The Genesis of Helper T cells and the Self-Nonself Model of Immune Regulation
Calculating the parameters and boundary conditions for the antigen independent pathway for the origin of effector T-helpers
Link to Math details.
We use the symbol "i" to describe the initial state of cells when they are incapable of expressing effector activity upon antigen binding. An i-state cell is capable of responding to antigen, but the response is not an expression of effector function. Antigen drives i-state cells to an anticipatory state (aTh), which in the absence of further signals collapses into cell death. Under appropriate conditions a-state cells can become functional effectors, or e-state cells. The initial decision to activate or inactivate is made when i-state cells fail to progress to the a-state. In the unique case of iTh cells after a sufficient lag, they become eTh. Of course the usual class of antigens unavailable to drive iTh to aTh are antigens that are not self. In contrast, persistent self antigens invariably drive iTh to aTh, and this leads to cell death because there are no eTh that can give permission for aTh to become eTh.
We propose an antigen-independent pathway in which there is an antigen-independent differentiation of iTh to eTh, meaning that prior and persistent self antigen would block this pathway. A kinetic formulation is modeled here.
There is a steady state N = (k1xG) of newly arriving iTh per unit time from the thymus. These cells are a mixture of anti-Self (S) and anti-Nonself (NS) such that SI is the proportion that are anti-S and (1.0 - SI) is the proportion anti-NS. There is, therefore, a steady state production of iTh anti-S, Is in number, that is equal to SIxN and of iTh anti-NS, Ins in number, that is equal to (1.0 - SI)xN. We consider here the steady state levels of each component in the absence of nonself and in the presence of self.
The model is that iTh cells are generated at a steady state. Those that are iTh anti-S interact with Self and are blocked (eventually die through k6 pathway) from entering the antigen-independent pathway to effectors. If the time to encounter Self is short compared to the time to become an effector then a S-NS discrimination at the level of primer eTh can be achieved in the absence of Nonself. The various pathways and their rate constants, k, are diagrammed.
The rate equations were derived and summed. This computer program allows the user to vary the rate constants (k), the size of the Protecton (T), the number of APCs per peptide, the repertoire size (R), and the probability that an iTh cell will be anti-S is SI. The rate constant, k per unit time, is related to the half life of the cell, k = (ln2) / (t1/2).
The output of the calculation is the steady state level of Is, As, Es, Ins, Ens, and N = K1xG in the absence of Nonself. The decision as to whether the choice of parameters are acceptable depends on the boundary conditions chosen to permit an effective response to Nonself and a negligible response to Self.
A set of default input values are provided to illustrate what appears
to be an acceptable output. The choice of SI, as well as the basis
for choosing boundary conditions is discussed in: