[Paper Review] Complex-balanced equilibria of generalized mass-action systems: Necessary conditions for linear stability

Balázs Boros, Stefan C. Müller+1 more|arXiv (Cornell University)|Jun 28, 2019

Nonlinear Dynamics and Pattern Formation24 references18 citations

TL;DR

This paper establishes necessary conditions for linear stability of complex-balanced equilibria in generalized mass-action systems, showing that linear stability implies uniqueness and characterizing it via D-stability and D-semistability of Jacobian matrices. For cyclic networks, linear stability is equivalent to D-stability; for weakly reversible networks, it requires D-semistability of all cycle Jacobians. Classical mass-action systems are shown to be diagonally stable, ensuring robust linear stability.

ABSTRACT

It is well known that, for mass-action systems, complex-balanced equilibria are asymptotically stable. For generalized mass-action systems, even if there exists a unique complex-balanced equilibrium (in every stoichiometric class and for all rate constants), it need not be stable. We first discuss several notions of matrix stability (on a linear subspace) such as D-stability and diagonal stability, and then we apply our abstract results to complex-balanced equilibria of generalized mass-action systems. In particular, we show that linear stability (on the stoichiometric subspace and for all rate constants) implies uniqueness. For cyclic networks, we characterize linear stability (in terms of D-stability of the Jacobian matrix); and for weakly reversible networks, we give necessary conditions for linear stability (in terms of D-semistability of the Jacobian matrices of all cycles in the network). Moreover, we show that, for classical mass-action systems, complex-balanced equilibria are not just asymptotically stable, but even diagonally stable (and hence linearly stable). Finally, we recall and extend characterizations of D-stability and diagonal stability for matrices of dimension up to three, and we illustrate our results by examples of irreversible cycles (of dimension up to three) and of reversible chains and S-systems (of arbitrary dimension).

Motivation & Objective

• To identify necessary conditions for linear stability of complex-balanced equilibria in generalized mass-action systems.
• To clarify the distinction between asymptotic stability and linear stability in generalized kinetics.
• To characterize linear stability via matrix stability concepts such as D-stability and diagonal stability on the stoichiometric subspace.
• To establish that linear stability implies uniqueness of complex-balanced equilibria.
• To extend known results on classical mass-action systems by proving diagonal stability of complex-balanced equilibria.

Proposed method

• Analyzing generalized mass-action systems using ODEs with stoichiometric and kinetic-order matrices.
• Defining complex-balanced equilibria as solutions to Ak x^Y = 0, where Ak is the Laplacian matrix of the reaction network.
• Employing linearization to study stability, focusing on the Jacobian matrix J(x) = Y Ak diag(x^Y) eY^T diag(x^{-1}).
• Applying matrix stability concepts—D-stability, diagonal stability, and D-semistability—on the stoichiometric subspace S.
• Using characterizations of D-stability and diagonal stability for matrices up to dimension three, including trace and determinant conditions.
• Applying results to specific network types: irreversible cycles, reversible chains, and S-systems, with explicit stability criteria derived from matrix inequalities.

Experimental results

Research questions

• RQ1 Under what conditions is a complex-balanced equilibrium linearly stable in a generalized mass-action system for all rate constants?
• RQ2 Does linear stability in generalized mass-action systems imply uniqueness of the complex-balanced equilibrium?
• RQ3 For cyclic networks, what matrix property (e.g., D-stability) characterizes linear stability of complex-balanced equilibria?
• RQ4 For weakly reversible networks, what necessary conditions on cycle Jacobian matrices ensure linear stability?
• RQ5 Are classical mass-action systems with complex-balanced equilibria diagonally stable, and what does this imply for robustness?

Key findings

• Linear stability of complex-balanced equilibria in generalized mass-action systems implies uniqueness of the equilibrium in every stoichiometric class.
• For cyclic networks, linear stability is equivalent to D-stability of the Jacobian matrix on the stoichiometric subspace.
• For weakly reversible networks, linear stability implies D-semistability of the Jacobian matrix for every cycle in the network.
• In classical mass-action systems, complex-balanced equilibria are diagonally stable, which implies linear stability and robustness under small perturbations of kinetic orders.
• For three-dimensional irreversible cycles, linear stability holds if and only if the determinant of the Jacobian matrix is positive and all diagonal entries are non-positive, with at least one strictly negative.
• In S-systems and reversible chains, linear instability can be detected if any reversible reaction has a kinetic order exceeding the stoichiometric coefficient in the corresponding species.