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Background

Module 3 is on cell killing. An increase in temperature, from 28°C to 42°C will kick off M3. Restriction enzymes DpnII and TaqI will be produced, and these will result in cell death.

We would like to model two things:

1) Thermoinduction with promoter characterisation

The cell death mechanism is triggered by an increase in temperature. An increase in temperature activates the production of restriction enzymes. In our model, we will look at how temperature correlates to restriction enzyme concentration, and seeing how it affects the population of live cells. This model will be linked to data obtained from the staining assay for tracking live and dead cells over time.

From the model, we hope to determine the temperature for optimum cell death.

In promoter characterisation, we would be modelling how the PoPS output of the promoters varies with time. This will help us characterise the promoters for future users' benefit.


2) Killing strategy

In our assays, we have determined quantitatively the enzymatic activity of our restriction enzymes and methylases. The balance between the amount of methylases needed to protect against the restriction enzymes, and the amount of methylase that will stifle restriction enzyme activity, can be modelled.

From our models, we will be able to determine the optimum amount of methylation for killing for different values of promoter leakiness.

Furthermore, from the graph of the percentage of live cells over time time, we can determine the rate of cell death. Hence, we would be able to model the efficiency of the restriction enzymes. We can then predict cell death over a certain a time period.


  • MB's FB:
    • OK but your model does nt include these pieces of info....
    • Drawings + more explanations please

Preliminary Model 1: Thermoinduction: How increase in temperature affects cell growth

Hypothesis

Assumptions

  • The production of restriction enzymes DpnII and TaqI is temperature dependent.
    • Therefore, cells will have a greater probability of dying at higher temperatures (37°C to 42°C).
  • Production is repressed at low temperatures.
  • When temperatures increase beyond threshold, repression on the production of restriction enzymes stops.
  • Hence, restriction enzymes will be produced at higher temperatures.
  • There is a methylase, Dam, to prevent cell death for basal levels of production of restriction enzymes. Dam is produced constitutively.
  • However, as the concentration of restriction enzymes increase at higher temperatures, Dam will no longer be sufficient to protect against cell death.
  • Restriction enzymes DpnII and TaqI will cut the cell DNA. Therefore, the increase in restriction enzyme concentration will increase the rate of cell death.
  • The death term is modelled by an activating hill function of temperature.
    • The hill function is chosen because when temperatures are low, there is no cell death
    • When temperatures increase above threshold, there is suddenly a large increase in cell death, akin to a switch

Predictions

  • If the temperature is below threshold, little restriction enzymes are being produced.
    • Population of live cells will increase exponentially
    • Population of dead cells will also increase exponentially, but should be magnitudes less than live cells.
    • This means that total dead cell population is negligible at low temperatures.
  • If the temperature is above threshold, the population growth of live cells is constrained by the concentration of restriction enzyme.
    • Initially, cells will grow exponentially as the restriction enzymes take time to be produced.
    • When enough restriction enzymes are being produced, cell death will begin.
    • This results in a "bump" in the cell number vs time graph, due to an initial increase in cell number and a subsequent killing of cells.
    • When we increase the temperature, we produce more restriction enzymes at a faster rate. Hence, the higher the temperature, the lower the "bump".


[N.B. We have used arbitrary values in our simulations]



  • MB'sFB:
    • make it clear that the results of promoter characterisation will inform this section and that in the meantime you are using the basic assumption of a Hill function for the effect of temperature
    • Make distinction between assumptions and predictions!!!!
    • Explain what your assumptions are regarding the killing - there are many...

Equations

same remarks as module 1 please explain to your reader what these equations mean

  • they're explained at the bottom. shld we move that up?
[math]\displaystyle{ \begin{alignat}{1} \frac{d[M]}{dt} & = \frac{k_1*{T}^{nt}}{{T}^{nt} + {K_{mt}}^{nt}} - d_m * [M] \qquad (1)\\ \frac{d[E]}{dt} & = k_2 * [M] - d_e * [E] \qquad (2)\\ \frac{d[N]}{dt} & = g * N - d * \frac{k_2*{[E]}^{ne}}{{[E]}^{ne} + {K_{me}}^{ne}} * N \qquad (3)\\ \frac{d[D]}{dt} & = d * \frac{k_2*{[E]}^{ne}}{{[E]}^{ne} + {K_{me}}^{ne}} * N \qquad (4)\\ \end{alignat} }[/math]


Legend

Term Variable/Parameter Meaning/description
M Variable mRNA of restriction enzymes DpnII or TaqI
E Variable Restriction enzymes DpnII or TaqI
D Variable Number of dead cells
g Parameter growth constant
d Parameter death constant
k1 Parameter Rate of production of mRNA
k2 Parameter Rate of production of enzyme
T Variable Temperature
nt Variable Hill coefficient for thermal induction
de Parameter Rate of decay of DpnII or TaqI
dm Parameter Rate of decay of mRNA
ne Parameter Hill coefficient for cell death
Kme Parameter Threshold of hill function for cell death
kmt Parameter Threshold of hill function for thermoinduction


Explanation of equations

  • In (1), we have an activating hill function of temperature. This is because an increase in temperature (T) will trigger the production of mRNA and restriction enzymes (DpnII and TaqI) needed for killing. A hill function is chosen because initially, there is very little killing with the increase of temperatures. When the threshold temperature is reached, we start killing cells.
    • At steady state:[math]\displaystyle{ [M]=\frac{k_{1}*\frac{T^n}{{[K_m]}^n+ T^n}}{d_{m}} }[/math], so when [math]\displaystyle{ T \lt K_m }[/math], [M]='off'.
    • When [math]\displaystyle{ T \gt K_mt }[/math] we produce more [M], which triggers production of restriction enzymes.
  • In (2), the rate of production of enzyme is a balance of rate of transcription (PoPS) and rate of degradation of the enzyme itself.
    • At steady state: [math]\displaystyle{ [E]=\frac{k_{m}*k_{p}}{d_{m}*d_{p}}*[M] }[/math], so once we produce MRNA, as triggered by the temperature threshold, we will also get restriction enzyme.
  • In (3), the growth term shows that exponential cell growth occurs when there is no increase in temperature. The death term is proportional to the population(N) and is a function of enzyme concentration. When we hit the threshold of enzyme necessary for killing, there will be a increase in the probability of cell death , hence an activating hill function is usedv(effectively, the enzyme hill function acts on increasing the death term so cells are killed over time past the threshold temperature).
    • Consider the death term :[math]\displaystyle{ d * \frac{k_2*{[E]}^{ne}}{{[E]}^{ne} + {K_{me}}^{ne}} * N }[/math].
    • When restriction enzyme concentrations are subthreshold, the death term is close to 0, so the population grows exponentially (as we see for T=1-4 in simulations)
    • When restriction enzyme concentrations are suprathreshold, this has the effect of increasing the influence of the death term, so now bacterial growth is bounded and saturates (see simulations for T=5-8)
  • In (4)we are describing the time trajectory of dead cells, which is simply the death term in equation (3), but now it is positive.


Simulations

Simulation 1: Effects of increased temperatures on cell population

Very ugly split your results over 2 rows (mRNS and Protein on top) if need be Explain what is special about T=5 and in general how you chose your parameters Explain both cases separately

Temperature below threshold
Temperature above threshold

Explanations

  • The killing term is set to be greater than the growth term in our parameters.

why? explain what this means etc....


  • In the first simulation, we set the arbitrary temperature to be low, varying from 0 to 4. From the simulation, we can tell that when the temperature is low, cell growth will be exponential, ie, there will be no cell death.

explain what T=4 means wrt your activation threshold Give qualitative explanation for system behaviour generalize your results for a temperature dependence that is not Hill-like Comment dead-popultion plot and live-population plot


  • In the second simulation, when we increase the temperature, ie, varying the arbitrary temperature from 5 to 9, cell start to die after an initial exponential cell death. Hence temperature indeed will induce cell death. Our hypothesis is proven right.
  • Also note that at the "bump", this is when cell death remain constant.

what does it mean? is is informative? etc...


N.B. Note that the values we have used for temperature are arbitrary, and they are for the purpose of illustrating the behaviour of the system. In reality, this will be linked to our data, and the parameters will change.

Data Fitting

Matlab files

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