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

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 \(C_0,\dots,C_N\) 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 \(N\le4\), where the source states the bound five for \(N=4\). For \(5\le N\le650\) 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.