Team:Aberdeen Scotland/Equations
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University of Aberdeen  ayeSwitch
Equations
Here we define the equations and parameters that describe the novel genetic toggle switch that works at the translational level. The switch allows mutually exclusive expression of either green fluorescent protein (GFP) or cyan fluorescent protein (CFP). The synthetic biological circuit is represented in Fig 1.
Figure 1: Translation of DNA to mRNA.
We can regulate the system when we add galactose or methionine. Galactose will bind to the GAL promoter and activate the transcription of M1, allowing the system to express GFP. If we remove methionine from the system instead of adding galactose, it will bind to the MET1 promoter, the transcription of M2 will be activated, leading to the expression of CFP.
From Fig 1 it can be seen that there is mutual inhibition of the translation of the two mRNAs. That is because the translated proteins can bind to the corresponding stem loop structures on the opposing construct.
For our initial conditions, we began with more GFP than CFP and thus the production of CFP was inhibited. When methionine was added removed from the system, the rate of CFP production will increase and decrease for GFP. Eventually, we will see more CFP than GFP so the system will have switched. Once we have more CFP than GFP, galactose can then be added to switch back to an expression of GFP.
The NPeptide and GFP strand has two MS2Stem loops as we discovered that one single loop would not inhibit the production of CFP enough to achieve our switch.
Equation 1
(1) 
This is the equation for the rate of change of the mRNA that is transcribed from the galactose promoter. The three terms represent production, degradation, and dilution respectively.
[GAL] represents the concentration of galactose that is added to the system. When galactose is added it binds to the promoter and activates the transcription of M1.
[M1] is the concentration of mRNA that translates the Npeptide and GFP.
Parameter 
Description 
λ_{1}: 
Constant representing rate of transcription of the DNA that encodes for the production of N peptide and GFP 
μ_{1}: 
Constant representing rate of degradation of mRNA 
n_{1}: 
Hill coefficient for the association between the galactose and the GAL promoter

K_{1}: 
Dissociation constant for the GAL promoter 
T: 
Time constant representing rate of cellular division 
Equation 2
(2) 
This is the equation for the rate of change of protein that is translated from the mRNA for GFP. The three terms represent production, degradation, and dilution respectively.
[M1] is the concentration of mRNA that translates the Npeptide GFP.
[GFP] represents the concentration of Npeptide and GFP.
[CFP] represents the concentration of the MS2protein and CFP.
Parameter: 
Description 
λ_{2}: 
Constant representing rate of translation of the mRNA that encodes for the production of Npeptide and GFP 
μ_{2}: 
Constant representing rate of degradation of the GFP 
n_{2}: 
Hill coefficient of the CFP/MS2 stem loop association

K_{2}: 
Dissociation constant for the MS2CFP protein to MS2 loop 
T: 
Time constant representing rate of cellular division 
Equation 3
(3) 
This is the equation for the rate of change of the mRNA that is transcribed from the copper promoter. The three terms represent production, degradation, and dilution respectively.
[Cu^{2+}] is the concentration of the copper added to the system that binds to the CUP1 promoter and activates the transcription of M2.
[M2] represents the concentration of mRNA that translates the MS2protein and CFP.
Parameter 
Description 
λ_{3:} 
Constant representing rate of transcription of the DNA that encodes for the production of the MS2protein and CFP 
μ_{3}: 
Constant representing rate of degradation of mRNA 
n_{3}: 
Hill coefficient of the association between copper and the CUP1 promoter

K_{3}: 
Dissociation constant for Copper promoter 
T: 
Time constant representing rate of cellular division 
Equation 4
(4) 
This is the equation for the rate of change of protein that is translated from the mRNA for CFP. The three terms represent production, degradation, and dilution respectively.
[M2] is the concentration of mRNA that translates to MS2protein and CFP.
[GFP] represents the concentration of the Npeptide and GFP.
[CFP] represents the concentration of the MS2protein and CFP.
Parameters 
Description 
λ_{4}: 
Constant representing rate of translation of the mRNA that encodes for the production of MS2protein and CFP 
μ_{4}: 
Constant representing rate of degradation of the CFP 
n_{4}: 
Hill coefficient of the GFP/Bbox stem loop association

K_{4}: 
Dissociation constant for the NPepGFP protein to the Bboxstem loop 
T: 
time constant representing rate of cellular division 
Parameter Study
The parameter values were first estimated based on the literature ^{[1]} and after the first estimation, a possible range of variation for each parameter was assigned, also based on literature. Then, we studied the bistability of the model depending on the parameter values that were varied in the above mentioned ranges. For more information, see Parameter Space Analysis and Directed Evolution.
Modification of the construct
Some experimental difficulties were encountered with the copper construct which led to the use of a methionine promoter to substitute it. Methionine acts as an inhibitor of the promoter, so that equation 3 had to be substituted by the following equation:
The behaviour of the switch can then be summarise in the following table:
What is present in the system 
Protein(s) produced 
Galactose and Methionine 
GFP 
Galactose only 
GFP, CFP (doses dependent) 
Methionine only 
No GFP or CFP 
No Galactose and no Methionine 
CFP 
References
^{[1]} Beyer A, Hollunder J, Nasheuer HP, Wilhelm T. (2004), Posttranscriptional expression regulation in the yeast Saccharomyces cerevisiae on a genomic scale, Mol Cell Proteomics., Vol. 3, No.11, pp. 10831092.
^{[2]} Alon, U. (2006), An Introduction to Systems Biology: Design Principles of Biological Circuits, Chapman and Hall.
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