Structural polyhedral stability of a biochemical network is equivalent to finiteness of the associated generalised Petri net

TL;DR

This paper establishes a link between biochemical network stability and Petri net boundedness, proposing a control method to enforce stability by pinning critical nodes with negative feedback.

Contribution

It demonstrates the equivalence between polyhedral Lyapunov stability of biochemical systems and Petri net boundedness, and introduces a control strategy to achieve stability via node pinning.

Findings

Polyhedral Lyapunov stability is equivalent to Petri net boundedness.

Controlling pinned nodes can enforce network stability.

Identifies critical nodes for local control to ensure overall stability.

Abstract

We consider biochemical systems associated with a generalised class of Petri nets with possibly negative token numbers. We show that the existence of a structural polyhedral Lyapunov function for the biochemical system is equivalent to the boundedness of the associated Petri net evolution or, equivalently, to the finiteness of the number of states reachable from each initial condition. For networks that do not admit a polyhedral Lyapunov function, we investigate whether it is possible to enforce polyhedral structural stability by applying a strong negative feedback on some pinned nodes: in terms of the Petri net, this is equivalent to turning pinned nodes into black holes that clear any positive or negative incoming token. If such nodes are chosen so that the transformed Petri net has bounded discrete trajectories, then there exists a stabilising pinning control: the biochemical network becomes Lyapunov stable if a sufficiently strong local negative feedback is applied to the pinned nodes. These results allow us to structurally identify the critical nodes to be locally controlled so as to ensure the stability of the whole network.