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

From 2010.igem.org

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|style="font-family: helvetica, arial, sans-serif;font-size:2em;color:#ea8828;"|Output Amplification Model
|style="font-family: helvetica, arial, sans-serif;font-size:2em;color:#ea8828;"|Output Amplification Model
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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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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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<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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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. Click on the button below to learn more about our models based on Michaelis-Menten kinetics.
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<html><h2>1. Law of Mass Action</h2></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.
 +
 
 +
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 <html>E<sub>0</sub> = E + E<sub>S</sub></html> does not hold for our system.
 +
*'''Substrate >> Enzyme'''
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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.
<div class="accordionButton">Abandoned Initial Attempts</div>
<div class="accordionButton">Abandoned Initial Attempts</div>
<div class="accordionContent">
<div class="accordionContent">
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<html><h2>Model based on Michaelis Menten Kinetics</h2></html>
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<b>Elements of the system</b><br/>
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[[Image:IC_Michaelis_Menten1.png|800px]]
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Depending on which amplification it there is several species appearing. The full list is:
 +
*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.
 +
*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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<html><h2>Improved Model which accounts for enzyme reactions</h2></html>
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<b>Simple models</b><br/>
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[[Image:IC_Michaelis_Menten2.png|800px]]
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Our first models only were accounting for the protein and mRNA production:
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{|style="width:825px;background:#eeeeee;text-align:justify;font-family: helvetica, arial, sans-serif;color:#555555;margin-top:5px;" cellspacing="5";
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|-
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|<div ALIGN=Center>
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{| style="width:270px;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.png|268px]]
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|-
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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: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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|}
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</div>
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|-
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|
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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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| <div ALIGN=Center>
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{| style="width:270px;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_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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{| 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_2_eqn.png|543px]]
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|}
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</div>
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|}
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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.
 +
{|style="width:825px;background:#eeeeee;text-align:justify;font-family: helvetica, arial, sans-serif;color:#555555;margin-top:5px;" cellspacing="5";
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|-
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|<div ALIGN=Center>
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{| style="width:470px;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.png|468px]]
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|-
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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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|[[Image:Initial_3_eqn.png|343px]]
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</div>
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<br/><br/>
<br/><br/>
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</html>
</html>
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<h2>Conclusion</h2>
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<b>Conclusion</b><br/>
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.
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.
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>
 +
 +
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>
<h2>References</h2>
#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]
#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><br/><br/><br/>
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</div>
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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:
+
<br/><br/>
-
*'''Vmax is proportional to the overall concentration of the enzyme.'''
+
-
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.
+
-
*'''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.'''
+
-
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'''
+
-
 
+
-
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.
+
 +
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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<html><h3>2. Model preA: Simple production of dioxygenase</h3></html>
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<html><h2>2. Model preA: Simple production of dioxygenase</h2></html>
<div ALIGN=CENTER>
<div ALIGN=CENTER>
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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.
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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<html><h3>3. Model A: Activation of Dioxygenase by TEV enzyme</h3></html>
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<html><h2>3. Model A: Activation of Dioxygenase by TEV enzyme</h2></html>
<div ALIGN=CENTER>
<div ALIGN=CENTER>
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</div>
</div>
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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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<html><h2>4. Model B: Activation of Dioxygenase by TEV or activated split TEV enzyme</h2></html>
<div ALIGN=CENTER>
<div ALIGN=CENTER>
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<html><h3>5. Model C: Further improvements</h3></html>
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<html><h2>5. Model C: Further improvements</h2></html>
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.  

Revision as of 00:30, 22 October 2010

Temporary sub-menu:

Objectives; Detailed Description; Parameters & Constants; Results & Conclusion;Download MatLab Files;

Output Amplification Model

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 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

IC Simple production.JPG
Transcription and translation
(simple production) of dioxygenase.

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.

3. Model A: Activation of Dioxygenase by TEV enzyme

IC 1-step amplification.JPG
1-step amplification.


This model consists of the basic enzymatic reaction:

Equations showing enzymatic reaction between TEV and split Dioxygenase

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:

PDEs describing the reaction presented above

These four equations were implemented in Matlab, using a built-in function (ode45) which solves ordinary differential equations.

Implementation in TinkerCell

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.

IC Tinkercell Model A.PNG
Model A implemented in TinkerCell.

4. Model B: Activation of Dioxygenase by TEV or activated split TEV enzyme

IC 2-step amplification.JPG
2-step amplification.

This version includes the following features:

  • 2 amplification steps (TEV and split TEV)
  • 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


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
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