Robustness and relative stability of multiple attractors in\\ a stochastic Boolean network

Abstract

Multiple phenotypic states and dynamic pathways always exist in biological networks, but their robustness against fluctuations and relative stability have not been fully recognized or carefully analyzed yet. Here we try to address these issues with a widely used stochastic Boolean network model. The stability of phenotypic states and dynamical pathways corresponds to the robustness and relative stability of attractors in such a stochastic Boolean network, which can be analyzed using the theory of exponentially perturbed Markov chains. It is already known that the logarithm of the expected exit time escaping from an attractor is proportional to the ``minimum activation energy barrier in the exponentially perturbed Markov chains. We first prove that within an exponentially perturbed Markov chain, all pathways between attractors with the minimum possible ``activation energy barrier have the same probability weights in the zero-noise limit, and therefore the relative stability just depends on the number of optimal paths. Another important implication of this theory is that, once the non-neglectable fluctuations are rather low, a phase transition phenomenon emerges: In one parameter region the probability weights of the coexisted phenotypic states and pathways are comparable with each other; whereas in some other parameter regions, the probability weight of certain phenotypic state or pathway can even dominate and become globally attractive. Finally, we apply the theory to an artificial model and the Siah-1/beta-catenin/p14/19 ARF loop of protein p53 dynamics. Our theory can also determine how the transition time and the number of optimal transition paths between the phenotypic states and parallel pathways depend on all the parameters, and help to identify those possibly more crucial nodes and the interactions between them in a biological network.