Atoms of multistationarity in chemical reaction networks

Details Atoms of multistationarity in chemical reaction networks Atoms of multistationarity in chemical reaction networks

2011-08-26

arxiv.org/abs/1108.5238v1

Mathematics

Dynamical Systems

3 figures

Scientific paper

Chemical reaction networks taken with mass-action kinetics are dynamical systems that arise in chemical engineering and systems biology. Deciding whether a chemical reaction network admits multiple positive steady states is to determine existence of multiple positive solutions to a system of polynomials with unknown coefficients. In this work, we consider the question of whether the minimal (in a precise sense) networks, which we propose to call `atoms of multistationarity,' characterize the entire set of multistationary networks. We show that if a subnetwork admits multiple nondegenerate positive steady states, then these steady states can be extended to establish multistationarity of a larger network, provided that the two networks share the same stoichiometric subspace. Our result provides the mathematical foundation for a technique used by Siegal-Gaskins et al. of establishing bistability by way of `network ancestry.' Here, our main application is for enumerating small multistationary continuous-flow stirred-tank reactors (CFSTRs), which are networks in which all chemical species take part in the inflow and outflow. Recent work of the first author presented a simple characterization of the one-reaction CFSTRs that admit multiple positive steady states. Here we consider the two-reaction CFSTRs. We enumerate all 386 bimolecular and reversible such networks. Of these, exactly 35 admit multiple positive steady states. Moreover, each admits a unique minimal multistationary subnetwork, and these subnetworks form a poset (with respect to the relation of `removing species') which has 11 minimal elements.

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