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Stoichiometric recipes for periodic oscillations in reaction networks

Alexander Blokhuis, Peter F. Stadler, Nicola Vassena

Abstract

Oscillatory chemical reactions are functional components in a variety of biological contexts. In chemistry, the construction and identification of even rudimentary oscillators remain elusive and lack a general framework. Using parameter-rich kinetics - a methodology enabling the disentanglement of parametric dependencies from structural analysis - we investigate the stoichiometry of chemical oscillators. We introduce the concept of oscillatory cores: minimal subnetworks that guarantee the potential for oscillations in any reaction network containing them. These cores fall into two classes, depending on whether they involve positive or negative feedback. In particular, the latter class unveils a family of oscillators - yet to be synthesized - that require a minimum number of reaction steps to exhibit oscillations, a phenomenon we refer to as the principle of length. We identify several mechanisms through which catalysis promotes oscillations: (I) furnishing instability (e.g. autocatalysis), (II) lifting dependencies, (III) lowering length thresholds. Notwithstanding this mechanistic ubiquity, we show that oscillators can also be realized without employing any catalysis. Our results highlight branches of chemistry where oscillators are likely to arise by chance, suggest new strategies for their design, and point to novel classes of oscillators yet to be realized experimentally.

Stoichiometric recipes for periodic oscillations in reaction networks

Abstract

Oscillatory chemical reactions are functional components in a variety of biological contexts. In chemistry, the construction and identification of even rudimentary oscillators remain elusive and lack a general framework. Using parameter-rich kinetics - a methodology enabling the disentanglement of parametric dependencies from structural analysis - we investigate the stoichiometry of chemical oscillators. We introduce the concept of oscillatory cores: minimal subnetworks that guarantee the potential for oscillations in any reaction network containing them. These cores fall into two classes, depending on whether they involve positive or negative feedback. In particular, the latter class unveils a family of oscillators - yet to be synthesized - that require a minimum number of reaction steps to exhibit oscillations, a phenomenon we refer to as the principle of length. We identify several mechanisms through which catalysis promotes oscillations: (I) furnishing instability (e.g. autocatalysis), (II) lifting dependencies, (III) lowering length thresholds. Notwithstanding this mechanistic ubiquity, we show that oscillators can also be realized without employing any catalysis. Our results highlight branches of chemistry where oscillators are likely to arise by chance, suggest new strategies for their design, and point to novel classes of oscillators yet to be realized experimentally.

Paper Structure

This paper contains 56 sections, 27 theorems, 245 equations, 12 figures, 2 tables.

Table of Contents

  1. Introduction
  2. Acknowledgments.
  3. Mathematical Results
  4. D-Hopf matrices and oscillations
  5. Parameter-rich kinetics and stoichiometric interpretations
  6. Child Selections and Cores
  7. Oscillatory Cores
  8. Recipe \ref{['recipe:0main']}: oscillations without employing oscillatory cores
  9. A minimalist template.
  10. Recipe \ref{['recipe:0main']} in the literature.
  11. The principle of length
  12. Roles of catalysis
  13. Outlook: building new oscillators
  14. Discussion
  15. Proof of the main results
  16. ...and 41 more sections

Key Result

Lemma 2.3

If $A$ is $D$-Hopf, then there exists positive diagonal matrix $D$ such that $AD$ has purely imaginary eigenvalues.

Figures (12)

  • Figure 1: a) Via Recipe \ref{['recipe:1main']}, oscillatory cores obtained by extending autocatalytic cores to stable subnetworks. b) Via Recipe \ref{['recipe:2main']}, oscillatory cores based on negative-feedback unstable cores, which are always nonautocatalytic. c) A simple setup to make an oscillator following Recipe \ref{['recipe:0main']}: to an autocatalytic core, a replenishment of an off-core reactant consumed by an autocatalyst is added, along with a degradation step for the same autocatalyst.
  • Figure 2: Recipe \ref{['recipe:1main']} illustrated for the motif $\ce{X}\rightarrow\ce{Y}+\ce{Z}$; $\ce{Y}\rightarrow \ce{Z}$; $\ce{Z}+\ce{W}\rightarrow \ce{X}$; $\ce{W}\rightarrow \ce{X}$. The first three reactions on species $(\ce{X},\ce{Y},\ce{Z})$ form an autocatalytic core with stoichiometric matrix as in \ref{['eq:autcorII']}. Any consistent reaction network that includes such motif has the capacity for periodic oscillations. The bifurcation process follows a parameter $\beta\in(0,1]$. At $\beta=1$ the associated CS-matrix is Hurwitz stable, while at $\beta\approx0$ is unstable due to the presence of the autocatalytic core. The product matrix is invertible throughout: change of stability happens at purely imaginary eigenvalues. a) The bifurcation process is intuitively illustrated in the network as a removal of the reactivity of one reaction. b) The trajectory of the eigenvalues is displayed where a purely imaginary crossing can be seen.
  • Figure 3: An example of Recipe \ref{['recipe:0main']}. a) A subnetwork of the Activation-Inhibition model from Nguyen18 with autocatalysis highlighted in magenta. b) The Jacobian $G$ is invertible for all parameter choices. c) Keeping the other reactivities fixed at 1, we vary $R_{7X}=\beta\in(0,5]$ d) the trajectories of the eigenvalues of $G$ illustrate the Hopf bifurcation.
  • Figure 4: b) Proposed reactions with molecular structures for the Groningenator Harmsel2023. a) A subnetwork, with autocatalytisis highlighted in magenta. A capacity to oscillate can be demonstrated through Recipe \ref{['recipe:0main']}. Under mass-action, both degradation steps are necessary (See the SM sec \ref{['sec:activatorinhibitor']}).
  • Figure 5: Recipe \ref{['recipe:2main']} illustrated for the motif $\ce{X}\rightarrow\ce{X}_1\rightarrow...\rightarrow \ce{X}_6\rightarrow \ce{Y}+\ce{Z}$; $\ce{Y}\rightarrow \ce{Z}$; $\ce{X}+\ce{Z}\rightarrow \;$. Any consistent reaction network that includes such motif has the capacity for periodic oscillations. The bifurcation process follows a parameter $\beta\in[0,1]$. At $\beta=1$ the associated CS-matrix is Hurwitz-unstable, while at $\beta\approx0$ is Hurwitz-stable. The product matrix is invertible throughout: change of stability happens at purely imaginary eigenvalues. a) The bifurcation process is intuitively illustrated in the network as a removal of the reactivity of one reaction. b) The trajectory of the eigenvalues is displayed, a purely imaginary crossing can be seen.
  • ...and 7 more figures

Theorems & Definitions (67)

  • Definition 2.1: Inertia of a matrix
  • Definition 2.2: D-Hopf matrix
  • Lemma 2.3
  • Theorem 2.4
  • Definition 2.5
  • Definition 2.6
  • Definition 2.7
  • Corollary 2.8
  • Definition 2.9: Oscillatory Cores of Class I
  • Definition 2.10: Oscillatory Cores of Class II
  • ...and 57 more