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.Comment: See http://www.math.rutgers.edu/~sontag/PUBDIR/index.html for related (p)reprint

Sign patterns for chemical reaction networks

Most differential equations found in chemical reaction networks (CRNs) have the form $dx/dt=f(x)= Sv(x)$, where $x$ lies in the nonnegative orthant, where $S$ is a real matrix (the stoichiometric matrix) and $v$ is a column vector consisting of real-valued functions having a special relationship to $S$. Our main interest will be in the Jacobian matrix, $f'(x)$, of $f(x)$, in particular in whether or not each entry $f'(x)_{ij}$ has the same sign for all $x$ in the orthant, i.e., the Jacobian respects a sign pattern. In other words species $x_j$ always acts on species $x_i$ in an inhibitory way or its action is always excitatory. In Helton, Klep, Gomez we gave necessary and sufficient conditions on the species-reaction graph naturally associated to $S$ which guarantee that the Jacobian of the associated CRN has a sign pattern. In this paper, given $S$ we give a construction which adds certain rows and columns to $S$, thereby producing a stoichiometric matrix $\widehat S$ corresponding to a new CRN with some added species and reactions. The Jacobian for this CRN based on $\hat S$ has a sign pattern. The equilibria for the $S$ and the $\hat S$ based CRN are in exact one to one correspondence with each equilibrium $e$ for the original CRN gotten from an equilibrium $\hat e$ for the new CRN by removing its added species. In our construction of a new CRN we are allowed to choose rate constants for the added reactions and if we choose them large enough the equilibrium $\hat e$ is locally asymptotically stable if and only if the equilibrium $e$ is locally asymptotically stable. Further properties of the construction are shown, such as those pertaining to conserved quantities and to how the deficiencies of the two CRNs compare.Comment: 23 page

Determinant Expansions of Signed Matrices and of Certain Jacobians

This paper treats two topics: matrices with sign patterns and Jacobians of certain mappings. The main topic is counting the number of plus and minus coefficients in the determinant expansion of sign patterns and of these Jacobians. The paper is motivated by an approach to chemical networks initiated by Craciun and Feinberg. We also give a graph-theoretic test for determining when the Jacobian of a chemical reaction dynamics has a sign pattern.Comment: 25 page

Disturbance Propagation in Interconnected Linear Dynamical Networks

We consider performance analysis of interconnected linear dynamical networks subject to external stochastic disturbances. For stable linear networks, we define scalar performance measures by considering weighted H2--norms of the underlying systems, which are defined from the disturbance input to a desired output. It is shown that the performance measure of a general stable linear network can be tightly bounded from above and below using some spectral functions of the state matrix of the network. This result is applied to a class of cyclic linear networks and shown that the performance measure of such networks scales quadratically with the network size. Next, we focus on first-- and second--order linear consensus networks and introduce the notion of Laplacian energy for such networks, which in fact measures the expected steady-state dispersion of the state of the entire network. We develop a graph-theoretic framework in order to relate graph characteristics to the Laplacian energy of the network and show that how the Laplacian energy scales asymptotically with the network size. We quantify several inherent fundamental limits on Laplacian energy in terms of graph diameter, node degrees, and the number of spanning trees, and several other graph specifications. Particularly we characterize several versions of fundamental tradeoffs between Laplacian energy and sparsity measures of a linear consensus network, showing that more sparse networks have higher levels of Laplacian energies. At the end, we show that several existing performance measures in real--world applications, such as total power loss in synchronous power networks and flock energy of a group of autonomous vehicles in a formation, are indeed special forms of Laplacian energies