Team:Imperial College London/Modelling/Protein Display
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<li>The surface protein consists of a cell wall binding domain, linker, AIP (Auto Inducing Peptide). It is expressed by constitutive gene expression. It was assumed that the bacteria would be fully grown before carrying out the detection so the cell wall would be covered with as many surface proteins as the cell can maintain.</li> | <li>The surface protein consists of a cell wall binding domain, linker, AIP (Auto Inducing Peptide). It is expressed by constitutive gene expression. It was assumed that the bacteria would be fully grown before carrying out the detection so the cell wall would be covered with as many surface proteins as the cell can maintain.</li> | ||
- | <li>Schistosoma elastase (enzyme released by the parasite) cleaves the AIP from the cell wall binding domain at the linker site. In the laboratory we used TEV protease as we could not get | + | <li>Schistosoma elastase (enzyme released by the parasite) cleaves the AIP from the cell wall binding domain at the linker site. In the laboratory we used TEV protease as we could not get hold of Schistosoma elastase, so the model was adjusted appropriately (TEV enzyme kinetic parameters were used).</li> |
- | <li>ComD receptor is | + | <li>The ComD receptor is activated by a sufficiently high AIP concentration. Once activated, ComD signals into the cell to activate the colour response.</li> |
</ol> | </ol> | ||
<h2>2. Interactions between elements </h2> | <h2>2. Interactions between elements </h2> | ||
- | Apart from proteins being expressed from genes, there was only one chemical reaction identified in this part of the system. | + | Apart from the proteins being expressed from genes, there was only one more chemical reaction identified in this part of the system. This is the cleavage of proteins, which is an enzymatic reaction: |
<ul> | <ul> | ||
<li>E+S <var>↔</var> ES <var>→</var> E+P</li> | <li>E+S <var>↔</var> ES <var>→</var> E+P</li> | ||
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<li>Product (P) = Peptide</li> | <li>Product (P) = Peptide</li> | ||
</ul><br /> | </ul><br /> | ||
- | This | + | This enzymatic reaction can be rewritten as partial differential equations (PDEs), which is of similar form as the 1-step amplification model. However, most of the constants and initial concentrations are different. For detailed description and derivation of PDEs, please refer to <a href="https://2010.igem.org/Team:Imperial_College_London/Modelling/Output">"Detailed Description"</a> part of Modelling Output.<br /> |
- | <h2>3. Threshold concentration of AIP | + | <h2>3. Threshold concentration of AIP</h2> |
The optimal peptide concentration required to activate ComD is 10 ng/ml <a href="http://ukpmc.ac.uk/backend/ptpmcrender.cgi?accid=PMC40587&blobtype=pdf">[1]</a>. This is the threshold value for ComD activation. However, the minimum concentration of peptide to give a detectable activation is 0.5ng/ml. | The optimal peptide concentration required to activate ComD is 10 ng/ml <a href="http://ukpmc.ac.uk/backend/ptpmcrender.cgi?accid=PMC40587&blobtype=pdf">[1]</a>. This is the threshold value for ComD activation. However, the minimum concentration of peptide to give a detectable activation is 0.5ng/ml. | ||
<br /> | <br /> | ||
- | The threshold for minimal activation of receptor is c<sub>th</sub>=4.4658x10<sup>-9</sup> mol/L. Click on the button below to | + | The threshold for the minimal activation of the receptor is c<sub>th</sub>=4.4658x10<sup>-9</sup> mol/L. Click on the button below to see the calculations. |
<div id="wrapper"> | <div id="wrapper"> | ||
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<h2>4. Control volume selection</h2> | <h2>4. Control volume selection</h2> | ||
- | Note that this enzymatic reaction is | + | Note that this enzymatic reaction is modelled outside the cell. Hence, it is important to take into account the cell boundaries. It is worth considering whether diffusion or fluid movements will play a significant role. |
- | Initially we defined control volume | + | Initially, we defined a control volume assuming that bacteria would grow in close colonies on the plate. The details can be seen by clicking on the button below. |
<br /> | <br /> | ||
<div id="wrapper"> | <div id="wrapper"> | ||
<div class="accordionButton">Initial Choice of Control Volume</div> | <div class="accordionButton">Initial Choice of Control Volume</div> | ||
<div class="accordionContent"> | <div class="accordionContent"> | ||
- | <b>Control volume initial choice | + | <b>Control volume initial choice</b><br/> |
- | This control volume is considered to be wrong | + | This control volume is considered to be wrong, but the details were kept for the reference.<br/> |
The control volume: | The control volume: | ||
- | The inner boundary is determined by the | + | The inner boundary is determined by the bacterium (proteins that have been displayed and cleaved cannot diffuse back into the bacterium). The outer boundary is time scale dependent. We have assumed that after mass cleavage of the display proteins by TEV, many of these AIPs will bind to the receptors quickly (e.g. 8 seconds). Our volume is determined by the distance that the AIPs could travel by diffusion within that short time. In this way, we can ensure that the concentration of AIPs outside our control volume is approximately 0 (after a given time interval). |
- | This approach is not very accurate and can lead | + | This approach is not very accurate and can lead to false negative conclusions (as in reality there will be a concentration gradient, with the highest conentration on the cell wall). |
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</html> | </html> | ||
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<br/><br/> | <br/><br/> | ||
- | <p>We realized that our initial choice of control volume was not accurate because | + | <p>We realized that our initial choice of control volume was not accurate because we had assumed that the bacteria was the medium. However, in reality bacteria live in colonies very close to each other. Since our bacteria was meant to be used in suspension we had to reconsider this issue.</p> |
<br/> | <br/> | ||
- | <b>Using CFU to estimate the spacing between cells | + | <b>Using CFU to estimate the spacing between cells</b><br/> |
CFU stands for Colony-forming unit. It is a measure of bacterial numbers. For liquids, CFU is measured per ml. | CFU stands for Colony-forming unit. It is a measure of bacterial numbers. For liquids, CFU is measured per ml. | ||
- | + | We already have data of CFU/ml from the Imperial iGEM 2008 team, so we could use this data to estimate the number of cells in a given volume using a spectrometer at 600nm wavelength. | |
The graph below is taken from the Imperial iGEM 2008 Wiki page <a hred="https://2008.igem.org/Imperial_College/18_September_2008">[4]</a>. | The graph below is taken from the Imperial iGEM 2008 Wiki page <a hred="https://2008.igem.org/Imperial_College/18_September_2008">[4]</a>. | ||
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The graph shows values of CFU/ml for different optical densities. The range of CFU/ml is therefore between 0.5x10^<sup>8</sup> - 5x10<sup>8</sup>. | The graph shows values of CFU/ml for different optical densities. The range of CFU/ml is therefore between 0.5x10^<sup>8</sup> - 5x10<sup>8</sup>. | ||
<br /> | <br /> | ||
- | In this calculation, we | + | In this calculation, we assumed that only one cell will grow and become one colony (i.e. no more than one cell will form no more than one colony). Therefore, the maximum number of cells in 1ml of solution is 5x10<sup>8</sup>. Taking the volume of 1 ml = 10<sup>-3</sup> dm<sup>3</sup> and dividing by the (maximum) number of cells in 1ml gives the average control volume (CV) around each cell: 2x10<sup>-12</sup> dm<sup>3</sup>/cell. For simplicity, we choose the control volume to be cubic. Taking the third root of this value gives the length of each side of the control volume. |
<p>Side length of CV = y = 1.26x10<sup>-4</sup> dm = 1.26x10<sup>-5</sup> m.</p> | <p>Side length of CV = y = 1.26x10<sup>-4</sup> dm = 1.26x10<sup>-5</sup> m.</p> | ||
- | <b>Choice of Control Volume allows simplifications | + | <b>Choice of Control Volume allows simplifications</b><br/> |
<ul> | <ul> | ||
- | <li>Firstly, assume that the cells will be placed in the centre of the CV. Hence, the protein | + | <li>Firstly, assume that the cells will be placed in the centre of the CV. Hence, after cleavage the protein will have an average distance of y/2 to travel in order to cross the boundary of the CV. This is calculated to happen within 0.18s. Even if the bacterium was not placed in the centre of the CV, the protein will travel from one end of the cube to the other in less than one second (~0.74s). Hence, it will take between 0.18 and 0.74s for the concentration of AIPs around the cell to be uniform. Noticing that these time values are very small, we can approximate our model to have a uniform concentration across the volume. Since we are underestimating the value of AIP concentration right next to the cell's surface, we are overestimating the time required for the AIP concentration to reach the threshold level.</li> |
<li>We can neglect the diffusive fluxes across the CV border (see figure below). Assuming that adjacent cells are producing the peptide at the same rate and that the concentration of TEV is the same around the cell, then the fluxes should be of the same value giving a net flux of zero. Hence, we can neglect diffusion and have our model limited to one bacterium.</li> | <li>We can neglect the diffusive fluxes across the CV border (see figure below). Assuming that adjacent cells are producing the peptide at the same rate and that the concentration of TEV is the same around the cell, then the fluxes should be of the same value giving a net flux of zero. Hence, we can neglect diffusion and have our model limited to one bacterium.</li> | ||
</ul> | </ul> | ||
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<br /><br /> | <br /><br /> | ||
<b>Limitations resulting from our assumptions</b><br/> | <b>Limitations resulting from our assumptions</b><br/> | ||
- | Most of our assumptions | + | Most of our assumptions concerning the control volume were plausible due to the careful choice of cell density = 5x10<sup>8</sup> CFU/ml. |
<br /> | <br /> | ||
- | If the density is | + | If the density is varied significantly, then our simplifications might not hold any longer. However, this does not mean that our system cannot function for lower cell densities. Our model might not be very accurate for predicting situations with cell densities that are much higher or lower than 5x10<sup>8</sup> CFU/ml. |
<br /> | <br /> | ||
- | It was decided that the model should not be used for cell densities lower 10<sup>7</sup> CFU/ml. Below | + | It was decided that the model should not be used for cell densities lower 10<sup>7</sup> CFU/ml. Below this value AIPs take more than 1 second to diffuse accross half of a side-length of the control volume (assuming that the cell is inside control volume). We agreed that below 10^<sup>7</sup> CFU/ml the approximation about uniform concentration throughout the control volume could be wrong and that the concentration gradients could become more significant. If our model was applied to this particular situation, it would possibly overestimate the time taken to activate the receptor. |
<br /> | <br /> | ||
- | It is not possible to increase cell density by more than 10<sup>9</sup> CFU/ml</math> because of the size of a cell. | + | It is not possible to increase the cell density by more than 10<sup>9</sup> CFU/ml</math> because of the size of a cell. |
<h2>5. Protein production (23/08/2010)</h2> | <h2>5. Protein production (23/08/2010)</h2> | ||
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</ul> | </ul> | ||
<br /> | <br /> | ||
- | Hence, we can deduce that the concentration that the protein expression will tend to: c = 1.4289x10<sup>-4</sup> mol/dm<sup>3</sup> = c<sub>final</sub>. | + | Hence, we can deduce that the final concentration that the protein expression will tend to is: c = 1.4289x10<sup>-4</sup> mol/dm<sup>3</sup> = c<sub>final</sub>. |
<br /><br /> | <br /><br /> | ||
- | Therefore, we can model the protein production by transcription and translation and adjust the production | + | Therefore, we can model the protein production by transcription and translation and adjust the production constants so that the concentration will tend towards c<sub>final</sub>. |
<br /> | <br /> | ||
Using a similar model to the simple production of Dioxygenase for the Output Amplification Model (<a href="https://2010.igem.org/Team:Imperial_College_London/Modelling/Output">Model preA</a>), we obtain the following graph: | Using a similar model to the simple production of Dioxygenase for the Output Amplification Model (<a href="https://2010.igem.org/Team:Imperial_College_London/Modelling/Output">Model preA</a>), we obtain the following graph: | ||
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- | <b>Protein production in Control Volume | + | <b>Protein production in Control Volume</b><br/> |
The previously determined constants of protein production in B.sub to obtain the concentration of proteins are not valid in the Control Volume. It has to be adjusted (multiplied) by the following factor: | The previously determined constants of protein production in B.sub to obtain the concentration of proteins are not valid in the Control Volume. It has to be adjusted (multiplied) by the following factor: | ||
<br /> | <br /> |
Revision as of 10:52, 15 October 2010
Surface Protein Model | ||||||||||||||||||||||||||
Objectives
The motivation to develop this model came from our design process. The idea of having surface proteins that could be cleaved and then activate the receptor was very innovatory. However, with this new approach questions arose that could not be answered without using computer models. Hence, the following aims for this model were specified:
Detailed Description
This model consists of 5 parts that had to be developed:
1. Elements of the system
2. Interactions between elementsApart from the proteins being expressed from genes, there was only one more chemical reaction identified in this part of the system. This is the cleavage of proteins, which is an enzymatic reaction:
This enzymatic reaction can be rewritten as partial differential equations (PDEs), which is of similar form as the 1-step amplification model. However, most of the constants and initial concentrations are different. For detailed description and derivation of PDEs, please refer to "Detailed Description" part of Modelling Output. 3. Threshold concentration of AIPThe optimal peptide concentration required to activate ComD is 10 ng/ml [1]. This is the threshold value for ComD activation. However, the minimum concentration of peptide to give a detectable activation is 0.5ng/ml.The threshold for the minimal activation of the receptor is cth=4.4658x10-9 mol/L. Click on the button below to see the calculations. Converting 10 ng/ml to 4.4658x10-9 mol/L
4. Control volume selectionNote that this enzymatic reaction is modelled outside the cell. Hence, it is important to take into account the cell boundaries. It is worth considering whether diffusion or fluid movements will play a significant role. Initially, we defined a control volume assuming that bacteria would grow in close colonies on the plate. The details can be seen by clicking on the button below.Initial Choice of Control Volume
Control volume initial choice
This control volume is considered to be wrong, but the details were kept for the reference. The control volume: The inner boundary is determined by the bacterium (proteins that have been displayed and cleaved cannot diffuse back into the bacterium). The outer boundary is time scale dependent. We have assumed that after mass cleavage of the display proteins by TEV, many of these AIPs will bind to the receptors quickly (e.g. 8 seconds). Our volume is determined by the distance that the AIPs could travel by diffusion within that short time. In this way, we can ensure that the concentration of AIPs outside our control volume is approximately 0 (after a given time interval). This approach is not very accurate and can lead to false negative conclusions (as in reality there will be a concentration gradient, with the highest conentration on the cell wall).
We realized that our initial choice of control volume was not accurate because we had assumed that the bacteria was the medium. However, in reality bacteria live in colonies very close to each other. Since our bacteria was meant to be used in suspension we had to reconsider this issue. Using CFU to estimate the spacing between cells CFU stands for Colony-forming unit. It is a measure of bacterial numbers. For liquids, CFU is measured per ml. We already have data of CFU/ml from the Imperial iGEM 2008 team, so we could use this data to estimate the number of cells in a given volume using a spectrometer at 600nm wavelength. The graph below is taken from the Imperial iGEM 2008 Wiki page [4].
Side length of CV = y = 1.26x10-4 dm = 1.26x10-5 m. Choice of Control Volume allows simplifications
5. Protein production (23/08/2010)
Hence, we can deduce that the final concentration that the protein expression will tend to is: c = 1.4289x10-4 mol/dm3 = cfinal. Therefore, we can model the protein production by transcription and translation and adjust the production constants so that the concentration will tend towards cfinal. Using a similar model to the simple production of Dioxygenase for the Output Amplification Model (Model preA), we obtain the following graph:
The degradation rate was kept constant, and the production rate was changed according to the final concentration. Protein production in Control VolumeThe previously determined constants of protein production in B.sub to obtain the concentration of proteins are not valid in the Control Volume. It has to be adjusted (multiplied) by the following factor: factor=Vbacillus/VCV = 5.7974x10-6 (for the particular numbers presented above)
References
Parameters & Constants
Constants for the Protein Display Model
References
Results & Conclusions
Risk of False positives (31/08/2010) It was pointed out that we should assess the risk of false positive activation of the receptor. We are particularly concerned about the display protein not binding to the cell wall, but instead diffusing into the extra-cellular environment. In order to be able to assess the risk of false positives, we need to do further research into the affinity of AIP with attached linker and transmembrane proteins for the receptor as compared to the affinity of the AIP itself for the receptor. This paper [1] might have some information on affinity comparison. We need to know how proteins are being transported from intracellular to transmembrane space. Understanding this concept could give us an idea of what could go wrong. References
Download: MatLab Files
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