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Unstable Cores are the source of instability in chemical reaction networks

Nicola Vassena, Peter F. Stadler

Abstract

In biochemical networks, complex dynamical features such as superlinear growth and oscillations are classically considered a consequence of autocatalysis. For the large class of parameter-rich kinetic models, which includes Generalized Mass Action kinetics and Michaelis-Menten kinetics, we show that certain submatrices of the stoichiometric matrix, so-called unstable cores, are sufficient for a reaction network to admit instability and potentially give rise to such complex dynamical behavior. The determinant of the submatrix distinguishes unstable-positive feedbacks, with a single real-positive eigenvalue, and unstable-negative feedbacks without real-positive eigenvalues. Autocatalytic cores turn out to be exactly the unstable-positive feedbacks that are Metzler matrices. Thus there are sources of dynamical instability in chemical networks that are unrelated to autocatalysis. We use such intuition to design non-autocatalytic biochemical networks with superlinear growth and oscillations.

Unstable Cores are the source of instability in chemical reaction networks

Abstract

In biochemical networks, complex dynamical features such as superlinear growth and oscillations are classically considered a consequence of autocatalysis. For the large class of parameter-rich kinetic models, which includes Generalized Mass Action kinetics and Michaelis-Menten kinetics, we show that certain submatrices of the stoichiometric matrix, so-called unstable cores, are sufficient for a reaction network to admit instability and potentially give rise to such complex dynamical behavior. The determinant of the submatrix distinguishes unstable-positive feedbacks, with a single real-positive eigenvalue, and unstable-negative feedbacks without real-positive eigenvalues. Autocatalytic cores turn out to be exactly the unstable-positive feedbacks that are Metzler matrices. Thus there are sources of dynamical instability in chemical networks that are unrelated to autocatalysis. We use such intuition to design non-autocatalytic biochemical networks with superlinear growth and oscillations.
Paper Structure (11 sections, 31 theorems, 100 equations, 8 figures)

This paper contains 11 sections, 31 theorems, 100 equations, 8 figures.

Table of Contents

  1. Introduction
  2. Chemical reaction networks, dynamics, and stability
  3. Parameter-rich kinetic models
  4. Child-Selections, CB-summands, elementary CB-components
  5. Unstable cores and D-unstable cores
  6. Unstable-positive and unstable-negative feedbacks
  7. Autocatalytic cores as unstable cores and Metzler matrices
  8. Examples
  9. NC-networks and what is special about autocatalysis
  10. Conclusion
  11. Proof of Theorem \ref{['thm:autocoremain']}

Key Result

Lemma 5

Michaelis--Menten kinetics, Eq.(MMeq), is parameter-rich.

Figures (8)

  • Figure 1: The system in \ref{['ex1mm']} possesses an unstable equilibrium $E_u$ at $x=(1,1,1,1)$. In the figure, nearby initial conditions $x(0)=(1.01,1,1,1)$ converge to the stable equilibrium $E_s=(2, 0.25, 1, 1).$ The trajectory of $x_A$ has a sigmoid shape with alternating regimes of superlinear, linear, and sublinear growth.
  • Figure 2: The system in \ref{['ex1b']} possesses an unstable equilibrium $(x_A,x_B,x_C,x_D,x_E)=(1,1,1,1,1)$. In the figure, nearby initial conditions $x(0)=(1.01,1,1,1,1)$ show superlinear growth and convergence to a limit stable equilibrium.
  • Figure 3: The system in \ref{['ex2mm']} possesses an unstable equilibrium $(x_A,x_B,x_C,x_D,x_E)=(1,1,1,1,1)$. In the figure, nearby initial conditions $x(0)=(1.01,1,1,1,1)$ shows convergence to a limit cycle.
  • Figure 4: The picture shows numerical simulations for the system in \ref{['ex2mm']}. The initial condition $x(0)=[1.10433115, 1.0969183, 1.15016837, 0.57719267, 1.05835769]$ is chosen in proximity of the periodic orbit. The plot on the left shows the time-evolution of the concentrations $x(t)$, while the plots in the center and on the right depict in 3d the periodic orbit for $(x_A,x_B,x_C)$ and $(x_A,x_D,x_E)$, respectively.
  • Figure 5: The system in \ref{['ex3mm']} possesses an unstable equilibrium at $x=(1,1,1,1,1,1)$. In the figure, nearby initial conditions $x(0)=(1,1,1,1,1.01,1)$ shows convergence to a limit cycle.
  • ...and 3 more figures

Theorems & Definitions (70)

  • Definition 1
  • Definition 2
  • Definition 3
  • Lemma 5
  • Lemma 6
  • proof
  • Definition 7
  • Definition 8
  • Definition 10: Cauchy--Binet summands, elementary Cauchy--Binet components
  • Lemma 11
  • ...and 60 more