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Multiple dynamic pathways always exist in biological networks, but theirrobustness against internal fluctuations and relative stability have not beenwell recognized and carefully analyzed yet. Here we try to address these issuesthrough an illustrative example, namely the Siah-1/beta-catenin/p14/19 ARF loopof protein p53 dynamics. Its deterministic Boolean network model predicts thattwo parallel pathways with comparable magnitudes of attractive basins shouldexist after the protein p53 is activated when a cell becomes harmfullydisturbed. Once the low but non-neglectable intrinsic fluctuations areincorporated into the model, we show that a phase transition phenomenon isemerged: in one parameter region the probability weights of the normal pathway,reported in experimental literature, are comparable with the other pathwaywhich is seemingly abnormal with the unknown functions, whereas, in some otherparameter regions, the probability weight of the abnormal pathway can evendominate and become globally attractive. The theory of exponentially perturbedMarkov chains is applied and further generalized in order to quantitativelyexplain such a phase transition phenomenon, in which the nonequilibrium"activation energy barriers" along each transiting trajectory between theparallel pathways and the number of "optimal transition paths" play a centralpart. Our theory can also determine how the transition time and the number ofoptimal transition paths between the parallel pathways depend on eachinteraction's strength, and help to identify those possibly more crucialinteractions in the biological network.

Stochastic robustness and relative stability of multiple pathways in biological networks