  Multisite phosphorylation plays an important role in intracellular signaling.
There has been much recent work aimed at understanding the dynamics of such
systems when the phosphorylation/dephosphorylation mechanism is distributive,
that is, when the binding of a substrate and an enzyme molecule results in
addition or removal of a single phosphate group and repeated binding therefore
is required for multisite phosphorylation. In particular, such systems admit
bistability. Here we analyze a different class of multisite systems, in which
the binding of a substrate and an enzyme molecule results in addition or
removal of phosphate groups at all phosphorylation sites. That is, we consider
systems in which the mechanism is processive, rather than distributive. We show
that in contrast with distributive systems, processive systems modeled with
mass-action kinetics do not admit bistability and, moreover, exhibit rigid
dynamics: each invariant set contains a unique equilibrium, which is a global
attractor. Additionally, we obtain a monomial parametrization of the steady
states. Our proofs rely on a technique of Johnston for using "translated"
networks to study systems with "toric steady states", recently given sign
conditions for injectivity of polynomial maps, and a result from monotone
systems theory due to Angeli and Sontag.
