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                     \[[๐‘ƒ๐‘Ÿ๐‘œ๐‘ก๐‘’๐‘–๐‘›_{gfp}]\rightarrow \emptyset\]
 
                     \[[๐‘ƒ๐‘Ÿ๐‘œ๐‘ก๐‘’๐‘–๐‘›_{gfp}]\rightarrow \emptyset\]
 
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<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.
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<br />Two equations, describing the functional binding and cleavage of Csy4 protein in biology, and three parameters k<sub>on</sub>, k<sub>off</sub>, k<sub>obs</sub>describing the same things in mathematics, are the core of our model.
 
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Revision as of 03:15, 11 October 2018

Team OUC-China: Main

Design

Reactions


We can describe our miniToe system to be followings:
\[\rightarrow [๐‘๐‘Ÿ๐‘…๐‘๐ด โˆ’ ๐‘…๐ต๐‘† โˆ’ ๐‘š๐‘…๐‘๐ด_{๐‘”๐‘“๐‘}]\] \[\rightarrow [๐‘š๐‘…๐‘๐ด_{Csy4}]\] \[[๐‘š๐‘…๐‘๐ด_{๐ถ๐‘ ๐‘ฆ4}]\rightarrow [๐‘š๐‘…๐‘๐ด_{๐ถ๐‘ ๐‘ฆ4}] + [๐‘ƒ๐‘Ÿ๐‘œ๐‘ก๐‘’๐‘–๐‘›_{๐ถ๐‘ ๐‘ฆ4}]\] \[[Protein_{๐ถ๐‘ ๐‘ฆ4}]+[crRNA-RBS-mRNA_{gfp}]\leftrightarrow [๐‘ƒ๐‘Ÿ๐‘œ๐‘ก๐‘’๐‘–๐‘›_{๐ถ๐‘ ๐‘ฆ4}-crRNA-RBS-๐‘š๐‘…๐‘๐ด_{๐‘”๐‘“๐‘}]\] \[[Protein_{๐ถ๐‘ ๐‘ฆ4}-crRNA-RBS-mRNA_{gfp}]\rightarrow [๐‘š๐‘…๐‘๐ด_{gfp}] + [๐‘ƒ๐‘Ÿ๐‘œ๐‘ก๐‘’๐‘–๐‘›_{๐ถ๐‘ ๐‘ฆ4}-๐‘๐‘Ÿ๐‘…๐‘๐ด]\] \[[๐‘š๐‘…๐‘๐ด_{gfp}]\rightarrow [๐‘š๐‘…๐‘๐ด_{gfp}] + [๐‘ƒ๐‘Ÿ๐‘œ๐‘ก๐‘’๐‘–๐‘›_{gfp}]\] \[[crRNA-RBS-mRNA_{gfp}]\rightarrow \emptyset\] \[[๐‘š๐‘…๐‘๐ด_{๐ถ๐‘ ๐‘ฆ4}]\rightarrow \emptyset\] \[[๐‘ƒ๐‘Ÿ๐‘œ๐‘ก๐‘’๐‘–๐‘›_{๐ถ๐‘ ๐‘ฆ4}]\rightarrow \emptyset\] \[[Protein_{๐ถ๐‘ ๐‘ฆ4}-crRNA-RBS-mRNA_{gfp}]\rightarrow \emptyset\] \[[๐‘š๐‘…๐‘๐ด_{gfp}]\rightarrow \emptyset\] \[[๐‘ƒ๐‘Ÿ๐‘œ๐‘ก๐‘’๐‘–๐‘›_{๐ถ๐‘ ๐‘ฆ4}-๐‘๐‘Ÿ๐‘…๐‘๐ด]\rightarrow \emptyset\] \[[๐‘ƒ๐‘Ÿ๐‘œ๐‘ก๐‘’๐‘–๐‘›_{gfp}]\rightarrow \emptyset\]

Two equations, describing the functional binding and cleavage of Csy4 protein in biology, and three parameters kon, koff, kobsdescribing 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:
\[\frac{d[๐‘๐‘Ÿ๐‘…๐‘๐ด โˆ’ ๐‘…๐ต๐‘† โˆ’ ๐‘š๐‘…๐‘๐ด_{๐‘”๐‘“๐‘}]}{dt}=๐‘˜_{1}-๐‘˜_{d1}[๐‘๐‘Ÿ๐‘…๐‘๐ด โˆ’ ๐‘…๐ต๐‘† โˆ’ ๐‘š๐‘…๐‘๐ด_{๐‘”๐‘“๐‘}]\] \[-๐‘˜_{on}[Protein_{๐ถ๐‘ ๐‘ฆ4}][๐‘๐‘Ÿ๐‘…๐‘๐ด โˆ’ ๐‘…๐ต๐‘† โˆ’ ๐‘š๐‘…๐‘๐ด_{๐‘”๐‘“๐‘}]\] \[+๐‘˜_{off}[Protein_{๐ถ๐‘ ๐‘ฆ4}-๐‘๐‘Ÿ๐‘…๐‘๐ด โˆ’ ๐‘…๐ต๐‘† โˆ’ ๐‘š๐‘…๐‘๐ด_{๐‘”๐‘“๐‘}]\] \[\frac{d[๐‘š๐‘…๐‘๐ด_{๐ถ๐‘ ๐‘ฆ4}]}{dt}=๐‘˜_{2}-๐‘˜_{d2}[๐‘š๐‘…๐‘๐ด_{๐ถ๐‘ ๐‘ฆ4}]\] \[\frac{d[Protein_{๐ถ๐‘ ๐‘ฆ4}]}{dt}=๐‘˜_{p2}[๐‘š๐‘…๐‘๐ด_{๐ถ๐‘ ๐‘ฆ4}]-๐‘˜_{dp2}[Protein_{๐ถ๐‘ ๐‘ฆ4}]\] \[-๐‘˜_{on}[Protein_{๐ถ๐‘ ๐‘ฆ4}][๐‘๐‘Ÿ๐‘…๐‘๐ด โˆ’ ๐‘…๐ต๐‘† โˆ’ ๐‘š๐‘…๐‘๐ด_{๐‘”๐‘“๐‘}]\] \[+๐‘˜_{off}[Protein_{๐ถ๐‘ ๐‘ฆ4}-๐‘๐‘Ÿ๐‘…๐‘๐ด โˆ’ ๐‘…๐ต๐‘† โˆ’ ๐‘š๐‘…๐‘๐ด_{๐‘”๐‘“๐‘}]\] \[\frac{d[๐‘ƒ๐‘Ÿ๐‘œ๐‘ก๐‘’๐‘–๐‘›_{๐ถ๐‘ ๐‘ฆ4}-crRNA-RBS-๐‘š๐‘…๐‘๐ด_{๐‘”๐‘“๐‘}]}{dt}=๐‘˜_{on}[Protein_{๐ถ๐‘ ๐‘ฆ4}][๐‘๐‘Ÿ๐‘…๐‘๐ด โˆ’ ๐‘…๐ต๐‘† โˆ’ ๐‘š๐‘…๐‘๐ด_{๐‘”๐‘“๐‘}]\] \[-๐‘˜_{of}[Protein_{๐ถ๐‘ ๐‘ฆ4}][Protein_{๐ถ๐‘ ๐‘ฆ4}-crRNA-RBS-mRNA_{gfp}]\] \[-๐‘˜_{d1}[Protein_{๐ถ๐‘ ๐‘ฆ4}][Protein_{๐ถ๐‘ ๐‘ฆ4}-crRNA-RBS-mRNA_{gfp}]\] \[-๐‘˜_{obs}[Protein_{๐ถ๐‘ ๐‘ฆ4}][Protein_{๐ถ๐‘ ๐‘ฆ4}-crRNA-RBS-mRNA_{gfp}]\] \[\frac{d[๐‘š๐‘…๐‘๐ด_{๐‘”๐‘“๐‘}]}{dt}=๐‘˜_{obs}[๐‘ƒ๐‘Ÿ๐‘œ๐‘ก๐‘’๐‘–๐‘›_{๐ถ๐‘ ๐‘ฆ4}-crRNA-RBS-๐‘š๐‘…๐‘๐ด_{๐‘”๐‘“๐‘}]-๐‘˜_{d3}[๐‘š๐‘…๐‘๐ด_{gfp}]\] \[\frac{d[๐‘ƒ๐‘Ÿ๐‘œ๐‘ก๐‘’๐‘–๐‘›_{๐ถ๐‘ ๐‘ฆ4}-crRNA]}{dt}=๐‘˜_{obs}[๐‘ƒ๐‘Ÿ๐‘œ๐‘ก๐‘’๐‘–๐‘›_{๐ถ๐‘ ๐‘ฆ4}-crRNA-RBS-๐‘š๐‘…๐‘๐ด_{๐‘”๐‘“๐‘}]-๐‘˜_{dc2}[๐‘ƒ๐‘Ÿ๐‘œ๐‘ก๐‘’๐‘–๐‘›_{๐ถ๐‘ ๐‘ฆ4}-๐‘๐‘Ÿ๐‘…๐‘๐ด]\] \[-๐‘˜_{on}[Protein_{๐ถ๐‘ ๐‘ฆ4}][๐‘๐‘Ÿ๐‘…๐‘๐ด โˆ’ ๐‘…๐ต๐‘† โˆ’ ๐‘š๐‘…๐‘๐ด_{๐‘”๐‘“๐‘}]\] \[+๐‘˜_{off}[Protein_{๐ถ๐‘ ๐‘ฆ4}-๐‘๐‘Ÿ๐‘…๐‘๐ด โˆ’ ๐‘…๐ต๐‘† โˆ’ ๐‘š๐‘…๐‘๐ด_{๐‘”๐‘“๐‘}]\] \[\frac{d[Protein_{gfp}]}{dt}=๐‘˜_{p1}[๐‘š๐‘…๐‘๐ด_{๐‘”๐‘“๐‘}]-๐‘˜_{dp1}[Protein_{gfp}]\]

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› \[Data(without leak)=Data(+IPTG)-Data(-IPTG)\]
By doing this, we can reduce some factor which may be influence estimation not just the leak, but also some background noise. So we can get more precise parameters of the Csy4.

Species, symbols and parameters

Species Symbol Initial value Units
\[๐‘๐‘Ÿ๐‘…๐‘๐ด โˆ’ ๐‘…๐ต๐‘† โˆ’ ๐‘š๐‘…๐‘๐ด_{๐‘”๐‘“๐‘}\] A 15 \[mol/L\]
\[๐‘š๐‘…๐‘๐ด_{Csy4}\] B 0 \[mol/L\]
\[Protein_{๐ถ๐‘ ๐‘ฆ4}\] C 0 \[mol/L\]
\[Protein_{๐ถ๐‘ ๐‘ฆ4}-crRNA-RBS-mRNA_{๐‘”๐‘“๐‘}\] D 0 \[mol/L\]
\[๐‘š๐‘…๐‘๐ด_{gfp}\] E 0 \[mol/L\]
\[Protein_{๐ถ๐‘ ๐‘ฆ4}-๐‘๐‘Ÿ๐‘…๐‘๐ด\] F 0 \[mol/L\]
\[Protein_{gfp}\] G 0 \[mol/L\]

Because the \[๐‘๐‘Ÿ๐‘…๐‘๐ด โˆ’ ๐‘…๐ต๐‘† โˆ’ ๐‘š๐‘…๐‘๐ด_{๐‘”๐‘“๐‘}\] is under controlled by a constitutive promoter, so we set the initial concentration to \[15mol/L\] .
The other parament we used in the ODEs is listed in the following table:
Parameters Definition Units Value
\[๐‘˜_{1}\]      
\[๐‘˜_{d1}\]      
\[๐‘˜_{2}\]      
\[๐‘˜_{d2}\]      
\[๐‘˜_{p2}\]      
\[๐‘˜_{dp2}\]      
\[๐‘˜_{on}\]      
\[๐‘˜_{off}\]      
\[๐‘˜_{dc1}\]      
\[๐‘˜_{obs}\]      
\[๐‘˜_{d3}\]      
\[๐‘˜_{dc2}\]      
\[๐‘˜_{p1}\]      
\[๐‘˜_{dp1}<\]/th>      

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