# American Institute of Mathematical Sciences

2008, 5(1): 1-19. doi: 10.3934/mbe.2008.5.1

## A passivity-based stability criterion for a class of biochemical reaction networks

 1 Department of Electrical, Computer, and Systems Engineering, Rensselaer Polytechnic Institute, 110 8th Street, Troy, NY, 12180, United States 2 Department of Mathematics, Rutgers, The State University of New Jersey, Hill Center, 110 Frelinghuysen Road, Piscataway, NJ 08854, United States

Received  May 2007 Revised  August 2007 Published  January 2008

This paper presents a stability test for a class of interconnected nonlinear systems motivated by biochemical reaction networks. The main result determines global asymptotic stability of the network from the diag- onal 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 encom- passes the secant criterion for cyclic networks presented in [1], and extends it to a general interconnection structure represented by a graph. The new stabil- ity test is illustrated on a mitogen-activated protein kinase (MAPK) cascade model, and on a branched interconnection structure motivated by metabolic networks. The next problem addressed is the robustness of stability in the presence of di®usion terms. A compartmental model is used to represent the localization of the reactions, and conditions are presented under which stability is preserved despite the di®usion terms between the compartments.
Citation: Murat Arcak, Eduardo D. Sontag. A passivity-based stability criterion for a class of biochemical reaction networks. Mathematical Biosciences & Engineering, 2008, 5 (1) : 1-19. doi: 10.3934/mbe.2008.5.1
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