  Deterministic and stochastic Boolean network models are build for the
dynamics of negative feedback loops of the p53 pathways. It is shown that the
main function of the negative feedback in the p53 pathways is to keep p53 at a
low steady state level, and each sequence of protein states in the negative
feedback loops, is globally attracted to a closed cycle of the p53 dynamics
after being perturbed by outside signal (e.g. DNA damage). Our theoretical and
numerical studies show that both the biological stationary state and the
biological oscillation after being perturbed are stable for a wide range of
noise level. Applying the mathematical circulation theory of Markov chains, we
investigate their stochastic synchronized dynamics and by comparing the network
dynamics of the stochastic model with its corresponding deterministic network
counterpart, a dominant circulation in the stochastic model is the natural
generalization of the deterministic limit cycle in the deterministic system.
Moreover, the period of the main peak in the power spectrum, which is in common
use to characterize the synchronized dynamics, perfectly corresponds to the
number of states in the main cycle with dominant circulation. Such a large
separation in the magnitude of the circulations, between a dominant, main cycle
and the rest, gives rise to the stochastic synchronization phenomenon.
