Deterministic and stochastic Boolean network models are build for thedynamics of negative feedback loops of the p53 pathways. It is shown that themain function of the negative feedback in the p53 pathways is to keep p53 at alow steady state level, and each sequence of protein states in the negativefeedback loops, is globally attracted to a closed cycle of the p53 dynamicsafter being perturbed by outside signal (e.g. DNA damage). Our theoretical andnumerical studies show that both the biological stationary state and thebiological oscillation after being perturbed are stable for a wide range ofnoise level. Applying the mathematical circulation theory of Markov chains, weinvestigate their stochastic synchronized dynamics and by comparing the networkdynamics of the stochastic model with its corresponding deterministic networkcounterpart, a dominant circulation in the stochastic model is the naturalgeneralization of the deterministic limit cycle in the deterministic system.Moreover, the period of the main peak in the power spectrum, which is in commonuse to characterize the synchronized dynamics, perfectly corresponds to thenumber of states in the main cycle with dominant circulation. Such a largeseparation in the magnitude of the circulations, between a dominant, main cycleand the rest, gives rise to the stochastic synchronization phenomenon.
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