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<br/>
<h1>Experimental model</h1>

<table class="summary">
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<tr class="summary">
<td class="summary">
<p><b>Question:</b> Are we able to somehow take advantage of the experimental measurements obtained 
from the wet lab? Does that allow us to predict the optimal input ratios of plasmids inserted?</p>
<b>Answer:</b> Measurements were exploited in their full potential as we have developed a brand 
new modeling approach. As it turned out, it well describes the real behavior of the switch. 
The results predicted bistable behavior for a range of input ratios. The optimal choice resulted
 to be the <b>p[B]_PMIN_TAL:KRAB</b> (repressors) plasmids being in around three times greater quantity
 than <b>p[B]_PMIN_TAL:VP16</b>  (activators).
</td></tr></tbody></table>





<p>
<ul style="margin-left:15px;">
<li><a href="#idea">Idea</a><br/></li>
<li><a href="#modeldev">Model derivation</a><br/></li>
<li><a href="#results">Results</a><br/></li>
<li><a href="#source">Source code</a><br/></li>
<li><a href="#references">References</a><br/></li>
</ul>
</p>



<h2 ><a name="idea"></a>Idea</h2>

As we discovered throughout the project, the switch design is very sensible to the
 masses of the different plasmids (parts) used in the experiment. 
 We thought, why not use the data we obtained for <i>individual constructs</i> and 
 try to predict the switch behavior as they are combined into <i>a single system?</i>
Thus, the data obtained from the wet lab was used to characterize each of the 
plasmid constructs in the positive feedback loop switch. The characterization 
was made in terms of how their products <i>regulate the transcription of the target genes.</i>
 That is the way how we connect the experimental work to the theory, as shown in Figure 1.

 Figure 1. Steps in connecting the experimental results to theoretical modeling. 
 
<h2 ><a name="modeldev"></a>Model derivation</h2>
Ordinary differential equations (ODE) model was used as a module to build around. 
We made use of the two basic relations for transcriptional regulation. 
The repressor and activator function, defined in (Alon, 2006), modify 
the basal transcription rate as follows:


Since precisely defining transcription rates was not the primary goal of our project, 
their <i>absolute values</i> are not of central importance in the model.  Rather, we focus on 
the <i>relative changes</i> in transcription levels, caused by activators and repressors. 
Furthermore, concentration levels of reporters will also be presented <i>relative to each other</i>. 
This way, we avoid the use of <i>any units we did not measure</i>.

<h3> Promoters </h3>
As we use two different kinds of promoters, PCMV and PMIN, to account for the difference in
 <i>basal transcription rates</i>. Experiments have shown that relative luciferase units under 
 PCMV promoter are 1000-fold greater than under PMIN (data not shown). Hence, PCMV 
 is assumed to have a 1000 times greater basal transcription rate than PMIN.

<h3> Transcriptional regulators </h3> 

To characterize activators, <b>pPCMV_TALA:VP16</b> and <b>p[A]_PMIN_fLuc</b> plasmids were tested. Various ratios of input masses produced different levels of luciferase activity (Figure). 
Following equation (1), activators increase the transcription rate up to some maximum factor of 
<i>fold induction</i>. The former is derived by comparing the resulting luciferase activity with that 
of the <i>inactivated</i> <b>p[A]_PMIN_fLuc</b>. Gathering data for various input ratios, we fitted it to an 
analytical function (Figure 2), enabling us to predict the fold induction (A) for <i>arbitrary</i> input
 ratios of DNA for the two plasmids.
Intuitively, more activator plasmids <i>increase</i> the maximum fold induction. On the other hand, 
more target plasmids <i>decrease it</i>. Relatively speaking, activator proteins thus have more plasmids 
to target. That leads to fold activation ratio being smaller from the <i>global point of view</i>.
In an analogue manner <b>TAL-B:VP16</b> constructs were characterized.
 
 
Figure 3. 
Fold induction rates, derived from relative luciferase activity for various ratios of 
<b>pPCMV_TALA:VP16</b> (activator) and <b>p[A]_PMIN_fLuc</b> (target) plasmids. 

Figure 4. 
Plot of fold induction function for a continuous  range of activator (<i>m</i>) and reporter (<i>n</i>)
 plasmid ratios, <i>f(m/n) = a * (m/n)**1/b</i>, where <i>a</i> and <i>b</i> are parameters subject to least square
 error fitting. 
 
 
<h3> Repressors </h3>
To characterize repressors, <b>pPCMV_TALA:KRAB </b> and <b>p[A]_PCMV_fLuc</b> plasmids were tested.
 Various ratios of input masses produced different levels of luciferase activity. 
Following equation (1), repressors <i>decrease</i> the basal transcription rate down to 
some minimum factor of <i>fold repression</i>.  The former is derived by comparing the luciferase
 activity with that of the <i>unrepressed</i> <b>p[A]_PCMV_fLuc</b>, resulting in percentage of remaining
 transcription intensity. Gathering data for various input ratios (Figure 6), we fitted it to 
 an analytical function (Figure 7). It enables us to predict the fold repression (Q) for arbitrary
 input rations of DNA.
Intuitively, more repressor plasmids <i>increase</i> the maximum fold induction. 
On the other hand, more target plasmids <i>decrease it</i>. Relatively speaking, 
repressor proteins thus have more plasmids to target. That leads to fold repression ratio
 being smaller from the <i>global point of view</i>.
In an analogue manner <b>TAL-B:KRAB</b> constructs were characterized.

Figure 6. 
Fold repression rates, derived from relative luciferase activity for various ratios of 
pPCMV_TALA:KRAB (repressor) and p[A]_PCMV_fLuc (target) plasmids. 

Figure 7. 
Plot of fold induction function for a continuous  range of activator (m)
 and reporter (n) plasmid ratios, f(m/n) = a *(e**b*(m/n))+c, where a, b and c are parameters subject to least square error fitting. 
 
<h3> Induction system </h3> 
To characterize the erythromycin repressor induction system we measured relative luciferase 
activity for various ratios of <b>pPCMV_E:KRAB</b> (inducible protein), <b>pPCMV_[ETR]_TALA:KRAB</b> (repressor)
 and <b>p[A]_PCMV_fLuc </b> (target) plasmids (Figure). 
As E:KRAB gets stimulated by erythromycin, it unbinds from the [etr] binding site, 
allowing the production of <b>TALA:KRAB</b>, which in turn represses the transcription of fLuc.
 By varying the ratios of the three input plasmids, we can observe the resulting difference in 
 luciferase activity. That directly determines the fold repression of the inducible system <i>as a
 whole</i>.
As inducible protein plasmids <i>increase</i> the luciferase activity <i>increases</i> as well.
 The increasing masses of repressor and target plasmids both <i>decrease</i> the luciferase activity 
 and consequently <i>decrease</i> the fold repression rate. 
In an analogue manner erithromycin activator, pristinamycin repressor and
 pristinamycin activator induction systems were characterized.
 
<h3> Noise </h3> 
Due to the noise in experimental measurements, we can directly apply it in the model as well. 
When determining fold repression ratios, <i>standard deviations</i> of up to 15 %  were observed. 
 
<h2 ><a name="results"></a>Results</h2>

<table>

<tr>
<td valign="top">
<table>
      <th><tr><td  style="background: #0099FF;"><b><font color="white">The Switch (top level)</font></b></td><td></td></tr></th>
      <th>
        <tr><td>Plasmid</td><td>Index</td></tr>
        </th>
      <tr><td><img align="left" src="https://static.igem.org/mediawiki/2012/0/09/Svn12_qm_interactive_p0.png" width="300px"/></td><td><font style="font-family:Times New Roman;">p0</font></td></tr>
      <tr><td><img align="left" src="https://static.igem.org/mediawiki/2012/2/28/Svn12_qm_interactive_p1.png" width="200px"/></td><td><font style="font-family:Times New Roman;">p1</font></td></tr>
      <tr><td><img align="left" src="https://static.igem.org/mediawiki/2012/f/f0/Svn12_qm_interactive_p2.png" width="300px"/></td><td><font style="font-family:Times New Roman;">p2</font></td></tr>
      <tr><td><img align="left" src="https://static.igem.org/mediawiki/2012/7/70/Svn12_qm_interactive_p3.png" width="200px"/></td><td><font style="font-family:Times New Roman;">p3</font></td></tr>
      <tr><td></td><td></td></tr>
</table>
</td>

<td valign="top">
<table>      
      <th><tr><td  style="background: #0099FF;"><b><font color="white">Inducible system (middle level)</font></b></td><td></td></tr></th>
       <th>
        <tr><td>Plasmid</td><td>Index</td></tr>
        </th>
      <tr><td><img align="left" src="https://static.igem.org/mediawiki/2012/c/cf/Svn12_qm_interactive_p4.png" width="200px"/><td><font style="font-family:Times New Roman;">p4</font></td></tr>
      <tr><td><img align="left" src="https://static.igem.org/mediawiki/2012/f/f7/Svn12_qm_interactive_p5.png" width="200px"/><td><font style="font-family:Times New Roman;">p5</font></td></tr>
      <tr><td><img align="left" src="https://static.igem.org/mediawiki/2012/9/96/Svn12_qm_interactive_p6.png" width="200px"/><td><font style="font-family:Times New Roman;">p6</font></td></tr>
      <tr><td><img align="left" src="https://static.igem.org/mediawiki/2012/9/9f/Svn12_qm_interactive_p7.png" width="200px"/><td><font style="font-family:Times New Roman;">p7</font></td></tr>
      <tr><td></td><td></td></tr>
</table>	  
</td>

<td valign="top">
<table>
	  
      <th><tr><td  style="background: #0099FF;"><b><font color="white">Inducible proteins (bottom level)</font></b></td><td></td></tr></th>
       <th>
        <tr><td>Plasmid</td><td>Index</td></tr>
        </th>
      <tr><td><img align="left" src="https://static.igem.org/mediawiki/2012/6/6a/Svn12_qm_interactive_p8.png" width="200px"/><td><font style="font-family:Times New Roman;">p8</font></td></tr>
      <tr><td><img align="left" src="https://static.igem.org/mediawiki/2012/b/b1/Svn12_qm_interactive_p9.png" width="200px"/><td><font style="font-family:Times New Roman;">p9</font></td></tr>
</table>
</td>
</tr>
</table>
Figure. The switch system level partitioning.

The main goal of this particular model was to help determine the input ratios for plasmids
 that make up the switch system. The first confirmation was that if the masses of all 
 plasmids will be equal, the switch will not be able to exhibit bistable behavior.
We partition the whole system into levels (Figure), with the Switch being the top level 
and the Pristinamycin/Erithromycin inducible proteins the bottom level. It is obvious that 
a plasmid on a given level influences another level only in the direction bottom -&gt; top, 
so as we go downwards, the masses shall increase to have more influence on the targeted 
upper level. Hence the relation:<br/><br/>
<center><font style="font-family:Times New Roman;">p0, p1, p2, p3 &lt; p4, p5, p6, p7 &lt; p8, p9</font></center><br/>
Next, the ratio between repressor and activator constructs was examined. To preserve symmetry,
 we assume: <br/><br/>
 <center><font style="font-family:Times New Roman;">p0 = p2, p1 = p3, p4  = p6, p5 = p7, p8 = p9</font> </center><br/> 
 All the following ratios will be given in the form p0 : p1 (repressor:activator).
Testing began at the 1:1 ratio, and we gradually increased both the repressor and activator to observe the qualitative behavioral change. The Switch exhibited bistable behavior for all ratios between 10:1 until 1:5, as can be seen in Scroll Box 1.

<table class="invisible"><tr class="invisible"><td class="invisible">
<img src="C:\Users\Strazkosan\Dropbox\igem-cellulator\Wiki\img\svn_12_qm_results-04.png"></img>
</td>
<td class="invisible">
<img src="C:\Users\Strazkosan\Dropbox\igem-cellulator\Wiki\img\svn_12_qm_results-03.png"></img>
<div style="height:380px;width:900px;border:1px solid #ccc;font:16px/26px Georgia, Garamond, Serif;overflow:auto;">
<img src="C:\Users\Strazkosan\Dropbox\igem-cellulator\Wiki\img\svn_12_qm_results-01.png"></img>
</div>
<img src="C:\Users\Strazkosan\Dropbox\igem-cellulator\Wiki\img\svn_12_qm_results-02.png"></img>
</td></tr></table>
<b>Scroll Box 1. </b> Results for valid range of activator:repressor plasmid ratios. 


The whole range in which the bistable behaviour is exhibited can be tested using our 
Interactive Simulations (click to test):<br /><br />
<center><font style="font-family:Times New Roman;">[10:1, 9:1, 8:1, 7:1, 6:1, 5:1, 4:1, 3:1, 2:1, 1:1, 1:2, 1:3, 1:4, 1:5] </font></center><br />

Taking the median of the above range, we conclude that 
the optimal ratio of input masses for wet lab experiments is at 
<b>around three to one in favour of the repressor plasmids</b>.

<h2 ><a name="source"></a>Source code</h2>
All source code for the model and simulations described above can be found <a href ="https://2012.igem.org/Team:Slovenia/SourceCode">here</a>;

<h2 style="color:grey;" ><a name="references"></a>References</h2>
<p style="color:grey;">
Alon U. (2007), An introduction to systems biology: design principles of biological circuits. Chapman & Hall/CRC.
<br/><br/>
Batard P., and Wurm F. (2001) Transfer of high copy number plasmid into mammalian cells by calcium phosphate transfection. <i>Gene: An international Journal on Genes and Genomes</i>, <b>270</b>, 61-68.
<br/><br/>
Chatterjee A., Kaznessis Y., and Hu W. (2006) Tweaking biological switches through a better understanding of bistability behavior. <i>Current opinion in biotechnology</i>, <b>19</b>, 475-81.
<br/><br/>
Gardner T., Cantor C., and Collins J. (2000) Construction of a genetic toggle switch in Escherichia coli. <i>Nature</i>, <b>403</b>, 339-42.
<br/><br/>
Batard P., and Wurm F. (2001) Transfer of high copy number plasmid into mammalian cells by calcium phosphate transfection. <i>Gene: An international Journal on Genes and Genomes</i>, <b>270</b>, 61-68.
<br/><br/>
Malphettes L., and Fussenegger M. (2006) Impact of RNA interference on gene networks. <i>Metabolic engineering</i>, <b>8</b>, 672-83.
<br/><br/>
Tigges M., Marquez-Lago T., Stelling J., and Fussenegger M. (2009) A tunable synthetic mammalian oscillator. <i>Nature</i>, <b>457</b>, 309-12.
<br/><br/>
Zakharova A., Kurths J., Vadivasova T., and Koseska A. (2011) Analysing dynamical behavior of cellular networks via stochastic bifurcations. <i>PloS one</i>, <b>6</b>, e19696.
</p>




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