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  .
Chemical reaction systems are dynamical systems that arise in chemical engineering and systems biology. In this work, we consider the question of whether the minimal (in a precise sense) multistationary chemical reaction networks, which we propose to call `atoms of multistationarity,' characterize the entire set of multistationary networks. Our main result states that the answer to this question is `yes' in the context of fully open continuous-flow stirred-tank reactors (CFSTRs), which are networks in which all chemical species take part in the inflow and outflow. In order to prove this result, we show that if a subnetwork admits multiple steady states, then these steady states can be lifted to a larger network, provided that the two networks share the same stoichiometric subspace. We also prove an analogous result when a smaller network is obtained from a larger network by `removing species.' Our results provide the mathematical foundation for a technique used by Siegal-Gaskins et al. of establishing bistability by way of `network ancestry.' Additionally, our work provides sufficient conditions for establishing multistationarity by way of atoms and moreover reduces the problem of classifying multistationary CFSTRs to that of cataloging atoms of multistationarity. As an application, we enumerate and classify all 386 bimolecular and reversible two-reaction 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 (the atoms of multistationarity) .
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networks taken with mass-action kinetics
systems
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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.
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multistationary chemical reaction
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We
Our main result states that the answer to this question is `yes' in the context of fully open continuous-flow stirred-tank reactors (CFSTRs), which are networks in which all chemical species take part in the inflow and outflow. In order to prove this result, we
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nondegenerate positive
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extended to establish multistationarity of
lifted to
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Our result provides
We also prove an analogous result when a smaller network is obtained from a larger network by `removing species.' Our results provide
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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
Additionally, our work provides sufficient conditions for establishing multistationarity by way of atoms and moreover reduces the problem of classifying multistationary CFSTRs to that of cataloging atoms of multistationarity. As an application, we enumerate and classify
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such
two-reaction
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(the atoms of multistationarity)