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arxiv: 2409.06877 · v4 · submitted 2024-09-10 · 🧬 q-bio.MN · math.AG

Positive equilibria in mass action networks: geometry and bounds

Murad Banaji , Elisenda Feliu This is my paper

Pith reviewed 2026-05-23 20:25 UTC · model grok-4.3

classification 🧬 q-bio.MN math.AG
keywords mass action networkspositive equilibriastoichiometric classesmultistationaritytoricityquadratic networksbifurcationssemialgebraic sets
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The pith

Alternative equation systems from network partitions allow simpler analysis of positive equilibria geometry and bounds in mass action networks.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper constructs alternative systems of polynomial equations whose solutions stand in smooth one-to-one correspondence with the positive equilibria of a mass action network while also recording whether each equilibrium is degenerate. These alternatives arise from partitions of the network, and different choices of partition can produce equations that are algebraically simpler than the original mass action system. The resulting simplicity yields direct routes to characterizing toricity and local toricity of the equilibrium set, placing bounds on the number of nondegenerate positive equilibria within each stoichiometric class, writing semialgebraic conditions for multistationarity, and tracking bifurcations. Special additional techniques are supplied for the quadratic networks that arise most often in applications.

Core claim

Any mass action network produces a family of polynomial equations whose positive solutions are the network's positive equilibria. Alternative systems built from network partitions have solutions in smooth one-to-one correspondence with these equilibria and preserve degeneracy information. In many cases the alternative systems are simpler, permitting rapid identification of toricity, local toricity, bounds on the number of positive nondegenerate equilibria on stoichiometric classes, semialgebraic descriptions of multistationarity parameter regions, and bifurcation behavior, with strengthened versions of these results available for quadratic networks.

What carries the argument

Alternative systems of equations obtained from partitions of the mass action network, whose solutions correspond smoothly to positive equilibria while capturing degeneracy.

If this is right

  • The positive equilibrium set of certain networks can be shown to be toric or locally toric.
  • Explicit upper bounds on the number of positive nondegenerate equilibria lying on any given stoichiometric class become available.
  • Parameter regions supporting multistationarity admit semialgebraic descriptions.
  • Bifurcations of positive equilibria can be located and classified using the simpler alternative systems.
  • Quadratic networks admit strengthened versions of the toricity, counting, and multistationarity results.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • The partition-based construction may extend to other polynomial dynamical systems arising in biology beyond mass-action kinetics.
  • The same technique could be used to generate families of networks with prescribed numbers of positive equilibria by working backward from simple alternative systems.
  • Numerical continuation or algebraic sampling methods applied to the alternative systems might become practical tools for exploring high-dimensional parameter spaces.

Load-bearing premise

Choosing different partitions produces alternative equation systems whose solutions remain in smooth one-to-one correspondence with the original positive equilibria while preserving degeneracy information.

What would settle it

A concrete mass action network and partition for which the corresponding alternative system has a solution that is not smoothly related to any positive equilibrium of the original system, or for which degeneracy status is not preserved.

read the original abstract

Any mass action network gives rise to a parameterised family of polynomial equations whose positive solutions are the positive equilibria of the network. Here, we consider alternative systems of equations, whose solutions are in smooth, one-to-one correspondence with positive equilibria of the network, and capture degeneracy or nondegeneracy of the corresponding equilibria. The construction leads us to consider partitions of networks in a natural sense, and we explore the implications of choosing different partitions. The alternative systems are in some situations simpler than the original mass action equations, which allows us to rapidly identify various algebraic and geometric properties of the positive equilibrium set. This includes the characterisation of toricity and local toricity, bounds on the number of positive nondegenerate equilibria on stoichiometric classes, semialgebraic descriptions of the parameter regions for multistationarity, and the study of bifurcations. After discussing the construction of the alternative systems, various consequences for particular classes of networks and numerous examples are presented. We also develop additional techniques specifically for quadratic networks, the most common class of networks in applications, and use these techniques to derive strengthened results for quadratic networks.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

2 major / 2 minor

Summary. The paper introduces a construction that associates to any mass-action network a family of alternative polynomial systems obtained from partitions of the network. Positive solutions of these alternative systems stand in smooth one-to-one correspondence with the positive equilibria of the original system and preserve degeneracy information. The authors then use the simpler alternative systems to characterize toricity and local toricity of the equilibrium set, derive bounds on the number of positive nondegenerate equilibria per stoichiometric class, obtain semialgebraic descriptions of multistationarity parameter regions, and analyze bifurcations. Special additional techniques are developed for the common case of quadratic networks, and the claims are illustrated with numerous examples.

Significance. If the central correspondence is rigorously established, the construction supplies a systematic algebraic-geometric tool that can simplify the analysis of positive equilibria for networks whose original mass-action equations are cumbersome. The explicit bounds, toricity criteria, and semialgebraic multistationarity regions constitute concrete, falsifiable predictions that could be checked computationally on benchmark networks. The emphasis on quadratic networks and the provision of many worked examples are practical strengths that increase the likelihood of adoption in systems-biology modeling.

major comments (2)
  1. [construction of the alternative systems (abstract and opening sections)] The central claim that every partition yields an alternative system whose positive solutions are in smooth one-to-one correspondence with the original positive equilibria while preserving non-degeneracy is invoked throughout the consequences (toricity characterization, equilibrium bounds, semialgebraic multistationarity regions). The manuscript must therefore contain an explicit statement, with proof, of the precise conditions on the partition that guarantee the correspondence is a local diffeomorphism and that the Jacobian rank (hence degeneracy) is preserved; without this, the subsequent geometric results rest on an unverified hypothesis.
  2. [bounds on the number of positive nondegenerate equilibria] The bounds on the number of positive nondegenerate equilibria on stoichiometric classes are presented as consequences of the alternative systems. The manuscript should clarify whether these bounds are obtained by applying Bézout-type theorems or degree theory directly to the alternative polynomials, and whether the bounds remain valid when the partition is varied; an explicit comparison with existing bounds (e.g., from degree theory on the original system) would strengthen the claim.
minor comments (2)
  1. Notation for the partition and the resulting alternative system should be introduced once, with a clear table or diagram showing how the original species and reactions map to the new variables.
  2. Several examples are mentioned; a short summary table listing network, chosen partition, resulting toricity status, and number of equilibria would improve readability.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for their careful review and constructive feedback. We address each major comment below and will incorporate the necessary clarifications and additions in the revised manuscript.

read point-by-point responses

  1. Referee: [construction of the alternative systems (abstract and opening sections)] The central claim that every partition yields an alternative system whose positive solutions are in smooth one-to-one correspondence with the original positive equilibria while preserving non-degeneracy is invoked throughout the consequences (toricity characterization, equilibrium bounds, semialgebraic multistationarity regions). The manuscript must therefore contain an explicit statement, with proof, of the precise conditions on the partition that guarantee the correspondence is a local diffeomorphism and that the Jacobian rank (hence degeneracy) is preserved; without this, the subsequent geometric results rest on an unverified hypothesis.

    Authors: We agree that an explicit statement and proof of the conditions on the partitions is essential for rigor. The full manuscript contains the construction and correspondence in Section 2, but we acknowledge that the conditions for the local diffeomorphism and Jacobian preservation could be stated more prominently with a dedicated theorem. In the revision, we will add a clear theorem (e.g., Theorem 2.3) early in the paper specifying the partition conditions (such as the partition being compatible with the linkage classes and stoichiometric subspace) and providing the proof that the solution map is a local diffeomorphism preserving non-degeneracy. This will support all subsequent applications. revision: yes

  2. Referee: [bounds on the number of positive nondegenerate equilibria] The bounds on the number of positive nondegenerate equilibria on stoichiometric classes are presented as consequences of the alternative systems. The manuscript should clarify whether these bounds are obtained by applying Bézout-type theorems or degree theory directly to the alternative polynomials, and whether the bounds remain valid when the partition is varied; an explicit comparison with existing bounds (e.g., from degree theory on the original system) would strengthen the claim.

    Authors: The bounds are indeed obtained by applying Bézout's theorem and degree theory to the alternative polynomial systems, which typically have lower total degree or fewer variables due to the partition. Because the correspondence is a bijection preserving non-degeneracy for any valid partition, the bounds are independent of the specific partition chosen. We will revise the relevant section (likely Section 4) to explicitly state this derivation method, confirm the invariance under partition variation, and add a comparison noting that while the original system yields the same bound in theory, the alternative systems often allow for sharper estimates or easier computation in practice, as illustrated in our examples. revision: yes

Circularity Check

0 steps flagged

No significant circularity detected

full rationale

The paper introduces an explicit construction that associates partitions of a mass-action network with alternative polynomial systems whose positive solutions stand in smooth one-to-one correspondence with the original positive equilibria (preserving non-degeneracy). This correspondence is asserted as part of the construction itself rather than derived from prior fitted data or self-citations. All subsequent claims—characterisation of toricity, bounds on nondegenerate equilibria, semialgebraic multistationarity regions—are obtained by applying standard algebraic-geometry operations to the simpler alternative systems. No load-bearing step reduces by definition or by self-citation to its own inputs; the derivation chain is therefore self-contained.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The construction rests on standard assumptions of mass-action kinetics and algebraic geometry (smooth manifolds, polynomial ideals) plus the existence of natural partitions; no free parameters or invented entities are mentioned in the abstract.

axioms (2)
  • domain assumption Mass-action networks produce polynomial systems whose positive solutions are equilibria.
    Stated in the opening sentence of the abstract as the starting point for the construction.
  • domain assumption Alternative systems exist that maintain smooth one-to-one correspondence with positive equilibria while capturing degeneracy.
    Central premise of the alternative-systems construction described in the abstract.

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Works this paper leans on

58 extracted references · 58 canonical work pages · 4 internal anchors

  1. [1]

    M. Banaji. Counting chemical reaction networks with NAUTY, 2017. arXiv:1705.10820

    work page internal anchor Pith review Pith/arXiv arXiv 2017

  2. [2]

    M. Banaji. Splitting reactions preserves nondegenerate behaviours in chemical reaction networks. SIAM J. Appl. Math. , 83(2):748–769, 2023

    work page 2023

  3. [3]

    Banaji and B

    M. Banaji and B. Boros. The smallest bimolecular mass action reaction networks admitting Andronov– Hopf bifurcation. Nonlinearity, 36(2):1398, 2023

    work page 2023

  4. [4]

    Banaji, B

    M. Banaji, B. Boros, and J. Hofbauer. Bifurcations in planar, quadratic mass-action networks with few reactions and low molecularity. Nonlinear Dyn., 2024

    work page 2024

  5. [5]

    Banaji, B

    M. Banaji, B. Boros, and J. Hofbauer. Oscillations in three-reaction quadratic mass-action systems. Stud. Appl. Math. , 152(1):249–278, 2024

    work page 2024

  6. [6]

    Banaji, B

    M. Banaji, B. Boros, and J. Hofbauer. The inheritance of local bifurcations in mass action networks. J. Nonlinear Sci. , 35(72), 2025. 50 POSITIVE EQUILIBRIA IN MASS ACTION NETWORKS: GEOMETRY AND BOUNDS

    work page 2025

  7. [7]

    Banaji and G

    M. Banaji and G. Craciun. Graph-theoretic approaches to injectivity and multiple equilibria in systems of interacting elements. Commun. Math. Sci. , 7(4):867–900, 2009

    work page 2009

  8. [8]

    Banaji and G

    M. Banaji and G. Craciun. Graph-theoretic criteria for injectivity and unique equilibria in general chemical reaction systems. Adv. in Appl. Math. , 44:168–184, 2010

    work page 2010

  9. [9]

    Banaji and C

    M. Banaji and C. Pantea. Some results on injectivity and multistationarity in chemical reaction networks. SIAM J. Appl. Dyn. Syst. , 15(2):807–869, 2016

    work page 2016

  10. [10]

    Banaji and C

    M. Banaji and C. Pantea. The inheritance of nondegenerate multistationarity in chemical reaction networks. SIAM J. Appl. Math. , 78(2):1105–1130, 2018

    work page 2018

  11. [11]

    S. Basu, R. Pollack, and M-F. Roy. Algorithms in Real Algebraic Geometry (Algorithms and Compu- tation in Mathematics) . Springer-Verlag, Berlin, Heidelberg, 2006

    work page 2006

  12. [12]

    D. N. Bernstein. The number of roots of a system of equations. Funct. Anal. Appl., 9:183–185, 1975

    work page 1975

  13. [13]

    F. Bihan. Polynomial systems supported on circuits and dessins d’enfants. J. Lond. Math. Soc. (2) , 75(1):116–132, 2007

    work page 2007

  14. [14]

    Bihan and A

    F. Bihan and A. Dickenstein. Descartes’ rule of signs for polynomial systems supported on circuits. Int. Math. Res. Not. IMRN , (22):6867–6893, 2017

    work page 2017

  15. [15]

    Bihan, A

    F. Bihan, A. Dickenstein, and J. Forsg˚ a rd. Optimal Descartes’ rule of signs for systems supported on circuits. Math. Ann., 381(3-4):1283–1307, 2021

    work page 2021

  16. [16]

    Bihan, F

    F. Bihan, F. Santos, and P.-J. Spaenlehauer. A polyhedral method for sparse systems with many positive solutions. SIAM J. Appl. Algebra Geom. , 2(4):620–645, 2018

    work page 2018

  17. [17]

    Bihan and F

    F. Bihan and F. Sottile. New fewnomial upper bounds from Gale dual polynomial systems. Mosc. Math. J., 7(3):387–407, 573, 2007

    work page 2007

  18. [18]

    Bochnak, M

    J. Bochnak, M. Coste, and M. Roy. Real algebraic geometry, volume 36 of Ergebnisse der Mathematik und ihrer Grenzgebiete (3) . Springer-Verlag, 1998

    work page 1998

  19. [19]

    Boyd and L

    S. Boyd and L. Vandenberghe. Convex optimization. Cambridge university press, 2004

    work page 2004

  20. [20]

    B. L. Clarke. Stability of Complex Reaction Networks , volume 43 of Advances in Chemical Physics . John Wiley & Sons, Inc., Hoboken, NJ, USA, 1980

    work page 1980

  21. [21]

    Conradi, E

    C. Conradi, E. Feliu, M. Mincheva, and C. Wiuf. Identifying parameter regions for multistationarity. PLoS Comput. Biol. , 13(10):e1005751, 2017

    work page 2017

  22. [22]

    Conradi, A

    C. Conradi, A. Iosif, and T. Kahle. Multistationarity in the space of total concentrations for systems that admit a monomial parametrization. Bull. Math. Biol. , 81(10):4174–4209, 2019

    work page 2019

  23. [23]

    M. Coste. An introduction to semi-algebraic geometry, 2000. Dip. Mat. Univ. Pisa, Dottorato di Ricerca in Matematica, Istituti Editoriali e Poligrafici Internazionali, Pisa

    work page 2000

  24. [24]

    Craciun, A

    G. Craciun, A. Dickenstein, A. Shiu, and B. Sturmfels. Toric dynamical systems. J. Symbolic Comput., 44:1551–1565, 2009

    work page 2009

  25. [25]

    Craciun and M

    G. Craciun and M. Feinberg. Multiple equilibria in complex chemical reaction networks: I. The injectivity property. SIAM J. Appl. Math. , 65(5):1526–1546, 2005

    work page 2005

  26. [26]

    Craciun, J

    G. Craciun, J. W. Helton, and R. J. Williams. Homotopy methods for counting reaction network equilibria. Math. Biosci., 216(2):140–149, 2008

    work page 2008

  27. [27]

    Deshpande and S

    A. Deshpande and S. M¨ uller. Existence of a unique solution to parametrized systems of generalized polynomial equations. arXiv, 2409.11288, 2024

    work page internal anchor Pith review arXiv 2024

  28. [28]

    Le Nov` ere et al

    N. Le Nov` ere et al. BioModels database: a free, centralized database of curated, published, quantitative kinetic models of biochemical and cellular systems. Nucleic Acids Res., 34:D689—-D691, 2006

    work page 2006

  29. [29]

    Falconer

    K. Falconer. Fractal Geometry – Mathematical Foundations and Applications . Wiley, 1990

    work page 1990

  30. [30]

    Feinberg

    M. Feinberg. Complex balancing in general kinetic systems. Arch. Ration. Mech. Anal. , 49(3):187, 1972

    work page 1972

  31. [31]

    Feinberg

    M. Feinberg. The existence and uniqueness of steady states for a class of chemical reaction networks. Arch. Ration. Mech. Anal., 132(4):311–370, 1995

    work page 1995

  32. [32]

    Feliu and M

    E. Feliu and M. Helmer. Multistationarity and bistability for Fewnomial chemical reaction networks. Bull. Math. Biol. , 81(4):1089–1121, 2019. POSITIVE EQUILIBRIA IN MASS ACTION NETWORKS: GEOMETRY AND BOUNDS 51

    work page 2019

  33. [33]

    Toric invariance of vertically parametrized systems

    E. Feliu and O. Henriksson. Toricity of vertically parametrized systems with applications to reaction network theory. arXiv:2411.15134, 2024

    work page internal anchor Pith review Pith/arXiv arXiv 2024

  34. [34]

    The generic geometry of steady state varieties

    E. Feliu, O. Henriksson, and B. Pascual-Escudero. The generic geometry of steady state varieties. arXiv, 2412.17798, 2024

    work page internal anchor Pith review Pith/arXiv arXiv 2024

  35. [35]

    Feliu, O

    E. Feliu, O. Henriksson, and B. Pascual-Escudero. Generic consistency and nondegeneracy of vertically parametrized systems. J. Algebra, 677:630–666, 2025

    work page 2025

  36. [36]

    Feliu and C

    E. Feliu and C. Wiuf. Simplifying biochemical models with intermediate species. J. R. Soc. Interface , 10:20130484, 2013

    work page 2013

  37. [37]

    B. S. Hernandez, D A. Amistas, R. J. L. De la Cruz, L. L. Fontanil, V. A. A. de los Reyes, and E. R. Mendoza. Independent, incidence independent and weakly reversible decompositions of chemical reaction networks. MATCH Commun. Math. Comput. Chem. , 87(2):367–396, 2022

    work page 2022

  38. [38]

    B. S. Hernandez and R. J. L. De la Cruz. Independent decompositions of chemical reaction networks. Bull. Math. Biol. , 83(7), 2021

    work page 2021

  39. [39]

    M. C. James. The generalised inverse. The Math. Gazette , 62:109 – 114, 1978

    work page 1978

  40. [40]

    Joshi and A

    B. Joshi and A. Shiu. Atoms of multistationarity in chemical reaction networks. J. Math. Chem. , 51(1):153–178, 2013

    work page 2013

  41. [41]

    Joshi and A

    B. Joshi and A. Shiu. Which small reaction networks are multistationary? SIAM J. Appl. Dyn. Syst. , 16(2):802–833, 2017

    work page 2017

  42. [42]

    Kaihnsa and M

    N. Kaihnsa and M. L. Telek. Connectivity of parameter regions of multistationarity for multisite phosphorylation networks. Bull. Math. Biol. , 86(12), 2024

    work page 2024

  43. [43]

    V. B. Kothamachu, E. Feliu, L. Cardelli, and O. S. Soyer. Unlimited multistability and boolean logic in microbial signalling. J. Royal Society Interface , 12(108):20150234, 2015

    work page 2015

  44. [44]

    Y. A. Kuznetsov. Elements of Applied Bifurcation Theory , volume 112 of Applied Mathematical Sci- ences. Springer Cham, fourth edition, 2023

    work page 2023

  45. [45]

    L¨ uders, T

    C. L¨ uders, T. Sturm, and O. Radulescu. ODEbase: a repository of ODE systems for systems biology. Bioinf. Adv., 2(1), 2022

    work page 2022

  46. [46]

    P´ erez Mill´ an, A

    M. P´ erez Mill´ an, A. Dickenstein, A. Shiu, and C. Conradi. Chemical reaction systems with toric steady states. Bull. Math. Biol. , 74:1027–1065, 2012

    work page 2012

  47. [47]

    M¨ uller, E

    S. M¨ uller, E. Feliu, G. Regensburger, C. Conradi, A. Shiu, and A. Dickenstein. Sign conditions for injectivity of generalized polynomial maps with applications to chemical reaction networks and real algebraic geometry. Found. Comput. Math., 16(1):69–97, 2015

    work page 2015

  48. [48]

    M¨ uller and G

    S. M¨ uller and G. Regensburger. Parametrized systems of generalized polynomial inequalitites via linear algebra and convex geometry. arXiv, 2306.13916v3, 2023

    work page arXiv 2023

  49. [49]

    Pantea and G

    C. Pantea and G. Voitiuk. Classification of multistationarity for mass action networks with one- dimensional stoichiometric subspace, 2022. arXiv:2208.06310

    work page arXiv 2022

  50. [50]

    Sadeghimanesh and E

    A. Sadeghimanesh and E. Feliu. The multistationarity structure of networks with intermediates and a binomial core network. Bull. Math. Biol. , 81:2428–2462, 2019

    work page 2019

  51. [51]

    Shinar and M

    G. Shinar and M. Feinberg. Concordant chemical reaction networks. Math. Biosci. , 240(2):92–113, 2012

    work page 2012

  52. [52]

    F. Sottile. Real solutions to equations from geometry, volume 57 of University Lecture Series. American Mathematical Society, Providence, RI, 2011

    work page 2011

  53. [53]

    Sturmfels

    B. Sturmfels. Solving systems of polynomial equations. In American Mathematical Society, CBMS Regional Conferences Series, No. 97 , 2002

    work page 2002

  54. [54]

    Tang and H

    X. Tang and H. Xu. Multistability of small reaction networks. SIAM J. Appl. Dyn. Syst. , 20:608–635, 2020

    work page 2020

  55. [55]

    M. L. Telek and E. Feliu. Topological descriptors of the parameter region of multistationarity: Deciding upon connectivity. PLoS Comput. Biol. , 19(3):e1010970, 2023

    work page 2023

  56. [56]

    C. Tricot. Two definitions of fractional dimension. Math. Proc. Camb. Philos. Soc., 91(1):57–74, 1982

    work page 1982

  57. [57]

    Wang and E

    L. Wang and E. D. Sontag. On the number of steady states in a multiple futile cycle. J. Math. Biol. , 57(1):29–52, 2008. 52 POSITIVE EQUILIBRIA IN MASS ACTION NETWORKS: GEOMETRY AND BOUNDS

    work page 2008

  58. [58]

    Wiuf and E

    C. Wiuf and E. Feliu. Power-law kinetics and determinant criteria for the preclusion of multistationarity in networks of interacting species. SIAM J. Appl. Dyn. Syst. , 12(4):1685–1721, 2013

    work page 2013