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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.

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 :

One NIP molecule (N) binds to a TCR (T) with the reaction rate k_on or a TCR-NIP complex (TN) dissociates with the reaction rate k_off :

Two TCR-NIP (TN) complexes dimerize to a TCR-NIP dimer (TND) with rate k_don ; the dissociation of a TCR-NIP dimer runs with rate k_doff :

In order to get active TCRs, the TCR-NIP dimer (TND) has to switch into two active TCRs (A) with rate k_a :

After activation, the TCR is internalised with rate k_i and does not take part anymore in the extracellular signaling :

ODEs derived from the kinetics (Details)

In the following equation T represents the free TCR in the T cell membrane where k_eff is a combination of k_don, k_doff and k_a. k_I is k_on/k_off :


The equation for the active TCR A is shown in the following:

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.

Figure 2: TCR densities in time	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

Ultrasensitivity

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.


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:
This logarithmic sensitivity means that the increasing of 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).

Biphasic kinetics and parameter analysis

Not all TCRs on a T cell membrane can be recruited to NIP binding as a cell´s membrane contains several transmembrane proteins whose size can avoid a TCR-NIP formation when they surround a TCR and make the approaching of the NIP to the binding side of the TCR impossible. Considerating this spatial barriers leads to the idea of a TCR which can switch between two states, one binding state and one non-binding state. Hence a introduction of two different pools of TCRs into the model is appropriate. If a TCR is not available for a NIP molecule, thus it is in the non-binding state,it belongs to the so called spare pool, whereby TCRs belonging to the so called interface pool are in the binding state and can be accessed by the NIP molecule. Moreover the spare pool is in dynamical exchange with the interface pool, so non-binding TCR can become binding TCRs. This exchange is regulated through the parameter λ, a ratio between the spare and the interface pool and φ, the exchange rate constant. S is the spare pool TCR density and T the TCR density of the interface pool. A represents the active TCR density. So the full model I equations are:
Spare pool:
Interface pool:
Active TCR:

TCR activity dependent on exchange rate φ and ratio λ :
(Extract from parameter analysis)
The dependency of the activity on λ and φ is shown in the following:

Figure 3: Active TCR density in time for different exchanges

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 = 1;  % internalization rate N = 0.2;  % NIP amount h = 2;  % reaction order initial integration conditions: S0 = 1;  % spare TCR density T0 = 1;  % interface TCR density TA0 = 0;  % active TCR density

The x-axis represents the timecourse of the activity, the y-axis represents both parameters φ ( = y) and λ ( = 2 - y). So each black line in the plot is a timecourse of the TCR activity for a different φ and λ. The z-axis is the response intensity. With increasing exchange between the interface and spare pool, more TCRs switch to the binding state, hence more TCRs can bind NIP. As a consequence the active TCR density is higher then for a low exchange.

Correctional terms

Regarding the reaction kinetics and considering the ODEs as a model for TCR dimerization (Bachmann, 1999) led to the realization of an error in the mentioned publication. The ODEs evolved from the reaction kinetics are not :

but :

Furthermore can be derived for the NIP density and the free TCR:

The NIP density is not a static value anymore but time dependent also. On that account the last five equations are the corrected model equations. Solving this five equations numerically for a set of parameters gives a similar timecourse of the active TCR density but the response strength is much lower than in the original version, although the initial NIP amount is five times higher:

Figure 4: Timecourse of NIP, free TCR and active TCR for a set of parameters

chosen parameters: s = 0.1;  % turnover rate kon = 3;  % TCR-NIP binding rate koff = 0.1;  % 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 initial integration conditions: N0 = 1;  % NIP amount T0 = 1  % free TCR density TN0 = 0  % TCR-NIP monomer density TND0 = 0  % TCR-NIP dimer density TA0 = 0;  % active TCR density

After NIP addition the TCR activity rises to a maximum and decreases as active receptors are internalized and the NIP amount is used up. The free receptor density recovers due to permanent expression of new receptors which are build into the membrane.

Receptor dimerization model II

A receptor dimerization can proceed in a different way aswell, especially when two NIP molecules are linked to a structure like our DNA-Origami:

Figure 5: Pathway of receptor dimerization due to bivalent ligand