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Atoms of multistationarity in chemical reaction networks

Badal Joshi, Anne Shiu

arXiv:1108.5238v2math.DSq-bio.MN

TL;DR

The paper asks whether minimal multistationary networks can characterize all such networks, and develops lifting results to study that question for CFSTRs. For fully open CFSTRs, it establishes characterization by atoms and classifies a set of bimolecular two-reaction networks, finding 35 multistationary networks and 11 minimal atoms.

  • Problem

    The paper asks whether minimal multistationary networks characterize the entire set of multistationary networks.

    Source: §1 · p. 2

  • Method

    The authors prove lifting results for subnetworks sharing a stoichiometric subspace and for embedded CFSTRs, then enumerate bimolecular two-reaction CFSTRs.

    Source: §1 · p. 2, §6 · p. 18

  • Results

    386 reversible bimolecular two-reaction networks were enumerated; exactly 35 admit multiple positive steady states, with 11 minimal atoms.

    Source: §1 · p. 3

  • Takeaways & Limitations

    For fully open CFSTRs, multistationarity is characterized by minimal multistationary networks under the embedded-network relation.

    Source: §1 · p. 2

    The embedded-network lifting result does not apply when the larger network lacks a flow-type subnetwork for a species, as in the example involving species B.

    Source: §4 · p. 16

Abstract

from arXiv · show

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).

1 Introduction

The paper asks whether minimal multistationary networks—“atoms of multistationarity”—characterize all multistationary networks, and proves this for fully open CFSTRs. It develops lifting results and applies them to classify bimolecular two-reaction CFSTRs.

1 Introduction

  • The motivating question is whether minimal multistationary networks characterize the full set of multistationary networks.The paper frames the question in the context of chemical reaction systems relevant to chemical engineering and systems biology.Source: §1 · pp. 1–2
  • Earlier criteria and software can determine multistationarity or rule it out in many cases, but the cited approaches can be inconclusive.The paper positions subnetwork-based lifting as an approach for such cases.Source: §1 · p. 2
  • Multiple steady states can be lifted from a subnetwork when both networks share a stoichiometric subspace and the steady states are nondegenerate.This result provides mathematical support for Siegal-Gaskins et al.’s “network ancestry” technique.Source: §1 · p. 2
  • For fully open CFSTRs, multistationarity is characterized by minimal networks called atoms, obtained under the embedded-network relation.Embedded networks can be formed by removing species as well as reactions.Source: §1 · p. 2
  • 386 reversible bimolecular two-reaction CFSTRs were enumerated; exactly 35 admit multiple positive steady states, with 11 minimal atoms.Each multistationary network has a unique minimal multistationary sub-CFSTR.Source: §1 · p. 3

2 Chemical reaction network theory

This section defines the network structures and dynamics used in the paper, then recalls a classification result for one-reaction CFSTRs.

2.1 Chemical reaction networks

  • A chemical reaction network is specified by finite sets of species, complexes, and reactions, with every complex participating in a reaction and every species appearing in a complex.An embedded network is obtained by selecting species and reactions, then restricting those reactions to the selected species.Source: §2.1 · pp. 3–4
  • A CFSTR contains every species’ outflow reaction, while a fully open CFSTR also contains every species’ inflow reaction.Flow reactions are either 0 → Xi or Xi → 0.Source: §2.1 · p. 5

2.2 Dynamics and steady states

  • Reaction kinetics define the system’s ODEs, and positive trajectories remain in their stoichiometric compatibility class, determined by the stoichiometric subspace.The subspace is spanned by reaction vectors; a positive steady state is a positive concentration vector where the system’s vector field is zero.Source: §2.2 · pp. 5–6
  • Under mass-action kinetics, each reaction rate is its positive rate constant multiplied by the reactant concentrations raised to their stoichiometric coefficients.A network is multistationary when some allowed kinetics and compatibility class have at least two positive steady states.Source: §2.2 · pp. 7–8

2.3 Classification of multistationary one-reaction CFSTRs

  • For one non-flow reaction in a CFSTR, multiple positive mass-action steady states occur exactly when the product of ai over species with bi > ai exceeds 1.The theorem also states that these multistationary CFSTRs have nondegenerate steady states.Source: §2.3 · p. 8
  • For a CFSTR with a pair of reversible non-flow reactions, the paper states an if-and-only-if criterion for multiple positive mass-action steady states.The supplied passage identifies the criterion but does not provide its formula.Source: §2.3 · p. 8

3 Lifting multistationarity from subnetworks

The section establishes when multistationarity can be lifted from subnetworks to larger networks. Its key conditions are nondegenerate steady states and a shared stoichiometric subspace.

3 Lifting multistationarity from subnetworks

  • Multiple nondegenerate positive steady states of a subnetwork lift to the larger network when their stoichiometric subspaces coincide.The proof continues each steady state to a nearby steady state as added reaction parameters are varied.Source: §3 · p. 9
  • The lifting argument uses a homotopy whose zero count is preserved when zeros avoid the domain boundary and the endpoint maps have image equal to the stoichiometric subspace.The authors identify this as a modified form of a homotopy result by Craciun, Helton, and Williams.Source: §3 · p. 9
  • Theorem 3.1 supplies a mathematical justification for Siegal-Gaskins et al.’s network-ancestry method for establishing bistability.That method identifies larger networks containing a bistable subnetwork with the same stoichiometric subspace.Source: §3 · p. 10
  • Nondegeneracy is necessary for the theorem: a subnetwork can have multiple degenerate positive steady states even when the larger network does not have multiple positive mass-action steady states.In the example, the subnetwork has steady states only when κ1 = κ2, and then each two-dimensional compatibility class contains infinitely many degenerate steady states.Source: §3 · p. 11
  • If a network is formed by making some irreversible reactions of another network reversible, its multiple nondegenerate positive steady states are preserved.If the smaller network has finitely many such steady states, the larger network has at least as many.Source: §3 · pp. 9, 11
  • A sub-CFSTR with multiple nondegenerate positive steady states implies the same for a fully open CFSTR containing it.The result also preserves exponentially stable steady states and, for finitely many, guarantees at least as many in the larger network.Source: §3 · pp. 9, 11

4 Lifting mass-action multistationarity from embedded CFSTRs

The section extends steady-state lifting from subnetworks to embedded networks, showing that multistationarity can transfer when species are removed under specified conditions. For fully open CFSTRs, multistationarity in an embedded CFSTR implies multistationarity in the full network.

Theorem 4.2

  • An embedded network lifts multiple nondegenerate positive steady states to the larger network when it is full-dimensional and each added species has a mass-action flow-type subnetwork.The larger network then has at least as many such steady states; the result also holds for exponentially stable steady states.Source: §4 · pp. 12–13

Lemma 4.3

  • For a one-species extension, the lifting lemma requires corresponding reactions to differ by removal of that species and the remaining reactions to form its flow-type subnetwork.Each nondegenerate steady state lifts to one near (u, 1), with exponential stability preserved when present.Source: §4 · p. 13

Proof of Theorem 4.2

  • The proof uses a large rate scale for the added species’ flow subnetwork to separate fast one-dimensional dynamics from slow dynamics of the smaller system.The construction places the lifted state near the smaller-system steady state with the added species concentration equal to 1.Source: §4 · pp. 13–15
  • A subnetwork sharing the larger network’s stoichiometric subspace can inherit lifted steady states, completing the extension from the embedded network to the full network.The proof first lifts to an intermediate network and then applies Theorem 3.1 to the full network.Source: §4 · p. 15

Example 4.5

  • Embedding alone is insufficient: a network lacking a flow-type subnetwork for the removed species can have one steady state even when its embedded network has multiple.The example shows why the flow-subnetwork condition is necessary for this lifting result.Source: §4 · pp. 15–16

Corollary 4.6

  • Embedded CFSTRs with multiple nondegenerate positive mass-action steady states imply the same for their fully open CFSTRs.The result applies when the embedded CFSTR is obtained by removing species and reactions.Source: §4 · p. 16

5 CFSTR atoms of multistationarity

The section defines CFSTR atoms as minimal multistationary fully open networks under the embedded-network relation. It establishes that every multistationary fully open CFSTR contains such an atom, and characterizes the one-reaction case.

CFSTR atoms of multistationarity

  • A fully open CFSTR is multistationary exactly when it contains an embedded CFSTR atom of multistationarity.An atom is minimal among fully open CFSTRs admitting multiple nondegenerate positive mass-action steady states.Source: §5 · pp. 16–17

One-reaction CFSTR atoms

  • A one-reaction CFSTR is an atom exactly when its sole non-flow reaction is a1X → a2X or X + Y → b1X + b2Y.The atom forms are the two reaction patterns given in equation (15).Source: §5 · p. 17
  • A one-reaction CFSTR has one such atom if and only if it admits multiple nondegenerate positive mass-action steady states.The section states that one-reaction atoms can be characterized by two parameters and contain at most two species.Source: §5 · p. 17

Questions

  • Whether atoms can be defined usefully outside the CFSTR setting remains an open question in the section.The authors note that some flow reactions can be removed from an atom while preserving multistationarity, whereas removing another can destroy it.Source: §5 · p. 17

6 Enumeration of reversible bimolecular two-reaction CFSTRs

The section asks which bimolecular two-reaction fully open CFSTRs admit multiple positive steady states and describes an enumeration strategy. It organizes candidate networks by total molecularity partitions, filters duplicates and trivial cases, and retains one representative per species-relabeling class.

6 Enumeration of reversible bimolecular two-reaction CFSTRs

  • The enumeration targets bimolecular two-reaction CFSTRs, where each complex has at most two molecules, and lists one network per species-relabeling equivalence class.The non-flow reactions may comprise two reversible pairs, one reversible and one irreversible reaction, or two irreversible reactions.Source: §6 · pp. 17–18

6.1 The total molecularity partition of a chemical reaction network

  • A network’s total molecularity partition groups species by the summed stoichiometric coefficients across reversible reactions.The partitions used for the enumeration have total m from 4 through 8.Source: §6.1 · pp. 18–19

6.2 Algorithm for enumerating networks

  • The enumeration lists partitions of m = 4, 5, 6, 7, 8, constructs networks for each partition, then removes trivial, repeated, and decoupled networks.It keeps one representative from each class related by species relabeling.Source: §6.2 · pp. 19–20
  • The algorithm yields 386 reversible bimolecular two-reaction networks for later testing for multiple positive steady states.The subsequent analysis also checks whether their sub-CFSTRs are multistationary.Source: §6 · p. 18, §6.2 · p. 20

6.3 The enumeration of small networks of Deckard, Bergmann, and Sauro

  • The cited earlier enumeration covered over 47 million small bimolecular networks, but used directed-reaction and species counts and excluded certain zero-complex and shared-species reactions.The authors distinguish their enumeration’s scope from that earlier catalog.Source: §6.3 · p. 20

7 Classification of multistationary two-reaction CFSTRs

The classification finds 35 multistationary networks among 386 reversible bimolecular two-reaction CFSTRs, with 11 minimal embedded networks serving as atoms. Jacobian screening and Toolbox analysis identify the multistationary cases, and the classification extends to irreversible networks through their minimal subnetworks and embedded atoms.

7 Classification of multistationary two-reaction CFSTRs

  • Exactly 35 of 386 reversible, bimolecular, two-reaction fully open CFSTRs admit multiple positive mass-action steady states; their embedded-network poset has 11 minimal atoms.Each of the 35 has a unique minimal multistationary subnetwork, and each admits multiple nondegenerate positive steady states.Source: §7 · p. 21, Figure 3 · p. 24

7.1 Ruling out multistationarity by the Jacobian Criterion

  • Of 386 networks, 142 are ruled out by the total-molecularity bound, 102 more pass the Jacobian Criterion, and the remaining 142 require further analysis.The first group has each partition part at most two; these networks pass the Jacobian Criterion as well.Source: §7.1 · p. 22, Table 2 · p. 22

7.2 Applying the CRN Toolbox to classify reversible two-reaction networks

  • Among the remaining 142 reversible networks that failed the Jacobian Criterion, CRN Toolbox analysis found exactly 35 admitting multiple positive mass-action steady states, while the other 107 did not.For each of the 35 multistationary CFSTRs, the Toolbox supplied rate constants, two positive steady states, and eigenvalues. Nondegeneracy was evident in all but one case; for that case, another multistationary instance with two nondegenerate steady states was found by hand.Source: §7.2 · p. 22

7.3 Classifying irreversible two-reaction networks

  • Each of the 35 multistationary reversible CFSTRs has a unique minimal multistationary sub-CFSTR: 29 have two directed non-flow reactions, while six have one reversible and one directed reaction.Among the 280 candidate irreversible subnetworks, these 35 minimal cases characterize multistationarity by subnetwork containment.Source: §7.3 · p. 23, Figure 2 · p. 23
  • The 35 minimal subnetworks form an embedded-network poset with 11 atoms, and a bimolecular two-reaction CFSTR is multistationary exactly when it contains one of those atoms.The study does not classify CFSTR atoms for bimolecular three-reaction networks because their number is prohibitively large.Source: §7.3 · pp. 23, 25
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