1. M.A. Al-Radhawi, D. Angeli, and E.D. Sontag. A computational framework for a Lyapunov-enabled analysis of biochemical reaction networks. PLoS Computational Biology, pp 16(2): e1007681, 2020. [PDF] Keyword(s): MAPK cascades, Lyapunov functions, stability, chemical networks, chemical rection networks, systems biology, RFM, ribosome flow model. Abstract:

   This paper deals with the analysis of the dynamics of chemical reaction networks, developing a theoretical framework based only on graphical knowledge and applying regardless of the particular form of kinetics. This paper introduces a class of networks that are "structurally (mono) attractive", by which we mean that they are incapable of exhibiting multiple steady states, oscillation, or chaos by the virtue of their reaction graphs. These networks are characterized by the existence of a universal energy-like function which we call a Robust Lyapunov function (RLF). To find such functions, a finite set of rank-one linear systems is introduced, which form the extremals of a linear convex cone. The problem is then reduced to that of finding a common Lyapunov function for this set of extremals. Based on this characterization, a computational package, Lyapunov-Enabled Analysis of Reaction Networks (LEARN), is provided that constructs such functions or rules out their existence. An extensive study of biochemical networks demonstrates that LEARN offers a new unified framework. We study basic motifs, three-body binding, and transcriptional networks. We focus on cellular signalling networks including various post-translational modification cascades, phosphotransfer and phosphorelay networks, T-cell kinetic proofreading, ERK signaling, and the Ribosome Flow Model.




2. D. Angeli, P. de Leenheer, and E.D. Sontag. Graph-theoretic characterizations of monotonicity of chemical networks in reaction coordinates. J. Mathematical Biology, 61:581-616, 2010. [PDF] Keyword(s): MAPK cascades, biochemical networks, fluxes, monotone systems, reaction cordinates, Petri nets, persistence, futile cycles. Abstract:

   This paper derives new results for certain classes of chemical reaction networks, linking structural to dynamical properties. In particular, it investigates their monotonicity and convergence without making assumptions on the form of the kinetics (e.g., mass-action) of the dynamical equations involved, and relying only on stoichiometric constraints. The key idea is to find an alternative representation under which the resulting system is monotone. As a simple example, the paper shows that a phosphorylation/dephosphorylation process, which is involved in many signaling cascades, has a global stability property.




3. D. Angeli, P. de Leenheer, and E.D. Sontag. Chemical networks with inflows and outflows: A positive linear differential inclusions approach. Biotechnology Progress, 25:632-642, 2009. [PDF] Keyword(s): biochemical networks, fluxes, differential inclusions, positive systems, Petri nets, persistence, switched systems. Abstract:

   Certain mass-action kinetics models of biochemical reaction networks, although described by nonlinear differential equations, may be partially viewed as state-dependent linear time-varying systems, which in turn may be modeled by convex compact valued positive linear differential inclusions. A result is provided on asymptotic stability of such inclusions, and applied to biochemical reaction networks with inflows and outflows. Included is also a characterization of exponential stability of general homogeneous switched systems




4. E.D. Sontag. Correction to: ``Structure and stability of certain chemical networks and applications to the kinetic proofreading model of T-cell receptor signal transduction'' [IEEE Trans. Automat. Control 46 (2001), no. 7, 1028--1047; MR1842137 (2002e:92006)]. IEEE Trans. Automat. Control, 47(4):705, 2002. [PDF] Keyword(s): zero-deficiency networks, systems biology, biochemical networks, nonlinear stability, dynamical systems. Abstract:

   errata for Structure and stability of certain chemical networks and applications to the kinetic proofreading model of T-cell receptor signal transduction




5. E.D. Sontag. Structure and stability of certain chemical networks and applications to the kinetic proofreading model of T-cell receptor signal transduction. IEEE Trans. Automat. Control, 46(7):1028-1047, 2001. [PDF] Keyword(s): zero-deficiency networks, systems biology, biochemical networks, nonlinear stability, dynamical systems, kinetic proofreading, T cells, immunology. Abstract:

   This paper deals with the theory of structure, stability, robustness, and stabilization for an appealing class of nonlinear systems which arises in the analysis of chemical networks. The results given here extend, but are also heavily based upon, certain previous work by Feinberg, Horn, and Jackson, of which a self-contained and streamlined exposition is included. The theoretical conclusions are illustrated through an application to the kinetic proofreading model proposed by McKeithan for T-cell receptor signal transduction.

1. D. Angeli, P. de Leenheer, and E.D. Sontag. Petri nets tools for the analysis of persistence in chemical networks. In Proc. 7th IFAC Symposium on Nonlinear Control Systems (NOLCOS 2007), Pretoria, South Africa, 22-24 August, 2007, 2007. Keyword(s): Petri nets, systems biology, biochemical networks, nonlinear stability, dynamical systems, futile cycles.



2. M. Chaves and E.D. Sontag. An alternative observer for zero deficiency chemical networks. In Proc. Nonlinear Control System Design Symposium, St. Petersburg, July 2001, pages 575-578, 2001. Keyword(s): observability, observers, zero-deficiency networks, systems biology, biochemical networks, nonlinear stability, dynamical systems.

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