Team:OUC-China/miniToe

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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,𝑘_on ,𝑘_off ,𝑘_obs , 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:

\[\frac{d[A]}{dt}=𝑘_{1}-𝑘_{d1}[A]-𝑘_{on}[C][A]+𝑘_{off}[D]\] \[\frac{d[B]}{dt}=𝑘_{2}-𝑘_{d2}[B]\] \[\frac{d[C]}{dt}=𝑘_{p2}[B]-𝑘_{dp2}[C]-𝑘_{on}[C][A]+𝑘_{off}[D]\] \[\frac{d[D]}{dt}=𝑘_{on}[C][A]-𝑘_{dp2}[C]-𝑘_{off}[D]-𝑘_{dc1}[D]-𝑘_{obs}[D]\] \[\frac{d[E]}{dt}=𝑘_{obs}[D]-𝑘_{d3}[E]\] \[\frac{d[F]}{dt}=𝑘_{obs}[D]-𝑘_{d3}[F]\] \[\frac{d[G]}{dt}=𝑘_{p1}[E]-𝑘_{dp1}[G]\]

    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.

Also, （crispr家族蛋白的背景：相较与其他crispr家族其他蛋白质的优点）