Team:SNU India/Mathematical Modelling
Bacterial Model
We assumed that 5µM estrogen is present outside. It is being degraded by laccase too. Some of the estrogen enters the cells and some of it can exit the cell to. We represented this by
where
v m
maximum velocityK m
Michaelis Menten constantsE out
Estrogen in waterFigure 1
Figure 2
The plot for the first equation (Figure 1) shows an exponential decay, in accordance with the findings of existing studies (Ref). As expected, in the plot for the second equation (Figure 2) the amount of estrogen that will enter the cell will increase at first but then decrease eventually based on the initial estrogen concentration outside the cell.
We then predicted the amount of estrogen sensitive RNA polymerase produced by each cell. A constitutive promoter codes for an estrogen sensitive RNA polymerase. We represent this transcription step, and the degradation of the mRNA using the following equation
Here
[mRNA_ES_RNAP]
concentration of mRNA coding for estrogen sensitive RNA polymerase[RNAP_Ecoli]
amount of natural E.coli RNA polymeraseFigure 3
Next, we captured the translation of polymerase
Here
[RNAP]
concentration of Estrogen-sensitive RNA polymeraseFigure 4
From the graph of mRNA of estrogen sensitive RNA Polymerase (Figure 3), we can see that that it reaches a saturation point at 1.85 × 10-13M, in 5000 seconds. RNA- Polymerase plot (Figure 4) is also sigmoidal with saturation at 10-11M in 12000 seconds.
This RNA polymerase transcribes for mRNA coding for RFP and TAL enzyme followed by their translation
Here
[mRNA_RFP]
concentration of mRNA coding for RFP[RFP]
concentration of RFP[mRNA_TAL]
concentration of mRNA coding for TAL[TAL]
concentration of TALpromoter2
amount of the T7 promoter available for transcriptionFigure 5
Figure 6
Note that the rates of translation for TAL and RFP will vary because of difference in nucleotide lengths and separation due to an RBS region.
From the plots (Figure 6), we can see that TAL reaches a maximum of 0.186 fM, in 12000 seconds and RFP reaches a maximum of 0.36 fM, in 12000 seconds (for given initial conditions). These are biologically reasonable concentrations.
Next, we calculate the amount of p-Coumaric Acid being produced due to the action of TAL on Tyrosine in the cell. Using a deterministic model of differential equations, we obtained the following equation for p-coumaric acid production using previous work done in 2013 by team iGEM Uppsala
Figure 7
Thus, we predict that at the end of 20000 seconds, 6 × 10-13 p-coumaric acid will be formed which is considerably low and will not be toxic for the cells.
Further it has been shown that p-coumaric acid enhances laccase activity 5 times. Thus, we expect this to be a good model system for prediction of RFP and p-coumaric acid production and levels of both needed for detection of estrogen and enhancement of laccase activity.
List of various constants used in the modal
Constant Name and Description
vm
Km
K0
K1
K2
K3
K4
K5
K6
K7
K8
K9
K10
K11
K12
K13
K14
K15
Description
Maximum Velocity
Michaelis Menten Constant for TAL
Effusion constant for estrogen across the cell barriers
Rate constant for degradation of estrogen
Rate constant for association of the constitutive promoter and polymerase
Rate constant of degradation of mRNA of estrogen sensitive RNA polymerase
Rate constant for translation of mRNA of Estrogen-sensitive RNA polymerase
Rate constant of degradation of estrogen-sensitive RNA polymerase
Rate constant for association of the T7 promoter and the estrogen sensitive polymerase
Rate constant for degradation of mRNA of RFP
Rate constant for translation of mRNA of RFP
Rate constant of degradation of RFP
Rate constant for association of the T7 promoter and the estrogen sensitive polymerase
Rate constant of degradation of mRNA of TAL
Rate constant for translation of mRNA of TAL
Rate constant of degradation of TAL
Kcat for kinetics of TAL
Michaelis Menten Constant for TAL
Value
Source (of value/formula)
https://doi.org/10.4061/2011/725172 https://doi.org/10.4061/2011/725172 https://2014.igem.org/Team:Carnegie_Mellon/SensorModel https://2014.igem.org/Team:Carnegie_Mellon/SensorModel https://2009.igem.org/Team:PKU_Beijing/Modeling/Parameters https://2013.igem.org/Team:Uppsala/P-Coumaric-acid-pathway https://2009.igem.org/Team:PKU_Beijing/Modeling/Parameters https://2013.igem.org/Team:Uppsala/P-Coumaric-acid-pathway https://2014.igem.org/Team:Carnegie_Mellon/SensorModel https://2014.igem.org/Team:Carnegie_Mellon/SensorModel https://2014.igem.org/Team:Carnegie_Mellon/SensorModel https://2014.igem.org/Team:Carnegie_Mellon/SensorModel https://2009.igem.org/Team:PKU_Beijing/Modeling/Parameters https://2009.igem.org/Team:PKU_Beijing/Modeling/Parameters https://2009.igem.org/Team:PKU_Beijing/Modeling/Parameters https://2013.igem.org/Team:Uppsala/P-Coumaric-acid-pathway https://2013.igem.org/Team:Uppsala/P-Coumaric-acid-pathway https://2013.igem.org/Team:Uppsala/P-Coumaric-acid-pathwayALGAL MODEL
Figure 8 (A)
Figure 8 (B)
We plotted the graph (figure 8A) by calculating the values of the band intensities of secreted protein as shown in the western blot gels [5]. We know the band intensity for 1µg protein. Using this, we extrapolated the expected protein concentration over time. The outcomes are shown in figure 8B
The equation of the plot for band intensity are
Where
y
protein band intensityx
time in daysThe equation of the plot for protein amount are
Where
y
amount of protein estimated (µg)x
time in daysUsing equation 2, we predict that after 3 days the amount of protein in the media will be around 0.228 µg/20 µl of 10X concentrated culture supernatant (as mentioned in reference 1). So, the concentration of laccase in the media after 3 days as estimated would be 1.14 x 10-3 g/l.
As per the study conducted by Fonseca AP, Cardoso M, Esteves V[6], our target range for estrogen mimicking compounds is 1 nM to 5 µM.
The Michaelis Menten equation is as follows
Where
dP/dt
the rate of progression of the reaction[P]
the product (i.e., degraded product)[S]
the substrate concentration (i.e., Estrogen concentration)vm
maximum rate of reactionKm
Michaelis Menten constantFigure 9 (A)
Using these results, we plot the degradation of a given amount of estrogen (figure 9B). We see that 20𝜇𝑀 of estrogen can be degraded in 70 minutes as has also been shown.
Hence, we conclude that under ideal situations, our model system will be able to degrade about 20𝜇M estrogen in around 1.2 hours.
Representation as circuit
Figure 8
Poisson Current Generator (Inducible Promoter)
Voltage Multplier
Poisson Current Generator (Constitutive Promoter)
Grounding
Resisistor
Part A orange
Detection of estrogen
This represents the binding of estrogen and estrogen sensitive RNA polymerase to induce transcription of the downstream proteins in estrogen dependent manner.
A constitutive poisson current generator produces estrogen sensitive RNA polymerase which is one of the inputs. The other input is the estrogen present in the circuit. These two inputs are multiplied via a voltage multiplier and the output of this is sent as an input to another poisson current flux generator. We have linked flux generators with a RC circuit in parallel in order to account for flux decay (i.e. enzyme degradation).
Part B light blue
Translation of protein
The current generated from the second generator gets splits into two. This represents the fact that the two proteins are translated at different rates from the transcribed mRNA. Thus, we get two outputs from one input in this case.
One of the outputs- TAL- is fed as an input to another voltage multiplier along with Tyrosine. These two inputs form the output of a substrate- enzyme bound state. This bound state can dissociate to give back the original inputs, or can be further degraded. The route chosen by the bound state depends on the capacitance and resistance offered in either loop.
Part C blue
Degradation by Laccase
There are three inputs for this part of the circuit- Estrogen, Laccase (produced by a constitutive current generator in Algae), and p-coumaric acid produced from Part B. These three inputs are again multiplied to give the final degraded product.
Representing synthetic genetic circuits in the form of an analog circuit provides a pictorial perspective to the molecular interactions. Here the flow of electrons is symbolic to the flow of molecules (enzymes) and the voltage gradient is analogous to the enzyme gradient. Further it enhances the process of quantifying the non-linear dynamics within a cell.
Currently this circuit is a pictorial representation (without quantifying the voltage inputs, resistance, and capacitance). By carrying out further experiments to find out the exact rate constants involved in the transcription and translation processes, we can quantify the resistance and capacitance, voltage drop etc. at each step. Constructing such a circuit enables us to predict the operation of a synthetic circuit and how tweaking some factors might affect its functioning.
Overall, analog circuits have now become an integral part in synthetic circuit design, analysis, simulation, and implementation [J. J. Y. Teo, S. S. Woo and R. Sarpeshkar ]; margin-right: 0.5rem
References
- Cardinal-Watkins, C., & Nicell, J. A. (2011). Enzyme-Catalyzed Oxidation of 17β-Estradiol Using Immobilized Laccase from Trametes versicolor. Enzyme Research, 2011, 1–11.https://doi.org/10.4061/2011/725172
- https://2013.igem.org/Team:Uppsala/P-Coumaric-acid-pathway
- https://2009.igem.org/Team:PKU_Beijing/Modeling/Parameters
- https://2014.igem.org/Team:Carnegie_Mellon/SensorModel
- Ramos-Martinez EM, Fimognari L, Sakuragi Y. High-yield secretion of recombinant proteins from the microalga Chlamydomonas reinhardtii. Plant Biotechnol J. 2017, 15(9): 1214-1224. doi: 10.1111/pbi.12710.
- Fonseca AP, Cardoso M, Esteves V (2013) Determination of Estrogens in Raw and Treated Wastewater by High-Performance Liquid Chromatography- Ultraviolet Detection. J Environ Anal Toxicol 4:203. doi: 10.4172/2161-0525.1000203
- Cardinal-Watkins, C., & Nicell, J. A. (2011). Enzyme-Catalyzed Oxidation of 17β-Estradiol Using Immobilized Laccase from Trametes versicolor. Enzyme Research, 2011, 1–11.https://doi.org/10.4061/2011/725172
- Engineering Genetic Circuits by Chris J Meyers
- J. J. Y. Teo, S. S. Woo and R. Sarpeshkar, "Synthetic Biology: A Unifying View and Review Using Analog Circuits," in IEEE Transactions on Biomedical Circuits and Systems, vol. 9, no. 4, pp. 453-474, Aug. 2015.