  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.
