{
  "schema_version": 1,
  "problem_number": "OWR-15436-004",
  "title": "Multistationarity of the Altan-Bonnet–Germain Kinetic-Proofreading Network: A Partial Answer to a Question of Rendall",
  "author": {
    "name": "Alper Ferudun",
    "affiliation": "Mercury Software GmbH",
    "url": "https://github.com/AlperTheKing"
  },
  "abstract": "In an Oberwolfach abstract of 2017, Rendall remarked that the kinetic-proofreading module of the Altan-Bonnet–Germain model of T-cell activation appears to have deficiency one even under the strongest simplifying assumptions, noted that it is unknown whether it admits multiple steady states, and asked about the asymptotics of solutions of the model. We give a partial answer for the module as published in SBML form (57 species, 158 irreversible reactions, deficiency 34), with mass-action kinetics and independent rate constants. Among 36 natural simplifications, those of deficiency one are exactly the four without CD8, with constant Lck and a single ZAP-70 docking level. For these, every positive stoichiometric class contains exactly one steady state, for all rate constants; the proof combines a flux-balance formula for the bound fraction with injectivity determinants. With two or three docking levels and constant kinases, and for the complete module, there are rate constants with several positive steady states. Exact rational certificates give, for each of these four reduced networks, a class with exactly three steady states, two locally exponentially stable and one unstable, and a class of the complete module with two locally exponentially stable steady states and at least one more. All 36 networks are persistent. The multistationary rate constants are reaction-specific, whereas Altan-Bonnet and Germain use one binding and one unbinding constant for all receptor states. Under that constraint the reduced networks with collapsed Michaelis–Menten steps are monostationary at every docking level; multistationarity of the complete module under this constraint, and global convergence in the deficiency-one case, remain open. This is an unrefereed note.",
  "result_type": "COMPLETE_SCOPED_PROOF",
  "categories": [
    "math.DS",
    "q-bio.MN"
  ],
  "keywords": [
    "chemical reaction network theory",
    "multistationarity",
    "deficiency",
    "kinetic proofreading",
    "T-cell activation",
    "Altan-Bonnet–Germain model",
    "injectivity",
    "persistence",
    "mass-action kinetics",
    "Oberwolfach Reports",
    "OWR-15436-004",
    "math.DS",
    "q-bio.MN",
    "open mathematics"
  ],
  "manuscript_version_date": "2026-09-30",
  "publication_date": "2026-09-30",
  "publication_date_kind": "first public online release",
  "version": "1.0",
  "date_modified": "2026-09-30",
  "presentation_revision_only": false,
  "doi_archived_file_version": "1.0",
  "status": "unrefereed preprint",
  "canonical_url": "https://eulersolve.org/papers/owr-15436-004/",
  "pdf_url": "https://eulersolve.org/papers/owr-15436-004/paper.pdf?v=ae4170c361e7",
  "doi": "10.5281/zenodo.23049794",
  "zenodo_record_url": "https://zenodo.org/records/23049794",
  "license": "https://creativecommons.org/licenses/by/4.0/",
  "scope_caveat": "Partial answer to Rendall's question (OWR 28/2017) for the published Altan-Bonnet–Germain kinetic-proofreading module with mass-action kinetics and independent rate constants: the deficiency-one simplifications are never multistationary, while the two- and three-level reductions and the complete module are multistationary for suitable reaction-specific rate constants (exact rational certificates). The identification of the intended simplification is our interpretation. Open: the complete module with rate constants shared across receptor states, and global convergence in the deficiency-one case. Unrefereed.",
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    "source.zip": {
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  "ai_use_disclosure": "AI-assisted tools supported research, computation, proof development, and manuscript preparation. The author remains responsible for all claims and the final text."
}