Multistationarity in Kinetic-Proofreading Networks

Definitions and mathematical statement

The supplied record concerns the kinetic-proofreading module of the Altan--Bonnet--Germain T-cell activation model. It reports that this module has deficiency one under its strongest simplifying assumptions, but does not specify the module's reactions, differential equations, rate laws, or simplifying assumptions.

For a finite reaction network, let $n_c$ denote the number of complexes, $\ell$ the number of linkage classes (connected components of the underlying undirected reaction graph), and $s$ the dimension of the span of its reaction vectors. Its deficiency is \[ \delta=n_c-\ell-s. \] The supplied qualification is that $\delta=1$ for the intended simplified module; the network needed to instantiate this formula is absent.

To describe the requested properties, write an autonomous concentration model schematically as \[ \dot{x}=f_\theta(x),\qquad x\in{\mathbb{R}}_{\geq 0}^{m}, \] where $m$, the vector field $f_\theta$, and the admissible parameter set $\Theta$ must be supplied by the intended model. This notation does not specify a rate law or assume that every nonnegative concentration vector is admissible. A positive steady state at fixed parameters $\theta\in\Theta$ is an admissible vector $x^*\in{\mathbb{R}}_{>0}^{m}$ satisfying $f_\theta(x^*)=0$. Two distinct positive steady states at the same parameters satisfy \[ x^*,y^*\in{\mathbb{R}}_{>0}^{m},\qquad x^*\ne y^*,\qquad f_\theta(x^*)=f_\theta(y^*)=0. \] If multistationarity is intended within one stoichiometric compatibility class, one must additionally require $y^*-x^*\in S$, where $S$ is the span of the reaction vectors; a compatibility class through $x_0$ is $(x_0+S)\cap{\mathbb{R}}_{\geq0}^{m}$.

Source problem. For the intended kinetic-proofreading module, under the strongest simplifying assumptions for which the source reports deficiency one, determine whether the dynamical system admits multiple positive steady states, and determine the asymptotic behavior of its solutions.

Unresolved qualifications. The record does not determine the vector field, the admissible parameter values, the conserved totals, or the admissible initial conditions. It also does not specify whether multiplicity is sought for some parameters or characterized across all parameters, or whether the steady states must belong to the same compatibility class. The asymptotic question has no specified initial-condition quantifier, boundary convention, or target conclusion. Discussion of $t\to\infty$ requires the relevant solutions to exist for all forward time, which the supplied record does not establish. Consequently, the displayed definitions clarify the requested properties, but the supplied evidence does not determine a complete model-specific mathematical statement.

Short English statement

For the Altan--Bonnet--Germain kinetic-proofreading module, reported to have deficiency one under the strongest simplifying assumptions, ask whether multiple positive steady states can occur and describe the long-time behavior of solutions. The equations, assumptions, parameter quantifiers, compatibility-class requirement, and initial-condition scope are missing, so a precise model-specific formulation requires further source information.

Sources

• Local UnsolvedMath registry, permanent ID $30003518$, title: Multistationarity in Kinetic-Proofreading Networks; exact source problem code: catalogue entry. The supplied collection is Oberwolfach Reports Open Problems. No individual proposer is supplied; the proposed year is recorded as 2017.
• Supplied report citation: Reaction Networks and Population Dynamics (2018), Oberwolfach Reports, pp. 1747--1804, DOI 10.4171/owr/2017/28. Supplied source URL and literature-source link, labeled "original OWR report": https://doi.org/10.4171/owr/2017/28. The record describes Oberwolfach Reports as documenting mathematical workshops and their open-problem discussions. The citation is supplied provenance and has not been independently checked here.
• Literature assessment dated 2026-08-21: the supplied curation labels the problem open and states that the report leaves multiplicity of steady states for the Altan--Bonnet--Germain proofreading module unknown. It further states that later results located concern simpler or modified proofreading networks and do not settle this exact model. This is supplied dated curation, not an independently verified literature assessment; its time-sensitive status and cited claims require independent verification. No separate references for those later results are supplied.