Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

From chemical reaction networks to algebraic and polyhedral geometry -- and back again

Elisenda Feliu, Anne Shiu

TL;DR

The chapter surveys how Bernd Sturmfels bridged chemical reaction network theory with algebraic and polyhedral geometry, unraveling complex-balanced steady states, global attractor conjectures, and invariant-based model discrimination. It develops toric dynamical system structure, algebraic matroids, and steady-state parametrizations to reveal when networks admit unique or multiple steady states, and uses mixed volumes and numerical algebraic geometry to bound and count these states. It also connects siphons, persistence, and boundary steady states to algebraic invariants, and explores attainable regions through convex geometry and patch methods. Together, these perspectives illustrate a deep, productive dialogue between biology and algebraic geometry with broad methodological impact across network analysis and systems biology.

Abstract

This is a chapter for a book in honor of Bernd Sturmfels and his contributions. We describe the contributions by Bernd Sturmfels and his collaborators in harnessing algebraic and combinatorial methods for analyzing chemical reaction networks. Topics explored include the steady-state variety, counting steady states, and the global attractor conjecture. We also recount some personal stories that highlight Sturmfels's long-lasting impact on this research area.

From chemical reaction networks to algebraic and polyhedral geometry -- and back again

TL;DR

The chapter surveys how Bernd Sturmfels bridged chemical reaction network theory with algebraic and polyhedral geometry, unraveling complex-balanced steady states, global attractor conjectures, and invariant-based model discrimination. It develops toric dynamical system structure, algebraic matroids, and steady-state parametrizations to reveal when networks admit unique or multiple steady states, and uses mixed volumes and numerical algebraic geometry to bound and count these states. It also connects siphons, persistence, and boundary steady states to algebraic invariants, and explores attainable regions through convex geometry and patch methods. Together, these perspectives illustrate a deep, productive dialogue between biology and algebraic geometry with broad methodological impact across network analysis and systems biology.

Abstract

This is a chapter for a book in honor of Bernd Sturmfels and his contributions. We describe the contributions by Bernd Sturmfels and his collaborators in harnessing algebraic and combinatorial methods for analyzing chemical reaction networks. Topics explored include the steady-state variety, counting steady states, and the global attractor conjecture. We also recount some personal stories that highlight Sturmfels's long-lasting impact on this research area.
Paper Structure (28 sections, 10 theorems, 59 equations, 2 figures, 2 tables)

This paper contains 28 sections, 10 theorems, 59 equations, 2 figures, 2 tables.

Table of Contents

  1. Background
  2. Reaction networks and associated models
  3. Reaction networks.
  4. Dynamical systems.
  5. Questions of interest and overview
  6. Complex-balanced steady states and toric dynamical systems
  7. The human story
  8. Steady-state structure
  9. Global attractor conjecture
  10. Siphons, persistence, and boundary steady states
  11. Invariants, matroids, and model selection
  12. The idea behind invariants
  13. The human story
  14. Algebraic matroids
  15. Steady-state parametrizations
  16. ...and 13 more sections

Key Result

Theorem 2.1

For $\kappa^* \in \mathbb{R}^{r}_{>0}$ and $x^* \in \mathbb{R}^n_{\geq 0}$, we have that $x^*$ is a complex-balanced steady state of the mass-action system of $\mathcal{G}$ with rate constants $\kappa^*$ if and only if $(x^*,\kappa^*)$ lies is in the positive part of the toric variety defined by $T_

Figures (2)

  • Figure 1: Group photo at 'Applications in Biology, Dynamics, and Statistics' workshop at IMA in March 2007. Sturmfels is seated in the first row, fifth from the left. Dickenstein is in the third row, seventh from the left; and Craciun is the unique person in the zeroth row.
  • Figure 2: The network on the left is endotactic, while the one on the right is strongly endotactic. Both networks are endotactic, because no reaction arrows point out of the Newton polytopes (the triangles) unless they first pass through the respective triangle. The network on the right is strongly endotactic, because every proper face of the triangle (that is, each edge and vertex) has a reaction arrow that starts on that face and exits the face. In contrast, the network on the left is not strongly endotactic: the edge of the triangle from $C$ to $3C+D$ has no reaction arrow that leaves that edge.

Theorems & Definitions (19)

  • Example 1.1
  • Example 1.2: Example \ref{['ex:mc']}, continued
  • Theorem 2.1: Complex-balanced steady states
  • Theorem 2.2: Rate constants for complex-balancing, part 1
  • Example 2.3: McKeithan network
  • Example 2.4: Extended McKeithan network
  • Theorem 2.5: Rate constants for complex-balancing, part 2
  • Example 2.6: Example \ref{['ex:extended-mckeithan']}, continued
  • Conjecture 2.7: Global attractor conjecture
  • Proposition 3.1
  • ...and 9 more