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Stability Analysis of Parameter Varying Genetic Toggle Switches Using Koopman Operators

Author

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  • Jamiree Harrison

    (Department of Mechanical Engineering, University of California, Santa Barbara, CA 93106, USA
    Current address: Biological Computing, Control, and Learning Laboratory, 1201 Bioengineering Center, University of California, Santa Barbara, CA 93106, USA.)

  • Enoch Yeung

    (Department of Mechanical Engineering, University of California, Santa Barbara, CA 93106, USA
    Center for Control, Dynamical Systems, and Computation, Biomolecular Science and Engineering Graduate Program, Center for Biological Engineering, University of California, Santa Barbara, CA 93106, USA
    Current address: Biological Computing, Control, and Learning Laboratory, 1201 Bioengineering Center, University of California, Santa Barbara, CA 93106, USA.)

Abstract

The genetic toggle switch is a well known model in synthetic biology that represents the dynamic interactions between two genes that repress each other. The mathematical models for the genetic toggle switch that currently exist have been useful in describing circuit dynamics in rapidly dividing cells, assuming fixed or time-invariant kinetic rates. There is a growing interest in being able to model and extend synthetic biological function for growth conditions such as stationary phase or during nutrient starvation. As cells transition from one growth phase to another, kinetic rates become time-varying parameters. In this paper, we propose a novel class of parameter varying nonlinear models that can be used to describe the dynamics of genetic circuits, including the toggle switch, as they transition from different phases of growth. We show that there exists unique solutions for this class of systems, as well as for a class of systems that incorporates the microbial phenomena of quorum sensing. Further, we show that the domain of these systems, which is the positive orthant, is positively invariant. We also showcase a theoretical control strategy for these systems that would grant asymptotic monostability of a desired fixed point. We then take the general form of these systems and analyze their stability properties through the framework of time-varying Koopman operator theory. A necessary condition for asymptotic stability is also provided as well as a sufficient condition for instability. A Koopman control strategy for the system is also proposed, as well as an analogous discrete time-varying Koopman framework for applications with regularly sampled measurements.

Suggested Citation

  • Jamiree Harrison & Enoch Yeung, 2021. "Stability Analysis of Parameter Varying Genetic Toggle Switches Using Koopman Operators," Mathematics, MDPI, vol. 9(23), pages 1-25, December.
    • Handle: RePEc:gam:jmathe:v:9:y:2021:i:23:p:3133-:d:695399
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    References listed on IDEAS

    as
    1. Evgeni V Nikolaev & Eduardo D Sontag, 2016. "Quorum-Sensing Synchronization of Synthetic Toggle Switches: A Design Based on Monotone Dynamical Systems Theory," PLOS Computational Biology, Public Library of Science, vol. 12(4), pages 1-33, April.
    2. Timothy S. Gardner & Charles R. Cantor & James J. Collins, 2000. "Construction of a genetic toggle switch in Escherichia coli," Nature, Nature, vol. 403(6767), pages 339-342, January.
    3. Steven L Brunton & Bingni W Brunton & Joshua L Proctor & J Nathan Kutz, 2016. "Koopman Invariant Subspaces and Finite Linear Representations of Nonlinear Dynamical Systems for Control," PLOS ONE, Public Library of Science, vol. 11(2), pages 1-19, February.
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    Cited by:

    1. Chun Zhang & Qiaoxia Tang & Zhixiang Wang, 2022. "Grazing and Symmetry-Breaking Bifurcations Induced Oscillations in a Switched System Composed of Duffing and van der Pol Oscillators," Mathematics, MDPI, vol. 10(5), pages 1-10, February.

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