Team:GeorgiaTech/Systems Modeling

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<center><img src="https://static.igem.org/mediawiki/2010/a/a4/Modelingpic_1.png" alt="modeling picture"</center>
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<p><center><font color="#FFFFFF" size=5><b>Modeling Bacterial Heat Production Due to AOX Pathway</b></font></center></p>
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Figure (A) depicts the approach taken to model E. coli heat production and localization using AOX in a 'dry' plated culture. In order to determine the steady state temperature of E. coli colonies expressing heat, colonies are approximated as a single layer composed of a certain amount of heat producing cells with thermal conductivity k1. The heat produced in this situation is transferred by conduction with the media (agar) below the colonies and by convection with the air and an equilibrium temperature will be reached.  
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    <td bgcolor="#964141" width="800"><font color="#FFFFFF"><p>AOX pathway is responsible for thermogenesis in various organisms. But to what extent it would be responsible for heat production in genetically engineered bacteria remains an interesting question. Georgia Tech modeling team aimed at theorizing an answer to this question using both analytical and computational methods. The primary goal was to suggest a calorimetric technique with optimal sensitivity, as well as to compare heat transfer in liquid culture and bacterial colonies. </p>
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Figure (B) presents the modeling of E. coli on a dry, plated culture in the form of a circuit diagram. The circuit diagram represents the CspA or HyBb promoter as a switch which would activate the expression of AOX pathway in the cell during cold shock. As the cell starts producing heat, mainly through AOX pathway, the heat starts to create a flux through the E. coli layer and release heat into the environment (basically in the media through conduction and in the air through convection). Since, the heat flows from E. coli in the media, it creates a temperature gradient across the whole plate due to the resistance encountered by it. These resistances are mainly due to conduction in media and convection in air which can be combined to single resistance as they are in parallel orientation to each other.   
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<a href="https://static.igem.org/mediawiki/2010/a/af/Modeling_heat_production_by_AOX.pdf"> Proposed model of heat generation via AOX in cells</a>
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    <td bgcolor="#964141" width="900"><font color="#FFFFFF" size=4><p><center>Heat Transfer Modeling Aims</font></center></p>
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<center><p><strong>The following models were devised:</strong></p>
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<p><strong>I. Rate of heat production via AOX pathway</strong></p>
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<p><strong>II. Heat transfer in liquid culture </strong></p>
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<p><strong>III. Heat transfer in bacterial colony (analytical solution  1D)</strong></p>
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<p><strong>IV. Heat  transfer in bacterial colony (computational solution 2D and 3D)</p></center>
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<p>I) Calculations for rate of heat production in E. coli:</strong></p>
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<font color="#FFFFFF"><ul>
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  <li>800 mV electric potential drop of 4  electrons  generates  5.12 x 10-19 Joules    </li>
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  <li>70% of electrons enter AOX pathway</li>
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  <li>Assume time scale of ATP cycle to calculate  power </li>
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  <li>Power  generated per cell is 1.6 x 10-13 Watt</li>
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</ul>
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<p><strong>II) Heat transfer in liquid  culture:</strong></p>
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  <li><u>Simplifying Assumptions</u></li>
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<p>1.Liquid  solution can be assumed water<br />
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  2.Complete  insulation <br />
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  3.Heat  accumulation within system<br />
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  4.Homogeneous  mixture<br />
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  5.No work done  on or performed by the system </p>
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<ul>
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  <li>Density of bacterial culture can vary by 2  orders of magnitude</li>
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  <li>Temperature  of system can be raised by 1K in 4 – 40 min. </li>
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<p><strong>III) Heat transfer in bacterial colony (analytical solution)</strong><br />
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  <u>Assumptions: </u><br />
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  1.   Petri dish is completely insulated, and kept at 288K<br />
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  2.   Ambient temperature is 288K<br />
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  3.   Conduction through E. coli is similar to that in water<br />
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  4.   Constant coefficients for conductivity in both media, constant  convective coefficient for air<br />
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  5.   Aspect ratio : width of colony &gt;&gt; height of colony<br />
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  Required : Insert Diagram of ecoli on agar  (green outlines)<br />
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  Required: Insert slides titled (steady  state temperature profile for agar and ecoli)<br />
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  <u>Solving for boundary conditions</u>: <br />
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  1. Heat flux at  the E. coli - air boundary was equal to the convective heat flux ( x = 0 ) <br />
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2. Heat flux and temperature were equated at E. coli - agarose boundary ( X = d )</p>
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<p>The following  information was also known: </p>
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<ul>
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  <li>Q: volumetric flow of heat generated by Ecoli</li>
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  <li>k : conductive coefficient of water at 288K</li>
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  <li>h : convective coefficient of air at 288K</li>
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  <li>Tambient: 298 K</li>
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  <li>Measurements of height of colony and agarose</li>
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  <li>C2 = Te , Temperature at  Ecoli- air boundary (unknown)</li>
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<p>3) We solved for boundary conditions by solving two temperature profile   equations simultaneously in MATLAB<br />
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<center><i>Figure 1. This figure shows the temperature profile of bacteria and  the solid growth media as a function of height. In E. coli, temperature drops  quadratically, and it drops linearly in agarose. This is because of the heat generation term included within the Poisson equation developed to describe heat  transfer in E. coli. The total drop of temperature at steady state across the height of bacterial colony and agarose is approximately 0.1 K. </p></i></center>
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<p><strong>IV) Heat transport in bacterial  colony 2D and 3D (using COMSOL) </strong><br />
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<center><img src="https://static.igem.org/mediawiki/2010/9/9c/2Dmodel.png" width="" height="" img style="border: 2px solid white"></center>
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<center><p>Figure 2.</p></center>
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<center><img src="https://static.igem.org/mediawiki/2010/3/34/3Dmodel.png" width="" height="" img style="border: 2px solid white"></center>
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<center><p>Figure 3.</p></center>
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<center><p>Figure 2 and 3 were developed in COMSOL. They depict 2D and 3D heat transfer in bacterial colony and agarose. The difference between peak temperatures in both scenarios did not differ by  more than 0.006K which indicates that a 2D control volume may provide  sufficiently accurate representation for heat transport modeling. In a 2D  control volume, heat is transferred radially to the environment. If high aspect  ratio is implemented, as in case of a uniform stretch of bacterial colony  formed on a petri dish, then 1D control volume will be sufficient. </p></center>
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<p><strong>V) Conclusions:</strong></p>
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  <li>Within solution 1K change in temperature in 4 – 40  minutes.</li>
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  <li>On agar, steady state temperature profile derived  analytically matches closely with those found computationally using COMSOL.</li>
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  <li>Using 1D control-volume is a good assumption, since 3D  temperature profile was not considerably different. </li>
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  <li>Derived analytically and computationally, the change in  temperature due to AOX expression should be approximately 0.1 K (on solid  growth media).</li>
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  <li>Due to better accumulation of energy in liquid media,  characterization of heat production may be more accessible using a liquid culture. </li>
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  <li>A highly sensitive (at least 0.1K) thermal imaging camera will be essential for measuring heat production of bacterial colony in  both liquid and solid growth media </li>
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<p>In order to better understand the extent of modeling work done please download the <a href="https://static.igem.org/mediawiki/2010/b/bf/JAMBOREE_6.pdf">modeling presentation</a> and <a href="https://static.igem.org/mediawiki/2010/9/9d/Wiki_modeling.pdf">written summary</a>.
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