Team:Freiburg/Modeling

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Introduction

The dimerization of the extracellular receptor domains is a important necessity for the functionality of our modular receptor system. Presenting the system a stimulus in the form of spatial arranged ligands, the extracellular domains dimerize, thus the corresponding intracellular parts such as the split lactamase halves or split fluorescent proteins complement to measureable output. To analyse the theoretical functionality due to dimerization, first two receptor dimerization models (one T cell receptor model and one general receptor model) are introduced and discussed and then a proper model for the modular receptor system is constructed.


Contents


T cell receptor dimerization model I


T cells are a special type of white blood cells (lymphocytes) and play a central role in cell-mediated immunity. They carry special receptors, so called T cell receptors (TCR) on their membrane. One part of the mechanisms to activate a TCR and thus to activate a T Cell is the binding of a ligand, also called antigen, to the TCR. As research showed, one single ligand-TCR complex does not lead to a T cell response yet, as at least two ligand-TCR complexes and their dimerization seem to be required for proper T cell activation (Schamel, 2006; Bachmann 1999). In our case the ligands are nitro-iodo-phenol (NIP) molecules attached to a DNA-Origami structure at a (variable) distance of ~6nm to each other. These NIPs are recognized and bound by the TCRs. As the distance is small enough for two TCRs to approach very closely when each of them binds a NIP, they can dimerize.

Extracellular signaling

A simplified pathway shows the extracellular sequence of TCR activation. After the NIP binding two complexes come together and form a dimer which then leads to activaton of the TCR and further intracellular signaling and T cell activation.

Figure 1 : Pathway of TCR dimerization


Reaction kinetics


The T cell maintains a pool of TCRs (T) which is dynamic. The continuous de novo expression, random internalization and degradation runs with constant rate S :
Freiburg2008 RKM1exp.png

One NIP molecule (N) binds to a TCR (T) with the reaction rate kon or a TCR-NIP complex (TN) dissociates with the reaction rate koff :
Freiburg2008 RKM1bin.png

Two TCR-NIP (TN) complexes dimerize to a TCR-NIP dimer (TND) with rate kdon ; the dissociation of a TCR-NIP dimer runs with rate kdoff :
Freiburg2008 RKM1dim.png

In order to get active TCRs, the TCR-NIP dimer (TND) has to switch into two active TCRs (A) with rate ka :
Freiburg2008 RKM1act.png

After activation, the TCR is internalised with rate ki and does not take part anymore in the extracellular signaling :
Freiburg2008 RKM1int.png

ODEs derived from the kinetics (Details)

In the following equation T represents the free TCR in the T cell membrane where keff is a combination of kdon, kdoff and ka. kI is kon/koff :
Freiburg2008 ODEM1fre.png

The equation for the active TCR A is shown in the following:
Freiburg2008 ODEM1act.png


TCR activity for a set of parameters

The two ODEs above of this first basic model for a set of parameters are solved numerically. They reveal the timecourse of the TCR activity aswell as the one of the unbound TCR.

Freiburg2008 basic1.png
chosen parameters:

s = 0.1;      % turnover rate
kon = 1;      % TCR-NIP binding rate
koff = 0.2;   % TCR-NIP dissociating rate
kdon = 1;     % TCR-NIP dimerization rate
kdoff = 0.2;  % dimer dissociating rate 
ka = 1;       % activation rate
ki = 0.8;     % internalization rate
N = 0.2;      % NIP amount

initial integration conditions:


T0 = 1    % free TCR density
TA0 = 0;  % active TCR density



Extensions: Ultrasensivity and biphasic kinetics<h1>

Ultrasensitivity<h3> The kinetics of the TCR activation can be generalised by substituting the second order kinetic of the ligand N and the receptor T by a parameter h, which then represents the kinetical order of the system. File:Freiburg2008 M1Sen.png Now the sensitivity of the model to ligand and receptor is altered aswell. It increases when the kinetical order increases. For a high kinetic order , small changes in ligand N or receptor T cause big changes in the TCR activation (A). Generally h can be described as: Freiburg2008 M1Senh.png This logarithmic sensitivity means the increasing the concentration of N with 10% will lead to an increase in the rate of TCR activation of 10000% for the 4th order kinetic (h=4).

Freiburg08 FT3.png