From this fact, the notion of phase can be extended to the entire phase space as described below. In this case, the oscillator recovers its amplitude and period, but generally, the phase of oscillation is shifted compared with that without perturbation. The oscillator deviates from the closed orbit after a perturbation is applied to the system, but subsequently returns to the closed orbit. First, the closed orbit is asymptotically stable ( Fig 1). Limit cycle oscillators have several properties. The system underlying this oscillation can be described as a limit-cycle oscillator, which forms an isolated closed orbit in a phase space. The negative regulations generate a stable periodic oscillation of the clock genes. PER and CRY are subsequently degraded by proteasomes. At the molecular level, E-box binding transcription factors such as Per (Per1/2) and Cry (Cry1/2) repress their own expression through directly inhibiting BMAL1 and CLOCK, which positively regulate Per and Cry. The oscillation of these circadian clock cells is regulated by a transcription-translation negative feedback loop. The mammalian cells of the biological clock oscillate with a stable circadian period. These findings support the existence of a circadian limit cycle oscillator in mammalian cells and suggest that a small number of variables determine the relaxation process after a perturbation. The average number of embedding dimensions at each dose of FK increased as the FK dose increased but most of the relaxation process was generally embedded within four dimensions. The trajectory of the limit cycle was embedded in two dimensions but with the perturbation of the state point moved out of the trajectory, the relaxation process was generally embedded in higher dimensions. Furthermore, we estimated the embedding dimension of the limit cycle before and after the perturbation. By broadly and finely changing the phase and strength of the perturbation, we detected the transition of the PRC from type 1 to type 0 and a possible singular transition point, the property of which agreed quite well with our numerical simulation of the noisy Goodwin model, a simple yet canonical model for the transcription-translation feedback loop of the core clock genes. As a perturbation, forskolin (FK) was administered to Rat-1 cells expressing the Per2:: luc gene. We detected the singular point and transition from a type 1 to type 0 phase response curve (PRC) and determined the embedding dimension to show how many variables are needed to describe the limit cycle oscillation and relaxation process after a perturbation. Here, we investigated a mammalian circadian oscillation geometrically before and after a perturbation. The circadian system has been regarded as a limit cycle oscillator constructed by the integrated interaction of clock genes and proteins.
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