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From Structure to Dynamics in Biological Networks

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Book cover Design and Analysis of Biomolecular Circuits

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Abstract

Biological systems often display behaviour that is robust to considerable perturbation. In fact, experimental and computational work suggests that some behaviours are ‘structural’ in that they occur in all systems with particular qualitative features. In this chapter, some relationships between structure and dynamics in biological networks are explored. The emphasis is on chemical reaction networks, regarded as special cases of more general classes of dynamical systems termed interaction networks. The mathematical approaches described involve relating patterns in the Jacobian matrix to the dynamics of a system. Via a series of examples, it is shown how simple computations on matrices and related graphs can lead to strong conclusions about allowed behaviours.

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Notes

  1. 1.

    Stoichiometry classes are also sometimes referred to as ‘stoichiometric compatibility classes’.

  2. 2.

    Although conclusions for CRNs with P 0 or P 0 ( − ) Jacobian are stated in terms of the absence of multiple positive nondegenerate equilibria, additional structure, for example involving inflow and outflow of substrates, can imply a P or P ( − ) Jacobian, and thus the existence of no more than one equilibrium on all of state space.

  3. 3.

    There is some ambiguity in terminology in different strands of the literature. Cones referred to as ‘pointed’ here and in [9] are termed ‘salient’ in some references, with the word ‘pointed’ referring to cones containing the zero vector. Since all cones discussed here are closed, they are all ‘pointed’ in this other sense too.

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Authors and Affiliations

  1. Department of Mathematics, University of Portsmouth, Lion Gate Building, Lion Terrace, Portsmouth, Hampshire, PO1 3HF, UK

    Murad Banaji

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  1. Murad Banaji
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Correspondence to Murad Banaji .

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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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Banaji, M. (2011). From Structure to Dynamics in Biological 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_4

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  • DOI: https://doi.org/10.1007/978-1-4419-6766-4_4

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