7 research outputs found
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
Oscillations in I/O monotone systems under negative feedback
Oscillatory behavior is a key property of many biological systems. The
Small-Gain Theorem (SGT) for input/output monotone systems provides a
sufficient condition for global asymptotic stability of an equilibrium and
hence its violation is a necessary condition for the existence of periodic
solutions. One advantage of the use of the monotone SGT technique is its
robustness with respect to all perturbations that preserve monotonicity and
stability properties of a very low-dimensional (in many interesting examples,
just one-dimensional) model reduction. This robustness makes the technique
useful in the analysis of molecular biological models in which there is large
uncertainty regarding the values of kinetic and other parameters. However,
verifying the conditions needed in order to apply the SGT is not always easy.
This paper provides an approach to the verification of the needed properties,
and illustrates the approach through an application to a classical model of
circadian oscillations, as a nontrivial ``case study,'' and also provides a
theorem in the converse direction of predicting oscillations when the SGT
conditions fail.Comment: Related work can be retrieved from second author's websit
Equilibria and stability analysis of a branched metabolic network with feedback inhibition
This paper deals with the analysis of a metabolic network with feedback inhibition. The considered system is an acyclic network of monomolecular enzymatic reactions in which metabolites can act as feedback regulators on enzymes located "at the beginning" of their own pathway, and in which one metabolite is the root of the whole network. We show, under mild assumptions, the uniqueness of the equilibrium. We then show that this equilibrium is globally attractive if we impose conditions on the kinetic parameters of the metabolic reactions. Finally, when these conditions are not satisfied, we show, with a specific fourth-order example, that the equilibrium may become unstable with an attracting limit cycle
Equilibria and stability analysis of a branched metabolic network with feedback inhibition
International audienceThis paper deals with the analysis of a metabolic network with feedback inhibition. The considered system is an acyclic network of mono-molecular enzymatic reactions in which metabolites can act as feedback regulators on "preceding" enzymes of their own pathway, and in which one metabolite is the root of the whole network. We show, under mild assumptions, the uniqueness of the equilibrium. In the simplified case where inhibition only acts on the reactions having the root of the network as substrate, we show that the equilibrium is globally attractive. This requires that we impose conditions on the kinetic parameters of the metabolic reactions