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Part:BBa_K3932011:Experience

Designed by: William Nathaniel   Group: iGEM21_UI_Indonesia   (2021-09-19)


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Applications of BBa_K3932011

Experience

Since there are no lab works done due to the COVID-19 pandemic restriction, here we present the mathematical model, parameters, and solution based on literature review to answer principally two most important questions revolving around the system: (1) how many products and (2) how long does it take to achieve the determined goal.

Here, we would like to create such a model that is a fusion between 2 models:
  1. The “Timer & AMP” model that models the production of AMP inside the Escherichia coli led by the copA promoter.
  2. The “Kill Switch” model that models the Escherichia coli’s lysis timing

Timer & AMP


Mathematically, this model can be expressed by an oversimplified set of differential equations as followed:1-3

See the parameters detail in our wiki modelling section

Figure 1. Timer & AMP model

As reflected by our model, the SUMO+AMP starts being produced almost at the same time as when the T7 promoter is set on, reaching a linear rate of production as early as the timer reaches 5 minutes. Slower rate of production will then start approximately at 40 minutes and will significantly affect the production at the one-hour mark (as soon as the production of ulp-1 becomes more significant), resulting in an instantaneous equilibrium around the 74 minutes mark with 3400 molecules of SUMO+AMP existing in the Escherichia coli as its highest ever.

It can be understood that the cleaving of SUMO+AMP into AMPs had already started since the beginning of the recorded time (with a near linear rate). The rate of cleaving will then increase as ulp-1 started being produced significantly at the one-hour mark; right before suddenly going back to another linear rate (but now with lower rate of cleaving than the previous rate) approximately at 103 minutes as the number of SUMO+AMP had almost drop to 0.

Kill switch

Mathematically, this model can be expressed by a simplified set of differential equations as followed1-3

See the parameters detail in our wiki modelling section

The lethal level of Holin for most Escherichia coli is believed to be around 170 molecules per cell, yet for some Escherichia coli that may contain unrecognizable “special conditions” the lethal level of Holin may deviate up to 1000 molecules per cell, with 300 molecules per cell being subjected as the most occurring observation between the abnormal subjects.

Figure 2. Holin production model


The graph implies a nearly-constant rate of holin growth up until the first hour of induction. During the first 20 minutes of induction it can be inferred that holin growth rate is increasing, and then it remain the most constant during the next 10 minutes, followed by a slight decline in holin growth rate afterwards, probably due to the fact that after some time reactions and transcriptions inside Escherichia coli will slow down as it adjust itself into equilibrium. From the graph, it can be deduced that for the level of holin to reach 170 molecules per cell, it would take approximately 11 minutes to do so since induction, and for reaching 300 and 1000 molecules per cell, it would take approximately 16 and 36 minutes to do so since induction respectively.

Due to the unknown spread and distribution between the categories of Escherichia coli factors that determine directly its holin threshold before approaching lysis, it is almost impossible for us to determine the average time of Escherichia coli lysis for our project unless we do a distribution modelling assumption for the results. We agree that the further the deviation of the time needed for lysis to happen, then the rarer the occurrence would be (because it depends on numerous alterations regarding the conditions in each Escherichia coli), therefore we would like to approximate the statistical distribution of the extra time taken for lysis to happen (compared to the usual 11 minutes taken) using a heavy-tailed distribution, such as the Weibull distribution with its shape parameter ranging from 0 to 1.

We assume that half of Escherichia coli will need less than 16 minutes to perform lysis on our model and we are sure that the event that an Escherichia coli will need more than 36 minutes on our model will be rare, hence using standard statistical confidence we set that 95% of Escherichia coli will need less than 36 minutes to perform lysis on our model. Therefore by applying the Weibull cumulative model, we can perform approximation of the shape (k) and scale (lambda) parameters of the Weibull distribution by solving these equations:


Which yield the the shape (k) and scale (lambda) parameters of the Weibull distribution with the exact answer and numerical approximation as followed:


Solving the equation provide us with the solution in which the time needed for Escherichia coli lysis (expressed as T) can be assumed with the following Weibull statistical distribution model; added by the constant 11 minutes inherited on normal lysis time.
T ~ 11 + W(7.48158, 0.909449)

And therefore the mean time for each Escherichia coli to achieve lysis in our model would be:
E[T] = E[11] + E[W(7.48158, 0.909449)]
E[T] = 11+7.48158(T(1+1/0.909449)
E [T] = 18.827723
Or approximately 18 minutes and 50 seconds.


Complete model Mathematically, this model can be expressed by the mixture of (slightly modified) differential equations mentioned in the previous 2 models as followed:


With each variable corresponding to points of interest as already described before with the addition of AMP as concentration of transcribed AMP. While parameters follow as what has been given in the previous 2 models with the addition of deAMP which is the degradation rate of AMP (0.0021).

The model is then simulated accordingly with similar alterations as mentioned in previous models, and thus resulted in the given graph:

Figure 3. Complete H. pylori eradication model


As can be inferred from our model, the moment when ulp-1 has started its significant production also indicates the moment when holin has started its slight production. Using the results inferred from the previous model, this moment will approximately start at the one-hour mark.

Our mathematical formulation and simulation suggests that the nature of holin production in the fused model is indifferent compared with its production nature on its stand-alone model. On its stand-alone model it would take approximately 12.5 minutes to reach a holin level of 200 molecules per cell and on this model it would take approximately 100 minutes (after significant production at 60 minutes) to reach a holin level of 200 molecules. Therefore, holin production is faster approximately 8 times on its stand-alone model rather than the last fused one.

Hence, on our latest fused model, it would take in average approximately 60 + 8(18.827723) minutes for lysis to happen in an Escherichia coli: 210 minutes and 37 seconds to do so. It is obvious that at this time, almost no SUMO+AMP particles will be left uncleaved; showing incredible efficiency of its use to provide the maximum number of AMPs.

As for the number of AMPs produced we will use a similar method of approximation as what we have done to approximate the time of lysis. As our graph infer, assuming near-linearity of AMP production since the 110 minute mark (start of time when almost no SUMO+AMP particles will be left uncleaved) of 120 AMP molecule produced each minute, therefore at the time of lysis that will occur approximately 100 minutes after, there would approximately be 12000+100(120) AMP molecules set to be released. Approximately an arsenal of 24.000 AMP molecules ready to fight H. pylori.

References
  1. T. Chen, H. L. He, & G. M. Church, “Modelling Gene Expression With Differential Equations”, 2019.
  2. http://2013.igem.org/Team:TU-Delft/Timer-Sumo-KillSwitch
  3. http://2008.igem.org/Team:UC_Berkeley/Modeling
  4. J. A. Bernstein, A. B. Khodursky, P. H. Lin, S. Lin-Chao, and S. N. Cohen, "Global analysis of mrna decay and abundance in escherichia coli at single-gene resolution using two-color fluorescent dna microarrays", 2002.
  5. M. Barrio, K. Burrage, A. Leier, and T. Tian, "Oscillatory regulation of hes1: Discrete stochastic delay modelling and simulation", 2006.
  6. R. Young and H. Bremer, "Polypeptide-chain-elongation rate in Escherichia coli B/r as a function of growth rate", 1976.
  7. C. Y. Chang, K. Nam, and R. Young, "S gene expression and the timing of lysis by bacteriophage lambda", 1995.
  8. MCLab, "Sumo protease", 2013.
  9. Team Aberdeen iGEM team 2009, "Dissociation constants," 2009.
  10. ExPASy - PeptideCutter [Internet]. [cited 2021 Oct 18]. Available from: https://web.expasy.org/cgi-bin/peptide_cutter/peptidecutter.pl
  11. Bajorath J, Hinrichs W, Saenger W. The enzymatic activity of proteinase K is controlled by calcium. Eur J Biochem. 1988;176(2):441–7.

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