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. A. Rendall and E. D. Sontag. Multiple steady states and the form of response functions to antigen in a model for the initiation of T cell activation. Royal Society Open Science, 4:170821-, 2017. [PDF] Keyword(s): kinetic proofreading, T cells, immunology, systems biology. Abstract:

   This paper analizes a model for the initial stage of T cell activation. The state variables in the model are the concentrations of phosphorylation states of the T cell receptor complex and the phosphatase SHP-1 in the cell. It is shown that these quantities cannot approach zero, and that there is more than one positive steady state for certain values of the parameters; in addition, damped oscillations are possible. It is also shown that the chemical concentration which represents the degree of activation of the cell, represented by the maximally phosphorylated form of the T cell receptor complex, is in general a non-monotone function of the activating signal. In particular there are cases where there is a value of the dissociation constant of the ligand from the receptor which produces an optimal activation of the T cell. In this way the results of certain simulations in the literature have been confirmed rigorously and new features are discovered.




3. 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, reaction 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




4. 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, reaction 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. P. Yu and E.D. Sontag. A necessary condition for non-monotonic dose response, with an application to a kinetic proofreading model. In Proc. 2024 63rd IEEE Conference on Decision and Control (CDC), 2024. Note: To appear. Note: there is an extended version in arXiv; journal paper in preparation.[PDF] Keyword(s): systems biology, IFFL, dose response. Abstract:

   Steady state non-monotonic ("biphasic") dose responses are often observed in experimental biology, which raises the control theoretic question of identifying which possible mechanisms might underlie such behaviors. It is well known that the presence of an incoherent feedforward loop (IFFL) in a network may give rise to a non-monotonic response, and it has been informally conjectured that this condition is also necessary. However, this conjecture has been disproved with an example of a system in which input and output nodes are the same. In this paper, we show that the converse implication does hold when the input and output are distinct. Towards this aim, we give necessary and sufficient conditions for when minors of a symbolic matrix have mixed signs. Finally, we study in full generality when a model of immune T-cell activation could exhibit a steady state non-monotonic dose response.

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