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Discussing the Proof of the Global Attractor Conjecture

By Elizabeth Gross, Matthew Johnston, and Nicolette Meshkat

The Global Attractor Conjecture (GAC) concerns an important class of ordinary differential equation models known as toric dynamical systems—originally called complex-balanced systems—that arise from chemical kinetics. Ludwig Boltzmann first introduced the complex-balanced condition [3] for modeling collisions in kinetic gas theory; Fritz Horn and Roy Jackson later applied the condition to chemical kinetics in their 1972 seminal paper [9], which is frequently credited with founding the research area popularly known as Chemical Reaction Network Theory. This area attempts to establish connections between the network properties of the reaction graph and the permissible behaviors of the resulting dynamical system. Related research has seen a swell of activity since the rise of systems biology over the last two decades, and has helped establish structural motifs underlying cellular regulation [1]. The adjective “toric” was proposed in [5] to underline the tight connection to the algebraic study of toric varieties.

Gheorghe Craciun discusses his construction of zero-separating surfaces, which are hypersurfaces in the positive orthant that play a key role in his proof of the Global Attractor Conjecture. Photo credit: Nida Kazi Obatake.
San José State University (SJSU), with cooperation from Santa Clara University and the University of California, Berkeley, hosted a weekend workshop in March about Gheorghe Craciun’s recent proof of the GAC [4]. Intensive reading seminars at UC Berkeley and SJSU, as well as a series of YouTube videos by Craciun, preceded the workshop, which included participants from nearly a dozen universities around the world.

A polynomial dynamical system arising from a reaction network (i.e. a weighted, directed graph) is said to be a toric dynamical system if the net inflow and net outflow across each vertex are equal at steady state. For example, consider the following reaction network:

Under mass-action kinetics, we can associate this network to the following system of polynomial differential equations:

\[\left[ \begin{array}{c} x'_1 \\ x'_2 \end{array} \right] = k_1 \left[ \begin{array}{c} -1 \\ 1 \end{array} \right] x_1^2 + k_2 \left[ \begin{array}{c} 1 \\ -1 \end{array} \right] x_1x_2+ k_3 \left[ \begin{array}{c} -1\\ 1 \end{array} \right] x_1x_2+ k_4 \left[ \begin{array}{c} 1\\ -1\end{array} \right] x_2^2. \qquad (1)\]

In order for this system to be a toric dynamical system, it must satisfy the steady state conditions implied by \((1)\) as well as \(k_2x_1x_2 = k_1x_1^2\) (vertex \(2X_1\)), \(k_1x_1^2+k_4x_2^2 = (k_2+k_3)x_1x_2\) (vertex \(X_1 + X_2\)), and \(k_3x_1x_2 = k_4x_2^2\) (vertex \(2X_2\). Given this network-balancing property, the authors of [9] were able to prove the following powerful dynamical result.

According to the theorem, "Within every strictly positive stoichiometric compatibility class (linear invariant space) of a toric dynamical system, there exists exactly one strictly positive steady state; this steady state is locally asymptotically stable with respect to its compatibility class” [9].

However, the theorem only guarantees local stability of the steady state, so one might naturally wonder whether this stability may be extended globally throughout the compatibility class. The GAC offers the resolution to this question:

The unique positive steady state \(\mathbf{x}^* \in \mathbb{R}_{>0}^m\) in every stoichiometric compatibility class of a toric dynamical system is the global attractor for its stoichiometric compatibility class.

The gap between local and global stability is more subtle than one might initially realize. The proof contained in [9] makes use of a Lyapunov function, which is strictly convex on the strictly positive orthant \(\mathbb{R}_{>0}^m.\) All trajectories consequently descend for all times through the contours of the Lyapunov function toward the positive steady state. This led the authors of the original paper [9] to errantly conclude that they had proved not only the above theorem, but the GAC as well! They only later realized that, since the Lyapunov function is bounded along \(\partial \mathbb{R}_{>0}^m,\) trajectories could possibly approach \(\partial \mathbb{R}_{>0}^m\) instead of the positive steady state. The claim was consequently retracted in 1974 in [8], which finally correctly states the conjecture as an open problem.

A blueprint for a zero-separating surface, an important construction in Craciun’s proof of the Global Attractor Conjecture. Photo credit: Nida Kazi Obatake.
The GAC has experienced a resurgence of interest since the 2000s. Published papers affirm the GAC in a number of special cases, including 2-dimensional systems [5], networks with a single linkage class [2], 3-dimensional systems [6], systems with 3-dimensional stoichiometric subspaces [10], and strongly endotactic networks [7]. In 2015, Gheorghe Craciun (University of Wisconsin-Madison) proposed a general proof, valid in all dimensions and without special conditions.

The purpose of the Global Attractor Conjecture Workshop was to bring together mathematicians to discuss the proof in detail and learn more about the ideas surrounding this significant breakthrough. The workshop began with an opening talk by Craciun. For the remainder of the weekend, participants gave presentations on the sections of [4], working through the proof step-by-step. The 41-page manuscript [4] introduces and utilizes sophisticated ideas from dynamical systems, polyhedral geometry, differential inclusions, and other areas. A main idea involves embedding toric dynamical systems into toric differential inclusions. Polyhedral geometry also plays an interesting role. The necessary differential inclusions are constructed by looking at the dual cones in the polyhedral fan formed by the reaction vectors. This application of polyhedral geometry allows the proof to very naturally extend to over four dimensions.

The GAC is related to several open problems about persistence and permanence of dynamical systems on the positive orthant. Most notably, the so-called Permanence Conjecture involves a much larger class of dynamical systems on the positive orthant. The GAC also has strong connections to open problems in thermodynamics and statistical mechanics, and in particular to global convergence problems derived from discrete approximations of the Boltzmann equation. So, while Craciun’s proof may bookend decades of work on global dynamics in chemical reaction network theory, the ideas of his proof will serve as the opening pages to new volumes of exploration.

[1] Alon, U. (2006). An Introduction to Systems Biology: Design Principles of Biological Circuits. Chapman & Hall/CRC Mathematical and Computational Biology Series. Boca Raton, FL: Taylor & Francis Group.

[2] Anderson, D. (2011). A proof of the global attractor conjecture in the single linkage class case. SIAM J. Appl. Math., 71(4), 1487-1508.

[3] Boltzmann, L. (1887). Neuer Beweis zweier Sätze über das Wärmegleichgewicht unter mehratomigen Gas-molekülen.” Sitzungsberichte der Kaiserlichen Akademie der Wissenschaften in Wien, 95, 153-164.

[4] Craciun, G. (2015). Toric Differential Inclusions and a Proof of the Global Attractor Conjecture. Cornell University Library. Preprint, arXiv:1501.02860.

[5] Craciun, G., Dickenstein, A., Shiu, A., & Sturmfels, B. (2009). Toric dynamical systems. J. Symbolic Comput., 44(11), 1551-1565.

[6] Craciun, G., Pantea, C., & Nazarov, F. (2013). Persistence and permanence of mass-action and power-law dynamical systems. SIAM J. Appl. Math., 73(1), 305-329.

[7] Gopalkrishnan, M., Miller, E., & Shiu, A. (2014). A geometric approach to the global attractor conjecture. SIAM J. Appl. Dyn. Syst., 13(2), 758-797.

[8] Horn. F. (1974). The dynamics of open reaction systems: Mathematical aspects of chemical and biochemical problems and quantum chemistry. SIAM-AMS Proceedings, 8, 125-137.

[9] Horn, F.J.M., & Jackson, R. (1972). General Mass Action Kinetics. Archive Rational Mech., 47, 81-116.

[10] Pantea, C. (2012). On the persistence and global stability of mass-action systems. SIAM J. Math. Anal., 44(3), 1636-1673.

Elizabeth Gross is an assistant professor at San José State University whose interests include algebraic statistics with applications to social and biological networks, phylogenetics, and neuroscience. Matthew Johnston is an assistant professor at San José State University whose interests include mathematical biology, dynamical systems, and stochastic processes. Nicolette Meshkat is an assistant professor at Santa Clara University whose interests include applied algebraic geometry and systems biology.

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