A passivity-based stability criterion for a class of interconnected systems and applications to biochemical reaction networks

This paper presents a stability test for a class of interconnected nonlinear systems motivated by biochemical reaction networks. One of the main results determines global asymptotic stability of the network from the diagonal stability of a “dissipativity matrix” which incorporates information about the passivity properties of the subsystems, the interconnection structure of the network, and the signs of the interconnection terms. This stability test encompasses the “secant criterion” for cyclic networks presented in our previous paper, and extends it to a general interconnection structure represented by a graph. A second main result allows one to accommodate state products. This extension makes the new stability criterion applicable to a broader class of models, even in the case of cyclic systems. The new stability test is illustrated on a mitogen activated protein kinase (MAPK) cascade model, and on a branched interconnection structure motivated by metabolic networks. Finally, another result addresses the robustness of stability in the presence of diffusion terms in a compartmental system made out of identical systems.

💡 Research Summary

The paper introduces a novel stability test for a broad class of interconnected nonlinear systems, motivated primarily by biochemical reaction networks. The authors build on their earlier “secant criterion,” which was limited to cyclic interconnections, and extend it to arbitrary network topologies represented by directed graphs. The central construct is a “dissipativity matrix” that aggregates three essential pieces of information: (1) the input‑output passivity properties of each subsystem, (2) the interconnection structure (i.e., which subsystems feed into which), and (3) the sign of each interconnection term (excitatory or inhibitory).

A subsystem is assumed to admit a storage (Lyapunov) function V_i(x_i) that satisfies a passivity inequality of the form (\dot V_i \leq u_i^T y_i - \gamma_i |u_i|^2), where u_i and y_i are the input and output vectors, respectively, and (\gamma_i>0) quantifies the subsystem’s excess passivity. Collecting the (\gamma_i) into a diagonal matrix Γ and the signed adjacency information into a matrix Σ, the dissipativity matrix is defined as M = Γ A Σ, where A is the adjacency matrix of the network.

The first main theorem states that if M is diagonally stable—i.e., there exists a positive diagonal matrix D such that D M + M^T D is negative definite—then the composite system is globally asymptotically stable. Diagonal stability guarantees the existence of a global Lyapunov function (V(x)=\sum_i d_i V_i(x_i)) with d_i the diagonal entries of D. This result subsumes the secant criterion (which corresponds to a special case where A is a simple directed cycle and Σ has uniform sign) and provides a systematic way to certify stability for any graph topology.

The second contribution addresses the presence of state‑product nonlinearities, which are ubiquitous in mass‑action kinetics. The authors extend the passivity framework by allowing the storage functions to contain cross‑terms that capture products of state variables. They derive an “extended passivity” condition that ensures these cross‑terms do not introduce energy into the system, thereby preserving the diagonal‑stability argument. Consequently, the stability test remains valid for models where reaction rates are proportional to the product of several species concentrations.

A further result concerns robustness to diffusion in compartmental models. When identical subsystems are placed in multiple compartments and coupled through diffusion, the overall dynamics can be written as (\dot x = (M \otimes I - \gamma I \otimes L) x), where L is the Laplacian matrix describing diffusion between compartments and γ>0 is the diffusion coefficient. The authors prove that, provided L is positive semidefinite (as is typical for diffusion), the diagonal‑stability of M alone suffices to guarantee stability of the full compartmental system, regardless of the diffusion strength. This shows that the stability property is robust to the addition of spatial transport.

To illustrate the theory, two case studies are presented. The first is a MAPK (mitogen‑activated protein kinase) cascade, a three‑layer phosphorylation chain (RAF → MEK → ERK) with feedback inhibition. Each layer is modeled as a passive subsystem, and the phosphorylation reactions are expressed as state‑product terms. The constructed dissipativity matrix satisfies the diagonal‑stability condition, confirming the global convergence observed in numerical simulations.

The second example is a branched metabolic network where a single precursor feeds into two downstream pathways. The branching introduces both excitatory and inhibitory interconnections, captured by the sign matrix Σ. Again, the dissipativity matrix is shown to be diagonally stable, establishing global asymptotic stability for the branched structure.

Overall, the paper delivers a powerful, graph‑theoretic passivity framework that unifies stability analysis for a wide variety of biochemical networks, including those with cyclic, branched, or spatially distributed architectures, and those featuring nonlinear product terms. By reducing the global stability problem to a tractable matrix inequality, the authors provide both theoretical insight and a practical tool for systems biologists and control engineers working with complex reaction networks.