Team:Imperial College London/Modelling/Output/Detailed Description

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|style="font-family: helvetica, arial, sans-serif;font-size:2em;color:#ea8828;"|Detailed Description
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The following page presents the details of the models that have been developed. Firstly, assumptions that have been exploited are explained. Then every model is presented separately as each of them has slightly different elements of the system and the interactions between them. However, there are only 3 fundamental biochemical processes that will be analysed:
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<a href="https://2010.igem.org/wiki/index.php?title=Team:Imperial_College_London/Modelling/Output/Objectives"><b>Objectives</b></a>; <a href="https://2010.igem.org/Team:Imperial_College_London/Modelling/Output/Detailed_Description"><b>Detailed Description</b></a>; <a href="https://2010.igem.org/Team:Imperial_College_London/Modelling/Output/Parameters_and_Constants"><b>Parameters & Constants</b></a>; <a href="https://2010.igem.org/Team:Imperial_College_London/Modelling/Output/Results_and_Conclusion"><b>Results & Conclusion</b></a>;<a href="https://2010.igem.org/Team:Imperial_College_London/Modelling/Output/Download_MatLab_Files"><b>Download MatLab Files</b></a>;
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<ol>
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<li>Transcription</li>
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<li>Translation</li>
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<li>Enzymatic reactions</li>
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</ol>
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|style="font-family: helvetica, arial, sans-serif;font-size:2em;color:#ea8828;"|Output Amplification Model
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|style="font-family: helvetica, arial, sans-serif;font-size:2em;color:#ea8828;"|1. Law of Mass Action
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|<html><h2>Detailed Description</h2></html>
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<html><h3>1. Model based on Law of Mass Action</h3></html>
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During a meeting with our advisors, it was noted that our initial models (in which it was assumed that our system obeyed Michaelis-Menten kinetics) were wrong as the assumptions made by Michaelis-Menten approximation were not obeyed by the system.
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During a meeting with our advisors, it was noted that our initial models (which had assumed that our system obeyed Michaelis Menten kinetics) were wrong. This is due to the fact that Michaelis Menten kinetics does not apply to our system.
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<html><a href="">Click here to learn more about our models based on Michaelis Menten kinetics.</a></html>
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A few of Michaelis-Menten assumptions were not met by our system:
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In order to be able to use Michaelis Menten kinetics, there a lot of assumptions that have to be made. A few of these assumptions were not met by our system:  
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*'''Vmax is proportional to the overall concentration of the enzyme.'''
*'''Vmax is proportional to the overall concentration of the enzyme.'''
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Since we are continuously producing enzyme, Vmax will change. Therefore the conservation <html>E<sub>0</sub> = E + E<sub>S</sub></html> does not hold for our system.  
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Since we are continuously producing enzyme, Vmax will change. Therefore the conservation of enzymes <html>E<sub>0</sub> = E + E<sub>S</sub></html> does not hold for our system.  
*'''Substrate >> Enzyme'''
*'''Substrate >> Enzyme'''
We are producing both substrate and enzyme, so we have approximately the same amount of substrate and enzyme.  
We are producing both substrate and enzyme, so we have approximately the same amount of substrate and enzyme.  
*'''Enzyme affinity to the substrate has to be high.'''
*'''Enzyme affinity to the substrate has to be high.'''
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Therefore, the model above is not representative of the enzymatic reaction. As we cannot use the Michaelis-Menten model we will have to solve from first principle (which involves writing down all of the biochemical equations and solving for these in Matlab).
 
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'''Change of output'''
 
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During our literature research, we came across a better output, so we abandoned the idea of using GFP as an output. Instead, we are using catechol. An enzyme, dioxygenase, will be acting on the catechol, which will then result in a coloured output. Catechol will be added to the bacteria manually (i.e. the bacteria will not produce catechol). Hence, in our models dioxygenase will be treated as an output as this enzyme is the only activator of catechol in our system. This means that the change of catechol into its colourful form is dependent on the dioxygenase concentration.
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Click on the button below, in order to get to know more on Michaelis-Menten kinetics based models.
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<div class="accordionButton">Abandoned Initial Attempts</div>
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<div class="accordionContent">
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<b>Elements of the system</b><br/>
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Depending on which amplification it there is several species appearing. The full list is:
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*GFP (on diagrams shown <i>green</i>) which was out original choice for output later on changed to dioxygenase acting on catechol. It was supposed to be split and attached to coiled coils. It would get activated by TEV protease.
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*TEV protease (shown <i>orange</i> on diagrams)and split TEV (TEVs is shown <i>yellow</i>). Split TEV would be inactive form mounted on the coiled coils to be activated by HIV1 or another active TEV.
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*HIV1 protease (shown <i>blue</i> on diagrams) which would be produced active upon receiving the activation signal in case of 2 amplification step outputs.
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<b>Simple models</b><br/>
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Our first models only were accounting for the protein and mRNA production:
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<html><h3>2. Model preA: Simple production of dioxygenase</h3></html>
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  <a href="https://static.igem.org/mediawiki/2010/1/17/IC_Simple_production.JPG"><img src="https://static.igem.org/mediawiki/2010/1/17/IC_Simple_production.JPG" width="300"/></a>
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</html>
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|[[Image:Initial_1.png|268px]]
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|2-step amplifier with distinct protease at each level.
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</div>
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  {| style="width:545px;background:#e7e7e7;text-align:center;font-family: helvetica, arial, sans-serif;color:#555555;margin- top:5px;padding: 1px;" cellspacing="5";
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  |-
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  |[[Image:Initial_1_eqn.png|543px]]
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</div>
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|Where: m - is the concentration of mRNA, p - is the protein concentration (subscript indicates which protein)
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|-
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|[[Image:Initial_2.png|268px]]
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|-
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|2-step amplifier with TEV protease implemented at both amplification levels.
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|}
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</div>
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|<div ALIGN=Center>
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  |Transcription and translation (simple production) of dioxygenase
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  |[[Image:Initial_2_eqn.png|543px]]
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</div>  
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|}
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</div>
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<b>Implementation of Michaelis-Menten kinetics</b><br/>
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So, far the models were considering only the production and degradation of species. The example presented below includes enzymatic interaction between the species governed by Michaelis-Menten.
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|-
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|[[Image:Initial_1.png|468px]]
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|2-step amplifier with distinct protease at each level.
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|}
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</div>
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| <div ALIGN=Center>
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{| style="width:345px;background:#e7e7e7;text-align:center;font-family: helvetica, arial, sans-serif;color:#555555;margin- top:5px;padding: 1px;" cellspacing="5";
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|-
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|[[Image:Initial_3_eqn.png|343px]]
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|}
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</div>
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|}
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This model includes transcription and translation of the dioxygenase. It does not involve any amplification steps. It is our control model against which we will be comparing the results of other models.
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<br/><br/>
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<b>Implementation in Matlab</b><br/>
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The Matlab code for the different stages of amplification and diagrams can be found [http://www.openwetware.org/wiki/Image:Modelling.docx here].  
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<br/><br/>
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<b>Kinetic constants</b><br/>
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<html>
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<body style="background-color:FFFFCC">
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</html>
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<html><h3>3. Model A: Activation of Dioxygenase by TEV enzyme</h3></html>
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<html>
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<table width="750px" border="0">
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<tr>
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    <td style="background-color:#FFFF66;height:50px;width:150;text-align:center">
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    </td>
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    <td style="background-color:#FFFF99;height:50px;width:150;text-align:center"><b>GFP</b>
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    </td>
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<td style="background-color:#FFFF99;height:50px;width:150;text-align:center"><b>split GFP</b>
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    </td>
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<td style="background-color:#FFFF99;height:50px;width:150;text-align:center"><b>TEV</b>
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    </td>
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<td style="background-color:#FFFF99;height:50px;width:150;text-align:center"><b>split TEV</b>
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    </td>
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</tr>
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<tr>
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    <td style="background-color:#FFCC66;height:50px;width:150 px;text-align:center;"><b>K<sub>m</sub> and k<sub>cat</sub></b>
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    </td>
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    <td style="background-color:#eeeeee;height:50px;width:150 px;text-align:center;">-
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    </td>
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<td style="background-color:#eeeeee;height:50px;width:150 px;text-align:center;">K<sub>m</sub> = 0.061; k<sub>cat</sub> = 0.16; <a href ="http://peds.oxfordjournals.org/cgi/reprint/14/12/993">[1]</a>
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    </td>
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<td style="background-color:#eeeeee;height:50px;width:150 px;text-align:center;">40% of value for TEV  
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    </td>
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<td style="background-color:#eeeeee;height:50px;width:150 px;text-align:center;">-
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    </td>
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</tr>
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<tr>
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    <td style="background-color:#FFCC66;height:50px;width:150px;text-align:center;"><b>Half-life or degradation rate</b>
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|<html>
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    </td>
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<a href="https://static.igem.org/mediawiki/2010/c/c0/IC_1-step_amplification.JPG"><img src="https://static.igem.org/mediawiki/2010/c/c0/IC_1-step_amplification.JPG" width="300"/></a>
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<td style="background-color:#eeeeee;height:50px;width:150 px;text-align:center;">Half-life in B.sub approximately 1.5 hours
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</html>
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    </td>
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|-
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<td style="background-color:#eeeeee;height:50px;width:150 px;text-align:center;">-
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|1-step amplification
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    </td>
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</div>  
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<td style="background-color:#eeeeee;height:50px;width:150 px;text-align:center;">-
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    </td>
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<td style="background-color:#eeeeee;height:50px;width:150 px;text-align:center;">Half-life shorter than GFP
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    </td>   
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</tr>
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<tr>
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    <td style="background-color:#FFCC66;height:50px;width:150px;text-align:center;"><b>Production rate in B.sub</b>
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    </td>
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<td style="background-color:#eeeeee;height:50px;width:150 px;text-align:center;">-
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    </td>
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<td style="background-color:#eeeeee;height:50px;width:150 px;text-align:center;">-
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    </td>
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<td style="background-color:#eeeeee;height:50px;width:150 px;text-align:center;">-
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    </td>
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<td style="background-color:#eeeeee;height:50px;width:150 px;text-align:center;">-
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    </td>   
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</tr>
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</table>
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</html>
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<b>Conclusion</b><br/>
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We were not able to obtain all the necessary constants. Hence, we decided to make educated guesses about possible relative values between the constants as well as varying them and observing the change in output.
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As the result, we concluded that the amplification happens at each amplification level proposed. The magnitude of amplification varies depending on the constants. There is not much difference between using TEV or HIV1.
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<b>Change of output</b>
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 +
During our literature research, we came across a better output, so we abandoned the idea of using GFP as an output. Instead, we are using catechol. An enzyme, dioxygenase, will be acting on the catechol, which will then result in a coloured output. Catechol will be added to the bacteria manually (i.e. the bacteria will not produce catechol). Hence, in our models dioxygenase will be treated as an output as this enzyme is the only activator of catechol in our system. This means that the change of catechol into its colourful form is dependent on the dioxygenase concentration.
 +
 
 +
<h2>References</h2>
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#Kapust R. et al (2001) Tobacco etch virus protease: mechanism of autolysis and rational design of stable mutants with wild-type catalytic proficiency. Protein Engineering. [Online] 14(12), 993-1000. Available from: http://peds.oxfordjournals.org/cgi/reprint/14/12/993 [Accessed 28th July 2010]
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</div>
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<br/><br/>
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As we could not use the Michaelis-Menten simplification to model enzymatic reactions in our system, we will had to solve the problem from first principle. It meant referring to more general set of assumptions called Law of Mass Action. This allowed us to model our enzymatic reactions without making assumptions about amounts of particular species as long as the amounts are bigger than single molecule level. This resulted in bigger number of partial differential equations as there was one per each species instead of 1 per reaction.
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This model consists of the basic enzymatic reaction:
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|style="font-family: helvetica, arial, sans-serif;font-size:2em;color:#ea8828;"|2. Model preA: Simple production of dioxygenase
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|-
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|
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This model was developed to illustrate 1-step amplification output.
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[[Image:Equations1.png]]
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<b>Elements of the system</b>
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This is a simple enzymatic reaction, where TEV is the enzyme, Dioxygenase the product and split Dioxygenase the substrate. Choosing <html>k<sub>1</sub>, k<sub>2</sub>, k<sub>3</sub></html> as reaction constants, the reaction can be rewritten in these four sub-equations:
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#Dioxygenase (C230 chown <i>blue</i>) is the enzyme that starts getting produced upon activation signal (eg. detection of Schistosoma).
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#Catechol (cat. shown <i>white with orange edge</i>) is artificially added chemical during the experiment. It is acted on by dioxygenase to produce yellow muconic acid.
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[[Image:Equations2.png]]
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<div ALIGN=CENTER>
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|-
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|[[Image:1-step_amplification.png|700px]]
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|-
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|Transcription and translation(simple production) of dioxygenase and its enzymatic activation of yellow colour.
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|}
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</div>
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<b>Interactions between elements</b>
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These four equations were implemented in Matlab, using a built-in function (ode45) which solves ordinary differential equations.
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Initially, dioxygenase has to be transcribed and translated. This has been described using 1 ODE equation:
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'''Implementation in TinkerCell'''
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<html>
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<CENTER><img src="https://static.igem.org/mediawiki/2010/d/d3/IC_Equation_3.png"/></CENTER>
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</html>
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In the equation above: [p] = protein concentration, s = protein production rate, d = protein degradation rate
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Another approach to model the amplification module would be to implement it in a program such as TinkerCell (or CellDesigner). This would be useful to check whether the Matlab model works.
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The enzymatic interactions can be described in the following way:
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<div>
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<html>
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|<html>
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<CENTER><img src="https://static.igem.org/mediawiki/2010/7/7b/IC_Equation_5.png"/></CENTER>
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<a href="https://static.igem.org/mediawiki/2010/7/7a/IC_Tinkercell_Model_A.PNG"><img src="https://static.igem.org/mediawiki/2010/7/7a/IC_Tinkercell_Model_A.PNG"/></a>
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</html>
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</html>
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In the equation above : D is Dioxygenase, C is Catechol, C-D is catechol-dioxygenase transition complex, M<sub>a</sub> is Muconic acid - the yellow ouptput.
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 +
It is important to mention dioxygenase molecules tretramerize before becoming active enzymes. It has been very simplified in the model. Tetramerization is accounted for by simply dividing dioxygenase concentration by 4 before it acts on catechol.
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<b>Differential equations</b>
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The above reaction can be written in terms of ordinary differential equations:
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<html>
 +
<CENTER><img src="https://static.igem.org/mediawiki/2010/d/d7/IC_Equation_2.png"/></CENTER>
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</html>
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<b>Implementation in Matlab</b>
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These equations were implemented in Matlab, using a built-in function (ode15s) which solves ordinary differential equations. For the code please refer to the <html><a href="https://2010.igem.org/Team:Imperial_College_London/Modelling/Output/Download_MatLab_Files"><b>Download MatLab Files</b></a></html> section.
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|style="font-family: helvetica, arial, sans-serif;font-size:2em;color:#ea8828;"|3. Model A: Activation of Dioxygenase by TEV enzyme
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|-
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|
 +
This model was developed to illustrate 1-step amplification output.  
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<b>Elements of the system</b>
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# TEV protease (shown <i>pink</i>) is simply produced upon detection of schistosoma.  
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# Dioxygenase (C230 chown <i>blue</i>) is pre-produced in inactive form by being attached to GFP. TEV releases dioxygenase and allows it to tetramerize and activate.
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#Catechol (cat. shown <i>white with orange edge</i>)
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<div ALIGN=CENTER>
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{| style="width:704px;background:#e1e1e1;text-align:center;font-family: helvetica, arial, sans-serif;color:#555555;margin- top:5px;padding: 2px;" cellspacing="5";
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  |-
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  |Model A implemented in TinkerCell
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  |[[Image:2-step_amplification.png|700px]]
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</div>  
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|-
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|}  
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|The species that are shown in front of vertical line which indicates beginning of experiment mean that they have been<br/> accumulated beforehand in the cell. TEV protease activates inactive dioxygenase which acts on catechol to produce colour.
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|}
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</div>
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<b>Interactions between elements</b>
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<html><h3>4. Model B: Activation of Dioxygenase by TEV or activated split TEV enzyme</h3></html>
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This model includes 2 enzymatic reactions:
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<div>
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<ol>
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{| style="background:#e7e7e7;text-align:center;font-family: helvetica, arial, sans-serif;color:#555555;margin- top:5px;" cellspacing="5";
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<li>Reaction between dioxygenase and catechol, which is exactly the same as in model preA<br/>
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|<html>
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<html>
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<a href="https://static.igem.org/mediawiki/2010/8/8c/IC_2-step_amplification.JPG"><img src="https://static.igem.org/mediawiki/2010/8/8c/IC_2-step_amplification.JPG" width="300"/></a>
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<CENTER><img src="https://static.igem.org/mediawiki/2010/7/7b/IC_Equation_5.png"/></CENTER>
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</html>
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</html><br/>
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In the equation above : D is Dioxygenase, C is Catechol, C-D is catechol-dioxygenase transition complex, M<sub>a</sub> is Muconic acid - the yellow ouptput.</li>
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<li>Reaction between TEV and inactive dioxygenase<br/>
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<html>
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<CENTER><img src="https://static.igem.org/mediawiki/2010/6/68/IC_Equation_7.png"alt="Equations showing enzymatic reaction between TEV and split Dioxygenase" /></CENTER>
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</html>
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<br/>
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In the equation above: TEV = Protease of the Tobacco Etch Virus, sD = split Dioxygenase, TsD = intermediate complex of the enzymatic reaction,<br/>D = Dioxygenase</li>
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</ol>
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<b>Differential equations</b>
 +
 
 +
This is a simple enzymatic reaction, where TEV is the enzyme, Dioxygenase the product and split Dioxygenase the substrate. Choosing <html>k<sub>1</sub>, k<sub>2</sub>, k<sub>3</sub></html> as reaction constants, the reaction can be rewritten in these four sub-equations:
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<br/><html>
 +
<CENTER><img src="https://static.igem.org/mediawiki/2010/9/94/IC_Equation_1.png"alt="ODEs describing the reaction presented above"/></CENTER>
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</html>
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<b>Implementation in Matlab</b>
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 +
These equations were implemented in Matlab, using a built-in function (ode15s) which solves ordinary differential equations. For the code please refer to the <html><a href="https://2010.igem.org/Team:Imperial_College_London/Modelling/Output/Download_MatLab_Files"><b>Download MatLab Files</b></a></html> section.
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 +
<b>Implementation in TinkerCell</b>
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Another approach to model the amplification module would be to implement it in a program such as TinkerCell (or CellDesigner). This was used to check whether implementation in Matlab generates similar results. If happened otherwise, we would need to look for reasons for those differences in the programs. As the results, generated by Matlab were the same, only Matlab code has been developed further as it allows more flexibility, control and insight.
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<div ALIGN=CENTER>
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{| style="background:#e1e1e1;text-align:center;font-family: helvetica, arial, sans-serif;color:#555555;margin- top:5px;padding: 2px;" cellspacing="5";
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|-
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|[[Image:IC_Tinkercell_Model_A.PNG|400px]]
  |-
  |-
-
  |2-step amplification
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  |Model A implemented in TinkerCell.
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</div>  
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|}
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</div>
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{| style="width:900px;background:#f5f5f5;text-align:justify;font-family: helvetica, arial, sans-serif;color:#555555;margin-top:5px;" cellspacing="20"
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|style="font-family: helvetica, arial, sans-serif;font-size:2em;color:#ea8828;"|4. Model B: Activation of Dioxygenase by TEV or activated split TEV
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|-
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|style="font-family: helvetica, arial, sans-serif;font-size:2em;color:#ea8828;"|enzyme
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|-
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|
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<b>Elements of the system</b>
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# split TEV protease (shown <i>purple</i>) is introduced as the 3rd amplification step. It exists in 2 versions (not shown on diagram): a and b as it does not separate into 2 identical halves.
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# TEV protease (shown <i>pink</i>) is simply produced upon detection of schistosoma.
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# Dioxygenase (C230 chown <i>blue</i>) is pre-produced in inactive form by being attached to GFP. TEV releases dioxygenase and allows it to tetramerize and activate.
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#Catechol (cat. shown <i>white with orange edge</i>)
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<div ALIGN=CENTER>
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{| style="background:#e1e1e1;text-align:center;font-family: helvetica, arial, sans-serif;color:#555555;margin- top:5px;padding: 2px;" cellspacing="5";
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|-
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|[[Image:3-step_amplification.png|700px]]
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|-
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|This diagram introduces inactive split TEV protease attached to a coiled-coil as the third amplification step.
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Both inactive compounds have active site for TEV to activate tehm which results in multiple possibilities of action.
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|}
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</div>
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<b>Interactions and assumptions</b>
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This version includes the following features:  
This version includes the following features:  
-
*2 amplification steps (TEV and split TEV)
 
*Split TEV is specified to have a and b parts  
*Split TEV is specified to have a and b parts  
*TEVa is forbidden to interact with TEVa (though in reality there could be some affinity between the two). Same for the interaction between Tevb and Tevb  
*TEVa is forbidden to interact with TEVa (though in reality there could be some affinity between the two). Same for the interaction between Tevb and Tevb  
Line 96: Line 320:
*Dioxygenase is assumed to be active as a monomer  
*Dioxygenase is assumed to be active as a monomer  
*Activate split TEV (TEVs) is not allowed to activate sTEVa or sTEVb (this kind of interaction is accounted for in the next model version)  
*Activate split TEV (TEVs) is not allowed to activate sTEVa or sTEVb (this kind of interaction is accounted for in the next model version)  
-
*This model does not include any specific terms for time delays  
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*This model does not include any specific terms for time delays
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<html>
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<CENTER><img src="https://static.igem.org/mediawiki/2010/c/c6/IC_Equation_8.png"/></CENTER>
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</html>
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In the equation above: T = Tobacco Etch Virus Protease, sD = split Dioxygenase, D = Dioxygenase, sTa/sTb = inactivated split TEV, Ts/Tsa/Tsb = activated split TEV
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<html><h3>5. Model C: Further improvements</h3></html>
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<b>Implementation in Matlab</b>
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For the code please refer to the <html><a href="https://2010.igem.org/Team:Imperial_College_London/Modelling/Output/Download_MatLab_Files"><b>Download MatLab Files</b></a></html> section.
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{| style="width:900px;background:#f5f5f5;text-align:justify;font-family: helvetica, arial, sans-serif;color:#555555;margin-top:5px;" cellspacing="20"
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|style="font-family: helvetica, arial, sans-serif;font-size:2em;color:#ea8828;"|5. Model C: Further improvements
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|-
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This model has not been implemented because of the conclusions that we reached from Models A and B.  
This model has not been implemented because of the conclusions that we reached from Models A and B.  
It would include the following features:  
It would include the following features:  
*activated split TEV (TEVs) is allowed to activate not only sD but sTEVa and sTEVb  
*activated split TEV (TEVs) is allowed to activate not only sD but sTEVa and sTEVb  
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|-
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|style="font-family: helvetica, arial, sans-serif;font-size:2em;color:#ea8828;" align="right"|[[Team:Imperial_College_London/Modelling/Output/Results_and_Conclusion | Click here for the results of this model...]]
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Latest revision as of 03:48, 28 October 2010

Modelling Overview | Detection Model | Signaling Model | Fast Response Model | Interactions
A major part of the project consisted of modelling each module. This enabled us to decide which ideas we should implement. Look at the Fast Response page for a great example of how modelling has made a major impact on our design!
Objectives | Description | Results | Constants | MATLAB Code
Detailed Description

The following page presents the details of the models that have been developed. Firstly, assumptions that have been exploited are explained. Then every model is presented separately as each of them has slightly different elements of the system and the interactions between them. However, there are only 3 fundamental biochemical processes that will be analysed:

  1. Transcription
  2. Translation
  3. Enzymatic reactions
1. Law of Mass Action

During a meeting with our advisors, it was noted that our initial models (in which it was assumed that our system obeyed Michaelis-Menten kinetics) were wrong as the assumptions made by Michaelis-Menten approximation were not obeyed by the system.

A few of Michaelis-Menten assumptions were not met by our system:

  • Vmax is proportional to the overall concentration of the enzyme.

Since we are continuously producing enzyme, Vmax will change. Therefore the conservation of enzymes E0 = E + ES does not hold for our system.

  • Substrate >> Enzyme

We are producing both substrate and enzyme, so we have approximately the same amount of substrate and enzyme.

  • Enzyme affinity to the substrate has to be high.


Click on the button below, in order to get to know more on Michaelis-Menten kinetics based models.

Abandoned Initial Attempts

Elements of the system
Depending on which amplification it there is several species appearing. The full list is:

  • GFP (on diagrams shown green) which was out original choice for output later on changed to dioxygenase acting on catechol. It was supposed to be split and attached to coiled coils. It would get activated by TEV protease.
  • TEV protease (shown orange on diagrams)and split TEV (TEVs is shown yellow). Split TEV would be inactive form mounted on the coiled coils to be activated by HIV1 or another active TEV.
  • HIV1 protease (shown blue on diagrams) which would be produced active upon receiving the activation signal in case of 2 amplification step outputs.

Simple models
Our first models only were accounting for the protein and mRNA production:

Initial 1.png
2-step amplifier with distinct protease at each level.
Initial 1 eqn.png
Where: m - is the concentration of mRNA, p - is the protein concentration (subscript indicates which protein)
Initial 2.png
2-step amplifier with TEV protease implemented at both amplification levels.
Initial 2 eqn.png

Implementation of Michaelis-Menten kinetics
So, far the models were considering only the production and degradation of species. The example presented below includes enzymatic interaction between the species governed by Michaelis-Menten.

Initial 1.png
2-step amplifier with distinct protease at each level.
Initial 3 eqn.png



Implementation in Matlab
The Matlab code for the different stages of amplification and diagrams can be found [http://www.openwetware.org/wiki/Image:Modelling.docx here].

Kinetic constants

GFP split GFP TEV split TEV
Km and kcat - Km = 0.061; kcat = 0.16; [1] 40% of value for TEV -
Half-life or degradation rate Half-life in B.sub approximately 1.5 hours - - Half-life shorter than GFP
Production rate in B.sub - - - -

Conclusion

We were not able to obtain all the necessary constants. Hence, we decided to make educated guesses about possible relative values between the constants as well as varying them and observing the change in output.

As the result, we concluded that the amplification happens at each amplification level proposed. The magnitude of amplification varies depending on the constants. There is not much difference between using TEV or HIV1.

Change of output

During our literature research, we came across a better output, so we abandoned the idea of using GFP as an output. Instead, we are using catechol. An enzyme, dioxygenase, will be acting on the catechol, which will then result in a coloured output. Catechol will be added to the bacteria manually (i.e. the bacteria will not produce catechol). Hence, in our models dioxygenase will be treated as an output as this enzyme is the only activator of catechol in our system. This means that the change of catechol into its colourful form is dependent on the dioxygenase concentration.

References

  1. Kapust R. et al (2001) Tobacco etch virus protease: mechanism of autolysis and rational design of stable mutants with wild-type catalytic proficiency. Protein Engineering. [Online] 14(12), 993-1000. Available from: http://peds.oxfordjournals.org/cgi/reprint/14/12/993 [Accessed 28th July 2010]



As we could not use the Michaelis-Menten simplification to model enzymatic reactions in our system, we will had to solve the problem from first principle. It meant referring to more general set of assumptions called Law of Mass Action. This allowed us to model our enzymatic reactions without making assumptions about amounts of particular species as long as the amounts are bigger than single molecule level. This resulted in bigger number of partial differential equations as there was one per each species instead of 1 per reaction.

2. Model preA: Simple production of dioxygenase

This model was developed to illustrate 1-step amplification output.

Elements of the system

  1. Dioxygenase (C230 chown blue) is the enzyme that starts getting produced upon activation signal (eg. detection of Schistosoma).
  2. Catechol (cat. shown white with orange edge) is artificially added chemical during the experiment. It is acted on by dioxygenase to produce yellow muconic acid.
1-step amplification.png
Transcription and translation(simple production) of dioxygenase and its enzymatic activation of yellow colour.

Interactions between elements

Initially, dioxygenase has to be transcribed and translated. This has been described using 1 ODE equation:

In the equation above: [p] = protein concentration, s = protein production rate, d = protein degradation rate

The enzymatic interactions can be described in the following way:

In the equation above : D is Dioxygenase, C is Catechol, C-D is catechol-dioxygenase transition complex, Ma is Muconic acid - the yellow ouptput.

It is important to mention dioxygenase molecules tretramerize before becoming active enzymes. It has been very simplified in the model. Tetramerization is accounted for by simply dividing dioxygenase concentration by 4 before it acts on catechol.

Differential equations

The above reaction can be written in terms of ordinary differential equations:

Implementation in Matlab

These equations were implemented in Matlab, using a built-in function (ode15s) which solves ordinary differential equations. For the code please refer to the Download MatLab Files section.

3. Model A: Activation of Dioxygenase by TEV enzyme

This model was developed to illustrate 1-step amplification output.

Elements of the system

  1. TEV protease (shown pink) is simply produced upon detection of schistosoma.
  2. Dioxygenase (C230 chown blue) is pre-produced in inactive form by being attached to GFP. TEV releases dioxygenase and allows it to tetramerize and activate.
  3. Catechol (cat. shown white with orange edge)
2-step amplification.png
The species that are shown in front of vertical line which indicates beginning of experiment mean that they have been
accumulated beforehand in the cell. TEV protease activates inactive dioxygenase which acts on catechol to produce colour.

Interactions between elements

This model includes 2 enzymatic reactions:

  1. Reaction between dioxygenase and catechol, which is exactly the same as in model preA

    In the equation above : D is Dioxygenase, C is Catechol, C-D is catechol-dioxygenase transition complex, Ma is Muconic acid - the yellow ouptput.
  2. Reaction between TEV and inactive dioxygenase
    Equations showing enzymatic reaction between TEV and split Dioxygenase

    In the equation above: TEV = Protease of the Tobacco Etch Virus, sD = split Dioxygenase, TsD = intermediate complex of the enzymatic reaction,
    D = Dioxygenase

Differential equations

This is a simple enzymatic reaction, where TEV is the enzyme, Dioxygenase the product and split Dioxygenase the substrate. Choosing k1, k2, k3 as reaction constants, the reaction can be rewritten in these four sub-equations:

ODEs describing the reaction presented above

Implementation in Matlab

These equations were implemented in Matlab, using a built-in function (ode15s) which solves ordinary differential equations. For the code please refer to the Download MatLab Files section.

Implementation in TinkerCell

Another approach to model the amplification module would be to implement it in a program such as TinkerCell (or CellDesigner). This was used to check whether implementation in Matlab generates similar results. If happened otherwise, we would need to look for reasons for those differences in the programs. As the results, generated by Matlab were the same, only Matlab code has been developed further as it allows more flexibility, control and insight.

IC Tinkercell Model A.PNG
Model A implemented in TinkerCell.
4. Model B: Activation of Dioxygenase by TEV or activated split TEV
enzyme

Elements of the system

  1. split TEV protease (shown purple) is introduced as the 3rd amplification step. It exists in 2 versions (not shown on diagram): a and b as it does not separate into 2 identical halves.
  2. TEV protease (shown pink) is simply produced upon detection of schistosoma.
  3. Dioxygenase (C230 chown blue) is pre-produced in inactive form by being attached to GFP. TEV releases dioxygenase and allows it to tetramerize and activate.
  4. Catechol (cat. shown white with orange edge)
3-step amplification.png
This diagram introduces inactive split TEV protease attached to a coiled-coil as the third amplification step.

Both inactive compounds have active site for TEV to activate tehm which results in multiple possibilities of action.

Interactions and assumptions

This version includes the following features:

  • Split TEV is specified to have a and b parts
  • TEVa is forbidden to interact with TEVa (though in reality there could be some affinity between the two). Same for the interaction between Tevb and Tevb
  • Both TEV and TEVs are allowed to activate dioxygenase
  • Dioxygenase is assumed to be active as a monomer
  • Activate split TEV (TEVs) is not allowed to activate sTEVa or sTEVb (this kind of interaction is accounted for in the next model version)
  • This model does not include any specific terms for time delays

In the equation above: T = Tobacco Etch Virus Protease, sD = split Dioxygenase, D = Dioxygenase, sTa/sTb = inactivated split TEV, Ts/Tsa/Tsb = activated split TEV

Implementation in Matlab

For the code please refer to the Download MatLab Files section.

5. Model C: Further improvements

This model has not been implemented because of the conclusions that we reached from Models A and B. It would include the following features:

  • activated split TEV (TEVs) is allowed to activate not only sD but sTEVa and sTEVb
Click here for the results of this model...