Stability and bifurcation analysis of a delayed genetic oscillator model

In this work, a sufficiently simple model allows one to perform detailed analytic studies to gain insights for the dynamical mechanisms in the general model of gene expression. Through theoretical analysis and numerical simulation, we find that the transcription and translation delays as a bifurcation parameter can drive the system to undergo a Hopf bifurcation, thereby producing oscillation behavior. Meanwhile, the length of these delays can determine the amplitude and period of the oscillations. Moreover, an optimal rate of model parameters is also essential to induce the limit-cycle oscillations. In particular, the transcription factor concentration is called a signal that not only induces twice Hopf bifurcation, but also affects the role of the delay in the system, that is, the varying time delay leads to subcritical Hopf bifurcation and bistability under the small signal strength. On the contrary, supercritical Hopf bifurcation is generated. Furthermore, two-parameter diagram is drawn to further explore how time delay and signal strength coordinate to regulate the oscillation behavior of the system. In addition, the stability and the direction of periodic solutions bifurcating from Hopf bifurcations are determined through the explicit formulae and the global existence of periodic solutions is discussed to verify the authenticity of the simulation. These results may be helpful for us to fully grasp oscillation dynamics and provide clues for further exploring the molecular mechanism of biological rhythms.

Automated summary

The study demonstrates that transcription and translation delays act as bifurcation parameters driving oscillation behavior in a gene expression model, with their length determining the amplitude and period of the oscillations. Optimal parameter rates are also crucial for inducing limit-cycle oscillations. Additionally, transcription factor concentration serves as a signal inducing bifurcations and affecting delay effects on the system, with subcritical Hopf bifurcation occurring under small signal strength.