Multistationarity in semi-open Phosphorylation-Dephosphorylation Cycles

TL;DR

This paper investigates how partial openness in phosphorylation-dephosphorylation cycles affects their ability to exhibit multiple stable states, revealing that opening substrates preserves multistationarity while opening enzymes destroys it.

Contribution

It provides structural results showing how opening specific species influences multistationarity in semi-open biochemical networks under mass action kinetics.

Findings

Opening substrate species preserves multistationarity.

Opening enzyme species destroys multistationarity.

Method combines ACR detection with projection techniques.

Abstract

Multistationarity, underlies biochemical switching and cellular decision-making. We study how multistationarity in the sequential n-site phosphorylation-dephosphorylation cycle is affected when only some species are open, meaning allowed to exchange with the environment (so-called semi-open networks). Working under mass action kinetics, we obtain two complementary structural results for every $n\geq 2$. First, opening any nonempty subset of the substrate species preserves the network's capacity for nondegenerate multistationarity. Second, opening the enzyme species (both kinase and phosphatase), possibly together with any subset of substrates, always destroys multistationarity. The latter result is proved by a general reduction framework combining the detection of absolute concentration robustness (ACR) with projection onto the remaining species; when the projection is monostationary, the full semi-open system is monostationary. We also illustrate the general method on multi-layer cascade variants and discuss biological implications: opening enzymes acts as a robust switch that converts a potentially multistationary phosphorylation module into a monostationary one, while substrate exchange preserves switching capacity and thus the ability to couple cycles to downstream processes.