Team:Tokyo Tech/Model
Model
Contents
Overview
To reproduce the story of ”Snow White”, we have designed the cell-cell communication system with improved or characterized parts and collected data from comprehensive experiments. Furthermore, we constructed the mathematical model in order to simulate the behavior of the whole system and to confirm the feasibility of our story. This simulation successfully contributed to give the suggestions to wet lab experiments and confirm the feasibility of our system. In addition, in order to utilize TA (Toxin-Antitoxin) system, we developed a new software in Java for adjusting the number of ACA sequences, which MazF dimer recognizes and cleaves in mRNAs.
Mathematical Model
To simulate our gene circuits, we developed an ordinary differential equation model.
[Detailed Description for Modeling]
Differencial Equations
Snow White
\begin{equation} \frac{d[mRNA_{RFP}]}{dt} = k - d[mRNA_{RFP}] - F_{DiMazF}(1-(1-f)^{f_{mRNA_{RFP}}})[mRNA_{RFP}][DiMazF] \end{equation} \begin{equation} \frac{d[mRNA_{RhlI}]}{dt} = leak_{P_{lux}} + \frac{\kappa_{Lux}[C12]^{n_{Lux}}}{K_{mLux}^{n_{Lux}} + [C12]^{n_{Lux}}} - d[mRNA_{RhlI}] - F_{DiMazF}f[mRNA_{RhlI}][DiMazF] \end{equation} \begin{equation} \frac{d[RFP]}{dt} = \alpha [mRNA_{RFP}] - d_{RFP}[RFP] \end{equation} \begin{equation} \frac{d[RhlI]}{dt} = \alpha [mRNA_{RhlI}] - d_{RhlI}[RhlI] \end{equation} \begin{equation} \frac{d[C4]}{dt} = p_{C4}[RhlI]P_{Snow White} - d_{C4}[C4] \end{equation} \begin{equation} \frac{d[mRNA_{MazF}]}{dt} = leak_{P_{lux}} + \frac{\kappa_{Lux}[C12]^{n_{Lux}}}{K_{mLux}^{n_{Lux}}+ [C12]^{n_{Lux}}} \\ - d[mRNA_{MazF}] - F_{DiMazF}(1-(1-f)^{f_{mRNA_{MazF}}})[mRNA_{MazF}][DiMazF] \end{equation} \begin{equation} \frac{d[mRNA_{MazE}]}{dt} = k - d[mRNA_{MazE}] - F_{DiMazF}(1-(1-f)^{f_{mRNA_{MazE}}})[mRNA_{MazE}][DiMazF] \end{equation} \begin{equation} \frac{d[MazF]}{dt} = \alpha [mRNA_{MazF}] - 2k_{Di_{MazF}}[MazF] + 2k_{-Di_{MazF}}[DiMazF] - d_{MazF}[MazF] \end{equation} \begin{equation} \frac{d[DiMazF]}{dt} = k_{Di_{MazF}}[MazF] - k_{-Di_{MazF}}[DiMazF] - 2k_{Hexa}[DiMazE][DiMazF]^2 \\ + 2k_{-Hexa}[MazHexamer] - d_{DiMazF}[DiMazF] \end{equation} \begin{equation} \frac{d[MazE]}{dt} = \alpha [mRNA_{MazE}] - 2k_{Di_{MazE}}[MazE] + 2k_{-Di_{MazE}}[DiMazE] - d_{MazE}[MazE] \end{equation} \begin{equation} \frac{d[DiMazE]}{dt} = k_{Di_{MazE}}[MazE] - k_{-Di_{MazE}}[DiMazE] - k_{Hexa}[DiMazE][DiMazF]^2 \\ + k_{-Hexa}[MazHexamer] - d_{DiMazE}[DiMazE] \end{equation} \begin{equation} \frac{d[MazHexa]}{dt} = k_{Hexa}[DiMazE][DiMazF]^2 - k_{-Hexa}[MazHexa] - d_{Hexa}[MazHexa] \end{equation} \begin{equation} \frac{dP_{Snow White}}{dt} = g \frac{E_{DiMazF}}{E_{DiMazF}+[DiMazF]}\left(1- \frac{P_{Snow White}+P_{Queen}+P_{Prince}}{P_{max}} \right) P_{Snow White} \end{equation}Queen
\begin{equation} \frac{d[mRNA_{GFP}]}{dt} = k - d[mRNA_{GFP}] - F_{DiMazF}(1-(1-f)^{f_{mRNA_{GFP}}})[mRNA_{GFP}][DiMazF] \end{equation} \begin{equation} \frac{d[mRNA_{LasI}]}{dt} = leak_{P_{rhl}} + \frac{\kappa_{Rhl}[C4]^{n_{Rhl}}}{K_{mRhl}^{n_{Rhl}} + [C4]^{n_{Rhl}}} \\ - d[mRNA_{LasI}] - F_{DiMazF}(1-(1-f)^{f_{mRNA_{LasI}}})[mRNA_{LasI}][DiMazF] \end{equation} \begin{equation} \frac{d[GFP]}{dt} = \alpha [mRNA_{GFP}] - d_{GFP}[GFP] \end{equation} \begin{equation} \frac{d[LasI]}{dt} = \alpha [mRNA_{LasI}] - d_{LasI}[LasI] \end{equation} \begin{equation} \frac{d[C12]}{dt} = p_{C12}[LasI]P_{Queen} - d_{C12}[C12] - D[C12][AmiE] \end{equation} \begin{equation} \frac{d[mRNA_{MazF}]}{dt} = leak_{P_{lux}} + \frac{\kappa_{Rhl}[C4]^{n_{Rhl}}}{K_{mRhl}^{n_{Rhl}} + [C4]^{n_{Rhl}}} \\ - d[mRNA_{MazF}] - F_{DiMazF}(1-(1-f)^{f_{mRNA_{MazF}}})[mRNA_{MazF}][DiMazF] \end{equation} \begin{equation} \frac{d[mRNA_{MazE}]}{dt} = k - d[mRNA_{MazE}] - F_{DiMazF}(1-(1-f)^{f_{mRNA_{MazE}}})[mRNA_{MazE}][DiMazF] \end{equation} \begin{equation} \frac{d[MazF]}{dt} = \alpha [mRNA_{MazF}] - 2k_{Di_{MazF}}[MazF] + 2k_{-Di_{MazF}}[DiMazF] - d_{MazF}[MazF] \end{equation} \begin{equation} \frac{d[DiMazF]}{dt} = k_{Di_{MazF}}[MazF] - k_{-Di_{MazF}}[DiMazF] - 2k_{Hexa}[DiMazE][DiMazF]^2 \\ + 2k_{-Hexa}[MazHexamer] - d_{DiMazF}[DiMazF] \end{equation} \begin{equation} \frac{d[MazE]}{dt} = \alpha [mRNA_{MazE}] - 2k_{Di_{MazE}}[MazE] + 2k_{-Di_{MazE}}[DiMazE] - d_{MazE}[MazE] \end{equation} \begin{equation} \frac{d[DiMazE]}{dt} = k_{Di_{MazE}}[MazE] - k_{-Di_{MazE}}[DiMazE] - k_{Hexa}[DiMazE][DiMazF]^2 \\ + k_{-Hexa}[MazHexamer] - d_{DiMazE}[DiMazE] \end{equation} \begin{equation} \frac{d[MazHexa]}{dt} = k_{Hexa}[DiMazE][DiMazF]^2 - k_{-Hexa}[MazHexa] - d_{Hexa}[MazHexa] \end{equation} \begin{equation} \frac{dP_{Queen}}{dt} = g \frac{E_{DiMazF}}{E_{DiMazF}+[DiMazF]}\left(1- \frac{P_{Snow White}+P_{Queen}+P_{Prince}}{P_{max}}\right) P_{Queen}\\ \end{equation}Prince
\begin{equation} \frac{d[mRNA_{AmiE}]}{dt} = leak_{P_{lux}} + \frac{\kappa_{Lux}[C12]^n}{K_{mLux}^n + [C12]^n} - d[mRNA_{AmiE}] \end{equation} \begin{equation} \frac{d[AmiE]}{dt} = \alpha [mRNA_{AmiE}]P_{Prince} - d_{AmiE}[AmiE] \end{equation} \begin{equation} \frac{dP_{Prince}}{dt} = g\left(1- \frac{P_{Snow White}+P_{Queen}+P_{Prince}}{P_{max}}\right) P_{Prince} \end{equation}Explanation about Parameters
| Parameter | Description |
| $$g$$ | Growth rate of each cells |
| $$P_{max}$$ | Carrying capacity |
| $$E_{DiMazF}$$ | Effect of MazF dimer on growth rate |
| $$k$$ | Transcription rate of mRNA under \(P_{tet}\) |
| $$leak_{P_{lux}}$$ | Leakage of \(P_{lux}\) |
| $$leak_{P_{rhl}}$$ | Leakage of \(P_{rhl}\) |
| $$\kappa_{Lux}$$ | Maximum transcription rate of mRNA under \(P_{lux}\) |
| $$\kappa_{Rhl}$$ | Maximum transcription rate of mRNA under \(P_{rhl}\) |
| $$n_{Lux}$$ | Hill coefficient for \(P_{lux}\) |
| $$n_{Rhl}$$ | Hill coefficient for \(P_{rhl}\) |
| $$K_{mLux}$$ | Lumped paremeter for the Lux System |
| $$K_{mRhl}$$ | Lumped paremeter for the Rhl System |
| $$F_{DiMazF}$$ | Cutting rate at ACA sequences on mRNA by MazF dimer |
| $$f$$ | The probability of distinction of ACA sequencess in each mRNA |
| $$f_{mRNA_{RFP}}$$ | The number of ACA sequences in \(mRNA_{RFP}\) |
| $$f_{mRNA_{GFP}}$$ | The number of ACA sequences in \(mRNA_{GFP}\) |
| $$f_{mRNA_{RhlI}}$$ | The number of ACA sequences in \(mRNA_{RhlI}\) |
| $$f_{mRNA_{LasI}}$$ | The number of ACA sequences in \(mRNA_{LasI}\) |
| $$f_{mRNA_{MazF}}$$ | The number of ACA sequences in \(mRNA_{MazF}\) |
| $$f_{mRNA_{MazE}}$$ | The number of ACA sequences in \(mRNA_{MazE}\) |
| $$\alpha$$ | Translation rate of Protein |
| $$k_{Di_{MazF}}$$ | Formation rate of MazF dimer |
| $$k_{-Di_{MazF}}$$ | Dissociation rate of MazF dimer |
| $$k_{Di_{MazE}}$$ | Formation rate of MazE dimer |
| $$k_{-Di_{MazE}}$$ | Dissociation rate of MazE dimer |
| $$k_{Hexa}$$ | Formation rate of Maz hexamer |
| $$k_{-Hexa}$$ | Dissociation rate of Maz hexamer |
| $$p_{C4}$$ | Production rate of C4HSL by RhlI |
| $$p_{C12}$$ | Production rate of 3OC12HSL by LuxI |
| $$D$$ | Decomposition rate of 3OC12HSL by AmiE |
| $$d$$ | Degradation rate of mRNA |
| $$d_{RFP}$$ | Degradation rate of RFP |
| $$d_{GFP}$$ | Degradation rate of GFP |
| $$d_{RhlI}$$ | Degradation rate of RhlI |
| $$d_{LasI}$$ | Degradation rate of LasI |
| $$d_{MazF}$$ | Degradation rate of MazF |
| $$d_{DiMazF}$$ | Degradation rate of MazF dimer |
| $$d_{MazE}$$ | Degradation rate of MazE |
| $$d_{DiMazE}$$ | Degradation rate of MazE dimer |
| $$d_{Hexa}$$ | Degradation rate of Maz Hexamer |
| $$d_{C4}$$ | Degradation rate of C4HSL |
| $$d_{C12}$$ | Degradation rate of 3OC12HSL |
| $$d_{AmiE}$$ | Degradation rate of AmiE |
Analysis
Lastly we got and verified the desirable behavior of the fluorescence by modifying and improving them. As described below, it is consistent with the development of story.
Software
ACA Sequence:
