Equations

Here we define the equation and parameters that describe the genetic toggle switch that works at the translational level. The switch allows mutually exclusive expression of protein production. For the purpose of testing, we tagged the proteins with markers, green fluorescent protein(GFP) for one and cyan fluorescent protein (CFP) for the other. This allows us to easily measure the protein expression by measuring the concentrations of GFP and CFP. For the sake of simplicity, we refer to the combination of protein and marker by the marker name (i.e. GFP and CFP). This synthetic biological circuit is represented in Fig 1.

Fig 1

We regulate the system by adding galactose or copper. Galactose binds to the GAL promoter and activates the transcription of mRNA1 (M1), allowing the system to express GFP. Copper binds to the CUP1 promoter, activating the transcription of mRNA2 (M2) and leading to the expression of CFP.

Fig 1 shows the mutual inhibition of the translation of the two mRNAs by the proteins binding to the corresponding stem loop structures on the opposing construct.

This means that if GFP is being expressed, the proteins will bind onto the M2 stem loops, thus preventing M2 from producing CFP. CFP exhibits the same behaviour, inhibiting the production of GFP.

Equation Terms

Each equation is composed of three terms: generation, degradation, and base rate

Generation: There are two forms of the generation term: one for the mRNAs and one for the proteins (GFP and CFP).
For the mRNAs, the generation term is in the form of the Michaelis-Menten equation with Hill coefficients to model the cooperativity of the binding affinities of the stem loops.
For the proteins (GFP and CFP), the Michaelis-Menten equation is modified to take into account the inhibition of one protein on the other. This describes how GFP inhibits the generation of CFP and vice-versa.

Degradation: This term describes the degradation the component within the cell and is a function of the reaction kinetics for the breakdown of the component over time and the dilution that occurs as the cell divides.

Base Rate: This is the concentration of molecules present in the cell when the promoter or inhibitor is not activated.

Equations and Parameters

	(1)
[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 N-peptide and GFP
λ1:	constant representing the rate of transcription of the DNA that encodes for the production of N-peptide and GFP
μ1:	constant representing the rate of degradation of mRNA
n1:	Hill coefficient
k1:	constant of association between galactose and DNA
T:	time constant representing rate of cellular division

This equation describes the rate of change of GFP that is translated from mRNA1.

This equation describes the rate of change of the mRNA (mRNA2) that is transcribed from the copper promoter.

This equation describes the rate of change of CFP translated from mRNA2.

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