{
  "schema_version": 1,
  "problem_number": "OWR-15993-009",
  "title": "Five Steady States in the Agonist-Only Rendall–Sontag Model for the Initiation of T Cell Activation: Certified Examples with Chain Lengths from 651 to 4000",
  "author": {
    "name": "Alper Ferudun",
    "affiliation": "Mercury Software GmbH",
    "url": "https://github.com/AlperTheKing"
  },
  "abstract": "Rendall asked, in an abstract on joint work with Sontag (Oberwolfach Report 8/2018), whether the ODE model for the initiation of T cell activation of François, Voisinne, Siggia, Altan-Bonnet and Vergassola has parameter values with more than three steady states. In the form studied by Rendall and Sontag the model is a kinetic proofreading chain C_0, …, C_N of receptor complexes with a negative feedback through the phosphatase SHP-1; they proved that its agonist-only case has exactly one positive steady state for N = 1, 2 and, for N = 3, at most three, and three for suitable parameters. We answer the question for the agonist-only system with the chain length N treated as a free parameter, as in the Oberwolfach abstracts: for N = 1000 and explicit positive rational values of all rate constants and totals the system has exactly five positive steady states in one stoichiometric class, all of them non-degenerate. The proof reduces the steady states to the roots, in an explicit interval, of an explicit polynomial of degree N + 1, whose sign is evaluated at six rational points in exact integer arithmetic; the exact count is computer-assisted as well (exact arithmetic; all programs and outputs are in the accompanying package). Fourteen further certified parameter points have N = 651, 652, 653, 655, 660, 680, 690, 700, 2000 and 4000. At each of the fifteen points the second and the fourth steady state are unstable; for N = 1000 the other three are hyperbolic sinks and these two are saddles (computer-assisted, interval arithmetic). In the other direction we prove that there are at most N positive steady states for odd N and at most N − 1 for even N, hence at most three for N ≤ 4, where the source states the bound five for N = 4. For 5 ≤ N ≤ 650 nothing is proved: numerical searches found no example and suggest that 651 is the smallest possible N (tested, not proved). The values of N that are biologically motivated in the sources, about 20 or less, are therefore not covered; the parameter windows are extremely thin; and no biological conclusion is drawn. The full model with antagonist and the parameter values of the biological literature are not considered. The method is elementary; a limited literature search found no earlier answer, and no priority is claimed. This is an unrefereed note.",
  "result_type": "COMPLETE_PROOF",
  "categories": [
    "math.DS",
    "q-bio.MN",
    "math.CA"
  ],
  "keywords": [
    "T cell activation",
    "kinetic proofreading",
    "SHP-1",
    "negative feedback",
    "multistationarity",
    "number of steady states",
    "agonist-only model of Rendall and Sontag",
    "chain length as a parameter",
    "computer-assisted proof",
    "exact integer arithmetic",
    "Descartes' rule of signs",
    "interval arithmetic",
    "Oberwolfach Reports open problems",
    "UnsolvedMath",
    "OWR-15993-009",
    "math.DS",
    "q-bio.MN",
    "math.CA",
    "open mathematics"
  ],
  "manuscript_version_date": "2026-10-11",
  "publication_date": "2026-10-11",
  "publication_date_kind": "first public online release",
  "version": "1.0",
  "date_modified": "2026-10-11",
  "presentation_revision_only": false,
  "doi_archived_file_version": "1.0",
  "status": "unrefereed preprint",
  "canonical_url": "https://eulersolve.org/papers/owr-15993-009/",
  "pdf_url": "https://eulersolve.org/papers/owr-15993-009/paper.pdf?v=907fc61c902b",
  "doi": "10.5281/zenodo.23298465",
  "zenodo_record_url": "https://zenodo.org/records/23298465",
  "license": "https://creativecommons.org/licenses/by/4.0/",
  "scope_caveat": "Answers the question of A. D. Rendall (joint work with E. D. Sontag; Oberwolfach Report 8/2018, p. 488) for the agonist-only system with the chain length N treated as a free parameter: certified examples with exactly five positive steady states for chain lengths from 651 to 4000. For 5 ≤ N ≤ 650, which includes every value of N that the sources regard as biologically motivated, nothing is proved; the parameter windows are extremely thin, and no biological conclusion is drawn. The full model with antagonist is not considered. The main theorems are computer-assisted (exact integer arithmetic). Not read: the supplement of François et al. (2013), which contains the equations of the model (they are taken from Rendall and Sontag), and the further sources listed in the paper.",
  "files": {
    "paper.pdf": {
      "sha256": "907fc61c902babc0adc382c39dfef0571bca477480ff08f39a6a2bf4a2b50a52"
    },
    "source.zip": {
      "sha256": "8df1d90938754e9fcd2e4b47b10a0a4909248a9252b254987cbea0eb2e2f30a4"
    },
    "verification_report.md": {
      "sha256": "7712b7150e427885ecde9b170875b303aa9d1be920d64c4bd4c4e1349b9148ed"
    }
  },
  "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."
}