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Revision as of 22:26, 11 October 2018

Team OUC-China: Main
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Design

Reactions


We can describe our miniToe system to be followings:
\[\rightarrow [𝑐𝑟𝑅𝑁𝐴 − 𝑅𝐵𝑆 − 𝑚𝑅𝑁𝐴_{𝑔𝑓𝑝}]\] \[\rightarrow [𝑚𝑅𝑁𝐴_{Csy4}]\] \[[𝑚𝑅𝑁𝐴_{𝐶𝑠𝑦4}]\rightarrow [𝑚𝑅𝑁𝐴_{𝐶𝑠𝑦4}] + [𝑃𝑟𝑜𝑡𝑒𝑖𝑛_{𝐶𝑠𝑦4}]\] \[[Protein_{𝐶𝑠𝑦4}]+[crRNA-RBS-mRNA_{gfp}]\leftrightarrow [𝑃𝑟𝑜𝑡𝑒𝑖𝑛_{𝐶𝑠𝑦4}-crRNA-RBS-𝑚𝑅𝑁𝐴_{𝑔𝑓𝑝}]\] \[[Protein_{𝐶𝑠𝑦4}-crRNA-RBS-mRNA_{gfp}]\rightarrow [𝑚𝑅𝑁𝐴_{gfp}] + [𝑃𝑟𝑜𝑡𝑒𝑖𝑛_{𝐶𝑠𝑦4}-𝑐𝑟𝑅𝑁𝐴]\] \[[𝑚𝑅𝑁𝐴_{gfp}]\rightarrow [𝑚𝑅𝑁𝐴_{gfp}] + [𝑃𝑟𝑜𝑡𝑒𝑖𝑛_{gfp}]\] \[[crRNA-RBS-mRNA_{gfp}]\rightarrow \emptyset\] \[[𝑚𝑅𝑁𝐴_{𝐶𝑠𝑦4}]\rightarrow \emptyset\] \[[𝑃𝑟𝑜𝑡𝑒𝑖𝑛_{𝐶𝑠𝑦4}]\rightarrow \emptyset\] \[[Protein_{𝐶𝑠𝑦4}-crRNA-RBS-mRNA_{gfp}]\rightarrow \emptyset\] \[[𝑚𝑅𝑁𝐴_{gfp}]\rightarrow \emptyset\] \[[𝑃𝑟𝑜𝑡𝑒𝑖𝑛_{𝐶𝑠𝑦4}-𝑐𝑟𝑅𝑁𝐴]\rightarrow \emptyset\] \[[𝑃𝑟𝑜𝑡𝑒𝑖𝑛_{gfp}]\rightarrow \emptyset\]

Two equations, describing the functional binding and cleavage of Csy4 protein in biology, and three parameters  kon  koff  kobs 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:
\[\frac{d[𝑐𝑟𝑅𝑁𝐴 − 𝑅𝐵𝑆 − 𝑚𝑅𝑁𝐴_{𝑔𝑓𝑝}]}{dt}=𝑘_{1}-𝑘_{d1}[𝑐𝑟𝑅𝑁𝐴 − 𝑅𝐵𝑆 − 𝑚𝑅𝑁𝐴_{𝑔𝑓𝑝}]\] \[-𝑘_{on}[Protein_{𝐶𝑠𝑦4}][𝑐𝑟𝑅𝑁𝐴 − 𝑅𝐵𝑆 − 𝑚𝑅𝑁𝐴_{𝑔𝑓𝑝}]\] \[+𝑘_{off}[Protein_{𝐶𝑠𝑦4}-𝑐𝑟𝑅𝑁𝐴 − 𝑅𝐵𝑆 − 𝑚𝑅𝑁𝐴_{𝑔𝑓𝑝}]\] \[\frac{d[𝑚𝑅𝑁𝐴_{𝐶𝑠𝑦4}]}{dt}=𝑘_{2}-𝑘_{d2}[𝑚𝑅𝑁𝐴_{𝐶𝑠𝑦4}]\] \[\frac{d[Protein_{𝐶𝑠𝑦4}]}{dt}=𝑘_{p2}[𝑚𝑅𝑁𝐴_{𝐶𝑠𝑦4}]-𝑘_{dp2}[Protein_{𝐶𝑠𝑦4}]\] \[-𝑘_{on}[Protein_{𝐶𝑠𝑦4}][𝑐𝑟𝑅𝑁𝐴 − 𝑅𝐵𝑆 − 𝑚𝑅𝑁𝐴_{𝑔𝑓𝑝}]\] \[+𝑘_{off}[Protein_{𝐶𝑠𝑦4}-𝑐𝑟𝑅𝑁𝐴 − 𝑅𝐵𝑆 − 𝑚𝑅𝑁𝐴_{𝑔𝑓𝑝}]\] \[\frac{d[𝑃𝑟𝑜𝑡𝑒𝑖𝑛_{𝐶𝑠𝑦4}-crRNA-RBS-𝑚𝑅𝑁𝐴_{𝑔𝑓𝑝}]}{dt}=𝑘_{on}[Protein_{𝐶𝑠𝑦4}][𝑐𝑟𝑅𝑁𝐴 − 𝑅𝐵𝑆 − 𝑚𝑅𝑁𝐴_{𝑔𝑓𝑝}]\] \[-𝑘_{of}[Protein_{𝐶𝑠𝑦4}][Protein_{𝐶𝑠𝑦4}-crRNA-RBS-mRNA_{gfp}]\] \[-𝑘_{d1}[Protein_{𝐶𝑠𝑦4}][Protein_{𝐶𝑠𝑦4}-crRNA-RBS-mRNA_{gfp}]\] \[-𝑘_{obs}[Protein_{𝐶𝑠𝑦4}][Protein_{𝐶𝑠𝑦4}-crRNA-RBS-mRNA_{gfp}]\] \[\frac{d[𝑚𝑅𝑁𝐴_{𝑔𝑓𝑝}]}{dt}=𝑘_{obs}[𝑃𝑟𝑜𝑡𝑒𝑖𝑛_{𝐶𝑠𝑦4}-crRNA-RBS-𝑚𝑅𝑁𝐴_{𝑔𝑓𝑝}]-𝑘_{d3}[𝑚𝑅𝑁𝐴_{gfp}]\] \[\frac{d[𝑃𝑟𝑜𝑡𝑒𝑖𝑛_{𝐶𝑠𝑦4}-crRNA]}{dt}=𝑘_{obs}[𝑃𝑟𝑜𝑡𝑒𝑖𝑛_{𝐶𝑠𝑦4}-crRNA-RBS-𝑚𝑅𝑁𝐴_{𝑔𝑓𝑝}]-𝑘_{dc2}[𝑃𝑟𝑜𝑡𝑒𝑖𝑛_{𝐶𝑠𝑦4}-𝑐𝑟𝑅𝑁𝐴]\] \[-𝑘_{on}[Protein_{𝐶𝑠𝑦4}][𝑐𝑟𝑅𝑁𝐴 − 𝑅𝐵𝑆 − 𝑚𝑅𝑁𝐴_{𝑔𝑓𝑝}]\] \[+𝑘_{off}[Protein_{𝐶𝑠𝑦4}-𝑐𝑟𝑅𝑁𝐴 − 𝑅𝐵𝑆 − 𝑚𝑅𝑁𝐴_{𝑔𝑓𝑝}]\] \[\frac{d[Protein_{gfp}]}{dt}=𝑘_{p1}[𝑚𝑅𝑁𝐴_{𝑔𝑓𝑝}]-𝑘_{dp1}[Protein_{gfp}]\]

For the readability, the complex symbol is simplified as:
\[\frac{d[A]}{dt}=𝑘_{1}-𝑘_{d1}[A]-𝑘_{on}[C][A]+𝑘_{off}[D]\] \[\frac{d[B]}{dt}=𝑘_{2}-𝑘_{d2}[B]\] \[\frac{d[C]}{dt}=𝑘_{p2}[B]-𝑘_{dp2}[C]-𝑘_{on}[C][A]+𝑘_{off}[D]\] \[\frac{d[D]}{dt}=𝑘_{on}[C][A]-𝑘_{dp2}[C]-𝑘_{off}[D]-𝑘_{dc1}[D]-𝑘_{obs}[D]\] \[\frac{d[E]}{dt}=𝑘_{obs}[D]-𝑘_{d3}[E]\] \[\frac{d[F]}{dt}=𝑘_{obs}[D]-𝑘_{d3}[F]\] \[\frac{d[G]}{dt}=𝑘_{p1}[E]-𝑘_{dp1}[G]\]

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； \[Data(without leak)=Data(+IPTG)-Data(-IPTG)\]
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
\[𝑐𝑟𝑅𝑁𝐴 − 𝑅𝐵𝑆 − 𝑚𝑅𝑁𝐴_{𝑔𝑓𝑝}\] A 15 \[mol/L\]
\[𝑚𝑅𝑁𝐴_{Csy4}\] B 0 \[mol/L\]
\[Protein_{𝐶𝑠𝑦4}\] C 0 \[mol/L\]
\[Protein_{𝐶𝑠𝑦4}-crRNA-RBS-mRNA_{𝑔𝑓𝑝}\] D 0 \[mol/L\]
\[𝑚𝑅𝑁𝐴_{gfp}\] E 0 \[mol/L\]
\[Protein_{𝐶𝑠𝑦4}-𝑐𝑟𝑅𝑁𝐴\] F 0 \[mol/L\]
\[Protein_{gfp}\] G 0 \[mol/L\]

Because the [𝑐𝑟𝑅𝑁𝐴 − 𝑅𝐵𝑆 − 𝑚𝑅𝑁𝐴𝑔𝑓𝑝] 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:

Simulation


With the help of Simbiology toolbox in Matlab，we simulate our miniToe system for 30h, the result can be seen in the Fig.1.

Fig.1 The dynamics of GFP by model prediction

We compare the experimental data to the simulation:

Fig.2 The comparison between experimental data and simulation data

Discussion


Combining the biology and math, we now discuss why the dynamics of GFP is like the curve in the Fig.1. In order to explain in detail, we plot the dynamics of all species in the miniToe system in Fig.3.

Fig.3 The dynamics of all species in the miniToe system

As we can see in the Fig.3, the red line refers to the dynamics of GFP, which is increasing in the beginning and then drop down to a stable level. The reason for this is that the cleavage rate of Csy4 is faster than the production rate of , which cause that the is decreasing after 10 hours. Before we add IPTG into our system to induce the Tac promoter, the is accumulated because it is under controlled by a constitutive promoter. After we add IPTG, the initial concentration of plays an important role in the production of GFP during the early 10 hours. Even the Csy4; cleavage rate is faster than the production rate of , the which is accumulated before make the keep increasing. But after the initial amount of is used out, the then drop down into a stable level with the balance of the product rate and decay rate, which result in the drop down of GFP curve.

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Parameters Definition Units Value
\[𝑘_{1}\]   \[h^{-1}\] 1.955
\[𝑘_{d1}\]   \[h^{-1}\] 0.002
\[𝑘_{2}\]   \[h^{-1}\] 1.116
\[𝑘_{d2}\]   \[h^{-1}\] 0.241
\[𝑘_{p2}\]   \[h^{-1}\] 1.134
\[𝑘_{dp2}\]   \[h^{-1}\] 6.547
\[𝑘_{on}\]   \[h^{-1}\] 23995.469
\[𝑘_{off}\]   \[h^{-1}\] 2.703
\[𝑘_{dc1}\]   \[h^{-1}\] 0.024
\[𝑘_{obs}\]   \[h^{-1}\] 0.327
\[𝑘_{d3}\]   \[h^{-1}\] 0.472
\[𝑘_{dc2}\]   \[h^{-1}\] 0.024
\[𝑘_{p1}\]   \[h^{-1}\] 1.711
\[𝑘_{dp1}\]   \[h^{-1}\] 0.479