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Network reduction and absence of Hopf Bifurcations in dual phosphorylation networks with three Intermediates

Elisenda Feliu, Nidhi Kaihnsa

TL;DR

The paper investigates oscillations in dual phosphorylation networks with three intermediates by establishing a network-reduction framework that preserves the absence of Hopf bifurcations. It leverages convex-parameter representations of Jacobians and a Hurwitz-determinant criterion to preclude simple Hopf bifurcations in reduced subnetworks, and uses extensive symbolic-computational analysis to show that three-intermediate subnetworks do not admit Hopf bifurcations (and thus the full four-intermediate network is unlikely to oscillate). A key contribution is Theorem \ref{thm:yes}, which provides a constructive reduction that maintains the Hopf-nonoccurrence property and explains how reduced polynomials capture the dynamics of the full system. The results strengthen evidence against oscillations in the investigated dual phosphorylation architecture and offer a practical reduction pathway for analyzing Hopf bifurcations in large biochemical networks.

Abstract

Phosphorylation networks, representing the mechanisms by which proteins are phosphorylated at one or multiple sites, are ubiquitous in cell signalling and display rich dynamics such as unlimited multistability. Dual-site phosphorylation networks are known to exhibit oscillations in the form of periodic trajectories, when phosphorylation and dephosphorylation occurs as a mixed mechanism: phosphorylation of the two sites requires one encounter of the kinase, while dephosphorylation of the two sites requires two encounters with the phosphatase. A still open question is whether a mechanism requiring two encounters for both phosphorylation and dephosphorylation also admits oscillations. In this work we provide evidence in favor of the absence of oscillations of this network by precluding Hopf bifurcations in any reduced network comprising three out of its four intermediate protein complexes. Our argument relies on a novel network reduction step that preserves the absence of Hopf bifurcations, and on a detailed analysis of the semi-algebraic conditions precluding Hopf bifurcations obtained from Hurwitz determinants of the characteristic polynomial of the Jacobian of the system. We conjecture that the removal of certain reverse reactions appearing in Michaelis-Menten-type mechanisms does not have an impact on the presence or absence of Hopf bifurcations. We prove an implication of the conjecture under certain favorable scenarios and support the conjecture with additional example-based evidence.

Network reduction and absence of Hopf Bifurcations in dual phosphorylation networks with three Intermediates

TL;DR

The paper investigates oscillations in dual phosphorylation networks with three intermediates by establishing a network-reduction framework that preserves the absence of Hopf bifurcations. It leverages convex-parameter representations of Jacobians and a Hurwitz-determinant criterion to preclude simple Hopf bifurcations in reduced subnetworks, and uses extensive symbolic-computational analysis to show that three-intermediate subnetworks do not admit Hopf bifurcations (and thus the full four-intermediate network is unlikely to oscillate). A key contribution is Theorem \ref{thm:yes}, which provides a constructive reduction that maintains the Hopf-nonoccurrence property and explains how reduced polynomials capture the dynamics of the full system. The results strengthen evidence against oscillations in the investigated dual phosphorylation architecture and offer a practical reduction pathway for analyzing Hopf bifurcations in large biochemical networks.

Abstract

Phosphorylation networks, representing the mechanisms by which proteins are phosphorylated at one or multiple sites, are ubiquitous in cell signalling and display rich dynamics such as unlimited multistability. Dual-site phosphorylation networks are known to exhibit oscillations in the form of periodic trajectories, when phosphorylation and dephosphorylation occurs as a mixed mechanism: phosphorylation of the two sites requires one encounter of the kinase, while dephosphorylation of the two sites requires two encounters with the phosphatase. A still open question is whether a mechanism requiring two encounters for both phosphorylation and dephosphorylation also admits oscillations. In this work we provide evidence in favor of the absence of oscillations of this network by precluding Hopf bifurcations in any reduced network comprising three out of its four intermediate protein complexes. Our argument relies on a novel network reduction step that preserves the absence of Hopf bifurcations, and on a detailed analysis of the semi-algebraic conditions precluding Hopf bifurcations obtained from Hurwitz determinants of the characteristic polynomial of the Jacobian of the system. We conjecture that the removal of certain reverse reactions appearing in Michaelis-Menten-type mechanisms does not have an impact on the presence or absence of Hopf bifurcations. We prove an implication of the conjecture under certain favorable scenarios and support the conjecture with additional example-based evidence.
Paper Structure (10 sections, 10 theorems, 111 equations)

This paper contains 10 sections, 10 theorems, 111 equations.

Table of Contents

  1. Introduction
  2. Background
  3. Reaction networks
  4. Convex parameters
  5. Hurwitz determinants
  6. Removing reversible reactions with intermediates and Hopf bifurcations
  7. Dual phosphorylation networks with three intermediates
  8. Network \ref{['eq:3-intmdnetwork2']}
  9. Network \ref{['eq:3-intmdnetwork1']}
  10. Proof of Theorem \ref{['thm:yes']}

Key Result

Proposition 2.1

For a reaction network with $s=\mathop{\mathrm{rk}}\nolimits(N)\geq 2$ and $(h,{\ell}) \in \mathbb{R}^n_{>0}\times\mathbb{R}_{> 0}^m$, let $H_i(h,{\ell})$ be the $i$-th Hurwitz matrix of the reduced characteristic polynomial $p_{(h,{\ell})}$ of $J(h,{\ell})$ as defined in eq:charpoly and let $a_s(h, Then for all $(h,\ell)\in \mathbb{R}^n_{>0}\times \mathbb{R}^m_{>0}$, $p_{(h,{\ell})}$ does not hav

Theorems & Definitions (22)

  • Proposition 2.1
  • Conjecture 3.1
  • Conjecture 3.2
  • Conjecture 3.3
  • Theorem 3.4
  • Corollary 3.5
  • Example 3.6
  • Example 3.7
  • Example 3.8
  • Theorem 4.1
  • ...and 12 more