Modeling > Parameters
Parameters
Constructing ODEs is only the first step of simulating our design. The parameters, actually, play a significant role in the modeling process. Here are two sets of parameters(T3 RNA polymerase and P2) we used.
We have done an systematic literature review to select parameters for our model. However, not every parameter can be found from existing data, which means we have to guess part of the parameters from trial and error. To check whether we have guessed correctly, we do the sensitivity test. The sensitivity test works like this: We first give both salicylate(food) and arabinose(bell) to make bistable turn to CI state(have memory). After a period of time, we give the E. coli arabinose stimulus and GFP output will raise after a short while. We select the highest concentration point of GFP in the second procedure. The sensitivity of a parameter is calculated by using the following equations. The closer the sensitivity is to zero, the more reasonable the parameter is.
Assumptions
Our model consists of 53 parameters, which makes the modeling process very difficult. To reduce the amount of work without losing quality, we proposed the following assumptions.
The average transcription speed in E.coli is 70nt/s. Assuming all the transcription in our circuit works in such speed, we can calculate the maximum transcription rate for each transcription equation by using this formula:
Maximum Transcription Rate = Transcription Speed(nt/min)/Gene Length(bp)=4200/Gene Length (nM/min)
Ref: http://gchelpdesk.ualberta.ca/CCDB/cgibin/STAT_NEW.cgi, NTUSingapore iGEM2008 Team
The average translation speed in E.coli is 40Aa/s. Also assuming all the translation in our circuit works in the same speed, we can calculate the theoretical transcription rate. However, in wetlab, we can use different rbs to regulate the translation process, thus, the translation rate can be written as:
Translation Rate = RBS * Translation Speed(Aa/min)/Protein Length(Aa) = 2400RBS/Protein Length (min^1)
This transformation does not change the degree of freedom of our system. However, this does limit the range of parameters since the strength of RBS can not be too extreme.
Ref: http://gchelpdesk.ualberta.ca/CCDB/cgibin/STAT_NEW.cgi, NTUSingapore iGEM2008 Team
We have assume the period of cell division is 30 mins, which means the "degradation rate" in our model is actually the sum of degradation rate of the substance(1/half life) and cell division rate(1/30 mins).
From Ref1(Belasco 1993) and Ref2(Genome Biology 2006, 7:R99), we have decided that all the mRNA in our system have a half life of 4.4 mins.
Modeling  T3 RNA polymerase
We have construct two models, the difference of which is in the AND Gate 2 module. In this section, we'll demonstrate the parameters of our first model, in which T3 RNA polymerase mRNA with amber mutation and AatRNA cooperate to produce T3 RNA polymerase protein.
Parameters  Brief Introduction  Value  Unit  Reference/Sensitivity

k_1  Max Transcription rate of tRNA  46.67  nM/min  Assumption, 0.19

k_2  Synthesis rate of AatRNA  0.08  min^1  0.09

k_3  Max Transcription rate of T7RNAP  1.5625  nM/min  Assumption, 0.00

k_4  Max Translation rate of T7RNAP  2.68*0.05  min^1  Assumption, 0.00

k_5  Max Transcription rate of trigger CI  5.6  nM/min  Assumption, 0.00

k_6  Transcription rate of bistable CI  5.6  nM/min  Assumption, 0.00

k_6'  Transcription rate of bistable CI  1  nM/min  0.00

k_7  Transcription rate of T3RNAP  1.75  nM/min  Assumption, 1.34

k_7'  Transcription rate of T3RNAP  1  nM/min  0.00

k_8  Translation rate of trigger CI  9.6*0.045  min^1  Assumption, 0.00

k_8'  Translation rate of bistable CI  9.6*0.3  min^1  Assumption, 0.00

k_9  Max Transcription rate of CI434  5.92  nM/min  Assumption, 0.00

k_10  Transcription rate of CI434  10.14*0.5  min^1  Assumption, 0.00

k_11  Max Translation rate of T3RNAP  3*0.15  min^1  Assumption, 1.34

k_12  Max Transcription rate of GFP from Sal  5.25  nM/min  Assumption, 0.00

k_12'  Max Trasncription rate of GFP from T3RNAP  5.25  nM/min  Assumption, 1.00

k_13  Translation rate of GFP  9*0.6  min^1  Assumption, 1.00

k_s  rate of AND Gate 1  0.3  nM^1  0.00

k_s'  rate of AND Gate 2  0.3  nM^1  0.18

K_1  dissociation constant of AraC,tRNA  14  nM  0.03

K_3  dissociation constant of Sal,T7RNAP  0.5  nM  0.00

K_5  dissociation constant of T7RNAP,trigger CI  3  nM  Ref

K_6  dissociation constant of CI,bistable CI  40  nM  Ref: iGEM2007 PKU Team

K_6'  dissociation constant of CI434,bistable CI  50  nM  Ref: iGEM2007 PKU Team

K_7  dissociation constant of CI,T3RNAP  40  nM  Ref: iGEM2007 PKU Team

K_7'  dissociation constant of CI434,T3RNAP  50  nM  Ref: iGEM2007 PKU Team

K_9  dissociation constant of CI,CI434  40  nM  Ref: iGEM2007 PKU Team

K_12  dissociation constant of Sal,GFP  0.5  nM  0.00

K_12'  dissociation constant of T3RNAP,GFP  55  nM  Ref: The FEBS journal 2006,273:17

n_1  Hill coeffiency of AraC,tRNA  2  

n_3  Hill coeffiency of Sal,T7RNAP  2  

n_5  Hill coeffiency of T7RNAP,CI  2  

n_6  Hill coeffiency of CI,bistable CI  4   Ref: iGEM2007 PKU Team

n_6'  Hill coeffiency of CI434,bistable CI  2   Ref: iGEM2007 PKU Team

n_7  Hill coeffiency of CI,T3RNAP  4   Ref: iGEM2007 PKU Team

n_7'  Hill coeffiency of CI434,T3RNAP  2   Ref: iGEM2007 PKU Team

n_9  Hill coeffiency of CI,CI434  4   Ref: iGEM2007 PKU Team

n_12  Hill coeffiency of Sal,GFP  2  

n_12'  Hill coeffiency of T3RNAP,GFP  2  

γ_1  Degradation rate of tRNA  1/30+1/60  min^1  Since half life of tRNA is very long, we decided to use 60 mins instead

γ_2  Degradation rate of AatRNA  1/30+1/40  min^1 

γ_2'  Real Degradation rate of AatRNA  1/40  min^1 

γ_3  Degradation rate of T7RNAP mRNA  1/30+1/4.4  min^1  Assumption

γ_4  Degradation rate of T7RNAP  1/30+1/40  min^1 

γ_5  Degradation rate of trigger CI mRNA  1/30+1/4.4  min^1  Assumption

γ_6  Degradation rate of bistable CI mRNA  1/30+1/4.4  min^1  Assumption

γ_7  Degradation rate of T3RNAP mRNA  1/30+1/4.4  min^1  Assumption

γ_8  Degradation rate of CI  1/30+1/44  min^1  Ref: iGEM2007 PKU Team

γ_9  Degradation rate of CI434 mRNA  1/30+1/4.4  min^1  Assumption

γ_10  Degradation rate of CI434  1/30+1/11  min^1  Ref: iGEM 2007 PKU Team

γ_11  Degradation rate of T3RNAP  1/30+1/30  min^1 

γ_12  Degradation rate of GFP mRNA  1/30+1/4.4  min^1  Assumption

γ_13  Degradation rate of GFP  1/30+1/60  min^1  Since half life of GFP is very long, we use 60 mins instead

Overall, the sensitivity of parameters from trial and error is generally low. The bistablerelated parameters' sensitivity indicates that the bistable model, which is the core in the circuit, is very stable. With all of these facts, we have concluded that this model is reasonable.
Modeling  P2
Here's our second model, the one with P2 instead of T3 RNA polymerase
Parameters  Brief Introduction  Value  Unit  Reference/Sensitivity

k_1  Max Transcription rate of tRNA  46.67  nM/min  Assumption, 1.41

k_2  Synthesis rate of AatRNA  0.08  min^1  0.81

k_3  Max Transcription rate of T7RNAP  1.5625  nM/min  Assumption, 0.00

k_4  Max Translation rate of T7RNAP  2.68*0.05  min^1  Assumption, 0.00

k_5  Max Transcription rate of trigger CI  5.6  nM/min  Assumption, 0.00

k_6  Transcription rate of bistable CI  5.6  nM/min  Assumption, 0.00

k_6'  Transcription rate of bistable CI  1  nM/min  0.00

k_7  Transcription rate of P2  16.8  nM/min  Assumption, 1.23

k_7'  Transcription rate of P2  1  nM/min  0.00

k_8  Translation rate of trigger CI  9.6*0.05  min^1  Assumption, 0.00

k_8'  Translation rate of bistable CI  9.6*0.5  min^1  Assumption, 0.00

k_9  Max Transcription rate of CI434  5.92  nM/min  Assumption, 0.00

k_10  Transcription rate of CI434  10.14*1  min^1  Assumption, 0.00

k_11  Max Translation rate of P2  28.8*0.0045  min^1  Assumption, 1.23

k_12  Max Transcription rate of GFP from Sal  5.25  nM/min  Assumption, 0.00

k_12'  Max Trasncription rate of GFP from P2  5.25  nM/min  Assumption, 1.00

k_13  Translation rate of GFP  9*0.6  min^1  Assumption, 1.00

k_s  rate of AND Gate 1  0.3  nM^1  0.00

k_s'  rate of AND Gate 2  0.01  nM^1  1.29

K_1  dissociation constant of AraC,tRNA  14  nM  0.20

K_3  dissociation constant of Sal,T7RNAP  0.5  nM  0.00

K_5  dissociation constant of T7RNAP,trigger CI  3  nM  Ref

K_6  dissociation constant of CI,bistable CI  40  nM  Ref: iGEM2007 PKU Team

K_6'  dissociation constant of CI434,bistable CI  50  nM  Ref: iGEM2007 PKU Team

K_7  dissociation constant of CI,P2  40  nM  Ref: iGEM2007 PKU Team

K_7'  dissociation constant of CI434,P2  50  nM  Ref: iGEM2007 PKU Team

K_9  dissociation constant of CI,CI434  40  nM  Ref: iGEM2007 PKU Team

K_12  dissociation constant of Sal,GFP  0.5  nM  0.00

K_12'  dissociation constant of P2,GFP  35  nM  1.29

n_1  Hill coeffiency of AraC,tRNA  2  

n_3  Hill coeffiency of Sal,T7RNAP  2  

n_5  Hill coeffiency of T7RNAP,CI  2  

n_6  Hill coeffiency of CI,bistable CI  4   Ref: iGEM2007 PKU Team

n_6'  Hill coeffiency of CI434,bistable CI  2   Ref: iGEM2007 PKU Team

n_7  Hill coeffiency of CI,P2  4   Ref: iGEM2007 PKU Team

n_7'  Hill coeffiency of CI434,P2  2   Ref: iGEM2007 PKU Team

n_9  Hill coeffiency of CI,CI434  4   Ref: iGEM2007 PKU Team

n_12  Hill coeffiency of Sal,GFP  2  

n_12'  Hill coeffiency of P2,GFP  2  

γ_1  Degradation rate of tRNA  1/30+1/60  min^1  Since half life of tRNA is very long, we decided to use 60 mins instead

γ_2  Degradation rate of AatRNA  1/30+1/40  min^1 

γ_2'  Real Degradation rate of AatRNA  1/40  min^1 

γ_3  Degradation rate of T7RNAP mRNA  1/30+1/4.4  min^1  Assumption

γ_4  Degradation rate of T7RNAP  1/30+1/40  min^1 

γ_5  Degradation rate of trigger CI mRNA  1/30+1/4.4  min^1  Assumption

γ_6  Degradation rate of bistable CI mRNA  1/30+1/4.4  min^1  Assumption

γ_7  Degradation rate of P2 mRNA  1/30+1/4.4  min^1  Assumption

γ_8  Degradation rate of CI  1/30+1/44  min^1  Ref: iGEM2007 PKU Team

γ_9  Degradation rate of CI434 mRNA  1/30+1/4.4  min^1  Assumption

γ_10  Degradation rate of CI434  1/30+1/11  min^1  Ref: iGEM 2007 PKU Team

γ_11  Degradation rate of P2  1/30+1/30  min^1 

γ_12  Degradation rate of GFP mRNA  1/30+1/4.4  min^1  Assumption

γ_13  Degradation rate of GFP  1/30+1/60  min^1  Since half life of GFP is very long, we use 60 mins instead

Overall, the sensitivity of parameters from trial and error is also low. The bistable is also shows great stability. However, we consider the value in AND Gate 2 is too extreme. Although this fact doesn't indicate that the circuit design is wrong or we can't finish the weblab project theoretically, we decided that it's very necessary to construct the circuit with T3 RNA polymerase.
With all necessities prepared, it's time to see the result!
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