Team:TU Munich/model/combined

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<div class="subcontent ui-corner-all sublab" style="clear:both;">
<h1>Combined Model</h1>
<h1>Combined Model</h1>
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<p> M-files: <a href="https://static.igem.org/mediawiki/2011/5/50/OmpReq.m">OmpReq.m</a> <a href="https://static.igem.org/mediawiki/2011/d/d0/YcgEeq.m">YcgEeq.m</a></p>
<p> M-files: <a href="https://static.igem.org/mediawiki/2011/5/50/OmpReq.m">OmpReq.m</a> <a href="https://static.igem.org/mediawiki/2011/d/d0/YcgEeq.m">YcgEeq.m</a></p>
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Revision as of 10:45, 16 September 2011

LyX Document

Combined Model

This page is also available as PDF. Please use Firefox 6.0 or later for best experience of this page.

M-files: OmpReq.m YcgEeq.m

Model

This model combines the work previously done for the red light sensor, the blue light sensor and the AND-Gate. See the respective pages for more details
image: 6_Users_schaluck_Desktop_igemmodel_completmodel.jpg

Equations

EnvZ x ˙ 1 = k ad x 2 - k ap x 1 RL+ k d2 x 4 - k b2 x 5 x 1 - k b3 * x 6 x 1 + k d3 x 7 EnvZ-P x ˙ 2 = k ap x 1 RL- k ad x 2 + k d1 x 3 - k b1 x 6 x 2 EnvZ-P.OmpR x ˙ 3 = -( k d1 + k pt ) x 3 + k b1 x 6 x 2 EnvZ.OmpR-P x ˙ 4 = k pt x 3 -( k ph + k d2 ) x 4 + k b2 x 5 x 1 OmpR-P x ˙ 5 = k d2 x 4 - k b2 x 5 x 1 OmpR x ˙ 6 = k d1 x 3 + k d3 x 7 - k b3 x 6 x 1 - k b1 x 6 x 2 EnvZ.OmpR x ˙ 7 = k ph x 4 - k d3 x 7 + k b3 x 6 x 1 Ycg F mRNA x ˙ 8 = k 1 - γ mRNA x 8 Ycg F inactive x ˙ 9 = k 3 x 8 - γ 2 x 9 -2 k dim x 9 2 B L 2 1 4 +( BL ) 2 +2 k dis x 10 - γ Protein x 9 Ycg F dimer x ˙ 10 = 2 k dim x 9 2 B L 2 1 4 +( BL ) 2 - k bind x 10 x 12 - k dis x 10 + k ubind x 13 - γ Protein x 10 Ycg E RNA x ˙ 11 = k 2 - γ mRNA x 11 Ycg E Protein x ˙ 12 = k 4 x 11 - γ 2 x 12 - k bind x 10 x 12 + k ubind x 13 - γ Protein x 12 YcgE.Ycg F complex x ˙ 13 = - k ubind x 13 + k bind x 10 x 12 tRNA x ˙ 14 = k t ( x 5 K1 ) 2 1+( x 5 K1 ) 2 -( γ 1 + k a ) x 14 + γ 2p x 15 +2 k 7p x 16 ( γ 3 k 7m ) ( x 14 γ 0 + x 14 ) 2 Aa-tRNA x ˙ 15 = k a x 14 -2 k 7p x 16 ( γ 3 k 7m ) ( x 14 γ 0 + x 14 ) 2 - γ 2 x 15 T7RNA P mRNA x ˙ 16 = k 7m ( 1- ( x 12 K1 ) 2 1+( x 12 K1 ) 2 )- γ 3 x 16 T7RNAP x ˙ 17 = k 7p x 16 ( γ 3 k 7m ) ( x 14 γ 0 + x 14 ) 2 - γ 4 x 17 lac Z mRNA x ˙ 18 = α M ( 1- ( x 17 K5 ) 2 1+( x 17 K5 ) 2 )- γ M x 18 β -Galactosidase x ˙ 19 = α B x 18 - γ B x 19 dye x ˙ 20 = α A x 19

Parameters

Parameter
Value
Unit
Name
Source
k ap
0.1
1 s
EnvZ autophosphorelation rate
[3]
k ad
0.001
1 s
EnvZ dephospholeration rate
[3]
k b1
0.5
1 s
binding rate EnvZ-P & OmpR
[3]
k d1
0.5
1 s
unbinding rate EnvZ-P.OmpR
[3]
k b2
0.05
1 s
binding rate EnvZ & OmpR-P
[3]
k d2
0.5
1 s
unbinding rate EnvZ.OmpR-P
[3]
k b3
0.5
1 s
binding rate EnvZ & OmpR
[3]
k d3
5
1 s
unbinding rate EnvZ.OmpR
[3]
k ph
0.05
1 s
dephosphorelation rate EnvZ.OmpR-P
[3]
k pt
1.5
1 s
phosphotransfer rate
[3]
k 1
1.54e-3
1 s
max transcription rate YcgF
[2]
k 2
0.848e-3
1 s
max transcription rate YcgE
[2]
k 3
0.167
1 s
max translation rate YcgF
[2]
k 4
0.167
1 s
max translation rate YcgE
[2]
k dim
0.008
1 s
dimerization rate YcgF
[2]
k dis
0.0058
1 s
dissociation rate YcgF dimer
[2]
k bind
100
1 s
binding rate YcgF dimer to YcgE
[2]
k ubind
1
1 s
unbinding rate YcgF.YcgE
[2]
γ mRNA
2.3105e-3
1 s
degradation mRNA YcgE/YcgF
[2]
γ Protein
1.9254e-5
1 s
degradation rate Protein YcgE/YcgF
[2]
k t
46.67 60
nM s
max transcription rate tRNA
[1]
k a
0.08 60
1 s
synthesis rate Aa-tRNA
[1]
k 7p
1.5625 60
nM s
max transcription rate T7RNAP
[1]
k 7m
268*0.05 60
1 s
max translation rate T7RNAP
[1]
k S
0.3
1 nM
AND Gate rate
[1]
γ 0
1
-
threshold Aa-tRNA
guessed
γ 1
1 60*60
1 s
degradation of tRNA
[1]
γ 2
1 40*60
1 s
degradation of Aa-tRNA
[1]
γ 3
1 4.4*60
1 s
degradation of T7RNAP mRNA
[1]
γ 4
46.67 40*60
1 s
degradation of T7RNAP
[1]
K1
5
nM
response param. OmpR-P,tRNA
guessed
K3
600
nM
response param. YcgE,T7RNAP
guessed
K5
k7p 4* γ 4
nM
response param T7RNAP,lacZ
guessed
α M
0.997 60
nM s
max transcription rate lacZ
[4]
α B
1.661e-5 60
1 s
max translation rate lacZ
[4]
α A
20 60
1 s
enzymatic reaction rate
[4]
γ M
0.411 60
1 s
degradation lacZ mRNA
[4]
γ B
8.331e-4 60
1 s
degradation β -Galactosidase
[4]

Initial Data

Name
Variable
Initial Value
Comment
Source
EnvZ
x 1
3500 0.60221
3500 molecules per cell
[3]
EnvZ-P
x 2
0
EnvZ-P.OmpR
x 3
0
EnvZ.OmpR-P
x 4
0
OmpR-P
x 5
0
OmpR
x 6
100 0.60221
100 molecules per cell
[3]
EnvZ.OmpR
x 7
0
Ycg F mRNA
x 8
k 1 γ mRNA
steady state
Ycg F inactive
x 9
k 3 γ Protein k 1 γ mRNA
steady state
Ycg F dimer
x 10
0
Ycg E mRNA
x 11
k 2 γ mRNA
steady state
YcgE
x 12
k 4 γ Protein k 2 γ mRNA
steady state
YcgE.YcgF
x 13
0
tRNA
x 14
0
Aa-tRNA
x 15
0
T7RNA P mRNA
x 16
0
T7RNAP
x 17
0
lac Z mRNA
x 18
0
β -Galactosidase
x 19
0
dye
x 20
0

Simulation

In all graphics the unit for time is seconds. Both intensities were varieed, but the intensities for blue and red light were scaled by a factor of 1 10 and 10 respectively. The duration for blue light was scaled by a factor of 1 5 . These scaling factors were used to have vary between the same intensities and exposure times that were used in the simulation of the seperate parts. Although it would be desirable only to vary only the characteristics of the light of the wavelength that affects the corresponding part of the system, this was not done since the equations intrinsically provide that each wavelength only affects one part of the system and computations were a lot faster like this.
Firstly the activation time of the blue light sensor part was simulated since the pathway was extended by the T7 Polymerase mRNA production. The threshold for the activation was 1nM T7pol mRNA.

image: 0_Users_schaluck_Desktop_cbactive.png
Values of -20 indicate that the threshold was not passed in the 60,000 seconds of simulation. Other values are in seconds. We see that if a minimum exposure time is exceeded the activation time only depends on the light intensity. This should coincide with real behavior. The spikes are due to numeric inaccuracies.
Secondly the deactivation time, the time at which the concentration dropped below the threshold, was simulated.
image: 1_Users_schaluck_Desktop_cbdeactive.png
We can observe that although the deactivation time depends on both intensity and exposure time but saturates very fast with respect to both variables.
The same procedure was done for the red light sensor part. Here the concentration of tRNA served as reference and a threshold of 30nM was used.
image: 3_Users_schaluck_Desktop_cractive.png
Here accurate predictions about the behavior is difficult since the values are monotonic neither in intensity nor exposure time. Still the activation time seems to be more or less independent of exposure time after a certain amount of time. The threshold is probably due to the fact that the signal needs to cascade down a pathway that involves slower reactions which dampen the signal speed if the original signal is no longer present. Wether this behavior is realistic is questionable.
image: 4_Users_schaluck_Desktop_crdeactive.png
The deactivation time of the red light sensor part also depends on intensity and exposure time but saturation is achieved a lot slower than for the blue light sensor part.
Finally the output of dye was simulated.
We can see that the output of dye depends on both intensity and exposure time. Of course here a simulation where both intensity and exposure time are varied independently for each wavelength would be desirable. This would mean an scaling in potency by 4 instead of 2 in the computation time and would lead to the question how to present the data in a good fashion. Also it would be questionable wether any real information would be gained due to the inaccuracy in the parameters.

conclusion

All simulations should be treated with extreme care, since some parameters were only guessed and the sensitivity to errors in the guessing increases with the complexity of the whole system. Hence the results should be only used as indicator for the qualitative behavior of the system and not the quantitative behavior. Unfortunately our assays did not provide enough data to make reasonable assumptions about the missing parameters, but if this data would be available the model could also be further refined.

References

1. PKU Beijing 2009, 'AND Gate 1' (2009).
2. KU Leuven 2009, 'Blue Light Receptor: Modeling' (2009).
3. Igoshin, Oleg A and Alves, Rui and Savageau, Michael A, 'Hysteretic and graded responses in bacterial two-component signal transduction', Mol Microbiol (2008), 1196-215.
4. Yildirim, N and Santillan, M and Horike, D and Mackey, MC, 'Dynamics and bistability in a reduced model of the lac operon', CHAOS (2004), 279-292.