KillerRed/KillerOrange model
Given that Grenoble 2013 built a kinetic model, using OD over time, that described their experimental data very well and Carnegie Mellon 2013 had taken an intracellular approach with kinetics, we decided to consider our system from a different angle. Once again, we looked at the system on an intracellular level but varied our approach by concentrating on a more physical point of view and assessing the rates of production on a case-by-case basis rather than system-wide.
At the outset, multiple assumptions were made to simplify the the system, enabling suitable definition of appropriate variables and rates. Mutations were also ignored to eradicate this further layer of complexity. Mindful that the timeframe considered was much larger than the replication time of E. coli, the amount of mRNA, Protein etc was adjusted accordingly by assuming that it splits evenly and halving the amount every time replication occurs, also affecting the amount of plasmids. However, as cited in (Nordström and Dasgupta, 2006), the frequency of replication is variable to maintain a constant amount. Therefore, in the case of our plasmid pSB1C3, which had a high copy number ~300, the amount was kept constant even after replication of the E. Coli.
We then isolated one cell and listed the major factors in the production of reactive oxygen species (ROS) starting from the transcription of mRNA.
- k1 is the rate of transcription
- k2 is the rate of mRNA degradation
- k2 is the rate of mRNA degradation
- k4 is the rate of degradation of the protein
- k5 is the rate of reactive oxygen species production
Protein production:
From this, we isolated the Transcription/Translation mechanism (k1 & k3) along with the degradation (k2 & k4) to obtain a protein production time.
K1 was calculated to be 22.175s-1. This was achieved by taking the total number of base pairs of the plasmid (887) and dividing through by the transcription rate of 40 base pairs per second for the T7 promoter (García and Molineux, 1995).
K3 was calculated in a similar way to be 28.690s-1. As the value found for the rate of 8.4 amino acids per seconds (Siwiak and Zielenkiewicz, 2013) is given in amino acids and not base pairs, this result was arrived at by taking the total number of base pairs of the protein coding region (723) and dividing by 3, given that a conversion of 3 base pairs = 1 amino acid is viable.
K2 was found to be 3-8 minutes for 80% of mrna (Bernstein et al., 2002). 5 minutes was used as an approximation of the median time as the actual number fluctuates.
K4 was initially estimated to be greater than 10 hours using protparam on expasy.org. 10 hours was used as the degradation time as this exceed the observed time taken for death of the cell.
Simbiology, a modelling app for matlab, was used to model the flow diagram with the above rate parameters. Running the simulation, a rate for mrna production and protein production could be found. A few issues were initially identified, these being that the degradation times affected the transcription/translation rate directly and also that as soon as mrna production started, protein production was initialised. This suggested that even when the mrna is in the process of being transcribed, translation has started and we know this not to be the case as you can’t start transcription with a non integer amount of mRNA.
To correct this mRNA needed to be a step function, limiting the amount to whole integers and meaning that production of the protein was restricted until the first mrna was produced. However, this only served to add a delay onto the overall protein production and multiply it by the mRNA amount as seen in Figure 1. This highlights another problem. As we can see in Figure 2 below, we know that it takes ~28 s to produce a protein (K3) after the mRNA has been made, suggesting that it should occur at ~54s. However, the second mRNA is produced in advance of this, increasing the rate at which the protein are produced and making the first protein at ~53s. This correctly describes the overall rate of the system in producing proteins but is incorrect for finding the time at which they are made, as each mRNA is independent of any other. This means that each mRNA required individual consideration based on each having a separate protein production mechanism that contributes to a total protein quantity.
To enable this we moved away from Simbiology and attempted to use Simulink. However, after careful consideration, we decided instead to write the process in C. This allowed us to handle each mRNA and its creation time separately and have it produce proteins up to the time in which the protein was induced. An array was used with each element representing a second; every time a protein was created the corresponding time element would increase by one. The total of all elements were then taken to find an overall amount of protein.
The next step was to include degradation. This was achieved by limiting the protein production of an individual mRNA to its respective degradation time. For protein degradation, the amount of time left from the time of creation until the overall run time (in our case the point at which we started shining light on the cultures) was compared to the degradation time; if the difference was greater, the protein degrades and is subtracted from the total.
Further, we took into account multiple ribosomes on one mRNA (polyribosome) and maturation. Polyribosomes were included in the code by restricting the first protein created by each mRNA to the the calculated translation rate found above (~28s). Then for the following proteins we used a modified rate where we divided the rate by the amount of ribosomes, which was found to be 3.46 per 100 codons (Siwiak and Zielenkiewicz, 2013). In this case it was calculated to be 8 if we rounded down to the nearest integer. As the degradation and the ability to produce ROS applies to only folded proteins, we wanted to know how many mature proteins there would be. This was achieved by calculating a new run time by taking away the maturation time and the time taken to make one mRNA and protein, suggesting that the proteins made in the new run time were created and matured and ready to produce ROS if they had not degraded.
Examining the degradation time for the protein we adjusted the previously estimated value to be that of green fluorescent protein (GFP), as Killerred is a homologue to provide a more accurate degradation time. However, we could only find a half life of the protein so to calculate the degradation we used equation (1).
$T = \frac{T_{\frac{1}{2}}}{ln(2)} (1)$(Hyperphysics.phy-astr.gsu.edu, 2016)
$T$: Degredation time [$\text{s}$]
$T_{\frac{1}{2}}$: Half life[$\text{s}$]
Running the initial model provides a overall protein quantity of under 3 million and mRNA at a maximum of 4200 (Thermofisher.com, 2016), which fits within the expected amount suggesting that the rates and process are a good, basic approximation of protein production.
Enzymatic Models
mRNA production in Enzymatic kill switches
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.
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 E. coli 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:
- Plasmid production
- Degradation of mRNA and protein
- Maturation of protein
- Duplication rate of E. coli
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 E. coli to the degradation rate of protein.
Assumptions
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 E. coli 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
E. coli 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 E. coli 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.
Features
Research showed that there is an upper limit of mRNA in E. coli,
therefore an upper limit of $4 \times 10^3$ mRNA per E. coli 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 E. coli - 17 minutes, at which point both the mRNA and protein is assumed to split among the two cells equally,.
$T_{(mRNA)} = \frac{T_{(mRNA) \frac{1}{2}}}{ln(2)} = \frac{300\text{s}}{ln(2)} = 430\text{s (2sf)}$(Hyperphysics.phy-astr.gsu.edu, 2016)
$T_{(mRNA)}$: Lifetime of mRNA [$\text{s}$]
$T_{(mRNA) \frac{1}{2}}$: Half life of mRNA [$\text{s}$]
Lysozyme Model
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 E. coli growth. The model will assume that when
E. coli splits, the contents of the cell and damage of the cell wall is shared equally among the two resulting E. coli. Lastly, it will assume that lysozyme does not degrade
throughout the simulation.
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 E. coli 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.
$t_{lysozyme} = \frac{L_{protein}}{V_{translation}} = \frac{\frac{507\text{bp}}{3}}{8.4\text{aas}^{-1}} = 20\text{s (2sf)}$
$t_{mRNA} = \frac{L_{plasmid}}{V_{transcription}} = \frac{507\text{bp} + 164\text{bp}}{40\text{bps}^{-1}} = 17\text{s (2sf)}$
$t_{lysozyme}$: Time to produce one lysozyme protein [$\text{s}$]
$t_{mRNA}$: Time to produce one mRNA [$\text{s}$]
$L_{plasmid}\text{, }L_{protein}$: Length of plasmid and protein coding region [$\text{bp}$]
To supplement the production of lysozyme, the program will implement the affects of multiple ribosomes on the coding site of lysozyme. For E. coli
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.
“Lysozyme c” is an enzyme that hydrolyses bonds holding together peptidoglycan in the cell wall, this is explained in detail on the project page. The
next task of the lysozyme model is to connect the amount of protein at each time to the degradation of the E. coli cell wall, to do this
Michaelis-Menten kinetics were applied, which gives the reaction rate of one enzyme (Berg et al., 2002).
$k_{(cat)} = \frac{[S]k_{(cat)max}}{[S] + K_M}$
$k_{(cat)}$: Reaction rate of one lysozyme [$\text{s}^{-1}$]
$k_{(cat)max}$: Maximum reaction rate of one lysozyme [$\text{s}^{-1}$]
$[S]$: Substrate concentration [$\text{M}$]
$K_M$: Michaelis constant [$\text{M}$]
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:
$K_M = 5.6\text{mM}$ (2sf)
$k_{(cat)avg} = \frac{k_{(cat)max}}{2} \approx \frac{k_{(cat)}^{obs}}{2} = \frac{0.86\text{s}^{-1}}{2} = 0.43\text{s}^{-1}$ (2sf)
Lastly, the initial concentration of the substrate peptidoglycan is calculated. An assumption is made that determines that all the peptidoglycan in
E. coli 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 E. coli (Vollmer and Höltje, 2004). Using an approximate volume of E. coli of $0.7 \mu \text{m}^3$, the concentration of peptidoglycan is:
$[Pep]_{int} = \frac{\frac{N_{pep}}{N_A}}{V_{E. coli}} = \frac{\frac{3.5\text{x}10^6}{6.02\text{x}10^{23}}}{0.7 \mu \text{m}^3} = 8.3\text{mM}$ (2sf)
$[Pep]_{int}$: Initial concentration of peptidoglycan [$\text{mM}$]
$N_{pep}$: Amount of peptidoglycan [molecules]
$N_A$: Avogadro's constant [molecules/mole]
$V_{\textit{E. coli}}$: Volume of E. coli [$\mu\text{m}^3$]
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.
Results
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 $0.51\text{s}^{-1}$.
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.
Fig. 2. The percentage of peptidoglycan compared to the original concentration plotted against time.
Fig. 3. Plots a smaller range of times as Fig. 2. To show the rapid decrease in peptidoglycan concentration.
The model predicted the complete degradation of the cell wall to be within the first generation
of E. coli, 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 E. coli 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.
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 E. coli of 17
minutes. Therefore it is a reasonable assumption that given no mutations were to occur that the
cell would be terminated before the E. coli could reproduce.
References
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