![banner](https://static.igem.org/mediawiki/2018/8/8a/T--OUC-China--model-minitoe.jpg)
Design
Reactions
We can describe our miniToe system to be followings:
\[\rightarrow [๐๐๐
๐๐ด โ ๐
๐ต๐ โ ๐๐
๐๐ด_{๐๐๐}]\]
\[\rightarrow [๐๐
๐๐ด_{Csy4}]\]
\[[๐๐
๐๐ด_{๐ถ๐ ๐ฆ4}]\rightarrow [๐๐
๐๐ด_{๐ถ๐ ๐ฆ4}] + [๐๐๐๐ก๐๐๐_{๐ถ๐ ๐ฆ4}]\]
\[[Protein_{๐ถ๐ ๐ฆ4}]+[crRNA-RBS-mRNA_{gfp}]\leftrightarrow [๐๐๐๐ก๐๐๐_{๐ถ๐ ๐ฆ4}-crRNA-RBS-๐๐
๐๐ด_{๐๐๐}]\]
\[[Protein_{๐ถ๐ ๐ฆ4}-crRNA-RBS-mRNA_{gfp}]\rightarrow [๐๐
๐๐ด_{gfp}] + [๐๐๐๐ก๐๐๐_{๐ถ๐ ๐ฆ4}-๐๐๐
๐๐ด]\]
\[[๐๐
๐๐ด_{gfp}]\rightarrow [๐๐
๐๐ด_{gfp}] + [๐๐๐๐ก๐๐๐_{gfp}]\]
\[[crRNA-RBS-mRNA_{gfp}]\rightarrow \emptyset\]
\[[๐๐
๐๐ด_{๐ถ๐ ๐ฆ4}]\rightarrow \emptyset\]
\[[๐๐๐๐ก๐๐๐_{๐ถ๐ ๐ฆ4}]\rightarrow \emptyset\]
\[[Protein_{๐ถ๐ ๐ฆ4}-crRNA-RBS-mRNA_{gfp}]\rightarrow \emptyset\]
\[[๐๐
๐๐ด_{gfp}]\rightarrow \emptyset\]
\[[๐๐๐๐ก๐๐๐_{๐ถ๐ ๐ฆ4}-๐๐๐
๐๐ด]\rightarrow \emptyset\]
\[[๐๐๐๐ก๐๐๐_{gfp}]\rightarrow \emptyset\]
Two equations, describing the functional binding and cleavage of Csy4 protein in biology, and three parameters,๐_{on} ,\[๐_{off}\] ,\[๐_{obs}\] , describing the same things in mathematics, are the core of our model.
![](https://static.igem.org/mediawiki/2018/9/90/T--OUC-China--design1-1.png)
![](https://static.igem.org/mediawiki/2018/1/16/T--OUC-China--design1-2.png)
![](https://static.igem.org/mediawiki/2018/3/3c/T--OUC-China--design1-3.png)
Fig.1-1 The structure of Csy4 Fig.1-2 The structure 22nt hairpin which can be recognized by Csy4. Fig.1-3 The Csy4/RNA complex.
ODEs
To simulate the dynamics of GFP, we use ordinary differential equations to model the reactions above. And ODEs are given as follows:
\[\frac{d[๐๐๐
๐๐ด โ ๐
๐ต๐ โ ๐๐
๐๐ด_{๐๐๐}]}{dt}=๐_{1}-๐_{d1}[๐๐๐
๐๐ด โ ๐
๐ต๐ โ ๐๐
๐๐ด_{๐๐๐}]\]
\[-๐_{on}[Protein_{๐ถ๐ ๐ฆ4}][๐๐๐
๐๐ด โ ๐
๐ต๐ โ ๐๐
๐๐ด_{๐๐๐}]\]
\[+๐_{off}[Protein_{๐ถ๐ ๐ฆ4}-๐๐๐
๐๐ด โ ๐
๐ต๐ โ ๐๐
๐๐ด_{๐๐๐}]\]
\[\frac{d[๐๐
๐๐ด_{๐ถ๐ ๐ฆ4}]}{dt}=๐_{2}-๐_{d2}[๐๐
๐๐ด_{๐ถ๐ ๐ฆ4}]\]
\[\frac{d[Protein_{๐ถ๐ ๐ฆ4}]}{dt}=๐_{p2}[๐๐
๐๐ด_{๐ถ๐ ๐ฆ4}]-๐_{dp2}[Protein_{๐ถ๐ ๐ฆ4}]\]
\[-๐_{on}[Protein_{๐ถ๐ ๐ฆ4}][๐๐๐
๐๐ด โ ๐
๐ต๐ โ ๐๐
๐๐ด_{๐๐๐}]\]
\[+๐_{off}[Protein_{๐ถ๐ ๐ฆ4}-๐๐๐
๐๐ด โ ๐
๐ต๐ โ ๐๐
๐๐ด_{๐๐๐}]\]
\[\frac{d[๐๐๐๐ก๐๐๐_{๐ถ๐ ๐ฆ4}-crRNA-RBS-๐๐
๐๐ด_{๐๐๐}]}{dt}=๐_{on}[Protein_{๐ถ๐ ๐ฆ4}][๐๐๐
๐๐ด โ ๐
๐ต๐ โ ๐๐
๐๐ด_{๐๐๐}]\]
\[-๐_{of}[Protein_{๐ถ๐ ๐ฆ4}][Protein_{๐ถ๐ ๐ฆ4}-crRNA-RBS-mRNA_{gfp}]\]
\[-๐_{d1}[Protein_{๐ถ๐ ๐ฆ4}][Protein_{๐ถ๐ ๐ฆ4}-crRNA-RBS-mRNA_{gfp}]\]
\[-๐_{obs}[Protein_{๐ถ๐ ๐ฆ4}][Protein_{๐ถ๐ ๐ฆ4}-crRNA-RBS-mRNA_{gfp}]\]
\[\frac{d[๐๐
๐๐ด_{๐๐๐}]}{dt}=๐_{obs}[๐๐๐๐ก๐๐๐_{๐ถ๐ ๐ฆ4}-crRNA-RBS-๐๐
๐๐ด_{๐๐๐}]-๐_{d3}[๐๐
๐๐ด_{gfp}]\]
\[\frac{d[๐๐๐๐ก๐๐๐_{๐ถ๐ ๐ฆ4}-crRNA]}{dt}=๐_{obs}[๐๐๐๐ก๐๐๐_{๐ถ๐ ๐ฆ4}-crRNA-RBS-๐๐
๐๐ด_{๐๐๐}]-๐_{dc2}[๐๐๐๐ก๐๐๐_{๐ถ๐ ๐ฆ4}-๐๐๐
๐๐ด]\]
\[-๐_{on}[Protein_{๐ถ๐ ๐ฆ4}][๐๐๐
๐๐ด โ ๐
๐ต๐ โ ๐๐
๐๐ด_{๐๐๐}]\]
\[+๐_{off}[Protein_{๐ถ๐ ๐ฆ4}-๐๐๐
๐๐ด โ ๐
๐ต๐ โ ๐๐
๐๐ด_{๐๐๐}]\]
\[\frac{d[Protein_{gfp}]}{dt}=๐_{p1}[๐๐
๐๐ด_{๐๐๐}]-๐_{dp1}[Protein_{gfp}]\]
For the readability, the complex symbol is simplified as:
\[\frac{d[A]}{dt}=๐_{1}-๐_{d1}[A]-๐_{on}[C][A]+๐_{off}[D]\]
\[\frac{d[B]}{dt}=๐_{2}-๐_{d2}[B]\]
\[\frac{d[C]}{dt}=๐_{p2}[B]-๐_{dp2}[C]-๐_{on}[C][A]+๐_{off}[D]\]
\[\frac{d[D]}{dt}=๐_{on}[C][A]-๐_{dp2}[C]-๐_{off}[D]-๐_{dc1}[D]-๐_{obs}[D]\]
\[\frac{d[E]}{dt}=๐_{obs}[D]-๐_{d3}[E]\]
\[\frac{d[F]}{dt}=๐_{obs}[D]-๐_{d3}[F]\]
\[\frac{d[G]}{dt}=๐_{p1}[E]-๐_{dp1}[G]\]
![](https://static.igem.org/mediawiki/2018/9/90/T--OUC-China--design1-1.png)
![](https://static.igem.org/mediawiki/2018/1/16/T--OUC-China--design1-2.png)
![](https://static.igem.org/mediawiki/2018/3/3c/T--OUC-China--design1-3.png)
Fig.1-1 The structure of Csy4 Fig.1-2 The structure 22nt hairpin which can be recognized by Csy4. Fig.1-3 The Csy4/RNA complex.
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