Transcription/lation Simulation

Abstract

We found some parameter sets on which E. coli changes gene expression by a generation. In addition, we also found ones on which circuit establishes the loops stably regardless of small fluctuation (noise in experiment). Therefore, our gene circuit seems likely to work as designed.

Introduction

We run a simulation about transcription and translation for two reasons. First is to make sure that our gene circuit functions as designed and establishes the loop of gene expression. Although the strain of E. coli used in our experiment, growing environment and sigma factors affect various parameter such as the speed of translation and transcription, and decomposition rate of factors, if there is no parameter sets which we need to establish the loop of gene expression of fluorescent protein, our project is theoretically impossible. In other words, to find the parameter set which establishes the loop means to show that our project is theoretically feasible. In addition, the simulation enables us to improve and redesign our gene circuit with reference to the parameter set.

The second purpose of simulation is to make sure whether the loop run correctly or not, when various parameters are found, such as the speed of translation and transcription, and decomposition rate of factors. If a tool is developed which calculate the movement of gene circuit according to the parameters found and outputs a graph of concentration change of fluorescent proteins, it enables us to decide whether the gene circuit run or not and choose the factor used in the experiment easily.

Method

Stability Analysis

Abstract

$$\frac{d[\sigma_A]}{dt}= \alpha \cdot\left(\frac{R_A+R_{td,A} [\mathrm{taRNA}]}{K^{\prime\prime}+[\mathrm{taRNA}]}\right)-\frac{\ln2}{\mathrm{HL}_{\sigma_A}}\cdot[\sigma_A]$$ $$\frac{d[\sigma_B]}{dt}= \alpha\cdot \left(\frac{R_B+R_{td,B}\cdot [\mathrm{taRNA}]}{K^{\prime\prime}+[\mathrm{taRNA}]}\right)-\frac{\ln2}{\mathrm{HL}_{\sigma_B}}\cdot[\sigma_B]$$ $$\frac{d[\sigma_C]}{dt}= \alpha\cdot \left(\frac{R_C+R_{td,C}\cdot [\mathrm{taRNA}]}{K^{\prime\prime}+[\mathrm{taRNA}]}\right)-\frac{\ln2}{\mathrm{HL}_{\sigma_C}}\cdot[\sigma_C]$$ $$\frac{d[\sigma_{\mbox{-}A}]}{dt}= \alpha\cdot R_{\mbox{-}A}-\frac{\ln2}{\mathrm{HL}_{\sigma_{\mbox{-}A}}}[\sigma_{\mbox{-}A}]$$ $$\frac{d[\sigma_{\mbox{-}B}]}{dt}= \alpha\cdot R_{\mbox{-}B}-\frac{\ln2}{\mathrm{HL}_{\sigma_{\mbox{-}B}}}[\sigma_{\mbox{-}B}]$$ $$\frac{d[\sigma_{\mbox{-}C}]}{dt}= \alpha\cdot R_{\mbox{-}C}-\frac{\ln2}{\mathrm{HL}_{\sigma_{\mbox{-}C}}}[\sigma_{\mbox{-}C}]$$ $$\frac{dR_A}{dt}=k\cdot \mathrm{C}_{low}\cdot [\sigma_A]\cdot\frac{1-\frac{[\sigma_{\mbox{-}A}]}{K^{\prime}_a+[\sigma_{\mbox{-}A}]}}{K_a+[\sigma_A]\cdot\left(1-\frac{[\sigma_{\mbox{-}A}]}{K^{\prime}_a+[\sigma_{\mbox{-}A}]}\right)}+1-c_{lacI}\cdot\frac{1-\frac{[\mathrm{IPTG}]}{K^{\prime}_{lacI}+[\mathrm{IPTG}]}}{K_{lacI}+c_{lacI}\left(1-\frac{[\mathrm{IPTG}]}{K^{\prime}_{lacI}+[\mathrm{IPTG}]}\right)}-\frac{\ln2}{\mathrm{HL}_{RNA}}\cdot R_A$$ $$\frac{dR_B}{dt}=k\cdot \mathrm{C}_{low}\cdot [\sigma_B]\cdot\frac{1-\frac{[\sigma_{\mbox{-}B}]}{K^{\prime}_b+[\sigma_{\mbox{-}B}]}}{K_b+[\sigma_B]\cdot\left(1-\frac{[\sigma_{\mbox{-}B}]}{K^{\prime}_b+[\sigma_{\mbox{-}B}]}\right)}-\frac{\ln2}{\mathrm{HL}_{RNA}}\cdot R_B$$ $$\frac{dR_C}{dt}=k\cdot \mathrm{C}_{low}\cdot [\sigma_C]\cdot\frac{1-\frac{[\sigma_{\mbox{-}C}]}{K^{\prime}_c+[\sigma_{\mbox{-}C}]}}{K_c+[\sigma_C]\cdot\left(1-\frac{[\sigma_{\mbox{-}C}]}{K^{\prime}_c+[\sigma_{\mbox{-}C}]}\right)}-\frac{\ln2}{\mathrm{HL}_{RNA}}\cdot R_C$$ $$\frac{dR_{\mbox{-}A}}{dt}=k\cdot \mathrm{C}_{high}\cdot [\sigma_B]\cdot\frac{1-\frac{[\sigma_{\mbox{-}B}]}{K^{\prime}_b+[\sigma_{\mbox{-}B}]}}{K_b+[\sigma_B]\cdot\left(1-\frac{[\sigma_{\mbox{-}B}]}{K^{\prime}_b+[\sigma_{\mbox{-}B}]}\right)}-\frac{\ln2}{\mathrm{HL}_{RNA}}\cdot R_{\mbox{-}A}$$ $$\frac{dR_{\mbox{-}B}}{dt}=k\cdot \mathrm{C}_{high}\cdot [\sigma_C]\cdot\frac{1-\frac{[\sigma_{\mbox{-}C}]}{K^{\prime}_c+[\sigma_{\mbox{-}C}]}}{K_c+[\sigma_C]\cdot\left(1-\frac{[\sigma_{\mbox{-}C}]}{K^{\prime}_c+[\sigma_{\mbox{-}C}]}\right)}-\frac{\ln2}{\mathrm{HL}_{RNA}}\cdot R_{\mbox{-}B}$$ $$\frac{dR_{\mbox{-}C}}{dt}=k\cdot \mathrm{C}_{high}\cdot [\sigma_A]\cdot\frac{1-\frac{[\sigma_{\mbox{-}A}]}{K^{\prime}_a+[\sigma_{\mbox{-}A}]}}{K_a+[\sigma_A]\cdot\left(1-\frac{[\sigma_{\mbox{-}A}]}{K^{\prime}_a+[\sigma_{\mbox{-}A}]}\right)}-\frac{\ln2}{\mathrm{HL}_{RNA}}\cdot R_{\mbox{-}C}$$ $$\frac{dR_{td,A}}{dt}=k\cdot \mathrm{C}_{low}\cdot [\sigma_C]\cdot\frac{1-\frac{[\sigma_{\mbox{-}C}]}{K^{\prime}_c+[\sigma_{\mbox{-}C}]}}{K_c+[\sigma_C]\cdot\left(1-\frac{[\sigma_{\mbox{-}C}]}{K^{\prime}_c+[\sigma_{\mbox{-}C}]}\right)}-\frac{\ln2}{\mathrm{HL}_{RNA}}\cdot R_{td,A}$$ $$\frac{dR_{td,B}}{dt}=k\cdot \mathrm{C}_{low}\cdot [\sigma_A]\cdot\frac{1-\frac{[\sigma_{\mbox{-}A}]}{K^{\prime}_a+[\sigma_{\mbox{-}A}]}}{K_a+[\sigma_A]\cdot\left(1-\frac{[\sigma_{\mbox{-}A}]}{K^{\prime}_a+[\sigma_{\mbox{-}A}]}\right)}-\frac{\ln2}{\mathrm{HL}_{RNA}}\cdot R_{td,B}$$ $$\frac{dR_{td,C}}{dt}=k\cdot \mathrm{C}_{low}\cdot [\sigma_B]\cdot\frac{1-\frac{[\sigma_{\mbox{-}B}]}{K^{\prime}_b+[\sigma_{\mbox{-}B}]}}{K_b+[\sigma_B]\cdot\left(1-\frac{[\sigma_{\mbox{-}B}]}{K^{\prime}_b+[\sigma_{\mbox{-}B}]}\right)}-\frac{\ln2}{\mathrm{HL}_{RNA}}\cdot R_{td,C}$$ $$\frac{d[\mathrm{taRNA}]}{dt}=-\frac{\ln2}{\mathrm{HL}_{RNA}}\cdot [\mathrm{taRNA}]\ \textrm{(when Pnrd is on)}$$ $$\frac{d[\mathrm{IPTG}]}{dt}=-\frac{\ln2}{\mathrm{HL}_{IPTG}}\cdot [\mathrm{IPTG}]$$ $$\frac{d[\mathrm{imGFP}]}{dt}= \alpha\cdot [R_{\mathrm{GFP}}]-\frac{\ln2}{\mathrm{MT}_{GFP}}\cdot [\mathrm{imGFP}]-\frac{\ln2}{\mathrm{HL}_{imGFP}}\cdot [\mathrm{imGFP}]$$ $$\frac{d[\mathrm{imRFP}]}{dt}= \alpha\cdot [R_{\mathrm{RFP}}]-\frac{\ln2}{\mathrm{MT}_{RFP}}\cdot [\mathrm{imRFP}]-\frac{\ln2}{\mathrm{HL}_{imRFP}}\cdot [\mathrm{imRFP}]$$ $$\frac{d[\mathrm{imCFP}]}{dt}= \alpha\cdot [R_{\mathrm{CFP}}]-\frac{\ln2}{\mathrm{MT}_{GFP}}\cdot [R_{\mathrm{CFP}}]-\frac{\ln2}{\mathrm{HL}_{imCFP}}\cdot [\mathrm{imCFP}]$$ $$\frac{d[\mathrm{GFP}]}{dt}=\frac{\ln2}{\mathrm{MT}_{GFP}}\cdot [\mathrm{imGFP}]-\frac{\ln2}{\mathrm{HL}_{GFP}}\cdot [\mathrm{GFP}]$$ $$\frac{d[\mathrm{RFP}]}{dt}=\frac{\ln2}{\mathrm{MT}_{RFP}}\cdot [\mathrm{imRFP}]-\frac{\ln2}{\mathrm{HL}_{RFP}}\cdot [\mathrm{RFP}]$$ $$\frac{d[\mathrm{CFP}]}{dt}=\frac{\ln2}{\mathrm{MT}_{CFP}}\cdot [\mathrm{imCFP}]-\frac{\ln2}{\mathrm{HL}_{CFP}}\cdot [\mathrm{CFP}]$$ $$\frac{dR_{\mathrm{GFP}}}{dt}=k\cdot \mathrm{C}_{low}\cdot [\sigma_A]\cdot\frac{1-\frac{[\sigma_{\mbox{-}A}]}{K^{\prime}_a+[\sigma_{\mbox{-}A}]}}{K_a+[\sigma_A]\cdot\left(1-\frac{[\sigma_{\mbox{-}A}]}{K^{\prime}_a+[\sigma_{\mbox{-}A}]}\right)}-\frac{\ln2}{\mathrm{HL}_{RNA}}\cdot R_{\mathrm{GFP}}$$ $$\frac{dR_{\mathrm{RFP}}}{dt}=k\cdot \mathrm{C}_{low}\cdot [\sigma_B]\cdot\frac{1-\frac{[\sigma_{\mbox{-}B}]}{K^{\prime}_b+[\sigma_{\mbox{-}B}]}}{K_b+[\sigma_B]\cdot\left(1-\frac{[\sigma_{\mbox{-}B}]}{K^{\prime}_b+[\sigma_{\mbox{-}B}]}\right)}-\frac{\ln2}{\mathrm{HL}_{RNA}}\cdot R_{\mathrm{RFP}}$$ $$\frac{dR_{\mathrm{CFP}}}{dt}=k\cdot \mathrm{C}_{low}\cdot [\sigma_C]\cdot\frac{1-\frac{[\sigma_{\mbox{-}C}]}{K^{\prime}_c+[\sigma_{\mbox{-}C}]}}{K_c+[\sigma_C]\cdot\left(1-\frac{[\sigma_{\mbox{-}C}]}{K^{\prime}_c+[\sigma_{\mbox{-}C}]}\right)}-\frac{\ln2}{\mathrm{HL}_{RNA}}\cdot R_{\mathrm{CFP}} $$