Team:OUC-China/miniToe

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 parament we used in the ODEs is listed in the following table:

Parameters	Definition	Units	Value
𝑘₁	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
𝑘₂	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 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