Difference between revisions of "Team:Exeter/Model"
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| − | + | <li><a id="links" style="margin:10px 0 30px 2px;padding:0;font-size:1.8vh;" href="https://2016.igem.org/Team:Exeter/Human_Practices">Human Practices Homepage</a></li> | |
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| − | <a href="#section_1" class="banner_link col-xs-6 col-sm-3"><span class="oneline"> | + | <a href="#section_1" class="banner_link col-xs-6 col-sm-3"><span class="oneline">Introduction</span></a> |
| − | <a href="#section_2" class="banner_link col-xs-6 col-sm-3"><span class="oneline"> | + | <a href="#section_2" class="banner_link col-xs-6 col-sm-3"><span class="oneline">Section 2</span></a> |
| − | <a href="#section_3" class="banner_link col-xs-6 col-sm-3"><span class=" | + | <a href="#section_3" class="banner_link col-xs-6 col-sm-3"><span class="twoline">Enzymatic <br /> Killswitches</span></a> |
| − | <a href="#section_4" class="banner_link col-xs-6 col-sm-3"><span class=" | + | <a href="#section_4" class="banner_link col-xs-6 col-sm-3"><span class="oneline">Section 4</span></a> |
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| − | + | Introduction | |
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| − | + | Section 2 | |
| + | </div> | ||
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<div> | <div> | ||
| − | + | <a id="Section_link" href="#section_3" style="display:block;margin:20px auto 0 auto;width:14px;"><span style="color:#47BCC2;font-size: 25px;" class="glyphicon glyphicon-menu-down" aria-hidden="true"></span></a> | |
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| − | + | <div id="contentTitle"> | |
| − | + | Enzymatic Models | |
| − | + | </div> | |
| − | < | + | <h5>mRNA production in Enzymatic kill switches</h5> |
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| − | < | + | |
| − | + | ||
| − | < | + | |
<p id="pp"> | <p id="pp"> | ||
| − | + | The change from software such as Simbiology and Simulink called for a more fundamental method of modelling cell death. Preliminary research | |
| − | + | showed kill switches producing the proteins “Lysozyme c” and “DNase 1” both had very similar mechanisms; as both are enzymes. Therefore, it | |
| − | + | was decided that the two models would use the same code to simulate mRNA and protein production. | |
| − | + | ||
| − | the | + | |
</p> | </p> | ||
<p id="pp"> | <p id="pp"> | ||
| − | <span class="equation">$ | + | Initially, following advice from biologists and biochemists on our team, the first code incorporated two step functions, the first of these |
| + | modelled mRNA production by a single <i>E. coli</i> cell. To calculate protein made by each mRNA a secondary step function triggered each time an mRNA | ||
| + | was produced, the sum of these functions gave the total amount of protein. This program was simple with the only input variables being the production time of | ||
| + | mRNA and the respective protein. | ||
| + | The initial model was presented to the rest of the team receiving plenty of feedback, the most prevalent point being that the code modelled a single | ||
| + | cell system with a single plasmid whilst it should model a single cell system that duplicates and has multiple plasmids. It was suggested the following factors were added to the model: | ||
| + | </p> | ||
| + | <ul> | ||
| + | <li>Plasmid production</li> | ||
| + | <li>Degradation of mRNA and protein</li> | ||
| + | <li>Maturation of protein</li> | ||
| + | <li>Duplication rate of <i>E. coli</i></li> | ||
| + | </ul> | ||
| + | <p id="pp"> | ||
| + | The initial feedback called for a re-write of the code as a considerable amount of the suggestions came in stages before mRNA production. | ||
| + | A second version of the code was written which incorporated all main steps between the splitting of <i>E. coli</i> to the degradation rate of protein. | ||
| + | </p> | ||
| + | <h6>Assumptions</h6> | ||
| + | <p id="pp"> | ||
| + | It is important to outline the factors that were overlooked due to research finding their effect on the model would be negligible. Firstly, after | ||
| + | researching the production time of plasmids in <i>E. coli</i> it was found that plasmids will reproduce at a variable rate to maintain a constant population | ||
| + | determined by their copy number (Nordström and Dasgupta, 2006). To address this, the production rate of plasmids was overlooked, allowing for a | ||
| + | constant value of plasmids to be maintained throughout the simulation. In experiments a pSB1C3 strain was used - a high copy number plasmid; | ||
| + | therefore the copy number was set to 300. | ||
| + | Secondly, both enzymatic models will assume that travel time of protein to the substrate is negligible. Protein diffusing through the cytoplasm of | ||
| + | <i>E. coli</i> have a diffusion coefficient on the scale of $10 \mu \text{m}^2 \text{s}^{-1}$ (Elowitz et al., 1998). Considering the surface area of <i>E. coli</i> is approximately | ||
| + | $10 \mu \text{m}^2$, the protein will reach the cell wall in a several seconds which is several magnitudes of order smaller than the simulation time. Lastly, the models | ||
| + | will assume no mutations occur; the aim of the simulations is to determine whether kill switches are a plausible method of biosafety, if the models show | ||
| + | a kill switch is not a reliable way to terminate GMO’s then accounting for mutations will only that enforce statement. | ||
| + | </p> | ||
| + | <h6>Features</h6> | ||
| + | <p id="pp"> | ||
| + | Research showed that there is an upper limit of mRNA in <i>E. coli</i>, | ||
| + | therefore an upper limit of $4 \times 10^3$ mRNA per <i>E. coli</i> cell (Thermofisher.com, 2016) has been included in the model. The lifetime of mRNA can be found | ||
| + | from the observed half life of approximately 5 minutes or $300\text{s}$ (Bernstein et al., 2002), resulting in an average lifetime of $430\text{s}$. The production rate of | ||
| + | mRNA along with ribosomes per coding region will be worked out for both the lysozyme and DNase models independently. In addition to this, both | ||
| + | enzymatic models use the well known duplication time of <i>E. coli</i> - 17 minutes, at which point both the mRNA and protein is assumed to split among the two cells equally,. | ||
| + | </p> | ||
| + | <p id="pp"> | ||
| + | <span class="equation">$T_{(mRNA)} = \frac{T_{(mRNA) \frac{1}{2}}}{ln(2)} = \frac{300\text{s}}{ln(2)} = 430\text{s (2sf)}$<span class="equation_ref">(Hyperphysics.phy-astr.gsu.edu, 2016)</span></span> | ||
<span class="equation_key"> | <span class="equation_key"> | ||
| − | $ | + | $T_{(mRNA)}$: Lifetime of mRNA [$\text{s}$]<br /> |
| − | + | $T_{(mRNA) \frac{1}{2}}$: Half life of mRNA [$\text{s}$] | |
| − | $ | + | |
| − | + | ||
</span> | </span> | ||
</p> | </p> | ||
| − | < | + | <h5>Lysozyme Model</h5> |
| − | + | <p id="pp"> | |
| − | + | In the case of the “Lysozyme c” kill switch the mechanisms beyond producing mRNA need to be modelled separately from the DNase model. There are several | |
| − | + | assumptions that will be made, the first of these is that enzymatic reactions can be modelled by Michaelis-Menten kinetics. Secondly, the temperature | |
| − | + | of the constants taken imply that this model is running in the range of $37-40^o\text{C}$ at an optimal pH for <i>E. coli</i> growth. The model will assume that when | |
| − | < | + | <i>E. coli</i> splits, the contents of the cell and damage of the cell wall is shared equally among the two resulting <i>E. coli</i>. Lastly, it will assume that lysozyme does not degrade |
| − | + | throughout the simulation. | |
| − | + | </p> | |
| − | + | ||
| − | </ | + | |
<p id="pp"> | <p id="pp"> | ||
| − | + | The plasmid used to produce lysozyme has a PCR with a length of $507\text{bp}$ with the promoter, RBS and terminator totalling a further $164\text{bp}$. Therefore the | |
| − | + | production rates of mRNA and lysozyme can be calculated using translation and transcription rates of $V_{translation} = 8.4\text{aas}^{-1}$ | |
| − | + | (Siwiak and Zielenkiewicz, 2013) and $V_{transcription} = 40\text{bps}^{-1}$ (García and Molineux, 1995) respectively. The translation time is for <i>E. coli</i> at $37^o\text{C}$, other values | |
| − | + | have been taken at $40^o\text{C}$ as this was the closest temperature that could be found. All reaction rates have been rounded to the nearest second as this | |
| − | + | reduces calculation times. | |
| − | + | ||
| − | + | ||
| − | + | ||
</p> | </p> | ||
<p id="pp"> | <p id="pp"> | ||
| − | <span class="equation">$ | + | <span class="equation">$t_{lysozyme} = \frac{L_{protein}}{V_{translation}} = \frac{\frac{507\text{bp}}{3}}{8.4\text{aas}^{-1}} = 20\text{s (2sf)}$</span><br /> |
| − | <span class="equation">$\ | + | <span class="equation">$t_{mRNA} = \frac{L_{plasmid}}{V_{transcription}} = \frac{507\text{bp} + 164\text{bp}}{40\text{bps}^{-1}} = 17\text{s (2sf)}$</span> |
<span class="equation_key"> | <span class="equation_key"> | ||
| − | $ | + | $t_{lysozyme}$: Time to produce one lysozyme protein [$\text{s}$]<br /> |
| − | $ | + | $t_{mRNA}$: Time to produce one mRNA [$\text{s}$]<br /> |
| − | $ | + | $L_{plasmid}\text{, }L_{protein}$: Length of plasmid and protein coding region [$\text{bp}$] |
</span> | </span> | ||
</p> | </p> | ||
<p id="pp"> | <p id="pp"> | ||
| − | + | To supplement the production of lysozyme, the program will implement the affects of multiple ribosomes on the coding site of lysozyme. For <i>E. coli</i> | |
| − | + | it has been found there are 3.46 codons per $100\text{bp}$ (Siwiak and Zielenkiewicz, 2013). The PCR used for lysozyme production has a length of $507\text{bp}$, meaning | |
| − | + | there are approximately 5.8 codons which will be rounded down to 5 as not to overproduce lysozyme. | |
| − | + | ||
| − | + | ||
| − | + | ||
</p> | </p> | ||
<p id="pp"> | <p id="pp"> | ||
| − | + | “Lysozyme c” is an enzyme that hydrolyses bonds holding together peptidoglycan in the cell wall, this is explained in detail on the <a href="https://2016.igem.org/Team:Exeter/Project">project page</a>. The | |
| − | + | next task of the lysozyme model is to connect the amount of protein at each time to the degradation of the <i>E. coli</i> cell wall, to do this | |
| − | + | Michaelis-Menten kinetics were applied, which gives the reaction rate of one enzyme (Berg et al., 2002). | |
| − | + | ||
| − | + | ||
| − | + | ||
| − | + | ||
| − | + | ||
</p> | </p> | ||
<p id="pp"> | <p id="pp"> | ||
| − | + | <span class="equation">$k_{(cat)} = \frac{[S]k_{(cat)max}}{[S] + K_M}$</span> | |
| − | + | <span class="equation_key"> | |
| − | + | $k_{(cat)}$: Reaction rate of one lysozyme [$\text{s}^{-1}$]<br /> | |
| − | + | $k_{(cat)max}$: Maximum reaction rate of one lysozyme [$\text{s}^{-1}$]<br /> | |
| − | + | $[S]$: Substrate concentration [$\text{M}$]<br /> | |
| − | + | $K_M$: Michaelis constant [$\text{M}$] | |
| − | + | </span> | |
| − | + | ||
</p> | </p> | ||
<p id="pp"> | <p id="pp"> | ||
| − | <span class="equation">$ | + | To use this model two constants are required, the Michaelis constant ($K_M$), the concentration of the substrate when the reaction rate is exactly one half of the maximum reaction rate. |
| + | The average reaction rate assumed to be $k_{(cat)avg} = (k_{(cat)max}/2$). The logarithms of both of these | ||
| + | values has been calculated at $40^oC$ to be $-log(K_M) = 5.18 \pm 0.3$M and $-log(k_{cat}^{obs}) = 0.15 \pm 0.005$s$^{-1}$ (Banerjee et al., 1975). Giving values of: | ||
| + | </p> | ||
| + | <p id="pp"> | ||
| + | <span class="equation">$K_M = 5.6\text{mM}$ (2sf)</span><br /> | ||
| + | <span class="equation">$k_{(cat)avg} = \frac{k_{(cat)max}}{2} \approx \frac{k_{(cat)}^{obs}}{2} = \frac{0.86\text{s}^{-1}}{2} = 0.43\text{s}^{-1}$ (2sf)</span> | ||
| + | </p> | ||
| + | <p id="pp"> | ||
| + | Lastly, the initial concentration of the substrate peptidoglycan is calculated. An assumption is made that determines that all the peptidoglycan in | ||
| + | <i>E. coli</i> is spread out over the entire volume of the cell. Peptidoglycan or murein amount has been calculated to be approximately $3.5\text{x}10^6$ molecules | ||
| + | per cell in a strain of <i>E. coli</i> (Vollmer and Höltje, 2004). Using an approximate volume of <i>E. coli</i> of $0.7 \mu \text{m}^3$, the concentration of peptidoglycan is: | ||
| + | </p> | ||
| + | <p id="pp"> | ||
| + | <span class="equation">$[Pep]_{int} = \frac{\frac{N_{pep}}{N_A}}{V_{E. coli}} = \frac{\frac{3.5\text{x}10^6}{6.02\text{x}10^{23}}}{0.7 \mu \text{m}^3} = 8.3\text{mM}$ (2sf)</span> | ||
<span class="equation_key"> | <span class="equation_key"> | ||
| − | $ | + | $[Pep]_{int}$: Initial concentration of peptidoglycan [$\text{mM}$]<br /> |
| − | $\ | + | $N_{pep}$: Amount of peptidoglycan [molecules]<br /> |
| + | $N_A$: Avogadro's constant [molecules/mole]<br /> | ||
| + | $V_{\textit{E. coli}}$: Volume of <i>E. coli</i> [$\mu\text{m}^3$] | ||
</span> | </span> | ||
</p> | </p> | ||
| + | <p id="pp"> | ||
| + | This calculation gives a value in the same order of magnitude as the Michaelis constant, which represents the concentration of substrate when the | ||
| + | reaction rate is at half of its maximum value. | ||
| + | </p> | ||
| + | <h5>Results</h5> | ||
<div class="col-xs-12" style="width:100%;position:relative;margin:auto;padding:0;"> | <div class="col-xs-12" style="width:100%;position:relative;margin:auto;padding:0;"> | ||
| − | <div class=" | + | <div class="graph_box_single col-xs-12"> |
| − | <img src=" | + | <img src="graph3_1.png"> |
| − | <span>Fig. | + | <span>Fig. 1. Using Michaelis-Menten kinetics the reaction rate of each lysozyme enzyme has |
| − | + | been plotted for each peptidoglycan substrate concentration. The | |
| − | + | average reaction rate of $0.43\text{s}^{-1}$ occurs when the concentration is equal to $K_M = 0.0056\text{M}$. | |
| − | + | The maximum or initial concentration $[Pep]_{int} = 0.0083\text{M}$ of the substrate causes a reaction rate of $0.51\text{s}^{-1}$.</span> | |
| − | + | ||
</div> | </div> | ||
</div> | </div> | ||
<p id="pp"> | <p id="pp"> | ||
| − | + | Enzyme reaction rates have been modelled by the Michaelis-Menten kinetics model in Fig. 1, therefore the reaction | |
| − | + | rate decreases as the substrate concentration decreases. This graph shows that the reaction | |
| − | + | rate will be greatest at the beginning of the simulation and approach zero when the cell wall is most damaged. | |
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</p> | </p> | ||
<div class="col-xs-12" style="width:100%;position:relative;margin:auto;padding:0;"> | <div class="col-xs-12" style="width:100%;position:relative;margin:auto;padding:0;"> | ||
<div class="graph_box col-xs-12"> | <div class="graph_box col-xs-12"> | ||
| − | <img src=" | + | <img src="graph1.png"> |
| − | <span>Fig. | + | <span>Fig. 2. The percentage of peptidoglycan compared to the original concentration plotted against time.</span> |
</div> | </div> | ||
<div class="graph_box col-xs-12"> | <div class="graph_box col-xs-12"> | ||
| − | <img src=" | + | <img src="graph2.png"> |
| − | <span>Fig. | + | <span>Fig. 3. Plots a smaller range of times as Fig. 2. To show the rapid decrease in peptidoglycan concentration.</span> |
</div> | </div> | ||
</div> | </div> | ||
<p id="pp"> | <p id="pp"> | ||
| − | + | The model predicted the complete degradation of the cell wall to be within the first generation | |
| − | < | + | of <i>E. coli</i>, Fig. 2. The reaction rate is slow at first due to the cell having no initial lysozyme, |
| − | + | this slowly increases, until the low concentration of the substrate casues | |
| − | + | the reaction rate of lysozyme to slow considerably. The peptidoglycan concentration in the cell is | |
| − | + | negligible until 17 minutes at which point the <i>E. coli</i> splits sharing the cell wall damage equally | |
| − | + | between the two child cells, hence cell damage drops from almost 100% to 50%. The immediate concern | |
| − | + | is that in this model cell death would occur far before it is able to duplicate, meaning that | |
| + | assuming no mutations the cell would terminate before 17 minutes. | ||
</p> | </p> | ||
<p id="pp"> | <p id="pp"> | ||
| − | + | The cell death threshold of peptidoglycan concentration in the cell wall is not well defined from | |
| − | + | research. Fig. 3 demonstrates that any threshold that is chosen is likely to fall in between 2 | |
| − | + | and 5 minutes of the simulation which is well before the reproduction rate of <i>E. coli</i> of 17 | |
| − | + | minutes. Therefore it is a reasonable assumption that given no mutations were to occur that the | |
| − | + | cell would be terminated before the <i>E. coli</i> could reproduce. | |
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</p> | </p> | ||
| + | <h5>References</h5> | ||
| + | <ol style="font-size:100%;"> | ||
| + | <li>Nordström, K. and Dasgupta, S. (2006). Copy-number control of the <i>Escherichia coli</i> chromosome: a plasmidologist's view. EMBO Rep, [online] 7(5), pp.484-489. Available at: http://www.ncbi.nlm.nih.gov/pmc/articles/PMC1479556/ [Accessed 6 Sep. 2016].</li> | ||
| + | <li>Elowitz, M., Surette, M., Wolf, P., Stock, J. and Leibler, S. (1998) ‘Protein mobility in the cytoplasm of Escherichia coli’, Journal of bacteriology., 181(1), pp. 197–203 [Accessed 6 Sep. 2016].</li> | ||
| + | <li>Thermofisher.com. (2016). Macromolecular Components of E. coli and HeLa Cells | Thermo Fisher Scientific. [online] Available at: https://www.thermofisher.com/uk/en/home/references/ambion-tech-support/rna-tools-and-calculators/macromolecular-components-of-e.html# [Accessed 6 Sep. 2016].</li> | ||
| + | <li>Bernstein, J., Khodursky, A., Lin, P., Lin-Chao, S. and Cohen, S. (2002). Global analysis of mRNA decay and abundance in Escherichia coli at single-gene resolution using two-color fluorescent DNA microarrays. Proceedings of the National Academy of Sciences, [online] 99(15), pp.9697-9702. Available at: http://www.pnas.org/content/99/15/9697.long [Accessed 6 Sep. 2016].</li> | ||
| + | <li>Berg, J., Tymoczko, J., Stryer, L. and Stryer, L. (2002). Biochemistry. New York: W.H. Freeman, pp.Section 8.4 - Equation (23).</li> | ||
| + | <li>Banerjee, S., Holler, E., Hess, G. and Rupley, J. (1975). Reaction of N-acetylglucosamine oligosaccharides with lysozyme. Temperature, pH, and solvent deuterium isotope effects; equilbrium, steady state, and pre-steady state measurements*. Journal of Biological Chemistry, [online] 250(11), pp.4357, Figure 2 and 4359, Table I. Available at: http://www.jbc.org/content/250/11/4355.long [Accessed 7 Sep. 2016].</li> | ||
| + | <li>Vollmer, W. and Höltje, J. (2004). The Architecture of the Murein (Peptidoglycan) in Gram-Negative Bacteria: Vertical Scaffold or Horizontal Layer(s)?. Journal of Bacteriology, [online] 186(18), p.5980. Available at: http://jb.asm.org/content/186/18/5978 [Accessed 7 Sep. 2016].</li> | ||
| + | <li>Siwiak, M. and Zielenkiewicz, P. (2013) ‘Transimulation - protein Biosynthesis web service’, PLoS ONE, 8(9), p. e73943. doi: 10.1371/journal.pone.0073943. [Accessed 7 Sep. 2016]</li> | ||
| + | <li>García, L. and Molineux, I. (1995) ‘Rate of translocation of bacteriophage T7 DNA across the membranes of Escherichia coli’, Journal of bacteriology., 177(14), pp. 4066–76.[Accessed 7 Sep. 2016]</li> | ||
| + | <li>Siwiak, M. and Zielenkiewicz, P. (2013). Transimulation - Protein Biosynthesis Web Service. PLoS ONE, [online] 8(9), p.3, left column, second paragraph. Available at: http://www.ncbi.nlm.nih.gov/pmc/articles/PMC3764131/ [Accessed 7 Sep. 2016].</li> | ||
| + | <li>Hyperphysics.phy-astr.gsu.edu. (2016). Mean Lifetime for Particle Decay. [online] Available at: http://hyperphysics.phy-astr.gsu.edu/hbase/nuclear/meanlif.html [Accessed 10 Sep. 2016].</li> | ||
| + | </ol> | ||
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<a id="Section_link" href="#section_4" style="display:block;margin:20px auto 0 auto;width:14px;"><span style="color:#47BCC2;font-size: 25px;" class="glyphicon glyphicon-menu-down" aria-hidden="true"></span></a> | <a id="Section_link" href="#section_4" style="display:block;margin:20px auto 0 auto;width:14px;"><span style="color:#47BCC2;font-size: 25px;" class="glyphicon glyphicon-menu-down" aria-hidden="true"></span></a> | ||
</div> | </div> | ||
</div> | </div> | ||
| − | <div class="col-xs-12 div_content"> | + | <div class="col-xs-12 div_content"> |
<div id="section_4" class="link_fix"></div> | <div id="section_4" class="link_fix"></div> | ||
<div id="contentTitle"> | <div id="contentTitle"> | ||
| − | + | Section 4 | |
</div> | </div> | ||
</div> | </div> | ||
Revision as of 16:20, 13 October 2016
