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66
A Petri net approach to the study of persistence in chemical reaction networks
 Mathematical Biosciences
, 2006
"... Persistence is the property, for differential equations in R n, that solutions starting in the positive orthant do not approach the boundary of the orthant. For chemical reactions and population models, this translates into the nonextinction property: provided that every species is present at the s ..."
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Cited by 32 (11 self)
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Persistence is the property, for differential equations in R n, that solutions starting in the positive orthant do not approach the boundary of the orthant. For chemical reactions and population models, this translates into the nonextinction property: provided that every species is present at the start of the reaction, no species will tend to be eliminated in the course of the reaction. This paper provides checkable conditions for persistence of chemical species in reaction networks, using concepts and tools from Petri net theory, and verifies these conditions on various systems which arise in the modeling of cell signaling pathways.
Molecular systems biology and control
 Control 11:396–435. of Boolean networks 15
, 2005
"... This paper, prepared for a tutorial at the 2005 IEEE Conference on Decision and Control, presents an introduction to molecular systems biology and some associated problems in control theory. It provides an introduction to basic biological concepts, describes several questions in dynamics and control ..."
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Cited by 27 (9 self)
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This paper, prepared for a tutorial at the 2005 IEEE Conference on Decision and Control, presents an introduction to molecular systems biology and some associated problems in control theory. It provides an introduction to basic biological concepts, describes several questions in dynamics and control that arise in the field, and argues that new theoretical problems arise naturally in this context. A final section focuses on the combined use of graphtheoretic, qualitative knowledge about monotone buildingblocks and steadystate step responses for components. 1
A SmallGain Theorem for Almost Global Convergence of Monotone Systems
"... A smallgain theorem is presented for almost global stability of monotone control systems which are openloop almost globally stable, when constant inputs are applied. The theorem assumes "negative feedback" interconnections. This typically destroys the monotonicity of the original flow and potentia ..."
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Cited by 19 (13 self)
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A smallgain theorem is presented for almost global stability of monotone control systems which are openloop almost globally stable, when constant inputs are applied. The theorem assumes "negative feedback" interconnections. This typically destroys the monotonicity of the original flow and potentially destabilizes the resulting closedloop system.
On the Stability of a Model of Testosterone Dynamics
 Journal of Mathematical Biology
"... We prove the global asymptotic stability of a wellknown delayed negativefeedback model of testosterone dynamics, which has been proposed as a model of oscillatory behavior. We establish stability (and hence the impossibility of oscillations) even in the presence of delays of arbitrary length. ..."
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Cited by 18 (9 self)
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We prove the global asymptotic stability of a wellknown delayed negativefeedback model of testosterone dynamics, which has been proposed as a model of oscillatory behavior. We establish stability (and hence the impossibility of oscillations) even in the presence of delays of arbitrary length.
On PredatorPrey Systems and SmallGain Theorems
 J. Mathematical Biosciences and Engineering
, 2002
"... This paper deals with an almost global attractivity result for LotkaVolterra systems with predatorprey interactions. These systems can be written as (negative) feedback systems. The subsystems of the feedback loop are monotone control systems, possessing particular inputoutput properties. We us ..."
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Cited by 18 (11 self)
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This paper deals with an almost global attractivity result for LotkaVolterra systems with predatorprey interactions. These systems can be written as (negative) feedback systems. The subsystems of the feedback loop are monotone control systems, possessing particular inputoutput properties. We use a smallgain theorem, adapted to a context of systems with multiple equilibrium points to obtain the desired almost global attractivity result. It provides su#cient conditions to rule out oscillatory or more complicated behavior which is often observed in predatorprey systems.
Monotone Systems Under Positive Feedback: Multistability and a Reduction Theorem
, 2004
"... For feedback loops involving single input, single output monotone systems with welldefined I/O characteristics, a recent paper by Angeli and Sontag provided an approach to determining the location and stability of steady states. A result on global convergence for multistable systems followed as a c ..."
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Cited by 17 (10 self)
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For feedback loops involving single input, single output monotone systems with welldefined I/O characteristics, a recent paper by Angeli and Sontag provided an approach to determining the location and stability of steady states. A result on global convergence for multistable systems followed as a consequence of the technique. The present paper extends the approach to multiple inputs and outputs. A key idea is the introduction of a reduced system which preserves local stability properties.
Nonmonotone systems decomposable into monotone systems with negative feedback
 the Journal of Differential Equations
"... Motivated by the work of Angeli and Sontag [1] and Enciso and Sontag [7] in control theory, we show that certain finite and infinite dimensional semidynamical systems with “negative feedback ” can be decomposed into a monotone “open loop” system with “inputs ” and a decreasing “output ” function. T ..."
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Cited by 15 (9 self)
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Motivated by the work of Angeli and Sontag [1] and Enciso and Sontag [7] in control theory, we show that certain finite and infinite dimensional semidynamical systems with “negative feedback ” can be decomposed into a monotone “open loop” system with “inputs ” and a decreasing “output ” function. The original system is reconstituted by “plugging the output into the input”. Employing a technique of Gouzé [9] and Cosner [5] of imbedding the system into a larger symmetric monotone system, we are able to obtain information on the asymptotic behavior of solutions, including existence of positively invariant sets and global convergence. 1
Algorithmic and complexity results for decompositions of biological networks into monotone subsystems
 IN LECTURE NOTES IN COMPUTER SCIENCE: EXPERIMENTAL ALGORITHMS: 5TH INTERNATIONAL WORKSHOP, WEA 2006, SPRINGERVERLAG, 253–264. (CALA GALDANA, MENORCA
, 2006
"... A useful approach to the mathematical analysis of largescale biological networks is based upon their decompositions into monotone dynamical systems. This paper deals with two computational problems associated to finding decompositions which are optimal in an appropriate sense. In graphtheoretic la ..."
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Cited by 14 (5 self)
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A useful approach to the mathematical analysis of largescale biological networks is based upon their decompositions into monotone dynamical systems. This paper deals with two computational problems associated to finding decompositions which are optimal in an appropriate sense. In graphtheoretic language, the problems can be recast in terms of maximal signconsistent subgraphs. The theoretical results include polynomialtime approximation algorithms as well as constantratio inapproximability results. One of the algorithms, which has a worstcase guarantee of 87.9 % from optimality, is based on the semidefinite programming relaxation approach of GoemansWilliamson [23]. The algorithm was implemented and tested on a Drosophila segmentation network and an Epidermal Growth Factor Receptor pathway model, and it was found to perform close to optimally.
Global attractivity, I/O monotone smallgain theorems, and biological delay systems, Discrete and Continuous Dynamical Systems
 Discrete Contin. Dyn. Syst
, 2006
"... Abstract. This paper further develops a method, originally introduced by Angeli and the second author, for proving global attractivity of steady states in certain classes of dynamical systems. In this approach, one views the given system as a negative feedback loop of a monotone controlled system. A ..."
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Cited by 13 (8 self)
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Abstract. This paper further develops a method, originally introduced by Angeli and the second author, for proving global attractivity of steady states in certain classes of dynamical systems. In this approach, one views the given system as a negative feedback loop of a monotone controlled system. An auxiliary discrete system, whose global attractivity implies that of the original system, plays a key role in the theory, which is presented in a general Banach space setting. Applications are given to delay systems, as well as to systems with multiple inputs and outputs, and the question of expressing a given system in the required negative feedback form is addressed. 1. Introduction. In their paper, Angeli and Sontag [2] introduced an approach for establishing sufficient conditions under which a dynamical system Φ, described by ordinary differential equations, is guaranteed to have a globally stable equilibrium. The method may be applied whenever Φ can be decomposed as a negative feedback loop around a monotone controlled system. A discrete system is associated to Φ,
A petri net approach to persistence analysis in chemical reaction networks
 Biology and Control Theory: Current Challenges, Lecture Notes in Control and Information Sciences
, 2007
"... Summary. A positive dynamical system is said to be persistent if every solution that starts in the interior of the positive orthant does not approach the boundary of this orthant. For chemical reaction networks and other models in biology, persistence represents a nonextinction property: if every s ..."
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Cited by 9 (0 self)
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Summary. A positive dynamical system is said to be persistent if every solution that starts in the interior of the positive orthant does not approach the boundary of this orthant. For chemical reaction networks and other models in biology, persistence represents a nonextinction property: if every species is present at the start of the reaction, then no species will tend to be eliminated in the course of the reaction. This paper provides checkable necessary as well as sufficient conditions for persistence of chemical species in reaction networks, and the applicability of these conditions is illustrated on some examples of relatively high dimension which arise in molecular biology. More specific results are also provided for reactions endowed with massaction kinetics. Overall, the results exploit concepts and tools from Petri net theory as well as ergodic and recurrence theory. 1