# Multiple steady states and the form of response functions to antigen in a model for the initiation of T cell activation

## TL;DR

This paper analyzes a mathematical model of T cell activation, revealing multiple steady states, oscillatory behavior, and non-monotonic response functions, which enhance understanding of immune response regulation.

## Contribution

It provides a rigorous analysis of the model's qualitative behavior, including the existence of multiple steady states and non-monotonic activation responses, confirming and extending previous simulation results.

## Key findings

- Multiple positive steady states exist under certain parameters.
- The model can exhibit damped oscillations.
- Activation degree is a non-monotone function of the ligand dissociation constant.

## Abstract

The aim of this paper is to study the qualitative behaviour predicted by a mathematical model for the initial stage of T cell activation. The state variables in the model are the concentrations of phosphorylation states of the T cell receptor complex and the phosphatase SHP-1 in the cell. It is shown that these quantities cannot approach zero and that the model possesses more than one positive steady state for certain values of the parameters. It can also exhibit damped oscillations. It is proved that the chemical concentration which represents the degree of activation of the cell, that of the maximally phosphorylated form of the T cell receptor complex, is in general a non-monotone function of the activating signal. In particular there are cases where there is a value of the dissociation constant of the ligand from the receptor which produces an optimal activation of the T cell. In this way the results of certain simulations in the literature have been confirmed rigorously and some important features which had not previously been seen have been discovered.