Multistationarity of the Altan-Bonnet–Germain Kinetic-Proofreading Network: A Partial Answer to a Question of Rendall

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.