Team:Freiburg/Modeling

{{Freiburg2008_Main|
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<font face="Arial Rounded MT Bold" style="color:#010369">_modeling</font></div>
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__NOTOC__
<h1>Introduction</h1>
Dimerization of the extracellular receptor domains is an important necessity for the functionality of our Modular Synthetic Receptor System. Presenting the system a stimulus in the form of spatial arranged ligands results in dimerization of the extracellular domains and thus the corresponding intracellular parts such as the split lactamase halves or split fluorescent proteins complement to a measurable output. To analyze the 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 [[Team:Freiburg/Modeling#Modular Synthetic Receptor System model|model for the Modular Synthetic Receptor System]] is constructed.
<br><br>
<font style="color:#333366; font-style:italic;">
As given by the diversity of our transmembrane fusion proteins, some of the constructs can bind fluorescein as ligand as well. As the model for the Modular Synthetic Receptor System is applicable for all our constructs, the ligand usage is extended with fluorescein. So either NIP or fluorescein can function as ligand, dependent on the choice of constructs for implementation.
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__TOC__
<br>

<h1>T cell receptor dimerization model I </h1>
[[media:Freiburg2008_M1_Basic.m|''matlab m-file'']]<br>
<br>
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|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.
<br><br>

<h3>Extracellular signaling </h3>

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 activation of the TCR and further intracellular signaling and T cell activation.
<table>
<tr>
<td>
[[Image:Freiburg2008_M1ePath.png|thumb|500px|'''Figure 1: '''Pathway of TCR dimerization]]
</td>
</tr>
</table>
<br>
<h3>Reaction kinetics </h3>
<br>
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''' :
<br>
[[Image:Freiburg2008_RKM1exp.png|200px]]
<br><br>

One NIP molecule ('''N''') binds to a TCR ('''T''') with the reaction rate '''k'''<sub>on</sub> or a TCR-NIP complex ('''TN''') dissociates with the reaction rate '''k'''<sub>off</sub> :
<br>
[[Image:Freiburg2008_RKM1bin.png|250px]]
<br><br>

Two TCR-NIP ('''TN''') complexes dimerize to a TCR-NIP dimer ('''TND''') with rate '''k'''<sub>don</sub> ; the dissociation of a TCR-NIP dimer runs with rate '''k'''<sub>doff</sub> :
<br>
[[Image:Freiburg2008_RKM1dim.png|295px]]
<br><br>

In order to get active TCRs, the TCR-NIP dimer ('''TND''') has to switch into two active TCRs ('''A''') with rate '''k'''<sub>a</sub> :
<br>
[[Image:Freiburg2008_RKM1act.png|210px]]
<br><br>

After activation, the TCR is internalized with rate '''k'''<sub>i</sub> and does not take part anymore in the extracellular signaling :
<br>
[[Image:Freiburg2008_RKM1int.png|150px]]
<br><br>
<h3>ODEs derived from the kinetics [[Team:Freiburg/Modeling ODE derivation|(Details)]]</h3>
In the following equation '''T''' represents the free TCR in the T cell membrane where '''k'''<sub>eff</sub> is a combination of  '''k'''<sub>don</sub>, '''k'''<sub>doff</sub> and '''k'''<sub>a</sub>. '''k'''<sub>I</sub> is '''k'''<sub>on</sub>/'''k'''<sub>off</sub> :<br>
[[Image:Freiburg2008_ODEM1fre.png|420px]]
<br><br>
The equation for the active TCR '''A''' is shown in the following:<br>
[[Image:Freiburg2008_ODEM1act.png|420px]]
<br><br><br>
<h3>TCR activity for a set of parameters</h3>
The two ODEs above of this first basic model for a set of parameters are solved numerically. They reveal the time course of the TCR activity and the time course of the unbound TCR aswell.
<table>
<tr>
<td>[[image:Freiburg2008_basic1.png|thumb|left|400px|'''Figure 2:''' TCR densities in time]]</td>
<td>chosen parameters:<br>
 <font color="#336600">
 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
 </font>
initial integration conditions:
 <font color="#336600">
 T0 = 1    % free TCR density
 TA0 = 0;  % active TCR density
 </font>
</td>
</tr>
</table>
The activity has a maximum a short time after NIP presentation and decreases as the active TCRs are internalized and as the free membrane TCR amount is decreasing. It is remarkable that the response never decreases to complete zero, even for very high internalization rates and in a long time course.
<br><br>
<h2>Extensions: Ultrasensitivity and biphasic kinetics</h2>
[[media:Freiburg2008_M1_Xten.m|''matlab m-file'']]<br>
<h3>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 kinetic order of the system.<br>
[[Image:Freiburg2008_M1odeT_h2.png|400px]]<br>
[[Image:Freiburg2008_M1odeA_h2.png|400px]]
<br>
Now the sensitivity of the model to ligand and receptor is altered aswell. It increases when the kinetic 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:
[[image:Freiburg2008_M1Senh.png|400px]]
<br>
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).
<br><br>
<h3>Biphasic kinetics and [[Team:Freiburg/Modeling/Parameter analysis|parameter analysis]]</h3>
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 TCRs can become binding TCRs.  This exchange is regulated through the parameter '''&lambda;''',  a ratio between the spare and the interface pool and '''&phi;''', 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:<br>
Spare pool:
[[Image:Freiburg2008_M1odeT_spare.png|380px]]<br>
Interface pool:
[[Image:Freiburg2008_M1odeT_interf.png|500px]]<br>
Active TCR:
[[Image:Freiburg2008_M1odeA_h.png|400px]]<br>

'''TCR activity dependent on exchange rate &phi; and ratio &lambda; :'''<br>
''(Extract from  [[Team:Freiburg/Modeling/Parameter analysis|parameter analysis]])''<br>
The dependency of the activity on &lambda; and &phi; is shown in the following:
<table>
<tr>
<td>
[[Image:Freiburg2008_TA_xRate_Dep.png|thumb|left|400px|'''Figure 3:''' Active TCR density in time for different exchanges]]
</td>
<td>
chosen parameters:<br>
 <font color="#336600">
 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;        % kinetic order
 </font>
initial integration conditions:
 <font color="#336600">
 S0 = 1;    % spare TCR density 
 T0 = 1;    % interface TCR density
 TA0 = 0;   % active TCR density
 </font>
</td>
</tr>
</table>
The x-axis represents the time course of the activity, the y-axis represents both parameters &phi; ( = y) and &lambda; ( = 2 - y). So each black line in the plot is a time course of the TCR activity for a different &phi; and &lambda;. 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. <br>
<h2>Correction terms</h2>
[[media:Freiburg2008_M1_Corrected.m|''matlab m-file'']]<br><br>
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''' :
<br>
[[image:Freiburg2008_M1err.png|left|500px]]
<br><br><br><br><br><br><br><br><br><br><br><br><br><br><br>

'''but :'''<br>

[[image:Freiburg2008_M1corr.png|left|620px]]
<br><br><br><br><br><br><br><br>
<br><br><br><br><br><br><br>

Furthermore can be derived for the NIP density and the free TCR:
<br>
[[image:Freiburg2008_M1corr2.png|left|460px]]

<br><br><br><br><br><br><br><br><br><br><br><br>
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 time course of the active TCR density but the response strength is lower than in [[Team:Freiburg/Modeling#ODEs derived from the kinetics (Details)|the original version]], although the initial NIP amount is five times higher:
<table>
<tr>
<td>
[[image:Freiburg2008_NIP_T_TA.png|thumb|left|400px|'''Figure 4:''' Time course of NIP, free TCR and active TCR for a set of parameters]]
</td>
<td>
chosen parameters:<br>
 <font color="#336600">
 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
 </font>
initial integration conditions:
 <font color="#336600">
 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
 </font>
</td>
</tr>
</table>
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.
<br>
<h1>Receptor dimerization model II</h1>
[[media:Freiburg2008_M_2.m|''matlab m-file'']]<br><br>
A receptor dimerization can proceed in a different way aswell, especially when two ligand molecules are linked to a structure like our DNA-Origami:
<table>
<tr>
<td>
[[Image:Freiburg2008_M2ePath.png|thumb|400px|'''Figure 5: '''Pathway of receptor dimerization due to bivalent ligand]]
</td>
</tr>
</table>
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. 
<br>
<h3>Reaction kinetics </h3>
The continuous production of the cell´s receptor '''R''' with rate '''s''' is:<br>

[[Image:Freiburg2008_RKPr_M2.png|180px|left]]

<br><br><br><br><br>

One receptor '''R''' binds to one of two NIP molecules '''NN''' of the DNA-Origami  with rate '''k'''<sub>on</sub> or a receptor-NIP complex '''RNN''' dissociates with '''k'''<sub>off</sub>:
[[Image:Freiburg2008_RKBind_M2.png|270px|left]]

<br>
<br>
<br><br><br><br><br>
The second receptor '''R''' binds to the second NIP molecule of the DNA-Origami and both receptors dimerize with rate '''k'''<sub>don</sub> or the dimer '''RNNR''' dissociates in one receptor '''R''' and one receptor-NIP complex '''RNN''' with rate '''k'''<sub>doff</sub>:
[[Image:Freiburg2008_RKdim_M2.png|300px|left]]

<br>
<br>
<br><br><br><br><br>
The two receptors get activated with rate '''k'''<sub>a</sub>:
[[Image:Freiburg2008_RKA_M2.png|220px|left]]

<br>
<br>
<br><br><br><br><br>
After activation the receptor is internalized with the '''k'''<sub>i</sub>:
[[Image:Freiburg2008_RKIntM2.png|140px|left]]

<br>
<br>
<br><br><br><br><br>
<br>
<h3>ODE derivation </h3>
The ODEs are derived from the reaction kinetics: <br>
[[Image:Freiburg2008_ODEs_M2.png|800px]]
<br>
<h3>Extracellular receptor activity</h3>
For a set of parameters the ODEs are solved and reveal amongst the other quantities the receptor densities in the time course:
<table>
<tr>
<td>
[[Image:Freiburg2008_RA_M2.png|400px|thumb|left|'''Figure 6:''' Receptor densities in time (model 2)]]
</td>
<td>
chosen parameters:
 <font color="#336600">
 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
 </font>
initial integration conditions:
 <font color="#336600">
 R0 = 1;      %receptor density
 NN0 = 0.5;   %NIP Amount
 RNN0 = 0;    %receptor-NIP monomer
 RNNR0 = 0;   %receptor-NIP dimer
 A0 = 0;      %active receptor
 </font>
</td>
</tr>
</table>
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. 
<br>
<h1>Discussion of model I and II</h1>
For comparison, the ODEs of the corrected model 1 and model 2 are taken.
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 . 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 ligand amounts are used in the solutions of the model I ODEs and model II ODEs.

<h3>Receptor densities dependent on binding rate k<sub>on</sub></h3>
The two model ODEs are solved for the  mentioned parameters below whereby the binding rate k<sub>on</sub> was variated.
So the course in time for the free receptor, the monomeric receptor-NIP complex, the dimer and the active receptor is plotted:<br>
<table>
<tr>
<td align="left">
chosen parameters:<br>
 <font color="#336600">
 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
 </font>
initial integration conditions:
 <font color="#336600">
 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.
 </font>
</td>
</tr>
</table>
'''k<sub>on</sub> = 0.2 :'''
<table>
<tr><th>Model I</th><th>Model II</th></tr>
<tr><td>[[image:Freiburg2008_M1lkon.png|thumb|380px|'''Figure 7.1:''' Receptor densities according to model 1 for binding rate = 0.2]] </td>
<td>[[image:Freiburg2008_M2lkon.png|thumb|380px|'''Figure 7.2:''' Receptor densities according to model 2 for binding rate = 0.2]]</td></tr>
</table>
<br><br>
''' k<sub>on</sub> = 3 : '''
<table>
<tr><th>Model I</th><th>Model II</th></tr>
<tr><td>[[image:Freiburg2008_Mlhkon.png|thumb|380px|'''Figure 8.1:''' Receptor densities according to model 1 for binding rate = 3]] </td>
<td>[[image:Freiburg2008_M2kon.png|thumb|380px|'''Figure 8.2:''' Receptor densities according to model 2 for binding rate = 3]]</td></tr>
</table>
<br><br>
'''receptor activity for different k<sub>on</sub> : '''
<table>
<tr><th>Model I</th><th>Model II</th></tr>
<tr><td>[[image:Freiburg2008_M1dkon.png|thumb|380px|'''Figure 9.1:''' Receptor activity according to model 1 for different binding rates]]</td>
<td>[[image:Freiburg2008_M2dkon.png|thumb|380px|'''Figure 9.2:''' Receptor activity according to model 2 for different binding rates]]</td></tr>
</table>
<br>
The production of receptor-NIP complexes is lower in model 2 than in model 1. Increasing k<sub>on</sub> 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 have approximately the same maxima 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<sub>on</sub> variations for the used parameters.
<br>
<br>
<h3>Receptor densities dependent on dimerization rate k<sub>don</sub></h3>
Applying different k<sub>don</sub> leads to the following result:
<table>
<tr>
<td align="left">
chosen parameters:<br>
 <font color="#336600">
 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
 </font>
initial integration conditions:
 <font color="#336600">
 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.
 </font>
</td>
</tr>
</table>

'''k<sub>don</sub> = 0.3 ''':
<table>
<tr><th>Model I</th><th>Model II</th></tr>
<tr><td>[[image:Freiburg2008_m1lowkd.png|thumb|380px|'''Figure 10.1:''' Receptor densities according to model 1 for dimerization rate = 0.3]] </td>
<td>[[image:Freiburg2008_m2lowkd.png|thumb|380px|'''Figure 10.2:''' Receptor densities according to model 2 for dimerization rate = 0.3]]</td></tr>
</table>
<br><br>
''' k<sub>don</sub> = 3 : '''
<table>
<tr><th>Model I</th><th>Model II</th></tr>
<tr><td>[[image:Freiburg08_M1hkd.png|thumb|380px|'''Figure 11.1:''' Receptor densities according to model 1 for dimerization rate = 3]] </td>
<td>[[image:Freiburg08_M2hkd.png|thumb|380px|'''Figure 11.2:''' Receptor densities according to model 2 for dimerization rate = 3]]</td></tr>
</table>
<br><br>
'''receptor activity for different k<sub>don</sub> : '''
<table>
<tr><th>Model I</th><th>Model II</th></tr>
<tr><td>[[image:Freiburg2008_M1dkdon.png|thumb|380px|'''Figure 12.1:''' Receptor activity according to model 1 for different dimerization rates]]</td>
<td>[[image:Freiburg2008_M2dkdon.png|thumb|380px|'''Figure 12.2:''' Receptor activity according to model 2 for different dimerization rates]]</td></tr>
</table>
<br>
A low k<sub>don</sub> 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<sub>don</sub> due to a higher dimerization and receptor activity. k<sub>don</sub> variations give similar changes in receptor activity in both models.
<br><br>

<h3>Receptor densities dependent on NIP amount</h3>
The use of different amounts of ligand is maybe the most important influence on the receptor activation. As the ligand is the stimulus in order to activate our system, it´s amount will determine strongly the activity of the receptor. So different amounts of NIP are used and a higher internalization rate k<sub>i</sub> is used aswell.
<table>
<tr>
<td align="left">
chosen parameters:<br>
 <font color="#336600">
 s = 0.1;        % turnover rate
 kon : 3         % 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
 </font>
initial integration conditions:
 <font color="#336600">
 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 
 </font>
</td>
</tr>
</table>
'''NIP =  0.1 :'''
<table>
<tr><th>Model I</th><th>Model II</th></tr>
<tr><td>[[image:Freiburg2008_M1lowNIP.png|thumb|380px|'''Figure 13.1:''' Receptor activity according to model 1 for a low amount of NIP]]</td>
<td>[[image:Freiburg2008_M2lowNIP.png|thumb|380px|'''Figure 13.2:''' Receptor activity according to model 2 for a low amount of NIP]]</td></tr>
</table>
<br><br>
'''NIP =  3:'''
<table>
<tr><th>Model I</th><th>Model II</th></tr>
<tr><td>[[image:Freiburg2008_M1highNIP.png|thumb|380px|'''Figure 14.1:''' Receptor activity according to model 1 for a high amount of NIP]]</td>
<td>[[image:Freiburg2008_M2highNIP.png|thumb|380px|'''Figure 14.2:''' Receptor activity according to model 2 for a high amount of NIP]]</td></tr>
</table>

<br><br>
'''receptor activity for different amounts of NIP:'''
<table>
<tr><th>Model I</th><th>Model II</th></tr>
<tr><td>[[image:Freiburg2008_M1dNIP_.png|thumb|380px|'''Figure 15.1:''' Receptor activity according to model 1 for different amounts of NIP (shown in % of initial free receptor amount)]]</td>
<td>[[image:Freiburg2008_M2dNIP.png|thumb|380px|'''Figure 15.2:''' Receptor activity according to model 2 for different amounts of NIP (shown in % of initial free receptor amount)]]</td></tr>
</table>
<br><br>
As k<sub>i</sub>=0.8, the receptor activity is decreasing 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 model 2 the active receptor density increases for increasing NIP amounts, but when the NIP amount reaches a certain level, it starts to decrease as too much receptor-NIP complexes are formed, leaving few receptors for dimerization. Aswell there can be observed a late activation for high NIP amounts in model 2.
<br>
<h1>Modular Synthetic Receptor System model</h1>
[[media:Freiburg2008_MSRS.m|''matlab m-file'']]<br><br>
The dimerization model II is used to form the base of the model for our Modular Synthetic Receptor System. It has two receptors '''R'''<sub>1</sub> and '''R'''<sub>2</sub>. The bivalent ligand are two NIP molecules '''NN''' which are linked to DNA-Origami. Aswell two linked fluorescein molecules, or one NIP and fluorescein can serve as the bivalent ligand. Each receptor consists of a extracellular detector domain, which is connected to an intracellular split protein half by a short transmembrane protein. Only when receptor '''R'''<sub>1</sub> and '''R'''<sub>2</sub> dimerize, the intracellular split protein halves complement to become the complete functional protein '''A''' in the form of a &beta;-lactamase or a fluorescent protein. A dimerization of '''R'''<sub>1</sub> and '''R'''<sub>1</sub> or '''R'''<sub>2</sub> and '''R'''<sub>2</sub> can happen to '''R'''<sub>1</sub>'''R'''<sub>1</sub> or '''R'''<sub>2</sub>'''R'''<sub>2</sub>, but does not lead to an active intracellular protein and measurable output. Of course all receptor dimers ('''R'''<sub>1</sub>'''R'''<sub>2</sub>, '''R'''<sub>1</sub>'''R'''<sub>1</sub>, '''R'''<sub>2</sub>'''R'''<sub>2</sub>) dissociate into monomers much easier than a receptor dimer which is active in the form of '''A''', as the connected split protein halves are bound to each other.
<br>
<h3>Reaction kinetics</h3>
[[image:Freiburg2008_RK_MSRS.png|650px]]
<br>
<h3>ODE derivation</h3>
The ODEs can be derived from the reaction kinetics again and are:<br>
[[image:Freiburg2008_ODEs_MSRS.png|800px]]
<br>
<h2>Involved quantities and split protein activity</h2>
The ODEs are solved for a set of parameters and reveal the time courses of the quantities:
<table>
<tr>
<td>
[[image:Freiburg2008_MSRSp1.png|thumb|left|400px|'''Figure 16:''' Involved receptor densities in time]]
Assuming equal parameters for R<sub>1</sub> and R<sub>2</sub> leads to equal densities of both; shown through one plot curve for each uniform density (free receptor density, monomeric density, dimeric density) of R<sub>1</sub> and R<sub>2</sub> 
</td>
<td>chosen parameters:<br>
 <font color="#336600">
 s1 = 0.1;   %turnover rate for R1
 s2 = 0.1;   %turnover rate for R2
 K1 = 3;     %binding rate 1
 k1 = 0.1;   %dissociating rate 1
 K2 = 3;     %binding rate 2
 k2 = 0.1;   %dissociating rate 2
 K3 = 2;     %R1R2 dimerization rate 1
 k3 = 0.1;   %R1R2 dimer dissociating rate 1
 K4 = 2;     %R1R2 dimerization rate 2
 k4 = 0.1;   %R1R2 dimer dissociating rate 2
 K5 = 2;     %R1R1 dimerization rate 
 k5 = 0.1;   %R1R1 dimer dissociating rate 
 K6 = 2;     %R2R2 dimerization rate 
 k6 = 0.1;   %R2R2 dimer dissociating rate 
 Ka = 1;     %activation rate
 Kda = 0.05; %deactivation rate
 </font>
initial integration conditions:
 <font color="#336600">
 R10 = 1;     %R1
 R20 = 1;     %R2
 NN0 = 1;     %ligand
 R1NN0 = 0;   %R1-Ligand monomer
 R2NN0 = 0;   %R2-Ligand monomer
 R1NNR20 = 0; %R1-R2 dimer
 R1NNR10 = 0; %R1-R1 dimer 
 R2NNR20 = 0; %R2-R2 dimer 
 A0 = 0;      %active protein
 </font>
<td>
<tr>
</table>
<h3>Active split protein density dependent on input ligand amount</h3>
In order to gain information about the split protein activity dependent on the input, the same parameters were used as above except the initial ligand parameter, which was variated.
<table>
<tr>
<td>
[[image:Freiburg2008_MSRSdNN.png|thumb|left|600px|'''Figure 17:''' Split protein activity dependent on ligand amount (% of initial free receptor amount)]]
</td>
</tr>
</table>
<br>

<h3>Split protein activity with two different kinetic receptors</h3>
As two different receptors are used, they can differ in the kinetics. Assuming that receptor R<sub>1</sub> is binding the ligand better, so its binding rate is higher and its dissociating rate is lower than of receptor R<sub>2</sub>. Choosing not only a higher turnover rate s<sub>1</sub> but also a higher dimerization and a lower dimer dissociating rate for receptor R<sub>1</sub> and a lower initial free receptor density for R<sub>2</sub>, gives the following theoretical result:
<table>
<tr>
<td>
[[image:Freiburg2008_MSRS_2.png|thumb|left|400px|'''Figure 18:''' Involved receptor densities in time with two different kinetic receptors involved]]
All densities connected to R<sub>2</sub> are shown dashed, except the dimer R<sub>1</sub>R<sub>2</sub>, which becomes the active protein. As stated in the parameters, R<sub>1</sub> is a better binding and dimerizing receptor than R<sub>2</sub> and is available in a higher concentration in the cell in the regarded time course. So all densities (free receptor density, monomeric density, dimeric density) connected to R<sub>1</sub> are higher than the according densities of R<sub>2</sub>.
</td>
<td>chosen parameters:<br>
 <font color="#336600">
 s1 = 0.2;    %turnover rate for R1
 s2 = 0.1;    %turnover rate for R2
 K1 = 4;      %binding rate 1
 k1 = 0.05;   %dissociating rate 1
 K2 = 3;      %binding rate 2
 k2 = 0.1;    %dissociating rate 2
 K3 = 2;      %R1R2 dimerization rate 1
 k3 = 0.1;    %R1R2 dimer dissociating rate 1
 K4 = 2.5;    %R1R2 dimerization rate 2
 k4 = 0.05;   %R1R2 dimer dissociating rate2
 K5 = 2.5;    %R1R1 dimerization rate 
 k5 = 0.05;   %R1R1 dimer dissociating rate 
 K6 = 2;      %R2R2 dimerization rate 
 k6 = 0.1;    %R2R2 dimer dissociating rate 
 Ka = 1;      %activation rate
 Kda = 0.05;  %deactivation rate
 </font>
initial integration conditions:
 <font color="#336600">
 R10 = 1;     %R1
 R20 = 0.7;   %R2
 NN0 = 1;     %ligand
 R1NN0 = 0;   %R1-Ligand monomer
 R2NN0 = 0;   %R2-Ligand monomer
 R1NNR20 = 0; %R1-R2 dimer
 R1NNR10 = 0; %R1-R1 dimer 
 R2NNR20 = 0; %R2-R2 dimer 
 A0 = 0;      %active protein
 </font>
</td>
</tr>
</table>
<table>
<tr>
<td>
[[image:Freiburg2008_MSRS_dNN_R1R2d.png|thumb|left|600px|'''Figure 19:''' Split protein activity dependent on ligand amount (% of initial free receptor amount) with two different kinetic receptors involved]]
</td>
</tr>
</table>
<br><br>
<h1>Matlab m-files</h1>
*[[media:Freiburg2008_M1_Basic.m|Basic dimerization model 1]]<br>
*[[media:Freiburg2008_M1_Xten.m|Model 1 with extensions]]<br>
*[[media:Freiburg2008_M1_Corrected.m|Corrected model 1 version]]<br>
*[[media:Freiburg2008_M_2.m|Dimerization Model 2]]<br>
*[[media:Freiburg2008_MSRS.m|Modular Synthetic Receptor System model]]<br>
<br><br>
<h1>[[Image:MO2.jpg|50px|]]Literature</h1>
*Martin F. Bachmann, Michael Salzmann, Annette Oxenius and Pamela S. Ohashi: "Formation of TCR dimers/trimers as a crucial step for T cell activation", Eur. J. Immunol., 1998
*Martin F. Bachmann and Pamela S. Ohashi: "The role of T-cell receptor dimerization in T-cell activation", Review Immunology Today, 1999
*João Sousa and Jorge Carneiro: "A mathematical analysis of TCR serial triggering and down-regulation", Eur. J. Immunol., 2000
*Susana Minguet, Mahima Swamy, Balbino Alarcón, Immanuel F. Luescher and Wolfgang W.A. Schamel: "Full Activation of the T Cell Receptor requires Both Clustering and Conformational Changes at CD3", Immunity, 2006
}}