Abstract
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.
Highlights
Phosphorylation and dephosphorylation, the processes in which proteins are modified by the addition or removal of phosphate groups, play an important role in biology
We show that in the context of this model multiple steady states can only occur when the phosphorylation increases the activity of the enzyme
It is shown that in a case where phosphorylation decreases the activity of the enzyme multiple steady states can occur but this requires a more complicated model with an external kinase which is operating well away from the Michaelis–Menten limit
Summary
Phosphorylation and dephosphorylation, the processes in which proteins are modified by the addition or removal of phosphate groups, play an important role in biology. The activity of an enzyme is influenced by its phosphorylation state, and these processes provide a way of switching enzymes on and off quickly. The enzymes which catalyse phosphorylation and dephosphorylation are called kinases and phosphatases, respectively. The phosphorylation of a protein X is usually catalysed by another protein Y. It may be catalysed by X itself, a process called autophosphorylation.
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