Switchable Genetic Oscillator Operating in Quasi-Stable Mode
📝 Abstract
Ring topologies of repressing genes have qualitatively different long-term dynamics if the number of genes is odd (they oscillate) or even (they exhibit bistability). However, these attractors may not fully explain the observed behavior in transient and stochastic environments such as the cell. We show here that even repressilators possess quasi-stable, travelling-wave periodic solutions that are reachable, long-lived and robust to parameter changes. These solutions underlie the sustained oscillations observed in even rings in the stochastic regime, even if these circuits are expected to behave as switches. The existence of such solutions can also be exploited for control purposes: operation of the system around the quasi-stable orbit allows us to turn on and off the oscillations reliably and on demand. We illustrate these ideas with a simple protocol based on optical interference that can induce oscillations robustly both in the stochastic and deterministic regimes.
💡 Analysis
Ring topologies of repressing genes have qualitatively different long-term dynamics if the number of genes is odd (they oscillate) or even (they exhibit bistability). However, these attractors may not fully explain the observed behavior in transient and stochastic environments such as the cell. We show here that even repressilators possess quasi-stable, travelling-wave periodic solutions that are reachable, long-lived and robust to parameter changes. These solutions underlie the sustained oscillations observed in even rings in the stochastic regime, even if these circuits are expected to behave as switches. The existence of such solutions can also be exploited for control purposes: operation of the system around the quasi-stable orbit allows us to turn on and off the oscillations reliably and on demand. We illustrate these ideas with a simple protocol based on optical interference that can induce oscillations robustly both in the stochastic and deterministic regimes.
📄 Content
Switchable quasi-stable gene oscillator 1 Switchable Genetic Oscillator Operating in Quasi-Stable Mode Natalja Strelkowa and Mauricio Barahona Department of Bioengineering & Institute for Mathematical Sciences, Imperial College London, South Kensington Campus, London SW7 2AZ, United Kingdom Abstract Ring topologies of repressing genes have qualitatively different long-term dynamics if the number of genes is odd (they oscillate) or even (they exhibit bistability). However, these attractors may not fully explain the observed behavior in transient and stochastic envi- ronments such as the cell. We show here that even re- pressilators possess quasi-stable, travelling-wave pe- riodic solutions that are reachable, long-lived and ro- bust to parameter changes. These solutions underlie the sustained oscillations observed in even rings in the stochastic regime, even if these circuits are ex- pected to behave as switches. The existence of such solutions can also be exploited for control purposes: operation of the system around the quasi-stable orbit allows us to turn on and offthe oscillations reliably and on demand. We illustrate these ideas with a sim- ple protocol based on optical interference that can in- duce oscillations robustly both in the stochastic and deterministic regimes. Key words: synthetic biology; oscillatory gene networks; generalized repressilator; traveling waves; stochastic dynamics Introduction Recent experimental advances in cellular and molec- ular biology have made it possible to engineer intri- cate gene regulatory circuits (1). Inspired in many cases by electronic elements, simple gene networks have been designed to perform reproducible, low-level functions. Some classic examples include the toggle switch (2), the genetic ring oscillator known as the repressilator (3), or a circuit that can exhibit both oscillatory and switching behavior through the alter- ation of biochemical interactions (4). Such simple cir- cuits could be potentially interconnected and built- up to form more elaborate ‘biological devices’ with large numbers of components. This trend is facili- tated by simulation software containing large number of genes (5) as well as libraries of composable biologi- cal parts for experimental realization (6). Simple syn- thetic modules can also be integrated into the com- plex machinery of the cell, as in the oscillator recently implemented in a mammalian cell (7), or interfaced with cellular pathways to induce particular responses, as in the construct where the toggle switch was con- nected to the SOS pathway to induce DNA protection mechanisms in E. coli when exposed to UV light (8). Similar principles have been exploited in the rational design of internal negative feedback operated in con- junction with external arabinose-driven positive feed- back to produce cell-synchronized oscillations (9). The central role played by oscillations in cellu- lar function has made oscillatory circuits a primary target for the analysis and design of synthetic net- works. A particular area of interest is the elucidation of strategies leading to robust timing and sequen- tial activation in the cell. For instance, key stages in developmental biology and in cell differentiation may be controlled by so-called master regulators—a small set of transcription factors sequentially acti- vating and driving several other genes with accurate timing (10, 11, 12). In addition, studies of both natu- ral (13, 14) and engineered circuits (15) indicate that the correct timing and order of gene activation is a key characteristic of balanced, optimal cell function, as it reduces the metabolic burdening that ensues from the continuous presence of heterologous sub- stances (16). In this paper, we consider the dynamics and con- trol of noisy genetic oscillatory circuits in quasi-stable mode operation. We exemplify our results with one of the simplest and most widely studied synthetic net- works: the N-gene ring repressilator (Fig. 1a). Some natural networks of master regulators (10) contain arXiv:0909.1935v2 [q-bio.MN] 19 Nov 2009 Switchable quasi-stable gene oscillator 2 such ring structures as subnetworks, making the ex- ploration of their dynamic behavior relevant for both naturally occurring and synthetic systems. The un- derlying idea is well-known: when observing the dy- namics of systems operating in highly variable envi- ronments, such as the cell, it might not be enough to characterize only the long-term attractors of the system since unstable solutions can play a significant role. For instance, quasi-stable transients might be so long-lived as to be the most significant feature of the observed noisy dynamics (17). Moreover, the presence of noise in nonlinear systems may induce non-stationary dynamics in systems with only fixed point attractors in the deterministic setting (18) or, conversely, noise may act as a stabilizer of unstable deterministic states (19). In the generalized repressilator, results due to Smith (
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