Boolean Network Approach to Negative Feedback Loops of the p53 Pathways: Synchronized Dynamics and Stochastic Limit Cycles

TL;DR

This paper models the negative feedback loops in p53 pathways using Boolean networks, revealing how they maintain low p53 levels and exhibit stable oscillations and stochastic synchronization under noise.

Contribution

It introduces deterministic and stochastic Boolean network models for p53 feedback loops, analyzing their synchronized dynamics and stochastic limit cycles with mathematical circulation theory.

Findings

Negative feedback maintains low p53 steady state.

Networks exhibit stable oscillations after perturbation.

Stochastic models show synchronization phenomena.

Abstract

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.