Difference between revisions of "Team:OUC-China/miniToe"

Revision as of 09:51, 16 October 2018

Team OUC-China: Main

miniToe

Reactions

We can describe our miniToe system to be followings:

\[\rightarrow [𝑐𝑟𝑅𝑁𝐴 − 𝑅𝐵𝑆 − 𝑚𝑅𝑁𝐴_{sfGFP}]\] \[\rightarrow [𝑚𝑅𝑁𝐴_{Csy4}]\] \[[𝑚𝑅𝑁𝐴_{𝐶𝑠𝑦4}]\rightarrow [𝑚𝑅𝑁𝐴_{𝐶𝑠𝑦4}] + [𝑃𝑟𝑜𝑡𝑒𝑖𝑛_{𝐶𝑠𝑦4}]\] \[[Protein_{𝐶𝑠𝑦4}]+[crRNA-RBS-mRNA_{sfGFP}]\leftrightarrow [𝑃𝑟𝑜𝑡𝑒𝑖𝑛_{𝐶𝑠𝑦4}-crRNA-RBS-𝑚𝑅𝑁𝐴_{sfGFP}]\] \[[Protein_{𝐶𝑠𝑦4}-crRNA-RBS-mRNA_{sfGFP}]\rightarrow [𝑚𝑅𝑁𝐴_{sfGFP}] + [𝑃𝑟𝑜𝑡𝑒𝑖𝑛_{𝐶𝑠𝑦4}-𝑐𝑟𝑅𝑁𝐴]\] \[[𝑚𝑅𝑁𝐴_{sfGFP}]\rightarrow [𝑚𝑅𝑁𝐴_{sfGFP}] + [𝑃𝑟𝑜𝑡𝑒𝑖𝑛_{sfGFP}]\] \[[crRNA-RBS-mRNA_{sfGFP}]\rightarrow \emptyset\] \[[𝑚𝑅𝑁𝐴_{𝐶𝑠𝑦4}]\rightarrow \emptyset\] \[[𝑃𝑟𝑜𝑡𝑒𝑖𝑛_{𝐶𝑠𝑦4}]\rightarrow \emptyset\] \[[Protein_{𝐶𝑠𝑦4}-crRNA-RBS-mRNA_{sfGFP}]\rightarrow \emptyset\] \[[𝑚𝑅𝑁𝐴_{sfGFP}]\rightarrow \emptyset\] \[[𝑃𝑟𝑜𝑡𝑒𝑖𝑛_{𝐶𝑠𝑦4}-𝑐𝑟𝑅𝑁𝐴]\rightarrow \emptyset\] \[[𝑃𝑟𝑜𝑡𝑒𝑖𝑛_{sfGFP}]\rightarrow \emptyset\]

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 sfGFP, we use ordinary differential equations to model the reactions above. And ODEs are given as follows:

\[\frac{d[𝑐𝑟𝑅𝑁𝐴 − 𝑅𝐵𝑆 − 𝑚𝑅𝑁𝐴_{sfGFP}]}{dt}=𝑘_{1}-𝑘_{d1}[𝑐𝑟𝑅𝑁𝐴 − 𝑅𝐵𝑆 − 𝑚𝑅𝑁𝐴_{sfGFP}]\] \[-𝑘_{on}[Protein_{𝐶𝑠𝑦4}][𝑐𝑟𝑅𝑁𝐴 − 𝑅𝐵𝑆 − 𝑚𝑅𝑁𝐴_{sfGFP}]\] \[+𝑘_{off}[Protein_{𝐶𝑠𝑦4}-𝑐𝑟𝑅𝑁𝐴 − 𝑅𝐵𝑆 − 𝑚𝑅𝑁𝐴_{sfGFP}]\] \[\frac{d[𝑚𝑅𝑁𝐴_{𝐶𝑠𝑦4}]}{dt}=𝑘_{2}-𝑘_{d2}[𝑚𝑅𝑁𝐴_{𝐶𝑠𝑦4}]\] \[\frac{d[Protein_{𝐶𝑠𝑦4}]}{dt}=𝑘_{p2}[𝑚𝑅𝑁𝐴_{𝐶𝑠𝑦4}]-𝑘_{dp2}[Protein_{𝐶𝑠𝑦4}]\] \[-𝑘_{on}[Protein_{𝐶𝑠𝑦4}][𝑐𝑟𝑅𝑁𝐴 − 𝑅𝐵𝑆 − 𝑚𝑅𝑁𝐴_{sfGFP}]\] \[+𝑘_{off}[Protein_{𝐶𝑠𝑦4}-𝑐𝑟𝑅𝑁𝐴 − 𝑅𝐵𝑆 − 𝑚𝑅𝑁𝐴_{sfGFP}]\] \[\frac{d[𝑃𝑟𝑜𝑡𝑒𝑖𝑛_{𝐶𝑠𝑦4}-crRNA-RBS-𝑚𝑅𝑁𝐴_{sfGFP}]}{dt}=𝑘_{on}[Protein_{𝐶𝑠𝑦4}][𝑐𝑟𝑅𝑁𝐴 − 𝑅𝐵𝑆 − 𝑚𝑅𝑁𝐴_{sfGFP}]\] \[-𝑘_{of}[Protein_{𝐶𝑠𝑦4}][Protein_{𝐶𝑠𝑦4}-crRNA-RBS-mRNA_{sfGFP}]\] \[-𝑘_{d1}[Protein_{𝐶𝑠𝑦4}][Protein_{𝐶𝑠𝑦4}-crRNA-RBS-mRNA_{sfGFP}]\] \[-𝑘_{obs}[Protein_{𝐶𝑠𝑦4}][Protein_{𝐶𝑠𝑦4}-crRNA-RBS-mRNA_{sfGFP}]\] \[\frac{d[𝑚𝑅𝑁𝐴_{sfGFP}]}{dt}=𝑘_{obs}[𝑃𝑟𝑜𝑡𝑒𝑖𝑛_{𝐶𝑠𝑦4}-crRNA-RBS-𝑚𝑅𝑁𝐴_{sfGFP}]-𝑘_{d3}[𝑚𝑅𝑁𝐴_{sfGFP}]\] \[\frac{d[𝑃𝑟𝑜𝑡𝑒𝑖𝑛_{𝐶𝑠𝑦4}-crRNA]}{dt}=𝑘_{obs}[𝑃𝑟𝑜𝑡𝑒𝑖𝑛_{𝐶𝑠𝑦4}-crRNA-RBS-𝑚𝑅𝑁𝐴_{sfGFP}]-𝑘_{dc2}[𝑃𝑟𝑜𝑡𝑒𝑖𝑛_{𝐶𝑠𝑦4}-𝑐𝑟𝑅𝑁𝐴]\] \[-𝑘_{on}[Protein_{𝐶𝑠𝑦4}][𝑐𝑟𝑅𝑁𝐴 − 𝑅𝐵𝑆 − 𝑚𝑅𝑁𝐴_{sfGFP}]\] \[+𝑘_{off}[Protein_{𝐶𝑠𝑦4}-𝑐𝑟𝑅𝑁𝐴 − 𝑅𝐵𝑆 − 𝑚𝑅𝑁𝐴_{sfGFP}]\] \[\frac{d[Protein_{sfGFP}]}{dt}=𝑘_{p1}[𝑚𝑅𝑁𝐴_{sfGFP}]-𝑘_{dp1}[Protein_{sfGFP}]\]

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 a 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 influenced 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
\[𝑐𝑟𝑅𝑁𝐴 − 𝑅𝐵𝑆 − 𝑚𝑅𝑁𝐴_{sfGFP}\]	A	15	\[mol/L\]
\[𝑚𝑅𝑁𝐴_{Csy4}\]	B	0	\[mol/L\]
\[Protein_{𝐶𝑠𝑦4}\]	C	0	\[mol/L\]
\[Protein_{𝐶𝑠𝑦4}-crRNA-RBS-mRNA_{sfGFP}\]	D	0	\[mol/L\]
\[𝑚𝑅𝑁𝐴_{sfGFP}\]	E	0	\[mol/L\]
\[Protein_{𝐶𝑠𝑦4}-𝑐𝑟𝑅𝑁𝐴\]	F	0	\[mol/L\]
\[Protein_{sfGFP}\]	G	0	\[mol/L\]

Because the [𝑐𝑟𝑅𝑁𝐴 − 𝑅𝐵𝑆 − 𝑚𝑅𝑁𝐴_sfGFP] 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
\[𝑘_{1}\]	The transcription rate of\[𝑐𝑟𝑅𝑁𝐴 − 𝑅𝐵𝑆 − 𝑚𝑅𝑁𝐴_{sfGFP}\]	\[h^{-1}\]	1.955
\[𝑘_{d1}\]	The degradation rate of\[𝑐𝑟𝑅𝑁𝐴 − 𝑅𝐵𝑆 − 𝑚𝑅𝑁𝐴_{sfGFP}\]	\[h^{-1}\]	0.002
\[𝑘_{2}\]	The transcription rate of\[𝑚𝑅𝑁𝐴_{Csy4}\]	\[h^{-1}\]	1.116
\[𝑘_{d2}\]	The degradation rate of\[𝑚𝑅𝑁𝐴_{Csy4}\]	\[h^{-1}\]	0.241
\[𝑘_{p2}\]	The translation rate of\[Protein_{𝐶𝑠𝑦4}\]	\[h^{-1}\]	1.134
\[𝑘_{dp2}\]	The degradation rate of\[Protein_{𝐶𝑠𝑦4}\]	\[h^{-1}\]	6.547
\[𝑘_{on}\]	The binding constant of \[Protein_{𝐶𝑠𝑦4}\] and \[𝑐𝑟𝑅𝑁𝐴 − 𝑅𝐵𝑆 − 𝑚𝑅𝑁𝐴_{sfGFP}\]	\[h^{-1}\]	23995.469
\[𝑘_{off}\]	The dissociation constant of \[Protein_{𝐶𝑠𝑦4}\] and \[𝑐𝑟𝑅𝑁𝐴 − 𝑅𝐵𝑆 − 𝑚𝑅𝑁𝐴_{sfGFP}\]	\[h^{-1}\]	2.703
\[𝑘_{dc1}\]	The degradation rate of\[Protein_{𝐶𝑠𝑦4}-crRNA-RBS-mRNA_{sfGFP}\]	\[h^{-1}\]	0.024
\[𝑘_{obs}\]	The cleavage rate of \[Protein_{𝐶𝑠𝑦4}\]	\[h^{-1}\]	0.327
\[𝑘_{d3}\]	The degradation rate of \[𝑚𝑅𝑁𝐴_{sfGFP}\]	\[h^{-1}\]	0.472
\[𝑘_{dc2}\]	The degradation rate of \[Protein_{𝐶𝑠𝑦4}-crRNA\]	\[h^{-1}\]	0.024
\[𝑘_{p1}\]	The translation rate of \[Protein_{sfGFP}\]	\[h^{-1}\]	1.711
\[𝑘_{dp1}\]	The degradation rate of \[Protein_{sfGFP}\]	\[h^{-1}\]	0.479

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 sfGFP 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 sfGFP 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 sfGFP, 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 [crRNA-RBS-mRNA_sfGFP] , which cause that the [mRNA_sfGFP] is decreasing after 10 hours. Before we add IPTG into our system to induce the Tac promoter, the [crRNA-RBS-mRNA_sfGFP] is accumulated because it is under control by a constitutive promoter. After we add IPTG, the initial concentration of [crRNA-RBS-mRNA_sfGFP] plays an important role in the production of sfGFP during the early 10 hours. Even the Csy4; cleavage rate is faster than the production rate of [crRNA-RBS-mRNA_sfGFP] , the [crRNA-RBS-mRNA_sfGFP] which is accumulated before make the keep increasing. But after the initial amount of [crRNA-RBS-mRNA_sfGFP] is used out, the [mRNA_sfGFP] then drop down into a stable level with the balance of the production rate and decay rate, which result in the drop down of sfGFP curve.

Sensitivities Analysis

After building the ODE model, we try to do something more deeply to our miniToe system by analyzing the sensitivity of parameters. Fig.4 shows the numerical integration of sensitivities of parameters in 30 hours.

Fig.4 The numerical integration of sensitivities of parameters in 30h

According to the sensitivity analysis, we can find that three core parameters, k_on ,k_off , k_obs , which is related to the protein Csy4, has different effect in the expression of sfGFP. The two binding parameters, k_on ,k_off , will not influence sfGFP while the cleavage parameter, , will influence the expression in sfGFP, which indicated if we change the wild-type Csy4 to some mutation then we can achieve the different expression of sfGFP.

Enlighten by the sensitivity analysis, we give a prediction curve that shows that what will happen in the sfGFP expression curve if we change the Csy4, and it can be seen in the Fig.5.

Fig.5 The curve of sfGFP with the changing cleavage rate

And the Fig.6 shows that the relationship between the stable expression level of sfGFP and the claevage rate,k_obs .

Fig.6 The relationship between the stable expression level of sfGFP and the claevage rate

@@ Line 154: / Line 154: @@
 <div>
-			            \[\rightarrow [𝑐𝑟𝑅𝑁𝐴 − 𝑅𝐵𝑆 − 𝑚𝑅𝑁𝐴_{𝑔𝑓𝑝}]\]
+			            \[\rightarrow [𝑐𝑟𝑅𝑁𝐴 − 𝑅𝐵𝑆 − 𝑚𝑅𝑁𝐴_{sfGFP}]\]
                          \[\rightarrow [𝑚𝑅𝑁𝐴_{Csy4}]\]
                         	\[[𝑚𝑅𝑁𝐴_{𝐶𝑠𝑦4}]\rightarrow [𝑚𝑅𝑁𝐴_{𝐶𝑠𝑦4}] + [𝑃𝑟𝑜𝑡𝑒𝑖𝑛_{𝐶𝑠𝑦4}]\]
-                         \[[Protein_{𝐶𝑠𝑦4}]+[crRNA-RBS-mRNA_{gfp}]\leftrightarrow [𝑃𝑟𝑜𝑡𝑒𝑖𝑛_{𝐶𝑠𝑦4}-crRNA-RBS-𝑚𝑅𝑁𝐴_{𝑔𝑓𝑝}]\]
+                         \[[Protein_{𝐶𝑠𝑦4}]+[crRNA-RBS-mRNA_{sfGFP}]\leftrightarrow [𝑃𝑟𝑜𝑡𝑒𝑖𝑛_{𝐶𝑠𝑦4}-crRNA-RBS-𝑚𝑅𝑁𝐴_{sfGFP}]\]
-                         \[[Protein_{𝐶𝑠𝑦4}-crRNA-RBS-mRNA_{gfp}]\rightarrow [𝑚𝑅𝑁𝐴_{gfp}] + [𝑃𝑟𝑜𝑡𝑒𝑖𝑛_{𝐶𝑠𝑦4}-𝑐𝑟𝑅𝑁𝐴]\]
+                         \[[Protein_{𝐶𝑠𝑦4}-crRNA-RBS-mRNA_{sfGFP}]\rightarrow [𝑚𝑅𝑁𝐴_{sfGFP}] + [𝑃𝑟𝑜𝑡𝑒𝑖𝑛_{𝐶𝑠𝑦4}-𝑐𝑟𝑅𝑁𝐴]\]
-                         \[[𝑚𝑅𝑁𝐴_{gfp}]\rightarrow [𝑚𝑅𝑁𝐴_{gfp}] + [𝑃𝑟𝑜𝑡𝑒𝑖𝑛_{gfp}]\]
+                         \[[𝑚𝑅𝑁𝐴_{sfGFP}]\rightarrow [𝑚𝑅𝑁𝐴_{sfGFP}] + [𝑃𝑟𝑜𝑡𝑒𝑖𝑛_{sfGFP}]\]
-  						\[[crRNA-RBS-mRNA_{gfp}]\rightarrow \emptyset\]
+  						\[[crRNA-RBS-mRNA_{sfGFP}]\rightarrow \emptyset\]
                       	\[[𝑚𝑅𝑁𝐴_{𝐶𝑠𝑦4}]\rightarrow \emptyset\]
                       	\[[𝑃𝑟𝑜𝑡𝑒𝑖𝑛_{𝐶𝑠𝑦4}]\rightarrow \emptyset\]
-                         \[[Protein_{𝐶𝑠𝑦4}-crRNA-RBS-mRNA_{gfp}]\rightarrow \emptyset\]
+                         \[[Protein_{𝐶𝑠𝑦4}-crRNA-RBS-mRNA_{sfGFP}]\rightarrow \emptyset\]
-						\[[𝑚𝑅𝑁𝐴_{gfp}]\rightarrow \emptyset\]
+						\[[𝑚𝑅𝑁𝐴_{sfGFP}]\rightarrow \emptyset\]
 						\[[𝑃𝑟𝑜𝑡𝑒𝑖𝑛_{𝐶𝑠𝑦4}-𝑐𝑟𝑅𝑁𝐴]\rightarrow \emptyset\]
-                      	\[[𝑃𝑟𝑜𝑡𝑒𝑖𝑛_{gfp}]\rightarrow \emptyset\]
+                      	\[[𝑃𝑟𝑜𝑡𝑒𝑖𝑛_{sfGFP}]\rightarrow \emptyset\]
 										</div>
 <br />Two equations, describing the functional binding and cleavage of Csy4 protein in biology, and three parameters&emsp;
@@ Line 176: / Line 176: @@
 <h3>ODEs</h3>
-<br />To simulate the dynamics of GFP, we use ordinary differential equations to model the reactions above. And ODEs are given as follows:
+<br />To simulate the dynamics of sfGFP, we use ordinary differential equations to model the reactions above. And ODEs are given as follows:
 <div>
-			\[\frac{d[𝑐𝑟𝑅𝑁𝐴 − 𝑅𝐵𝑆 − 𝑚𝑅𝑁𝐴_{𝑔𝑓𝑝}]}{dt}=𝑘_{1}-𝑘_{d1}[𝑐𝑟𝑅𝑁𝐴 − 𝑅𝐵𝑆 − 𝑚𝑅𝑁𝐴_{𝑔𝑓𝑝}]\]
+			\[\frac{d[𝑐𝑟𝑅𝑁𝐴 − 𝑅𝐵𝑆 − 𝑚𝑅𝑁𝐴_{sfGFP}]}{dt}=𝑘_{1}-𝑘_{d1}[𝑐𝑟𝑅𝑁𝐴 − 𝑅𝐵𝑆 − 𝑚𝑅𝑁𝐴_{sfGFP}]\]
-			\[-𝑘_{on}[Protein_{𝐶𝑠𝑦4}][𝑐𝑟𝑅𝑁𝐴 − 𝑅𝐵𝑆 − 𝑚𝑅𝑁𝐴_{𝑔𝑓𝑝}]\]
+			\[-𝑘_{on}[Protein_{𝐶𝑠𝑦4}][𝑐𝑟𝑅𝑁𝐴 − 𝑅𝐵𝑆 − 𝑚𝑅𝑁𝐴_{sfGFP}]\]
-			\[+𝑘_{off}[Protein_{𝐶𝑠𝑦4}-𝑐𝑟𝑅𝑁𝐴 − 𝑅𝐵𝑆 − 𝑚𝑅𝑁𝐴_{𝑔𝑓𝑝}]\]
+			\[+𝑘_{off}[Protein_{𝐶𝑠𝑦4}-𝑐𝑟𝑅𝑁𝐴 − 𝑅𝐵𝑆 − 𝑚𝑅𝑁𝐴_{sfGFP}]\]
@@ Line 189: / Line 189: @@
 			\[\frac{d[Protein_{𝐶𝑠𝑦4}]}{dt}=𝑘_{p2}[𝑚𝑅𝑁𝐴_{𝐶𝑠𝑦4}]-𝑘_{dp2}[Protein_{𝐶𝑠𝑦4}]\]
-			\[-𝑘_{on}[Protein_{𝐶𝑠𝑦4}][𝑐𝑟𝑅𝑁𝐴 − 𝑅𝐵𝑆 − 𝑚𝑅𝑁𝐴_{𝑔𝑓𝑝}]\]
+			\[-𝑘_{on}[Protein_{𝐶𝑠𝑦4}][𝑐𝑟𝑅𝑁𝐴 − 𝑅𝐵𝑆 − 𝑚𝑅𝑁𝐴_{sfGFP}]\]
-			\[+𝑘_{off}[Protein_{𝐶𝑠𝑦4}-𝑐𝑟𝑅𝑁𝐴 − 𝑅𝐵𝑆 − 𝑚𝑅𝑁𝐴_{𝑔𝑓𝑝}]\]
+			\[+𝑘_{off}[Protein_{𝐶𝑠𝑦4}-𝑐𝑟𝑅𝑁𝐴 − 𝑅𝐵𝑆 − 𝑚𝑅𝑁𝐴_{sfGFP}]\]
-			\[\frac{d[𝑃𝑟𝑜𝑡𝑒𝑖𝑛_{𝐶𝑠𝑦4}-crRNA-RBS-𝑚𝑅𝑁𝐴_{𝑔𝑓𝑝}]}{dt}=𝑘_{on}[Protein_{𝐶𝑠𝑦4}][𝑐𝑟𝑅𝑁𝐴 − 𝑅𝐵𝑆 − 𝑚𝑅𝑁𝐴_{𝑔𝑓𝑝}]\]
+			\[\frac{d[𝑃𝑟𝑜𝑡𝑒𝑖𝑛_{𝐶𝑠𝑦4}-crRNA-RBS-𝑚𝑅𝑁𝐴_{sfGFP}]}{dt}=𝑘_{on}[Protein_{𝐶𝑠𝑦4}][𝑐𝑟𝑅𝑁𝐴 − 𝑅𝐵𝑆 − 𝑚𝑅𝑁𝐴_{sfGFP}]\]
-			\[-𝑘_{of}[Protein_{𝐶𝑠𝑦4}][Protein_{𝐶𝑠𝑦4}-crRNA-RBS-mRNA_{gfp}]\]
+			\[-𝑘_{of}[Protein_{𝐶𝑠𝑦4}][Protein_{𝐶𝑠𝑦4}-crRNA-RBS-mRNA_{sfGFP}]\]
-			\[-𝑘_{d1}[Protein_{𝐶𝑠𝑦4}][Protein_{𝐶𝑠𝑦4}-crRNA-RBS-mRNA_{gfp}]\]
+			\[-𝑘_{d1}[Protein_{𝐶𝑠𝑦4}][Protein_{𝐶𝑠𝑦4}-crRNA-RBS-mRNA_{sfGFP}]\]
-			\[-𝑘_{obs}[Protein_{𝐶𝑠𝑦4}][Protein_{𝐶𝑠𝑦4}-crRNA-RBS-mRNA_{gfp}]\]
+			\[-𝑘_{obs}[Protein_{𝐶𝑠𝑦4}][Protein_{𝐶𝑠𝑦4}-crRNA-RBS-mRNA_{sfGFP}]\]
-			\[\frac{d[𝑚𝑅𝑁𝐴_{𝑔𝑓𝑝}]}{dt}=𝑘_{obs}[𝑃𝑟𝑜𝑡𝑒𝑖𝑛_{𝐶𝑠𝑦4}-crRNA-RBS-𝑚𝑅𝑁𝐴_{𝑔𝑓𝑝}]-𝑘_{d3}[𝑚𝑅𝑁𝐴_{gfp}]\]
+			\[\frac{d[𝑚𝑅𝑁𝐴_{sfGFP}]}{dt}=𝑘_{obs}[𝑃𝑟𝑜𝑡𝑒𝑖𝑛_{𝐶𝑠𝑦4}-crRNA-RBS-𝑚𝑅𝑁𝐴_{sfGFP}]-𝑘_{d3}[𝑚𝑅𝑁𝐴_{sfGFP}]\]
-			\[\frac{d[𝑃𝑟𝑜𝑡𝑒𝑖𝑛_{𝐶𝑠𝑦4}-crRNA]}{dt}=𝑘_{obs}[𝑃𝑟𝑜𝑡𝑒𝑖𝑛_{𝐶𝑠𝑦4}-crRNA-RBS-𝑚𝑅𝑁𝐴_{𝑔𝑓𝑝}]-𝑘_{dc2}[𝑃𝑟𝑜𝑡𝑒𝑖𝑛_{𝐶𝑠𝑦4}-𝑐𝑟𝑅𝑁𝐴]\]
+			\[\frac{d[𝑃𝑟𝑜𝑡𝑒𝑖𝑛_{𝐶𝑠𝑦4}-crRNA]}{dt}=𝑘_{obs}[𝑃𝑟𝑜𝑡𝑒𝑖𝑛_{𝐶𝑠𝑦4}-crRNA-RBS-𝑚𝑅𝑁𝐴_{sfGFP}]-𝑘_{dc2}[𝑃𝑟𝑜𝑡𝑒𝑖𝑛_{𝐶𝑠𝑦4}-𝑐𝑟𝑅𝑁𝐴]\]
-			\[-𝑘_{on}[Protein_{𝐶𝑠𝑦4}][𝑐𝑟𝑅𝑁𝐴 − 𝑅𝐵𝑆 − 𝑚𝑅𝑁𝐴_{𝑔𝑓𝑝}]\]
+			\[-𝑘_{on}[Protein_{𝐶𝑠𝑦4}][𝑐𝑟𝑅𝑁𝐴 − 𝑅𝐵𝑆 − 𝑚𝑅𝑁𝐴_{sfGFP}]\]
-			\[+𝑘_{off}[Protein_{𝐶𝑠𝑦4}-𝑐𝑟𝑅𝑁𝐴 − 𝑅𝐵𝑆 − 𝑚𝑅𝑁𝐴_{𝑔𝑓𝑝}]\]
+			\[+𝑘_{off}[Protein_{𝐶𝑠𝑦4}-𝑐𝑟𝑅𝑁𝐴 − 𝑅𝐵𝑆 − 𝑚𝑅𝑁𝐴_{sfGFP}]\]
-			\[\frac{d[Protein_{gfp}]}{dt}=𝑘_{p1}[𝑚𝑅𝑁𝐴_{𝑔𝑓𝑝}]-𝑘_{dp1}[Protein_{gfp}]\]
+			\[\frac{d[Protein_{sfGFP}]}{dt}=𝑘_{p1}[𝑚𝑅𝑁𝐴_{sfGFP}]-𝑘_{dp1}[Protein_{sfGFP}]\]
 										</div>
 <br />For the readability, the complex symbol is simplified as:
@@ Line 251: / Line 251: @@
      </tr>
      <tr>
-       <th scope="row"> \[𝑐𝑟𝑅𝑁𝐴 − 𝑅𝐵𝑆 − 𝑚𝑅𝑁𝐴_{𝑔𝑓𝑝}\]</th>
+       <th scope="row"> \[𝑐𝑟𝑅𝑁𝐴 − 𝑅𝐵𝑆 − 𝑚𝑅𝑁𝐴_{sfGFP}\]</th>
        <td align="center">A</td>
        <td align="center">15</td>
@@ Line 269: / Line 269: @@
      </tr>
      <tr>
-       <th scope="row">\[Protein_{𝐶𝑠𝑦4}-crRNA-RBS-mRNA_{𝑔𝑓𝑝}\]</th>
+       <th scope="row">\[Protein_{𝐶𝑠𝑦4}-crRNA-RBS-mRNA_{sfGFP}\]</th>
        <td align="center">D</td>
        <td align="center">0</td>
@@ Line 275: / Line 275: @@
      </tr>
      <tr>
-       <th scope="row">\[𝑚𝑅𝑁𝐴_{gfp}\]</th>
+       <th scope="row">\[𝑚𝑅𝑁𝐴_{sfGFP}\]</th>
        <td align="center">E</td>
        <td align="center">0</td>
@@ Line 287: / Line 287: @@
      </tr>
      <tr>
-       <th scope="row">\[Protein_{gfp}\]</th>
+       <th scope="row">\[Protein_{sfGFP}\]</th>
        <td align="center">G</td>
        <td align="center">0</td>
@@ Line 295: / Line 295: @@
 </table>
-<br /> Because the [𝑐𝑟𝑅𝑁𝐴 − 𝑅𝐵𝑆 − 𝑚𝑅𝑁𝐴<sub>𝑔𝑓𝑝</sub>] is under controlled by a constitutive promoter, so we set the initial concentration to 15mol/L .
+<br /> Because the [𝑐𝑟𝑅𝑁𝐴 − 𝑅𝐵𝑆 − 𝑚𝑅𝑁𝐴<sub>sfGFP</sub>] is under controlled by a constitutive promoter, so we set the initial concentration to 15mol/L .
 <br />  The other parament we used in the ODEs is listed in the following table:												<table width="200" border="1">
@@ Line 307: / Line 307: @@
      <tr>
        <th scope="row">\[𝑘_{1}\]</th>
-       <td align="center">The transcription rate of\[𝑐𝑟𝑅𝑁𝐴 − 𝑅𝐵𝑆 − 𝑚𝑅𝑁𝐴_{𝑔𝑓𝑝}\] </td>
+       <td align="center">The transcription rate of\[𝑐𝑟𝑅𝑁𝐴 − 𝑅𝐵𝑆 − 𝑚𝑅𝑁𝐴_{sfGFP}\] </td>
        <td align="center">\[h^{-1}\]</td>
        <td align="center">1.955</td>
@@ Line 313: / Line 313: @@
      <tr>
        <th scope="row">\[𝑘_{d1}\]</th>
-       <td align="center">The degradation rate of\[𝑐𝑟𝑅𝑁𝐴 − 𝑅𝐵𝑆 − 𝑚𝑅𝑁𝐴_{𝑔𝑓𝑝}\] </td>
+       <td align="center">The degradation rate of\[𝑐𝑟𝑅𝑁𝐴 − 𝑅𝐵𝑆 − 𝑚𝑅𝑁𝐴_{sfGFP}\] </td>
        <td align="center">\[h^{-1}\]</td>
        <td align="center">0.002</td>
@@ Line 343: / Line 343: @@
      <tr>
        <th scope="row">\[𝑘_{on}\]</th>
-       <td align="center">The binding constant of \[Protein_{𝐶𝑠𝑦4}\] and \[𝑐𝑟𝑅𝑁𝐴 − 𝑅𝐵𝑆 − 𝑚𝑅𝑁𝐴_{𝑔𝑓𝑝}\]</td>
+       <td align="center">The binding constant of \[Protein_{𝐶𝑠𝑦4}\] and \[𝑐𝑟𝑅𝑁𝐴 − 𝑅𝐵𝑆 − 𝑚𝑅𝑁𝐴_{sfGFP}\]</td>
        <td align="center">\[h^{-1}\]</td>
        <td align="center">23995.469</td>
@@ Line 349: / Line 349: @@
      <tr>
        <th scope="row">\[𝑘_{off}\]</th>
-       <td align="center">The dissociation constant of \[Protein_{𝐶𝑠𝑦4}\] and \[𝑐𝑟𝑅𝑁𝐴 − 𝑅𝐵𝑆 − 𝑚𝑅𝑁𝐴_{𝑔𝑓𝑝}\]</td>
+       <td align="center">The dissociation constant of \[Protein_{𝐶𝑠𝑦4}\] and \[𝑐𝑟𝑅𝑁𝐴 − 𝑅𝐵𝑆 − 𝑚𝑅𝑁𝐴_{sfGFP}\]</td>
        <td align="center">\[h^{-1}\]</td>
        <td align="center">2.703</td>
@@ Line 355: / Line 355: @@
      <tr>
        <th scope="row">\[𝑘_{dc1}\]</th>
-       <td align="center">The degradation rate of\[Protein_{𝐶𝑠𝑦4}-crRNA-RBS-mRNA_{𝑔𝑓𝑝}\] </td>
+       <td align="center">The degradation rate of\[Protein_{𝐶𝑠𝑦4}-crRNA-RBS-mRNA_{sfGFP}\] </td>
        <td align="center">\[h^{-1}\]</td>
        <td align="center">0.024</td>
@@ Line 367: / Line 367: @@
      <tr>
        <th scope="row">\[𝑘_{d3}\]</th>
-       <td align="center">The degradation rate of \[𝑚𝑅𝑁𝐴_{gfp}\] </td>
+       <td align="center">The degradation rate of \[𝑚𝑅𝑁𝐴_{sfGFP}\] </td>
        <td align="center">\[h^{-1}\]</td>
        <td align="center">0.472</td>
@@ Line 379: / Line 379: @@
      <tr>
        <th scope="row">\[𝑘_{p1}\]</th>
-       <td align="center">The translation rate of \[Protein_{gfp}\]</td>
+       <td align="center">The translation rate of \[Protein_{sfGFP}\]</td>
        <td align="center">\[h^{-1}\]</td>
        <td align="center">1.711</td>
@@ Line 385: / Line 385: @@
      <tr>
        <th scope="row">\[𝑘_{dp1}\]</th>
-       <td align="center">The degradation rate of \[Protein_{gfp}\] </td>
+       <td align="center">The degradation rate of \[Protein_{sfGFP}\] </td>
        <td align="center">\[h^{-1}\]</td>
        <td align="center">0.479</td>
@@ Line 398: / Line 398: @@
 </div>
 <br />
-  <div align="center"><p >Fig.1 The dynamics of GFP by model prediction</p></div>
+  <div align="center"><p >Fig.1 The dynamics of sfGFP by model prediction</p></div>
 			We compare the experimental data to the simulation:
 	<div align="center"><img src="https://static.igem.org/mediawiki/2018/8/89/T--OUC-China--min2.png" width="600" >
@@ Line 406: / Line 406: @@
 	<h3>Discussion</h3>
-<br />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.
+<br />Combining the biology and math, we now discuss why the dynamics of sfGFP 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.
 <div align="center"><img src="https://static.igem.org/mediawiki/2018/2/2e/T--OUC-China--res6.png" width="600" >
@@ Line 414: / Line 414: @@
-	  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 [crRNA-RBS-mRNA<sub>gfp</sub>] , which cause that the [mRNA<sub>gfp</sub>] is decreasing after 10 hours. Before we add IPTG into our system to induce the Tac promoter, the [crRNA-RBS-mRNA<sub>gfp</sub>] is accumulated because it is under control by a constitutive promoter. After we add IPTG, the initial concentration of [crRNA-RBS-mRNA<sub>gfp</sub>] 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 [crRNA-RBS-mRNA<sub>gfp</sub>] ,  the [crRNA-RBS-mRNA<sub>gfp</sub>]  which is accumulated before make the  keep increasing. But after the initial amount of [crRNA-RBS-mRNA<sub>gfp</sub>] is used out, the [mRNA<sub>gfp</sub>] then drop down into a stable level with the balance of the production rate and decay rate, which result in the drop down of GFP curve.
+	  As we can see in the Fig.3, the red line refers to the dynamics of sfGFP, 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 [crRNA-RBS-mRNA<sub>sfGFP</sub>] , which cause that the [mRNA<sub>sfGFP</sub>] is decreasing after 10 hours. Before we add IPTG into our system to induce the Tac promoter, the [crRNA-RBS-mRNA<sub>sfGFP</sub>] is accumulated because it is under control by a constitutive promoter. After we add IPTG, the initial concentration of [crRNA-RBS-mRNA<sub>sfGFP</sub>] plays an important role in the production of sfGFP during the early 10 hours. Even the Csy4; cleavage rate is faster than the production rate of [crRNA-RBS-mRNA<sub>sfGFP</sub>] ,  the [crRNA-RBS-mRNA<sub>sfGFP</sub>]  which is accumulated before make the  keep increasing. But after the initial amount of [crRNA-RBS-mRNA<sub>sfGFP</sub>] is used out, the [mRNA<sub>sfGFP</sub>] then drop down into a stable level with the balance of the production rate and decay rate, which result in the drop down of sfGFP curve.
 <br /><br />
 	<h3>Sensitivities Analysis</h3>
@@ Line 423: / Line 423: @@
 <br />
   <div align="center"><p >Fig.4 The numerical integration of sensitivities of parameters in 30h</p></div>
-According to the sensitivity analysis, we can find that three core parameters, k<sub>on</sub> ,k<sub>off</sub> , k<sub>obs</sub> , which is related to the protein Csy4, has different effect in the expression of GFP. The two binding parameters, k<sub>on</sub> ,k<sub>off</sub>  , will not influence GFP while the cleavage parameter, , will influence the expression in GFP, which indicated if we change the wild-type Csy4 to some mutation then we can achieve the different expression of GFP.
+According to the sensitivity analysis, we can find that three core parameters, k<sub>on</sub> ,k<sub>off</sub> , k<sub>obs</sub> , which is related to the protein Csy4, has different effect in the expression of sfGFP. The two binding parameters, k<sub>on</sub> ,k<sub>off</sub>  , will not influence sfGFP while the cleavage parameter, , will influence the expression in sfGFP, which indicated if we change the wild-type Csy4 to some mutation then we can achieve the different expression of sfGFP.
-<br /><br />Enlighten by the sensitivity analysis, we give a prediction curve that shows that what will happen in the GFP expression curve if we change the Csy4, and it can be seen in the Fig.5.
+<br /><br />Enlighten by the sensitivity analysis, we give a prediction curve that shows that what will happen in the sfGFP expression curve if we change the Csy4, and it can be seen in the Fig.5.
    	<div align="center"><img src="https://static.igem.org/mediawiki/2018/c/ca/T--OUC-China--res23.png" width="600" >
 </div>
 <br />
-  <div align="center"><p >Fig.5 The curve of GFP with the changing cleavage rate</p></div>
+  <div align="center"><p >Fig.5 The curve of sfGFP with the changing cleavage rate</p></div>
-And the Fig.6 shows that the relationship between the stable expression level of GFP and the claevage rate,k<sub>obs</sub>  .
+And the Fig.6 shows that the relationship between the stable expression level of sfGFP and the claevage rate,k<sub>obs</sub>  .
 	<div align="center"><img src="https://static.igem.org/mediawiki/2018/5/55/T--OUC-China--min6.png" width="600" >
 </div>
 <br />
-  <div align="center"><p >Fig.6 The relationship between the stable expression level of GFP and the claevage rate</p></div>
+  <div align="center"><p >Fig.6 The relationship between the stable expression level of sfGFP and the claevage rate</p></div>
 </p>
 <br /><br /><br /><br />