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Design and Analysis of Biomolecular Circuits pp 63–72Cite as

Graph-Theoretic Analysis of Multistability and Monotonicity for Biochemical Reaction Networks

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Abstract

Mathematical models of biochemical reaction networks are usually high dimensional, nonlinear, and have many unknown parameters, such as reaction rate constants, or unspecified types of chemical kinetics (such as mass-action, Michaelis-Menten, or Hill kinetics). On the other hand, important properties of these dynamical systems are often determined by the network structure, and do not depend on the unknown parameter values or kinetics. For example, some reaction networks may give rise to multiple equilibria (i.e., they may function as a biochemical switch) while other networks have unique equilibria for any parameter values. Or, some reaction networks may give rise to monotone systems, which renders their dynamics especially stable. We describe how the species-reaction graph (SR graph) can be used to analyze both multistability and monotonicity of networks.

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Notes

  1. 1.

    In [5] cycles are called ‘loops’.

  2. 2.

    The original definition of e-cycles and o-cycles in [11] describes these types of cycles in terms of ‘c-pairs’: e-cycles have an even number of c-pairs and o-cycles have an odd number of c-pairs. The two definitions are equivalent for networks that do not have one-step catalysis, which are our main focus here. Compare also with Lemma 4.4 in [5].

  3. 3.

    Often much more can be said, e.g., under some additional assumptions it follows that all positive solutions converge to an equilibrium. See Theorem 2 in [5] for details.

  4. 4.

    On the other hand, note that the notion of ‘directed SR graph’ in [5] is different from the notion of ‘DSR graph’ (also called directed SR graph) introduced in [10].

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Acknowledgements

We thank David Angeli, Murad Banaji, and Patrick DeLeenheer for very useful discussions about various topics described in this chapter. The authors acknowledge support from NIH grant R01GM086881, and from the DOE BACTER Institute.

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Authors and Affiliations

  1. Department of Mathematics and Department of Biomolecular Chemistry, University of Wisconsin, Madison, USA

    Gheorghe Craciun

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  1. Gheorghe Craciun
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  2. Casian Pantea
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  3. Eduardo D. Sontag
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Correspondence to Gheorghe Craciun .

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Editors and Affiliations

  1. , Automatic Control Laboratory, ETH Zurich, Physikstrasse 3, Zurich, 8092, Switzerland

    Heinz Koeppl

  2. ARCES, Università di Bologna, Via Toffano 2/2, Bologna, 40125, Italy

    Gianluca Setti

  3. Università di Napoli Federico II, Via Claudio 21, Napoli, 80125, Italy

    Mario di Bernardo

  4. , Department of Electrical and Computer En, Boston University, Saint Mary's St. 8, Boston, 02446, Massachusetts, USA

    Douglas Densmore

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Craciun, G., Pantea, C., Sontag, E.D. (2011). Graph-Theoretic Analysis of Multistability and Monotonicity for Biochemical Reaction Networks. In: Koeppl, H., Setti, G., di Bernardo, M., Densmore, D. (eds) Design and Analysis of Biomolecular Circuits. Springer, New York, NY. https://doi.org/10.1007/978-1-4419-6766-4_3

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