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    Equilibria and Stability of a Class of Positive Feedback Loops : Mathematical analysis and its application to caspase-dependent apoptosis

    Lopez-Caamal, Fernando and Middleton, Richard H. and Huber, Heinrich J. (2014) Equilibria and Stability of a Class of Positive Feedback Loops : Mathematical analysis and its application to caspase-dependent apoptosis. Journal of Mathematical Biology, 68 (3). pp. 609-645. ISSN 0303-6812

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    Positive feedback loops are common regulatory elements in metabolic and protein signalling pathways. The length of such feedback loops deter- mines stability and sensitivity to network perturbations. Here we provide a mathematical analysis of arbitrary length positive feedback loops with protein production and degradation. These loops serve as an abstraction of typical regulation patterns in protein signalling pathways. We first perform a steady state analysis and, independently of the chain length, identify exactly two steady states that represent either biological activity or inactivity. We thereby provide two formulas for the steady state protein concentrations as a function of feedback length, strength of feedback, as well as protein production and degradation rates. Using a control theory approach, analysing the frequency response of the linearisation of the system and exploiting the Small Gain The- orem, we provide conditions for local stability for both steady states. Our results demonstrate that, under some parameter relationships, once a biolog- ical meaningful on steady state arises, it is stable, while the off steady state, where all proteins are inactive, becomes unstable. We apply our results to a three-tier feedback of caspase activation in apoptosis and demonstrate how an intermediary protein in such a loop may be used as a signal amplifier within the cascade. Our results provide a rigorous mathematical analysis of positive feedback chains of arbitrary length, thereby relating pathway structure and stability.

    Item Type: Article
    Additional Information: This is the postprint version of the published article, which is available at DOI: I 10.1007/s00285-013-0644-z
    Keywords: Positive feedback; Recursive protein activation; Apoptosis; ODE; Equilibrium points; Local stability;
    Academic Unit: Faculty of Science and Engineering > Research Institutes > Hamilton Institute
    Item ID: 6963
    Identification Number:
    Depositing User: Hamilton Editor
    Date Deposited: 09 Feb 2016 12:30
    Journal or Publication Title: Journal of Mathematical Biology
    Publisher: Springer Verlag
    Refereed: Yes
    Funders: Programme for Research in Third Level Institutions, Cycle 4, Ireland EU Structural Funds Programmes 2007-2013, Science Foundation Ireland (SFI)

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