  This work investigates the emergence of oscillations in one of the simplest
cellular signaling networks exhibiting oscillations, namely, the dual-site
phosphorylation and dephosphorylation network (futile cycle), in which the
mechanism for phosphorylation is processive while the one for dephosphorylation
is distributive (or vice-versa). The fact that this network yields oscillations
was shown recently by Suwanmajo and Krishnan. Our results, which significantly
extend their analyses, are as follows. First, in the three-dimensional space of
total amounts, the border between systems with a stable versus unstable steady
state is a surface defined by the vanishing of a single Hurwitz determinant.
Second, this surface consists generically of simple Hopf bifurcations. Next,
simulations suggest that when the steady state is unstable, oscillations are
the norm. Finally, the emergence of oscillations via a Hopf bifurcation is
enabled by the catalytic and association constants of the distributive part of
the mechanism: if these rate constants satisfy two inequalities, then the
system generically admits a Hopf bifurcation. Our proofs are enabled by the
Routh-Hurwitz criterion, a Hopf-bifurcation criterion due to Yang, and a
monomial parametrization of steady states.
