IDEAS home Printed from https://ideas.repec.org/a/gam/jmathe/v9y2021i23p3133-d695399.html
   My bibliography  Save this article

Stability Analysis of Parameter Varying Genetic Toggle Switches Using Koopman Operators

Author

Listed:
  • 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
    as

    Download full text from publisher

    File URL: https://www.mdpi.com/2227-7390/9/23/3133/pdf
    Download Restriction: no

    File URL: https://www.mdpi.com/2227-7390/9/23/3133/
    Download Restriction: no
    ---><---

    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.
    Full references (including those not matched with items on IDEAS)

    Citations

    Citations are extracted by the CitEc Project, subscribe to its RSS feed for this item.
    as


    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.

    Most related items

    These are the items that most often cite the same works as this one and are cited by the same works as this one.
    1. Avraham E Mayo & Yaakov Setty & Seagull Shavit & Alon Zaslaver & Uri Alon, 2006. "Plasticity of the cis-Regulatory Input Function of a Gene," PLOS Biology, Public Library of Science, vol. 4(4), pages 1-1, March.
    2. Tomas Tokar & Jozef Ulicny, 2013. "The Mathematical Model of the Bcl-2 Family Mediated MOMP Regulation Can Perform a Non-Trivial Pattern Recognition," PLOS ONE, Public Library of Science, vol. 8(12), pages 1-8, December.
    3. Weiyue Ji & Handuo Shi & Haoqian Zhang & Rui Sun & Jingyi Xi & Dingqiao Wen & Jingchen Feng & Yiwei Chen & Xiao Qin & Yanrong Ma & Wenhan Luo & Linna Deng & Hanchi Lin & Ruofan Yu & Qi Ouyang, 2013. "A Formalized Design Process for Bacterial Consortia That Perform Logic Computing," PLOS ONE, Public Library of Science, vol. 8(2), pages 1-9, February.
    4. Chen Jia & Ramon Grima, 2024. "Holimap: an accurate and efficient method for solving stochastic gene network dynamics," Nature Communications, Nature, vol. 15(1), pages 1-14, December.
    5. T. Ochiai & J. C. Nacher, 2007. "Stochastic analysis of autoregulatory gene expression dynamics," Mathematical and Computer Modelling of Dynamical Systems, Taylor & Francis Journals, vol. 14(4), pages 377-388, November.
    6. Shivang Hina-Nilesh Joshi & Chentao Yong & Andras Gyorgy, 2022. "Inducible plasmid copy number control for synthetic biology in commonly used E. coli strains," Nature Communications, Nature, vol. 13(1), pages 1-16, December.
    7. Thomas B. Kepler & Timothy C. Elston, 2001. "Stochasticity in Transcriptional Regulation: Origins, Consequences and Mathematical Representations," Working Papers 01-06-033, Santa Fe Institute.
    8. Luis Mier-y-Terán-Romero & Mary Silber & Vassily Hatzimanikatis, 2010. "The Origins of Time-Delay in Template Biopolymerization Processes," PLOS Computational Biology, Public Library of Science, vol. 6(4), pages 1-15, April.
    9. Ashty S. Karim & Dylan M. Brown & Chloé M. Archuleta & Sharisse Grannan & Ludmilla Aristilde & Yogesh Goyal & Josh N. Leonard & Niall M. Mangan & Arthur Prindle & Gabriel J. Rocklin & Keith J. Tyo & L, 2024. "Deconstructing synthetic biology across scales: a conceptual approach for training synthetic biologists," Nature Communications, Nature, vol. 15(1), pages 1-14, December.
    10. Bonassi Fernando V. & You Lingchong & West Mike, 2011. "Bayesian Learning from Marginal Data in Bionetwork Models," Statistical Applications in Genetics and Molecular Biology, De Gruyter, vol. 10(1), pages 1-27, October.
    11. Zhdanov, Vladimir P., 2011. "Periodic perturbation of the bistable kinetics of gene expression," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 390(1), pages 57-64.
    12. Tatiana Baumuratova & Simona Dobre & Thierry Bastogne & Thomas Sauter, 2013. "Switch of Sensitivity Dynamics Revealed with DyGloSA Toolbox for Dynamical Global Sensitivity Analysis as an Early Warning for System's Critical Transition," PLOS ONE, Public Library of Science, vol. 8(12), pages 1-12, December.
    13. Xu, Yong & Wu, Juan & Du, Lin & Yang, Hui, 2016. "Stochastic resonance in a genetic toggle model with harmonic excitation and Lévy noise," Chaos, Solitons & Fractals, Elsevier, vol. 92(C), pages 91-100.
    14. Simeon D. Castle & Michiel Stock & Thomas E. Gorochowski, 2024. "Engineering is evolution: a perspective on design processes to engineer biology," Nature Communications, Nature, vol. 15(1), pages 1-10, December.
    15. Alicia Sanchez-Gorostiaga & Djordje Bajić & Melisa L Osborne & Juan F Poyatos & Alvaro Sanchez, 2019. "High-order interactions distort the functional landscape of microbial consortia," PLOS Biology, Public Library of Science, vol. 17(12), pages 1-34, December.
    16. Tai-Yin Chiu & Hui-Ju K Chiang & Ruei-Yang Huang & Jie-Hong R Jiang & François Fages, 2015. "Synthesizing Configurable Biochemical Implementation of Linear Systems from Their Transfer Function Specifications," PLOS ONE, Public Library of Science, vol. 10(9), pages 1-27, September.
    17. Nagarajan, Radhakrishnan, 2007. "Delay estimation in a two-node acyclic network," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 376(C), pages 725-737.
    18. Takako Kaneko-Kawano & Fugo Takasu & Honda Naoki & Yuichi Sakumura & Shin Ishii & Takahiro Ueba & Akinori Eiyama & Aiko Okada & Yoji Kawano & Kenji Suzuki, 2012. "Dynamic Regulation of Myosin Light Chain Phosphorylation by Rho-kinase," PLOS ONE, Public Library of Science, vol. 7(6), pages 1-10, June.
    19. Liu, Xian & Wang, Jinzhi & Huang, Lin, 2007. "Global synchronization for a class of dynamical complex networks," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 386(1), pages 543-556.
    20. Tobias May & Lee Eccleston & Sabrina Herrmann & Hansjörg Hauser & Jorge Goncalves & Dagmar Wirth, 2008. "Bimodal and Hysteretic Expression in Mammalian Cells from a Synthetic Gene Circuit," PLOS ONE, Public Library of Science, vol. 3(6), pages 1-7, June.

    Corrections

    All material on this site has been provided by the respective publishers and authors. You can help correct errors and omissions. When requesting a correction, please mention this item's handle: RePEc:gam:jmathe:v:9:y:2021:i:23:p:3133-:d:695399. See general information about how to correct material in RePEc.

    If you have authored this item and are not yet registered with RePEc, we encourage you to do it here. This allows to link your profile to this item. It also allows you to accept potential citations to this item that we are uncertain about.

    If CitEc recognized a bibliographic reference but did not link an item in RePEc to it, you can help with this form .

    If you know of missing items citing this one, you can help us creating those links by adding the relevant references in the same way as above, for each refering item. If you are a registered author of this item, you may also want to check the "citations" tab in your RePEc Author Service profile, as there may be some citations waiting for confirmation.

    For technical questions regarding this item, or to correct its authors, title, abstract, bibliographic or download information, contact: MDPI Indexing Manager (email available below). General contact details of provider: https://www.mdpi.com .

    Please note that corrections may take a couple of weeks to filter through the various RePEc services.

    IDEAS is a RePEc service. RePEc uses bibliographic data supplied by the respective publishers.