THE EFFECT OF TIME SCALE DIFFERENCES AND TIME DELAYS ON THE STRUCTURAL STABILITY OF OSCILLATIONS IN A TWO-GENE NETWORK
Biological networks are often modeled by systems of ordinary differential equations. In chemical reaction kinetics, for instance, sigmoid functions represent the regulation of gene expression via transcription factors. Solutions of these models tend to converge to a unique steady state, and feedback control mechanisms are required for a more complex dynamic behavior.This paper focuses on periodic behavior in two-component regulatory networks. Here, a key issue is that oscillations in chemical reaction systems are usually not robust with respect to parameter variations. Small variations lead to bifurcations that change the system's overall qualitative dynamic behavior. This concerns the mechanisms stabilizing periodic behavior in living cells. Using a small sample network, we demonstrate that oscillations can efficiently be stabilized by large time scale differences that correspond to reactions with different velocities. Furthermore, the inclusion of a time delay, reflecting transport and diffusion processes, has a similar effect. This suggests that processes of this kind potentially play a crucial role in biological oscillators.
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