The role of T cells in the immune system is to recognize foreign substances and then take appropriate action. In order for this to happen information must be propagated from the surface of the cell, where the T cell receptor is, to the nucleus in order to initiate DNA transcription. The last step in this process is the binding of a suitable combination of transcription factors to the DNA. NFAT (nuclear factor of activated T cells) is one of these transcription factors. The fact that the associated signalling pathway plays an important role in the activation of T cells explains the name. In fact this substance (or class of substances – there are actually five different ones) are important for signalling in many cells of the immune system. I already mentioned the NFAT signalling pathway, its connection to calcium and a paper on the subject by Salazar and Höfer in a previous post. Now I have written a paper where I look into mathematical aspects of the activation of NFAT by means of dephosphorylation and the role of calcium in this process. Salazar and Höfer introduced a high-dimensional dynamical system and computed stationary solutions in a slightly simplified version of that system. I now proved, using chemical reaction network theory, that for each choice of the many parameters in the system there exists exactly one stationary solution of the full system for each value of the total amount of NFAT in the cell. Every solution with that total amount of NFAT converges to the stationary solution at late times. Furthermore, this solution is well approximated by the explicit solution of the simplified system under a biologically motivated assumption that certain parameters are small enough. The main tool in the proof is the Deficiency Zero Theorem.
The result just mentioned concerns the model for the dephosphorylation process with the stimulation of the cell expressed through fixed choices of the parameters. In reality the stimulation is communicated through the calcium concentration in the cytosol. This means that the parameters in the model for desphosphorylation should be replaced by time-dependent functions which themselves are the result of a dynamical process. The situation is described by Salazar and Höfer with the help of a two-dimensional dynamical system closely related to one introduced by Somogyi and Stucki to describe calcium oscillations in liver cells. In the paper I did some analysis of the model, giving criteria for the stability of the unique stationary solution for given parameter values and the existence of periodic solutions. Hopf bifurcations play a role. The model is closely related to the Brusselator and techniques of proof can be imported from that case. In particular it is important to identify explicit invariant regions for the flow. When a solution of the model for the calcium concentration is such that it tends to a constant at late times then it can be shown that the resulting configuration of the phosphorylation states of NFAT also converges to the situation with constant coefficients previously analysed. When a solution converges to a periodic solution at late times it is not clear what can be said.