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Chemical Reaction Systems with Toric Steady States

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Abstract

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.

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Authors and Affiliations

  1. Dto. de Matemática, FCEN, Universidad de Buenos Aires, Ciudad Universitaria, Pab. I, C1428EGA, Buenos Aires, Argentina

    Mercedes Pérez Millán & Alicia Dickenstein

  2. IMAS/CONICET, Universidad de Buenos Aires, Buenos Aires, Argentina

    Alicia Dickenstein

  3. Dept. of Mathematics, Duke University, Box 90320, Durham, NC, 27708-0320, USA

    Anne Shiu

  4. Max-Planck-Institut Dynamik komplexer technischer Systeme, Sandtorstr. 1, 39106, Magdeburg, Germany

    Carsten Conradi

Authors
  1. Mercedes Pérez Millán
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  2. Alicia Dickenstein
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  3. Anne Shiu
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  4. Carsten Conradi
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Correspondence to Carsten Conradi.

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Pérez Millán, M., Dickenstein, A., Shiu, A. et al. Chemical Reaction Systems with Toric Steady States. Bull Math Biol 74, 1027–1065 (2012). https://doi.org/10.1007/s11538-011-9685-x

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  • DOI: https://doi.org/10.1007/s11538-011-9685-x

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