# A Proof of the Global Attractor Conjecture for General Reaction Networks

Friday, November 20, 2015 - 10:45am - 11:00am

Keller 3-180

Gheorghe Craciun (University of Wisconsin, Madison)

In a groundbreaking 1972 paper Fritz Horn and Roy Jackson showed that a complex balanced mass-action network must have a unique (locally stable) equilibrium within any compatibility class [*]. In 1974 Horn conjectured that this equilibrium is actually a global attractor, i.e., all solutions in the same compatibility class must converge to this equilibrium. Later, this claim was called the Global Attractor Conjecture, and it was shown that it has remarkable implications for the dynamics of large classes of polynomial and power-law dynamical systems, even if they are not derived from mass-action kinetics.

Several special cases of this conjecture have been proved during the last decade. We describe a proof of the global attractor conjecture in full generality. In particular, it will follow that all detailed balanced systems and all deficiency zero weakly reversible networks have the global attractor property. Also, it follows that there exists no bounded control mechanism that can drive to zero (e.g., to extinction) one or more of the variables of a weakly reversible reaction network [**].

We will also describe some implications for biochemical networks that implement noise filtering and cellular homeostasis.

References:

[*] F. Horn, R. Jackson, General Mass Action Kinetics, Archive of Rational Mechanics and Analysis, 1972.

[**] G. Craciun, Toric Differential Inclusions and a Proof of the Global Attractor Conjecture, http://arxiv.org/abs/1501.02860 , 2015.

Several special cases of this conjecture have been proved during the last decade. We describe a proof of the global attractor conjecture in full generality. In particular, it will follow that all detailed balanced systems and all deficiency zero weakly reversible networks have the global attractor property. Also, it follows that there exists no bounded control mechanism that can drive to zero (e.g., to extinction) one or more of the variables of a weakly reversible reaction network [**].

We will also describe some implications for biochemical networks that implement noise filtering and cellular homeostasis.

References:

[*] F. Horn, R. Jackson, General Mass Action Kinetics, Archive of Rational Mechanics and Analysis, 1972.

[**] G. Craciun, Toric Differential Inclusions and a Proof of the Global Attractor Conjecture, http://arxiv.org/abs/1501.02860 , 2015.