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Multistationarity in semi-open Phosphorylation-Dephosphorylation Cycles

Praneet Nandan, Beatriz Pascual-Escudero, Diego Rojas La Luz

TL;DR

The paper addresses how semipermeable exchange with the environment affects multistationarity in sequential $n$-site phosphorylation cycles. It develops a dual approach: substrate opening preserves multistationarity through an inductive construction of steady states, while enzyme opening enforces a robust monostationary state via an ACR-based projection framework and deficiency-zero arguments. The main contributions are (i) a constructive induction proving nondegenerate multistationarity persists under substrate opening for all $n\,\ge\,2$, and (ii) a general reduction method showing enzyme openness leads to ACR and often monostationarity, extended to cascade networks like MAPK. These results illuminate how environmental exchange shapes biochemical switching and provide practical tools for diagnosing when semi-open modifications preserve or eliminate multistationarity.

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.

Multistationarity in semi-open Phosphorylation-Dephosphorylation Cycles

TL;DR

The paper addresses how semipermeable exchange with the environment affects multistationarity in sequential -site phosphorylation cycles. It develops a dual approach: substrate opening preserves multistationarity through an inductive construction of steady states, while enzyme opening enforces a robust monostationary state via an ACR-based projection framework and deficiency-zero arguments. The main contributions are (i) a constructive induction proving nondegenerate multistationarity persists under substrate opening for all , and (ii) a general reduction method showing enzyme openness leads to ACR and often monostationarity, extended to cascade networks like MAPK. These results illuminate how environmental exchange shapes biochemical switching and provide practical tools for diagnosing when semi-open modifications preserve or eliminate multistationarity.

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 . 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.

Paper Structure

This paper contains 16 sections, 18 theorems, 52 equations, 1 figure, 1 table.

Table of Contents

  1. Introduction
  2. Preliminaries and General Toolbox
  3. Notation and Definitions
  4. Network modifications
  5. Deficiency Theory
  6. Sequential $n$-site double phosphorylation cycle
  7. Substrate‐Opening in the $n$‐Site Cycle
  8. Overview and Main Statement
  9. Inductive Construction of Steady States
  10. Numerical illustration and discussion
  11. Enzyme‐Opening and monostationarity
  12. Overview and main statement
  13. ACR, projection and inheritance
  14. Application to the $n$‐site double phosphorylation cycle
  15. Extension to cascade networks
  16. ...and 1 more sections

Key Result

Theorem 2.17

Let $G=(\mathcal{X},\mathcal{C},\mathcal{R})$ be a network, let Y be a list of $k$ new species, and let $G'$ arise from $G$ by replacing each of the reactions: Suppose further that the new species Y enters nondegenerately into $G'$ in the sense that $\beta :=(\beta_1|\beta_2|\dots|\beta_r)$ has rank $r$ ($k\geq r$). $a_i,b_i$ and $c_i$ are nonnegative vectors and any or all may coincide. If $G$ a

Figures (1)

  • Figure 1: The MAPK cascade network

Theorems & Definitions (51)

  • Definition 2.1
  • Definition 2.2: Craciun2019
  • Example 2.3
  • Definition 2.4
  • Definition 2.5
  • Remark 2.6
  • Example 2.7
  • Definition 2.8
  • Remark 2.9
  • Remark 2.10
  • ...and 41 more