## Mathematical models for T cell activation

The proper functioning of our immune system is heavily dependent on the ability of T cells to identify foreign substances and take appropriate action. For this they need to be able to distinguish the foreign substances (non-self) from those coming from substances belonging to the host (self). In the first case the T cell should be activated, in the second not. The process of activation is very complicated and takes days. On the other hand it seems that an important part of the distinction between self and non-self only takes a few seconds. A T cell must scan the surface of huge numbers of dendritic cells for the presence of the antigen it is specific for and it can only spare very little time for each one. Within that time the cell must register that there is something relevant there and be induced to stay longer, instead of continuing with its search.

A mathematical model for the initial stages of T cell activation (the first few minutes) was formulated and studied by Altan-Bonnet and Germain (PloS Biol. 3(11), e356). They were able to use it successfully to make experimental predictions, which they could then confirm. The predictions were made with the help of numerical simulations. From the point of view of the mathematician a disadvantage of this model is its great complexity. It is a system of more than 250 ordinary differential equations with numerous parameters. It is difficult to even write the definition of the model on paper or to describe it completely in words. It is clear that such a system is difficult to study analytically. Later Francois et. el. (PNAS 110, E888) introduced a radically simplified model for the same biological situation which seemed to show a comparable degree of effectiveness to the original model in fitting the experimental data. In fact the simplicity of the model even led to some new successful experimental predictions. (Altan-Bonnet was among the authors of the second paper.) This is the kind of situation I enjoy, where a relatively simple mathematical model suffices for interesting biological applications.

In the paper of Francois et. al. they not only do simulations but also carry out interesting analytical calculations for their model. On the other hand they do not follow the direction of attempting to use these calculations to formulate and prove mathematical theorems about the solutions of the model. Together with Eduardo Sontag we have now written a paper where we obtain some rigorous results about the solutions of this system. In the original paper the only situation considered is that where the system has a unique steady state and any other solution converges to that steady state at late times. We have proved that there are parameters for which there exist three steady states. A numerical study of these indicates that two of them are stable. A parameter in the system is the number $N$ of phosphorylation sites on the T cell receptor complex which are included in the model. The results just mentioned on steady states were obtained for $N=3$.

An object of key importance is the response function. The variable which measures the degree of activation of the T cell in this model is the concentration $C_N$ of the maximally phosphorylated state of the T cell receptor. The response function describes how $C_N$ depends on the important input variables of the system. These are the concentration $L$ of the ligand and the constant $\nu$ describing the rate at which the ligand unbinds from the T cell receptor. A widespread idea (the lifetime dogma) is that the quantity $\nu^{-1}$, the dissociation time, determines how strongly an antigen signals to a T cell. It might naively be thought that the response should be an increasing function of $L$ (the more antigen present the stronger the stimulation) and a decreasing function of $\nu$ (the longer the binding the stronger the stimulation). However both theoretical and experimental results lead to the conclusion that this is not always the case.

We proved analytically that for certain values of the parameters $C_N$ is a decreasing function of $L$ and an increasing function of $\nu$. Since these rigorous results give rather poor information on the concrete values of the parameters leading to this behaviour and on the global form of the function we complemented this analytical work by simulations. These show how $C_N$ can have a maximum as a function of $\nu$ within this model and that as a function of $L$ it can have the following form in a log-log plot. For $L$ small the graph is a straight line of slope one. As $L$ increases it switches to being a straight line of slope $1-N/2$ and for still larger values it once again becomes a line of slope one, shifted with respect to the original one. Finally the curve levels out as it must do, since the function is bounded. The proofs do not make heavy use of general theorems and are in general based on doing certain estimates by hand.

All of these results were of the general form ‘there exist parameter values for the system such that $X$ happens’. Of course this is just a first step. In the future we would like to understand better to what extent biologically motivated restrictions on the parameters lead to restrictions on the dynamical behaviour.