  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.
