Team:Technion-Israel/Modeling

From 2014.igem.org

(Difference between revisions)
m
m
 
(30 intermediate revisions not shown)
Line 356: Line 356:
<div id="logo">
<div id="logo">
<h1 style="color: #ebebeb">Modeling</h1>
<h1 style="color: #ebebeb">Modeling</h1>
-
<p style="color: #ebebeb"></p>
+
<p style="color: #ebebeb">Click <a href="https://static.igem.org/mediawiki/2014/3/3d/Modeling-Everything_Ever.pdf" target="_blank" style="color:white;"><u>here</u></a> for full modeling file</p>
</div>
</div>
Line 375: Line 375:
<li id="child1"><a href="https://2014.igem.org/Team:Technion-Israel/Project#hk">Histidine Kinase</a></li>
<li id="child1"><a href="https://2014.igem.org/Team:Technion-Israel/Project#hk">Histidine Kinase</a></li>
<li id="child1"><a href="https://2014.igem.org/Team:Technion-Israel/Project#newmethod">New Method</a></li>
<li id="child1"><a href="https://2014.igem.org/Team:Technion-Israel/Project#newmethod">New Method</a></li>
-
<li id="child1"><a href="https://2014.igem.org/Team:Technion-Israel/Project#experiments">Experiments</a></li>
 
<li id="child1"><a href="https://2014.igem.org/Team:Technion-Israel/Project#protocol">Protocols</a></li>
<li id="child1"><a href="https://2014.igem.org/Team:Technion-Israel/Project#protocol">Protocols</a></li>
<li id="child1"><a href="https://2014.igem.org/Team:Technion-Israel/Project#notebook">Lab Notebook</a></li>
<li id="child1"><a href="https://2014.igem.org/Team:Technion-Israel/Project#notebook">Lab Notebook</a></li>
Line 387: Line 386:
     <li id="child1"><a href="https://2014.igem.org/Team:Technion-Israel/Modeling#splint">RNA Splint</a></li>
     <li id="child1"><a href="https://2014.igem.org/Team:Technion-Israel/Modeling#splint">RNA Splint</a></li>
     <li id="child1"><a href="https://2014.igem.org/Team:Technion-Israel/Modeling#biofilm">Synthetic Biofilm<br>Formation</a></li>
     <li id="child1"><a href="https://2014.igem.org/Team:Technion-Israel/Modeling#biofilm">Synthetic Biofilm<br>Formation</a></li>
 +
  </ul>
 +
</li>
 +
 +
<li id="parent"><a href="https://2014.igem.org/Team:Technion-Israel/Experiments">Experiments</a>
 +
  <ul class="sub1">
 +
    <li id="child1"><a href="https://2014.igem.org/Team:Technion-Israel/Experiments#gate1">Gate 1</a></li>
 +
    <li id="child1"><a href="https://2014.igem.org/Team:Technion-Israel/Experiments#gate2">Gate 2</a></li>
 +
    <li id="child1"><a href="https://2014.igem.org/Team:Technion-Israel/Experiments#rna">RNA Splint</a></li>
 +
    <li id="child1"><a href="https://2014.igem.org/Team:Technion-Israel/Experiments#pompc">Pompc-RFP</a></li>
 +
    <li id="child1"><a href="https://2014.igem.org/Team:Technion-Israel/Experiments#taz">TaZ</a></li>
 +
    <li id="child1"><a href="https://2014.igem.org/Team:Technion-Israel/Experiments#mcherry">mCherry</a></li>
 +
    <li id="child1"><a href="https://2014.igem.org/Team:Technion-Israel/Experiments#amilcp">amilCP</a></li>
 +
    <li id="child1"><a href="https://2014.igem.org/Team:Technion-Israel/Experiments#azo">Azobenzene</a></li>
   </ul>
   </ul>
</li>
</li>
Line 422: Line 434:
                                     <ul class="sub1">
                                     <ul class="sub1">
<li id="child1"><a href="https://igem.org/2014_Judging_Form?id=1343" target="_blank">Judging Form</a></li>
<li id="child1"><a href="https://igem.org/2014_Judging_Form?id=1343" target="_blank">Judging Form</a></li>
-
<li id="child1"><a href="https://2014.igem.org/Team:Technion-Israel/Judging#results">Results</a></li>
 
<li id="child1"><a href="https://2014.igem.org/Team:Technion-Israel/Judging#biobrick">BioBricks</a></li>
<li id="child1"><a href="https://2014.igem.org/Team:Technion-Israel/Judging#biobrick">BioBricks</a></li>
-
<li id="child1"><a href="https://2014.igem.org/Team:Technion-Israel/Judging#criteria">Judging Criteria</a></li>
+
<li id="child1"><a href="https://2014.igem.org/Team:Technion-Israel/Judging#results">Results</a></li>
                                     </ul>
                                     </ul>
                                 </li>
                                 </li>
Line 438: Line 449:
<div class="wrapper style2">
<div class="wrapper style2">
-
<div class="title" id="whyworks">Why Should it Work</div>
+
<div class="title" id="whyworks">Why it Should Work</div>
<div id="main" class="container">
<div id="main" class="container">
Line 463: Line 474:
</div>
</div>
</div>
</div>
-
<p style="font-size: 1.1em;">It is clear from the goals of our system, that we want to have some sort of bi-stability in the result, when Gate I is small (see [2]). The answer to whether this condition is met, would obviously depend on the constants of the system for which we could not find a reliable source, but using a simple geometric analysis of the phase space (see [3]), we were able to produce a graph showing for which values of (v_A,v_B) we could configure the system (by changing the IPTG concentration and the OD) to show bi-stability:</p>
+
<p style="font-size: 1.1em;">It is clear from the goals of our system, that we want to have some sort of bi-stability in the result, when the term Gate I is small (see [2]). The answer to whether this condition is met, would obviously depend on the constants of the system for which we could not find a reliable source, but using a simple geometric analysis of the phase space (see [3]), we were able to produce a graph showing for which values of (v<sub>1</sub>,v<sub>2</sub>) we could configure the system (by changing the IPTG concentration and the OD) to show bi-stability:</p>
-
<p style="font-size: 1.1em;">*map of bi-stability in the alpha system*</p>
+
<p style="font-size: 1.1em;">
 +
<div class="Unindented">
 +
<div class="float">
 +
<a class="Label" name="Figure-1"> </a><div class="figure">
 +
<img class="embedded" src="https://static.igem.org/mediawiki/2014/e/ec/Technion-Israel-AlphaBool.jpg" style="width: 540px; max-width: 1200px; height: 405px; max-height: 900px;">
 +
<img class="embedded" src="https://static.igem.org/mediawiki/2014/f/fe/Technion-Israel-AlphaNorm.jpg" alt="figure AlphaNorm.jpg" style="width: 540px; max-width: 1200px; height: 405px; max-height: 900px;">
 +
<div class="caption">
 +
Figure 1 On the left:a map of the values of the parameters of the system for which it is bi-stable (bi-stable in red, mono-stable in blue). On the right: a map of the normalized bi-stability parameter we have defined as a function of its parameters.
 +
</div>
 +
</p>
 +
<p style="font-size: 1.1em;">It is clear from this graph, that the alpha system is bi-stable for a large part of the range of possible inputs.</p>
</center>
</center>
</header>
</header>
Line 476: Line 497:
         <div class="wrapper style3">
         <div class="wrapper style3">
-
<div class="title" id="whyfail">Why Should it Fail</div>
+
<div class="title" id="whyfail">Why it Should Fail</div>
<div id="highlights" class="container">
<div id="highlights" class="container">
<center>
<center>
<h1  style="font-size: 2em;">Why Alpha System Should Fail – a stochastic model of alpha system</h1>
<h1  style="font-size: 2em;">Why Alpha System Should Fail – a stochastic model of alpha system</h1>
-
<p>The above model assumes a low-noise system (as do all rate equation models), but especially when constructing a bi-stable network, it is important to consider the noise. To do this we need to create a stochastic model, which in our case, we based upon the commonly used Fokker Planck equation. Using the derivation found in [4] (book on stochastic models from Roee), we produced the Fokker Planck variant of the equation for the AHL concentration derived in *(link to “Why Alpha System Should Work”)*</p>
+
<p>The above model assumes a low-noise system (as do all rate equation models), but especially when constructing a bi-stable network, it is important to consider the noise. To do this we need to create a stochastic model, which in our case, we based upon the commonly used Fokker Planck equation. Using the derivation found in [4] (Van Kampen "Stochastic Processes in Physics and Chemistry", Third Edition), we produced the Fokker Planck variant of the equation for the AHL concentration shown above</p>
-
<p>After analyzing this equation as explained in [5], we produced the following results (using a point on the (v_A,v_B) plane which the previous analysis showed would be bi-stable)</p>
+
<p>After analyzing this equation as explained in [5], we produced the following results (using a point on the (v<sub>1</sub>,v<sub>2</sub>) plane which the previous analysis showed would be bi-stable)</p>
-
<p>*Fokker Planck Results*</p>
+
<div class="Unindented">
 +
<div class="float">
 +
<a class="Label" name="Figure-1"> </a><div class="figure">
 +
<img class="embedded" src="https://static.igem.org/mediawiki/2014/3/3b/1%2C2%2C0.jpg" alt="figure 1,2,0.jpg" style="width: 540px; max-width: 1200px; height: 405px; max-height: 900px;">
 +
<img class="embedded" src="https://static.igem.org/mediawiki/2014/2/26/1-2-0.gif" alt="figure 1-2-0.gif" style="width: 540px; max-width: 1200px; height: 405px; max-height: 900px;">
 +
<img class="embedded" src="https://static.igem.org/mediawiki/2014/e/e4/1%2C2%2C1.jpg" alt="figure 1,2,1.jpg" style="width: 540px; max-width: 1200px; height: 405px; max-height: 900px;">
 +
<img class="embedded" src="https://static.igem.org/mediawiki/2014/b/bf/1-2-1.gif" alt="figure 1-2-1.gif" style="width: 540px; max-width: 1200px; height: 405px; max-height: 900px;">
 +
<div class="caption">
 +
Figure 1 From Top To Bottom: Fokker Planck in the Alpha System when the system begins off, and then when it begins off: On the left is a heat map of the probability distribution function, as a function of time. On the right is a gif showing the probability distribution function over 100 timelapses
 +
</div>
 +
 
 +
</div>
 +
 
 +
</div>
 +
 
 +
</div>
 +
<div class="Indented">
 +
Clearly the “on” state (high AHL concentration - low on the graph) of our system is the more stable state of our system - so much so that it can spontaneously switch to the on state. This means that our system is bound to have a high likelihood of false positives.
 +
</div>
</center>
</center>
</div>
</div>
Line 494: Line 533:
<h1  style="font-size: 2em;">The RNA Splint – Deterministic and Stochastic Models of this Noise Reduction Method</h1>
<h1  style="font-size: 2em;">The RNA Splint – Deterministic and Stochastic Models of this Noise Reduction Method</h1>
<p style="color:#919499">When considering an additional design for our system, we thought of the idea for the RNA splint:
<p style="color:#919499">When considering an additional design for our system, we thought of the idea for the RNA splint:
-
Basically, instead of directly producing the AHL, the cell will use a system called an RNA Splint (*link to RNA Splint*) to build the mRNA encoding for the production of  AHL, in two parts and also produce a third component which would combine the two.<br>Adjusting the equation for the production of AHL <div class="formula">
+
Basically, instead of directly producing the AHL, the cell will use a system called an <a href="https://2014.igem.org/Team:Technion-Israel/Project#rna" style="color:#919499">RNA Splint</a> to build the mRNA encoding for the production of  AHL, in two parts and also produce a third component which would combine the two.<br>Adjusting the equation for the production of AHL <div class="formula">
<span  style="color:#919499"><span class="fraction"><span class="ignored">(</span><span class="numerator"><i>d</i><span class="symbol">[</span><i>AHL</i><span class="symbol">]</span></span><span class="ignored">)/(</span><span class="denominator"><i>dt</i></span><span class="ignored">)</span></span> = <span class="fraction"><span class="ignored">(</span><span class="numerator"><i>v</i><sub><i>B</i></sub> + <i>v</i><sub><i>A</i></sub><i>k</i><sub><i>A</i></sub><span class="symbol">[</span><i>AHL</i><span class="symbol">]</span><sup>2</sup></span><span class="ignored">)/(</span><span class="denominator">1 + <i>k</i><sub><i>A</i></sub><span class="symbol">[</span><i>AHL</i><span class="symbol">]</span><sup>2</sup></span><span class="ignored">)</span></span> − <i>γ</i><sub><i>AHL</i></sub><span class="symbol">[</span><i>AHL</i><span class="symbol">]</span> + <i>GateI</i>
<span  style="color:#919499"><span class="fraction"><span class="ignored">(</span><span class="numerator"><i>d</i><span class="symbol">[</span><i>AHL</i><span class="symbol">]</span></span><span class="ignored">)/(</span><span class="denominator"><i>dt</i></span><span class="ignored">)</span></span> = <span class="fraction"><span class="ignored">(</span><span class="numerator"><i>v</i><sub><i>B</i></sub> + <i>v</i><sub><i>A</i></sub><i>k</i><sub><i>A</i></sub><span class="symbol">[</span><i>AHL</i><span class="symbol">]</span><sup>2</sup></span><span class="ignored">)/(</span><span class="denominator">1 + <i>k</i><sub><i>A</i></sub><span class="symbol">[</span><i>AHL</i><span class="symbol">]</span><sup>2</sup></span><span class="ignored">)</span></span> − <i>γ</i><sub><i>AHL</i></sub><span class="symbol">[</span><i>AHL</i><span class="symbol">]</span> + <i>GateI</i>
-
</div></span></p> <p  style="color:#919499">for this change (see [6]), we obtain:</p>
+
</div></span></p> <p  style="color:#919499">for this change (see [6]), we obtain two new potential models:</p>
 +
 
<div class="formula" style="color:#919499">
<div class="formula" style="color:#919499">
<span class="fraction"><span class="ignored">(</span><span class="numerator"><i>d</i><span class="symbol">[</span><i>AHL</i><span class="symbol">]</span></span><span class="ignored">)/(</span><span class="denominator"><i>dt</i></span><span class="ignored">)</span></span> = <span class="fraction"><span class="ignored">(</span><span class="numerator"><i>v</i><sub><i>B</i></sub> + <i>v</i><sub><i>A</i></sub><i>k</i><sub><i>A</i></sub><span class="symbol">[</span><i>AHL</i><span class="symbol">]</span><sup>6</sup></span><span class="ignored">)/(</span><span class="denominator">1 + <i>k</i><sub><i>A</i></sub><span class="symbol">[</span><i>AHL</i><span class="symbol">]</span><sup>6</sup></span><span class="ignored">)</span></span> − <i>γ</i><sub><i>AHL</i></sub><span class="symbol">[</span><i>AHL</i><span class="symbol">]</span> + <i>GateI</i>
<span class="fraction"><span class="ignored">(</span><span class="numerator"><i>d</i><span class="symbol">[</span><i>AHL</i><span class="symbol">]</span></span><span class="ignored">)/(</span><span class="denominator"><i>dt</i></span><span class="ignored">)</span></span> = <span class="fraction"><span class="ignored">(</span><span class="numerator"><i>v</i><sub><i>B</i></sub> + <i>v</i><sub><i>A</i></sub><i>k</i><sub><i>A</i></sub><span class="symbol">[</span><i>AHL</i><span class="symbol">]</span><sup>6</sup></span><span class="ignored">)/(</span><span class="denominator">1 + <i>k</i><sub><i>A</i></sub><span class="symbol">[</span><i>AHL</i><span class="symbol">]</span><sup>6</sup></span><span class="ignored">)</span></span> − <i>γ</i><sub><i>AHL</i></sub><span class="symbol">[</span><i>AHL</i><span class="symbol">]</span> + <i>GateI</i>
</div>
</div>
-
<p style="color:#919499">We thought these changes would improve the bi-stability of the system (thereby reducing the odds of a false positive), because they enhance the non-linearity inherent in the system which has been shown to play a vital role in the bi-stability of the system (see [9] – Gardner et. al. [10] – Cold Spring Harbor Vernalization, other sources we copied from [10]). When producing a similar analysis for the phase plane of this gate as we did for the phase plane of the original equation (see [7]), we found the values of (v_A,v_B) for which the system is bi-stable, and compared this analysis to the results of the analysis of the original analysis.</p>
+
 
-
<p style="color:#919499">*map of bi-stability with RNA Splint*<br>
+
<div class="formula" style="color:#919499">
-
*map of bi-stability without RNA Splint*
+
<span class="fraction"><span class="ignored">(</span><span class="numerator"><i>d</i><span class="symbol">[</span><i>AHL</i><span class="symbol">]</span></span><span class="ignored">)/(</span><span class="denominator"><i>dt</i></span><span class="ignored">)</span></span> = <span class="array"><span class="arrayrow"><span class="bracket align-left">⎛</span></span><span class="arrayrow"><span class="bracket align-left">⎝</span></span></span><span class="fraction"><span class="ignored">(</span><span class="numerator"><i>v</i><sub><i>B</i></sub> + <i>v</i><sub><i>A</i></sub><i>k</i><sub><i>A</i></sub><span class="symbol">[</span><i>AHL</i><span class="symbol">]</span><sup>2</sup></span><span class="ignored">)/(</span><span class="denominator">1 + <i>k</i><sub><i>A</i></sub><span class="symbol">[</span><i>AHL</i><span class="symbol">]</span><sup>2</sup></span><span class="ignored">)</span></span><span class="array"><span class="arrayrow"><span class="bracket align-right">⎞</span></span><span class="arrayrow"><span class="bracket align-right">⎠</span></span></span><sup>3</sup> − <i>γ</i><sub><i>AHL</i></sub><span class="symbol">[</span><i>AHL</i><span class="symbol">]</span> + <i>GateI</i>
 +
</div>
 +
 
 +
<p style="color:#919499">We thought these changes would improve the bi-stability of the system (thereby reducing the odds of a false positive), because they enhance the non-linearity inherent in the system which has been shown to play a vital role in the bi-stability of the system ([9],[10]). When producing a similar analysis for the phase plane of this gate as we did for the phase plane of the original equation (see [7]), we found the values of (v<sub>1</sub>,v<sub>2</sub>) for which the system is bi-stable, and compared this analysis to the results of the analysis of the original analysis.</p>
 +
<p style="color:#919499">
 +
<div class="Unindented">
 +
<div class="float">
 +
<a class="Label" name="Figure-2"> </a><div class="figure">
 +
<img class="embedded" src="https://static.igem.org/mediawiki/2014/e/ec/Technion-Israel-AlphaBool.jpg" style="width: 540px; max-width: 1200px; height: 405px; max-height: 900px;">
 +
<img class="embedded" src="https://static.igem.org/mediawiki/2014/f/fe/Technion-Israel-AlphaNorm.jpg" style="width: 540px; max-width: 1200px; height: 405px; max-height: 900px;">
 +
<img class="embedded" src="https://static.igem.org/mediawiki/2014/f/fe/Technion-Israel-RNA1Bool.jpg" style="width: 540px; max-width: 1200px; height: 405px; max-height: 900px;">
 +
<img class="embedded" src="https://static.igem.org/mediawiki/2014/c/c0/Technion-Israel-RNA1Norm.jpg" style="width: 540px; max-width: 1200px; height: 405px; max-height: 900px;">
 +
<img class="embedded" src="https://static.igem.org/mediawiki/2014/c/cb/Technion-Israel-RNA2Bool.jpg" style="width: 540px; max-width: 1200px; height: 405px; max-height: 900px;">
 +
<img class="embedded" src="https://static.igem.org/mediawiki/2014/6/6a/Technion-Israel-RNA2Norm.jpg" style="width: 540px; max-width: 1200px; height: 405px; max-height: 900px;">
 +
 
 +
</div>
 +
 
 +
</div>
 +
 
 +
</div>
</p>
</p>
 +
 +
<p style="color:#919499">
 +
Figure 2 From top to bottom: The Alpha System, the first model for the RNA Splint, and the second model for the RNA Splint. On the left:a map of the values of the parameters of the system for which it is bi-stable (bi-stable in red, mono-stable in blue). On the right: a map of the normalized bi-stability parameter we have defined as a function of its parameters.
 +
</p>
 +
 +
 +
<p style="color:#919499">
 +
It is clear from this graph, that the RNA Splint is more bi-stable than the original alpha system. Moreover, it is clear that in both models of the RNA Splint, it is bi-stable in the same points, but not to the same degree.
 +
</p>
 +
 +
<p style="color:#919499">We then proceeded to produce the Fokker Planck equation for the new system (see [8]), and we got the following results:</p>
<p style="color:#919499">We then proceeded to produce the Fokker Planck equation for the new system (see [8]), and we got the following results:</p>
-
<p style="color:#919499">*Fokker Planck Results With and without RNA Splint*</p>
+
<a class="Label" name="Figure-2"> </a><div class="figure">
 +
<img class="embedded" src="https://static.igem.org/mediawiki/2014/3/3b/1%2C2%2C0.jpg" alt="figure 1,2,0.jpg" style="width: 360px; max-width: 1200px; height: 270px; max-height: 900px;">
 +
<img class="embedded" src="https://static.igem.org/mediawiki/2014/2/2f/1%2C6%2C0.jpg" alt="figure 1,6,0.jpg" style="width: 360px; max-width: 1200px; height: 270px; max-height: 900px;">
 +
<img class="embedded" src="https://static.igem.org/mediawiki/2014/6/68/3%2C2%2C0.jpg" alt="figure 3,2,0.jpg" style="width: 360px; max-width: 1200px; height: 270px; max-height: 900px;">
 +
<img class="embedded" src="https://static.igem.org/mediawiki/2014/2/26/1-2-0.gif" alt="figure 1-2-0.gif" style="width: 360px; max-width: 1200px; height: 270px; max-height: 900px;">
 +
<img class="embedded" src="https://static.igem.org/mediawiki/2014/f/f9/1-6-0.gif" alt="figure 1-6-0.gif" style="width: 360px; max-width: 1200px; height: 270px; max-height: 900px;">
 +
<img class="embedded" src="https://static.igem.org/mediawiki/2014/e/e0/3-2-0.gif" alt="figure 3-2-0.gif" style="width: 360px; max-width: 1200px; height: 270px; max-height: 900px;">
 +
<img class="embedded" src="https://static.igem.org/mediawiki/2014/e/e4/1%2C2%2C1.jpg" alt="figure 1,2,1.jpg" style="width: 360px; max-width: 1200px; height: 270px; max-height: 900px;">
 +
<img class="embedded" src="https://static.igem.org/mediawiki/2014/4/49/1%2C6%2C1.jpg" alt="figure 1,6,1.jpg" style="width: 360px; max-width: 1200px; height: 270px; max-height: 900px;">
 +
<img class="embedded" src="https://static.igem.org/mediawiki/2014/0/04/3%2C2%2C1.jpg" alt="figure 3,2,1.jpg" style="width: 360px; max-width: 1200px; height: 270px; max-height: 900px;">
 +
<img class="embedded" src="https://static.igem.org/mediawiki/2014/b/bf/1-2-1.gif" alt="figure 1-2-1.gif" style="width: 360px; max-width: 1200px; height: 270px; max-height: 900px;">
 +
<img class="embedded" src="https://static.igem.org/mediawiki/2014/3/38/1-6-1.gif" alt="figure 1-6-1.gif" style="width: 360px; max-width: 1200px; height: 270px; max-height: 900px;">
 +
<img class="embedded" src="https://static.igem.org/mediawiki/2014/8/87/3-2-1.gif" alt="figure 3-2-1.gif" style="width: 360px; max-width: 1200px; height: 270px; max-height: 900px;">
 +
<div class="caption">
 +
<p style="color:#919499">Figure 2 Left to Right: The Alpha System, The <span class="formula">1<sup><i>st</i></sup></span> Model for the RNA Splint, The <span class="formula">2<sup><i>nd</i></sup></span> Model. Top to Bottom: the Heat Map and then the Time-Lapse of the system starting at each of the modes of actiavtion.
 +
</p></div>
 +
<div class="Indented">
 +
<p style="color:#919499">It is clear that the “on” (high AHL concentration - low on the graph) state of all of these systems is more stable than their off state, and that there is a high likelihood of the systems spontaneously mocing to the “on” state (i.e. false-positive). However, we may notice that for both models of RNA Splint, the time for a shift from the “off” state to the “on” state is longer than for the Alpha System which is to say, that there is a far lower chance of recieving a false positive.
 +
</p></div>
</center>
</center>
</header>
</header>
Line 523: Line 611:
</article>
</article>
<center>
<center>
-
<h1>A Simulated Model for the Azo-Benzene</h1>
+
<p style:="line-height:1.75em;">
-
<p><br>We aimed to create a dynamic simulation of bacteria with Azobenzene molecules attached to their membranes. These molecules, once activated by an outside stimulus (usually a certain wavelength of photons) - will act as a sort of “Velcro” between the bacteria; they attach to other bacteria upon contact forming clusters. <br>
+
<h1 style="font-size:2em;">A Simulated Model for the Azobenzene</h1>
-
The clusters of bacteria will thereafter act as one unit - a biofim.<br>
+
<br>
-
With this model we opted for a "brute-force" simulation of particles in a fluid under the following terms:<br><br>
+
We aimed to create a dynamic simulation of bacteria with Azobenzene molecules attached to their membranes. These molecules, once activated by an outside stimulus (usually a certain wavelength of photons) - will act as a sort of “Velcro” between the bacteria; they attach to other bacteria upon contact forming clusters.  
-
&#149; The simulation “Playground” will be a discreet matrix of the dimentions x × y × z.<br>
+
<br>
-
&#149; Each bacterium will occupy a 1 × 1 × 1 point in in space.<br>
+
The clusters of bacteria will thereafter act as one unit - a biofim.
-
&#149; For every t=t+1 passage of time, each bacterium “tumbles” a random amount of steps in a random direction, we called this a "Tumble Vector"<br>
+
<br>
-
&#149; Each bacterium can have either a “sticky” or “non-sticky” value corresponding to it. This is equivalent of assuming that all azobenzene molecules “switch on” at once in all directions.<br>
+
With this model we opted for a "brute-force" simulation of particles in a fluid under the following terms:
-
&#149; Each sticky bacterium (i.e. with a “sticky” value) will “attach” to any “neighbor” (i.e. a bacterium with a location of 0, ± 1 in either direction), after which they will “tumble” together as one cluster, with their direction being determined by summing up all the bacteria's "Tumble Vectors" together.<br>
+
<br>
-
&#149; Once a bacterium has a neighbor attached to it, they cannot separate and that neighbor's location is forever occupied by the same bacterium, it cannot be overridden.<br>
+
 
-
&#149; A sticky bacterium on the edge of a cluster can stick to any neighboring bacterium. If said neighbor is already a part of a cluster we now have two clusters joining to form a "super-cluster" – which does not vary in definition from a normal cluster programming-wise.<br><br>
+
-
The simulation was written using C++, using tumble and playground sizes values to simulate the world of actual bacteria. The results were then rendered in MATLAB:
+
</p>
</p>
-
<p>We improved this simulation by adding options for these clusters to degrade, initially by giving the connections the ability to be torn apart, from simple stress, but later on, when the size of these clusters became clear from the model, we decided to add another degradation due to problematic structural integrity of the cell clusters.</p>
+
</center>
-
<p>*Graph of large Cell Cluster with structurally weak point/s highlighted*</p>
+
<div style="margin-left:15%; width:70%;">
-
<p>We define a cluster as having a problematic structural integrity, if it is very large, and can be divided into two large enough parts which are held together by only a small number of cells. When formulating the condition for which we want to search over the graph of cells (defined as connected if they have an Azo-Benzene link between them), we obtain an algorithmic problem which (while we have not proven is NP-hard), we do not know how to solve (or how to polynomialy verify).</p>
+
<ul>
-
<p>As a result, we had to try to create a probabilistic algorithm which would solve the problem of locating these weak points in the cluster (see [11]). When running this algorithm several times on the above cluster, we were capable of finding the weak points indicated in the following graph:</p>
+
&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<li>&#149;&nbsp;&nbsp;The simulation “Playground” will be a discreet matrix of the dimentions x × y × z.</li>
-
<p>*Graph of weak points*</p>
+
 
-
<p>When combining these degradation effects with our original model, we got the following behavior:</p>
+
&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<li>&#149;&nbsp;&nbsp; Each bacterium will occupy a 1 × 1 × 1 point in in space.</li>
-
<p>*Results of new AB simulation*</p>
+
 
-
</center>
+
&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<li>&#149;&nbsp;&nbsp; For every t=t+1 passage of time, each bacterium “tumbles” a random amount of steps in a random direction, we called this a "Tumble Vector"</li>
 +
 
 +
&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<li>&#149;&nbsp;&nbsp; Each bacterium can have either a “sticky” or “non-sticky” value corresponding to it. This is equivalent of assuming that all azobenzene molecules “switch on” at once in all directions.</li>
 +
 
 +
&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<li>&#149;&nbsp;&nbsp; Each sticky bacterium (i.e. with a “sticky” value) will “attach” to any “neighbor” (i.e. a bacterium with a location of 0, ± 1 in either direction), after which they will “tumble” together as one cluster, with their direction being determined by summing up all the bacteria's "Tumble Vectors" together.</li>
 +
 
 +
&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<li>&#149;&nbsp;&nbsp; Once a bacterium has a neighbor attached to it, they cannot separate and that neighbor's location is forever occupied by the same bacterium, it cannot be overridden.</li>
 +
 
 +
&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<li>&#149;&nbsp;&nbsp; A sticky bacterium on the edge of a cluster can stick to any neighboring bacterium. If said neighbor is already a part of a cluster we now have two clusters joining to form a "super-cluster" – which does not vary in definition from a normal cluster programming-wise.</li>
 +
</ul>
 +
</div>
 +
<br>
 +
 
 +
<center>
 +
<p style:="line-height:1.75em;">
 +
The simulation was written using C++, using tumble and playground sizes values to simulate the world of actual bacteria. The results were then rendered in MATLAB:<br>
 +
</p>
 +
<div class="float">
 +
<a class="Label" name="Figure-3"> </a><div class="figure">
 +
<img class="embedded" src="https://static.igem.org/mediawiki/2014/a/aa/Nucleation.gif" alt="figure Nucleation.gif" style="width: 642px; height: 304px; margin-left: auto; margin-right: auto;">
 +
<div class="caption">
 +
Figure 3 A simulation of the clustering of cells in the presence of AB. The simulation contains 10,000 cells of which 2,000 are sticky simulated over 400 secs, with a time-lapse of 4 seconds per image. We can clearly see that over half of the cells are joined into 1 cluster at the end of the simulation, leading us to believe that the clustering would have a visible effect on the OD of the sample.
 +
</div>
 +
 
 +
</div>
 +
 
 +
</div>
 +
</center>
 +
</div>
</div>

Latest revision as of 03:57, 18 October 2014


Safie by Technion-Israel

Why it Should Work

Why Alpha System Should Work – a deterministic model of alpha system

When modelling our system, we began with the simplest method known – deterministic rate equations. Moreover, from the design it was clear that the most important benchmark for the signal within the system would be the concentration of AHL as a function of time, so we began by modelling this part of our system. It took only a simple derivation (see [1]) to obtain these equations which characterize this part of the system:

(d[mRNALuxI])/(dt) = (vB + vAkA[AHL]2)/(1 + kA[AHL]2) − γmRNALuxI[mRNALuxI] + GateI
(d[LuxI])/(dt) = αLuxI[mRNALuxI] − γLuxI[LuxI]
(d[AHL])/(dt) = αAHL[AHL] − γAHL[AHL]

(For a glossary see [1]).

We began to analyze this system by attempting to simplify it, by assuming a steady state solution wherever possible. Using this method (see [2]) we managed to obtain this equation:

(d[AHL])/(dt) = (vB + vAkA[AHL]2)/(1 + kA[AHL]2) − γAHL[AHL] + GateI

It is clear from the goals of our system, that we want to have some sort of bi-stability in the result, when the term Gate I is small (see [2]). The answer to whether this condition is met, would obviously depend on the constants of the system for which we could not find a reliable source, but using a simple geometric analysis of the phase space (see [3]), we were able to produce a graph showing for which values of (v1,v2) we could configure the system (by changing the IPTG concentration and the OD) to show bi-stability:

figure AlphaNorm.jpg
Figure 1 On the left:a map of the values of the parameters of the system for which it is bi-stable (bi-stable in red, mono-stable in blue). On the right: a map of the normalized bi-stability parameter we have defined as a function of its parameters.

It is clear from this graph, that the alpha system is bi-stable for a large part of the range of possible inputs.

Why it Should Fail

Why Alpha System Should Fail – a stochastic model of alpha system

The above model assumes a low-noise system (as do all rate equation models), but especially when constructing a bi-stable network, it is important to consider the noise. To do this we need to create a stochastic model, which in our case, we based upon the commonly used Fokker Planck equation. Using the derivation found in [4] (Van Kampen "Stochastic Processes in Physics and Chemistry", Third Edition), we produced the Fokker Planck variant of the equation for the AHL concentration shown above

After analyzing this equation as explained in [5], we produced the following results (using a point on the (v1,v2) plane which the previous analysis showed would be bi-stable)

figure 1,2,0.jpg figure 1-2-0.gif figure 1,2,1.jpg figure 1-2-1.gif
Figure 1 From Top To Bottom: Fokker Planck in the Alpha System when the system begins off, and then when it begins off: On the left is a heat map of the probability distribution function, as a function of time. On the right is a gif showing the probability distribution function over 100 timelapses
Clearly the “on” state (high AHL concentration - low on the graph) of our system is the more stable state of our system - so much so that it can spontaneously switch to the on state. This means that our system is bound to have a high likelihood of false positives.
Synthetic Biofilm Formation

A Simulated Model for the Azobenzene


We aimed to create a dynamic simulation of bacteria with Azobenzene molecules attached to their membranes. These molecules, once activated by an outside stimulus (usually a certain wavelength of photons) - will act as a sort of “Velcro” between the bacteria; they attach to other bacteria upon contact forming clusters.
The clusters of bacteria will thereafter act as one unit - a biofim.
With this model we opted for a "brute-force" simulation of particles in a fluid under the following terms:

         
  • •  The simulation “Playground” will be a discreet matrix of the dimentions x × y × z.
  •      
  • •   Each bacterium will occupy a 1 × 1 × 1 point in in space.
  •      
  • •   For every t=t+1 passage of time, each bacterium “tumbles” a random amount of steps in a random direction, we called this a "Tumble Vector"
  •      
  • •   Each bacterium can have either a “sticky” or “non-sticky” value corresponding to it. This is equivalent of assuming that all azobenzene molecules “switch on” at once in all directions.
  •      
  • •   Each sticky bacterium (i.e. with a “sticky” value) will “attach” to any “neighbor” (i.e. a bacterium with a location of 0, ± 1 in either direction), after which they will “tumble” together as one cluster, with their direction being determined by summing up all the bacteria's "Tumble Vectors" together.
  •      
  • •   Once a bacterium has a neighbor attached to it, they cannot separate and that neighbor's location is forever occupied by the same bacterium, it cannot be overridden.
  •      
  • •   A sticky bacterium on the edge of a cluster can stick to any neighboring bacterium. If said neighbor is already a part of a cluster we now have two clusters joining to form a "super-cluster" – which does not vary in definition from a normal cluster programming-wise.

The simulation was written using C++, using tumble and playground sizes values to simulate the world of actual bacteria. The results were then rendered in MATLAB:

figure Nucleation.gif
Figure 3 A simulation of the clustering of cells in the presence of AB. The simulation contains 10,000 cells of which 2,000 are sticky simulated over 400 secs, with a time-lapse of 4 seconds per image. We can clearly see that over half of the cells are joined into 1 cluster at the end of the simulation, leading us to believe that the clustering would have a visible effect on the OD of the sample.