Difference between revisions of "Team:Exeter/Model"

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Enzymatic Models
 
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<h5>mRNA production in Enzymatic kill switches</h5>
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<p id="pp">
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The change from software such as Simbiology and Simulink called for a more fundamental method of modelling cell death. Preliminary research
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showed kill switches producing the proteins “Lysozyme c” and “DNase 1” both had very similar mechanisms; as both are enzymes. Therefore, it
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was decided that the two models would use the same code to simulate mRNA and protein production.
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</p>
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<p id="pp">
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Initially, following advice from biologists and biochemists on our team, the first code incorporated two step functions, the first of these
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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
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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
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mRNA and the respective protein.
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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
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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:
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</p>
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<ul>
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<li>Plasmid production</li>
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<li>Degradation of mRNA and protein</li>
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<li>Maturation of protein</li>
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<li>Duplication rate of <i>E. coli</i></li>
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</ul>
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<p id="pp">
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The initial feedback called for a re-write of the code as a considerable amount of the suggestions came in stages before mRNA production.
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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.
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</p>
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<h6>Assumptions</h6>
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<p id="pp">
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It is important to outline the factors that were overlooked due to research finding their effect on the model would be negligible. Firstly, after
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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
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constant value of plasmids to be maintained throughout the simulation. In experiments a pSB1C3 strain was used - a high copy number plasmid;
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therefore the copy number was set to 300.
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Secondly, both enzymatic models will assume that travel time of protein to the substrate is negligible. Protein diffusing through the cytoplasm of
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<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
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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
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a kill switch is not a reliable way to terminate GMO’s then accounting for mutations will only that enforce statement.
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</p>
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<h6>Features</h6>
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<p id="pp">
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Research showed that there is an upper limit of mRNA in <i>E. coli</i>,
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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
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mRNA along with ribosomes per coding region will be worked out for both the lysozyme and DNase models independently. In addition to this, both
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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,.
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</p>
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<p id="pp">
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<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>
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<span class="equation_key">
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$T_{(mRNA)}$: Lifetime of mRNA [$\text{s}$]<br />
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$T_{(mRNA) \frac{1}{2}}$: Half life of mRNA [$\text{s}$]
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</span>
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</p>
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<h5>Lysozyme Model</h5>
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<p id="pp">
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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
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assumptions that will be made, the first of these is that enzymatic reactions can be modelled by Michaelis-Menten kinetics. Secondly, the temperature
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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
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<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.
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</p>
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<p id="pp">
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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
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production rates of mRNA and lysozyme can be calculated using translation and transcription rates of $V_{translation} = 8.4\text{aas}^{-1}$
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(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
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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
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reduces calculation times.
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</p>
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<p id="pp">
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<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 />
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<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>
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<span class="equation_key">
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$t_{lysozyme}$: Time to produce one lysozyme protein [$\text{s}$]<br />
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$t_{mRNA}$: Time to produce one mRNA [$\text{s}$]<br />
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$L_{plasmid}\text{, }L_{protein}$: Length of plasmid and protein coding region [$\text{bp}$]
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</span>
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</p>
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<p id="pp">
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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>
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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
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there are approximately 5.8 codons which will be rounded down to 5 as not to overproduce lysozyme.
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</p>
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<p id="pp">
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“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
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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
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Michaelis-Menten kinetics were applied, which gives the reaction rate of one enzyme (Berg et al., 2002).
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</p>
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<p id="pp">
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<span class="equation">$k_{(cat)} = \frac{[S]k_{(cat)max}}{[S] + K_M}$</span>
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<span class="equation_key">
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$k_{(cat)}$: Reaction rate of one lysozyme [$\text{s}^{-1}$]<br />
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$k_{(cat)max}$: Maximum reaction rate of one lysozyme [$\text{s}^{-1}$]<br />
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$[S]$: Substrate concentration [$\text{M}$]<br />
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$K_M$: Michaelis constant [$\text{M}$]
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</span>
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</p>
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<p id="pp">
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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.
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The average reaction rate assumed to be $k_{(cat)avg} = (k_{(cat)max}/2$). The logarithms of both of these
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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:
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</p>
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<p id="pp">
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<span class="equation">$K_M = 5.6\text{mM}$ (2sf)</span><br />
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<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>
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</p>
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<p id="pp">
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Lastly, the initial concentration of the substrate peptidoglycan is calculated. An assumption is made that determines that all the peptidoglycan in
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<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
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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:
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</p>
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<p id="pp">
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<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>
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<span class="equation_key">
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$[Pep]_{int}$: Initial concentration of peptidoglycan [$\text{mM}$]<br />
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$N_{pep}$: Amount of peptidoglycan [molecules]<br />
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$N_A$: Avogadro's constant [molecules/mole]<br />
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$V_{\textit{E. coli}}$: Volume of <i>E. coli</i> [$\mu\text{m}^3$]
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</span>
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</p>
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<p id="pp">
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This calculation gives a value in the same order of magnitude as the Michaelis constant, which represents the concentration of substrate when the
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reaction rate is at half of its maximum value.
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</p>
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<h5>Results</h5>
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<div class="col-xs-12" style="width:100%;position:relative;margin:auto;padding:0;">
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<div class="graph_box_single col-xs-12">
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<img src="graph3_1.png">
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<span>Fig. 1. Using Michaelis-Menten kinetics the reaction rate of each lysozyme enzyme has
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been plotted for each peptidoglycan substrate concentration. The
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average reaction rate of $0.43\text{s}^{-1}$ occurs when the concentration is equal to $K_M = 0.0056\text{M}$.
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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>
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</div>
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</div>
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<p id="pp">
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Enzyme reaction rates have been modelled by the Michaelis-Menten kinetics model in Fig. 1, therefore the reaction
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rate decreases as the substrate concentration decreases. This graph shows that the reaction
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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>
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<div class="col-xs-12" style="width:100%;position:relative;margin:auto;padding:0;">
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<div class="graph_box col-xs-12">
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<img src="graph1.png">
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<span>Fig. 2. The percentage of peptidoglycan compared to the original concentration plotted against time.</span>
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</div>
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<div class="graph_box col-xs-12">
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<img src="graph2.png">
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<span>Fig. 3. Plots a smaller range of times as Fig. 2. To show the rapid decrease in peptidoglycan concentration.</span>
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</div>
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</div>
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<p id="pp">
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The model predicted the complete degradation of the cell wall to be within the first generation
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of <i>E. coli</i>, Fig. 2. The reaction rate is slow at first due to the cell having no initial lysozyme,
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this slowly increases, until the low concentration of the substrate casues
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the reaction rate of lysozyme to slow considerably. The peptidoglycan concentration in the cell is
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negligible until 17 minutes at which point the <i>E. coli</i> splits sharing the cell wall damage equally
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between the two child cells, hence cell damage drops from almost 100% to 50%. The immediate concern
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is that in this model cell death would occur far before it is able to duplicate, meaning that
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assuming no mutations the cell would terminate before 17 minutes.
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</p>
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<p id="pp">
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The cell death threshold of peptidoglycan concentration in the cell wall is not well defined from
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research. Fig. 3 demonstrates that any threshold that is chosen is likely to fall in between 2
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and 5 minutes of the simulation which is well before the reproduction rate of <i>E. coli</i> of 17
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minutes. Therefore it is a reasonable assumption that given no mutations were to occur that the
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cell would be terminated before the <i>E. coli</i> could reproduce.
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</p>
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<h5>References</h5>
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<ol style="font-size:100%;">
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<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>
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<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>
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<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>
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<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>
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<li>Berg, J., Tymoczko, J., Stryer, L. and Stryer, L. (2002). Biochemistry. New York: W.H. Freeman, pp.Section 8.4 - Equation (23).</li>
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<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>
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<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>
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<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>
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<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>
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<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>
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<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>
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</ol>
  
  

Revision as of 17:50, 5 October 2016