Team:Ecuador/Model
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<a name="MOD" style="text-decoration: none;">
MATHEMATICAL MODELLING
</a>
Index
- <a href="#LOV" style="color:black; text-decoration: none">List of variables and constants</a>
- I<a href="#IND3" style="color:black; text-decoration: none">PTG-inducible model</a>
- <a href="#IND4" style="color:black; text-decoration: none">Entry of IPTG into the cell</a>
- <a href="#IND5" style="color:black; text-decoration: none">CBD cipA-BMP2 expression</a>
- <a href="#IND6" style="color:black; text-decoration: none">Kinetic constants</a>
- <a href="#IND7" style="color:black; text-decoration: none">Results of the IPTG-inducible model</a>
- <a href="#IND8" style="color:black; text-decoration: none">Calculation of entry of IPTG into the cell</a>
- I<a href="#IND9" style="color:black; text-decoration: none">PTG-LacI repressor interaction</a>
- <a href="#IND10" style="color:black; text-decoration: none">Calculation and validation of CBD cipA-BMP2 expression</a>
- <a href="#IND11" style="color:black; text-decoration: none">Conclusions</a>
<a name="INT" style="text-decoration:none;">
INTRODUCTION
</a>
This is an IPTG-inducible model for the expression of the fusion protein CBD cipA-BMP2 under the control of Plac promoter and cloned in psb1c3. The vector does not contain the ORF of LacI and thus the LacI repressor protein concentration is that in E. coli strains (10nM per cell).1
<a name="LOV" style="text-decoration:none;"></a>The model consists of systems of differential equations that were solved in Python using Spyder 3.6 (Anaconda).
<a name="LOV2" style="text-decoration:none;">
LIST OF VARIABLES AND CONSTANTS
</a>
Variable/Constant |
Definition |
Unit/Value |
[IPTG] TOTAL |
Total IPTG concentration in the medium |
nM |
[IPTG]ext |
IPTG concentration outside E. coli |
nM |
[IPTG]int |
IPTG concentration inside E. coli |
nM |
[IPTG]TOTAL |
Total LacI concentration in E. coli |
nM |
[LacI]act |
Active LacI concentration in E. coli |
nM |
[CB]mRNA |
CBD-BMP2 mRNA concentration in E. coli |
nM |
[CB] |
CBD-BMP2 protein concentration in E. coli |
nM |
δ[CB] |
Degradation rate of CBD-BMP2 mRNA |
min-1 |
β[CB] |
Translational rate of CBD-BMP2 protein |
min-1 |
σ[CB] |
Degradation rate of CBD-BMP2 protein |
min-1 |
t |
time |
min |
α[CB] |
Transcriptional leakage of Plac promoter |
% |
kPlac |
Maximum transcription rate of Plac promoter |
0.5 nM min-1 |
kIPTGupt |
Rate constant of IPTG uptake |
0.92 min-1 |
kIPTGout |
Rate constant of IPTG output |
0.05 min-1 |
KIPTG |
Michaelis constant of IPTG-LacI binding |
600 nM |
<a name="IND3"></a>hIPTG |
Hill coefficient of IPTG-LacI binding |
2 |
HLacI |
Hill coefficient of LacI-Plac promoter binding |
2 |
Mimicking lactose, IPTG is a chemical widely used in scientific research because it cannot be metabolized. IPGT induces the expression of genes under the control of Plac promoter.
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Eq 1.
The first term on the right side of Eq 1 represents the forward rate where kIPTGupt is the rate constant of IPTG uptake (0.92 min-1)3 while the second term has negative sign because it represents the backward rate where kIPTGout is the rate constant of IPTG output (0.05 min-1).3 Because of [IPTG]ext = [IPTG]TOTAL - [IPTG]int, it can be replaced in equation 1 in order to have IPTGint as the unique independent variable:
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Eq 2.
The LacI-IPTG interaction is assumed to be so fast and when the IPTG concentration largely exceeds that of lacI. The LacI active ([LacI]act), i.e that unbounded to IPTG, can be described with the Hill repression function.4,2
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Eq 3.
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Eq 4.
Because there is not constitutive expression of LacI protein from the plasmid cloned in E. coli carrying the CBD cipA-BMP2 construct and based on our preliminary measurements of total protein concentration using BCA method, the transcriptional leakage can be taken to be 0.8 and thus the balance for [CB]mRNA becomes:
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Eq 5.
The concentration of CBD cipA-BMP2 protein ([CB]) can be described as follows:
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Eq 6.
The time cell division of E. coli is function of factors such as pressure, temperature and culture medium and ranging from 20-30 min under optimal conditions. Culturing in universal common medium such as Nutrient Broth and Luria-Bertani at 35 ℃, the time cell division can be assumed to be 30 min.8,9,10In E. coli , the translation takes place at 12-21 amino acids per second (aa s-1).12 We choose 19 aa s-1 because the codon optimization. <a name="IND7"></a><a name="IND8"></a>Because CBD cipA-BMP2 contains 292 aa, β[CB] can be taken to be 3.904 min-1.The half life of CBD cipA-BMP2 protein can be assumed to be 60 min-1, 13 consequently, σ[CB]= 0.05min-1:
The half life of mRNAs in E. coli has been well reported by Bernstein and coworkers who determined that it depends of culture medium. In, LB 99% of mRNAs had a half-life time between 1-15 min with a mean of 5.2 min.11 Therefore, 5.2 min can be assumed as the half-life time of CBD cipA -BMP2 mRNA. Consequently, δ[CB]= 0.226 min-1.
Figure 1 is a rate balance plot derived from Equation 1 whith both forward and backward rates as functions of the normalized [IPTG]int concentration whose value in the equilibrium (steady state) can be obtained where the two functions intersect. Analitically, this point is [IPTG]int =0.9485[IPTG]TOTAL
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<a name="IND9"></a>
Because of IPTG is the inducer, it is important to evaluate its interaction with LacI repressor at different initial concentrations added to the system ([IPTG]TOTAL). For these values the [LacI]act can be calculated with Equation 3 (Table 1). From table 1 and figure 4, it can be deduced that a concentration of IPTG = 0.1 mM is saturating because the solutions derived from higher IPTG values converge.
|
[IPTG]TOTAL (mM) |
[LacI]act (nM) |
0,001 |
9.7562 |
0,02 |
0.9095 |
0,03 |
0.4257 |
0.1 |
0.0399 |
1.0 |
0.0004 |
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Eq 7.
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A reasonable error less than 0.01 for both mRNA and protein concentrations was obtained with a time step Dt=0.1/8. Therefore, this Dt was considered to evaluate a phase plane plot which indicates the equilibrium points of the mathematical model. To determine the maximum concentrations of CBD cipA-BMP2 mRNA and CBD cipA-BMP2 protein, a saturating concentration of IPTG (IPTG=0.1 mM) was evaluated. This is a monostable system where the equilibrium point at the steady state (Figure 2) is 2.21 nM and 172.69nM for CBD cipA-BMP2 mRNA and CBD cipA-BMP2 protein respectively.
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Different values of [LacI]act cause a different effect in the mathematical model for expression of [CB]mRNA and [CB]. Figure 3 and Figure 4 show the concentrations of [CB]mRNA and [CB], respectively, as functions of time with a time step size Dt=0.1/8. It is worth to note that the higher IPTG dosage the higher [CB]mRNA and [CB] until the saturating level of IPTG = 0.1 mM.
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<a name="IND11"></a>
1. Oehler, S. (2009). Feedback Regulation of Lac Repressor Expression in Escherichia coli. Journal of Bacteriology, 191(16), 5301–5303. http://doi.org/10.1128/JB.00427-093. Team TUDelft. (2009).Bacterial Relay Race. International Genetically Enginereed Machines. Retrieved on june 25th, 2018 from https://2009.igem.org/Team:TUDelft.
2. Ukkonen, K. Vasala, A. (2015). Protein expression by IPTG autoinduction in EnPresso B:Protocol with minimal manual work and superior yields compared to other media.BioSilta. Retrieved on june 24th, 2018 from https://bioscience.co.uk/userles/pdf/Application%20Note%20-%20Protein%20expression%20by%20IPTG%20autoinduction%20in%20EnPresso%20B.pdf.
6. Ozbudak, E., Thattai, M., Lim, H., Shraiman, B. Van Oudenaarden, A. (2004).Multistability in the lactose utilization network of Escherichia coli. Nature, 427 (1476-4687), 737?740.https://www.nature.com/articles/nature02298.
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