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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

Before the first step the receptors are unbound and diffuse in the cell´s membrane. The DNA-Origami binds to a receptor in the way that only one NIP molecule is connecting to one receptor. In the second step the second NIP molecule binds to the second receptor and both receptors dimerize.

Reaction kinetics

The continuous production of the cell´s receptor R with rate s is:

One receptor R binds to one of two NIP molecules NN of the DNA-Origami with rate k_on or a receptor-NIP complex RNN dissociates with k_off:

The second receptor R binds to the second NIP molecule of the DNA-Origami and both receptors dimerize with rate k_don or the dimer RNNR dissociates in one receptor R and one receptor-NIP complex RNN with rate k_doff:

The two receptors get activated with rate k_a:

After activation the receptor is internalized with the k_i:

ODE derivation

The ODEs can be derived from the reaction kinetics again and are:

Extracellular receptor activity

For a set of parameters the ODEs were solved and reveal amongst the other quantities the receptor densities in the time course:

Figure 6: Receptor densities in time (model 2)

chosen parameters: s = 0.1;  %turnover rate Kon = 3;  %binding rate Koff = 0.1;  %dissociating rate Kdon = 1;  %dimerization rate Kdoff = 0.2;  %dimer dissociating rate Ka = 1;  %activation rate Ki = 0.8;  %internalisation rate initial integration conditions: R0 = 1;  %receptor density NN0 = 0.5;  %NIP Amount RNN0 = 0;  %receptor-NIP monomer RNNR0 = 0;  %receptor-NIP dimer A0 = 0;  %active receptor

The receptor activity rises after NIP addition, has a maximum and decreases due to the internalization of the active receptor. The dimer density increases fast after addition of NIP and decreases as dimers fade into active receptors.

Discussion of model I and II

For the comparison, the corrected model I is taken and model II. The receptor NIP binding mechanism differs in the two models. Model I assumes the production of two independent receptor-NIP complexes which can dimerize after formation. In model II, dimerization happens after one of two receptors connects to the bivalent ligand consisting of two NIP molecules. Setting several conditions unravels the consequences of this differences for the active receptor, the dimer and the receptor recovery. Different binding rates, dimerization rates and NIP amounts are used in the solutions of the model I ODEs and model II ODEs.

receptor densities dependent on binding rate k_on

The two model ODEs were solved for the mentioned parameters below whereby the binding rate k_on was variated. So the course in time for the free receptor, the monomeric receptor-NIP complex, the dimer and the active receptor is plotted:

s = 0.1;        % turnover rate
kon : variated  % receptor-NIP binding rate
koff = 0.1;     % receptor-NIP dissociating rate
kdon = 1;       % receptor-NIP dimerization rate
kdoff = 0.2;    % dimer dissociating rate 
ka = 1;         % activation rate
ki = 0.1;       % internalization rate

free receptor : 1;
receptor-NIP monomer, dimer : 0;
Ligand/NIP density:
 Model 1: 0.5;
 Model 2: 0.25;
 As the ligand in model 2 is bivalent, the amount 
 of initial NIP molecules is equal in both models.

k_on = 0.2 :

Model I	Model II
Figure 7.1: Receptor densities of model 1 for binding rate = 0.2	Figure 7.2: Receptor densities of model 2 for binding rate = 0.2

k_on = 3 :

Model I	Model II
File:Freiburg2008 Mlhkdon.png Figure 8.1: Receptor densities of model 1 for binding rate = 3	Figure 8.2: Receptor densities of model 2 for binding rate = 3

receptor activity for different k_on :

The production of receptor-NIP complexes is lower in model 2 than in model 1. Increasing k_on leads to an increase in the formation of receptor NIP-complexes thus to a higher dimerization and receptor activation in both models. In model 1 receptor-NIP densities are as high as the active receptor densities and in model 2 receptor-NIP densities are much lower than active receptor densities. Active receptor densities react in both models similar to k_on variations for the used parameters.

receptor densities dependent on dimerization rate k_don

Applying different k_don leads to the following result:

s = 0.1;        % turnover rate
kon = 3;        % receptor-NIP binding rate
koff = 0.1;     % receptor-NIP dissociating rate 
kdon : variated % dimerization rate
kdoff = 0.2;    % dimer dissociating rate 
ka = 1;         % activation rate
ki = 0.1;       % internalization rate

free receptor : 1;
receptor-NIP monomer, dimer : 0;
Ligand/NIP density:
 Model 1: 0.5;
 Model 2: 0.25;
 As the ligand in model 2 is bivalent, the amount 
 of initial NIP molecules is equal in both models.

k_don = 0.3 :

Model I	Model II
Figure 10.1: Receptor densities of model 1 dimerization = 0.3	Figure 10.2: Receptor densities of model 2 for dimerization rate = 0.3

k_don = 3 :

Model I	Model II
Figure 11.1: Receptor densities of model 1 for dimerization rate = 3	Figure 11.2: Receptor densities of model 2 for dimerization rate = 3

receptor activity for different k_don :

Model I	Model II
Figure 12.1: Receptor activity of model 1 for different dimerization rates	Figure 12.2: Receptor activity of model 2 for different dimerization rates

A low k_don gives in both models a high receptor-NIP monomer density as few of this complexes are recruited to build dimers and remain receptor-NIP monomers. This density decreases for high k_don due to a higher more dimerization and then receptor activity. k_don variations give similar changes in receptor activity in both models.

receptor densities dependent on NIP Amount

The use of different ligand inputs in form of NIP amount is maybe the most important influence on the active receptor density. As the ligand is the stimulus in order to activate our system, it´s amount will determine strongly the activity of the receptor. Aswell a higher internalization rate k_i is used.

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

free receptor : 1;
receptor-NIP monomer, dimer : 0;
Ligand/NIP density:
 Model 1: variated;
 Model 2: variated;
 Variating in a way that the parameter in model 1 is always 
 double as high as the parameter in model 2 gives equal amounts of
 NIP molecules as stimulus, because the ligand in model 2 is bivalent 

NIP = 0.1

Model I	Model II
Figure 13.1: Receptor activity of model 1 for a low amount of NIP	Figure 13.2: Receptor activity of model 2 for a low amount of NIP

NIP = 3

Model I	Model II
Figure 14.1: Receptor activity of model 1 for a high amount of NIP	Figure 14.2: Receptor activity of model 2 for a high amount of NIP

receptor activity for different amounts of NIP:

Model I	Model II
Figure 15.1: Receptor activity of model 1 for different amounts of NIP (shown in % of initial receptor amount	Figure 15.2: Receptor activity of model 2 for different amounts of NIP (shown in % of initial receptor amount

As k_i is decreased the receptor activity is decreased aswell due to a fast internalization of active receptors. For a little stimulus the active receptor activity is almost zero. Increasing the amount of NIP leads in both models to an increase of the active receptor density. It saturates for increasing amounts of NIP in model 1. In model2 the active receptor density increases for increasing NIP amounts, but when the NIP amount is too high, it starts to decrease as too much receptor-NIP complexes are formed leaving few recepors for dimerization. Aswell there can be observed a late activation for high NIP amounts in model 2.