An all-encompassing global convergence result for processive multisite phosphorylation systems

Abstract

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, which require an enzyme and substrate to bind several times in order to add multiple phosphate groups, can be bistable. Processive systems, in contrast, require only one binding to add all phosphate groups, and 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 first defining an all-encompassing processive network that encapsulates all of these schemes, and then appealing to recent results of Marcondes de Freitas et al. that assert global convergence by way of monotone systems theory and network/graph reductions (corresponding to removing 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.

A computational framework for a Lyapunov-enabled analysis of biochemical reaction networks

“…Our class of networks is distinct from the one identified by the HJF theory [10,13] and it has wider applications to biology as we have shown. Furthermore, our results include all networks that have been studied via compartmental system techniques [18,19] and via cooperativity in reaction coordinates [4,20,29]. In fact, showing that the latter class of network always admits an RLF is a subject of a forthcoming paper.…”

“…A generalized processive cycle An "all-encompassing" processive cycle has been studied in [4] which allows multiple enzymes and is depicted in Figure 9-c. It takes the following form:…”

“…Both networks above have been studied in [4,29] by establishing monotonicity in reaction coordinates. Such techniques require checking persistence a priori and do not provide Lyapunov functions.…”

“…We call such networks attractive, meaning specifically in our context that they do not have the potential for exhibiting complex behavior such as multiple-steady states (e.g, toggle switches), oscillations (e.g, repressilator), or chaos. For instance, within the class of post-translational modification (PTM) cycles, some cascades are "attractive" but others can exhibit oscillations and multistability [4]. Figure 1 illustrates the typical behavior of an attractive network vs a multistable network for two PTM cycles that have been proposed as models for double phosphorylation.…”

Intermediates and Generic Convergence to Equilibria

“…See also [6] for a class of reaction networks generalising this example where the reduction approach presented here can be applied.…”