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Stochastic robustness and relative stability of multiple pathways in biological networks

Multiple dynamic pathways always exist in biological networks, but their robustness against internal fluctuations and relative stability have not been well recognized and carefully analyzed yet. Here we try to address these issues through an illustrative example, namely the Siah-1/beta-catenin/p14/19 ARF loop of protein p53 dynamics. Its deterministic Boolean network model predicts that two parallel pathways with comparable magnitudes of attractive basins should exist after the protein p53 is activated when a cell becomes harmfully disturbed. Once the low but non-neglectable intrinsic fluctuations are incorporated into the model, we show that a phase transition phenomenon is emerged: in one parameter region the probability weights of the normal pathway, reported in experimental literature, are comparable with the other pathway which is seemingly abnormal with the unknown functions, whereas, in some other parameter regions, the probability weight of the abnormal pathway can even dominate and become globally attractive. The theory of exponentially perturbed Markov chains is applied and further generalized in order to quantitatively explain such a phase transition phenomenon, in which the nonequilibrium "activation energy barriers" along each transiting trajectory between the parallel pathways and the number of "optimal transition paths" play a central part. Our theory can also determine how the transition time and the number of optimal transition paths between the parallel pathways depend on each interaction's strength, and help to identify those possibly more crucial interactions in the biological network.

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