Example: Molecular Switch

This is a model of T cell activation by antigen presenting cells such as Dendritic cells which has been developed and studied by Cliburn Chan in his 2002 PhD thesis[1]. It involves three complexes known to be important in early signal transduction events involved in T cell activation: (a) Lck a src family protein tyrosine kinase, (b) ZAP-70 a syk family protein kinase and (c) SHP-1 a protein tyrosine phosphatase.

In the model each species is assumed to exist in two states — a basal state and an activated state indicated by an asterisk as in Lck*. The interactions amongst these proteins is illustrated in the diagram below.

As can be seen, activated Lck* phosphorylates and activates ZAP-70 as well as the phosphatase and may autophosphorylate itself. Activated ZAP-70* can activate Lck while activated phosphatase may inactivate Lck* and ZAP-70*.

The basal and feedback reactions are

Basal:

Feedback:

The corresponding differential system is:

where

This system is completed by adding a phenomenological equation

which couples the total concentration of Lck i.e. LckT to the duration time of binding of a T cell receptor (TCR) by a peptide-MHC class II complex (p-MHC II). In this equation binding begins at t=t₀ and endures for a time of t=t₁ - t₀. The first term on the righthand side is the binding term where r_o denotes the amplitude of the binding effect and H(t) is the Heaviside step function. The effect of this equation is to recruit more Lck into the region as long as there is binding and to reduce the levels of Lck concentration to its prebinding value (pi₀Lck(0)) following loss of contact between the TCR and p-MHC II complex.

Chan has shown the model exhibits hysteresis over a large parameter range. When the Lck concentration available to the TCR reaches a threshold, there is a sudden dramatic state change in which most of the Lck and ZAP-70 are activated. Below this threshold, almost all Lck and ZAP-70 are present in an inactive state.

The stochastic analysis for low copies has been analyzed by SigTran and presented at ASCB (American Society for Cell Biology).

[1] Chan, Cliburn (2002), Modeling of T Cell Activation, PhD thesis, University College London, London, UK.