Difference between revisions of "Team:Technion Israel/Model"

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{{Technion_Israel}}
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{{Team:Technion_Israel/navbar}}
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<html>
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<html lang="en">
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    <head>
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        <!-- Title for the page & metadata  -->       
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        <title>S.tar, by iGEM Technion 2016</title>
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    <meta charset="utf-8">
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    <meta name="navbar" content="width=device-width, initial-scale=1">
  
  
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<div class="column full_size judges-will-not-evaluate">
 
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<h3>★  ALERT! </h3>
 
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<p>This page is used by the judges to evaluate your team for the <a href="https://2016.igem.org/Judging/Awards#SpecialPrizes">Best Model award</a>. </p>
 
  
 +
<style>/* inline page CSS */
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body {
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}
  
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<p> Delete this box in order to be evaluated for this medal. See more information at <a href="https://2016.igem.org/Judging/Pages_for_Awards/Instructions"> Instructions for Pages for awards</a>.</p>
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/* ==========Background and effects ========== */
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</div>
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/*Make sure the div is unuiqe to each page*/
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.model_wrapper {
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position: relative;
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background: white; /* For browsers that do not support gradients */
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background: -webkit-linear-gradient(white, #ecf7fb, white); /* For Safari 5.1 to 6.0 */
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background: -o-linear-gradient(white, #ecf7fb, white); /* For Opera 11.1 to 12.0 */
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background: -moz-linear-gradient(white, #ecf7fb, white); /* For Firefox 3.6 to 15 */
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background: linear-gradient(white, #ecf7fb, white); /* Standard syntax */
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/*background-image:url('addres');
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background-size:cover;
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filter: blur(5px) grayscale(80%) opacity(10%);*/
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}
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/* ========== END: and effects ========== */
  
  
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/*
 +
Object in external CSS sheet:
  
 +
nav-tabs, cont_tabs:
 +
Open diffrent tabs, we uses imgs.
  
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back-to-top:
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go back to top. Apears only when going down the page.
  
 +
cont_box:
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The containers (box) which hold the texts and imgs in the page.
  
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img_cont:
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Every in-content-page img needs to have this class of img.
  
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<div class="column full_size">
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no-title-col:
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<h2> Modeling</h2>
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Every in-content-page img needs to have this class of col.
−
<p>Mathematical models and computer simulations provide a great way to describe the function and operation of BioBrick Parts and Devices. Synthetic Biology is an engineering discipline, and part of engineering is simulation and modeling to determine the behavior of your design before you build it. Designing and simulating can be iterated many times in a computer before moving to the lab. This award is for teams who build a model of their system and use it to inform system design or simulate expected behavior in conjunction with experiments in the wetlab.</p>
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*/
  
 +
/*canceling wiki bug (inlarge imgs stuck the page)*/
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.modal-backdrop {
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position: relative;
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}
  
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<h5> Inspiration </h5>
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.ocenter .img-responsive {
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<p>
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margin: 0 auto;
−
Here are a few examples from previous teams:
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}
 +
 
 +
.vcenter {
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display: inline-block;
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vertical-align: middle;
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float: none;
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}
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.references {
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  font-size: 16px;
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padding: 20px;
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}
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</style>
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<script>
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//up arrow:
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$(document).ready(function(){
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$(window).scroll(function () {
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if ($(this).scrollTop() > 350) {
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$('#back-to-top').fadeIn();
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} else {
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$('#back-to-top').fadeOut();
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}
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});
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// scroll body to 0px on click
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$('#back-to-top').click(function () {
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$('#back-to-top').tooltip('hide');
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$('body,html').animate({
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scrollTop: 0
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}, 800);
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return false;
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});
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$('#back-to-top').tooltip('show');
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});
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////////////////////////
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//Code: inlarge img on click:
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$(function() {
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$('.pop').on('click', function() {
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$('.imagepreview').attr('src', $(this).find('img').attr('src'));
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$('#imagemodal').modal('show'); 
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});
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});
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//////////////////////
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///pop-ups:
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$(document).ready(function(){
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$('[data-toggle="popover"]').popover();
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$(function () {
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$('button').popover();
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});
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</script>
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</head>
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<body>
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<header>
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</header>
 +
 +
<!-- ===== One overall container (wrapper) ===== -->
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<div class="desk_wrapper">
 +
 
 +
<!-- ======== Cover photo: ======== -->
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<div class="row">
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<div class="col-xs-12">
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<img src="https://static.igem.org/mediawiki/2016/d/df/T--Technion_Israel--modelingcov.jpg" class="img-responsive img-center cont_cover" width="100%">
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</div>
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</div>
 +
<!-- ===== END: Cover photo: ===== -->
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 +
 
 +
 
 +
<!-- =============== Tabs names: =============== -->
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<div class="row">
 +
<div class="col-sm-8 col-sm-offset-2"><!-- 8/12 -->
 +
 
 +
 +
<ul class="nav nav-tabs" role="tablist">
 +
 
 +
<li role="presentation" class="col-sm-3 col-xs-6">
 +
<a href="#111" aria-controls="111" role="tab" data-toggle="tab">
 +
<img src="https://static.igem.org/mediawiki/2016/d/db/T--Technion_Israel--icon_intro.png" class="img-responsive img-center cont_tabs" width="75" height="75">
 +
<br><h4 class="text-center">Introduction</h4>
 +
</a>
 +
</li>
 +
 
 +
<li role="presentation" class="col-sm-3 col-xs-6">
 +
<a href="#222" aria-controls="222" role="tab" data-toggle="tab">
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<img src="https://static.igem.org/mediawiki/2016/4/49/T--Technion_Israel--icon_lab.png" class="img-responsive img-center cont_tabs" width="75" height="75">
 +
<br><h4 class="text-center">Chemo-Repellent</h4>
 +
</a>
 +
</li>
 +
 
 +
<li role="presentation" class="col-sm-3 col-xs-6">
 +
<a href="#333" aria-controls="333" role="tab" data-toggle="tab">
 +
<img src="https://static.igem.org/mediawiki/2016/4/45/T--Technion_Israel--icon_results.png" class="img-responsive img-center cont_tabs" width="75" height="75">
 +
<br><h4 class="text-center">Bacterial Concentration</h4>
 +
</a>
 +
</li>
 +
 
 +
<li role="presentation" class="col-sm-3 col-xs-6">
 +
<a href="#444" aria-controls="444" role="tab" data-toggle="tab">
 +
<img src="https://static.igem.org/mediawiki/2016/4/47/T--Technion_Israel--icon_outlook.png" class="img-responsive img-center cont_tabs" width="75" height="75">
 +
<br><h4 class="text-center">Conclusions</h4>
 +
</a>
 +
</li>
 +
 
 +
</ul>
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</div>
 +
</div>
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<!-- ========== End: Tabs panel ========== -->
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<div class="tab-content">
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 +
 
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<!-- 111111111111111111111111111111111111111111111111111111111 -->
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<div role="tabpanel" class="tab-pane fade in active" id="111">
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<div class="cont_box">
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 +
 
 +
<div class="row"><!-- #1 row -->
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<div class="col-sm-8 col-sm-offset-2"><!-- 8/12 -->
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 +
<div class="cont_box">
 +
<div class="row">
 +
<div class="col-sm-12">
 +
 +
<div class="col-md-12 col-sm-12">
 +
<h2 class="">Introduction</h2>
 +
<p class="text-justify">
 +
The goal of this model is to describe the processes inside the Flash Lab system:<br>
 +
<p style="margin-left: 40px">
 +
- Change in the concentration of chemo-repellent.<br>
 +
- Change in the concentration of bacteria.<br>
 
</p>
 
</p>
−
<ul>
+
<p class="text-justify">
−
<li><a href="https://2014.igem.org/Team:ETH_Zurich/modeling/overview">ETH Zurich 2014</a></li>
+
This model is based on the Keller – Segal equation of chemotaxis <b>(1)</b> in
−
<li><a href="https://2014.igem.org/Team:Waterloo/Math_Book">Waterloo 2014</a></li>
+
a one dimensional problem (Thin channel).<br>
−
</ul>
+
<br>
 +
It is important to notice that this model can show the overall behavior and not exact
 +
values. The final system is supposed to detect a variety of materials in many different
 +
solvents, each of them has its own diffusion properties. Also, some aspects such
 +
as working conditions (temperature, humidity etc.) might change in widespread commercial
 +
use and affect the results. Taking those into account, further fitting will be necessary.
 +
                                                                                                        <br>
 +
<br>
 +
                                                                                                      For PDF version of this model press <a href="https://static.igem.org/mediawiki/igem.org/1/1c/T--Technion_Israel-ModelPDF.pdf">here</a>
 +
</p>
 +
</div>
 +
</div>
 +
</div>
 +
</div>
 +
</div>
 +
</div><!-- END: #1 row -->
  
  
 +
</div>
 +
</div>
 +
<!--===============-->
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<!-- 222222222222222222222222222222222222222222222222222222222 -->
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<div role="tabpanel" class="tab-pane fade" id="222">
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<div class="cont_box">
 +
<div class="row"><!-- #1 row -->
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<div class="col-sm-8 col-sm-offset-2"><!-- 8/12 -->
 +
 +
<div class="cont_box">
 +
<div class="row">
 +
 +
<div class="col-md-12 col-sm-12">
 +
<h2 class="">Chemo-Repellent Concentration</h2>
 +
 +
<div class="col-md-12 col-sm-12">
 +
<h3 class="">Model</h3>
 +
<p class="text-justify"> </p>
 +
</div>
 +
 +
 +
<br>
 +
 +
<p class="text-justify">
 +
The basic assumptions of the model for the chemo-repellent are:<br>
 +
<br>
 +
<p style="margin-left: 40px">
 +
<b>-</b>There are no forces except for diffusion:
 +
<p style="margin-left: 40px">
 +
<b>-</b> Chemo-repellent concentration in the sample is relatively low and does not cause osmotic pressure.<br>
 +
<b>-</b> The changes in pressure due to loading the sample is negligible.<br>
 +
<b>-</b> No other significant external forces (for example, moving the chip while using).<br><br>
 +
</p>
 +
<p class="text-justify">
 +
<p style="margin-left: 40px">
 +
<b>-</b> The bacteria do not consume the chemo-repellent and its concentration does not change with time. In contrast, in case of a chemo-attractant this assumption is not valid.
 +
<br>
 +
<br>
 +
<b>-</b> Expect to detect small proteins and molecules (the ligands). The diffusion coefficient for such materials is about 10<sup>-9 </sup>[m<sup>2</sup>/s].
 +
<br>
 +
<br>
 +
<b>-</b> Because of the geometric properties of the channel and the expected diffusion coefficient, this is approximately a semi-infinite one dimensional problem.
 +
<br>
 +
<br>
 +
<b>-</b> Initial condition: no chemo-repellent is present in the chip at time zero (a).
 +
<br>
 +
<br>
 +
 +
<b>-</b> Boundary condition: at infinite distance the concentration is zero (b) and the there is conservation of dissolved material mass (c).
 +
<br>
 +
<br>
 +
</p>
 +
<p class="text-justify">
 +
 +
We modelled the change in concentration of the chemo-repellent based on a "Top Hat Function" solution for the diffusion problem:
 +
<br>
 +
 +
</p>
 +
 +
<!--6 text - 6 img div-->
 +
<div class="row">
 +
<div class="col-md-6 col-sm-12 vcenter"><!--6 text-->
 +
<p class="text-justify">
 +
<b>v [M]</b> is the chemo-repellent concentration.<br>
 +
<b>D[m<sup>2</sup>/s]</b> is the chemo-repellent diffusion coefficient.<br>
 +
<b>N [mol]</b> is the number of moles of the repellent .<br>
 +
<b>A [m<sup>2</sup>]</b> is the cross-section area of the channel. <br>
 +
<b>h [m]</b> is length of the sample in the channel.<br>
 +
<b>x [m]</b> is the distance on the channel.<br>
 +
<b>t [s]</b> is time.
 +
<br>
 +
<br>
 +
The solution for this problem is:
 +
</p>
 +
</div><!--
 +
--><div class="col-md-6 col-sm-12 vcenter"><!--6 img div-->
 +
<a class="pop ocenter">
 +
<img src="https://static.igem.org/mediawiki/igem.org/f/ff/T--Technion_Israel--Model40.png" class="img-responsive img-center" width="450" style="cursor: pointer;">
 +
</a>
 +
<p class="text-center"><b>Equation 1:</b> Chemo-repellent diffusion problem.</p>
 +
</div>
 +
</div>
 +
 +
 +
<div class="col-md-12 col-sm-12 new-row no-title-col">
 +
<div class="row">
 +
<div class="col-md-12">
 +
<a class="pop">
 +
<img src="https://static.igem.org/mediawiki/2016/0/0d/T--Technion_Israel--model_2.2.PNG" class="img-responsive img-center img-cont" style="cursor: pointer;" width="300"><br>
 +
</a>
 +
<p class="text-center"><b>Equation 2:</b> General solution for chemo-repellent diffusion problem.</p>
 +
</div>
 +
</div>
 +
</div>
 +
 +
<div class="col-md-12 col-sm-12 new-row no-title-col">
 +
<div class="row">
 +
<div class="col-md-12">
 +
<a class="pop">
 +
<img src="https://static.igem.org/mediawiki/2016/5/50/T--Technion_Israel--graph2.1.PNG" class="img-responsive img-center img-cont" style="cursor: pointer;" width="300"><br>
 +
</a>
 +
<p class="text-center"><b>Graph 1:</b> Chemo-repellent concentration: top hat diffusion.</p>
 +
<br>
 +
</div>
 +
</div>
 +
</div>
 +
 +
<div class="col-md-12 col-sm-12">
 +
<p class="text-justify">
 +
If x=0 [m] is defined as the point where the sample is loaded to the chip, then the solution for the diffusion problem is:
 +
 +
</p>
 +
</div>
 +
 +
<div class="col-md-12 col-sm-12 new-row no-title-col">
 +
<div class="row">
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<div class="col-md-12">
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<a class="pop">
 +
<img src="https://static.igem.org/mediawiki/2016/d/de/T--Technion_Israel--model_2.3.PNG" class="img-responsive img-center img-cont" style="cursor: pointer;" width="300"><br>
 +
</a>
 +
<p class="text-center"><b>Equation 3:</b> Chemo-repellent concentration.
 +
<br>
 +
</p>
 +
</div>
 +
</div>
 +
</div>
 +
</div>
 +
</div>
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<div class="cont_box">
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<div class="row">
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<div class="col-sm-12">
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<div class="col-md-12 col-sm-12">
 +
<h3 class="">Model Results</h3>
 +
<p class="text-justify">
 +
We ran the chemo-repellent concentration equation in Matlab - Our code can be found <a href="https://static.igem.org/mediawiki/igem.org/5/5a/T--Technion_Israel--ModelRepCode.zip">here</a>. The parameters used:
 +
</p>
 +
 +
 +
<div class="col-md-12 col-sm-12 new-row no-title-col">
 +
<a class="pop">
 +
<img src="https://static.igem.org/mediawiki/2016/9/9d/T--Technion_Israel--table2.1.JPG" class="img-responsive img-center img-cont" width="300" style="cursor: pointer;"><br>
 +
</a>
 +
<p class="text-center"><b>Table 1:</b> Parameters for diffusion model.</p>
 +
</div>
 +
</div>
 +
 +
<p class="text-justify">
 +
*This is the diffusion coefficient for potassium permanganate (see "Comparison to Experiment")<br>
 +
** h = (Sample volume)/(Reservoir cross section).<br>
 +
</p>
 +
 +
 +
 +
<div class="col-sm-12">
 +
<div class="col-md-12 col-sm-12">
 +
<p class="text-justify">
 +
The output for different times:
 +
</p>
 +
</div>
 +
 +
<div class="col-md-12 col-sm-12 new-row no-title-col">
 +
<a class="pop">
 +
<img src="https://static.igem.org/mediawiki/2016/4/47/T--Technion_Israel--graph2.2.PNG" class="img-responsive img-center img-cont" width="300" style="cursor: pointer;"><br>
 +
</a>
 +
<p class="text-center"><b>Graph 2:</b> Model for repellent concentration.</p>
 +
</div>
 +
</div>
 +
 +
<div class="col-md-12 col-sm-12">
 +
<p class="text-justify">
 +
The change in distance of the diffusion limit between 0 to 15 minutes, is relatively big. As the time passes the diffusion limit's speed lowers significantly, and the concentration profile becomes more linear.
 +
<br>
 +
<br>
 +
</p>
 +
<h3 class="">Comparison to Experiment</h3>
 +
<p class="text-justify">
 +
Most diffusion experiments need a dedicated system that is based on the diffusion of an isotope or a fluorescent material that can be detected easily and very precisely. we chose a more basic system since this is only a preliminary testing for showing that the overall system behaves as we expect.
 +
<br>
 +
<br>
 +
The experiment ran as the setup shown in the "Overview" section: The channel was filled with bacteria in motility buffer and then the sample was inserted. We replaced the motility buffer with water and the chemo-repellent with potassium permanganate in the following amounts:
 +
</p>
 +
</div>
 +
<div class="col-md-12 col-sm-12 new-row no-title-col">
 +
<a class="pop">
 +
<img src="https://static.igem.org/mediawiki/2016/9/9b/T--Technion_Israel--table2.2.JPG" class="img-responsive img-center img-cont" width="500" style="cursor: pointer;"><br>
 +
</a>
 +
<p class="text-center"><b>Table 2:</b> Substance for diffusion experiment.</p>
 +
</div>
 +
 +
<div class="col-md-12 col-sm-12">
 +
<p class="text-justify">
 +
 +
 +
Modeling was done under the assumption that the motility buffer acts as water, due to the fact that it is mostly water (~98%). Potassium permanganate is a salt with a known diffusion limit and acts as most of the materials we want to detect using our system (small molecules). Also, it has an intense pink color in low concentration which makes diffusion limit visible. <br>
 +
<br>
 +
We ran the experiment 4 times, with a standard ruler to measure the distance of the diffusion limit.
 +
 +
</p>
 +
</div>
 +
 +
<div class="col-md-12 col-sm-12 new-row no-title-col">
 +
<a class="pop">
 +
<img src="https://static.igem.org/mediawiki/2016/a/a1/T--Technion_Israel--figure2.1.PNG" class="img-responsive img-center img-cont" width="300" style="cursor: pointer;"><br>
 +
</a>
 +
<p class="text-center"><b>Fig. 2:</b> Diffusion of potassium permanganate in water in different times (enhanced picture).</p>
 +
</div>
 +
 +
<div class="col-md-12 col-sm-12">
 +
<p class="text-justify">
 +
As expected by the mathematical model, the diffusion limit starts moving relatively fast and its speed decreases rapidly. The difference in distance between the models and the experiment can be explained by:
 +
<br> <br>
 +
<p style="margin-left: 40px">
 +
<b>-</b>The actual diffusion limit is in too low concentration of potassium permanganate to be seen in the naked eye. If the visible concentration is about  1.5&#42;10<sup>-5</sup> [M] the experimental results line up with the model (graph 3). <br>
 +
<br>
 +
 +
<b>-</b>The ruler is a crude measuring tool. Its mistake is &plusmn; 0.5 [mm].
 +
<br>
 +
<br>
 +
 +
<b>-</b>Difficulties in loading uniformly the sample, especially in low volumes. Mistakes in loading the sample inside the bacterial fluid and not on, or sticking the drop of sample to one of the entry slot walls will cause uneven diffusion.
 +
<br>
 +
</p>
 +
</div>
 +
 +
<div class="col-md-12 col-sm-12 new-row no-title-col">
 +
<a class="pop">
 +
<img src="https://static.igem.org/mediawiki/2016/b/b1/T--Technion_Israel--graph2.3.PNG" class="img-responsive img-center img-cont" width="496" style="cursor: pointer;"><br>
 +
</a>
 +
<p class="text-center"><b>Graph 3:</b> Comparison of diffusion model (c=0.00015[M]) to experiment.</p>
 +
</div>
 +
</div>
 +
</div>
 +
</div>
 +
</div>
 +
</div>
 +
</div>
 +
 +
 +
</div>
 +
</div>
 +
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<h2>Bacterial Concentration</h2>
 +
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 +
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<div class="col-md-12 col-sm-12">
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<p class="text-justify">
 +
The basic mathematical model for bacterial chemotaxis is the Keller-Segal equations (1) of chemotaxis:<br>
 +
 +
</p>
 +
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<img src="https://static.igem.org/mediawiki/igem.org/4/45/T--Technion_Israel--Model18.png" class="img-responsive img-center img-cont" width="450" style="cursor: pointer;">
 +
</a>
 +
<p class="text-center"><b>Equation 1:</b> Keller-Segal equation</p>
 +
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 +
 +
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<p class="text-justify">
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                                                                                where:
 +
                                                                                <br>
 +
                                                                                <br>
 +
                                      <b>u[M]</b> is the bacteria concentration.<br>
 +
                                          <b>v[M]</b> is the chemo-repellent concentration.<br>
 +
                                          <b>k<sub>1</sub>[m<sup>2</sup>/s]</b> is the bacteria diffusion coefficient. <br>
 +
                                          <b>k<sub>2</sub>[m<sup>2</sup>/M &#42; s]</b> is the bacteria chemotactic coefficient. <br>
 +
                                          <b>k<sub>3</sub>[M/s]</b> is bacteria life and death coefficient. <br>
 +
                                          <b>x[m]</b> is the distance in the channel. <br>
 +
                                          <b>t[s]</b> is time. <br>
 +
<br>
 +
The basic assumptions of the chemotaxis model are:
 +
                                      <p style="margin-left: 40px">
 +
<br>
 +
                                        <b>-</b>k<sub>2</sub> describes how sensitive is the bacteria to changes in chemo-repellent concentration.
 +
In other words, as its negative value decreases the bacteria will react more violently to same repellent exposure. 
 +
<br>
 +
<br>
 +
<b>-</b>Because of the geometric properties of the channel, this is approximately a one dimensional problem.
 +
<br>
 +
<br>
 +
 +
<b>-</b>We ran our tests in a short time scale (t&#60;30[min]) so we presume that the change in concentration
 +
of bacteria due to life and death is negligible, k<sub>3</sub>=0.
 +
<br>
 +
<br>
 +
<b>-</b>The chemo-repellent concentration is known for every x and t.
 +
<br>
 +
<br>
 +
<b>-</b>There are no changes in the flux of bacterial concentration at the start (a) and at end of the channel (b).
 +
<br>
 +
<br>
 +
<br>
 +
</p>
 +
</div>
 +
</div>
 +
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<div class="row">
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<h3>Mathematical Model</h3>
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</div>
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<div class="col-md-12 col-sm-12">
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<p class="text-justify">
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Under these assumptions the equation takes the form:
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</p>
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<a class="pop ocenter">
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<img src="https://static.igem.org/mediawiki/igem.org/d/d6/T--Technion_Israel--Model11.png" class="img-responsive img-center img-cont" style="cursor: pointer;"><br>
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</a>
 +
<p class="text-center"><b>Equation 2:</b> The Bacterial concentration problem.</p>
 +
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<div class="col-md-12 col-sm-12">
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<p class="text-justify">
 +
The above partial differential equation cannot be solved analytically, so we must turn to
 +
numerical analysis tools. The implicit Euler method is one of the most basic numerical
 +
methods for the solution of ordinary and partial differential equations. This method is
 +
guaranteed to be stable and gives us the solution for the entire space in a single point in time.<br>
 +
<br>
 +
Using the following discretization:
 +
</p>
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<img src="https://static.igem.org/mediawiki/igem.org/b/ba/T--Technion_Israel--Model8.png" class="img-responsive img-center img-cont" style="cursor: pointer;"><br>
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</a>
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<a class="pop ocenter">
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<img src="https://static.igem.org/mediawiki/igem.org/d/d2/T--Technion_Israel--Model3.png" class="img-responsive img-center img-cont" style="cursor: pointer;"><br>
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</a>
 +
<p class="text-center"><b>Equation 3:</b> Bacteria concentration problem – discrete form.</p>
 +
</div>
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</div>
 +
 +
<br>
 +
<br>
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<br>
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<div class="row">
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<div class="col-md-12 col-sm-12">
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<p class="text-justify">
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And we can write it as follows:
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</p>
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</div>
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</div>
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<br>
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<div class="col-md-12 col-sm-12">
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<img src="https://static.igem.org/mediawiki/igem.org/2/27/T--Technion_Israel--Model4.png" class="img-responsive img-center img-cont" style="cursor: pointer;"><br>
 +
</a>
 +
<p class="text-center"><b>Equation 4:</b> Bacteria concentration problem – Final form</p>
 +
</div>
 +
</div>
 +
 +
<br>
 +
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<div class="row">
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<div class="col-md-12 col-sm-12">
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<p class="text-justify">
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Given the initial condition:
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</p>
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</div>
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</div>
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<br>
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<div class="row">
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<div class="col-md-12 col-sm-12">
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<a class="pop ocenter">
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<img src="https://static.igem.org/mediawiki/igem.org/9/95/T--Technion_Israel--Model32.png" class="img-responsive img-center img-cont" style="cursor: pointer;"><br>
 +
</a>
 +
<p class="text-center"><b>Equation 5:</b> Bacteria concentration problem- initial condition.</p>
 +
</div>
 +
</div>
 +
 +
<br>
 +
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<div class="col-md-12 col-sm-12">
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<p class="text-justify">
 +
And the boundary conditions which are translated to the following discrete conditions:
 +
</p>
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</div>
 +
</div>
 +
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<br>
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<div class="row">
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<div class="col-md-12 col-sm-12">
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<a class="pop ocenter">
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<img src="https://static.igem.org/mediawiki/igem.org/8/84/T--Technion_Israel--Model21.png" class="img-responsive img-center img-cont" style="cursor: pointer;"><br>
 +
</a>
 +
<p class="text-center"><b>Equation 6:</b> Bacteria concentration problem- boundary conditions</p>
 +
</div>
 +
</div>
 +
 +
<br>
 +
 +
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<div class="col-md-12 col-sm-12">
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<p class="text-justify">
 +
With <I>I</I> being the final value of i – index of location.<br>
 +
The above conditions keep the flux of bacteria zero on both sides of the chip as occurs in the actual chip.
 +
Hence, no bacteria enter or exit the chip.
 +
<br>
 +
The above equation was entered into the Thomas-Three-Diagonal algorithm for solving matrix equations,
 +
giving us the solution for the entire space of the problem in a specific point in time. By advancing
 +
in time as we solve the equation at each time point we get the solution for the bacterial concentration
 +
for every x,t.
 +
</p>
 +
</div>
 +
</div>
 +
 +
<!--small headline-->
 +
<div class="row">
 +
<div class="col-md-12 col-sm-12">
 +
<h3>Model results</h3>
 +
</div>
 +
</div>
 +
 +
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 +
<div class="row">
 +
<div class="col-md-12 col-sm-12">
 +
<p class="text-justify">
 +
We ran the chemo-repellent concentration equation in Matlab -
 +
Our code can be found <a href="https://static.igem.org/mediawiki/igem.org/8/8f/T--Technion_Israel--ModelCode.zip">
 +
here</a>.<br>
 +
The parameters used:
 +
</p>
 +
</div>
 +
</div>
 +
 +
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<div class="row">
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<div class="col-md-12 col-sm-12">
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<a class="pop ocenter">
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<img src="https://static.igem.org/mediawiki/igem.org/5/57/T--Technion_Israel--Model12.png" class="img-responsive img-center img-cont" style="cursor: pointer;"><br>
 +
</a>
 +
<p class="text-center"><b>Table 1:</b> Parameters for chemotaxis model</p>
 +
</div>
 +
</div>
 +
 +
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 +
<div class="row">
 +
<div class="col-md-12 col-sm-12">
 +
<p class="text-justify">
 +
Please notice, the results are normalized to enable us to show them on the same scale. <br>
 +
<br>
 +
The results are as follows:
 +
</p>
 +
</div>
 +
</div>
 +
 +
<!-- 12 img div -->
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<div class="row">
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<div class="col-md-12 col-sm-12">
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<a class="pop ocenter">
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<img src="https://static.igem.org/mediawiki/igem.org/9/99/T--Technion_Israel--Model9.png" class="img-responsive img-center img-cont" width="450" style="cursor: pointer;"><br>
 +
</a>
 +
<p class="text-center"><b>Graph 1:</b> Results of bacterial chemotaxis model</p>
 +
</div>
 +
</div>
 +
 +
<!-- 12 text div -->
 +
<div class="row">
 +
<div class="col-md-12 col-sm-12">
 +
<p class="text-justify">
 +
<b>Model conclusions:</b><br>
 +
                                <p style="margin-left: 40px">
 +
<b>- </b> The peak of bacterial concentration is caused by the bacterial chemotactic
 +
response, moving away from the chemo-repellent, and concentrating.<br>
 +
<b>- </b>The "wave" of bacterial concentration starts moving relatively fast, and slows
 +
down quickly. This is due to the change in repellent concentration. The "wave"
 +
converges to ~7[mm].<br>
 +
<b>- </b>The bacteria react significantly less to a normalized repellent concentration of
 +
less than ~3[mm]. This is approximately where the two graphs intersect. <br>
 +
<b>- </b>As the concentration of repellent goes down, the bacteria are less reactive.
 +
This continues as the bacteria's diffusion speed surpasses the chemotaxis rate. In other
 +
words, more bacteria move away from the concentration peak than into it for t>15[mim].<br>
 +
<b>- </b>Projecting this on the chip color experiment, we can predict there will be three shades
 +
of color: weak where the bacteria moved from (low concentration), strong where the bacteria
 +
moved to (high concentration) and on the far end, unchanged as the bacteria were not exposed
 +
to the repellent.<br>
 +
</p>
 +
</div>
 +
</div>
 +
 +
<br>
 +
 +
<!--small headline-->
 +
<div class="row">
 +
<div class="col-md-12 col-sm-12">
 +
<h3>Comparison to Experiments</h3>
 +
</div>
 +
</div>
 +
 +
<div class="row">
 +
<div class="col-md-12 col-sm-12">
 +
<p class="text-justify">
 +
The experiment ran as the setup shown in the "Overview" section: The channel was filled with
 +
bacteria in motility buffer and then the sample was inserted. We used engineered E. coli with a
 +
<a href="https://2016.igem.org/Team:Technion_Israel/S.Tar_intro" target="_blank">S.Tar</a> PctA
 +
receptor taken from a plate and suspended in motility buffer. The chemo-repellent used is TCE.
 +
</p>
 +
</div>
 +
</div>
 +
 +
<br>
 +
 +
<!-- 12 img div -->
 +
<div class="row">
 +
<div class="col-md-12 col-sm-12">
 +
<a class="pop ocenter">
 +
<img src="https://static.igem.org/mediawiki/igem.org/c/c7/T--Technion_Israel--Model1.png" class="img-responsive img-center img-cont" width="450" style="cursor: pointer;"><br>
 +
</a>
 +
<p class="text-center"><b>Table 2:</b> Substance for chemotaxis experiment</p>
 +
</div>
 +
</div>
 +
 +
<br>
 +
 +
<div class="row">
 +
<div class="col-sm-8 col-sm-offset-2"><!-- 8/12 -->
 +
<div class="col-md-6 col-sm-12">
 +
<p class="text-justify">a.</p>
 +
<a class="pop">
 +
<img src="https://static.igem.org/mediawiki/igem.org/c/ca/T--Technion_Israel--Model2.png" class="img-responsive img-center img-cont" width="290" style="cursor: pointer;"><br>
 +
</a>
 +
</div>
 +
<div class="col-md-6 col-sm-12">
 +
<p class="text-justify">b.</p>
 +
<a class="pop">
 +
<img src="https://static.igem.org/mediawiki/2016/7/76/T--Technion_Israel--ModelControl.png" class="img-responsive img-center img-cont" width="220" style="cursor: pointer;"><br>
 +
</a>
 +
</div>
 +
<div class="col-md-12 col-sm-12">
 +
<p class="text-center"><b>Fig. 1:</b>
 +
<b>a.</b> Chemotaxis of <i>E. coli</i> with a S.Tar PctA receptor due to exposure to TCE (enhanced picture).
 +
<b>b.</b> <i>E. coli</i> with a S.Tar PctA receptor exposed to motility buffer (control).
 +
</p>
 +
</div>
 +
</div>
 +
</div>
 +
 +
<br>
 +
 +
<!-- 12 text div -->
 +
<div class="row">
 +
<div class="col-md-12 col-sm-12">
 +
<p class="text-justify">
 +
As expected, a visible cluster of strong dark blue has formed next to a lighter shade due to
 +
chemotactic activity. Furthermore, the distance the bacteria passed is only a few millimeter
 +
as the model predicted.<br>
 +
The time scale does not line up: The color darkens as the experiment continues. This will
 +
probably be corrected by using more accurate diffusion and chemotactic coefficients.<br>
 +
Some of the inconsistencies between the model and the experiment (like the uneven cluster
 +
of colored bacteria) can be explained by problems loading the chemo-repellent as shown in
 +
the chemo-repellent concentration experiment.
 +
</p>
 +
</div>
 +
</div>
 +
 +
</div>
 +
</div>
 +
</div>
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 +
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 +
<h1>Conclusions</h1>
 +
</div>
 +
</div>
 +
 +
 +
<br>
 +
<br>
 +
 +
 +
<!-- Mini headline -->
 +
<div class="row">
 +
<div class="col-md-12 col-sm-12">
 +
<h3>Results</h3>
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</div>
 +
</div>
 +
 +
<br>
 +
 +
<!-- 12 text div -->
 +
<div class="row">
 +
<div class="col-md-12 col-sm-12">
 +
<p class="text-justify">
 +
This model predicts the overall behavior of our system. In our experiments we were able to
 +
show that the concentration of the repellent acts as we expected in terms of changes in the
 +
diffusion limit's velocity. When compared to the bacterial concentration, again, the experiments
 +
showed a similar behavior as the numeric solution of Keller-Segal equation.
 +
<br>
 +
<br>
 +
As explained before, this model requires further fitting to get more accurate results. Not only
 +
by using more accurate coefficients, but with improving the system itself.   
 +
</p>
 +
</div>
 +
</div>
 +
 +
 +
<br>
 +
<br>
 +
 +
 +
<!-- Mini headline -->
 +
<div class="row">
 +
<div class="col-md-12 col-sm-12">
 +
<h3>Future development</h3>
 +
</div>
 +
</div>
 +
 +
<!-- 12 text div -->
 +
<div class="row">
 +
<div class="col-md-12 col-sm-12">
 +
<p class="text-justify">
 +
We would like to improve the model and design new ones based on it. First, develop
 +
models for the different coefficients in the Keller-Segal equation: bacteria diffusion
 +
coefficient, chemotactic coefficient and bacteria life and death. Theoretically,                                        those coefficients can be controlled by changing some parameters like
 +
        the number of flagella
 +
or receptors a cell has, or even manipulating the biological tracks of the bacteria.
 +
This will enable us to get even more accurate results. Secondly, to build a library of
 +
different receptors and ligands. Thirdly, expanding our
 +
model so it could predict movement of bacteria in different geometric constructs such
 +
as funnels or U bends. This research can be the basis of a prototype commercial device.<br>
 +
<br>
 +
        As for the first part, we worked with <a href="https://static.igem.org/mediawiki/igem.org/1/11/T--Technion_Israel-HardModelFrieborg.pdf" target="_blank">iGEM Freiburg</a> (2) in developing a function for the
 +
chemotactic coefficient. Also, we designed a new assay for experimentally measuring it.<br>
 +
<br>
 +
This assay, named Trap & Track, is a novel way to detect chemotaxis on the nanometric level.
 +
By using it we can measure the exact repellent concentration that induces chemotaxis and
 +
calculate the chemotactic coefficient accurately for every material. A detailed explanation
 +
about the assay can be found
 +
<a href="https://2016.igem.org/Team:Technion_Israel/Measurement" target="_blank">here</a>.<br>
 +
<br>
 +
As for the second part, the S.Tar system enables us to change the receptors a bacterium
 +
has and by that, change the materials it repels from. In the future, this system can be
 +
expanded to control the efficiency of these receptors and even control other aspects of
 +
the chemotaxis pathway such as the flagella.<br>
 +
 +
<br>
 +
The third and final part is to improve the device itself. We designed a new fluidic chip
 +
and fabricated it in different <a href="https://2016.igem.org/Team:Technion_Israel/Design" target="_blank">methods</a>.
 +
This new design will give us a more controlled diffusion by cancelling out most of the
 +
flow and fixing the diffusion source. Also, by changing the geometry of the channel,
 +
the bacteria concentration will increase and cause a more noticeable signal. This will improve
 +
the accuracy of the experiments we run, and in turn, our overall model. <br>
 +
<br>
 +
</p>
 +
</div>
 +
</div>
 +
 +
</div>
 +
</div>
 +
</div>
 +
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</div><!-- End: tabs -->
 +
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<div class="row">
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<div class="col-sm-8 col-sm-offset-2"><!-- 8/12 -->
 +
 +
<p class="referances">
 +
References:<br>
 +
 +
1. KELLER, Evelyn F.; SEGEL, Lee A. Model for chemotaxis. Journal of theoretical biology, 1971, 30.2: 225-234.<br>
 +
2. MAZZAG, B. C.; ZHULIN, I. B.; MOGILNER, Alexander. Model of bacterial band formation in aerotaxis. Biophysical journal, 2003, 85.6: 3558-3574..<br>
 +
<br>
 +
</p>
 +
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</div>
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</div>
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<br>
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<br>
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<br>
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</div>
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Latest revision as of 00:32, 20 October 2016

S.tar, by iGEM Technion 2016

S.tar, by iGEM Technion 2016

Introduction

The goal of this model is to describe the processes inside the Flash Lab system:

- Change in the concentration of chemo-repellent.
- Change in the concentration of bacteria.

This model is based on the Keller – Segal equation of chemotaxis (1) in a one dimensional problem (Thin channel).

It is important to notice that this model can show the overall behavior and not exact values. The final system is supposed to detect a variety of materials in many different solvents, each of them has its own diffusion properties. Also, some aspects such as working conditions (temperature, humidity etc.) might change in widespread commercial use and affect the results. Taking those into account, further fitting will be necessary.

For PDF version of this model press here

Chemo-Repellent Concentration

Model


The basic assumptions of the model for the chemo-repellent are:

-There are no forces except for diffusion:

- Chemo-repellent concentration in the sample is relatively low and does not cause osmotic pressure.
- The changes in pressure due to loading the sample is negligible.
- No other significant external forces (for example, moving the chip while using).

- The bacteria do not consume the chemo-repellent and its concentration does not change with time. In contrast, in case of a chemo-attractant this assumption is not valid.

- Expect to detect small proteins and molecules (the ligands). The diffusion coefficient for such materials is about 10-9 [m2/s].

- Because of the geometric properties of the channel and the expected diffusion coefficient, this is approximately a semi-infinite one dimensional problem.

- Initial condition: no chemo-repellent is present in the chip at time zero (a).

- Boundary condition: at infinite distance the concentration is zero (b) and the there is conservation of dissolved material mass (c).

We modelled the change in concentration of the chemo-repellent based on a "Top Hat Function" solution for the diffusion problem:

v [M] is the chemo-repellent concentration.
D[m2/s] is the chemo-repellent diffusion coefficient.
N [mol] is the number of moles of the repellent .
A [m2] is the cross-section area of the channel.
h [m] is length of the sample in the channel.
x [m] is the distance on the channel.
t [s] is time.

The solution for this problem is:

Equation 1: Chemo-repellent diffusion problem.

Equation 2: General solution for chemo-repellent diffusion problem.

Graph 1: Chemo-repellent concentration: top hat diffusion.


If x=0 [m] is defined as the point where the sample is loaded to the chip, then the solution for the diffusion problem is:


Equation 3: Chemo-repellent concentration.

Model Results

We ran the chemo-repellent concentration equation in Matlab - Our code can be found here. The parameters used:


Table 1: Parameters for diffusion model.

*This is the diffusion coefficient for potassium permanganate (see "Comparison to Experiment")
** h = (Sample volume)/(Reservoir cross section).

The output for different times:


Graph 2: Model for repellent concentration.

The change in distance of the diffusion limit between 0 to 15 minutes, is relatively big. As the time passes the diffusion limit's speed lowers significantly, and the concentration profile becomes more linear.

Comparison to Experiment

Most diffusion experiments need a dedicated system that is based on the diffusion of an isotope or a fluorescent material that can be detected easily and very precisely. we chose a more basic system since this is only a preliminary testing for showing that the overall system behaves as we expect.

The experiment ran as the setup shown in the "Overview" section: The channel was filled with bacteria in motility buffer and then the sample was inserted. We replaced the motility buffer with water and the chemo-repellent with potassium permanganate in the following amounts:


Table 2: Substance for diffusion experiment.

Modeling was done under the assumption that the motility buffer acts as water, due to the fact that it is mostly water (~98%). Potassium permanganate is a salt with a known diffusion limit and acts as most of the materials we want to detect using our system (small molecules). Also, it has an intense pink color in low concentration which makes diffusion limit visible.

We ran the experiment 4 times, with a standard ruler to measure the distance of the diffusion limit.


Fig. 2: Diffusion of potassium permanganate in water in different times (enhanced picture).

As expected by the mathematical model, the diffusion limit starts moving relatively fast and its speed decreases rapidly. The difference in distance between the models and the experiment can be explained by:

-The actual diffusion limit is in too low concentration of potassium permanganate to be seen in the naked eye. If the visible concentration is about 1.5*10-5 [M] the experimental results line up with the model (graph 3).

-The ruler is a crude measuring tool. Its mistake is ± 0.5 [mm].

-Difficulties in loading uniformly the sample, especially in low volumes. Mistakes in loading the sample inside the bacterial fluid and not on, or sticking the drop of sample to one of the entry slot walls will cause uneven diffusion.


Graph 3: Comparison of diffusion model (c=0.00015[M]) to experiment.

Bacterial Concentration

The basic mathematical model for bacterial chemotaxis is the Keller-Segal equations (1) of chemotaxis:

Equation 1: Keller-Segal equation

where:

u[M] is the bacteria concentration.
v[M] is the chemo-repellent concentration.
k1[m2/s] is the bacteria diffusion coefficient.
k2[m2/M * s] is the bacteria chemotactic coefficient.
k3[M/s] is bacteria life and death coefficient.
x[m] is the distance in the channel.
t[s] is time.

The basic assumptions of the chemotaxis model are:


-k2 describes how sensitive is the bacteria to changes in chemo-repellent concentration. In other words, as its negative value decreases the bacteria will react more violently to same repellent exposure.

-Because of the geometric properties of the channel, this is approximately a one dimensional problem.

-We ran our tests in a short time scale (t<30[min]) so we presume that the change in concentration of bacteria due to life and death is negligible, k3=0.

-The chemo-repellent concentration is known for every x and t.

-There are no changes in the flux of bacterial concentration at the start (a) and at end of the channel (b).


Mathematical Model

Under these assumptions the equation takes the form:


Equation 2: The Bacterial concentration problem.

The above partial differential equation cannot be solved analytically, so we must turn to numerical analysis tools. The implicit Euler method is one of the most basic numerical methods for the solution of ordinary and partial differential equations. This method is guaranteed to be stable and gives us the solution for the entire space in a single point in time.

Using the following discretization:



Equation 3: Bacteria concentration problem – discrete form.




And we can write it as follows:



Equation 4: Bacteria concentration problem – Final form


Given the initial condition:



Equation 5: Bacteria concentration problem- initial condition.


And the boundary conditions which are translated to the following discrete conditions:



Equation 6: Bacteria concentration problem- boundary conditions


With I being the final value of i – index of location.
The above conditions keep the flux of bacteria zero on both sides of the chip as occurs in the actual chip. Hence, no bacteria enter or exit the chip.
The above equation was entered into the Thomas-Three-Diagonal algorithm for solving matrix equations, giving us the solution for the entire space of the problem in a specific point in time. By advancing in time as we solve the equation at each time point we get the solution for the bacterial concentration for every x,t.

Model results

We ran the chemo-repellent concentration equation in Matlab - Our code can be found here.
The parameters used:


Table 1: Parameters for chemotaxis model

Please notice, the results are normalized to enable us to show them on the same scale.

The results are as follows:


Graph 1: Results of bacterial chemotaxis model

Model conclusions:

- The peak of bacterial concentration is caused by the bacterial chemotactic response, moving away from the chemo-repellent, and concentrating.
- The "wave" of bacterial concentration starts moving relatively fast, and slows down quickly. This is due to the change in repellent concentration. The "wave" converges to ~7[mm].
- The bacteria react significantly less to a normalized repellent concentration of less than ~3[mm]. This is approximately where the two graphs intersect.
- As the concentration of repellent goes down, the bacteria are less reactive. This continues as the bacteria's diffusion speed surpasses the chemotaxis rate. In other words, more bacteria move away from the concentration peak than into it for t>15[mim].
- Projecting this on the chip color experiment, we can predict there will be three shades of color: weak where the bacteria moved from (low concentration), strong where the bacteria moved to (high concentration) and on the far end, unchanged as the bacteria were not exposed to the repellent.


Comparison to Experiments

The experiment ran as the setup shown in the "Overview" section: The channel was filled with bacteria in motility buffer and then the sample was inserted. We used engineered E. coli with a S.Tar PctA receptor taken from a plate and suspended in motility buffer. The chemo-repellent used is TCE.



Table 2: Substance for chemotaxis experiment


a.

b.

Fig. 1: a. Chemotaxis of E. coli with a S.Tar PctA receptor due to exposure to TCE (enhanced picture). b. E. coli with a S.Tar PctA receptor exposed to motility buffer (control).


As expected, a visible cluster of strong dark blue has formed next to a lighter shade due to chemotactic activity. Furthermore, the distance the bacteria passed is only a few millimeter as the model predicted.
The time scale does not line up: The color darkens as the experiment continues. This will probably be corrected by using more accurate diffusion and chemotactic coefficients.
Some of the inconsistencies between the model and the experiment (like the uneven cluster of colored bacteria) can be explained by problems loading the chemo-repellent as shown in the chemo-repellent concentration experiment.

Conclusions



Results


This model predicts the overall behavior of our system. In our experiments we were able to show that the concentration of the repellent acts as we expected in terms of changes in the diffusion limit's velocity. When compared to the bacterial concentration, again, the experiments showed a similar behavior as the numeric solution of Keller-Segal equation.

As explained before, this model requires further fitting to get more accurate results. Not only by using more accurate coefficients, but with improving the system itself.



Future development

We would like to improve the model and design new ones based on it. First, develop models for the different coefficients in the Keller-Segal equation: bacteria diffusion coefficient, chemotactic coefficient and bacteria life and death. Theoretically, those coefficients can be controlled by changing some parameters like the number of flagella or receptors a cell has, or even manipulating the biological tracks of the bacteria. This will enable us to get even more accurate results. Secondly, to build a library of different receptors and ligands. Thirdly, expanding our model so it could predict movement of bacteria in different geometric constructs such as funnels or U bends. This research can be the basis of a prototype commercial device.

As for the first part, we worked with iGEM Freiburg (2) in developing a function for the chemotactic coefficient. Also, we designed a new assay for experimentally measuring it.

This assay, named Trap & Track, is a novel way to detect chemotaxis on the nanometric level. By using it we can measure the exact repellent concentration that induces chemotaxis and calculate the chemotactic coefficient accurately for every material. A detailed explanation about the assay can be found here.

As for the second part, the S.Tar system enables us to change the receptors a bacterium has and by that, change the materials it repels from. In the future, this system can be expanded to control the efficiency of these receptors and even control other aspects of the chemotaxis pathway such as the flagella.

The third and final part is to improve the device itself. We designed a new fluidic chip and fabricated it in different methods. This new design will give us a more controlled diffusion by cancelling out most of the flow and fixing the diffusion source. Also, by changing the geometry of the channel, the bacteria concentration will increase and cause a more noticeable signal. This will improve the accuracy of the experiments we run, and in turn, our overall model.

References:
1. KELLER, Evelyn F.; SEGEL, Lee A. Model for chemotaxis. Journal of theoretical biology, 1971, 30.2: 225-234.
2. MAZZAG, B. C.; ZHULIN, I. B.; MOGILNER, Alexander. Model of bacterial band formation in aerotaxis. Biophysical journal, 2003, 85.6: 3558-3574..




S.tar, by iGEM Technion 2016