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.
Highlights
Switching behavior in biological networks is of interest to many researchers across a variety of disciplines, such as epigenetics [1], synthetic biology [2], dynamical systems [3]and more [4,5,6]
In order to show that the genetic toggle switch with quorum sensing (GTSQS) meets some basic criteria, and is in line with existing genetic toggle switch theory, we provide Proposition 2 to ensure that our model has unique solutions on our domain
We extended the classical genetic toggle switch model to a parameter varying toggle switch that maintained the key characteristics of the original model while allowing for the kinetic rates of synthesis to vary with time
Summary
Switching behavior in biological networks is of interest to many researchers across a variety of disciplines, such as epigenetics [1], synthetic biology [2], dynamical systems [3]. In every batch growth condition, bacterial cells will slowly deplete all nutrients in a growth medium, giving rise to late log-phase or early stationary phase dynamics [10] In this setting, genetic circuits can be modeled as being subject to time-varying kinetic rates, as the cells are no longer in a state of dynamic, chemical equilibrium. If we relax this assumption for the effective rates of synthesis and the decay rates, we could have in our model that the kinetic parameters α1 , α2 , δ1 , δ2 become time-varying functions, i.e., parameters that change over time This generalization of the functional form of our system parameters, from time-invariant to time varying, provides a more accurate framework to model the genetic toggle switch in cells transitioning growth phases. Motivated by the parametervarying nature of growth-phase transitions, here we define and extend the Koopman operator framework for the analysis of parameter varying nonlinear system models
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