  This work concerns the question of how two important dynamical properties,
oscillations and bistability, emerge in an important biological signaling
network. Specifically, we consider a model for dual-site phosphorylation and
dephosphorylation of extracellular signal-regulated kinase (ERK). We prove that
oscillations persist even as the model is greatly simplified (reactions are
made irreversible and intermediates are removed). Bistability, however, is much
less robust -- this property is lost when intermediates are removed or even
when all reactions are made irreversible. Moreover, bistability is
characterized by the presence of two reversible, catalytic reactions: as other
reactions are made irreversible, bistability persists as long as one or both of
the specified reactions is preserved. Finally, we investigate the maximum
number of steady states, aided by a network's "mixed volume" (a concept from
convex geometry). Taken together, our results shed light on the question of how
oscillations and bistability emerge from a limiting network of the ERK network
-- namely, the fully processive dual-site network -- which is known to be
globally stable and therefore lack both oscillations and bistability. Our
proofs are enabled by a Hopf bifurcation criterion due to Yang, analyses of
Newton polytopes arising from Hurwitz determinants, and recent
characterizations of multistationarity for networks having a steady-state
parametrization.
