Line 155: | Line 155: | ||
\[[๐๐๐๐ก๐๐๐_{gfp}]\rightarrow \emptyset\] | \[[๐๐๐๐ก๐๐๐_{gfp}]\rightarrow \emptyset\] | ||
</div> | </div> | ||
โ | + | <br />Two equations, describing the functional binding and cleavage of Csy4 protein in biology, and three parameters,๐<sub>on</sub> ,๐<sub>off</sub> ,๐<sub>obs</sub> , describing the same things in mathematics, are the core of our model. | |
</p> | </p> | ||
Line 210: | Line 210: | ||
\[\frac{d[G]}{dt}=๐_{p1}[E]-๐_{dp1}[G]\] | \[\frac{d[G]}{dt}=๐_{p1}[E]-๐_{dp1}[G]\] | ||
โ | </div> | + | </div> |
โ | + | ||
โ | + | ||
โ | + | ||
โ | + | ||
โ | + | ||
โ | + | ||
โ | + | ||
โ | + | ||
โ | + | ||
</p> | </p> | ||
+ | <p> | ||
+ | <h3>Data Processing</h3> | ||
+ | <br /> The leak in the experiment is an big problem in estimating parameters in our ODEs model, so we processing the data by following formula๏ผ | ||
+ | <div align="center"> <i>Data(without leak)=Data(+IPTG)-Data(-IPTG) </i> | ||
+ | </div> | ||
+ | |||
+ | Data(without leak)=Data(+IPTG)-Data(-IPTG) | ||
+ | |||
+ | |||
+ | |||
+ | <div> | ||
+ | \[\frac{d[๐๐๐
๐๐ด โ ๐
๐ต๐ โ ๐๐
๐๐ด_{๐๐๐}]}{dt}=๐_{1}-๐_{d1}[๐๐๐
๐๐ด โ ๐
๐ต๐ โ ๐๐
๐๐ด_{๐๐๐}]\] | ||
+ | \[-๐_{on}[Protein_{๐ถ๐ ๐ฆ4}][๐๐๐
๐๐ด โ ๐
๐ต๐ โ ๐๐
๐๐ด_{๐๐๐}]\] | ||
+ | \[+๐_{off}[Protein_{๐ถ๐ ๐ฆ4}-๐๐๐
๐๐ด โ ๐
๐ต๐ โ ๐๐
๐๐ด_{๐๐๐}]\] | ||
+ | |||
+ | |||
+ | |||
+ | \[\frac{d[๐๐
๐๐ด_{๐ถ๐ ๐ฆ4}]}{dt}=๐_{2}-๐_{d2}[๐๐
๐๐ด_{๐ถ๐ ๐ฆ4}]\] | ||
+ | |||
+ | |||
+ | \[\frac{d[Protein_{๐ถ๐ ๐ฆ4}]}{dt}=๐_{p2}[๐๐
๐๐ด_{๐ถ๐ ๐ฆ4}]-๐_{dp2}[Protein_{๐ถ๐ ๐ฆ4}]\] | ||
+ | \[-๐_{on}[Protein_{๐ถ๐ ๐ฆ4}][๐๐๐
๐๐ด โ ๐
๐ต๐ โ ๐๐
๐๐ด_{๐๐๐}]\] | ||
+ | \[+๐_{off}[Protein_{๐ถ๐ ๐ฆ4}-๐๐๐
๐๐ด โ ๐
๐ต๐ โ ๐๐
๐๐ด_{๐๐๐}]\] | ||
+ | |||
+ | |||
+ | \[\frac{d[๐๐๐๐ก๐๐๐_{๐ถ๐ ๐ฆ4}-crRNA-RBS-๐๐
๐๐ด_{๐๐๐}]}{dt}=๐_{on}[Protein_{๐ถ๐ ๐ฆ4}][๐๐๐
๐๐ด โ ๐
๐ต๐ โ ๐๐
๐๐ด_{๐๐๐}]\] | ||
+ | \[-๐_{of}[Protein_{๐ถ๐ ๐ฆ4}][Protein_{๐ถ๐ ๐ฆ4}-crRNA-RBS-mRNA_{gfp}]\] | ||
+ | \[-๐_{d1}[Protein_{๐ถ๐ ๐ฆ4}][Protein_{๐ถ๐ ๐ฆ4}-crRNA-RBS-mRNA_{gfp}]\] | ||
+ | \[-๐_{obs}[Protein_{๐ถ๐ ๐ฆ4}][Protein_{๐ถ๐ ๐ฆ4}-crRNA-RBS-mRNA_{gfp}]\] | ||
+ | |||
+ | |||
+ | \[\frac{d[๐๐
๐๐ด_{๐๐๐}]}{dt}=๐_{obs}[๐๐๐๐ก๐๐๐_{๐ถ๐ ๐ฆ4}-crRNA-RBS-๐๐
๐๐ด_{๐๐๐}]-๐_{d3}[๐๐
๐๐ด_{gfp}]\] | ||
+ | |||
+ | |||
+ | \[\frac{d[๐๐๐๐ก๐๐๐_{๐ถ๐ ๐ฆ4}-crRNA]}{dt}=๐_{obs}[๐๐๐๐ก๐๐๐_{๐ถ๐ ๐ฆ4}-crRNA-RBS-๐๐
๐๐ด_{๐๐๐}]-๐_{dc2}[๐๐๐๐ก๐๐๐_{๐ถ๐ ๐ฆ4}-๐๐๐
๐๐ด]\] | ||
+ | \[-๐_{on}[Protein_{๐ถ๐ ๐ฆ4}][๐๐๐
๐๐ด โ ๐
๐ต๐ โ ๐๐
๐๐ด_{๐๐๐}]\] | ||
+ | \[+๐_{off}[Protein_{๐ถ๐ ๐ฆ4}-๐๐๐
๐๐ด โ ๐
๐ต๐ โ ๐๐
๐๐ด_{๐๐๐}]\] | ||
+ | |||
+ | |||
+ | \[\frac{d[Protein_{gfp}]}{dt}=๐_{p1}[๐๐
๐๐ด_{๐๐๐}]-๐_{dp1}[Protein_{gfp}]\] | ||
+ | </div> | ||
+ | <br />For the readability, the complex symbol is simplified as: | ||
+ | |||
+ | <div> \[\frac{d[A]}{dt}=๐_{1}-๐_{d1}[A]-๐_{on}[C][A]+๐_{off}[D]\] | ||
+ | |||
+ | \[\frac{d[B]}{dt}=๐_{2}-๐_{d2}[B]\] | ||
+ | |||
+ | \[\frac{d[C]}{dt}=๐_{p2}[B]-๐_{dp2}[C]-๐_{on}[C][A]+๐_{off}[D]\] | ||
+ | |||
+ | \[\frac{d[D]}{dt}=๐_{on}[C][A]-๐_{dp2}[C]-๐_{off}[D]-๐_{dc1}[D]-๐_{obs}[D]\] | ||
+ | |||
+ | \[\frac{d[E]}{dt}=๐_{obs}[D]-๐_{d3}[E]\] | ||
+ | |||
+ | \[\frac{d[F]}{dt}=๐_{obs}[D]-๐_{d3}[F]\] | ||
+ | |||
+ | \[\frac{d[G]}{dt}=๐_{p1}[E]-๐_{dp1}[G]\] | ||
+ | </div> | ||
+ | </p> | ||
Revision as of 23:40, 10 October 2018
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,๐on ,๐off ,๐obs , describing the same things in mathematics, are the core of our model.
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:
\[\frac{d[A]}{dt}=๐_{1}-๐_{d1}[A]-๐_{on}[C][A]+๐_{off}[D]\]
\[\frac{d[B]}{dt}=๐_{2}-๐_{d2}[B]\]
\[\frac{d[C]}{dt}=๐_{p2}[B]-๐_{dp2}[C]-๐_{on}[C][A]+๐_{off}[D]\]
\[\frac{d[D]}{dt}=๐_{on}[C][A]-๐_{dp2}[C]-๐_{off}[D]-๐_{dc1}[D]-๐_{obs}[D]\]
\[\frac{d[E]}{dt}=๐_{obs}[D]-๐_{d3}[E]\]
\[\frac{d[F]}{dt}=๐_{obs}[D]-๐_{d3}[F]\]
\[\frac{d[G]}{dt}=๐_{p1}[E]-๐_{dp1}[G]\]
Data Processing
The leak in the experiment is an big problem in estimating parameters in our ODEs model, so we processing the data by following formula๏ผ
Data(without leak)=Data(+IPTG)-Data(-IPTG)
Data(without leak)=Data(+IPTG)-Data(-IPTG)
\[\frac{d[๐๐๐
๐๐ด โ ๐
๐ต๐ โ ๐๐
๐๐ด_{๐๐๐}]}{dt}=๐_{1}-๐_{d1}[๐๐๐
๐๐ด โ ๐
๐ต๐ โ ๐๐
๐๐ด_{๐๐๐}]\]
\[-๐_{on}[Protein_{๐ถ๐ ๐ฆ4}][๐๐๐
๐๐ด โ ๐
๐ต๐ โ ๐๐
๐๐ด_{๐๐๐}]\]
\[+๐_{off}[Protein_{๐ถ๐ ๐ฆ4}-๐๐๐
๐๐ด โ ๐
๐ต๐ โ ๐๐
๐๐ด_{๐๐๐}]\]
\[\frac{d[๐๐
๐๐ด_{๐ถ๐ ๐ฆ4}]}{dt}=๐_{2}-๐_{d2}[๐๐
๐๐ด_{๐ถ๐ ๐ฆ4}]\]
\[\frac{d[Protein_{๐ถ๐ ๐ฆ4}]}{dt}=๐_{p2}[๐๐
๐๐ด_{๐ถ๐ ๐ฆ4}]-๐_{dp2}[Protein_{๐ถ๐ ๐ฆ4}]\]
\[-๐_{on}[Protein_{๐ถ๐ ๐ฆ4}][๐๐๐
๐๐ด โ ๐
๐ต๐ โ ๐๐
๐๐ด_{๐๐๐}]\]
\[+๐_{off}[Protein_{๐ถ๐ ๐ฆ4}-๐๐๐
๐๐ด โ ๐
๐ต๐ โ ๐๐
๐๐ด_{๐๐๐}]\]
\[\frac{d[๐๐๐๐ก๐๐๐_{๐ถ๐ ๐ฆ4}-crRNA-RBS-๐๐
๐๐ด_{๐๐๐}]}{dt}=๐_{on}[Protein_{๐ถ๐ ๐ฆ4}][๐๐๐
๐๐ด โ ๐
๐ต๐ โ ๐๐
๐๐ด_{๐๐๐}]\]
\[-๐_{of}[Protein_{๐ถ๐ ๐ฆ4}][Protein_{๐ถ๐ ๐ฆ4}-crRNA-RBS-mRNA_{gfp}]\]
\[-๐_{d1}[Protein_{๐ถ๐ ๐ฆ4}][Protein_{๐ถ๐ ๐ฆ4}-crRNA-RBS-mRNA_{gfp}]\]
\[-๐_{obs}[Protein_{๐ถ๐ ๐ฆ4}][Protein_{๐ถ๐ ๐ฆ4}-crRNA-RBS-mRNA_{gfp}]\]
\[\frac{d[๐๐
๐๐ด_{๐๐๐}]}{dt}=๐_{obs}[๐๐๐๐ก๐๐๐_{๐ถ๐ ๐ฆ4}-crRNA-RBS-๐๐
๐๐ด_{๐๐๐}]-๐_{d3}[๐๐
๐๐ด_{gfp}]\]
\[\frac{d[๐๐๐๐ก๐๐๐_{๐ถ๐ ๐ฆ4}-crRNA]}{dt}=๐_{obs}[๐๐๐๐ก๐๐๐_{๐ถ๐ ๐ฆ4}-crRNA-RBS-๐๐
๐๐ด_{๐๐๐}]-๐_{dc2}[๐๐๐๐ก๐๐๐_{๐ถ๐ ๐ฆ4}-๐๐๐
๐๐ด]\]
\[-๐_{on}[Protein_{๐ถ๐ ๐ฆ4}][๐๐๐
๐๐ด โ ๐
๐ต๐ โ ๐๐
๐๐ด_{๐๐๐}]\]
\[+๐_{off}[Protein_{๐ถ๐ ๐ฆ4}-๐๐๐
๐๐ด โ ๐
๐ต๐ โ ๐๐
๐๐ด_{๐๐๐}]\]
\[\frac{d[Protein_{gfp}]}{dt}=๐_{p1}[๐๐
๐๐ด_{๐๐๐}]-๐_{dp1}[Protein_{gfp}]\]
For the readability, the complex symbol is simplified as:
\[\frac{d[A]}{dt}=๐_{1}-๐_{d1}[A]-๐_{on}[C][A]+๐_{off}[D]\]
\[\frac{d[B]}{dt}=๐_{2}-๐_{d2}[B]\]
\[\frac{d[C]}{dt}=๐_{p2}[B]-๐_{dp2}[C]-๐_{on}[C][A]+๐_{off}[D]\]
\[\frac{d[D]}{dt}=๐_{on}[C][A]-๐_{dp2}[C]-๐_{off}[D]-๐_{dc1}[D]-๐_{obs}[D]\]
\[\frac{d[E]}{dt}=๐_{obs}[D]-๐_{d3}[E]\]
\[\frac{d[F]}{dt}=๐_{obs}[D]-๐_{d3}[F]\]
\[\frac{d[G]}{dt}=๐_{p1}[E]-๐_{dp1}[G]\]
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