Switchable Genetic Oscillator Operating in Quasi-Stable Mode

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.

💡 Research Summary

This paper investigates the dynamics of the generalized repressilator—a synthetic gene circuit in which each gene’s transcription is repressed by the protein product of the preceding gene—focusing on the often‑overlooked behavior of rings with an even number of genes. Classical theory, based on the work of Smith, predicts that odd‑gene rings exhibit a globally attracting limit‑cycle oscillation (via a Hopf bifurcation) while even‑gene rings display bistability (via a pitch‑fork bifurcation) and therefore behave as switches. The authors challenge this binary view by demonstrating that even‑gene repressilators also possess long‑lived, quasi‑stable periodic solutions that can dominate the observable dynamics in both deterministic and stochastic regimes.

Model formulation
The deterministic model consists of 2N ordinary differential equations for mRNA (m_i) and protein (p_i) concentrations:
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