Team:Aberdeen Scotland/Equations

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University of Aberdeen - ayeSwitch - iGEM 2010

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 copper. Galactose will bind to the GAL promoter and activate the transcription of M1, allowing the system to express GFP. If we add copper instead of galactose, it will bind to the CUP1 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 copper was added to 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 N-Peptide and GFP strand has two MS2-Stem 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 N-peptide and GFP.


Parameter

Description

λ1:

Constant representing rate of transcription of the DNA that encodes for the production of N peptide and GFP

μ1:

Constant representing rate of degradation of mRNA

n1:

Hill coefficient for the association between the galactose and the GAL promoter

k1:

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 N-peptide GFP.
[GFP] represents the concentration of N-peptide and GFP.
[CFP] represents the concentration of the MS2-protein and CFP.


Parameter:

Description

[GFP]:

represents the concentration of N-peptide and GFP

[CFP]:

represents the concentration of the MS2-protein and CFP

λ2:

constant representing the rate of translation of the DNA that encodes
for the production of N-peptide and GFP

μ2:

constant representing the rate of degradation of GFP

n2:

Hill coefficient of the CFP/MS2 stem loop association

k2:

dissociation constant of association between mRNA and amino acids

T:

time constant representing rate of cellular division


Equation (3) describes the rate of change of the mRNA (mRNA2) that is transcribed from the copper promoter.

(3)

[Cu2+]:

is the concentration of the copper added to the
system that binds to the CUP1 promoter and activates
the transcription of M2

[M2]:

is the concentration of mRNA that translates the N-peptide and GFP

λ3:

constant representing the rate of transcription of the DNA that encodes
for the production of MS2-peptide and CFP

μ3:

constant representing the rate of degradation of mRNA

n3:

Hill coefficient

k3:

constant of association between copper and DNA

T:

time constant representing rate of cellular division


Equation (4) describes the rate of change of CFP that is translated from mRNA2.

(4)

[M2]:

is the concentration of mRNA that translates the MS2-protein and CFP

[GFP]:

represents the concentration of N-peptide and GFP

[CFP]:

represents the concentration of the MS2-protein and CFP

λ4:

constant representing the rate of translation of the DNA that encodes
for the production of MS2-protein and CFP

μ4:

constant representing the rate of degradation of CFP

n4:

Hill coefficient

k4:

constant of association between mRNA and amino acids

T:

time constant representing rate of cellular division


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