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We aim to create a genetic circuit functional in E.coli which allows the
gene expressed to change after each cell division. Specifically, the E. coli
will be expressing GFP, RFP, and CFP after subsequent cell divisions and will
loop back to expressing GFP again.
This will be achieved by the use of the Pnrd promoter to sense to cell division,
sigma factor/anti-sigma pairs control transcription, and a toehold switch to create
an AND gate required in the circuit.
Our project would be able to exhibit, in E. coli, that despite carrying the same
genetic information, different phenotypes can be expressed across generations.
Much like the phenomenon seen in epigenetics. Moreover, this technology could
potentially be applied in field to carry out certain tasks automatically without
the need of external input eg. to make engineered cells automatically carry
programmed cell death after a certain number of divisions.
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$$\frac{d[\sigma_A]}{dt} = \alpha\cdot (\frac{R_A+R_{td,A} [taRNA]}{K+[taRNA]})$$-\frac{\ln2}{\mathrm{HL}_{\sigma_A}}\cdot[\sigma_A]\\
\frac{d[\sigma_B]}{dt}&=& \alpha\cdot \left(\frac{R_B+R_{td,B}\cdot [\mathrm{taRNA}]}{K+[\mathrm{taRNA}]}\right)-\frac{\ln2}{\mathrm{HL}_{\sigma_B}}\cdot[\sigma_B]\\
\frac{d[\sigma_C]}{dt}&=& \alpha\cdot \left(\frac{R_C+R_{td,C}\cdot [\mathrm{taRNA}]}{K+[\mathrm{taRNA}]}\right)-\frac{\ln2}{\mathrm{HL}_{\sigma_C}}\cdot[\sigma_C]\\
\\
\frac{d[\sigma_{\rm\mathchar `-A}]}{dt}&=& \alpha\cdot R_{\rm\mathchar `-A}-\frac{\ln2}{\mathrm{HL}_{\sigma_{\rm\mathchar `-A}}}[\sigma_{\rm\mathchar `-A}]\\
\frac{d[\sigma_{\rm\mathchar `-B}]}{dt}&=& \alpha\cdot R_{\rm\mathchar `-B}-\frac{\ln2}{\mathrm{HL}_{\sigma_{\rm\mathchar `-B}}}[\sigma_{\rm\mathchar `-B}]\\
\frac{d[\sigma_{\rm\mathchar `-C}]}{dt}&=& \alpha\cdot R_{\rm\mathchar `-C}-\frac{\ln2}{\mathrm{HL}_{\sigma_{\rm\mathchar `-C}}}[\sigma_{\rm\mathchar `-C}]\\
\\
\frac{dR_A}{dt}&=&k\cdot \mathrm{C}_{low}\cdot [\sigma_A]\cdot\frac{1-\frac{[\sigma_{\rm\mathchar `-A}]}{K'_a+[\sigma_{\rm\mathchar `-A}]}}{K_a+[\sigma_A]\cdot\left(1-\frac{[\sigma_{\rm\mathchar `-A}]}{K'_a+[\sigma_{\rm\mathchar `-A}]}\right)}+1-c_{lacI}\cdot\frac{1-\frac{[\mathrm{IPTG}]}{K'_{lacI}+[\mathrm{IPTG}]}}{K_{lacI}+c_{lacI}\left(1-\frac{[\mathrm{IPTG}]}{K'_{lacI}+[\mathrm{IPTG}]}\right)}-\frac{\ln2}{\mathrm{HL}_{RNA}}\cdot R_A\\
\\
\frac{dR_B}{dt}&=&k\cdot \mathrm{C}_{low}\cdot [\sigma_B]\cdot\frac{1-\frac{[\sigma_{\rm\mathchar `-B}]}{K'_b+[\sigma_{\rm\mathchar `-B}]}}{K_b+[\sigma_B]\cdot\left(1-\frac{[\sigma_{\rm\mathchar `-B}]}{K'_b+[\sigma_{\rm\mathchar `-B}]}\right)}-\frac{\ln2}{\mathrm{HL}_{RNA}}\cdot R_B\\
\frac{dR_C}{dt}&=&k\cdot \mathrm{C}_{low}\cdot [\sigma_C]\cdot\frac{1-\frac{[\sigma_{\rm\mathchar `-C}]}{K'_c+[\sigma_{\rm\mathchar `-C}]}}{K_c+[\sigma_C]\cdot\left(1-\frac{[\sigma_{\rm\mathchar `-C}]}{K'_c+[\sigma_{\rm\mathchar `-C}]}\right)}-\frac{\ln2}{\mathrm{HL}_{RNA}}\cdot R_C\\
\frac{dR_{\rm\mathchar `-A}}{dt}&=&k\cdot \mathrm{C}_{high}\cdot [\sigma_B]\cdot\frac{1-\frac{[\sigma_{\rm\mathchar `-B}]}{K'_b+[\sigma_{\rm\mathchar `-B}]}}{K_b+[\sigma_B]\cdot\left(1-\frac{[\sigma_{\rm\mathchar `-B}]}{K'_b+[\sigma_{\rm\mathchar `-B}]}\right)}-\frac{\ln2}{\mathrm{HL}_{RNA}}\cdot R_{\rm\mathchar `-A}\\
\frac{dR_{\rm\mathchar `-B}}{dt}&=&k\cdot \mathrm{C}_{high}\cdot [\sigma_C]\cdot\frac{1-\frac{[\sigma_{\rm\mathchar `-C}]}{K'_c+[\sigma_{\rm\mathchar `-C}]}}{K_c+[\sigma_C]\cdot\left(1-\frac{[\sigma_{\rm\mathchar `-C}]}{K'_c+[\sigma_{\rm\mathchar `-C}]}\right)}-\frac{\ln2}{\mathrm{HL}_{RNA}}\cdot R_{\rm\mathchar `-B}\\
\\
\frac{dR_{\rm\mathchar `-C}}{dt}&=&k\cdot \mathrm{C}_{high}\cdot [\sigma_A]\cdot\frac{1-\frac{[\sigma_{\rm\mathchar `-A}]}{K'_a+[\sigma_{\rm\mathchar `-A}]}}{K_a+[\sigma_A]\cdot\left(1-\frac{[\sigma_{\rm\mathchar `-A}]}{K'_a+[\sigma_{\rm\mathchar `-A}]}\right)}-\frac{\ln2}{\mathrm{HL}_{RNA}}\cdot R_{\rm\mathchar `-C}\\
\frac{dR_{td,A}}{dt}&=&k\cdot \mathrm{C}_{low}\cdot [\sigma_C]\cdot\frac{1-\frac{[\sigma_{\rm\mathchar `-C}]}{K'_c+[\sigma_{\rm\mathchar `-C}]}}{K_c+[\sigma_C]\cdot\left(1-\frac{[\sigma_{\rm\mathchar `-C}]}{K'_c+[\sigma_{\rm\mathchar `-C}]}\right)}-\frac{\ln2}{\mathrm{HL}_{RNA}}\cdot R_{td,A}\\
\\
\frac{dR_{td,B}}{dt}&=&k\cdot \mathrm{C}_{low}\cdot [\sigma_A]\cdot\frac{1-\frac{[\sigma_{\rm\mathchar `-A}]}{K'_a+[\sigma_{\rm\mathchar `-A}]}}{K_a+[\sigma_A]\cdot\left(1-\frac{[\sigma_{\rm\mathchar `-A}]}{K'_a+[\sigma_{\rm\mathchar `-A}]}\right)}-\frac{\ln2}{\mathrm{HL}_{RNA}}\cdot R_{td,B}\\
\frac{dR_{td,C}}{dt}&=&k\cdot \mathrm{C}_{low}\cdot [\sigma_B]\cdot\frac{1-\frac{[\sigma_{\rm\mathchar `-B}]}{K'_b+[\sigma_{\rm\mathchar `-B}]}}{K_b+[\sigma_B]\cdot\left(1-\frac{[\sigma_{\rm\mathchar `-B}]}{K'_b+[\sigma_{\rm\mathchar `-B}]}\right)}-\frac{\ln2}{\mathrm{HL}_{RNA}}\cdot R_{td,C}\\
\\
\frac{d[\mathrm{taRNA}]}{dt}&=&-\frac{\ln2}{\mathrm{HL}_{RNA}}\cdot [\mathrm{taRNA}]\ \textrm{(when Pnrd is on)}\\
\frac{d[\mathrm{IPTG}]}{dt}&=&-\frac{\ln2}{\mathrm{HL}_{IPTG}}\cdot [\mathrm{IPTG}]\\
\\
\frac{d[\mathrm{imGFP}]}{dt}&=& \alpha\cdot [R_{\mathrm{GFP}}]-\frac{\ln2}{\mathrm{MT}_{GFP}}\cdot [\mathrm{imGFP}]-\frac{\ln2}{\mathrm{HL}_{imGFP}}\cdot [\mathrm{imGFP}]\\
\frac{d[\mathrm{imRFP}]}{dt}&=& \alpha\cdot [R_{\mathrm{RFP}}]-\frac{\ln2}{\mathrm{MT}_{RFP}}\cdot [\mathrm{imRFP}]-\frac{\ln2}{\mathrm{HL}_{imRFP}}\cdot [\mathrm{imRFP}]\\
\frac{d[\mathrm{imCFP}]}{dt}&=& \alpha\cdot [R_{\mathrm{CFP}}]-\frac{\ln2}{\mathrm{MT}_{GFP}}\cdot [R_{\mathrm{CFP}}]-\frac{\ln2}{\mathrm{HL}_{imCFP}}\cdot [\mathrm{imCFP}]\\
\\
\frac{d[\mathrm{GFP}]}{dt}&=&\frac{\ln2}{\mathrm{MT}_{GFP}}\cdot [\mathrm{imGFP}]-\frac{\ln2}{\mathrm{HL}_{GFP}}\cdot [\mathrm{GFP}]\\
\frac{d[\mathrm{RFP}]}{dt}&=&\frac{\ln2}{\mathrm{MT}_{RFP}}\cdot [\mathrm{imRFP}]-\frac{\ln2}{\mathrm{HL}_{RFP}}\cdot [\mathrm{RFP}]\\
\frac{d[\mathrm{CFP}]}{dt}&=&\frac{\ln2}{\mathrm{MT}_{CFP}}\cdot [\mathrm{imCFP}]-\frac{\ln2}{\mathrm{HL}_{CFP}}\cdot [\mathrm{CFP}]\\
\\
\frac{dR_{\mathrm{GFP}}}{dt}&=&k\cdot \mathrm{C}_{low}\cdot [\sigma_A]\cdot\frac{1-\frac{[\sigma_{\rm\mathchar `-A}]}{K'_a+[\sigma_{\rm\mathchar `-A}]}}{K_a+[\sigma_A]\cdot\left(1-\frac{[\sigma_{\rm\mathchar `-A}]}{K'_a+[\sigma_{\rm\mathchar `-A}]}\right)}-\frac{\ln2}{\mathrm{HL}_{RNA}}\cdot R_{\mathrm{GFP}}\\
\frac{dR_{\mathrm{RFP}}}{dt}&=&k\cdot \mathrm{C}_{low}\cdot [\sigma_B]\cdot\frac{1-\frac{[\sigma_{\rm\mathchar `-B}]}{K'_b+[\sigma_{\rm\mathchar `-B}]}}{K_b+[\sigma_B]\cdot\left(1-\frac{[\sigma_{\rm\mathchar `-B}]}{K'_b+[\sigma_{\rm\mathchar `-B}]}\right)}-\frac{\ln2}{\mathrm{HL}_{RNA}}\cdot R_{\mathrm{RFP}}\\
\frac{dR_{\mathrm{CFP}}}{dt}&=&k\cdot \mathrm{C}_{low}\cdot [\sigma_C]\cdot\frac{1-\frac{[\sigma_{\rm\mathchar `-C}]}{K'_c+[\sigma_{\rm\mathchar `-C}]}}{K_c+[\sigma_C]\cdot\left(1-\frac{[\sigma_{\rm\mathchar `-C}]}{K'_c+[\sigma_{\rm\mathchar `-C}]}\right)}-\frac{\ln2}{\mathrm{HL}_{RNA}}\cdot R_{\mathrm{CFP}}
$$
