  Lck (lymphocyte-specific protein tyrosine kinase) is an enzyme which plays a
number of important roles in the function of immune cells. It belongs to the
Src family of kinases which are known to undergo autophosphorylation. It turns
out that this leads to a remarkable variety of dynamical behaviour which can
occur during their activation. We prove that in the presence of
autophosphorylation one phenomenon, bistability, already occurs in a
mathematical model for a protein with a single phosphorylation site. We further
show that a certain model of Lck exhibits oscillations. Finally we discuss the
relations of these results to models in the literature which involve Lck and
describe specific biological processes, such as the early stages of T cell
activation and the stimulation of T cell responses resulting from the
suppression of PD-1 signalling which is important in immune checkpoint therapy
for cancer.
