Endotactic Reaction Networks

My recent paper “A computational approach to persistence, permanence, and endotacticity of biochemical reaction systems,” with Casian Pantea and Pete Donnell, was recently accepted in the Journal of Mathematical Biology. A preprint is available here.

The paper considers a class of reaction networks called known as “endotactic” networks, which was introduced by Gheorghe Craciun, Casian Pantea, and Fedor Nazarov in [1]. To envision what endotacticity means, we construct a polytope in species space consisting of the convex hull of all source complexes (i.e. every combination of species which appears at the tail of a reaction). We imagine reactions like A + B –> … , for instance, as starting from the point (1,1) in species space. Roughly speaking, a network is endotactic if every reaction points “inward” relative to this polytope of source complexes, as in the following network (polytope in gray):

Endotactic networks have been studied recently in a number of papers due to the hypothesized (but difficult to prove!) connection with persistence, which is the property that no species tends toward extinction (either asymptotically, or subsequentially). The intuition is that, near the boundary of the state space (i.e. where one or more species are near zero), the dominant reaction terms are those corresponding to reactions on the boundary of the aforementioned polytope. Since the corresponding reaction vectors, which guide our solutions through the vector field, point toward the interior of the space (that is, they increase the numbers of those species!), it should follow that no species can approach an extinction state. The proof is known in several special cases, including when the stoichiometric space is 2 or fewer dimensions, and when the network is strongly endotactic, but a full proof is to date unknown.

Our paper addresses a related but equally important (and surprisingly unstudied) problem: Given an arbitrary reaction network, can it be determined whether the network is endotactic (or strongly endotactic)? That’s right—for all the effort over the past three years proving results which follow if a network is endotactic networks, it was still unknown (beyond two-dimensional systems) how to check whether a given network actually satisfies the conditions in the first place!

In this paper, we address this deficiency by formulating the problem as a mixed-integer linear programming (MILP) problem. The algorithm is able to determine whether a network is endotactic or strongly endotactic, regardless of dimension or number of reactions, and is implemented in the open-source online software package CoNtRol. We are also able to show that some previously unconsidered biochemical reaction networks are strongly endotactic, including a common circadian rhythm mechanism [2] which can be accessed in the European Bioinformatics Institute’s BioModels database here.


[1] G. Craciun, F. Nazarov, and Casian Pantea. Persistence and permanence of mass-action and power-law dynamical systems. SIAM Journal on Applied Mathematics, 73(1): 305-329, 2013.

[2] J.C. Leloup and A. Goldbeter. Chaos and birhythmicity in a model for circadian oscillations of the PER and TIM proteins in drosophila. J. Theor. Biol. 198(3): 445-459, 1999.