickenstein
  • Anne Shiu
  • Carsten ConradiEmail author
Original Article


Mass-action chemical reaction systems are frequently used in computational biology. The corresponding polynomial dynamical systems are often large (consisting of tens or even hundreds of ordinary differential equations) and poorly parameterized (due to noisy measurement data and a small number of data points and repetitions). Therefore, it is often difficult to establish the existence of (positive) steady states or to determine whether more complicated phenomena such as multistationarity exist. If, however, the steady state ideal of the system is a binomial ideal, then we show that these questions can be answered easily. The focus of this work is on systems with this property, and we say that such systems have toric steady states. Our main result gives sufficient conditions for a chemical reaction system to have toric steady states. Furthermore, we analyze the capacity of such a system to exhibit positive steady states and multistationarity. Examples of systems with toric steady states include weakly-reversible zero-deficiency chemical reaction systems. An important application of our work concerns the networks that describe the multisite phosphorylation of a protein by a kinase/phosphatase pair in a sequential and distributive mechanism.


Chemical reaction networks Mass-action kinetics Multistationarity Multisite phosphorylation Binomial ideal 


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Copyright information

© Society for Mathematical Biology 2011

Authors and Affiliations

  • Mercedes Pérez Millán
    • 1
  • Alicia Dickenstein
    • 1
    • 2
  • Anne Shiu
    • 3
  • Carsten Conradi
    • 4
    Email author
  1. 1.Dto. de Matemática, FCENUniversidad de Buenos AiresBuenos AiresArgentina
  2. 2.IMAS/CONICETUniversidad de Buenos AiresBuenos AiresArgentina
  3. 3.Dept. of MathematicsDuke UniversityDurhamUSA
  4. 4.Max-Planck-Institut Dynamik komplexer technischer SystemeMagdeburgGermany

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