In the cell, there are intrinsic statistical fluctuations from finite numbers of molecules and the external fluctuations from highly dynamical and inhomogeneous environments ( 1–8). They have been extensively explored, including cell cycles, circadian clocks, calcium oscillations, and glycolytic oscillations. Finally, we performed the global sensitivity analysis of the underlying parameters to find the key network wirings responsible for the stability of the oscillation system.īiological oscillations widely exist in living organisms. We found that when barrier height is higher, escape time is longer and phase coherence of oscillation is higher. Furthermore, we explored the escape time from the oscillation ring to outside at different molecular number. As a consequence of nonzero probability flux, we show that the three-point correlations from the time traces show irreversibility, providing a possible way to explore the flux from the observations. The peak is less prominent for smaller number of molecules and less barrier height and therefore can be used as another measure of stability of oscillations. We also found that the power spectrum of autocorrelation functions show resonance peak at the frequency of coherent oscillations. We also investigated the phase coherence and the entropy production rate of the system at different fluctuations and found that dissipations are less and the coherence is higher for larger number of molecules. Landscape attracts the systems down to the oscillation ring while flux drives the coherent oscillation on the ring. We found that in nonequilibrium dynamic systems, not only the potential landscape but also the probability flux are important to the dynamics of the system under intrinsic noise. The height of the Mexican-hat provides a quantitative measure to evaluate the robustness and coherence of the oscillation. The potential landscape of the network is uncovered and has a global Mexican-hat shape. We developed a landscape and flux framework to explore the global stability and robustness of a circadian oscillation system. However, the global and physical natures of the oscillation under fluctuations are still not very clear. The underlying natures of the rhythmic behavior have been explored by experimental and theoretical approaches. They are broadly distributed in living organisms, such as Neurospora, Drosophila, and mammals. Circadian rhythms with a period of ∼24 h, are natural timing machines.
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