  Phosphorylation, the enzyme-mediated addition of a phosphate group to a
molecule, is a ubiquitous chemical mechanism in biology. Multisite
phosphorylation, the addition of phosphate groups to multiple sites of a single
molecule, may be distributive or processive. Distributive systems can be
bistable, while processive systems were recently shown to be globally stable.
However, this global convergence result was proven only for a specific
mechanism of processive phosphorylation/dephosphorylation (namely, all
catalytic reactions are reversible). Accordingly, we generalize this result to
allow for processive phosphorylation networks in which each reaction may be
irreversible, and also to account for possible product inhibition. We
accomplish this by defining an all-encompassing processive network that
encapsulates all of these schemes, and then appealing to recent results of
Marcondes de Freitas, Wiuf, and Feliu that assert global convergence by way of
monontone systems theory and network/graph reductions (which correspond to
removal of intermediate complexes). Our results form a case study into the
question of when global convergence is preserved when reactions and/or
intermediate complexes are added to or removed from a network.
