Purpose

The bacterial cell model was written in the BioNetGen Language, a rule-based modeling language that is useful for generating differential equations from a description of how various biological components interact with one another. The model was constructed from both data found in the literature and experimental data from the lab. This allows the model to run various simulations of the experiment under different conditions in order to find the optimal conditions for running the experiment. Also if the experiments are not producing any measurable results, the model can be used to identify which component of the experiment could be the problem. The model was run in Rule-Bender, an environment which is dedicated to running, analyzing, visualizing, and debugging BioNetGen Language models.

The Outline

Rule-Bender provides users with a visual contact map of how each molecular component interfaces with one another. The above figure is a contact map for the T7-Intein complex, and was generated by Rule-Bender prior to the simulation of the model.The T7-Intein complex is composed of the N terminal T7 RNAP - N terminal intein – Estrogen Ligand Binding Domain – C terminal intein – C terminal T7 RNAP.

The model captures a total of 17 different interactions:

                       1. The rate at which mRNA T7-Intein and YFP is transcribed.
                       2. The rate at which mRNA T7-Intein is degraded.
                       3. The rate at which T7-Intein is translated.
                       4. The rate at which T7-Intein is degraded.
                       5. The rate at which T7 RNAP polymerase is degraded.
                       6. The rate at which estrogen enters the cell.
                       7. The rate at which estrogen leaves the cell.
                       8. The rate at which estrogen binds to the intein-LBD.
                       9. The rate at which estrogen disassociates from the intein-LBD.
                       10. The rate at which the intein is spliced out and T7 RNAP is formed.
                       11. The rate at which T7 RNAP binds to the T7 promoter of plasmid 2.
                       12. The rate at which mRNA RFP is transcribed.
                       13. The rate at which mRNA RFP is degraded.
                       14. The rate at which RFP is translated.
                       15. The rate at which RFP is degraded.
                       16. The rate at which YFP is translated.
                       17. The rate at which YFP is degraded.

To run the model simply open the latest version of Rulebender, go to the simulation tab, provide the file path for the file you wish to run, and hit run.

The Parameters

Parameter	Value	Reasoning
Temp	Variable (default value of 37 Celsius)	The experiment can be carried out under various temperatures. Temperature affects the rate of intein splicing and RFP levels of fluorescence.
ppt	Variable (default value of 1000)	Parts per trillion of estrogen in the water you are testing.
RFP_copies	15 copies/cell	Number of RFP containing plasmids per cell (pSB3K3).
Int_copies	2984 copies/cell	Number of intein containing plasmids per cell (pSB1C3).
Estrogen_out	ppt * 5.6*10^-2 nM	Concentration of estrogen outside the cell.
Estrogen_in	0 nM	Concentration of estrogen inside the cell. Since e. coli is a prokaryote assume it is 0 nM.
T7-Intein	0 nM	Assume initial concentration of T7-Intein complex is 0 nM.
T7-Estrogen(U)	0 nM	Assume initial concentration of T7-Intein complex bound to estrogen is 0 nM.
T7-Estrogen(S)	0 nM	Assume initial concentration of spliced out Intein-Estrogen complex is 0 nM.
T7-NC	0 nM	Assume initial concentration of T7 polymerase is 0 nM.
Plas_Int	Int_copies * 1.0*10^-3 nM	Concentration of T7-Intein plasmids (pSB1C3).
Plas_RFP	RFP_copies * 1.0*10^-3 nM	Concentration of T7-RFP plasmids (pSB3K3).
T7-RFP	0 nM	Assume initial concentration of T7 RNAP bound to plasmid 2 is 0 nM.
mRNA_RFP	0 nM	Assume initial concentration of mRNA RFP is 0 nM.
Prot_RFP	0 nM	Assume initial concentration of protein RFP is 0 nM.
mRNA_Int	0 nM	Assume initial concentration of mRNA intein is 0 nM.
Prot_YFP	0 nM	Assume initial concentration of protein YFP is 0 nM.
k_e	1.7*10^-2 s^-1	Rate at which estrogen diffuses through membrane. The diffusion coefficient of a steroid hormone in an aqueous phase = 10^-13 m ²/s [1] and the surface area of a bacteria is 6 * 10^-12m²[2]. Thus (10^-13 m²/s) / (6 * 10^-12m²) = 1.7*10^-2 s^-1
k_e-T7(on)	1.3*10^-3 nM^-1s^-1	Rate at which estrogen binds to the intein. The intein contains the ligand binding domain of the human estrogen receptor, so the value is from the literature [3].
k_e-T7(off)	1.2*10^-3 s^-1	Rate at which estrogen dissociates from the intein. Based on literature value of human estrogen receptor [3].
k_splice	7.1*10^-4 s^-1	Rate at which intein splices out. Based on literature values of other inteins’ splicing kinetics [4].
k_T7-RFP(on)	3.3*10^-1 nM^-1s^-1	Rate at which T7 RNAP binds to the T7 promoter of the RFP plasmid. Value obtained from literature [5].
k_T7-RFP(off)	1.0*10^-1 s^-1	Rate at which T7 RNAP dissociates from the T7 promoter of the RFP plasmid. Value obtained from literature [5].
k_{RNA_RFP}	8.8*10^-1 nM s^-1	Rate at which mRNA is synthesized from RFP plasmid. From iGEM team PKU ’09 [6].
k_{RNA_RFP(deg)}	4.3*10^-3 s^-1	Rate at which mRNA RFP degrades. From iGEM team PKU ’09 [6].
k_{Prot_RFP}	9.0*10^-3 s^-1	Rate at which RFP protein is made. From iGEM team PKU ’09 [6].
k_{Prot_RFP(deg)}	8.3*10^-4 s^-1	Rate at which RFP protein degrades. From iGEM team PKU ’09 [6].
k_{RNA_Int}	2.6*10^-2 nM s^-1	Rate at which mRNA is synthesized from T7-Intein plasmid. From iGEM team PKU ’09 [6].
k_{RNA_Int(deg)}	4.3*10^-3 s^-1	Rate at which mRNA of T7-Intein degrades. From iGEM team PKU ’09 [6].
k_{Prot_Int}	2.2*10^-3 s^-1	Rate at which T7-Intein is made from mRNA. From iGEM team PKU ’09 [6].
k_{Prot_Int(deg)}	9.7*10^-4 s^-1	Rate at which T7 polymerase is degraded. From iGEM team PKU ’09 [6].

Insights

Insight: In order to accurately describe the behavior of certain interactions in the cell, it is necessary to use more than just simple rate constants. New functions which modify the rate constant based on cellular conditions must be introduced.

The rate at which mRNA RFP is synthesized is the product of the rate constant and the number of T7 polymerases bound to the RFP plasmid. Thus:

mRNA_RFP Synthesis = k_{RNA_RFP} * T7_RFP

The rate at which RFP protein is produced increases proportionally with the concentration of mRNA RFP present and decreases proportionally with the square root of the concentration of RFP currently present in the cell. Essentially the cell will make less RFP if there is already a lot of RFP currently present. Thus:

Protein_RFP Synthesis = k_{Prot_RFP} * mRNA_RFP * (1 + Prot_RFP) ^-½

The rate at which estrogen enters and exits the cell also depends on the concentration gradient of estrogen. If the concentration of estrogen outside the cell is much greater than the concentration of estrogen inside the cell, then estrogen will enter the cell at a much faster rate than if the concentration gradient is not as large. Thus:

E_in = k_e * (Estrogen_out) / (Estrogen_out + Estrogen_in)
E_out = k_e * (Estrogen_in) / (Estrogen_out + Estrogen_in)

The rate at which the T7-Intein complex is produced increases proportionally with the concentration of mRNA T7-Intein present and decreases proportionally with the square root of the concentration of T7-Intein complex currently present in the cell. Essentially the cell will make less T7-Intein complex if there is already a lot of T7-Intein currently present. Thus:

Protein_Int Synthesis = k_{Prot_Int} * mRNA_Int * (1 + T7_unbound) ^-½

The rate at which YFP is produced increases proportionally with the concentration of mRNA T7-Intein present (as the YFP sequence is located on the same mRNA) and decreases proportionally with the square root of the concentration of YFP currently present in the cell. Essentially the cell will make less YFP if there is already a lot of YFP currently present. Thus:

Protein_YFP Synthesis = k_{Prot_Int} * mRNA_Int * (1 + Prot_YFP) ^-½

The rate at which the intein splices out is dependent on the temperature of environment surrounding the cell. Since the cell is an E. coli cell, the optimal temperature for the surrounding environment is 37 degrees Celsius. This was confirmed in the wet-lab. Thus:

Int Splice() = k_splice * (1 + |37 - Temp| / °C) ^-½

Results

The units of concentration in all of the graphs are in nM. The units of time in all of the graphs are in seconds. According to the literature enhanced GFP can be detected fairly reasonably at concentrations greater than 1 µm [7]. Thus we can assume that regular RFP will be detected at concentrations of around 100 µm. All of the simulations were carried out at 37 degrees Celsius for the optimal results. The sensor should detect estrogen within a couple of hours, thus the simulation ends in a couple of hours.

Scenario 1: Low amount of estrogen in the environment (< 100 ppt)

The sensor is not sensitive enough to detect such low concentrations of estrogen in the environment, as the concentration of RFP produced never exceeds 100 µm. Also eventually the rate at which RFP degrades eclipses the rate at which RFP is produced thus making the sensor not very valuable at detecting incredibly low estrogenic concentrations.

Scenario 2: Ideal scenario. 1000 ppt of estrogen in the environment and all of the splicing kinetics work at a reasonably fast rate. Essentially the same scenario as if the T7 polymerase was constitutively expressed.

This scenario is based on the wet-lab data we received for the constitutively expressed T7 polymerase. RFP is produced at detectable amounts and the sensor does its job of detecting estrogen in water.

Scenario 3: The intein is not splicing properly. This is representative of what is occurring in the wet-lab at the moment when T7 is not constitutively expressed and its formation solely depends on the splicing of the intein.

Currently in the lab the sensor does not produce any measurable amount of RFP in the presence of estrogen. This graph represents what could possibly be occurring. Using the model, the bottleneck was identified to be the rate at which the intein splices. Thus the model saves the wet-lab biologists’ time by showing exactly which part of the system needs to be improved.

Code

A commented, working version of the code can be found at E. coli Cell Model

References

1. E. Oren et al. Free Diffusion of Steroid Hormones Across Biomembranes: A Simplex Search with Implicit Solvent Model Calculations. Biophys J. Vol. 87, pp. 768–779. August 2004.
2. http://bionumbers.hms.harvard.edu/search.aspx?log=y&task=searchbytrmorg&trm=surface+area+of+cell&org=&rpp=100
3. R. L. Rich et al. Kinetic analysis of estrogen receptor/ligand interactions. PNAS. Vol. 99. No. 13. pp. 8562–8567, June 2002.
4. S. Brenzel, T. Kurpiers, H. D. Mootz. Engineering artificially split inteins for applications in protein chemistry: biochemical characterization of the split Ssp DnaB intein and comparison to the split Sce VMA intein. Biochemistry. Vol 45(6), pp. 1571-8, February 2006.
5. http://www.jbc.org/content/271/48/30451/T1.expansion.html
6. iGEM ’09 PKU-Beijing. https://2009.igem.org/Team:PKU_Beijing/Modeling/Parameters
7. A. Furtado, and R. Henry. Measurement of green fluorescent protein concentration in single cells by image analysis. Vol. 310, pp. 84–92, May 2002.
8. J. R. Faeder et al. Rule-based modeling of biochemical systems with BioNetGen. Methods Mol Biol. 500:113-67 2009.
9. RuleBender. http://visualizlab.org/rulebender
10. BioNetGen. http://bionetgen.org/index.php/Main_Page

Team:Carnegie Mellon/SensorModel

From 2014.igem.org