http://2010.igem.org/wiki/index.php?title=Special:Contributions/Lena&feed=atom&limit=50&target=Lena&year=&month=2010.igem.org - User contributions [en]2024-03-29T07:31:39ZFrom 2010.igem.orgMediaWiki 1.16.5http://2010.igem.org/Team:UT-Tokyo/Sudoku_modelingTeam:UT-Tokyo/Sudoku modeling2010-11-29T08:29:40Z<p>Lena: /* Whole system */</p>
<hr />
<div>__NOTOC__{{UT-Tokyo_CSS3}}{{UT-Tokyo_CSS_p}}{{UT-Tokyo_Head}}<br />
<br />
= '''Sudoku''' =<br />
<html><br />
<div><br />
<ul id="inpagemenu"><br />
<li><a href="/Team:UT-Tokyo/Sudoku_abstract" id="abstract">Introduction</a></li><br />
<li><a href="/Team:UT-Tokyo/Sudoku_construct" id="construct">System</a></li><br />
<li><span>Modeling</span></li><br />
<li><a href="/Team:UT-Tokyo/Sudoku_experiments" id="experiment">Experiments</a></li><br />
<li><a href="/Team:UT-Tokyo/Sudoku_perspective" id="perspective">Perspective</a></li><br />
<li><a href="/Team:UT-Tokyo/Sudoku_reference" id="reference">Reference</a></li><br />
</ul><br />
</div><br />
<div id="clear"></div><br />
</html><br />
== '''4C3leak switch''' ==<br />
<br />
We modeled our 4C3 leak switch as deterministic ODE system.<br />
<br />
==='''Modelling'''===<br />
<br />
===='''Assumption'''====<br />
<br />
1.We treated DNA as continuous variable, since there could be hundreds of plasmid in one E.coli (multicopy plasmid).<br />
There are 32 kinds of variables that stand for DNA concentration corresponding to their different internal states.<br />
<br />
2.We assumed that cre recombinase operate as tetramer, and other recombinase as dimer (but this seems not to be relevant).<br />
<br />
3.We assumed that association and dissociation of DNA recombinase to DNA is sufficiently fast so that equilibrate in the timescale of the whole switch.<br />
<br />
4.We ignored the reverse reaction of DNA recombinase because we used irreversible sequence (lox66, lox71 etc).<br />
<br />
===='''Variable'''====<br />
<br />
Basically, "pn" denotes protein concentration and "rn" denotes mRNA concentration<br />
<br />
p1,p2,p3,p4 : concentration of 4 recombinase protein<br />
<br />
r1,r2,r3,r4 : concentration of 4 recombinass mRNA<br />
<br />
u1,u2,u3,u4 : concentration of input mRNA<br />
<br />
pc,rc : concentration of cre recombinase protein and mRNA<br />
<br />
ps,rs : concentration of SP6 polymerase protein and mRNA<br />
<br />
Variables that denote concentration of DNA in each state.<br />
"a","b","c","d","e" means there are four,three,two,one,zero recombinases in the DNA, respectively.<br />
And the subscripts specify which recobinases remain, while the numbers in bracket mean whether the DNA has been cut by cre recombinase( (1) has not been cut, (2) has been cut).<br />
<br />
{| border="1"<br />
|<br />
|all<br />
|123<br />
|124<br />
|134<br />
|234<br />
<br />
|-<br />
|not cut by cre<br />
|a(1)<br />
|b123(1)<br />
|b124(1)<br />
|b134(1)<br />
|b234(1)<br />
|-<br />
|cut by cre<br />
|a(2)<br />
|b123(2)<br />
|b124(2)<br />
|b134(2)<br />
|b234(2)<br />
|}<br />
<br />
<br />
{| border = "1"<br />
|12<br />
|13<br />
|14<br />
|23<br />
|24<br />
|34<br />
|1<br />
|2<br />
|3<br />
|4<br />
|none<br />
|-<br />
|c12(1)<br />
|c13(1)<br />
|c14(1)<br />
|c23(1)<br />
|c24(1)<br />
|c34(1)<br />
|d1(1)<br />
|d2(1)<br />
|d3(1)<br />
|d4(1)<br />
|e(1)<br />
|-<br />
|c12(2)<br />
|c13(2)<br />
|c14(2)<br />
|c23(2)<br />
|c24(2)<br />
|c34(2)<br />
|d1(2)<br />
|d2(2)<br />
|d3(2)<br />
|d4(2)<br />
|e(2)<br />
|}<br />
<br />
====''' Parameters '''====<br />
k0 : mRNA translation rate<br />
<br />
k1 : protein degradation rate<br />
<br />
k2 : mRNA translation rate<br />
<br />
k3 : cre protein degradation rate<br />
<br />
l0 : transcription speed per unit concentration of RNA polymearase<br />
<br />
l1 : mRNA degradation rate<br />
<br />
l2 : cre mRNA degradation rate<br />
<br />
K : recombinase binding constant<br />
<br />
v : recombinase reaction rate<br />
<br />
pT7 : concentration o T7 polymerase<br />
<br />
p : terminator leak probability<br />
<br />
<br />
<br />
===='''Equations'''====<br />
<br />
[[Image:Eqs1.jpg]]<br />
<br />
[[Image:eqs_cs.jpg]]<br />
<br />
[[Image:eqsdna1.jpg]]<br />
<br />
[[Image:eqsdna2.jpg]]<br />
<br />
==='''Methods'''===<br />
<br />
We used our original python program (4 th order explicit Runge-Kutta algorithm) to solve thess equations.<br />
<br />
==='''Results'''===<br />
<br />
We input signal 1,2,4 and simulated the time evolution of the system.<br />
<br />
The time course of DNA concentration<br />
<br />
[[Image:Sim_res.JPG]]<br />
<br />
The time course of mRNA concentration. Note that the output of this system is not protein but mRNA.<br />
<br />
[[Image:Rna.JPG]]<br />
<br />
==='''Discussion'''===<br />
<br />
It is confirmed that the system works as expected with a certain combination of parameters.<br />
<br />
In addition, we changed many parameters, especially,l0,v(whose values are unclear) and p(leakiness of terminator) , to see whether the device is able to operate within a large part of parameter space.<br />
<br />
Although increasing p basically increases the concentration of correct output mRNA, it also increases incorrect mRNA. <br />
On the other hand, decreasing p generally decreases both the concentration of correct mRNA and incorrect mRNA. Besides, too low value of p cannot leads to failure operation because cre protein concentration cannot get high enough.<br />
Careful test have shown that there is certain value of p that [correct mRNA]/[incorrect mRNA] takes maximum value. In this case, the optimal value is approximately p = 0.001 and the ratio is 10^(-6).<br />
<br />
Changing the value of v and l0 doesn't affect the function of the circuits qualitatively although it changed the time the systems takes to reach steady state.<br />
<br />
As a result, the circuit can function correctly within the range of 0.001< p < 0.1, 0.001 < v <1.0, 0.01 < l0 < 1.0 , demonstrating its robustness.<br />
This robustness may come from the assumption that DNA recombination is irreversible, which is supported by our design of the system.<br />
<br />
==='''conclusion'''===<br />
<br />
The correct operation of the switch was confirmed. By changing parameter exponentially, the robustness of the system was shown.<br />
<br />
=='''An indifferent sudoku cell simulation '''==<br />
<br />
-------including MS2 phage infection-------<br />
<br />
[[Image:What_Sudoku_99.png|200px|thumb|What's Sudoku?]]<br />
<br />
<br />
We made computer simulations to confirm that an empty sudoku cell get signals from other cells and differentiate to the correct answer in 4*4 sudoku.<br />
<br />
<br />
===='''Modeling'''====<br />
<br />
'''Variable'''<br />
<br />
U: uninfected coli<br />
<br />
I: coli which got '1' signal<br />
<br />
Io1: other once infected coli<br />
<br />
I2: coli which got '1''2' signal<br />
<br />
Io2: other second infected coli<br />
<br />
I3: coli which got '1''2''3' signal<br />
<br />
Io3: other third infected coli<br />
<br />
B4: '4' coli which began phage making<br />
<br />
Bo: other coli which began phage making<br />
<br />
P4: free phage of '4'<br />
<br />
Po: other free phage<br />
<br />
S1: signal virus with information of number '1' which come from other sudoku cell<br />
<br />
S2: signal virus with information of number '2' which come from other sudoku cell <br />
<br />
S3: signal virus with information of number '3' which come from other sudoku cell<br />
<br />
S4: signal virus with information of number '4' which come from other sudoku cell<br />
<br />
<br />
<br />
'''Parameters'''<br />
<br />
g: growth rate of coli <br />
<br />
k1: rate of infection<br />
<br />
k2: death rate of uninfected<br />
<br />
k3: lysis rate of infected<br />
<br />
k4: rate of phage producing<br />
<br />
k5: phage death rate<br />
<br />
k6: terminator leak rate<br />
<br />
Given parameters are<br />
unit value <br />
*g /min 0.0113 <br />
reference:Investigation of Bacteriophage MS2 Viral Dynamics Using Model Discrimination <br />
Analysis and the Implications for Phage Therapy<br />
Rishi Jain, Andrea L. Knorr, Joseph Bernacki, and Ranjan Srivastava(2006)<br />
<br />
*k2 /min 0 <br />
reference:Investigation of Bacteriophage MS2 Viral Dynamics Using Model Discrimination <br />
Analysis and the Implications for Phage Therapy<br />
Rishi Jain, Andrea L. Knorr, Joseph Bernacki, and Ranjan Srivastava(2006) <br />
<br />
*k3 /min 0.0167 <br />
reference:Investigation of Bacteriophage MS2 Viral Dynamics Using Model Discrimination <br />
Analysis and the Implications for Phage Therapy<br />
Rishi Jain, Andrea L. Knorr, Joseph Bernacki, and Ranjan Srivastava(2006)<br />
<br />
*k5 /min 0.25 <br />
reference: Dynamical Determinants of Drug-Inducible Gene Expression in a Single Bacterium<br />
Thuc T. Le, Thierry Emonet, Sebastien Harlepp, Călin C. Guet, and Philippe Cluzel(2006) <br />
<br />
<br />
<br />
Arbitrary parameters we can change are<br />
<br />
* k1(rate of infection) : k1 relates to the specificity of antisenseRNA. <br />
<br />
* k4(rate of phage producing) : k4 relates to the strength of promoter. <br />
<br />
* k6(terminator leak rate) : k6 express terminator leak. <br />
<br />
We change these parameters, and confirmed its effect.<br />
<br />
<br />
<br />
'''Equations'''<br />
<br />
We assume that cell lysis caused by MS2 phage is sufficiently rapid than coli death rate.<br />
<br />
<br />
[[Image:Model-eq.png|680px|Equations used in this modeling]]<br />
<br />
===='''Method'''====<br />
<br />
We used original matlab program.<br />
<br />
We set conditions as following.<br />
-when t = 10(min), signal of number '1' comes.(step input)<br />
-when t = 30(min), signal of number '2' comes.(step input)<br />
-when t = 50(min), signal of number '3' comes.(step input)<br />
<br />
<br />
===='''Results'''====<br />
<br />
The initial condition of uninfected coli concentration is set to be 1, and observed time evolution.<br />
<br />
<br />
[[Image:Whole1.png]]<br />
<br />
brue(U): initial concentration is 1. After got signal '1', it decrease suddenly. <br />
<br />
green: I1 begins increasing after getting first signal '1'.<br />
<br />
<br />
<br />
<br />
[[Image:MS2-0.5,0.6,0.png]]<br />
<br />
k1:0.4, k4:0.7, k6:0.1<br />
red: The time course of virus concentration which has the information of number '4'.<br />
<br />
blue: The time course of virus concentration which has the information of number '1'or'2'or'3'.<br />
<br />
<br />
<br />
<br />
[[Image:MS2 2.png]]<br />
<br />
k1:0.4, k4:0.7, k6:0.01<br />
<br />
red: The time course of virus concentration which has the information of number '4'.<br />
<br />
blue: The time course of virus concentration which has the information of number '1'or'2'or'3'.<br />
<br />
As terminator leak rate is less than the previous trial, it takes longer time to start increasing. the concentration of Virus '4' clearly increases comparing to others.<br />
<br />
<br />
<br />
[[Image:MS2 4.png]]<br />
<br />
k1:0.4, k4:0.1, k6:0.01<br />
<br />
<br />
<br />
[[Image:MS2 3.png ]]<br />
<br />
k1:0.8, k4:0.7, k6:0.01<br />
<br />
<br />
===='''Discussion & Conclusion'''====<br />
<br />
It is confirmed that when one cell get '1''2''3'signals in order, the cell differentiate into '4'.<br />
<br />
The rate of infection(k1) corresponds to antisense RNA specificity. <br />
Intuitively, when k1 is too high, coli become sensitive to not only target signals but also wrong signals. Our simulation also indicate when k1 becomes higher, the concentration of Virus'4' decrease.<br />
<br />
The rate of phage producing(k4) corresponds to the strength of promoter. <br />
If promoter is too strong, the amount of the phage increase but it also raises terminator leak. However in our simulation, we confirm the dependence between the rate of terminator leak and k1 is rather small. <br />
<br />
The rate of terminator leak relates to the time from receiving signals to the differentiation, and to the accuracy of finding the correct answer.<br />
According to the simulation, we find the clear difference between Pharge'4' and other Pharge'1''2''3'in 100-300 minutes, when the terminator leak rate is around 0.01.<br />
<br />
<br />
<br />
<br />
== '''Whole system''' ==<br />
<br />
finally, we made a simulation of solving whole 4*4 sudoku.<br />
<br />
This simulation is based on 1 cell simulation written above.<br />
<br />
The result of this simulation can be seen below.<br />
<br />
<br />
#video(http://www.youtube.com/watch?v=ZFwbP4zg-H4)<br />
<br />
<br />
<br />
<br />
<br />
{{UT-Tokyo_Foot}}</div>Lenahttp://2010.igem.org/Team:UT-Tokyo/Sudoku_modelingTeam:UT-Tokyo/Sudoku modeling2010-11-29T08:23:04Z<p>Lena: /* Whole system */</p>
<hr />
<div>__NOTOC__{{UT-Tokyo_CSS3}}{{UT-Tokyo_CSS_p}}{{UT-Tokyo_Head}}<br />
<br />
= '''Sudoku''' =<br />
<html><br />
<div><br />
<ul id="inpagemenu"><br />
<li><a href="/Team:UT-Tokyo/Sudoku_abstract" id="abstract">Introduction</a></li><br />
<li><a href="/Team:UT-Tokyo/Sudoku_construct" id="construct">System</a></li><br />
<li><span>Modeling</span></li><br />
<li><a href="/Team:UT-Tokyo/Sudoku_experiments" id="experiment">Experiments</a></li><br />
<li><a href="/Team:UT-Tokyo/Sudoku_perspective" id="perspective">Perspective</a></li><br />
<li><a href="/Team:UT-Tokyo/Sudoku_reference" id="reference">Reference</a></li><br />
</ul><br />
</div><br />
<div id="clear"></div><br />
</html><br />
== '''4C3leak switch''' ==<br />
<br />
We modeled our 4C3 leak switch as deterministic ODE system.<br />
<br />
==='''Modelling'''===<br />
<br />
===='''Assumption'''====<br />
<br />
1.We treated DNA as continuous variable, since there could be hundreds of plasmid in one E.coli (multicopy plasmid).<br />
There are 32 kinds of variables that stand for DNA concentration corresponding to their different internal states.<br />
<br />
2.We assumed that cre recombinase operate as tetramer, and other recombinase as dimer (but this seems not to be relevant).<br />
<br />
3.We assumed that association and dissociation of DNA recombinase to DNA is sufficiently fast so that equilibrate in the timescale of the whole switch.<br />
<br />
4.We ignored the reverse reaction of DNA recombinase because we used irreversible sequence (lox66, lox71 etc).<br />
<br />
===='''Variable'''====<br />
<br />
Basically, "pn" denotes protein concentration and "rn" denotes mRNA concentration<br />
<br />
p1,p2,p3,p4 : concentration of 4 recombinase protein<br />
<br />
r1,r2,r3,r4 : concentration of 4 recombinass mRNA<br />
<br />
u1,u2,u3,u4 : concentration of input mRNA<br />
<br />
pc,rc : concentration of cre recombinase protein and mRNA<br />
<br />
ps,rs : concentration of SP6 polymerase protein and mRNA<br />
<br />
Variables that denote concentration of DNA in each state.<br />
"a","b","c","d","e" means there are four,three,two,one,zero recombinases in the DNA, respectively.<br />
And the subscripts specify which recobinases remain, while the numbers in bracket mean whether the DNA has been cut by cre recombinase( (1) has not been cut, (2) has been cut).<br />
<br />
{| border="1"<br />
|<br />
|all<br />
|123<br />
|124<br />
|134<br />
|234<br />
<br />
|-<br />
|not cut by cre<br />
|a(1)<br />
|b123(1)<br />
|b124(1)<br />
|b134(1)<br />
|b234(1)<br />
|-<br />
|cut by cre<br />
|a(2)<br />
|b123(2)<br />
|b124(2)<br />
|b134(2)<br />
|b234(2)<br />
|}<br />
<br />
<br />
{| border = "1"<br />
|12<br />
|13<br />
|14<br />
|23<br />
|24<br />
|34<br />
|1<br />
|2<br />
|3<br />
|4<br />
|none<br />
|-<br />
|c12(1)<br />
|c13(1)<br />
|c14(1)<br />
|c23(1)<br />
|c24(1)<br />
|c34(1)<br />
|d1(1)<br />
|d2(1)<br />
|d3(1)<br />
|d4(1)<br />
|e(1)<br />
|-<br />
|c12(2)<br />
|c13(2)<br />
|c14(2)<br />
|c23(2)<br />
|c24(2)<br />
|c34(2)<br />
|d1(2)<br />
|d2(2)<br />
|d3(2)<br />
|d4(2)<br />
|e(2)<br />
|}<br />
<br />
====''' Parameters '''====<br />
k0 : mRNA translation rate<br />
<br />
k1 : protein degradation rate<br />
<br />
k2 : mRNA translation rate<br />
<br />
k3 : cre protein degradation rate<br />
<br />
l0 : transcription speed per unit concentration of RNA polymearase<br />
<br />
l1 : mRNA degradation rate<br />
<br />
l2 : cre mRNA degradation rate<br />
<br />
K : recombinase binding constant<br />
<br />
v : recombinase reaction rate<br />
<br />
pT7 : concentration o T7 polymerase<br />
<br />
p : terminator leak probability<br />
<br />
<br />
<br />
===='''Equations'''====<br />
<br />
[[Image:Eqs1.jpg]]<br />
<br />
[[Image:eqs_cs.jpg]]<br />
<br />
[[Image:eqsdna1.jpg]]<br />
<br />
[[Image:eqsdna2.jpg]]<br />
<br />
==='''Methods'''===<br />
<br />
We used our original python program (4 th order explicit Runge-Kutta algorithm) to solve thess equations.<br />
<br />
==='''Results'''===<br />
<br />
We input signal 1,2,4 and simulated the time evolution of the system.<br />
<br />
The time course of DNA concentration<br />
<br />
[[Image:Sim_res.JPG]]<br />
<br />
The time course of mRNA concentration. Note that the output of this system is not protein but mRNA.<br />
<br />
[[Image:Rna.JPG]]<br />
<br />
==='''Discussion'''===<br />
<br />
It is confirmed that the system works as expected with a certain combination of parameters.<br />
<br />
In addition, we changed many parameters, especially,l0,v(whose values are unclear) and p(leakiness of terminator) , to see whether the device is able to operate within a large part of parameter space.<br />
<br />
Although increasing p basically increases the concentration of correct output mRNA, it also increases incorrect mRNA. <br />
On the other hand, decreasing p generally decreases both the concentration of correct mRNA and incorrect mRNA. Besides, too low value of p cannot leads to failure operation because cre protein concentration cannot get high enough.<br />
Careful test have shown that there is certain value of p that [correct mRNA]/[incorrect mRNA] takes maximum value. In this case, the optimal value is approximately p = 0.001 and the ratio is 10^(-6).<br />
<br />
Changing the value of v and l0 doesn't affect the function of the circuits qualitatively although it changed the time the systems takes to reach steady state.<br />
<br />
As a result, the circuit can function correctly within the range of 0.001< p < 0.1, 0.001 < v <1.0, 0.01 < l0 < 1.0 , demonstrating its robustness.<br />
This robustness may come from the assumption that DNA recombination is irreversible, which is supported by our design of the system.<br />
<br />
==='''conclusion'''===<br />
<br />
The correct operation of the switch was confirmed. By changing parameter exponentially, the robustness of the system was shown.<br />
<br />
=='''An indifferent sudoku cell simulation '''==<br />
<br />
-------including MS2 phage infection-------<br />
<br />
[[Image:What_Sudoku_99.png|200px|thumb|What's Sudoku?]]<br />
<br />
<br />
We made computer simulations to confirm that an empty sudoku cell get signals from other cells and differentiate to the correct answer in 4*4 sudoku.<br />
<br />
<br />
===='''Modeling'''====<br />
<br />
'''Variable'''<br />
<br />
U: uninfected coli<br />
<br />
I: coli which got '1' signal<br />
<br />
Io1: other once infected coli<br />
<br />
I2: coli which got '1''2' signal<br />
<br />
Io2: other second infected coli<br />
<br />
I3: coli which got '1''2''3' signal<br />
<br />
Io3: other third infected coli<br />
<br />
B4: '4' coli which began phage making<br />
<br />
Bo: other coli which began phage making<br />
<br />
P4: free phage of '4'<br />
<br />
Po: other free phage<br />
<br />
S1: signal virus with information of number '1' which come from other sudoku cell<br />
<br />
S2: signal virus with information of number '2' which come from other sudoku cell <br />
<br />
S3: signal virus with information of number '3' which come from other sudoku cell<br />
<br />
S4: signal virus with information of number '4' which come from other sudoku cell<br />
<br />
<br />
<br />
'''Parameters'''<br />
<br />
g: growth rate of coli <br />
<br />
k1: rate of infection<br />
<br />
k2: death rate of uninfected<br />
<br />
k3: lysis rate of infected<br />
<br />
k4: rate of phage producing<br />
<br />
k5: phage death rate<br />
<br />
k6: terminator leak rate<br />
<br />
Given parameters are<br />
unit value <br />
*g /min 0.0113 <br />
reference:Investigation of Bacteriophage MS2 Viral Dynamics Using Model Discrimination <br />
Analysis and the Implications for Phage Therapy<br />
Rishi Jain, Andrea L. Knorr, Joseph Bernacki, and Ranjan Srivastava(2006)<br />
<br />
*k2 /min 0 <br />
reference:Investigation of Bacteriophage MS2 Viral Dynamics Using Model Discrimination <br />
Analysis and the Implications for Phage Therapy<br />
Rishi Jain, Andrea L. Knorr, Joseph Bernacki, and Ranjan Srivastava(2006) <br />
<br />
*k3 /min 0.0167 <br />
reference:Investigation of Bacteriophage MS2 Viral Dynamics Using Model Discrimination <br />
Analysis and the Implications for Phage Therapy<br />
Rishi Jain, Andrea L. Knorr, Joseph Bernacki, and Ranjan Srivastava(2006)<br />
<br />
*k5 /min 0.25 <br />
reference: Dynamical Determinants of Drug-Inducible Gene Expression in a Single Bacterium<br />
Thuc T. Le, Thierry Emonet, Sebastien Harlepp, Călin C. Guet, and Philippe Cluzel(2006) <br />
<br />
<br />
<br />
Arbitrary parameters we can change are<br />
<br />
* k1(rate of infection) : k1 relates to the specificity of antisenseRNA. <br />
<br />
* k4(rate of phage producing) : k4 relates to the strength of promoter. <br />
<br />
* k6(terminator leak rate) : k6 express terminator leak. <br />
<br />
We change these parameters, and confirmed its effect.<br />
<br />
<br />
<br />
'''Equations'''<br />
<br />
We assume that cell lysis caused by MS2 phage is sufficiently rapid than coli death rate.<br />
<br />
<br />
[[Image:Model-eq.png|680px|Equations used in this modeling]]<br />
<br />
===='''Method'''====<br />
<br />
We used original matlab program.<br />
<br />
We set conditions as following.<br />
-when t = 10(min), signal of number '1' comes.(step input)<br />
-when t = 30(min), signal of number '2' comes.(step input)<br />
-when t = 50(min), signal of number '3' comes.(step input)<br />
<br />
<br />
===='''Results'''====<br />
<br />
The initial condition of uninfected coli concentration is set to be 1, and observed time evolution.<br />
<br />
<br />
[[Image:Whole1.png]]<br />
<br />
brue(U): initial concentration is 1. After got signal '1', it decrease suddenly. <br />
<br />
green: I1 begins increasing after getting first signal '1'.<br />
<br />
<br />
<br />
<br />
[[Image:MS2-0.5,0.6,0.png]]<br />
<br />
k1:0.4, k4:0.7, k6:0.1<br />
red: The time course of virus concentration which has the information of number '4'.<br />
<br />
blue: The time course of virus concentration which has the information of number '1'or'2'or'3'.<br />
<br />
<br />
<br />
<br />
[[Image:MS2 2.png]]<br />
<br />
k1:0.4, k4:0.7, k6:0.01<br />
<br />
red: The time course of virus concentration which has the information of number '4'.<br />
<br />
blue: The time course of virus concentration which has the information of number '1'or'2'or'3'.<br />
<br />
As terminator leak rate is less than the previous trial, it takes longer time to start increasing. the concentration of Virus '4' clearly increases comparing to others.<br />
<br />
<br />
<br />
[[Image:MS2 4.png]]<br />
<br />
k1:0.4, k4:0.1, k6:0.01<br />
<br />
<br />
<br />
[[Image:MS2 3.png ]]<br />
<br />
k1:0.8, k4:0.7, k6:0.01<br />
<br />
<br />
===='''Discussion & Conclusion'''====<br />
<br />
It is confirmed that when one cell get '1''2''3'signals in order, the cell differentiate into '4'.<br />
<br />
The rate of infection(k1) corresponds to antisense RNA specificity. <br />
Intuitively, when k1 is too high, coli become sensitive to not only target signals but also wrong signals. Our simulation also indicate when k1 becomes higher, the concentration of Virus'4' decrease.<br />
<br />
The rate of phage producing(k4) corresponds to the strength of promoter. <br />
If promoter is too strong, the amount of the phage increase but it also raises terminator leak. However in our simulation, we confirm the dependence between the rate of terminator leak and k1 is rather small. <br />
<br />
The rate of terminator leak relates to the time from receiving signals to the differentiation, and to the accuracy of finding the correct answer.<br />
According to the simulation, we find the clear difference between Pharge'4' and other Pharge'1''2''3'in 100-300 minutes, when the terminator leak rate is around 0.01.<br />
<br />
<br />
<br />
<br />
== '''Whole system''' ==<br />
<br />
finally, we made a simulation of solving whole 4*4 sudoku.<br />
<br />
This simulation is based on 1 cell simulation written above.<br />
<br />
The result of this simulation can be seen below.<br />
<br />
<br />
http://www.youtube.com/watch?v=ZFwbP4zg-H4<br />
<br />
&u2b(http://www.youtube.com/watch?v=ZFwbP4zg-H4){640,385}<br />
<br />
<br />
<br />
{{UT-Tokyo_Foot}}</div>Lenahttp://2010.igem.org/Team:UT-Tokyo/Sudoku_modelingTeam:UT-Tokyo/Sudoku modeling2010-11-29T08:07:56Z<p>Lena: /* Whole system */</p>
<hr />
<div>__NOTOC__{{UT-Tokyo_CSS3}}{{UT-Tokyo_CSS_p}}{{UT-Tokyo_Head}}<br />
<br />
= '''Sudoku''' =<br />
<html><br />
<div><br />
<ul id="inpagemenu"><br />
<li><a href="/Team:UT-Tokyo/Sudoku_abstract" id="abstract">Introduction</a></li><br />
<li><a href="/Team:UT-Tokyo/Sudoku_construct" id="construct">System</a></li><br />
<li><span>Modeling</span></li><br />
<li><a href="/Team:UT-Tokyo/Sudoku_experiments" id="experiment">Experiments</a></li><br />
<li><a href="/Team:UT-Tokyo/Sudoku_perspective" id="perspective">Perspective</a></li><br />
<li><a href="/Team:UT-Tokyo/Sudoku_reference" id="reference">Reference</a></li><br />
</ul><br />
</div><br />
<div id="clear"></div><br />
</html><br />
== '''4C3leak switch''' ==<br />
<br />
We modeled our 4C3 leak switch as deterministic ODE system.<br />
<br />
==='''Modelling'''===<br />
<br />
===='''Assumption'''====<br />
<br />
1.We treated DNA as continuous variable, since there could be hundreds of plasmid in one E.coli (multicopy plasmid).<br />
There are 32 kinds of variables that stand for DNA concentration corresponding to their different internal states.<br />
<br />
2.We assumed that cre recombinase operate as tetramer, and other recombinase as dimer (but this seems not to be relevant).<br />
<br />
3.We assumed that association and dissociation of DNA recombinase to DNA is sufficiently fast so that equilibrate in the timescale of the whole switch.<br />
<br />
4.We ignored the reverse reaction of DNA recombinase because we used irreversible sequence (lox66, lox71 etc).<br />
<br />
===='''Variable'''====<br />
<br />
Basically, "pn" denotes protein concentration and "rn" denotes mRNA concentration<br />
<br />
p1,p2,p3,p4 : concentration of 4 recombinase protein<br />
<br />
r1,r2,r3,r4 : concentration of 4 recombinass mRNA<br />
<br />
u1,u2,u3,u4 : concentration of input mRNA<br />
<br />
pc,rc : concentration of cre recombinase protein and mRNA<br />
<br />
ps,rs : concentration of SP6 polymerase protein and mRNA<br />
<br />
Variables that denote concentration of DNA in each state.<br />
"a","b","c","d","e" means there are four,three,two,one,zero recombinases in the DNA, respectively.<br />
And the subscripts specify which recobinases remain, while the numbers in bracket mean whether the DNA has been cut by cre recombinase( (1) has not been cut, (2) has been cut).<br />
<br />
{| border="1"<br />
|<br />
|all<br />
|123<br />
|124<br />
|134<br />
|234<br />
<br />
|-<br />
|not cut by cre<br />
|a(1)<br />
|b123(1)<br />
|b124(1)<br />
|b134(1)<br />
|b234(1)<br />
|-<br />
|cut by cre<br />
|a(2)<br />
|b123(2)<br />
|b124(2)<br />
|b134(2)<br />
|b234(2)<br />
|}<br />
<br />
<br />
{| border = "1"<br />
|12<br />
|13<br />
|14<br />
|23<br />
|24<br />
|34<br />
|1<br />
|2<br />
|3<br />
|4<br />
|none<br />
|-<br />
|c12(1)<br />
|c13(1)<br />
|c14(1)<br />
|c23(1)<br />
|c24(1)<br />
|c34(1)<br />
|d1(1)<br />
|d2(1)<br />
|d3(1)<br />
|d4(1)<br />
|e(1)<br />
|-<br />
|c12(2)<br />
|c13(2)<br />
|c14(2)<br />
|c23(2)<br />
|c24(2)<br />
|c34(2)<br />
|d1(2)<br />
|d2(2)<br />
|d3(2)<br />
|d4(2)<br />
|e(2)<br />
|}<br />
<br />
====''' Parameters '''====<br />
k0 : mRNA translation rate<br />
<br />
k1 : protein degradation rate<br />
<br />
k2 : mRNA translation rate<br />
<br />
k3 : cre protein degradation rate<br />
<br />
l0 : transcription speed per unit concentration of RNA polymearase<br />
<br />
l1 : mRNA degradation rate<br />
<br />
l2 : cre mRNA degradation rate<br />
<br />
K : recombinase binding constant<br />
<br />
v : recombinase reaction rate<br />
<br />
pT7 : concentration o T7 polymerase<br />
<br />
p : terminator leak probability<br />
<br />
<br />
<br />
===='''Equations'''====<br />
<br />
[[Image:Eqs1.jpg]]<br />
<br />
[[Image:eqs_cs.jpg]]<br />
<br />
[[Image:eqsdna1.jpg]]<br />
<br />
[[Image:eqsdna2.jpg]]<br />
<br />
==='''Methods'''===<br />
<br />
We used our original python program (4 th order explicit Runge-Kutta algorithm) to solve thess equations.<br />
<br />
==='''Results'''===<br />
<br />
We input signal 1,2,4 and simulated the time evolution of the system.<br />
<br />
The time course of DNA concentration<br />
<br />
[[Image:Sim_res.JPG]]<br />
<br />
The time course of mRNA concentration. Note that the output of this system is not protein but mRNA.<br />
<br />
[[Image:Rna.JPG]]<br />
<br />
==='''Discussion'''===<br />
<br />
It is confirmed that the system works as expected with a certain combination of parameters.<br />
<br />
In addition, we changed many parameters, especially,l0,v(whose values are unclear) and p(leakiness of terminator) , to see whether the device is able to operate within a large part of parameter space.<br />
<br />
Although increasing p basically increases the concentration of correct output mRNA, it also increases incorrect mRNA. <br />
On the other hand, decreasing p generally decreases both the concentration of correct mRNA and incorrect mRNA. Besides, too low value of p cannot leads to failure operation because cre protein concentration cannot get high enough.<br />
Careful test have shown that there is certain value of p that [correct mRNA]/[incorrect mRNA] takes maximum value. In this case, the optimal value is approximately p = 0.001 and the ratio is 10^(-6).<br />
<br />
Changing the value of v and l0 doesn't affect the function of the circuits qualitatively although it changed the time the systems takes to reach steady state.<br />
<br />
As a result, the circuit can function correctly within the range of 0.001< p < 0.1, 0.001 < v <1.0, 0.01 < l0 < 1.0 , demonstrating its robustness.<br />
This robustness may come from the assumption that DNA recombination is irreversible, which is supported by our design of the system.<br />
<br />
==='''conclusion'''===<br />
<br />
The correct operation of the switch was confirmed. By changing parameter exponentially, the robustness of the system was shown.<br />
<br />
=='''An indifferent sudoku cell simulation '''==<br />
<br />
-------including MS2 phage infection-------<br />
<br />
[[Image:What_Sudoku_99.png|200px|thumb|What's Sudoku?]]<br />
<br />
<br />
We made computer simulations to confirm that an empty sudoku cell get signals from other cells and differentiate to the correct answer in 4*4 sudoku.<br />
<br />
<br />
===='''Modeling'''====<br />
<br />
'''Variable'''<br />
<br />
U: uninfected coli<br />
<br />
I: coli which got '1' signal<br />
<br />
Io1: other once infected coli<br />
<br />
I2: coli which got '1''2' signal<br />
<br />
Io2: other second infected coli<br />
<br />
I3: coli which got '1''2''3' signal<br />
<br />
Io3: other third infected coli<br />
<br />
B4: '4' coli which began phage making<br />
<br />
Bo: other coli which began phage making<br />
<br />
P4: free phage of '4'<br />
<br />
Po: other free phage<br />
<br />
S1: signal virus with information of number '1' which come from other sudoku cell<br />
<br />
S2: signal virus with information of number '2' which come from other sudoku cell <br />
<br />
S3: signal virus with information of number '3' which come from other sudoku cell<br />
<br />
S4: signal virus with information of number '4' which come from other sudoku cell<br />
<br />
<br />
<br />
'''Parameters'''<br />
<br />
g: growth rate of coli <br />
<br />
k1: rate of infection<br />
<br />
k2: death rate of uninfected<br />
<br />
k3: lysis rate of infected<br />
<br />
k4: rate of phage producing<br />
<br />
k5: phage death rate<br />
<br />
k6: terminator leak rate<br />
<br />
Given parameters are<br />
unit value <br />
*g /min 0.0113 <br />
reference:Investigation of Bacteriophage MS2 Viral Dynamics Using Model Discrimination <br />
Analysis and the Implications for Phage Therapy<br />
Rishi Jain, Andrea L. Knorr, Joseph Bernacki, and Ranjan Srivastava(2006)<br />
<br />
*k2 /min 0 <br />
reference:Investigation of Bacteriophage MS2 Viral Dynamics Using Model Discrimination <br />
Analysis and the Implications for Phage Therapy<br />
Rishi Jain, Andrea L. Knorr, Joseph Bernacki, and Ranjan Srivastava(2006) <br />
<br />
*k3 /min 0.0167 <br />
reference:Investigation of Bacteriophage MS2 Viral Dynamics Using Model Discrimination <br />
Analysis and the Implications for Phage Therapy<br />
Rishi Jain, Andrea L. Knorr, Joseph Bernacki, and Ranjan Srivastava(2006)<br />
<br />
*k5 /min 0.25 <br />
reference: Dynamical Determinants of Drug-Inducible Gene Expression in a Single Bacterium<br />
Thuc T. Le, Thierry Emonet, Sebastien Harlepp, Călin C. Guet, and Philippe Cluzel(2006) <br />
<br />
<br />
<br />
Arbitrary parameters we can change are<br />
<br />
* k1(rate of infection) : k1 relates to the specificity of antisenseRNA. <br />
<br />
* k4(rate of phage producing) : k4 relates to the strength of promoter. <br />
<br />
* k6(terminator leak rate) : k6 express terminator leak. <br />
<br />
We change these parameters, and confirmed its effect.<br />
<br />
<br />
<br />
'''Equations'''<br />
<br />
We assume that cell lysis caused by MS2 phage is sufficiently rapid than coli death rate.<br />
<br />
<br />
[[Image:Model-eq.png|680px|Equations used in this modeling]]<br />
<br />
===='''Method'''====<br />
<br />
We used original matlab program.<br />
<br />
We set conditions as following.<br />
-when t = 10(min), signal of number '1' comes.(step input)<br />
-when t = 30(min), signal of number '2' comes.(step input)<br />
-when t = 50(min), signal of number '3' comes.(step input)<br />
<br />
<br />
===='''Results'''====<br />
<br />
The initial condition of uninfected coli concentration is set to be 1, and observed time evolution.<br />
<br />
<br />
[[Image:Whole1.png]]<br />
<br />
brue(U): initial concentration is 1. After got signal '1', it decrease suddenly. <br />
<br />
green: I1 begins increasing after getting first signal '1'.<br />
<br />
<br />
<br />
<br />
[[Image:MS2-0.5,0.6,0.png]]<br />
<br />
k1:0.4, k4:0.7, k6:0.1<br />
red: The time course of virus concentration which has the information of number '4'.<br />
<br />
blue: The time course of virus concentration which has the information of number '1'or'2'or'3'.<br />
<br />
<br />
<br />
<br />
[[Image:MS2 2.png]]<br />
<br />
k1:0.4, k4:0.7, k6:0.01<br />
<br />
red: The time course of virus concentration which has the information of number '4'.<br />
<br />
blue: The time course of virus concentration which has the information of number '1'or'2'or'3'.<br />
<br />
As terminator leak rate is less than the previous trial, it takes longer time to start increasing. the concentration of Virus '4' clearly increases comparing to others.<br />
<br />
<br />
<br />
[[Image:MS2 4.png]]<br />
<br />
k1:0.4, k4:0.1, k6:0.01<br />
<br />
<br />
<br />
[[Image:MS2 3.png ]]<br />
<br />
k1:0.8, k4:0.7, k6:0.01<br />
<br />
<br />
===='''Discussion & Conclusion'''====<br />
<br />
It is confirmed that when one cell get '1''2''3'signals in order, the cell differentiate into '4'.<br />
<br />
The rate of infection(k1) corresponds to antisense RNA specificity. <br />
Intuitively, when k1 is too high, coli become sensitive to not only target signals but also wrong signals. Our simulation also indicate when k1 becomes higher, the concentration of Virus'4' decrease.<br />
<br />
The rate of phage producing(k4) corresponds to the strength of promoter. <br />
If promoter is too strong, the amount of the phage increase but it also raises terminator leak. However in our simulation, we confirm the dependence between the rate of terminator leak and k1 is rather small. <br />
<br />
The rate of terminator leak relates to the time from receiving signals to the differentiation, and to the accuracy of finding the correct answer.<br />
According to the simulation, we find the clear difference between Pharge'4' and other Pharge'1''2''3'in 100-300 minutes, when the terminator leak rate is around 0.01.<br />
<br />
<br />
<br />
<br />
== '''Whole system''' ==<br />
<br />
finally, we made a simulation of solving whole 4*4 sudoku.<br />
<br />
This simulation is based on 1 cell simulation written above.<br />
<br />
The result of this simulation can be seen below.<br />
<br />
<br />
http://www.youtube.com/watch?v=ZFwbP4zg-H4<br />
<br />
<br />
<html><br />
<object width="640" height="385"><param name="movie" value="http://www.youtube.com/watch?v=ZFwbP4zg-H4"></param><param name="allowFullScreen" value="true"></param><param name="allowscriptaccess" value="always"></param><embed src="http://www.youtube.com/watch?v=ZFwbP4zg-H4" type="application/x-shockwave-flash" allowscriptaccess="always" allowfullscreen="true" width="640" height="385"></embed></object><br />
</html><br />
<br />
<br />
<br />
{{UT-Tokyo_Foot}}</div>Lenahttp://2010.igem.org/Team:UT-Tokyo/Sudoku_modelingTeam:UT-Tokyo/Sudoku modeling2010-11-29T06:18:41Z<p>Lena: /* Discussion & Conclusion */</p>
<hr />
<div>__NOTOC__{{UT-Tokyo_CSS3}}{{UT-Tokyo_CSS_p}}{{UT-Tokyo_Head}}<br />
<br />
= '''Sudoku''' =<br />
<html><br />
<div><br />
<ul id="inpagemenu"><br />
<li><a href="/Team:UT-Tokyo/Sudoku_abstract" id="abstract">Introduction</a></li><br />
<li><a href="/Team:UT-Tokyo/Sudoku_construct" id="construct">System</a></li><br />
<li><span>Modeling</span></li><br />
<li><a href="/Team:UT-Tokyo/Sudoku_experiments" id="experiment">Experiments</a></li><br />
<li><a href="/Team:UT-Tokyo/Sudoku_perspective" id="perspective">Perspective</a></li><br />
<li><a href="/Team:UT-Tokyo/Sudoku_reference" id="reference">Reference</a></li><br />
</ul><br />
</div><br />
<div id="clear"></div><br />
</html><br />
== '''4C3leak switch''' ==<br />
<br />
We modeled our 4C3 leak switch as deterministic ODE system.<br />
<br />
==='''Modelling'''===<br />
<br />
===='''Assumption'''====<br />
<br />
1.We treated DNA as continuous variable, since there could be hundreds of plasmid in one E.coli (multicopy plasmid).<br />
There are 32 kinds of variables that stand for DNA concentration corresponding to their different internal states.<br />
<br />
2.We assumed that cre recombinase operate as tetramer, and other recombinase as dimer (but this seems not to be relevant).<br />
<br />
3.We assumed that association and dissociation of DNA recombinase to DNA is sufficiently fast so that equilibrate in the timescale of the whole switch.<br />
<br />
4.We ignored the reverse reaction of DNA recombinase because we used irreversible sequence (lox66, lox71 etc).<br />
<br />
===='''Variable'''====<br />
<br />
Basically, "pn" denotes protein concentration and "rn" denotes mRNA concentration<br />
<br />
p1,p2,p3,p4 : concentration of 4 recombinase protein<br />
<br />
r1,r2,r3,r4 : concentration of 4 recombinass mRNA<br />
<br />
u1,u2,u3,u4 : concentration of input mRNA<br />
<br />
pc,rc : concentration of cre recombinase protein and mRNA<br />
<br />
ps,rs : concentration of SP6 polymerase protein and mRNA<br />
<br />
Variables that denote concentration of DNA in each state.<br />
"a","b","c","d","e" means there are four,three,two,one,zero recombinases in the DNA, respectively.<br />
And the subscripts specify which recobinases remain, while the numbers in bracket mean whether the DNA has been cut by cre recombinase( (1) has not been cut, (2) has been cut).<br />
<br />
{| border="1"<br />
|<br />
|all<br />
|123<br />
|124<br />
|134<br />
|234<br />
<br />
|-<br />
|not cut by cre<br />
|a(1)<br />
|b123(1)<br />
|b124(1)<br />
|b134(1)<br />
|b234(1)<br />
|-<br />
|cut by cre<br />
|a(2)<br />
|b123(2)<br />
|b124(2)<br />
|b134(2)<br />
|b234(2)<br />
|}<br />
<br />
<br />
{| border = "1"<br />
|12<br />
|13<br />
|14<br />
|23<br />
|24<br />
|34<br />
|1<br />
|2<br />
|3<br />
|4<br />
|none<br />
|-<br />
|c12(1)<br />
|c13(1)<br />
|c14(1)<br />
|c23(1)<br />
|c24(1)<br />
|c34(1)<br />
|d1(1)<br />
|d2(1)<br />
|d3(1)<br />
|d4(1)<br />
|e(1)<br />
|-<br />
|c12(2)<br />
|c13(2)<br />
|c14(2)<br />
|c23(2)<br />
|c24(2)<br />
|c34(2)<br />
|d1(2)<br />
|d2(2)<br />
|d3(2)<br />
|d4(2)<br />
|e(2)<br />
|}<br />
<br />
====''' Parameters '''====<br />
k0 : mRNA translation rate<br />
<br />
k1 : protein degradation rate<br />
<br />
k2 : mRNA translation rate<br />
<br />
k3 : cre protein degradation rate<br />
<br />
l0 : transcription speed per unit concentration of RNA polymearase<br />
<br />
l1 : mRNA degradation rate<br />
<br />
l2 : cre mRNA degradation rate<br />
<br />
K : recombinase binding constant<br />
<br />
v : recombinase reaction rate<br />
<br />
pT7 : concentration o T7 polymerase<br />
<br />
p : terminator leak probability<br />
<br />
<br />
<br />
===='''Equations'''====<br />
<br />
[[Image:Eqs1.jpg]]<br />
<br />
[[Image:eqs_cs.jpg]]<br />
<br />
[[Image:eqsdna1.jpg]]<br />
<br />
[[Image:eqsdna2.jpg]]<br />
<br />
==='''Methods'''===<br />
<br />
We used our original python program (4 th order explicit Runge-Kutta algorithm) to solve thess equations.<br />
<br />
==='''Results'''===<br />
<br />
We input signal 1,2,4 and simulated the time evolution of the system.<br />
<br />
The time course of DNA concentration<br />
<br />
[[Image:Sim_res.JPG]]<br />
<br />
The time course of mRNA concentration. Note that the output of this system is not protein but mRNA.<br />
<br />
[[Image:Rna.JPG]]<br />
<br />
==='''Discussion'''===<br />
<br />
It is confirmed that the system works as expected with a certain combination of parameters.<br />
<br />
In addition, we changed many parameters, especially,l0,v(whose values are unclear) and p(leakiness of terminator) , to see whether the device is able to operate within a large part of parameter space.<br />
<br />
Although increasing p basically increases the concentration of correct output mRNA, it also increases incorrect mRNA. <br />
On the other hand, decreasing p generally decreases both the concentration of correct mRNA and incorrect mRNA. Besides, too low value of p cannot leads to failure operation because cre protein concentration cannot get high enough.<br />
Careful test have shown that there is certain value of p that [correct mRNA]/[incorrect mRNA] takes maximum value. In this case, the optimal value is approximately p = 0.001 and the ratio is 10^(-6).<br />
<br />
Changing the value of v and l0 doesn't affect the function of the circuits qualitatively although it changed the time the systems takes to reach steady state.<br />
<br />
As a result, the circuit can function correctly within the range of 0.001< p < 0.1, 0.001 < v <1.0, 0.01 < l0 < 1.0 , demonstrating its robustness.<br />
This robustness may come from the assumption that DNA recombination is irreversible, which is supported by our design of the system.<br />
<br />
==='''conclusion'''===<br />
<br />
The correct operation of the switch was confirmed. By changing parameter exponentially, the robustness of the system was shown.<br />
<br />
=='''An indifferent sudoku cell simulation '''==<br />
<br />
-------including MS2 phage infection-------<br />
<br />
[[Image:What_Sudoku_99.png|200px|thumb|What's Sudoku?]]<br />
<br />
<br />
We made computer simulations to confirm that an empty sudoku cell get signals from other cells and differentiate to the correct answer in 4*4 sudoku.<br />
<br />
<br />
===='''Modeling'''====<br />
<br />
'''Variable'''<br />
<br />
U: uninfected coli<br />
<br />
I: coli which got '1' signal<br />
<br />
Io1: other once infected coli<br />
<br />
I2: coli which got '1''2' signal<br />
<br />
Io2: other second infected coli<br />
<br />
I3: coli which got '1''2''3' signal<br />
<br />
Io3: other third infected coli<br />
<br />
B4: '4' coli which began phage making<br />
<br />
Bo: other coli which began phage making<br />
<br />
P4: free phage of '4'<br />
<br />
Po: other free phage<br />
<br />
S1: signal virus with information of number '1' which come from other sudoku cell<br />
<br />
S2: signal virus with information of number '2' which come from other sudoku cell <br />
<br />
S3: signal virus with information of number '3' which come from other sudoku cell<br />
<br />
S4: signal virus with information of number '4' which come from other sudoku cell<br />
<br />
<br />
<br />
'''Parameters'''<br />
<br />
g: growth rate of coli <br />
<br />
k1: rate of infection<br />
<br />
k2: death rate of uninfected<br />
<br />
k3: lysis rate of infected<br />
<br />
k4: rate of phage producing<br />
<br />
k5: phage death rate<br />
<br />
k6: terminator leak rate<br />
<br />
Given parameters are<br />
unit value <br />
*g /min 0.0113 <br />
reference:Investigation of Bacteriophage MS2 Viral Dynamics Using Model Discrimination <br />
Analysis and the Implications for Phage Therapy<br />
Rishi Jain, Andrea L. Knorr, Joseph Bernacki, and Ranjan Srivastava(2006)<br />
<br />
*k2 /min 0 <br />
reference:Investigation of Bacteriophage MS2 Viral Dynamics Using Model Discrimination <br />
Analysis and the Implications for Phage Therapy<br />
Rishi Jain, Andrea L. Knorr, Joseph Bernacki, and Ranjan Srivastava(2006) <br />
<br />
*k3 /min 0.0167 <br />
reference:Investigation of Bacteriophage MS2 Viral Dynamics Using Model Discrimination <br />
Analysis and the Implications for Phage Therapy<br />
Rishi Jain, Andrea L. Knorr, Joseph Bernacki, and Ranjan Srivastava(2006)<br />
<br />
*k5 /min 0.25 <br />
reference: Dynamical Determinants of Drug-Inducible Gene Expression in a Single Bacterium<br />
Thuc T. Le, Thierry Emonet, Sebastien Harlepp, Călin C. Guet, and Philippe Cluzel(2006) <br />
<br />
<br />
<br />
Arbitrary parameters we can change are<br />
<br />
* k1(rate of infection) : k1 relates to the specificity of antisenseRNA. <br />
<br />
* k4(rate of phage producing) : k4 relates to the strength of promoter. <br />
<br />
* k6(terminator leak rate) : k6 express terminator leak. <br />
<br />
We change these parameters, and confirmed its effect.<br />
<br />
<br />
<br />
'''Equations'''<br />
<br />
We assume that cell lysis caused by MS2 phage is sufficiently rapid than coli death rate.<br />
<br />
<br />
[[Image:Model-eq.png|680px|Equations used in this modeling]]<br />
<br />
===='''Method'''====<br />
<br />
We used original matlab program.<br />
<br />
We set conditions as following.<br />
-when t = 10(min), signal of number '1' comes.(step input)<br />
-when t = 30(min), signal of number '2' comes.(step input)<br />
-when t = 50(min), signal of number '3' comes.(step input)<br />
<br />
<br />
===='''Results'''====<br />
<br />
The initial condition of uninfected coli concentration is set to be 1, and observed time evolution.<br />
<br />
<br />
[[Image:Whole1.png]]<br />
<br />
brue(U): initial concentration is 1. After got signal '1', it decrease suddenly. <br />
<br />
green: I1 begins increasing after getting first signal '1'.<br />
<br />
<br />
<br />
<br />
[[Image:MS2-0.5,0.6,0.png]]<br />
<br />
k1:0.4, k4:0.7, k6:0.1<br />
red: The time course of virus concentration which has the information of number '4'.<br />
<br />
blue: The time course of virus concentration which has the information of number '1'or'2'or'3'.<br />
<br />
<br />
<br />
<br />
[[Image:MS2 2.png]]<br />
<br />
k1:0.4, k4:0.7, k6:0.01<br />
<br />
red: The time course of virus concentration which has the information of number '4'.<br />
<br />
blue: The time course of virus concentration which has the information of number '1'or'2'or'3'.<br />
<br />
As terminator leak rate is less than the previous trial, it takes longer time to start increasing. the concentration of Virus '4' clearly increases comparing to others.<br />
<br />
<br />
<br />
[[Image:MS2 4.png]]<br />
<br />
k1:0.4, k4:0.1, k6:0.01<br />
<br />
<br />
<br />
[[Image:MS2 3.png ]]<br />
<br />
k1:0.8, k4:0.7, k6:0.01<br />
<br />
<br />
===='''Discussion & Conclusion'''====<br />
<br />
It is confirmed that when one cell get '1''2''3'signals in order, the cell differentiate into '4'.<br />
<br />
The rate of infection(k1) corresponds to antisense RNA specificity. <br />
Intuitively, when k1 is too high, coli become sensitive to not only target signals but also wrong signals. Our simulation also indicate when k1 becomes higher, the concentration of Virus'4' decrease.<br />
<br />
The rate of phage producing(k4) corresponds to the strength of promoter. <br />
If promoter is too strong, the amount of the phage increase but it also raises terminator leak. However in our simulation, we confirm the dependence between the rate of terminator leak and k1 is rather small. <br />
<br />
The rate of terminator leak relates to the time from receiving signals to the differentiation, and to the accuracy of finding the correct answer.<br />
According to the simulation, we find the clear difference between Pharge'4' and other Pharge'1''2''3'in 100-300 minutes, when the terminator leak rate is around 0.01.<br />
<br />
<br />
<br />
<br />
== '''Whole system''' ==<br />
<br />
finally, we made a simulation of solving whole 4*4 sudoku.<br />
<br />
This simulation is based on 1 cell simulation written above.<br />
<br />
The result of this simulation can be seen below.<br />
<br />
<br />
http://www.youtube.com/watch?v=ZFwbP4zg-H4<br />
<br />
<br />
{{UT-Tokyo_Foot}}</div>Lenahttp://2010.igem.org/Team:UT-Tokyo/Sudoku_modelingTeam:UT-Tokyo/Sudoku modeling2010-11-29T06:02:45Z<p>Lena: /* Modeling */</p>
<hr />
<div>__NOTOC__{{UT-Tokyo_CSS3}}{{UT-Tokyo_CSS_p}}{{UT-Tokyo_Head}}<br />
<br />
= '''Sudoku''' =<br />
<html><br />
<div><br />
<ul id="inpagemenu"><br />
<li><a href="/Team:UT-Tokyo/Sudoku_abstract" id="abstract">Introduction</a></li><br />
<li><a href="/Team:UT-Tokyo/Sudoku_construct" id="construct">System</a></li><br />
<li><span>Modeling</span></li><br />
<li><a href="/Team:UT-Tokyo/Sudoku_experiments" id="experiment">Experiments</a></li><br />
<li><a href="/Team:UT-Tokyo/Sudoku_perspective" id="perspective">Perspective</a></li><br />
<li><a href="/Team:UT-Tokyo/Sudoku_reference" id="reference">Reference</a></li><br />
</ul><br />
</div><br />
<div id="clear"></div><br />
</html><br />
== '''4C3leak switch''' ==<br />
<br />
We modeled our 4C3 leak switch as deterministic ODE system.<br />
<br />
==='''Modelling'''===<br />
<br />
===='''Assumption'''====<br />
<br />
1.We treated DNA as continuous variable, since there could be hundreds of plasmid in one E.coli (multicopy plasmid).<br />
There are 32 kinds of variables that stand for DNA concentration corresponding to their different internal states.<br />
<br />
2.We assumed that cre recombinase operate as tetramer, and other recombinase as dimer (but this seems not to be relevant).<br />
<br />
3.We assumed that association and dissociation of DNA recombinase to DNA is sufficiently fast so that equilibrate in the timescale of the whole switch.<br />
<br />
4.We ignored the reverse reaction of DNA recombinase because we used irreversible sequence (lox66, lox71 etc).<br />
<br />
===='''Variable'''====<br />
<br />
Basically, "pn" denotes protein concentration and "rn" denotes mRNA concentration<br />
<br />
p1,p2,p3,p4 : concentration of 4 recombinase protein<br />
<br />
r1,r2,r3,r4 : concentration of 4 recombinass mRNA<br />
<br />
u1,u2,u3,u4 : concentration of input mRNA<br />
<br />
pc,rc : concentration of cre recombinase protein and mRNA<br />
<br />
ps,rs : concentration of SP6 polymerase protein and mRNA<br />
<br />
Variables that denote concentration of DNA in each state.<br />
"a","b","c","d","e" means there are four,three,two,one,zero recombinases in the DNA, respectively.<br />
And the subscripts specify which recobinases remain, while the numbers in bracket mean whether the DNA has been cut by cre recombinase( (1) has not been cut, (2) has been cut).<br />
<br />
{| border="1"<br />
|<br />
|all<br />
|123<br />
|124<br />
|134<br />
|234<br />
<br />
|-<br />
|not cut by cre<br />
|a(1)<br />
|b123(1)<br />
|b124(1)<br />
|b134(1)<br />
|b234(1)<br />
|-<br />
|cut by cre<br />
|a(2)<br />
|b123(2)<br />
|b124(2)<br />
|b134(2)<br />
|b234(2)<br />
|}<br />
<br />
<br />
{| border = "1"<br />
|12<br />
|13<br />
|14<br />
|23<br />
|24<br />
|34<br />
|1<br />
|2<br />
|3<br />
|4<br />
|none<br />
|-<br />
|c12(1)<br />
|c13(1)<br />
|c14(1)<br />
|c23(1)<br />
|c24(1)<br />
|c34(1)<br />
|d1(1)<br />
|d2(1)<br />
|d3(1)<br />
|d4(1)<br />
|e(1)<br />
|-<br />
|c12(2)<br />
|c13(2)<br />
|c14(2)<br />
|c23(2)<br />
|c24(2)<br />
|c34(2)<br />
|d1(2)<br />
|d2(2)<br />
|d3(2)<br />
|d4(2)<br />
|e(2)<br />
|}<br />
<br />
====''' Parameters '''====<br />
k0 : mRNA translation rate<br />
<br />
k1 : protein degradation rate<br />
<br />
k2 : mRNA translation rate<br />
<br />
k3 : cre protein degradation rate<br />
<br />
l0 : transcription speed per unit concentration of RNA polymearase<br />
<br />
l1 : mRNA degradation rate<br />
<br />
l2 : cre mRNA degradation rate<br />
<br />
K : recombinase binding constant<br />
<br />
v : recombinase reaction rate<br />
<br />
pT7 : concentration o T7 polymerase<br />
<br />
p : terminator leak probability<br />
<br />
<br />
<br />
===='''Equations'''====<br />
<br />
[[Image:Eqs1.jpg]]<br />
<br />
[[Image:eqs_cs.jpg]]<br />
<br />
[[Image:eqsdna1.jpg]]<br />
<br />
[[Image:eqsdna2.jpg]]<br />
<br />
==='''Methods'''===<br />
<br />
We used our original python program (4 th order explicit Runge-Kutta algorithm) to solve thess equations.<br />
<br />
==='''Results'''===<br />
<br />
We input signal 1,2,4 and simulated the time evolution of the system.<br />
<br />
The time course of DNA concentration<br />
<br />
[[Image:Sim_res.JPG]]<br />
<br />
The time course of mRNA concentration. Note that the output of this system is not protein but mRNA.<br />
<br />
[[Image:Rna.JPG]]<br />
<br />
==='''Discussion'''===<br />
<br />
It is confirmed that the system works as expected with a certain combination of parameters.<br />
<br />
In addition, we changed many parameters, especially,l0,v(whose values are unclear) and p(leakiness of terminator) , to see whether the device is able to operate within a large part of parameter space.<br />
<br />
Although increasing p basically increases the concentration of correct output mRNA, it also increases incorrect mRNA. <br />
On the other hand, decreasing p generally decreases both the concentration of correct mRNA and incorrect mRNA. Besides, too low value of p cannot leads to failure operation because cre protein concentration cannot get high enough.<br />
Careful test have shown that there is certain value of p that [correct mRNA]/[incorrect mRNA] takes maximum value. In this case, the optimal value is approximately p = 0.001 and the ratio is 10^(-6).<br />
<br />
Changing the value of v and l0 doesn't affect the function of the circuits qualitatively although it changed the time the systems takes to reach steady state.<br />
<br />
As a result, the circuit can function correctly within the range of 0.001< p < 0.1, 0.001 < v <1.0, 0.01 < l0 < 1.0 , demonstrating its robustness.<br />
This robustness may come from the assumption that DNA recombination is irreversible, which is supported by our design of the system.<br />
<br />
==='''conclusion'''===<br />
<br />
The correct operation of the switch was confirmed. By changing parameter exponentially, the robustness of the system was shown.<br />
<br />
=='''An indifferent sudoku cell simulation '''==<br />
<br />
-------including MS2 phage infection-------<br />
<br />
[[Image:What_Sudoku_99.png|200px|thumb|What's Sudoku?]]<br />
<br />
<br />
We made computer simulations to confirm that an empty sudoku cell get signals from other cells and differentiate to the correct answer in 4*4 sudoku.<br />
<br />
<br />
===='''Modeling'''====<br />
<br />
'''Variable'''<br />
<br />
U: uninfected coli<br />
<br />
I: coli which got '1' signal<br />
<br />
Io1: other once infected coli<br />
<br />
I2: coli which got '1''2' signal<br />
<br />
Io2: other second infected coli<br />
<br />
I3: coli which got '1''2''3' signal<br />
<br />
Io3: other third infected coli<br />
<br />
B4: '4' coli which began phage making<br />
<br />
Bo: other coli which began phage making<br />
<br />
P4: free phage of '4'<br />
<br />
Po: other free phage<br />
<br />
S1: signal virus with information of number '1' which come from other sudoku cell<br />
<br />
S2: signal virus with information of number '2' which come from other sudoku cell <br />
<br />
S3: signal virus with information of number '3' which come from other sudoku cell<br />
<br />
S4: signal virus with information of number '4' which come from other sudoku cell<br />
<br />
<br />
<br />
'''Parameters'''<br />
<br />
g: growth rate of coli <br />
<br />
k1: rate of infection<br />
<br />
k2: death rate of uninfected<br />
<br />
k3: lysis rate of infected<br />
<br />
k4: rate of phage producing<br />
<br />
k5: phage death rate<br />
<br />
k6: terminator leak rate<br />
<br />
Given parameters are<br />
unit value <br />
*g /min 0.0113 <br />
reference:Investigation of Bacteriophage MS2 Viral Dynamics Using Model Discrimination <br />
Analysis and the Implications for Phage Therapy<br />
Rishi Jain, Andrea L. Knorr, Joseph Bernacki, and Ranjan Srivastava(2006)<br />
<br />
*k2 /min 0 <br />
reference:Investigation of Bacteriophage MS2 Viral Dynamics Using Model Discrimination <br />
Analysis and the Implications for Phage Therapy<br />
Rishi Jain, Andrea L. Knorr, Joseph Bernacki, and Ranjan Srivastava(2006) <br />
<br />
*k3 /min 0.0167 <br />
reference:Investigation of Bacteriophage MS2 Viral Dynamics Using Model Discrimination <br />
Analysis and the Implications for Phage Therapy<br />
Rishi Jain, Andrea L. Knorr, Joseph Bernacki, and Ranjan Srivastava(2006)<br />
<br />
*k5 /min 0.25 <br />
reference: Dynamical Determinants of Drug-Inducible Gene Expression in a Single Bacterium<br />
Thuc T. Le, Thierry Emonet, Sebastien Harlepp, Călin C. Guet, and Philippe Cluzel(2006) <br />
<br />
<br />
<br />
Arbitrary parameters we can change are<br />
<br />
* k1(rate of infection) : k1 relates to the specificity of antisenseRNA. <br />
<br />
* k4(rate of phage producing) : k4 relates to the strength of promoter. <br />
<br />
* k6(terminator leak rate) : k6 express terminator leak. <br />
<br />
We change these parameters, and confirmed its effect.<br />
<br />
<br />
<br />
'''Equations'''<br />
<br />
We assume that cell lysis caused by MS2 phage is sufficiently rapid than coli death rate.<br />
<br />
<br />
[[Image:Model-eq.png|680px|Equations used in this modeling]]<br />
<br />
===='''Method'''====<br />
<br />
We used original matlab program.<br />
<br />
We set conditions as following.<br />
-when t = 10(min), signal of number '1' comes.(step input)<br />
-when t = 30(min), signal of number '2' comes.(step input)<br />
-when t = 50(min), signal of number '3' comes.(step input)<br />
<br />
<br />
===='''Results'''====<br />
<br />
The initial condition of uninfected coli concentration is set to be 1, and observed time evolution.<br />
<br />
<br />
[[Image:Whole1.png]]<br />
<br />
brue(U): initial concentration is 1. After got signal '1', it decrease suddenly. <br />
<br />
green: I1 begins increasing after getting first signal '1'.<br />
<br />
<br />
<br />
<br />
[[Image:MS2-0.5,0.6,0.png]]<br />
<br />
k1:0.4, k4:0.7, k6:0.1<br />
red: The time course of virus concentration which has the information of number '4'.<br />
<br />
blue: The time course of virus concentration which has the information of number '1'or'2'or'3'.<br />
<br />
<br />
<br />
<br />
[[Image:MS2 2.png]]<br />
<br />
k1:0.4, k4:0.7, k6:0.01<br />
<br />
red: The time course of virus concentration which has the information of number '4'.<br />
<br />
blue: The time course of virus concentration which has the information of number '1'or'2'or'3'.<br />
<br />
As terminator leak rate is less than the previous trial, it takes longer time to start increasing. the concentration of Virus '4' clearly increases comparing to others.<br />
<br />
<br />
<br />
[[Image:MS2 4.png]]<br />
<br />
k1:0.4, k4:0.1, k6:0.01<br />
<br />
<br />
<br />
[[Image:MS2 3.png ]]<br />
<br />
k1:0.8, k4:0.7, k6:0.01<br />
<br />
<br />
===='''Discussion & Conclusion'''====<br />
<br />
It is confirmed that when one cell get '1''2''3'signals in order, the cell differentiate into '4'.<br />
<br />
The rate of infection(k1) corresponds to antisense RNA specificity. <br />
Intuitively, when k1 is too high, coli become sensitive to not only target signals but also wrong signals. Our simulation also indicate when k1 becomes higher, the concentration of Virus'4' decrease.<br />
<br />
The rate of phage producing(k4) corresponds to the strength of promoter. <br />
If promoter is too strong, the amount of the phage increase but it also raises terminator leak. However in our simulation, we confirm the dependence between the rate of terminator leak and k1 is rather small. <br />
<br />
The rate of terminator leak relates to the time from receiving signals to the differentiation, and to the accuracy of finding the correct answer.<br />
According to the simulation, we find the clear difference between Pharge'4' and other Pharge'1''2''3'in 100-300 minutes, when the terminator leak rate is around 0.01.<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
{{UT-Tokyo_Foot}}</div>Lenahttp://2010.igem.org/Team:UT-Tokyo/Sudoku_modelingTeam:UT-Tokyo/Sudoku modeling2010-10-28T03:59:00Z<p>Lena: /* A indifferent sudoku cell simulation */</p>
<hr />
<div>__NOTOC__{{UT-Tokyo_CSS3}}{{UT-Tokyo_CSS_p}}{{UT-Tokyo_Head}}<br />
<br />
= '''Sudoku''' =<br />
<html><br />
<div><br />
<ul id="inpagemenu"><br />
<li><a href="/Team:UT-Tokyo/Sudoku_abstract" id="abstract">Introduction</a></li><br />
<li><a href="/Team:UT-Tokyo/Sudoku_construct" id="construct">System</a></li><br />
<li><span>Modeling</span></li><br />
<li><a href="/Team:UT-Tokyo/Sudoku_experiments" id="experiment">Experiments</a></li><br />
<li><a href="/Team:UT-Tokyo/Sudoku_perspective" id="perspective">Perspective</a></li><br />
<li><a href="/Team:UT-Tokyo/Sudoku_reference" id="reference">Reference</a></li><br />
</ul><br />
</div><br />
<div id="clear"></div><br />
</html><br />
== '''4C3leak switch''' ==<br />
<br />
We modeled our 4C3 leak switch as deterministic ODE system.<br />
<br />
==='''Modelling'''===<br />
<br />
===='''Assumption'''====<br />
<br />
1.We treated DNA as continuous variable, since there could be hundreds of plasmid in one E.coli (multicopy plasmid).<br />
There are 32 kinds of variables that stand for DNA concentration corresponding to their different internal states.<br />
<br />
2.We assumed that cre recombinase operate as tetramer, and other recombinase as dimer (but this seems not to be relevant).<br />
<br />
3.We assumed that association and dissociation of DNA recombinase to DNA is sufficiently fast so that equilibrate in the timescale of the whole switch.<br />
<br />
4.We ignored the reverse reaction of DNA recombinase because we used irreversible sequence (lox66, lox71 etc).<br />
<br />
===='''Variable'''====<br />
<br />
Basically, "pn" denotes protein concentration and "rn" denotes mRNA concentration<br />
<br />
p1,p2,p3,p4 : concentration of 4 recombinase protein<br />
<br />
r1,r2,r3,r4 : concentration of 4 recombinass mRNA<br />
<br />
u1,u2,u3,u4 : concentration of input mRNA<br />
<br />
pc,rc : concentration of cre recombinase protein and mRNA<br />
<br />
ps,rs : concentration of SP6 polymerase protein and mRNA<br />
<br />
Variables that denote concentration of DNA in each state.<br />
"a","b","c","d","e" means there are four,three,two,one,zero recombinases in the DNA, respectively.<br />
And the subscripts specify which recobinases remain, while the numbers in bracket mean whether the DNA has been cut by cre recombinase( (1) has not been cut, (2) has been cut).<br />
<br />
{| border="1"<br />
|<br />
|all<br />
|123<br />
|124<br />
|134<br />
|234<br />
<br />
|-<br />
|not cut by cre<br />
|a(1)<br />
|b123(1)<br />
|b124(1)<br />
|b134(1)<br />
|b234(1)<br />
|-<br />
|cut by cre<br />
|a(2)<br />
|b123(2)<br />
|b124(2)<br />
|b134(2)<br />
|b234(2)<br />
|}<br />
<br />
<br />
{| border = "1"<br />
|12<br />
|13<br />
|14<br />
|23<br />
|24<br />
|34<br />
|1<br />
|2<br />
|3<br />
|4<br />
|none<br />
|-<br />
|c12(1)<br />
|c13(1)<br />
|c14(1)<br />
|c23(1)<br />
|c24(1)<br />
|c34(1)<br />
|d1(1)<br />
|d2(1)<br />
|d3(1)<br />
|d4(1)<br />
|e(1)<br />
|-<br />
|c12(2)<br />
|c13(2)<br />
|c14(2)<br />
|c23(2)<br />
|c24(2)<br />
|c34(2)<br />
|d1(2)<br />
|d2(2)<br />
|d3(2)<br />
|d4(2)<br />
|e(2)<br />
|}<br />
<br />
====''' Parameters '''====<br />
k0 : mRNA translation rate<br />
<br />
k1 : protein degradation rate<br />
<br />
k2 : mRNA translation rate<br />
<br />
k3 : cre protein degradation rate<br />
<br />
l0 : transcription speed per unit concentration of RNA polymearase<br />
<br />
l1 : mRNA degradation rate<br />
<br />
l2 : cre mRNA degradation rate<br />
<br />
K : recombinase binding constant<br />
<br />
v : recombinase reaction rate<br />
<br />
pT7 : concentration o T7 polymerase<br />
<br />
p : terminator leak probability<br />
<br />
<br />
<br />
===='''Equations'''====<br />
<br />
[[Image:Eqs1.jpg]]<br />
<br />
[[Image:eqs_cs.jpg]]<br />
<br />
[[Image:eqsdna1.jpg]]<br />
<br />
[[Image:eqsdna2.jpg]]<br />
<br />
==='''Methods'''===<br />
<br />
We used our original python program (4 th order explicit Runge-Kutta algorithm) to solve thess equations.<br />
<br />
==='''Results'''===<br />
<br />
We input signal 1,2,4 and simulated the time evolution of the system.<br />
<br />
The time course of DNA concentration<br />
<br />
[[Image:Sim_res.JPG]]<br />
<br />
The time course of mRNA concentration. Note that the output of this system is not protein but mRNA.<br />
<br />
[[Image:Rna.JPG]]<br />
<br />
==='''Discussion'''===<br />
<br />
It is confirmed that the system works as expected with a certain combination of parameters.<br />
<br />
In addition, we changed many parameters, especially,l0,v(whose values are unclear) and p(leakiness of terminator) , to see whether the device is able to operate within a large part of parameter space.<br />
<br />
Although increasing p basically increases the concentration of correct output mRNA, it also increases incorrect mRNA. <br />
On the other hand, decreasing p generally decreases both the concentration of correct mRNA and incorrect mRNA. Besides, too low value of p cannot leads to failure operation because cre protein concentration cannot get high enough.<br />
Careful test have shown that there is certain value of p that [correct mRNA]/[incorrect mRNA] takes maximum value. In this case, the optimal value is approximately p = 0.001 and the ratio is 10^(-6).<br />
<br />
Changing the value of v and l0 doesn't affect the function of the circuits qualitatively although it changed the time the systems takes to reach steady state.<br />
<br />
As a result, the circuit can function correctly within the range of 0.001< p < 0.1, 0.001 < v <1.0, 0.01 < l0 < 1.0 , demonstrating its robustness.<br />
This robustness may come from the assumption that DNA recombination is irreversible, which is supported by our design of the system.<br />
<br />
==='''conclusion'''===<br />
<br />
The correct operation of the switch was confirmed. By changing parameter exponentially, the robustness of the system was shown.<br />
<br />
=='''An indifferent sudoku cell simulation '''==<br />
<br />
-------including MS2 phage infection-------<br />
<br />
[[Image:What_Sudoku_99.png|200px|thumb|What's Sudoku?]]<br />
<br />
<br />
We made computer simulations to confirm that an empty sudoku cell get signals from other cells and differentiate to the correct answer in 4*4 sudoku.<br />
<br />
<br />
===='''Modeling'''====<br />
<br />
'''Variable'''<br />
<br />
U: uninfected coli<br />
<br />
I: coli which got '1' signal<br />
<br />
Io1: other once infected coli<br />
<br />
I2: coli which got '1''2' signal<br />
<br />
Io2: other second infected coli<br />
<br />
I3: coli which got '1''2''3' signal<br />
<br />
Io3: other third infected coli<br />
<br />
B4: '4' coli which began phage making<br />
<br />
Bo: other coli which began phage making<br />
<br />
P4: free phage of '4'<br />
<br />
Po: other free phage<br />
<br />
S1: signal virus with information of number '1' which come from other sudoku cell<br />
<br />
S2: signal virus with information of number '2' which come from other sudoku cell <br />
<br />
S3: signal virus with information of number '3' which come from other sudoku cell<br />
<br />
S4: signal virus with information of number '4' which come from other sudoku cell<br />
<br />
<br />
<br />
'''Parameters'''<br />
<br />
g: growth rate of coli <br />
<br />
k1: rate of infection<br />
<br />
k2: death rate of uninfected<br />
<br />
k3: lysis rate of infected<br />
<br />
k4: rate of phage producing<br />
<br />
k5: phage death rate<br />
<br />
k6: terminator leak rate<br />
<br />
Given parameters are<br />
unit value <br />
*g /min 0.0113 <br />
reference:Investigation of Bacteriophage MS2 Viral Dynamics Using Model Discrimination <br />
Analysis and the Implications for Phage Therapy<br />
Rishi Jain, Andrea L. Knorr, Joseph Bernacki, and Ranjan Srivastava(2006)<br />
<br />
*k2 /min 0 <br />
reference:Investigation of Bacteriophage MS2 Viral Dynamics Using Model Discrimination <br />
Analysis and the Implications for Phage Therapy<br />
Rishi Jain, Andrea L. Knorr, Joseph Bernacki, and Ranjan Srivastava(2006) <br />
<br />
*k3 /min 0.0167 <br />
reference:Investigation of Bacteriophage MS2 Viral Dynamics Using Model Discrimination <br />
Analysis and the Implications for Phage Therapy<br />
Rishi Jain, Andrea L. Knorr, Joseph Bernacki, and Ranjan Srivastava(2006)<br />
<br />
*k5 /min 0.25 <br />
reference: Dynamical Determinants of Drug-Inducible Gene Expression in a Single Bacterium<br />
Thuc T. Le, Thierry Emonet, Sebastien Harlepp, Călin C. Guet, and Philippe Cluzel(2006) <br />
<br />
<br />
<br />
Arbitrary parameters we can change are<br />
<br />
* k1(rate of infection) : k1 relates to the specificity of antisenseRNA. <br />
<br />
* k4(rate of phage producing) : k4 relates to the strength of promoter. <br />
<br />
* k6(terminator leak rate) : k6 express terminator leak. <br />
<br />
We change these parameters, and confirmed its effect.<br />
<br />
<br />
<br />
'''Equations'''<br />
<br />
We assume that cell lysis caused by MS2 phage is sufficiently rapid than coli death rate.<br />
<br />
<br />
[[Image:Model-eq.png|680px|Equations used in this modeling]]<br />
<br />
<br />
===='''Method'''====<br />
<br />
We used original matlab program.<br />
<br />
We set conditions as following.<br />
-when t = 10(min), signal of number '1' comes.(step input)<br />
-when t = 30(min), signal of number '2' comes.(step input)<br />
-when t = 50(min), signal of number '3' comes.(step input)<br />
<br />
<br />
===='''Results'''====<br />
<br />
The initial condition of uninfected coli concentration is set to be 1, and observed time evolution.<br />
<br />
<br />
[[Image:Whole1.png]]<br />
<br />
brue(U): initial concentration is 1. After got signal '1', it decrease suddenly. <br />
<br />
green: I1 begins increasing after getting first signal '1'.<br />
<br />
<br />
<br />
<br />
[[Image:MS2-0.5,0.6,0.png]]<br />
<br />
k1:0.4, k4:0.7, k6:0.1<br />
red: The time course of virus concentration which has the information of number '4'.<br />
<br />
blue: The time course of virus concentration which has the information of number '1'or'2'or'3'.<br />
<br />
<br />
<br />
<br />
[[Image:MS2 2.png]]<br />
<br />
k1:0.4, k4:0.7, k6:0.01<br />
<br />
red: The time course of virus concentration which has the information of number '4'.<br />
<br />
blue: The time course of virus concentration which has the information of number '1'or'2'or'3'.<br />
<br />
As terminator leak rate is less than the previous trial, it takes longer time to start increasing. the concentration of Virus '4' clearly increases comparing to others.<br />
<br />
<br />
<br />
[[Image:MS2 4.png]]<br />
<br />
k1:0.4, k4:0.1, k6:0.01<br />
<br />
<br />
<br />
[[Image:MS2 3.png ]]<br />
<br />
k1:0.8, k4:0.7, k6:0.01<br />
<br />
<br />
===='''Discussion & Conclusion'''====<br />
<br />
It is confirmed that when one cell get '1''2''3'signals in order, the cell differentiate into '4'.<br />
<br />
The rate of infection(k1) corresponds to antisense RNA specificity. <br />
Intuitively, when k1 is too high, coli become sensitive to not only target signals but also wrong signals. Our simulation also indicate when k1 becomes higher, the concentration of Virus'4' decrease.<br />
<br />
The rate of phage producing(k4) corresponds to the strength of promoter. <br />
If promoter is too strong, the amount of the phage increase but it also raises terminator leak. However in our simulation, we confirm the dependence between the rate of terminator leak and k1 is rather small. <br />
<br />
The rate of terminator leak relates to the time from receiving signals to the differentiation, and to the accuracy of finding the correct answer.<br />
According to the simulation, we find the clear difference between Pharge'4' and other Pharge'1''2''3'in 100-300 minutes, when the terminator leak rate is around 0.01.<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
{{UT-Tokyo_Foot}}</div>Lenahttp://2010.igem.org/Team:UT-Tokyo/Sudoku_modelingTeam:UT-Tokyo/Sudoku modeling2010-10-28T03:03:32Z<p>Lena: /* A indifferent sudoku cell simulation */</p>
<hr />
<div>__NOTOC__{{UT-Tokyo_CSS3}}{{UT-Tokyo_CSS_p}}{{UT-Tokyo_Head}}<br />
<br />
= '''Sudoku''' =<br />
<html><br />
<div><br />
<ul id="inpagemenu"><br />
<li><a href="/Team:UT-Tokyo/Sudoku_abstract" id="abstract">Introduction</a></li><br />
<li><a href="/Team:UT-Tokyo/Sudoku_construct" id="construct">System</a></li><br />
<li><span>Modeling</span></li><br />
<li><a href="/Team:UT-Tokyo/Sudoku_experiments" id="experiment">Experiments</a></li><br />
<li><a href="/Team:UT-Tokyo/Sudoku_perspective" id="perspective">Perspective</a></li><br />
<li><a href="/Team:UT-Tokyo/Sudoku_reference" id="reference">Reference</a></li><br />
</ul><br />
</div><br />
<div id="clear"></div><br />
</html><br />
== '''4C3leak switch''' ==<br />
<br />
We modeled our 4C3 leak switch as deterministic ODE system.<br />
<br />
<br />
==='''Modelling'''===<br />
<br />
<br />
===='''Assumption'''====<br />
<br />
1.We treated DNA as continuous variable, since there could be hundreds of plasmid in one E.coli (multicopy plasmid).<br />
There are 32 kinds of variables that stand for DNA concentration corresponding to their different internal states.<br />
<br />
2.We assumed that cre recombinase operate as tetramer, and other recombinase as dimer (but this seems not to be relevant).<br />
<br />
3.We assumed that association and dissociation of DNA recombinase to DNA is sufficiently fast so that equilibrate in the timescale of the whole switch.<br />
<br />
4.We ignored the reverse reaction of DNA recombinase because we used irreversible sequence (lox66, lox71 etc).<br />
<br />
===='''Variable'''====<br />
<br />
Basically, "pn" denotes protein concentration and "rn" denotes mRNA concentration<br />
<br />
p1,p2,p3,p4 : concentration of 4 recombinase protein<br />
<br />
r1,r2,r3,r4 : concentration of 4 recombinass mRNA<br />
<br />
u1,u2,u3,u4 : concentration of input mRNA<br />
<br />
pc,rc : concentration of cre recombinase protein and mRNA<br />
<br />
ps,rs : concentration of SP6 polymerase protein and mRNA<br />
<br />
Variables that denote concentration of DNA in each state.<br />
"a","b","c","d","e" means there are four,three,two,one,zero recombinases in the DNA, respectively.<br />
And the subscripts specify which recobinases remain, while the numbers in bracket mean whether the DNA has been cut by cre recombinase( (1) has not been cut, (2) has been cut).<br />
<br />
{| border="1"<br />
|<br />
|all<br />
|123<br />
|124<br />
|134<br />
|234<br />
<br />
|-<br />
|not cut by cre<br />
|a(1)<br />
|b123(1)<br />
|b124(1)<br />
|b134(1)<br />
|b234(1)<br />
|-<br />
|cut by cre<br />
|a(2)<br />
|b123(2)<br />
|b124(2)<br />
|b134(2)<br />
|b234(2)<br />
|}<br />
<br />
<br />
{| border = "1"<br />
|12<br />
|13<br />
|14<br />
|23<br />
|24<br />
|34<br />
|1<br />
|2<br />
|3<br />
|4<br />
|none<br />
|-<br />
|c12(1)<br />
|c13(1)<br />
|c14(1)<br />
|c23(1)<br />
|c24(1)<br />
|c34(1)<br />
|d1(1)<br />
|d2(1)<br />
|d3(1)<br />
|d4(1)<br />
|e(1)<br />
|-<br />
|c12(2)<br />
|c13(2)<br />
|c14(2)<br />
|c23(2)<br />
|c24(2)<br />
|c34(2)<br />
|d1(2)<br />
|d2(2)<br />
|d3(2)<br />
|d4(2)<br />
|e(2)<br />
|}<br />
<br />
====''' Parameters '''====<br />
k0 : mRNA translation rate<br />
<br />
k1 : protein degradation rate<br />
<br />
k2 : mRNA translation rate<br />
<br />
k3 : cre protein degradation rate<br />
<br />
l0 : transcription speed per unit concentration of RNA polymearase<br />
<br />
l1 : mRNA degradation rate<br />
<br />
l2 : cre mRNA degradation rate<br />
<br />
K : recombinase binding constant<br />
<br />
v : recombinase reaction rate<br />
<br />
pT7 : concentration o T7 polymerase<br />
<br />
p : terminator leak probability<br />
<br />
<br />
<br />
===='''Equations'''====<br />
<br />
[[Image:Eqs1.jpg]]<br />
<br />
[[Image:eqs_cs.jpg]]<br />
<br />
[[Image:eqsdna1.jpg]]<br />
<br />
[[Image:eqsdna2.jpg]]<br />
<br />
==='''Methods'''===<br />
<br />
We used our original python program (4 th order explicit Runge-Kutta algorithm) to solve thess equations.<br />
<br />
==='''Results'''===<br />
<br />
We input signal 1,2,4 and simulated the time evolution of the system.<br />
<br />
The time course of DNA concentration<br />
<br />
[[Image:Sim_res.JPG]]<br />
<br />
The time course of mRNA concentration. Note that the output of this system is not protein but mRNA.<br />
<br />
[[Image:Rna.JPG]]<br />
<br />
==='''Discussion'''===<br />
<br />
It is confirmed that the system works as expected with a certain combination of parameters.<br />
<br />
In addition, we changed many parameters, especially,l0,v(whose values are unclear) and p(leakiness of terminator) , to see whether the device is able to operate within a large part of parameter space.<br />
<br />
Although increasing p basically increases the concentration of correct output mRNA, it also increases incorrect mRNA. <br />
On the other hand, decreasing p generally decreases both the concentration of correct mRNA and incorrect mRNA. Besides, too low value of p cannot leads to failure operation because cre protein concentration cannot get high enough.<br />
Careful test have shown that there is certain value of p that [correct mRNA]/[incorrect mRNA] takes maximum value. In this case, the optimal value is approximately p = 0.001 and the ratio is 10^(-6).<br />
<br />
Changing the value of v and l0 doesn't affect the function of the circuits qualitatively although it changed the time the systems takes to reach steady state.<br />
<br />
As a result, the circuit can function correctly within the range of 0.001< p < 0.1, 0.001 < v <1.0, 0.01 < l0 < 1.0 , demonstrating its robustness.<br />
This robustness may come from the assumption that DNA recombination is irreversible, which is supported by our design of the system.<br />
<br />
==='''conclusion'''===<br />
<br />
The correct operation of the switch was confirmed. By changing parameter exponentially, the robustness of the system was shown.<br />
<br />
<br />
<br />
<br />
<br />
=='''A indifferent sudoku cell simulation '''==<br />
<br />
-------including MS2 phage infection-------<br />
<br />
[[Image:What_Sudoku_99.png|200px|thumb|What's Sudoku?]]<br />
<br />
<br />
We made a model for a indifferent sudoku cell.<br />
<br />
We traverse the amount of phages and see wether the cell react properly against given signals.<br />
<br />
<br />
===='''Modeling'''====<br />
<br />
'''Variable'''<br />
<br />
U: uninfected coli<br />
<br />
I: coli which got '1' signal<br />
<br />
Io1: other once infected coli<br />
<br />
I2: coli which got '1''2' signal<br />
<br />
Io2: other second infected coli<br />
<br />
I3: coli which got '1''2''3' signal<br />
<br />
Io3: other third infected coli<br />
<br />
B4: '4' coli which began phage making<br />
<br />
Bo: other coli which began phage making<br />
<br />
P4: free phage of '4'<br />
<br />
Po: other free phage<br />
<br />
S1: signal virus with information of number '1' which come from other sudoku cell<br />
<br />
S2: signal virus with information of number '2' which come from other sudoku cell <br />
<br />
S3: signal virus with information of number '3' which come from other sudoku cell<br />
<br />
S4: signal virus with information of number '4' which come from other sudoku cell<br />
<br />
<br />
<br />
'''Parameters'''<br />
<br />
g: growth rate of coli <br />
<br />
k1: rate of infection<br />
<br />
k2: death rate of uninfected<br />
<br />
k3: lysis rate of infected<br />
<br />
k4: rate of phage producing<br />
<br />
k5: phage death rate<br />
<br />
k6: terminator leak rate<br />
<br />
<br />
Arbitrary parameters we can change are<br />
<br />
* k1(rate of infection) : k1 relates to the specificity of antisenseRNA. <br />
<br />
* k4(rate of phage producing) : k4 relates to the strength of promoter. <br />
<br />
* k6(terminator leak rate) : k6 express terminator leak. <br />
<br />
We change these parameters, and confirmed its effect.<br />
<br />
<br />
<br />
'''Equations'''<br />
<br />
We assume that cell lysis caused by MS2 phage is sufficiently rapid than coli death rate.<br />
<br />
<br />
[[Image:Model-eq.png|680px|Equations used in this modeling]]<br />
<br />
<br />
===='''Method'''====<br />
<br />
We used original matlab program.<br />
<br />
We set conditions as following.<br />
-when t = 10(min), signal of number '1' comes.(step input)<br />
-when t = 30(min), signal of number '2' comes.(step input)<br />
-when t = 50(min), signal of number '3' comes.(step input)<br />
<br />
<br />
===='''Results'''====<br />
<br />
The initial condition of uninfected coli concentration is set to be 1, and observed time evolution.<br />
<br />
<br />
[[Image:Whole1.png]]<br />
<br />
brue(U): initial concentration is 1. After got signal '1', it decrease suddenly. <br />
<br />
green: I1 begins increasing after getting first signal '1'.<br />
<br />
<br />
<br />
[[Image:MS2-0.5,0.6,0.png]]<br />
<br />
k1:0.4, k4:0.7, k6:0.1<br />
red: The time course of virus concentration which has the information of number '4'.<br />
<br />
blue: The time course of virus concentration which has the information of number '1'or'2'or'3'.<br />
<br />
<br />
<br />
[[Image:MS2 2.png]]<br />
<br />
k1:0.4, k4:0.7, k6:0.01<br />
<br />
red: The time course of virus concentration which has the information of number '4'.<br />
<br />
blue: The time course of virus concentration which has the information of number '1'or'2'or'3'.<br />
<br />
As terminator leak rate is less than the previous trial, it takes longer time to start increasing. the concentration of Virus '4' clearly increases comparing to others.<br />
<br />
<br />
[[Image:MS2 4.png]]<br />
<br />
k1:0.4, k4:0.1, k6:0.01<br />
<br />
<br />
[[Image:MS2 3.png ]]<br />
<br />
k1:0.8, k4:0.7, k6:0.01<br />
<br />
<br />
===='''Discussion & Conclusion'''====<br />
<br />
It is confirmed that when one cell get '1''2''3'signals in order, the cell differentiate into '4'.<br />
<br />
The rate of infection(k1) corresponds to antisense RNA specificity. <br />
Intuitively, when k1 is too high, coli become sensitive to not only target signals but also wrong signal. Our simulation also indicate when k1 becomes higher, the concentration of Virus'4' decrease.<br />
<br />
The rate of phage producing(k4) corresponds to the strength of promoter. <br />
If promoter is too strong, the amount of the phage increase but it also raise terminator leak. However in our simulation, the effect to the rate of terminator leak error is lower. <br />
<br />
The rate of terminator leak relates to the time of the differentiation, and the accuracy of the signal.<br />
According to the simulation, we can see the clear difference between Pharge'4' and other Pharge'1''2''3'in 100-300 minutes, when the terminator leak rate is around 0.01.<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
{{UT-Tokyo_Foot}}</div>Lenahttp://2010.igem.org/Team:UT-Tokyo/Sudoku_modelingTeam:UT-Tokyo/Sudoku modeling2010-10-28T03:02:15Z<p>Lena: /* conclusion */</p>
<hr />
<div>__NOTOC__{{UT-Tokyo_CSS3}}{{UT-Tokyo_CSS_p}}{{UT-Tokyo_Head}}<br />
<br />
= '''Sudoku''' =<br />
<html><br />
<div><br />
<ul id="inpagemenu"><br />
<li><a href="/Team:UT-Tokyo/Sudoku_abstract" id="abstract">Introduction</a></li><br />
<li><a href="/Team:UT-Tokyo/Sudoku_construct" id="construct">System</a></li><br />
<li><span>Modeling</span></li><br />
<li><a href="/Team:UT-Tokyo/Sudoku_experiments" id="experiment">Experiments</a></li><br />
<li><a href="/Team:UT-Tokyo/Sudoku_perspective" id="perspective">Perspective</a></li><br />
<li><a href="/Team:UT-Tokyo/Sudoku_reference" id="reference">Reference</a></li><br />
</ul><br />
</div><br />
<div id="clear"></div><br />
</html><br />
== '''4C3leak switch''' ==<br />
<br />
We modeled our 4C3 leak switch as deterministic ODE system.<br />
<br />
<br />
==='''Modelling'''===<br />
<br />
<br />
===='''Assumption'''====<br />
<br />
1.We treated DNA as continuous variable, since there could be hundreds of plasmid in one E.coli (multicopy plasmid).<br />
There are 32 kinds of variables that stand for DNA concentration corresponding to their different internal states.<br />
<br />
2.We assumed that cre recombinase operate as tetramer, and other recombinase as dimer (but this seems not to be relevant).<br />
<br />
3.We assumed that association and dissociation of DNA recombinase to DNA is sufficiently fast so that equilibrate in the timescale of the whole switch.<br />
<br />
4.We ignored the reverse reaction of DNA recombinase because we used irreversible sequence (lox66, lox71 etc).<br />
<br />
===='''Variable'''====<br />
<br />
Basically, "pn" denotes protein concentration and "rn" denotes mRNA concentration<br />
<br />
p1,p2,p3,p4 : concentration of 4 recombinase protein<br />
<br />
r1,r2,r3,r4 : concentration of 4 recombinass mRNA<br />
<br />
u1,u2,u3,u4 : concentration of input mRNA<br />
<br />
pc,rc : concentration of cre recombinase protein and mRNA<br />
<br />
ps,rs : concentration of SP6 polymerase protein and mRNA<br />
<br />
Variables that denote concentration of DNA in each state.<br />
"a","b","c","d","e" means there are four,three,two,one,zero recombinases in the DNA, respectively.<br />
And the subscripts specify which recobinases remain, while the numbers in bracket mean whether the DNA has been cut by cre recombinase( (1) has not been cut, (2) has been cut).<br />
<br />
{| border="1"<br />
|<br />
|all<br />
|123<br />
|124<br />
|134<br />
|234<br />
<br />
|-<br />
|not cut by cre<br />
|a(1)<br />
|b123(1)<br />
|b124(1)<br />
|b134(1)<br />
|b234(1)<br />
|-<br />
|cut by cre<br />
|a(2)<br />
|b123(2)<br />
|b124(2)<br />
|b134(2)<br />
|b234(2)<br />
|}<br />
<br />
<br />
{| border = "1"<br />
|12<br />
|13<br />
|14<br />
|23<br />
|24<br />
|34<br />
|1<br />
|2<br />
|3<br />
|4<br />
|none<br />
|-<br />
|c12(1)<br />
|c13(1)<br />
|c14(1)<br />
|c23(1)<br />
|c24(1)<br />
|c34(1)<br />
|d1(1)<br />
|d2(1)<br />
|d3(1)<br />
|d4(1)<br />
|e(1)<br />
|-<br />
|c12(2)<br />
|c13(2)<br />
|c14(2)<br />
|c23(2)<br />
|c24(2)<br />
|c34(2)<br />
|d1(2)<br />
|d2(2)<br />
|d3(2)<br />
|d4(2)<br />
|e(2)<br />
|}<br />
<br />
====''' Parameters '''====<br />
k0 : mRNA translation rate<br />
<br />
k1 : protein degradation rate<br />
<br />
k2 : mRNA translation rate<br />
<br />
k3 : cre protein degradation rate<br />
<br />
l0 : transcription speed per unit concentration of RNA polymearase<br />
<br />
l1 : mRNA degradation rate<br />
<br />
l2 : cre mRNA degradation rate<br />
<br />
K : recombinase binding constant<br />
<br />
v : recombinase reaction rate<br />
<br />
pT7 : concentration o T7 polymerase<br />
<br />
p : terminator leak probability<br />
<br />
<br />
<br />
===='''Equations'''====<br />
<br />
[[Image:Eqs1.jpg]]<br />
<br />
[[Image:eqs_cs.jpg]]<br />
<br />
[[Image:eqsdna1.jpg]]<br />
<br />
[[Image:eqsdna2.jpg]]<br />
<br />
==='''Methods'''===<br />
<br />
We used our original python program (4 th order explicit Runge-Kutta algorithm) to solve thess equations.<br />
<br />
==='''Results'''===<br />
<br />
We input signal 1,2,4 and simulated the time evolution of the system.<br />
<br />
The time course of DNA concentration<br />
<br />
[[Image:Sim_res.JPG]]<br />
<br />
The time course of mRNA concentration. Note that the output of this system is not protein but mRNA.<br />
<br />
[[Image:Rna.JPG]]<br />
<br />
==='''Discussion'''===<br />
<br />
It is confirmed that the system works as expected with a certain combination of parameters.<br />
<br />
In addition, we changed many parameters, especially,l0,v(whose values are unclear) and p(leakiness of terminator) , to see whether the device is able to operate within a large part of parameter space.<br />
<br />
Although increasing p basically increases the concentration of correct output mRNA, it also increases incorrect mRNA. <br />
On the other hand, decreasing p generally decreases both the concentration of correct mRNA and incorrect mRNA. Besides, too low value of p cannot leads to failure operation because cre protein concentration cannot get high enough.<br />
Careful test have shown that there is certain value of p that [correct mRNA]/[incorrect mRNA] takes maximum value. In this case, the optimal value is approximately p = 0.001 and the ratio is 10^(-6).<br />
<br />
Changing the value of v and l0 doesn't affect the function of the circuits qualitatively although it changed the time the systems takes to reach steady state.<br />
<br />
As a result, the circuit can function correctly within the range of 0.001< p < 0.1, 0.001 < v <1.0, 0.01 < l0 < 1.0 , demonstrating its robustness.<br />
This robustness may come from the assumption that DNA recombination is irreversible, which is supported by our design of the system.<br />
<br />
==='''conclusion'''===<br />
<br />
The correct operation of the switch was confirmed. By changing parameter exponentially, the robustness of the system was shown.<br />
<br />
=='''A indifferent sudoku cell simulation '''==<br />
<br />
-------including MS2 phage infection-------<br />
<br />
[[Image:What_Sudoku_99.png|200px|thumb|What's Sudoku?]]<br />
<br />
<br />
We made a model for a indifferent sudoku cell.<br />
<br />
We traverse the amount of phages and see wether the cell react properly against given signals.<br />
<br />
<br />
===='''Modeling'''====<br />
<br />
'''Variable'''<br />
<br />
U: uninfected coli<br />
<br />
I: coli which got '1' signal<br />
<br />
Io1: other once infected coli<br />
<br />
I2: coli which got '1''2' signal<br />
<br />
Io2: other second infected coli<br />
<br />
I3: coli which got '1''2''3' signal<br />
<br />
Io3: other third infected coli<br />
<br />
B4: '4' coli which began phage making<br />
<br />
Bo: other coli which began phage making<br />
<br />
P4: free phage of '4'<br />
<br />
Po: other free phage<br />
<br />
S1: signal virus with information of number '1' which come from other sudoku cell<br />
<br />
S2: signal virus with information of number '2' which come from other sudoku cell <br />
<br />
S3: signal virus with information of number '3' which come from other sudoku cell<br />
<br />
S4: signal virus with information of number '4' which come from other sudoku cell<br />
<br />
<br />
<br />
'''Parameters'''<br />
<br />
g: growth rate of coli <br />
<br />
k1: rate of infection<br />
<br />
k2: death rate of uninfected<br />
<br />
k3: lysis rate of infected<br />
<br />
k4: rate of phage producing<br />
<br />
k5: phage death rate<br />
<br />
k6: terminator leak rate<br />
<br />
<br />
Arbitrary parameters we can change are<br />
<br />
* k1(rate of infection) : k1 relates to the specificity of antisenseRNA. <br />
<br />
* k4(rate of phage producing) : k4 relates to the strength of promoter. <br />
<br />
* k6(terminator leak rate) : k6 express terminator leak. <br />
<br />
We change these parameters, and confirmed its effect.<br />
<br />
<br />
<br />
'''Equations'''<br />
<br />
We assume that cell lysis caused by MS2 phage is sufficiently rapid than coli death rate.<br />
<br />
<br />
[[Image:Model-eq.png|680px|Equations used in this modeling]]<br />
<br />
<br />
===='''Method'''====<br />
<br />
We used original matlab program.<br />
<br />
We set conditions as following.<br />
-when t = 10(min), signal of number '1' comes.(step input)<br />
-when t = 30(min), signal of number '2' comes.(step input)<br />
-when t = 50(min), signal of number '3' comes.(step input)<br />
<br />
<br />
===='''Results'''====<br />
<br />
The initial condition of uninfected coli concentration is set to be 1, and observed time evolution.<br />
<br />
<br />
[[Image:Whole1.png]]<br />
<br />
brue(U): initial concentration is 1. After got signal '1', it decrease suddenly. <br />
<br />
green: I1 begins increasing after getting first signal '1'.<br />
<br />
<br />
<br />
[[Image:MS2-0.5,0.6,0.png]]<br />
<br />
k1:0.4, k4:0.7, k6:0.1<br />
red: The time course of virus concentration which has the information of number '4'.<br />
<br />
blue: The time course of virus concentration which has the information of number '1'or'2'or'3'.<br />
<br />
<br />
<br />
[[Image:MS2 2.png]]<br />
<br />
k1:0.4, k4:0.7, k6:0.01<br />
<br />
red: The time course of virus concentration which has the information of number '4'.<br />
<br />
blue: The time course of virus concentration which has the information of number '1'or'2'or'3'.<br />
<br />
As terminator leak rate is less than the previous trial, it takes longer time to start increasing. the concentration of Virus '4' clearly increases comparing to others.<br />
<br />
<br />
[[Image:MS2 4.png]]<br />
<br />
k1:0.4, k4:0.1, k6:0.01<br />
<br />
<br />
[[Image:MS2 3.png ]]<br />
<br />
k1:0.8, k4:0.7, k6:0.01<br />
<br />
<br />
===='''Discussion & Conclusion'''====<br />
<br />
It is confirmed that when one cell get '1''2''3'signals in order, the cell differentiate into '4'.<br />
<br />
The rate of infection(k1) corresponds to antisense RNA specificity. <br />
Intuitively, when k1 is too high, coli become sensitive to not only target signals but also wrong signal. Our simulation also indicate when k1 becomes higher, the concentration of Virus'4' decrease.<br />
<br />
The rate of phage producing(k4) corresponds to the strength of promoter. <br />
If promoter is too strong, the amount of the phage increase but it also raise terminator leak. However in our simulation, the effect to the rate of terminator leak error is lower. <br />
<br />
The rate of terminator leak relates to the time of the differentiation, and the accuracy of the signal.<br />
According to the simulation, we can see the clear difference between Pharge'4' and other Pharge'1''2''3'in 100-300 minutes, when the terminator leak rate is around 0.01.<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
{{UT-Tokyo_Foot}}</div>Lenahttp://2010.igem.org/Team:UT-Tokyo/Sudoku_modelingTeam:UT-Tokyo/Sudoku modeling2010-10-28T02:58:09Z<p>Lena: /* A indifferent sudoku cell simulation including MS2 phage infection */</p>
<hr />
<div>__NOTOC__{{UT-Tokyo_CSS3}}{{UT-Tokyo_CSS_p}}{{UT-Tokyo_Head}}<br />
<br />
= '''Sudoku''' =<br />
<html><br />
<div><br />
<ul id="inpagemenu"><br />
<li><a href="/Team:UT-Tokyo/Sudoku_abstract" id="abstract">Introduction</a></li><br />
<li><a href="/Team:UT-Tokyo/Sudoku_construct" id="construct">System</a></li><br />
<li><span>Modeling</span></li><br />
<li><a href="/Team:UT-Tokyo/Sudoku_experiments" id="experiment">Experiments</a></li><br />
<li><a href="/Team:UT-Tokyo/Sudoku_perspective" id="perspective">Perspective</a></li><br />
<li><a href="/Team:UT-Tokyo/Sudoku_reference" id="reference">Reference</a></li><br />
</ul><br />
</div><br />
<div id="clear"></div><br />
</html><br />
== '''4C3leak switch''' ==<br />
<br />
We modeled our 4C3 leak switch as deterministic ODE system.<br />
<br />
<br />
==='''Modelling'''===<br />
<br />
<br />
===='''Assumption'''====<br />
<br />
1.We treated DNA as continuous variable, since there could be hundreds of plasmid in one E.coli (multicopy plasmid).<br />
There are 32 kinds of variables that stand for DNA concentration corresponding to their different internal states.<br />
<br />
2.We assumed that cre recombinase operate as tetramer, and other recombinase as dimer (but this seems not to be relevant).<br />
<br />
3.We assumed that association and dissociation of DNA recombinase to DNA is sufficiently fast so that equilibrate in the timescale of the whole switch.<br />
<br />
4.We ignored the reverse reaction of DNA recombinase because we used irreversible sequence (lox66, lox71 etc).<br />
<br />
===='''Variable'''====<br />
<br />
Basically, "pn" denotes protein concentration and "rn" denotes mRNA concentration<br />
<br />
p1,p2,p3,p4 : concentration of 4 recombinase protein<br />
<br />
r1,r2,r3,r4 : concentration of 4 recombinass mRNA<br />
<br />
u1,u2,u3,u4 : concentration of input mRNA<br />
<br />
pc,rc : concentration of cre recombinase protein and mRNA<br />
<br />
ps,rs : concentration of SP6 polymerase protein and mRNA<br />
<br />
Variables that denote concentration of DNA in each state.<br />
"a","b","c","d","e" means there are four,three,two,one,zero recombinases in the DNA, respectively.<br />
And the subscripts specify which recobinases remain, while the numbers in bracket mean whether the DNA has been cut by cre recombinase( (1) has not been cut, (2) has been cut).<br />
<br />
{| border="1"<br />
|<br />
|all<br />
|123<br />
|124<br />
|134<br />
|234<br />
<br />
|-<br />
|not cut by cre<br />
|a(1)<br />
|b123(1)<br />
|b124(1)<br />
|b134(1)<br />
|b234(1)<br />
|-<br />
|cut by cre<br />
|a(2)<br />
|b123(2)<br />
|b124(2)<br />
|b134(2)<br />
|b234(2)<br />
|}<br />
<br />
<br />
{| border = "1"<br />
|12<br />
|13<br />
|14<br />
|23<br />
|24<br />
|34<br />
|1<br />
|2<br />
|3<br />
|4<br />
|none<br />
|-<br />
|c12(1)<br />
|c13(1)<br />
|c14(1)<br />
|c23(1)<br />
|c24(1)<br />
|c34(1)<br />
|d1(1)<br />
|d2(1)<br />
|d3(1)<br />
|d4(1)<br />
|e(1)<br />
|-<br />
|c12(2)<br />
|c13(2)<br />
|c14(2)<br />
|c23(2)<br />
|c24(2)<br />
|c34(2)<br />
|d1(2)<br />
|d2(2)<br />
|d3(2)<br />
|d4(2)<br />
|e(2)<br />
|}<br />
<br />
====''' Parameters '''====<br />
k0 : mRNA translation rate<br />
<br />
k1 : protein degradation rate<br />
<br />
k2 : mRNA translation rate<br />
<br />
k3 : cre protein degradation rate<br />
<br />
l0 : transcription speed per unit concentration of RNA polymearase<br />
<br />
l1 : mRNA degradation rate<br />
<br />
l2 : cre mRNA degradation rate<br />
<br />
K : recombinase binding constant<br />
<br />
v : recombinase reaction rate<br />
<br />
pT7 : concentration o T7 polymerase<br />
<br />
p : terminator leak probability<br />
<br />
<br />
<br />
===='''Equations'''====<br />
<br />
[[Image:Eqs1.jpg]]<br />
<br />
[[Image:eqs_cs.jpg]]<br />
<br />
[[Image:eqsdna1.jpg]]<br />
<br />
[[Image:eqsdna2.jpg]]<br />
<br />
==='''Methods'''===<br />
<br />
We used our original python program (4 th order explicit Runge-Kutta algorithm) to solve thess equations.<br />
<br />
==='''Results'''===<br />
<br />
We input signal 1,2,4 and simulated the time evolution of the system.<br />
<br />
The time course of DNA concentration<br />
<br />
[[Image:Sim_res.JPG]]<br />
<br />
The time course of mRNA concentration. Note that the output of this system is not protein but mRNA.<br />
<br />
[[Image:Rna.JPG]]<br />
<br />
==='''Discussion'''===<br />
<br />
It is confirmed that the system works as expected with a certain combination of parameters.<br />
<br />
In addition, we changed many parameters, especially,l0,v(whose values are unclear) and p(leakiness of terminator) , to see whether the device is able to operate within a large part of parameter space.<br />
<br />
Although increasing p basically increases the concentration of correct output mRNA, it also increases incorrect mRNA. <br />
On the other hand, decreasing p generally decreases both the concentration of correct mRNA and incorrect mRNA. Besides, too low value of p cannot leads to failure operation because cre protein concentration cannot get high enough.<br />
Careful test have shown that there is certain value of p that [correct mRNA]/[incorrect mRNA] takes maximum value. In this case, the optimal value is approximately p = 0.001 and the ratio is 10^(-6).<br />
<br />
Changing the value of v and l0 doesn't affect the function of the circuits qualitatively although it changed the time the systems takes to reach steady state.<br />
<br />
As a result, the circuit can function correctly within the range of 0.001< p < 0.1, 0.001 < v <1.0, 0.01 < l0 < 1.0 , demonstrating its robustness.<br />
This robustness may come from the assumption that DNA recombination is irreversible, which is supported by our design of the system.<br />
<br />
==='''conclusion'''===<br />
<br />
The correct operation of the switch was confirmed. By changing parameter exponentially, the robustness of the system was shown.<br />
<br />
=='''A indifferent sudoku cell simulation <br />
including MS2 phage infection'''==<br />
<br />
<br />
[[Image:What_Sudoku_99.png|200px|thumb|What's Sudoku?]]<br />
<br />
<br />
We made a model for a indifferent sudoku cell.<br />
<br />
We traverse the amount of phages and see wether the cell react properly against given signals.<br />
<br />
<br />
===='''Modeling'''====<br />
<br />
'''Variable'''<br />
<br />
U: uninfected coli<br />
<br />
I: coli which got '1' signal<br />
<br />
Io1: other once infected coli<br />
<br />
I2: coli which got '1''2' signal<br />
<br />
Io2: other second infected coli<br />
<br />
I3: coli which got '1''2''3' signal<br />
<br />
Io3: other third infected coli<br />
<br />
B4: '4' coli which began phage making<br />
<br />
Bo: other coli which began phage making<br />
<br />
P4: free phage of '4'<br />
<br />
Po: other free phage<br />
<br />
S1: signal virus with information of number '1' which come from other sudoku cell<br />
<br />
S2: signal virus with information of number '2' which come from other sudoku cell <br />
<br />
S3: signal virus with information of number '3' which come from other sudoku cell<br />
<br />
S4: signal virus with information of number '4' which come from other sudoku cell<br />
<br />
<br />
<br />
'''Parameters'''<br />
<br />
g: growth rate of coli <br />
<br />
k1: rate of infection<br />
<br />
k2: death rate of uninfected<br />
<br />
k3: lysis rate of infected<br />
<br />
k4: rate of phage producing<br />
<br />
k5: phage death rate<br />
<br />
k6: terminator leak rate<br />
<br />
<br />
Arbitrary parameters we can change are<br />
<br />
* k1(rate of infection) : k1 relates to the specificity of antisenseRNA. <br />
<br />
* k4(rate of phage producing) : k4 relates to the strength of promoter. <br />
<br />
* k6(terminator leak rate) : k6 express terminator leak. <br />
<br />
We change these parameters, and confirmed its effect.<br />
<br />
<br />
<br />
'''Equations'''<br />
<br />
We assume that cell lysis caused by MS2 phage is sufficiently rapid than coli death rate.<br />
<br />
<br />
[[Image:Model-eq.png|680px|Equations used in this modeling]]<br />
<br />
<br />
===='''Method'''====<br />
<br />
We used original matlab program.<br />
<br />
We set conditions as following.<br />
-when t = 10(min), signal of number '1' comes.(step input)<br />
-when t = 30(min), signal of number '2' comes.(step input)<br />
-when t = 50(min), signal of number '3' comes.(step input)<br />
<br />
<br />
===='''Results'''====<br />
<br />
The initial condition of uninfected coli concentration is set to be 1, and observed time evolution.<br />
<br />
<br />
[[Image:Whole1.png]]<br />
<br />
brue(U): initial concentration is 1. After got signal '1', it decrease suddenly. <br />
<br />
green: I1 begins increasing after getting first signal '1'.<br />
<br />
<br />
<br />
[[Image:MS2-0.5,0.6,0.png]]<br />
<br />
k1:0.4, k4:0.7, k6:0.1<br />
red: The time course of virus concentration which has the information of number '4'.<br />
<br />
blue: The time course of virus concentration which has the information of number '1'or'2'or'3'.<br />
<br />
<br />
<br />
[[Image:MS2 2.png]]<br />
<br />
k1:0.4, k4:0.7, k6:0.01<br />
<br />
red: The time course of virus concentration which has the information of number '4'.<br />
<br />
blue: The time course of virus concentration which has the information of number '1'or'2'or'3'.<br />
<br />
As terminator leak rate is less than the previous trial, it takes longer time to start increasing. the concentration of Virus '4' clearly increases comparing to others.<br />
<br />
<br />
[[Image:MS2 4.png]]<br />
<br />
k1:0.4, k4:0.1, k6:0.01<br />
<br />
<br />
[[Image:MS2 3.png ]]<br />
<br />
k1:0.8, k4:0.7, k6:0.01<br />
<br />
<br />
===='''Discussion & Conclusion'''====<br />
<br />
It is confirmed that when one cell get '1''2''3'signals in order, the cell differentiate into '4'.<br />
<br />
The rate of infection(k1) corresponds to antisense RNA specificity. <br />
Intuitively, when k1 is too high, coli become sensitive to not only target signals but also wrong signal. Our simulation also indicate when k1 becomes higher, the concentration of Virus'4' decrease.<br />
<br />
The rate of phage producing(k4) corresponds to the strength of promoter. <br />
If promoter is too strong, the amount of the phage increase but it also raise terminator leak. However in our simulation, the effect to the rate of terminator leak error is lower. <br />
<br />
The rate of terminator leak relates to the time of the differentiation, and the accuracy of the signal.<br />
According to the simulation, we can see the clear difference between Pharge'4' and other Pharge'1''2''3'in 100-300 minutes, when the terminator leak rate is around 0.01.<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
{{UT-Tokyo_Foot}}</div>Lenahttp://2010.igem.org/Team:UT-Tokyo/Sudoku_modelingTeam:UT-Tokyo/Sudoku modeling2010-10-28T02:57:08Z<p>Lena: /* Whole system */</p>
<hr />
<div>__NOTOC__{{UT-Tokyo_CSS3}}{{UT-Tokyo_CSS_p}}{{UT-Tokyo_Head}}<br />
<br />
= '''Sudoku''' =<br />
<html><br />
<div><br />
<ul id="inpagemenu"><br />
<li><a href="/Team:UT-Tokyo/Sudoku_abstract" id="abstract">Introduction</a></li><br />
<li><a href="/Team:UT-Tokyo/Sudoku_construct" id="construct">System</a></li><br />
<li><span>Modeling</span></li><br />
<li><a href="/Team:UT-Tokyo/Sudoku_experiments" id="experiment">Experiments</a></li><br />
<li><a href="/Team:UT-Tokyo/Sudoku_perspective" id="perspective">Perspective</a></li><br />
<li><a href="/Team:UT-Tokyo/Sudoku_reference" id="reference">Reference</a></li><br />
</ul><br />
</div><br />
<div id="clear"></div><br />
</html><br />
== '''4C3leak switch''' ==<br />
<br />
We modeled our 4C3 leak switch as deterministic ODE system.<br />
<br />
<br />
==='''Modelling'''===<br />
<br />
<br />
===='''Assumption'''====<br />
<br />
1.We treated DNA as continuous variable, since there could be hundreds of plasmid in one E.coli (multicopy plasmid).<br />
There are 32 kinds of variables that stand for DNA concentration corresponding to their different internal states.<br />
<br />
2.We assumed that cre recombinase operate as tetramer, and other recombinase as dimer (but this seems not to be relevant).<br />
<br />
3.We assumed that association and dissociation of DNA recombinase to DNA is sufficiently fast so that equilibrate in the timescale of the whole switch.<br />
<br />
4.We ignored the reverse reaction of DNA recombinase because we used irreversible sequence (lox66, lox71 etc).<br />
<br />
===='''Variable'''====<br />
<br />
Basically, "pn" denotes protein concentration and "rn" denotes mRNA concentration<br />
<br />
p1,p2,p3,p4 : concentration of 4 recombinase protein<br />
<br />
r1,r2,r3,r4 : concentration of 4 recombinass mRNA<br />
<br />
u1,u2,u3,u4 : concentration of input mRNA<br />
<br />
pc,rc : concentration of cre recombinase protein and mRNA<br />
<br />
ps,rs : concentration of SP6 polymerase protein and mRNA<br />
<br />
Variables that denote concentration of DNA in each state.<br />
"a","b","c","d","e" means there are four,three,two,one,zero recombinases in the DNA, respectively.<br />
And the subscripts specify which recobinases remain, while the numbers in bracket mean whether the DNA has been cut by cre recombinase( (1) has not been cut, (2) has been cut).<br />
<br />
{| border="1"<br />
|<br />
|all<br />
|123<br />
|124<br />
|134<br />
|234<br />
<br />
|-<br />
|not cut by cre<br />
|a(1)<br />
|b123(1)<br />
|b124(1)<br />
|b134(1)<br />
|b234(1)<br />
|-<br />
|cut by cre<br />
|a(2)<br />
|b123(2)<br />
|b124(2)<br />
|b134(2)<br />
|b234(2)<br />
|}<br />
<br />
<br />
{| border = "1"<br />
|12<br />
|13<br />
|14<br />
|23<br />
|24<br />
|34<br />
|1<br />
|2<br />
|3<br />
|4<br />
|none<br />
|-<br />
|c12(1)<br />
|c13(1)<br />
|c14(1)<br />
|c23(1)<br />
|c24(1)<br />
|c34(1)<br />
|d1(1)<br />
|d2(1)<br />
|d3(1)<br />
|d4(1)<br />
|e(1)<br />
|-<br />
|c12(2)<br />
|c13(2)<br />
|c14(2)<br />
|c23(2)<br />
|c24(2)<br />
|c34(2)<br />
|d1(2)<br />
|d2(2)<br />
|d3(2)<br />
|d4(2)<br />
|e(2)<br />
|}<br />
<br />
====''' Parameters '''====<br />
k0 : mRNA translation rate<br />
<br />
k1 : protein degradation rate<br />
<br />
k2 : mRNA translation rate<br />
<br />
k3 : cre protein degradation rate<br />
<br />
l0 : transcription speed per unit concentration of RNA polymearase<br />
<br />
l1 : mRNA degradation rate<br />
<br />
l2 : cre mRNA degradation rate<br />
<br />
K : recombinase binding constant<br />
<br />
v : recombinase reaction rate<br />
<br />
pT7 : concentration o T7 polymerase<br />
<br />
p : terminator leak probability<br />
<br />
<br />
<br />
===='''Equations'''====<br />
<br />
[[Image:Eqs1.jpg]]<br />
<br />
[[Image:eqs_cs.jpg]]<br />
<br />
[[Image:eqsdna1.jpg]]<br />
<br />
[[Image:eqsdna2.jpg]]<br />
<br />
==='''Methods'''===<br />
<br />
We used our original python program (4 th order explicit Runge-Kutta algorithm) to solve thess equations.<br />
<br />
==='''Results'''===<br />
<br />
We input signal 1,2,4 and simulated the time evolution of the system.<br />
<br />
The time course of DNA concentration<br />
<br />
[[Image:Sim_res.JPG]]<br />
<br />
The time course of mRNA concentration. Note that the output of this system is not protein but mRNA.<br />
<br />
[[Image:Rna.JPG]]<br />
<br />
==='''Discussion'''===<br />
<br />
It is confirmed that the system works as expected with a certain combination of parameters.<br />
<br />
In addition, we changed many parameters, especially,l0,v(whose values are unclear) and p(leakiness of terminator) , to see whether the device is able to operate within a large part of parameter space.<br />
<br />
Although increasing p basically increases the concentration of correct output mRNA, it also increases incorrect mRNA. <br />
On the other hand, decreasing p generally decreases both the concentration of correct mRNA and incorrect mRNA. Besides, too low value of p cannot leads to failure operation because cre protein concentration cannot get high enough.<br />
Careful test have shown that there is certain value of p that [correct mRNA]/[incorrect mRNA] takes maximum value. In this case, the optimal value is approximately p = 0.001 and the ratio is 10^(-6).<br />
<br />
Changing the value of v and l0 doesn't affect the function of the circuits qualitatively although it changed the time the systems takes to reach steady state.<br />
<br />
As a result, the circuit can function correctly within the range of 0.001< p < 0.1, 0.001 < v <1.0, 0.01 < l0 < 1.0 , demonstrating its robustness.<br />
This robustness may come from the assumption that DNA recombination is irreversible, which is supported by our design of the system.<br />
<br />
==='''conclusion'''===<br />
<br />
The correct operation of the switch was confirmed. By changing parameter exponentially, the robustness of the system was shown.<br />
<br />
=='''A indifferent sudoku cell simulation including MS2 phage infection'''==<br />
<br />
We made a model for a indifferent sudoku cell.<br />
<br />
We traverse the amount of phages and see wether the cell react properly against given signals.<br />
<br />
<br />
===='''Modeling'''====<br />
<br />
'''Variable'''<br />
<br />
U: uninfected coli<br />
<br />
I: coli which got '1' signal<br />
<br />
Io1: other once infected coli<br />
<br />
I2: coli which got '1''2' signal<br />
<br />
Io2: other second infected coli<br />
<br />
I3: coli which got '1''2''3' signal<br />
<br />
Io3: other third infected coli<br />
<br />
B4: '4' coli which began phage making<br />
<br />
Bo: other coli which began phage making<br />
<br />
P4: free phage of '4'<br />
<br />
Po: other free phage<br />
<br />
S1: signal virus with information of number '1' which come from other sudoku cell<br />
<br />
S2: signal virus with information of number '2' which come from other sudoku cell <br />
<br />
S3: signal virus with information of number '3' which come from other sudoku cell<br />
<br />
S4: signal virus with information of number '4' which come from other sudoku cell<br />
<br />
<br />
<br />
'''Parameters'''<br />
<br />
g: growth rate of coli <br />
<br />
k1: rate of infection<br />
<br />
k2: death rate of uninfected<br />
<br />
k3: lysis rate of infected<br />
<br />
k4: rate of phage producing<br />
<br />
k5: phage death rate<br />
<br />
k6: terminator leak rate<br />
<br />
<br />
Arbitrary parameters we can change are<br />
<br />
* k1(rate of infection) : k1 relates to the specificity of antisenseRNA. <br />
<br />
* k4(rate of phage producing) : k4 relates to the strength of promoter. <br />
<br />
* k6(terminator leak rate) : k6 express terminator leak. <br />
<br />
We change these parameters, and confirmed its effect.<br />
<br />
<br />
<br />
'''Equations'''<br />
<br />
We assume that cell lysis caused by MS2 phage is sufficiently rapid than coli death rate.<br />
<br />
<br />
[[Image:Model-eq.png|680px|Equations used in this modeling]]<br />
<br />
<br />
===='''Method'''====<br />
<br />
We used original matlab program.<br />
<br />
We set conditions as following.<br />
-when t = 10(min), signal of number '1' comes.(step input)<br />
-when t = 30(min), signal of number '2' comes.(step input)<br />
-when t = 50(min), signal of number '3' comes.(step input)<br />
<br />
<br />
===='''Results'''====<br />
<br />
The initial condition of uninfected coli concentration is set to be 1, and observed time evolution.<br />
<br />
<br />
[[Image:Whole1.png]]<br />
brue(U): initial concentration is 1. After got signal '1', it decrease suddenly. <br />
<br />
green: I1 begins increasing after getting first signal '1'.<br />
<br />
<br />
<br />
[[Image:MS2-0.5,0.6,0.png]]<br />
k1:0.4, k4:0.7, k6:0.1<br />
red: The time course of virus concentration which has the information of number '4'.<br />
<br />
blue: The time course of virus concentration which has the information of number '1'or'2'or'3'.<br />
<br />
<br />
<br />
[[Image:MS2 2.png]]<br />
k1:0.4, k4:0.7, k6:0.01<br />
<br />
red: The time course of virus concentration which has the information of number '4'.<br />
<br />
blue: The time course of virus concentration which has the information of number '1'or'2'or'3'.<br />
<br />
As terminator leak rate is less than the previous trial, it takes longer time to start increasing. the concentration of Virus '4' clearly increases comparing to others.<br />
<br />
<br />
[[Image:MS2 4.png]]<br />
k1:0.4, k4:0.1, k6:0.01<br />
<br />
<br />
[[Image:MS2 3.png ]]<br />
k1:0.8, k4:0.7, k6:0.01<br />
<br />
<br />
===='''Discussion & Conclusion'''====<br />
<br />
It is confirmed that when one cell get '1''2''3'signals in order, the cell differentiate into '4'.<br />
<br />
The rate of infection(k1) corresponds to antisense RNA specificity. <br />
Intuitively, when k1 is too high, coli become sensitive to not only target signals but also wrong signal. Our simulation also indicate when k1 becomes higher, the concentration of Virus'4' decrease.<br />
<br />
The rate of phage producing(k4) corresponds to the strength of promoter. <br />
If promoter is too strong, the amount of the phage increase but it also raise terminator leak. However in our simulation, the effect to the rate of terminator leak error is lower. <br />
<br />
The rate of terminator leak relates to the time of the differentiation, and the accuracy of the signal.<br />
According to the simulation, we can see the clear difference between Pharge'4' and other Pharge'1''2''3'in 100-300 minutes, when the terminator leak rate is around 0.01.<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
{{UT-Tokyo_Foot}}</div>Lenahttp://2010.igem.org/Team:UT-Tokyo/Sudoku_modelingTeam:UT-Tokyo/Sudoku modeling2010-10-28T02:56:13Z<p>Lena: /* MS2 phage infection */</p>
<hr />
<div>__NOTOC__{{UT-Tokyo_CSS3}}{{UT-Tokyo_CSS_p}}{{UT-Tokyo_Head}}<br />
<br />
= '''Sudoku''' =<br />
<html><br />
<div><br />
<ul id="inpagemenu"><br />
<li><a href="/Team:UT-Tokyo/Sudoku_abstract" id="abstract">Introduction</a></li><br />
<li><a href="/Team:UT-Tokyo/Sudoku_construct" id="construct">System</a></li><br />
<li><span>Modeling</span></li><br />
<li><a href="/Team:UT-Tokyo/Sudoku_experiments" id="experiment">Experiments</a></li><br />
<li><a href="/Team:UT-Tokyo/Sudoku_perspective" id="perspective">Perspective</a></li><br />
<li><a href="/Team:UT-Tokyo/Sudoku_reference" id="reference">Reference</a></li><br />
</ul><br />
</div><br />
<div id="clear"></div><br />
</html><br />
== '''4C3leak switch''' ==<br />
<br />
We modeled our 4C3 leak switch as deterministic ODE system.<br />
<br />
<br />
==='''Modelling'''===<br />
<br />
<br />
===='''Assumption'''====<br />
<br />
1.We treated DNA as continuous variable, since there could be hundreds of plasmid in one E.coli (multicopy plasmid).<br />
There are 32 kinds of variables that stand for DNA concentration corresponding to their different internal states.<br />
<br />
2.We assumed that cre recombinase operate as tetramer, and other recombinase as dimer (but this seems not to be relevant).<br />
<br />
3.We assumed that association and dissociation of DNA recombinase to DNA is sufficiently fast so that equilibrate in the timescale of the whole switch.<br />
<br />
4.We ignored the reverse reaction of DNA recombinase because we used irreversible sequence (lox66, lox71 etc).<br />
<br />
===='''Variable'''====<br />
<br />
Basically, "pn" denotes protein concentration and "rn" denotes mRNA concentration<br />
<br />
p1,p2,p3,p4 : concentration of 4 recombinase protein<br />
<br />
r1,r2,r3,r4 : concentration of 4 recombinass mRNA<br />
<br />
u1,u2,u3,u4 : concentration of input mRNA<br />
<br />
pc,rc : concentration of cre recombinase protein and mRNA<br />
<br />
ps,rs : concentration of SP6 polymerase protein and mRNA<br />
<br />
Variables that denote concentration of DNA in each state.<br />
"a","b","c","d","e" means there are four,three,two,one,zero recombinases in the DNA, respectively.<br />
And the subscripts specify which recobinases remain, while the numbers in bracket mean whether the DNA has been cut by cre recombinase( (1) has not been cut, (2) has been cut).<br />
<br />
{| border="1"<br />
|<br />
|all<br />
|123<br />
|124<br />
|134<br />
|234<br />
<br />
|-<br />
|not cut by cre<br />
|a(1)<br />
|b123(1)<br />
|b124(1)<br />
|b134(1)<br />
|b234(1)<br />
|-<br />
|cut by cre<br />
|a(2)<br />
|b123(2)<br />
|b124(2)<br />
|b134(2)<br />
|b234(2)<br />
|}<br />
<br />
<br />
{| border = "1"<br />
|12<br />
|13<br />
|14<br />
|23<br />
|24<br />
|34<br />
|1<br />
|2<br />
|3<br />
|4<br />
|none<br />
|-<br />
|c12(1)<br />
|c13(1)<br />
|c14(1)<br />
|c23(1)<br />
|c24(1)<br />
|c34(1)<br />
|d1(1)<br />
|d2(1)<br />
|d3(1)<br />
|d4(1)<br />
|e(1)<br />
|-<br />
|c12(2)<br />
|c13(2)<br />
|c14(2)<br />
|c23(2)<br />
|c24(2)<br />
|c34(2)<br />
|d1(2)<br />
|d2(2)<br />
|d3(2)<br />
|d4(2)<br />
|e(2)<br />
|}<br />
<br />
====''' Parameters '''====<br />
k0 : mRNA translation rate<br />
<br />
k1 : protein degradation rate<br />
<br />
k2 : mRNA translation rate<br />
<br />
k3 : cre protein degradation rate<br />
<br />
l0 : transcription speed per unit concentration of RNA polymearase<br />
<br />
l1 : mRNA degradation rate<br />
<br />
l2 : cre mRNA degradation rate<br />
<br />
K : recombinase binding constant<br />
<br />
v : recombinase reaction rate<br />
<br />
pT7 : concentration o T7 polymerase<br />
<br />
p : terminator leak probability<br />
<br />
<br />
<br />
===='''Equations'''====<br />
<br />
[[Image:Eqs1.jpg]]<br />
<br />
[[Image:eqs_cs.jpg]]<br />
<br />
[[Image:eqsdna1.jpg]]<br />
<br />
[[Image:eqsdna2.jpg]]<br />
<br />
==='''Methods'''===<br />
<br />
We used our original python program (4 th order explicit Runge-Kutta algorithm) to solve thess equations.<br />
<br />
==='''Results'''===<br />
<br />
We input signal 1,2,4 and simulated the time evolution of the system.<br />
<br />
The time course of DNA concentration<br />
<br />
[[Image:Sim_res.JPG]]<br />
<br />
The time course of mRNA concentration. Note that the output of this system is not protein but mRNA.<br />
<br />
[[Image:Rna.JPG]]<br />
<br />
==='''Discussion'''===<br />
<br />
It is confirmed that the system works as expected with a certain combination of parameters.<br />
<br />
In addition, we changed many parameters, especially,l0,v(whose values are unclear) and p(leakiness of terminator) , to see whether the device is able to operate within a large part of parameter space.<br />
<br />
Although increasing p basically increases the concentration of correct output mRNA, it also increases incorrect mRNA. <br />
On the other hand, decreasing p generally decreases both the concentration of correct mRNA and incorrect mRNA. Besides, too low value of p cannot leads to failure operation because cre protein concentration cannot get high enough.<br />
Careful test have shown that there is certain value of p that [correct mRNA]/[incorrect mRNA] takes maximum value. In this case, the optimal value is approximately p = 0.001 and the ratio is 10^(-6).<br />
<br />
Changing the value of v and l0 doesn't affect the function of the circuits qualitatively although it changed the time the systems takes to reach steady state.<br />
<br />
As a result, the circuit can function correctly within the range of 0.001< p < 0.1, 0.001 < v <1.0, 0.01 < l0 < 1.0 , demonstrating its robustness.<br />
This robustness may come from the assumption that DNA recombination is irreversible, which is supported by our design of the system.<br />
<br />
==='''conclusion'''===<br />
<br />
The correct operation of the switch was confirmed. By changing parameter exponentially, the robustness of the system was shown.<br />
<br />
=='''A indifferent sudoku cell simulation including MS2 phage infection'''==<br />
<br />
We made a model for a indifferent sudoku cell.<br />
<br />
We traverse the amount of phages and see wether the cell react properly against given signals.<br />
<br />
<br />
===='''Modeling'''====<br />
<br />
'''Variable'''<br />
<br />
U: uninfected coli<br />
<br />
I: coli which got '1' signal<br />
<br />
Io1: other once infected coli<br />
<br />
I2: coli which got '1''2' signal<br />
<br />
Io2: other second infected coli<br />
<br />
I3: coli which got '1''2''3' signal<br />
<br />
Io3: other third infected coli<br />
<br />
B4: '4' coli which began phage making<br />
<br />
Bo: other coli which began phage making<br />
<br />
P4: free phage of '4'<br />
<br />
Po: other free phage<br />
<br />
S1: signal virus with information of number '1' which come from other sudoku cell<br />
<br />
S2: signal virus with information of number '2' which come from other sudoku cell <br />
<br />
S3: signal virus with information of number '3' which come from other sudoku cell<br />
<br />
S4: signal virus with information of number '4' which come from other sudoku cell<br />
<br />
<br />
<br />
'''Parameters'''<br />
<br />
g: growth rate of coli <br />
<br />
k1: rate of infection<br />
<br />
k2: death rate of uninfected<br />
<br />
k3: lysis rate of infected<br />
<br />
k4: rate of phage producing<br />
<br />
k5: phage death rate<br />
<br />
k6: terminator leak rate<br />
<br />
<br />
Arbitrary parameters we can change are<br />
<br />
* k1(rate of infection) : k1 relates to the specificity of antisenseRNA. <br />
<br />
* k4(rate of phage producing) : k4 relates to the strength of promoter. <br />
<br />
* k6(terminator leak rate) : k6 express terminator leak. <br />
<br />
We change these parameters, and confirmed its effect.<br />
<br />
<br />
<br />
'''Equations'''<br />
<br />
We assume that cell lysis caused by MS2 phage is sufficiently rapid than coli death rate.<br />
<br />
<br />
[[Image:Model-eq.png|680px|Equations used in this modeling]]<br />
<br />
<br />
===='''Method'''====<br />
<br />
We used original matlab program.<br />
<br />
We set conditions as following.<br />
-when t = 10(min), signal of number '1' comes.(step input)<br />
-when t = 30(min), signal of number '2' comes.(step input)<br />
-when t = 50(min), signal of number '3' comes.(step input)<br />
<br />
<br />
===='''Results'''====<br />
<br />
The initial condition of uninfected coli concentration is set to be 1, and observed time evolution.<br />
<br />
<br />
[[Image:Whole1.png]]<br />
brue(U): initial concentration is 1. After got signal '1', it decrease suddenly. <br />
<br />
green: I1 begins increasing after getting first signal '1'.<br />
<br />
<br />
<br />
[[Image:MS2-0.5,0.6,0.png]]<br />
k1:0.4, k4:0.7, k6:0.1<br />
red: The time course of virus concentration which has the information of number '4'.<br />
<br />
blue: The time course of virus concentration which has the information of number '1'or'2'or'3'.<br />
<br />
<br />
<br />
[[Image:MS2 2.png]]<br />
k1:0.4, k4:0.7, k6:0.01<br />
<br />
red: The time course of virus concentration which has the information of number '4'.<br />
<br />
blue: The time course of virus concentration which has the information of number '1'or'2'or'3'.<br />
<br />
As terminator leak rate is less than the previous trial, it takes longer time to start increasing. the concentration of Virus '4' clearly increases comparing to others.<br />
<br />
<br />
[[Image:MS2 4.png]]<br />
k1:0.4, k4:0.1, k6:0.01<br />
<br />
<br />
[[Image:MS2 3.png ]]<br />
k1:0.8, k4:0.7, k6:0.01<br />
<br />
<br />
===='''Discussion & Conclusion'''====<br />
<br />
It is confirmed that when one cell get '1''2''3'signals in order, the cell differentiate into '4'.<br />
<br />
The rate of infection(k1) corresponds to antisense RNA specificity. <br />
Intuitively, when k1 is too high, coli become sensitive to not only target signals but also wrong signal. Our simulation also indicate when k1 becomes higher, the concentration of Virus'4' decrease.<br />
<br />
The rate of phage producing(k4) corresponds to the strength of promoter. <br />
If promoter is too strong, the amount of the phage increase but it also raise terminator leak. However in our simulation, the effect to the rate of terminator leak error is lower. <br />
<br />
The rate of terminator leak relates to the time of the differentiation, and the accuracy of the signal.<br />
According to the simulation, we can see the clear difference between Pharge'4' and other Pharge'1''2''3'in 100-300 minutes, when the terminator leak rate is around 0.01.<br />
<br />
=='''Whole system'''==<br />
<br />
[[Image:What_Sudoku_99.png|200px|thumb|What's Sudoku?]]<br />
<br />
<br />
<br />
<br />
<br />
<br />
{{UT-Tokyo_Foot}}</div>Lenahttp://2010.igem.org/File:MS2_3.pngFile:MS2 3.png2010-10-28T02:54:25Z<p>Lena: </p>
<hr />
<div></div>Lenahttp://2010.igem.org/File:MS2_4.pngFile:MS2 4.png2010-10-28T02:51:59Z<p>Lena: </p>
<hr />
<div></div>Lenahttp://2010.igem.org/Team:UT-Tokyo/Sudoku_modelingTeam:UT-Tokyo/Sudoku modeling2010-10-28T02:30:25Z<p>Lena: /* Results */</p>
<hr />
<div>__NOTOC__{{UT-Tokyo_CSS3}}{{UT-Tokyo_CSS_p}}{{UT-Tokyo_Head}}<br />
<br />
= '''Sudoku''' =<br />
<html><br />
<div><br />
<ul id="inpagemenu"><br />
<li><a href="/Team:UT-Tokyo/Sudoku_abstract" id="abstract">Introduction</a></li><br />
<li><a href="/Team:UT-Tokyo/Sudoku_construct" id="construct">System</a></li><br />
<li><span>Modeling</span></li><br />
<li><a href="/Team:UT-Tokyo/Sudoku_experiments" id="experiment">Experiments</a></li><br />
<li><a href="/Team:UT-Tokyo/Sudoku_perspective" id="perspective">Perspective</a></li><br />
<li><a href="/Team:UT-Tokyo/Sudoku_reference" id="reference">Reference</a></li><br />
</ul><br />
</div><br />
<div id="clear"></div><br />
</html><br />
== '''4C3leak switch''' ==<br />
<br />
We modeled our 4C3 leak switch as deterministic ODE system.<br />
<br />
<br />
==='''Modelling'''===<br />
<br />
<br />
===='''Assumption'''====<br />
<br />
1.We treated DNA as continuous variabl, since there could be hundreds of plasmid in one E.coli (multicopy plasmid).<br />
There are 32 kinds of variables that stand for DNA concentration corresponding to their different internal states.<br />
<br />
2.We assumed that cre recombinase operate as tetramer, and other recombinase as dimer (but this seems not to be relevant).<br />
<br />
3.We assumed that association and dissociation of DNA recombinase to DNA is sufficiently fast so that equilibrate in the timescale of the whole switch.<br />
<br />
4.We ignored the reverse reaction of DNA recombinase because we used irreversible sequence (lox66, lox71 etc).<br />
<br />
===='''Variable'''====<br />
<br />
Basically, "pn" denotes protein concentration and "rn" denotes mRNA concentration<br />
<br />
p1,p2,p3,p4 : concentration of 4 recombinase protein<br />
<br />
r1,r2,r3,r4 : concentration of 4 recombinass mRNA<br />
<br />
u1,u2,u3,u4 : concentration of input mRNA<br />
<br />
pc,rc : concentration of cre recombinase protein and mRNA<br />
<br />
ps,rs : concentration of SP6 polymerase protein and mRNA<br />
<br />
Variables that denote concentration of DNA in each state.<br />
"a","b","c","d","e" means there are four,three,two,one,zero recombinases in the DNA, respectively.<br />
And the subscripts specify which recobinases remain, while the numbers in bracket mean whether the DNA has been cut by cre recombinase( (1) has not been cut, (2) has been cut).<br />
<br />
{| border="1"<br />
|<br />
|all<br />
|123<br />
|124<br />
|134<br />
|234<br />
<br />
|-<br />
|not cut by cre<br />
|a(1)<br />
|b123(1)<br />
|b124(1)<br />
|b134(1)<br />
|b234(1)<br />
|-<br />
|cut by cre<br />
|a(2)<br />
|b123(2)<br />
|b124(2)<br />
|b134(2)<br />
|b234(2)<br />
|}<br />
<br />
<br />
{| border = "1"<br />
|12<br />
|13<br />
|14<br />
|23<br />
|24<br />
|34<br />
|1<br />
|2<br />
|3<br />
|4<br />
|none<br />
|-<br />
|c12(1)<br />
|c13(1)<br />
|c14(1)<br />
|c23(1)<br />
|c24(1)<br />
|c34(1)<br />
|d1(1)<br />
|d2(1)<br />
|d3(1)<br />
|d4(1)<br />
|e(1)<br />
|-<br />
|c12(2)<br />
|c13(2)<br />
|c14(2)<br />
|c23(2)<br />
|c24(2)<br />
|c34(2)<br />
|d1(2)<br />
|d2(2)<br />
|d3(2)<br />
|d4(2)<br />
|e(2)<br />
|}<br />
<br />
====''' Parameters '''====<br />
k0 : mRNA translation rate<br />
<br />
k1 : protein degradation rate<br />
<br />
k2 : mRNA translation rate<br />
<br />
k3 : cre protein degradation rate<br />
<br />
l0 : transcription speed per unit concentration of RNA polymearase<br />
<br />
l1 : mRNA degradation rate<br />
<br />
l2 : cre mRNA degradation rate<br />
<br />
K : recombinase binding constant<br />
<br />
v : recombinase reaction rate<br />
<br />
pT7 : concentration o T7 polymerase<br />
<br />
p : terminator leak probability<br />
<br />
<br />
<br />
===='''Equations'''====<br />
<br />
[[Image:Eqs1.jpg]]<br />
<br />
[[Image:eqs_cs.jpg]]<br />
<br />
==='''Methods'''===<br />
<br />
We used our original python program (4 th order explicit Runge-Kutta algorithm) to solve thess equations.<br />
<br />
==='''Results'''===<br />
<br />
We input signal 1,2,4 and simulated the time evolution of the system.<br />
<br />
The time course of DNA concentration<br />
<br />
[[Image:Sim_res.JPG]]<br />
<br />
The time course of mRNA concentration. Note that the output of this system is not protein but mRNA.<br />
<br />
[[Image:Rna.JPG]]<br />
<br />
==='''Discussion'''===<br />
<br />
It is confirmed that the system works as expected with a certain combination of parameters.<br />
<br />
In addition, we changed many parameters, especially,l0,v(whose values are unclear) and p(leakiness of terminator) , to see whether the device is able to operate within a large part of parameter space.<br />
<br />
Although increasing p basically increases the concentration of correct output mRNA, it also increases incorrect mRNA. <br />
On the other hand, decreasing p generally decreases both the concentration of correct mRNA and incorrect mRNA. Besides, too low value of p cannot leads to failure operation because cre protein concentration cannot get high enough.<br />
Careful test have shown that there is certain value of p that [correct mRNA]/[incorrect mRNA] takes minimum value. In this case it is p = 0.001 and the ratio is 10^(-6).<br />
<br />
Changing the value of v and l0 doesn't affect the function of the circuits qualitatively although it changed the time the systems takes to reach steady state.<br />
<br />
As a result, the circuit can function correctly within the range of 0.001< p < 0.1, 0.001 < v <1.0, 0.01 < l0 < 1.0 , demonstrating its robustness.<br />
This robustness may come from the assumption that DNA recombination is irreversible, which is supported by our design of the system.<br />
<br />
==='''conclusion'''===<br />
<br />
The correct operation of the switch was confirmed. By changing parameter exponentially, the robustness of the system was shown.<br />
<br />
=='''MS2 phage infection'''==<br />
<br />
We made a model for a indifferent sudoku cell.<br />
<br />
We traverse the amount of phages and see wether the cell react properly against given signals.<br />
<br />
<br />
===='''Modeling'''====<br />
<br />
'''Variable'''<br />
<br />
U: uninfected coli<br />
<br />
I: coli which got '1' signal<br />
<br />
Io1: other once infected coli<br />
<br />
I2: coli which got '1''2' signal<br />
<br />
Io2: other second infected coli<br />
<br />
I3: coli which got '1''2''3' signal<br />
<br />
Io3: other third infected coli<br />
<br />
B4: '4' coli which began phage making<br />
<br />
Bo: other coli which began phage making<br />
<br />
P4: free phage of '4'<br />
<br />
Po: other free phage<br />
<br />
S1: signal virus with information of number '1' which come from other sudoku cell<br />
<br />
S2: signal virus with information of number '2' which come from other sudoku cell <br />
<br />
S3: signal virus with information of number '3' which come from other sudoku cell<br />
<br />
S4: signal virus with information of number '4' which come from other sudoku cell<br />
<br />
<br />
<br />
'''Parameters'''<br />
<br />
g: growth rate of coli <br />
<br />
k1: rate of infection<br />
<br />
k2: death rate of uninfected<br />
<br />
k3: lysis rate of infected<br />
<br />
k4: rate of phage producing<br />
<br />
k5: phage death rate<br />
<br />
k6: terminator leak rate<br />
<br />
<br />
'''Equations'''<br />
<br />
We assume that cell lysis caused by MS2 phage is sufficiently rapid than coli death rate.<br />
<br />
<br />
<br />
We assume that cell lysis caused by MS2 phage is sufficiently rapid than coli death rate.<br />
<br />
[[Image:Model-eq.png|680px|Equations used in this modeling]]<br />
<br />
===='''Method'''====<br />
<br />
We used original matlab program.<br />
<br />
We set conditions as following.<br />
-when t = 10(min), signal of number '1' comes.(step input)<br />
-when t = 30(min), signal of number '2' comes.(step input)<br />
-when t = 50(min), signal of number '3' comes.(step input)<br />
<br />
===='''Results'''====<br />
<br />
The initial condition of uninfected coli concentration is set to be 1, and observed time evolution.<br />
<br />
<br />
[[Image:Whole1.png]]<br />
brue(U): initial concentration is 1. After got signal '1', it decrease suddenly. <br />
green: I1 begins increasing after getting first signal '1'.<br />
<br />
<br />
[[Image:MS2-0.5,0.6,0.png]]<br />
k1:0.4, k4:0.7, k6:0.1<br />
red: The time course of virus concentration which has the information of number '4'.<br />
blue: The time course of virus concentration which has the information of number '1'or'2'or'3'.<br />
<br />
<br />
[[Image:MS2 2.png]]<br />
k1:0.4, k4:0.7, k6:0.01<br />
red: The time course of virus concentration which has the information of number '4'.<br />
blue: The time course of virus concentration which has the information of number '1'or'2'or'3'.<br />
As terminator leak rate is less than the previous trial, it takes longer time to start increasing. the concentration of Virus '4' clearly increases comparing to others.<br />
<br />
===='''Discussion'''====<br />
<br />
<br />
<br />
'''conclusion'''<br />
<br />
=='''Whole system'''==<br />
<br />
[[Image:What_Sudoku_99.png|200px|thumb|What's Sudoku?]]<br />
<br />
<br />
<br />
<br />
<br />
<br />
{{UT-Tokyo_Foot}}</div>Lenahttp://2010.igem.org/Team:UT-Tokyo/Sudoku_modelingTeam:UT-Tokyo/Sudoku modeling2010-10-28T02:25:33Z<p>Lena: /* Modeling */</p>
<hr />
<div>__NOTOC__{{UT-Tokyo_CSS3}}{{UT-Tokyo_CSS_p}}{{UT-Tokyo_Head}}<br />
<br />
= '''Sudoku''' =<br />
<html><br />
<div><br />
<ul id="inpagemenu"><br />
<li><a href="/Team:UT-Tokyo/Sudoku_abstract" id="abstract">Introduction</a></li><br />
<li><a href="/Team:UT-Tokyo/Sudoku_construct" id="construct">System</a></li><br />
<li><span>Modeling</span></li><br />
<li><a href="/Team:UT-Tokyo/Sudoku_experiments" id="experiment">Experiments</a></li><br />
<li><a href="/Team:UT-Tokyo/Sudoku_perspective" id="perspective">Perspective</a></li><br />
<li><a href="/Team:UT-Tokyo/Sudoku_reference" id="reference">Reference</a></li><br />
</ul><br />
</div><br />
<div id="clear"></div><br />
</html><br />
== '''4C3leak switch''' ==<br />
<br />
We modeled our 4C3 leak switch as deterministic ODE system.<br />
<br />
<br />
==='''Modelling'''===<br />
<br />
<br />
===='''Assumption'''====<br />
<br />
1.We treated DNA as continuous variabl, since there could be hundreds of plasmid in one E.coli (multicopy plasmid).<br />
There are 32 kinds of variables that stand for DNA concentration corresponding to their different internal states.<br />
<br />
2.We assumed that cre recombinase operate as tetramer, and other recombinase as dimer (but this seems not to be relevant).<br />
<br />
3.We assumed that association and dissociation of DNA recombinase to DNA is sufficiently fast so that equilibrate in the timescale of the whole switch.<br />
<br />
4.We ignored the reverse reaction of DNA recombinase because we used irreversible sequence (lox66, lox71 etc).<br />
<br />
===='''Variable'''====<br />
<br />
Basically, "pn" denotes protein concentration and "rn" denotes mRNA concentration<br />
<br />
p1,p2,p3,p4 : concentration of 4 recombinase protein<br />
<br />
r1,r2,r3,r4 : concentration of 4 recombinass mRNA<br />
<br />
u1,u2,u3,u4 : concentration of input mRNA<br />
<br />
pc,rc : concentration of cre recombinase protein and mRNA<br />
<br />
ps,rs : concentration of SP6 polymerase protein and mRNA<br />
<br />
Variables that denote concentration of DNA in each state.<br />
"a","b","c","d","e" means there are four,three,two,one,zero recombinases in the DNA, respectively.<br />
And the subscripts specify which recobinases remain, while the numbers in bracket mean whether the DNA has been cut by cre recombinase( (1) has not been cut, (2) has been cut).<br />
<br />
{| border="1"<br />
|<br />
|all<br />
|123<br />
|124<br />
|134<br />
|234<br />
<br />
|-<br />
|not cut by cre<br />
|a(1)<br />
|b123(1)<br />
|b124(1)<br />
|b134(1)<br />
|b234(1)<br />
|-<br />
|cut by cre<br />
|a(2)<br />
|b123(2)<br />
|b124(2)<br />
|b134(2)<br />
|b234(2)<br />
|}<br />
<br />
<br />
{| border = "1"<br />
|12<br />
|13<br />
|14<br />
|23<br />
|24<br />
|34<br />
|1<br />
|2<br />
|3<br />
|4<br />
|none<br />
|-<br />
|c12(1)<br />
|c13(1)<br />
|c14(1)<br />
|c23(1)<br />
|c24(1)<br />
|c34(1)<br />
|d1(1)<br />
|d2(1)<br />
|d3(1)<br />
|d4(1)<br />
|e(1)<br />
|-<br />
|c12(2)<br />
|c13(2)<br />
|c14(2)<br />
|c23(2)<br />
|c24(2)<br />
|c34(2)<br />
|d1(2)<br />
|d2(2)<br />
|d3(2)<br />
|d4(2)<br />
|e(2)<br />
|}<br />
<br />
====''' Parameters '''====<br />
k0 : mRNA translation rate<br />
<br />
k1 : protein degradation rate<br />
<br />
k2 : mRNA translation rate<br />
<br />
k3 : cre protein degradation rate<br />
<br />
l0 : transcription speed per unit concentration of RNA polymearase<br />
<br />
l1 : mRNA degradation rate<br />
<br />
l2 : cre mRNA degradation rate<br />
<br />
K : recombinase binding constant<br />
<br />
v : recombinase reaction rate<br />
<br />
pT7 : concentration o T7 polymerase<br />
<br />
p : terminator leak probability<br />
<br />
<br />
<br />
===='''Equations'''====<br />
<br />
[[Image:Eqs1.jpg]]<br />
<br />
[[Image:eqs_cs.jpg]]<br />
<br />
==='''Methods'''===<br />
<br />
We used our original python program (4 th order explicit Runge-Kutta algorithm) to solve thess equations.<br />
<br />
==='''Results'''===<br />
<br />
We input signal 1,2,4 and simulated the time evolution of the system.<br />
<br />
The time course of DNA concentration<br />
<br />
[[Image:Sim_res.JPG]]<br />
<br />
The time course of mRNA concentration. Note that the output of this system is not protein but mRNA.<br />
<br />
[[Image:Rna.JPG]]<br />
<br />
==='''Discussion'''===<br />
<br />
It is confirmed that the system works as expected with a certain combination of parameters.<br />
<br />
In addition, we changed many parameters, especially,l0,v(whose values are unclear) and p(leakiness of terminator) , to see whether the device is able to operate within a large part of parameter space.<br />
<br />
Although increasing p basically increases the concentration of correct output mRNA, it also increases incorrect mRNA. <br />
On the other hand, decreasing p generally decreases both the concentration of correct mRNA and incorrect mRNA. Besides, too low value of p cannot leads to failure operation because cre protein concentration cannot get high enough.<br />
Careful test have shown that there is certain value of p that [correct mRNA]/[incorrect mRNA] takes minimum value. In this case it is p = 0.001 and the ratio is 10^(-6).<br />
<br />
Changing the value of v and l0 doesn't affect the function of the circuits qualitatively although it changed the time the systems takes to reach steady state.<br />
<br />
As a result, the circuit can function correctly within the range of 0.001< p < 0.1, 0.001 < v <1.0, 0.01 < l0 < 1.0 , demonstrating its robustness.<br />
This robustness may come from the assumption that DNA recombination is irreversible, which is supported by our design of the system.<br />
<br />
==='''conclusion'''===<br />
<br />
The correct operation of the switch was confirmed. By changing parameter exponentially, the robustness of the system was shown.<br />
<br />
=='''MS2 phage infection'''==<br />
<br />
We made a model for a indifferent sudoku cell.<br />
<br />
We traverse the amount of phages and see wether the cell react properly against given signals.<br />
<br />
<br />
===='''Modeling'''====<br />
<br />
'''Variable'''<br />
<br />
U: uninfected coli<br />
<br />
I: coli which got '1' signal<br />
<br />
Io1: other once infected coli<br />
<br />
I2: coli which got '1''2' signal<br />
<br />
Io2: other second infected coli<br />
<br />
I3: coli which got '1''2''3' signal<br />
<br />
Io3: other third infected coli<br />
<br />
B4: '4' coli which began phage making<br />
<br />
Bo: other coli which began phage making<br />
<br />
P4: free phage of '4'<br />
<br />
Po: other free phage<br />
<br />
S1: signal virus with information of number '1' which come from other sudoku cell<br />
<br />
S2: signal virus with information of number '2' which come from other sudoku cell <br />
<br />
S3: signal virus with information of number '3' which come from other sudoku cell<br />
<br />
S4: signal virus with information of number '4' which come from other sudoku cell<br />
<br />
<br />
<br />
'''Parameters'''<br />
<br />
g: growth rate of coli <br />
<br />
k1: rate of infection<br />
<br />
k2: death rate of uninfected<br />
<br />
k3: lysis rate of infected<br />
<br />
k4: rate of phage producing<br />
<br />
k5: phage death rate<br />
<br />
k6: terminator leak rate<br />
<br />
<br />
'''Equations'''<br />
<br />
We assume that cell lysis caused by MS2 phage is sufficiently rapid than coli death rate.<br />
<br />
<br />
<br />
We assume that cell lysis caused by MS2 phage is sufficiently rapid than coli death rate.<br />
<br />
[[Image:Model-eq.png|680px|Equations used in this modeling]]<br />
<br />
===='''Method'''====<br />
<br />
We used original matlab program.<br />
<br />
We set conditions as following.<br />
-when t = 10(min), signal of number '1' comes.(step input)<br />
-when t = 30(min), signal of number '2' comes.(step input)<br />
-when t = 50(min), signal of number '3' comes.(step input)<br />
<br />
===='''Results'''====<br />
<br />
Parameters we can vary voluntarily are <br />
<br />
*k1(rate of infection) : k1 relates to the specificity of antisenseRNA.<br />
<br />
*k4(rate of phage producing) : k4 relates to the strength of promoter.<br />
<br />
*k6(terminator leak rate) : k6 express terminator leak.<br />
<br />
[[Image:Whole1.png]]<br />
<br />
[[Image:MS2-0.5,0.6,0.png]]<br />
<br />
[[Image:MS2 2.png]]<br />
<br />
===='''Discussion'''====<br />
<br />
<br />
<br />
'''conclusion'''<br />
<br />
=='''Whole system'''==<br />
<br />
[[Image:What_Sudoku_99.png|200px|thumb|What's Sudoku?]]<br />
<br />
<br />
<br />
<br />
<br />
<br />
{{UT-Tokyo_Foot}}</div>Lenahttp://2010.igem.org/Team:UT-Tokyo/Sudoku_modelingTeam:UT-Tokyo/Sudoku modeling2010-10-27T19:01:29Z<p>Lena: /* Results */</p>
<hr />
<div>__NOTOC__{{UT-Tokyo_CSS3}}{{UT-Tokyo_CSS_p}}{{UT-Tokyo_Head}}<br />
<br />
= '''Sudoku''' =<br />
<br />
[https://2010.igem.org/Team:UT-Tokyo/Sudoku_abstract Introduction] <br />
[https://2010.igem.org/Team:UT-Tokyo/Sudoku_construct System] <br />
Modeling <br />
[https://2010.igem.org/Team:UT-Tokyo/Sudoku_experiments Experiments] <br />
[https://2010.igem.org/Team:UT-Tokyo/Sudoku_perspective Perspective]<br />
[https://2010.igem.org/Team:UT-Tokyo/Sudoku_reference Reference]<br />
<br />
== '''4C3leak switch''' ==<br />
<br />
We modeled our 4C3 leak switch as deterministic ODE system.<br />
<br />
<br />
==='''Modelling'''===<br />
<br />
<br />
===='''Assumption'''====<br />
<br />
1.We treated DNA as continuous variabl, since there could be hundreds of plasmid in one E.coli (multicopy plasmid).<br />
There are 32 kinds of variables that stand for DNA concentration corresponding to their different internal states.<br />
<br />
2.We assumed that cre recombinase operate as tetramer, and other recombinase as dimer (but this seems not to be relevant).<br />
<br />
3.We assumed that association and dissociation of DNA recombinase to DNA is sufficiently fast so that equilibrate in the timescale of the whole switch.<br />
<br />
4.We ignored the reverse reaction of DNA recombinase because we used irreversible sequence (lox66, lox71 etc).<br />
<br />
===='''Variable'''====<br />
<br />
Basically, "pn" denotes for protein concentration and "rn" denotes for mRNA concentration<br />
<br />
p1,p2,p3,p4 : concentration of 4 recombinase protein<br />
<br />
r1,r2,r3,r4 : concentration of 4 recombinass mRNA<br />
<br />
u1,u2,u3,u4 : concentration of input mRNA<br />
<br />
pc,rc : concentration of cre recombinase protein and mRNA<br />
<br />
ps,rs : concentration of SP6 polymerase protein and mRNA<br />
<br />
<br />
====''' Parameters '''====<br />
k0 : mRNA translation rate<br />
<br />
k1 : protein degradation rate<br />
<br />
k2 : mRNA translation rate<br />
<br />
k3 : cre protein degradation rate<br />
<br />
l0 : transcription speed per unit concentration of RNA polymearase<br />
<br />
l1 : mRNA degradation rate<br />
<br />
l2 : cre mRNA degradation rate<br />
<br />
K : recombinase binding constant<br />
<br />
v : recombinase reaction rate<br />
<br />
pT7 : concentration o T7 polymerase<br />
<br />
p : terminator leak probability<br />
<br />
<br />
<br />
===='''Equations'''====<br />
<br />
<br />
<br />
==='''Methods'''===<br />
<br />
We used our original python program (4 th order explicit Runge-Kutta algorithm) to solve thess equations.<br />
<br />
<br />
==='''Results'''===<br />
<br />
We input signal 1,2,4 and simulated the time evolution of the system.<br />
<br />
The time course of DNA concentration<br />
<br />
[[Image:Sim_res.JPG]]<br />
<br />
The time course of mRNA concentration. Note that the output of this system is not protein but mRNA.<br />
<br />
[[Image:Rna.JPG]]<br />
<br />
==='''Discussion'''===<br />
<br />
It is confirmed that the system works as expected with a certain combination of parameters.<br />
<br />
In addition, we changed many parameters, especially,l0,v(whose values are unclear) and p(leakiness of terminator) , to see whether the device is able to operate within a large part of parameter space.<br />
<br />
As a result, the circuit can function correctly within the range of 0.001< p < 0.1, 0.001 < v <1.0, 0.01 < l0 < 1.0 , demonstrating its robustness.<br />
This robustness may come from the assumption that DNA recombination is irreversible, which is supported by our design of the system.<math>\frac{2}{2}</math><br />
<br />
==='''conclusion'''===<br />
<br />
The correct operation of the switch was confirmed. By changing parameter exponentially, the robustness of the system was shown.<br />
<br />
=='''MS2 phage infection'''==<br />
<br />
We made a model for a indifferent sudoku cell.<br />
<br />
We traverse the amount of phages and see wether the cell react properly against given signals.<br />
<br />
<br />
===='''Modeling'''====<br />
<br />
'''Variable'''<br />
<br />
U: uninfected coli<br />
<br />
I: coli which got '1' signal<br />
<br />
Io1: other once infected coli<br />
<br />
I2: coli which got '1''2' signal<br />
<br />
Io2: other second infected coli<br />
<br />
I3: coli which got '1''2''3' signal<br />
<br />
Io3: other third infected coli<br />
<br />
B4: '4' coli which began phage making<br />
<br />
Bo: other coli which began phage making<br />
<br />
P4: free phage of '4'<br />
<br />
Po: other free phage<br />
<br />
<br />
'''Parameters'''<br />
<br />
g: growth rate of coli <br />
<br />
k1: rate of infection<br />
<br />
k2: death rate of uninfected<br />
<br />
k3: lysis rate of infected<br />
<br />
k4: rate of phage producing<br />
<br />
k5: phage death rate<br />
<br />
k6: terminator leak rate<br />
<br />
<br />
'''Equations'''<br />
<br />
We assume that cell lysis caused by MS2 phage is sufficiently rapid than coli death rate.<br />
<br />
===='''Method'''====<br />
<br />
We used original matlab program.<br />
<br />
We set conditions as following.<br />
-when t = 10(min), signal of number '1' comes.(step input)<br />
-when t = 30(min), signal of number '2' comes.(step input)<br />
-when t = 50(min), signal of number '3' comes.(step input)<br />
<br />
===='''Results'''====<br />
<br />
Parameters we can vary voluntarily are <br />
<br />
*k1(rate of infection) : k1 relates to the specificity of antisenseRNA.<br />
<br />
*k4(rate of phage producing) : k4 relates to the strength of promoter.<br />
<br />
*k6(terminator leak rate) : k6 express terminator leak.<br />
<br />
[[Image:Whole1.png]]<br />
<br />
[[Image:MS2-0.5,0.6,0.png]]<br />
<br />
[[Image:MS2 2.png]]<br />
<br />
===='''Discussion'''====<br />
<br />
<br />
<br />
'''conclusion'''<br />
<br />
=='''Whole system'''==<br />
<br />
[[Image:What_Sudoku_99.png|200px|thumb|What's Sudoku?]]<br />
<br />
<br />
<br />
<br />
<br />
<br />
{{UT-Tokyo_Foot}}</div>Lenahttp://2010.igem.org/Team:UT-Tokyo/Sudoku_modelingTeam:UT-Tokyo/Sudoku modeling2010-10-27T18:48:00Z<p>Lena: /* MS2 phage infection */</p>
<hr />
<div>__NOTOC__{{UT-Tokyo_CSS3}}{{UT-Tokyo_CSS_p}}{{UT-Tokyo_Head}}<br />
<br />
= '''Sudoku''' =<br />
<br />
[https://2010.igem.org/Team:UT-Tokyo/Sudoku_abstract Introduction] <br />
[https://2010.igem.org/Team:UT-Tokyo/Sudoku_construct System] <br />
Modeling <br />
[https://2010.igem.org/Team:UT-Tokyo/Sudoku_experiments Experiments] <br />
[https://2010.igem.org/Team:UT-Tokyo/Sudoku_perspective Perspective]<br />
[https://2010.igem.org/Team:UT-Tokyo/Sudoku_reference Reference]<br />
<br />
== '''4C3leak switch''' ==<br />
<br />
We modeled our 4C3 leak switch as deterministic ODE system.<br />
<br />
<br />
==='''Modelling'''===<br />
<br />
<br />
===='''Assumption'''====<br />
<br />
1.We treated DNA as continuous variabl, since there could be hundreds of plasmid in one E.coli (multicopy plasmid).<br />
There are 32 kinds of variables that stand for DNA concentration corresponding to their different internal states.<br />
<br />
2.We assumed that cre recombinase operate as tetramer, and other recombinase as dimer (but this seems not to be relevant).<br />
<br />
3.We assumed that association and dissociation of DNA recombinase to DNA is sufficiently fast so that equilibrate in the timescale of the whole switch.<br />
<br />
4.We ignored the reverse reaction of DNA recombinase because we used irreversible sequence (lox66, lox71 etc).<br />
<br />
===='''Variable'''====<br />
<br />
Basically, "pn" denotes for protein concentration and "rn" denotes for mRNA concentration<br />
<br />
p1,p2,p3,p4 : concentration of 4 recombinase protein<br />
<br />
r1,r2,r3,r4 : concentration of 4 recombinass mRNA<br />
<br />
u1,u2,u3,u4 : concentration of input mRNA<br />
<br />
pc,rc : concentration of cre recombinase protein and mRNA<br />
<br />
ps,rs : concentration of SP6 polymerase protein and mRNA<br />
<br />
<br />
====''' Parameters '''====<br />
k0 : mRNA translation rate<br />
<br />
k1 : protein degradation rate<br />
<br />
k2 : mRNA translation rate<br />
<br />
k3 : cre protein degradation rate<br />
<br />
l0 : transcription speed per unit concentration of RNA polymearase<br />
<br />
l1 : mRNA degradation rate<br />
<br />
l2 : cre mRNA degradation rate<br />
<br />
K : recombinase binding constant<br />
<br />
v : recombinase reaction rate<br />
<br />
pT7 : concentration o T7 polymerase<br />
<br />
p : terminator leak probability<br />
<br />
<br />
<br />
===='''Equations'''====<br />
<br />
<br />
<br />
==='''Methods'''===<br />
<br />
We used our original python program (4 th order explicit Runge-Kutta algorithm) to solve thess equations.<br />
<br />
<br />
==='''Results'''===<br />
<br />
We input signal 1,2,4 and simulated the time evolution of the system.<br />
<br />
The time course of DNA concentration<br />
<br />
[[Image:Sim_res.JPG]]<br />
<br />
The time course of mRNA concentration. Note that the output of this system is not protein but mRNA.<br />
<br />
[[Image:Rna.JPG]]<br />
<br />
==='''Discussion'''===<br />
<br />
It is confirmed that the system works as expected with a certain combination of parameters.<br />
<br />
In addition, we changed many parameters, especially,l0,v(whose values are unclear) and p(leakiness of terminator) , to see whether the device is able to operate within a large part of parameter space.<br />
<br />
As a result, the circuit can function correctly within the range of 0.001< p < 0.1, 0.001 < v <1.0, 0.01 < l0 < 1.0 , demonstrating its robustness.<br />
This robustness may come from the assumption that DNA recombination is irreversible, which is supported by our design of the system.<math>\frac{2}{2}</math><br />
<br />
==='''conclusion'''===<br />
<br />
The correct operation of the switch was confirmed. By changing parameter exponentially, the robustness of the system was shown.<br />
<br />
=='''MS2 phage infection'''==<br />
<br />
We made a model for a indifferent sudoku cell.<br />
<br />
We traverse the amount of phages and see wether the cell react properly against given signals.<br />
<br />
<br />
===='''Modeling'''====<br />
<br />
'''Variable'''<br />
<br />
U: uninfected coli<br />
<br />
I: coli which got '1' signal<br />
<br />
Io1: other once infected coli<br />
<br />
I2: coli which got '1''2' signal<br />
<br />
Io2: other second infected coli<br />
<br />
I3: coli which got '1''2''3' signal<br />
<br />
Io3: other third infected coli<br />
<br />
B4: '4' coli which began phage making<br />
<br />
Bo: other coli which began phage making<br />
<br />
P4: free phage of '4'<br />
<br />
Po: other free phage<br />
<br />
<br />
'''Parameters'''<br />
<br />
g: growth rate of coli <br />
<br />
k1: rate of infection<br />
<br />
k2: death rate of uninfected<br />
<br />
k3: lysis rate of infected<br />
<br />
k4: rate of phage producing<br />
<br />
k5: phage death rate<br />
<br />
k6: terminator leak rate<br />
<br />
<br />
'''Equations'''<br />
<br />
We assume that cell lysis caused by MS2 phage is sufficiently rapid than coli death rate.<br />
<br />
===='''Method'''====<br />
<br />
We used original matlab program.<br />
<br />
We set conditions as following.<br />
-when t = 10(min), signal of number '1' comes.(step input)<br />
-when t = 30(min), signal of number '2' comes.(step input)<br />
-when t = 50(min), signal of number '3' comes.(step input)<br />
<br />
===='''Results'''====<br />
<br />
<br />
[[Image:Whole1.png]]<br />
<br />
[[Image:MS2-0.5,0.6,0.png]]<br />
<br />
[[Image:MS2 2.png]]<br />
<br />
===='''Discussion'''====<br />
<br />
<br />
<br />
'''conclusion'''<br />
<br />
=='''Whole system'''==<br />
<br />
[[Image:What_Sudoku_99.png|200px|thumb|What's Sudoku?]]<br />
<br />
<br />
<br />
<br />
<br />
<br />
{{UT-Tokyo_Foot}}</div>Lenahttp://2010.igem.org/Team:UT-Tokyo/Sudoku_modelingTeam:UT-Tokyo/Sudoku modeling2010-10-27T18:46:26Z<p>Lena: /* Results */</p>
<hr />
<div>__NOTOC__{{UT-Tokyo_CSS3}}{{UT-Tokyo_CSS_p}}{{UT-Tokyo_Head}}<br />
<br />
= '''Sudoku''' =<br />
<br />
[https://2010.igem.org/Team:UT-Tokyo/Sudoku_abstract Introduction] <br />
[https://2010.igem.org/Team:UT-Tokyo/Sudoku_construct System] <br />
Modeling <br />
[https://2010.igem.org/Team:UT-Tokyo/Sudoku_experiments Experiments] <br />
[https://2010.igem.org/Team:UT-Tokyo/Sudoku_perspective Perspective]<br />
[https://2010.igem.org/Team:UT-Tokyo/Sudoku_reference Reference]<br />
<br />
== '''4C3leak switch''' ==<br />
<br />
We modeled our 4C3 leak switch as deterministic ODE system.<br />
<br />
<br />
==='''Modelling'''===<br />
<br />
<br />
===='''Assumption'''====<br />
<br />
1.We treated DNA as continuous variabl, since there could be hundreds of plasmid in one E.coli (multicopy plasmid).<br />
There are 32 kinds of variables that stand for DNA concentration corresponding to their different internal states.<br />
<br />
2.We assumed that cre recombinase operate as tetramer, and other recombinase as dimer (but this seems not to be relevant).<br />
<br />
3.We assumed that association and dissociation of DNA recombinase to DNA is sufficiently fast so that equilibrate in the timescale of the whole switch.<br />
<br />
4.We ignored the reverse reaction of DNA recombinase because we used irreversible sequence (lox66, lox71 etc).<br />
<br />
===='''Variable'''====<br />
<br />
Basically, "pn" denotes for protein concentration and "rn" denotes for mRNA concentration<br />
<br />
p1,p2,p3,p4 : concentration of 4 recombinase protein<br />
<br />
r1,r2,r3,r4 : concentration of 4 recombinass mRNA<br />
<br />
u1,u2,u3,u4 : concentration of input mRNA<br />
<br />
pc,rc : concentration of cre recombinase protein and mRNA<br />
<br />
ps,rs : concentration of SP6 polymerase protein and mRNA<br />
<br />
<br />
====''' Parameters '''====<br />
k0 : mRNA translation rate<br />
<br />
k1 : protein degradation rate<br />
<br />
k2 : mRNA translation rate<br />
<br />
k3 : cre protein degradation rate<br />
<br />
l0 : transcription speed per unit concentration of RNA polymearase<br />
<br />
l1 : mRNA degradation rate<br />
<br />
l2 : cre mRNA degradation rate<br />
<br />
K : recombinase binding constant<br />
<br />
v : recombinase reaction rate<br />
<br />
pT7 : concentration o T7 polymerase<br />
<br />
p : terminator leak probability<br />
<br />
<br />
<br />
===='''Equations'''====<br />
<br />
<br />
<br />
==='''Methods'''===<br />
<br />
We used our original python program (4 th order explicit Runge-Kutta algorithm) to solve thess equations.<br />
<br />
<br />
==='''Results'''===<br />
<br />
We input signal 1,2,4 and simulated the time evolution of the system.<br />
<br />
The time course of DNA concentration<br />
<br />
[[Image:Sim_res.JPG]]<br />
<br />
The time course of mRNA concentration. Note that the output of this system is not protein but mRNA.<br />
<br />
[[Image:Rna.JPG]]<br />
<br />
==='''Discussion'''===<br />
<br />
It is confirmed that the system works as expected with a certain combination of parameters.<br />
<br />
In addition, we changed many parameters, especially,l0,v(whose values are unclear) and p(leakiness of terminator) , to see whether the device is able to operate within a large part of parameter space.<br />
<br />
As a result, the circuit can function correctly within the range of 0.001< p < 0.1, 0.001 < v <1.0, 0.01 < l0 < 1.0 , demonstrating its robustness.<br />
This robustness may come from the assumption that DNA recombination is irreversible, which is supported by our design of the system.<math>\frac{2}{2}</math><br />
<br />
==='''conclusion'''===<br />
<br />
The correct operation of the switch was confirmed. By changing parameter exponentially, the robustness of the system was shown.<br />
<br />
=='''MS2 phage infection'''==<br />
<br />
<br />
===='''Modeling'''====<br />
<br />
'''Variable'''<br />
<br />
U: uninfected coli<br />
<br />
I: coli which got '1' signal<br />
<br />
Io1: other once infected coli<br />
<br />
I2: coli which got '1''2' signal<br />
<br />
Io2: other second infected coli<br />
<br />
I3: coli which got '1''2''3' signal<br />
<br />
Io3: other third infected coli<br />
<br />
B4: '4' coli which began phage making<br />
<br />
Bo: other coli which began phage making<br />
<br />
P4: free phage of '4'<br />
<br />
Po: other free phage<br />
<br />
<br />
'''Parameters'''<br />
<br />
g: growth rate of coli <br />
<br />
k1: rate of infection<br />
<br />
k2: death rate of uninfected<br />
<br />
k3: lysis rate of infected<br />
<br />
k4: rate of phage producing<br />
<br />
k5: phage death rate<br />
<br />
k6: terminator leak rate<br />
<br />
<br />
'''Equations'''<br />
<br />
We assume that cell lysis caused by MS2 phage is sufficiently rapid than coli death rate.<br />
<br />
===='''Method'''====<br />
<br />
We used original matlab program.<br />
<br />
We set conditions as following.<br />
-when t = 10(min), signal of number '1' comes.(step input)<br />
-when t = 30(min), signal of number '2' comes.(step input)<br />
-when t = 50(min), signal of number '3' comes.(step input)<br />
<br />
===='''Results'''====<br />
<br />
<br />
[[Image:Whole1.png]]<br />
<br />
[[Image:MS2-0.5,0.6,0.png]]<br />
<br />
[[Image:MS2 2.png]]<br />
<br />
===='''Discussion'''====<br />
<br />
<br />
<br />
'''conclusion'''<br />
<br />
=='''Whole system'''==<br />
<br />
[[Image:What_Sudoku_99.png|200px|thumb|What's Sudoku?]]<br />
<br />
<br />
<br />
<br />
<br />
<br />
{{UT-Tokyo_Foot}}</div>Lenahttp://2010.igem.org/File:MS2_2.pngFile:MS2 2.png2010-10-27T18:45:49Z<p>Lena: </p>
<hr />
<div></div>Lenahttp://2010.igem.org/Team:UT-Tokyo/Sudoku_modelingTeam:UT-Tokyo/Sudoku modeling2010-10-27T18:40:43Z<p>Lena: /* Results */</p>
<hr />
<div>__NOTOC__{{UT-Tokyo_CSS3}}{{UT-Tokyo_CSS_p}}{{UT-Tokyo_Head}}<br />
<br />
= '''Sudoku''' =<br />
<br />
[https://2010.igem.org/Team:UT-Tokyo/Sudoku_abstract Introduction] <br />
[https://2010.igem.org/Team:UT-Tokyo/Sudoku_construct System] <br />
Modeling <br />
[https://2010.igem.org/Team:UT-Tokyo/Sudoku_experiments Experiments] <br />
[https://2010.igem.org/Team:UT-Tokyo/Sudoku_perspective Perspective]<br />
[https://2010.igem.org/Team:UT-Tokyo/Sudoku_reference Reference]<br />
<br />
== '''4C3leak switch''' ==<br />
<br />
We modeled our 4C3 leak switch as deterministic ODE system.<br />
<br />
<br />
==='''Modelling'''===<br />
<br />
<br />
===='''Assumption'''====<br />
<br />
1.We treated DNA as continuous variabl, since there could be hundreds of plasmid in one E.coli (multicopy plasmid).<br />
There are 32 kinds of variables that stand for DNA concentration corresponding to their different internal states.<br />
<br />
2.We assumed that cre recombinase operate as tetramer, and other recombinase as dimer (but this seems not to be relevant).<br />
<br />
3.We assumed that association and dissociation of DNA recombinase to DNA is sufficiently fast so that equilibrate in the timescale of the whole switch.<br />
<br />
4.We ignored the reverse reaction of DNA recombinase because we used irreversible sequence (lox66, lox71 etc).<br />
<br />
===='''Variable'''====<br />
<br />
Basically, "pn" denotes for protein concentration and "rn" denotes for mRNA concentration<br />
<br />
p1,p2,p3,p4 : concentration of 4 recombinase protein<br />
<br />
r1,r2,r3,r4 : concentration of 4 recombinass mRNA<br />
<br />
u1,u2,u3,u4 : concentration of input mRNA<br />
<br />
pc,rc : concentration of cre recombinase protein and mRNA<br />
<br />
ps,rs : concentration of SP6 polymerase protein and mRNA<br />
<br />
<br />
====''' Parameters '''====<br />
k0 : mRNA translation rate<br />
<br />
k1 : protein degradation rate<br />
<br />
k2 : mRNA translation rate<br />
<br />
k3 : cre protein degradation rate<br />
<br />
l0 : transcription speed per unit concentration of RNA polymearase<br />
<br />
l1 : mRNA degradation rate<br />
<br />
l2 : cre mRNA degradation rate<br />
<br />
K : recombinase binding constant<br />
<br />
v : recombinase reaction rate<br />
<br />
pT7 : concentration o T7 polymerase<br />
<br />
p : terminator leak probability<br />
<br />
<br />
<br />
===='''Equations'''====<br />
<br />
<br />
<br />
==='''Methods'''===<br />
<br />
We used our original python program (4 th order explicit Runge-Kutta algorithm) to solve thess equations.<br />
<br />
<br />
==='''Results'''===<br />
<br />
We input signal 1,2,4 and simulated the time evolution of the system.<br />
<br />
The time course of DNA concentration<br />
<br />
[[Image:Sim_res.JPG]]<br />
<br />
The time course of mRNA concentration. Note that the output of this system is not protein but mRNA.<br />
<br />
[[Image:Rna.JPG]]<br />
<br />
==='''Discussion'''===<br />
<br />
It is confirmed that the system works as expected with a certain combination of parameters.<br />
<br />
In addition, we changed many parameters, especially,l0,v(whose values are unclear) and p(leakiness of terminator) , to see whether the device is able to operate within a large part of parameter space.<br />
<br />
As a result, the circuit can function correctly within the range of 0.001< p < 0.1, 0.001 < v <1.0, 0.01 < l0 < 1.0 , demonstrating its robustness.<br />
This robustness may come from the assumption that DNA recombination is irreversible, which is supported by our design of the system.<math>\frac{2}{2}</math><br />
<br />
==='''conclusion'''===<br />
<br />
The correct operation of the switch was confirmed. By changing parameter exponentially, the robustness of the system was shown.<br />
<br />
=='''MS2 phage infection'''==<br />
<br />
<br />
===='''Modeling'''====<br />
<br />
'''Variable'''<br />
<br />
U: uninfected coli<br />
<br />
I: coli which got '1' signal<br />
<br />
Io1: other once infected coli<br />
<br />
I2: coli which got '1''2' signal<br />
<br />
Io2: other second infected coli<br />
<br />
I3: coli which got '1''2''3' signal<br />
<br />
Io3: other third infected coli<br />
<br />
B4: '4' coli which began phage making<br />
<br />
Bo: other coli which began phage making<br />
<br />
P4: free phage of '4'<br />
<br />
Po: other free phage<br />
<br />
<br />
'''Parameters'''<br />
<br />
g: growth rate of coli <br />
<br />
k1: rate of infection<br />
<br />
k2: death rate of uninfected<br />
<br />
k3: lysis rate of infected<br />
<br />
k4: rate of phage producing<br />
<br />
k5: phage death rate<br />
<br />
k6: terminator leak rate<br />
<br />
<br />
'''Equations'''<br />
<br />
We assume that cell lysis caused by MS2 phage is sufficiently rapid than coli death rate.<br />
<br />
===='''Method'''====<br />
<br />
We used original matlab program.<br />
<br />
We set conditions as following.<br />
-when t = 10(min), signal of number '1' comes.(step input)<br />
-when t = 30(min), signal of number '2' comes.(step input)<br />
-when t = 50(min), signal of number '3' comes.(step input)<br />
<br />
===='''Results'''====<br />
<br />
[[Image:MS2-0.5,0.6,0.png]]<br />
<br />
[[Image:Whole1.png]]<br />
<br />
===='''Discussion'''====<br />
<br />
<br />
<br />
'''conclusion'''<br />
<br />
=='''Whole system'''==<br />
<br />
[[Image:What_Sudoku_99.png|200px|thumb|What's Sudoku?]]<br />
<br />
<br />
<br />
<br />
<br />
<br />
{{UT-Tokyo_Foot}}</div>Lenahttp://2010.igem.org/File:Whole1.pngFile:Whole1.png2010-10-27T18:39:57Z<p>Lena: </p>
<hr />
<div></div>Lenahttp://2010.igem.org/Team:UT-Tokyo/Sudoku_modelingTeam:UT-Tokyo/Sudoku modeling2010-10-27T18:32:23Z<p>Lena: /* Results */</p>
<hr />
<div>__NOTOC__{{UT-Tokyo_CSS3}}{{UT-Tokyo_CSS_p}}{{UT-Tokyo_Head}}<br />
<br />
= '''Sudoku''' =<br />
<br />
[https://2010.igem.org/Team:UT-Tokyo/Sudoku_abstract Introduction] <br />
[https://2010.igem.org/Team:UT-Tokyo/Sudoku_construct System] <br />
Modeling <br />
[https://2010.igem.org/Team:UT-Tokyo/Sudoku_experiments Experiment] <br />
[https://2010.igem.org/Team:UT-Tokyo/Sudoku_perspective Perspective]<br />
[https://2010.igem.org/Team:UT-Tokyo/Sudoku_reference Reference]<br />
<br />
== '''4C3leak switch''' ==<br />
<br />
We modeled our 4C3 leak switch as deterministic ODE system.<br />
<br />
<br />
==='''Modelling'''===<br />
<br />
<br />
===='''Assumption'''====<br />
<br />
1.We treated DNA as continuous variabl, since there could be hundreds of plasmid in one E.coli (multicopy plasmid).<br />
There are 32 kinds of variables that stand for DNA concentration corresponding to their different internal states.<br />
<br />
2.We assumed that cre recombinase operate as tetramer, and other recombinase as dimer (but this seems not to be relevant).<br />
<br />
3.We assumed that association and dissociation of DNA recombinase to DNA is sufficiently fast so that equilibrate in the timescale of the whole switch.<br />
<br />
4.We ignored the reverse reaction of DNA recombinase because we used irreversible sequence (lox66, lox71 etc).<br />
<br />
===='''Variable'''====<br />
<br />
Basically, "pn" denotes for protein concentration and "rn" denotes for mRNA concentration<br />
<br />
p1,p2,p3,p4 : concentration of 4 recombinase protein<br />
<br />
r1,r2,r3,r4 : concentration of 4 recombinass mRNA<br />
<br />
u1,u2,u3,u4 : concentration of input mRNA<br />
<br />
pc,rc : concentration of cre recombinase protein and mRNA<br />
<br />
ps,rs : concentration of SP6 polymerase protein and mRNA<br />
<br />
<br />
====''' Parameters '''====<br />
k0 : mRNA translation rate<br />
<br />
k1 : protein degradation rate<br />
<br />
k2 : mRNA translation rate<br />
<br />
k3 : cre protein degradation rate<br />
<br />
l0 : transcription speed per unit concentration of RNA polymearase<br />
<br />
l1 : mRNA degradation rate<br />
<br />
l2 : cre mRNA degradation rate<br />
<br />
K : recombinase binding constant<br />
<br />
v : recombinase reaction rate<br />
<br />
pT7 : concentration o T7 polymerase<br />
<br />
p : terminator leak probability<br />
<br />
<br />
<br />
===='''Equations'''====<br />
<br />
<br />
<br />
==='''Methods'''===<br />
<br />
We used our original python program (4 th order explicit Runge-Kutta algorithm) to solve thess equations.<br />
<br />
<br />
==='''Results'''===<br />
<br />
We input signal 1,2,4 and simulated the time evolution of the system.<br />
<br />
The time course of DNA concentration<br />
<br />
[[Image:Sim_res.JPG]]<br />
<br />
The time course of mRNA concentration. Note that the output of this system is not protein but mRNA.<br />
<br />
[[Image:Rna.JPG]]<br />
<br />
==='''Discussion'''===<br />
<br />
It is confirmed that the system works as expected with a certain combination of parameters.<br />
<br />
In addition, we changed many parameters, especially,l0,v(whose values are unclear) and p(leakiness of terminator) , to see whether the device is able to operate within a large part of parameter space.<br />
<br />
As a result, the circuit can function correctly within the range of 0.001< p < 0.1, 0.001 < v <1.0, 0.01 < l0 < 1.0 , demonstrating its robustness.<br />
This robustness may come from the assumption that DNA recombination is irreversible, which is supported by our design of the system.<math>\frac{2}{2}</math><br />
<br />
==='''conclusion'''===<br />
<br />
The correct operation of the switch was confirmed. By changing parameter exponentially, the robustness of the system was shown.<br />
<br />
=='''MS2 phage infection'''==<br />
<br />
<br />
===='''Modeling'''====<br />
<br />
'''Variable'''<br />
<br />
U: uninfected coli<br />
<br />
I: coli which got '1' signal<br />
<br />
Io1: other once infected coli<br />
<br />
I2: coli which got '1''2' signal<br />
<br />
Io2: other second infected coli<br />
<br />
I3: coli which got '1''2''3' signal<br />
<br />
Io3: other third infected coli<br />
<br />
B4: '4' coli which began phage making<br />
<br />
Bo: other coli which began phage making<br />
<br />
P4: free phage of '4'<br />
<br />
Po: other free phage<br />
<br />
<br />
'''Parameters'''<br />
<br />
g: growth rate of coli <br />
<br />
k1: rate of infection<br />
<br />
k2: death rate of uninfected<br />
<br />
k3: lysis rate of infected<br />
<br />
k4: rate of phage producing<br />
<br />
k5: phage death rate<br />
<br />
k6: terminator leak rate<br />
<br />
<br />
'''Equations'''<br />
<br />
We assume that cell lysis caused by MS2 phage is sufficiently rapid than coli death rate.<br />
<br />
===='''Method'''====<br />
<br />
We used original matlab program.<br />
<br />
We set conditions as following.<br />
-when t = 10(min), signal of number '1' comes.(step input)<br />
-when t = 30(min), signal of number '2' comes.(step input)<br />
-when t = 50(min), signal of number '3' comes.(step input)<br />
<br />
===='''Results'''====<br />
<br />
[[Image:MS2-0.5,0.6,0.png]]<br />
<br />
===='''Discussion'''====<br />
<br />
<br />
<br />
'''conclusion'''<br />
<br />
=='''Whole system'''==<br />
<br />
[[Image:What_Sudoku_99.png|200px|thumb|What's Sudoku?]]<br />
<br />
<br />
<br />
<br />
<br />
<br />
{{UT-Tokyo_Foot}}</div>Lenahttp://2010.igem.org/File:MS2-0.5,0.6,0.pngFile:MS2-0.5,0.6,0.png2010-10-27T18:30:35Z<p>Lena: uploaded a new version of "Image:MS2-0.5,0.6,0.png"</p>
<hr />
<div></div>Lenahttp://2010.igem.org/Team:UT-Tokyo/Sudoku_modelingTeam:UT-Tokyo/Sudoku modeling2010-10-27T17:54:35Z<p>Lena: /* Results */</p>
<hr />
<div>__NOTOC__{{UT-Tokyo_CSS3}}{{UT-Tokyo_CSS_p}}{{UT-Tokyo_Head}}<br />
<br />
= '''Sudoku''' =<br />
<br />
[https://2010.igem.org/Team:UT-Tokyo/Sudoku_abstract Introduction] <br />
[https://2010.igem.org/Team:UT-Tokyo/Sudoku_construct System] <br />
Modeling <br />
[https://2010.igem.org/Team:UT-Tokyo/Sudoku_experiments Experiment] <br />
[https://2010.igem.org/Team:UT-Tokyo/Sudoku_perspective Perspective]<br />
[https://2010.igem.org/Team:UT-Tokyo/Sudoku_reference Reference]<br />
<br />
== '''4C3leak switch''' ==<br />
<br />
We modeled our 4C3 leak switch as deterministic ODE system.<br />
<br />
<br />
==='''Modelling'''===<br />
<br />
<br />
===='''Assumption'''====<br />
<br />
1.We treated DNA as continuous variabl, since there could be hundreds of plasmid in one E.coli (multicopy plasmid).<br />
There are 32 kinds of variables that stand for DNA concentration corresponding to their different internal states.<br />
<br />
2.We assumed that cre recombinase operate as tetramer, and other recombinase as dimer (but this seems not to be relevant).<br />
<br />
3.We assumed that association and dissociation of DNA recombinase to DNA is sufficiently fast so that equilibrate in the timescale of the whole switch.<br />
<br />
4.We ignored the reverse reaction of DNA recombinase because we used irreversible sequence (lox66, lox71 etc).<br />
<br />
===='''Variable'''====<br />
<br />
Basically, "pn" denotes for protein concentration and "rn" denotes for mRNA concentration<br />
<br />
p1,p2,p3,p4 : concentration of 4 recombinase protein<br />
<br />
r1,r2,r3,r4 : concentration of 4 recombinass mRNA<br />
<br />
u1,u2,u3,u4 : concentration of input mRNA<br />
<br />
pc,rc : concentration of cre recombinase protein and mRNA<br />
<br />
ps,rs : concentration of SP6 polymerase protein and mRNA<br />
<br />
<br />
====''' Parameters '''====<br />
k0 : mRNA translation rate<br />
<br />
k1 : protein degradation rate<br />
<br />
k2 : mRNA translation rate<br />
<br />
k3 : cre protein degradation rate<br />
<br />
l0 : transcription speed per unit concentration of RNA polymearase<br />
<br />
l1 : mRNA degradation rate<br />
<br />
l2 : cre mRNA degradation rate<br />
<br />
K : recombinase binding constant<br />
<br />
v : recombinase reaction rate<br />
<br />
pT7 : concentration o T7 polymerase<br />
<br />
p : terminator leak probability<br />
<br />
<br />
<br />
===='''Equations'''====<br />
<br />
<br />
<br />
==='''Methods'''===<br />
<br />
We used our original python program (4 th order explicit Runge-Kutta algorithm) to solve thess equations.<br />
<br />
<br />
==='''Results'''===<br />
<br />
We input signal 1,2,4 and simulated the time evolution of the system.<br />
<br />
The time course of DNA concentration<br />
<br />
[[Image:Sim_res.JPG]]<br />
<br />
The time course of mRNA concentration. Note that the output of this system is not protein but mRNA.<br />
<br />
[[Image:Rna.JPG]]<br />
<br />
==='''Discussion'''===<br />
<br />
It is confirmed that the system works as expected with a certain combination of parameters.<br />
<br />
In addition, we changed many parameters, especially,l0,v(whose values are unclear) and p(leakiness of terminator) , to see whether the device is able to operate within a large part of parameter space.<br />
<br />
As a result, the circuit can function correctly within the range of {{<math> 0.001< p < 0.1, 0.001 < v <1.0, 0.01 < l0 < 1.0</math>}} , demonstrating its robustness.<br />
This robustness may come from the assumption that DNA recombination is irreversible, which is supported by our design of the system.<br />
<br />
==='''conclusion'''===<br />
<br />
The correct operation of the switch was confirmed. By changing parameter exponentially, the robustness of the system was shown.<br />
<br />
=='''MS2 phage infection'''==<br />
<br />
<br />
===='''Modeling'''====<br />
<br />
'''Variable'''<br />
<br />
U: uninfected coli<br />
<br />
I: coli which got '1' signal<br />
<br />
Io1: other once infected coli<br />
<br />
I2: coli which got '1''2' signal<br />
<br />
Io2: other second infected coli<br />
<br />
I3: coli which got '1''2''3' signal<br />
<br />
Io3: other third infected coli<br />
<br />
B4: '4' coli which began phage making<br />
<br />
Bo: other coli which began phage making<br />
<br />
P4: free phage of '4'<br />
<br />
Po: other free phage<br />
<br />
<br />
'''Parameters'''<br />
<br />
g: growth rate of coli <br />
<br />
k1: rate of infection<br />
<br />
k2: death rate of uninfected<br />
<br />
k3: lysis rate of infected<br />
<br />
k4: rate of phage producing<br />
<br />
k5: phage death rate<br />
<br />
k6: terminator leak rate<br />
<br />
<br />
'''Equations'''<br />
<br />
We assume that cell lysis caused by MS2 phage is sufficiently rapid than coli death rate.<br />
<br />
===='''Method'''====<br />
<br />
We used original matlab program.<br />
<br />
We set conditions as following.<br />
-when t = 10(min), signal of number '1' comes.(step input)<br />
-when t = 30(min), signal of number '2' comes.(step input)<br />
-when t = 50(min), signal of number '3' comes.(step input)<br />
<br />
===='''Results'''====<br />
<br />
Image:MS2-0.5,0.6,0.png<br />
<br />
===='''Discussion'''====<br />
<br />
<br />
<br />
'''conclusion'''<br />
<br />
=='''Whole system'''==<br />
<br />
[[Image:What_Sudoku_99.png|200px|thumb|What's Sudoku?]]<br />
<br />
<br />
<br />
<br />
<br />
<br />
{{UT-Tokyo_Foot}}</div>Lenahttp://2010.igem.org/File:MS2-0.5,0.6,0.pngFile:MS2-0.5,0.6,0.png2010-10-27T17:52:30Z<p>Lena: </p>
<hr />
<div></div>Lenahttp://2010.igem.org/Team:UT-Tokyo/Sudoku_modelingTeam:UT-Tokyo/Sudoku modeling2010-10-27T17:40:06Z<p>Lena: /* Discussion */</p>
<hr />
<div>__NOTOC__{{UT-Tokyo_CSS3}}{{UT-Tokyo_CSS_p}}{{UT-Tokyo_Head}}<br />
<br />
= '''Sudoku''' =<br />
<br />
[https://2010.igem.org/Team:UT-Tokyo/Sudoku_abstract Introduction] <br />
[https://2010.igem.org/Team:UT-Tokyo/Sudoku_construct System] <br />
Modeling <br />
[https://2010.igem.org/Team:UT-Tokyo/Sudoku_experiments Experiment] <br />
[https://2010.igem.org/Team:UT-Tokyo/Sudoku_result Result]<br />
[https://2010.igem.org/Team:UT-Tokyo/Sudoku_reference Reference]<br />
<br />
== '''4C3leak switch''' ==<br />
<br />
We modeled our 4C3 leak switch as deterministic ODE system.<br />
<br />
<br />
==='''Modelling'''===<br />
<br />
<br />
===='''Assumption'''====<br />
<br />
1.We treated DNA as continuous variabl, since there could be hundreds of plasmid in one E.coli (multicopy plasmid).<br />
There are 32 kinds of variables that stand for DNA concentration corresponding to their different internal states.<br />
<br />
2.We assumed that cre recombinase operate as tetramer, and other recombinase as dimer (but this seems not to be relevant).<br />
<br />
3.We assumed that association and dissociation of DNA recombinase to DNA is sufficiently fast so that equilibrate in the timescale of the whole switch.<br />
<br />
4.We ignored the reverse reaction of DNA recombinase because we used irreversible sequence (lox66, lox71 etc).<br />
<br />
===='''Variable'''====<br />
<br />
Basically, "pn" denotes for protein concentration and "rn" denotes for mRNA concentration<br />
<br />
p1,p2,p3,p4 : concentration of 4 recombinase protein<br />
<br />
r1,r2,r3,r4 : concentration of 4 recombinass mRNA<br />
<br />
u1,u2,u3,u4 : concentration of input mRNA<br />
<br />
pc,rc : concentration of cre recombinase protein and mRNA<br />
<br />
ps,rs : concentration of SP6 polymerase protein and mRNA<br />
<br />
<br />
====''' Parameters '''====<br />
k0 : mRNA translation rate<br />
<br />
k1 : protein degradation rate<br />
<br />
k2 : mRNA translation rate<br />
<br />
k3 : cre protein degradation rate<br />
<br />
l0 : transcription speed per unit concentration of RNA polymearase<br />
<br />
l1 : mRNA degradation rate<br />
<br />
l2 : cre mRNA degradation rate<br />
<br />
K : recombinase binding constant<br />
<br />
v : recombinase reaction rate<br />
<br />
pT7 : concentration o T7 polymerase<br />
<br />
p : terminator leak probability<br />
<br />
<br />
<br />
===='''Equations'''====<br />
<br />
<br />
<br />
==='''Methods'''===<br />
<br />
We used our original python program (4 th order explicit Runge-Kutta algorithm) to solve thess equations.<br />
<br />
<br />
==='''Results'''===<br />
<br />
We input signal 1,2,4 and simulated the time evolution of the system.<br />
<br />
The time course of DNA concentration<br />
<br />
[[Image:Sim_res.JPG]]<br />
<br />
The time course of mRNA concentration. Note that the output of this system is not protein but mRNA.<br />
<br />
[[Image:Rna.JPG]]<br />
<br />
==='''Discussion'''===<br />
<br />
It is confirmed that the system works as expected with a certain combination of parameters.<br />
<br />
In addition, we changed many parameters, especially,l0,v(whose values are unclear) and p(leakiness of terminator) , to see whether the device is able to operate within a large part of parameter space.<br />
<br />
As a result, the circuit can function correctly within the range of 0.001< p < 0.1, 0.001 < v <1.0, 0.01 < l0 < 1.0 , demonstrating its robustness.<br />
This robustness may come from the assumption that DNA recombination is irreversible, which is supported by our design of the system.<br />
<br />
==='''conclusion'''===<br />
<br />
The correct operation of the switch was confirmed. By changing parameter exponentially, the robustness of the system was shown.<br />
<br />
=='''MS2 phage infection'''==<br />
<br />
<br />
===='''Modeling'''====<br />
<br />
'''Variable'''<br />
<br />
U: uninfected coli<br />
<br />
I: coli which got '1' signal<br />
<br />
Io1: other once infected coli<br />
<br />
I2: coli which got '1''2' signal<br />
<br />
Io2: other second infected coli<br />
<br />
I3: coli which got '1''2''3' signal<br />
<br />
Io3: other third infected coli<br />
<br />
B4: '4' coli which began phage making<br />
<br />
Bo: other coli which began phage making<br />
<br />
P4: free phage of '4'<br />
<br />
Po: other free phage<br />
<br />
<br />
'''Parameters'''<br />
<br />
g: growth rate of coli <br />
<br />
k1: rate of infection<br />
<br />
k2: death rate of uninfected<br />
<br />
k3: lysis rate of infected<br />
<br />
k4: rate of phage producing<br />
<br />
k5: phage death rate<br />
<br />
k6: terminator leak rate<br />
<br />
<br />
'''Equations'''<br />
<br />
We assume that cell lysis caused by MS2 phage is sufficiently rapid than coli death rate.<br />
<br />
===='''Method'''====<br />
<br />
We used original matlab program.<br />
<br />
We set conditions as following.<br />
-when t = 10(min), signal of number '1' comes.(step input)<br />
-when t = 30(min), signal of number '2' comes.(step input)<br />
-when t = 50(min), signal of number '3' comes.(step input)<br />
<br />
===='''Results'''====<br />
<br />
===='''Discussion'''====<br />
<br />
<br />
<br />
'''conclusion'''<br />
<br />
=='''Whole system'''==<br />
<br />
[[Image:What_Sudoku_99.png|200px|thumb|What's Sudoku?]]<br />
<br />
<br />
<br />
<br />
<br />
<br />
{{UT-Tokyo_Foot}}</div>Lenahttp://2010.igem.org/Team:UT-Tokyo/Sudoku_modelingTeam:UT-Tokyo/Sudoku modeling2010-10-27T17:39:19Z<p>Lena: /* Results */</p>
<hr />
<div>__NOTOC__{{UT-Tokyo_CSS3}}{{UT-Tokyo_CSS_p}}{{UT-Tokyo_Head}}<br />
<br />
= '''Sudoku''' =<br />
<br />
[https://2010.igem.org/Team:UT-Tokyo/Sudoku_abstract Introduction] <br />
[https://2010.igem.org/Team:UT-Tokyo/Sudoku_construct System] <br />
Modeling <br />
[https://2010.igem.org/Team:UT-Tokyo/Sudoku_lab_note Lab note] <br />
[https://2010.igem.org/Team:UT-Tokyo/Sudoku_experiments Experiment] <br />
[https://2010.igem.org/Team:UT-Tokyo/Sudoku_result Result]<br />
[https://2010.igem.org/Team:UT-Tokyo/Sudoku_reference Reference]<br />
<br />
== '''4C3leak switch''' ==<br />
<br />
We modeled our 4C3 leak switch as deterministic ODE system.<br />
<br />
<br />
==='''Modelling'''===<br />
<br />
<br />
===='''Assumption'''====<br />
<br />
1.We treated DNA as continuous variabl, since there could be hundreds of plasmid in one E.coli (multicopy plasmid).<br />
There are 32 kinds of variables that stand for DNA concentration corresponding to their different internal states.<br />
<br />
2.We assumed that cre recombinase operate as tetramer, and other recombinase as dimer (but this seems not to be relevant).<br />
<br />
3.We assumed that association and dissociation of DNA recombinase to DNA is sufficiently fast so that equilibrate in the timescale of the whole switch.<br />
<br />
4.We ignored the reverse reaction of DNA recombinase because we used irreversible sequence (lox66, lox71 etc).<br />
<br />
===='''Variable'''====<br />
<br />
Basically, "pn" denotes for protein concentration and "rn" denotes for mRNA concentration<br />
<br />
p1,p2,p3,p4 : concentration of 4 recombinase protein<br />
<br />
r1,r2,r3,r4 : concentration of 4 recombinass mRNA<br />
<br />
u1,u2,u3,u4 : concentration of input mRNA<br />
<br />
pc,rc : concentration of cre recombinase protein and mRNA<br />
<br />
ps,rs : concentration of SP6 polymerase protein and mRNA<br />
<br />
<br />
====''' Parameters '''====<br />
k0 : mRNA translation rate<br />
<br />
k1 : protein degradation rate<br />
<br />
k2 : mRNA translation rate<br />
<br />
k3 : cre protein degradation rate<br />
<br />
l0 : transcription speed per unit concentration of RNA polymearase<br />
<br />
l1 : mRNA degradation rate<br />
<br />
l2 : cre mRNA degradation rate<br />
<br />
K : recombinase binding constant<br />
<br />
v : recombinase reaction rate<br />
<br />
pT7 : concentration o T7 polymerase<br />
<br />
p : terminator leak probability<br />
<br />
<br />
<br />
===='''Equations'''====<br />
<br />
<br />
<br />
==='''Methods'''===<br />
<br />
We used our original python program (4 th order explicit Runge-Kutta algorithm) to solve thess equations.<br />
<br />
<br />
==='''Results'''===<br />
<br />
We input signal 1,2,4 and simulated the time evolution of the system.<br />
<br />
The time course of DNA concentration<br />
<br />
[[Image:Sim_res.JPG]]<br />
<br />
The time course of mRNA concentration. Note that the output of this system is not protein but mRNA.<br />
<br />
[[Image:Rna.JPG]]<br />
<br />
==='''Discussion'''===<br />
<br />
It is confirmed that the system works as expected with a certain combination of parameters.<br />
<br />
In addition, we changed many parameters, especially,l0,v(whose values are unclear) and p(leakiness of terminator) , to see whether the device is able to operate within a large part of parameter space.<br />
<br />
As a result, the circuit can function correctly within the range of 0.001< p < 0.1, 0.001 < v <1.0, 0.01 < l0 < 1.0 , demonstrating its robustness.<br />
This robustness may come from the assumption that DNA recombination is irreversible, which is supported by our design of the system.<br />
<br />
==='''conclusion'''===<br />
<br />
The correct operation of the switch was confirmed. By changing parameter exponentially, the robustness of the system was shown.<br />
<br />
=='''MS2 phage infection'''==<br />
<br />
<br />
===='''Modeling'''====<br />
<br />
'''Variable'''<br />
<br />
U: uninfected coli<br />
<br />
I: coli which got '1' signal<br />
<br />
Io1: other once infected coli<br />
<br />
I2: coli which got '1''2' signal<br />
<br />
Io2: other second infected coli<br />
<br />
I3: coli which got '1''2''3' signal<br />
<br />
Io3: other third infected coli<br />
<br />
B4: '4' coli which began phage making<br />
<br />
Bo: other coli which began phage making<br />
<br />
P4: free phage of '4'<br />
<br />
Po: other free phage<br />
<br />
<br />
'''Parameters'''<br />
<br />
g: growth rate of coli <br />
<br />
k1: rate of infection<br />
<br />
k2: death rate of uninfected<br />
<br />
k3: lysis rate of infected<br />
<br />
k4: rate of phage producing<br />
<br />
k5: phage death rate<br />
<br />
k6: terminator leak rate<br />
<br />
<br />
'''Equations'''<br />
<br />
We assume that cell lysis caused by MS2 phage is sufficiently rapid than coli death rate.<br />
<br />
===='''Method'''====<br />
<br />
We used original matlab program.<br />
<br />
We set conditions as following.<br />
-when t = 10(min), signal of number '1' comes.(step input)<br />
-when t = 30(min), signal of number '2' comes.(step input)<br />
-when t = 50(min), signal of number '3' comes.(step input)<br />
<br />
===='''Results'''====<br />
<br />
=='''Discussion'''==<br />
<br />
<br />
<br />
●conclusion<br />
<br />
=='''Whole system'''==<br />
<br />
[[Image:What_Sudoku_99.png|200px|thumb|What's Sudoku?]]<br />
<br />
<br />
<br />
<br />
<br />
<br />
{{UT-Tokyo_Foot}}</div>Lenahttp://2010.igem.org/Team:UT-Tokyo/Sudoku_modelingTeam:UT-Tokyo/Sudoku modeling2010-10-27T17:38:50Z<p>Lena: /* MS2 phage infection */</p>
<hr />
<div>__NOTOC__{{UT-Tokyo_CSS3}}{{UT-Tokyo_CSS_p}}{{UT-Tokyo_Head}}<br />
<br />
= '''Sudoku''' =<br />
<br />
[https://2010.igem.org/Team:UT-Tokyo/Sudoku_abstract Introduction] <br />
[https://2010.igem.org/Team:UT-Tokyo/Sudoku_construct System] <br />
Modeling <br />
[https://2010.igem.org/Team:UT-Tokyo/Sudoku_lab_note Lab note] <br />
[https://2010.igem.org/Team:UT-Tokyo/Sudoku_experiments Experiment] <br />
[https://2010.igem.org/Team:UT-Tokyo/Sudoku_result Result]<br />
[https://2010.igem.org/Team:UT-Tokyo/Sudoku_reference Reference]<br />
<br />
== '''4C3leak switch''' ==<br />
<br />
We modeled our 4C3 leak switch as deterministic ODE system.<br />
<br />
<br />
==='''Modelling'''===<br />
<br />
<br />
===='''Assumption'''====<br />
<br />
1.We treated DNA as continuous variabl, since there could be hundreds of plasmid in one E.coli (multicopy plasmid).<br />
There are 32 kinds of variables that stand for DNA concentration corresponding to their different internal states.<br />
<br />
2.We assumed that cre recombinase operate as tetramer, and other recombinase as dimer (but this seems not to be relevant).<br />
<br />
3.We assumed that association and dissociation of DNA recombinase to DNA is sufficiently fast so that equilibrate in the timescale of the whole switch.<br />
<br />
4.We ignored the reverse reaction of DNA recombinase because we used irreversible sequence (lox66, lox71 etc).<br />
<br />
===='''Variable'''====<br />
<br />
Basically, "pn" denotes for protein concentration and "rn" denotes for mRNA concentration<br />
<br />
p1,p2,p3,p4 : concentration of 4 recombinase protein<br />
<br />
r1,r2,r3,r4 : concentration of 4 recombinass mRNA<br />
<br />
u1,u2,u3,u4 : concentration of input mRNA<br />
<br />
pc,rc : concentration of cre recombinase protein and mRNA<br />
<br />
ps,rs : concentration of SP6 polymerase protein and mRNA<br />
<br />
<br />
====''' Parameters '''====<br />
k0 : mRNA translation rate<br />
<br />
k1 : protein degradation rate<br />
<br />
k2 : mRNA translation rate<br />
<br />
k3 : cre protein degradation rate<br />
<br />
l0 : transcription speed per unit concentration of RNA polymearase<br />
<br />
l1 : mRNA degradation rate<br />
<br />
l2 : cre mRNA degradation rate<br />
<br />
K : recombinase binding constant<br />
<br />
v : recombinase reaction rate<br />
<br />
pT7 : concentration o T7 polymerase<br />
<br />
p : terminator leak probability<br />
<br />
<br />
<br />
===='''Equations'''====<br />
<br />
<br />
<br />
==='''Methods'''===<br />
<br />
We used our original python program (4 th order explicit Runge-Kutta algorithm) to solve thess equations.<br />
<br />
<br />
==='''Results'''===<br />
<br />
We input signal 1,2,4 and simulated the time evolution of the system.<br />
<br />
The time course of DNA concentration<br />
<br />
[[Image:Sim_res.JPG]]<br />
<br />
The time course of mRNA concentration. Note that the output of this system is not protein but mRNA.<br />
<br />
[[Image:Rna.JPG]]<br />
<br />
==='''Discussion'''===<br />
<br />
It is confirmed that the system works as expected with a certain combination of parameters.<br />
<br />
In addition, we changed many parameters, especially,l0,v(whose values are unclear) and p(leakiness of terminator) , to see whether the device is able to operate within a large part of parameter space.<br />
<br />
As a result, the circuit can function correctly within the range of 0.001< p < 0.1, 0.001 < v <1.0, 0.01 < l0 < 1.0 , demonstrating its robustness.<br />
This robustness may come from the assumption that DNA recombination is irreversible, which is supported by our design of the system.<br />
<br />
==='''conclusion'''===<br />
<br />
The correct operation of the switch was confirmed. By changing parameter exponentially, the robustness of the system was shown.<br />
<br />
=='''MS2 phage infection'''==<br />
<br />
<br />
===='''Modeling'''====<br />
<br />
'''Variable'''<br />
<br />
U: uninfected coli<br />
<br />
I: coli which got '1' signal<br />
<br />
Io1: other once infected coli<br />
<br />
I2: coli which got '1''2' signal<br />
<br />
Io2: other second infected coli<br />
<br />
I3: coli which got '1''2''3' signal<br />
<br />
Io3: other third infected coli<br />
<br />
B4: '4' coli which began phage making<br />
<br />
Bo: other coli which began phage making<br />
<br />
P4: free phage of '4'<br />
<br />
Po: other free phage<br />
<br />
<br />
'''Parameters'''<br />
<br />
g: growth rate of coli <br />
<br />
k1: rate of infection<br />
<br />
k2: death rate of uninfected<br />
<br />
k3: lysis rate of infected<br />
<br />
k4: rate of phage producing<br />
<br />
k5: phage death rate<br />
<br />
k6: terminator leak rate<br />
<br />
<br />
'''Equations'''<br />
<br />
We assume that cell lysis caused by MS2 phage is sufficiently rapid than coli death rate.<br />
<br />
===='''Method'''====<br />
<br />
We used original matlab program.<br />
<br />
We set conditions as following.<br />
-when t = 10(min), signal of number '1' comes.(step input)<br />
-when t = 30(min), signal of number '2' comes.(step input)<br />
-when t = 50(min), signal of number '3' comes.(step input)<br />
<br />
=='''Results'''==<br />
<br />
<br />
=='''Discussion'''==<br />
<br />
<br />
<br />
●conclusion<br />
<br />
=='''Whole system'''==<br />
<br />
[[Image:What_Sudoku_99.png|200px|thumb|What's Sudoku?]]<br />
<br />
<br />
<br />
<br />
<br />
<br />
{{UT-Tokyo_Foot}}</div>Lenahttp://2010.igem.org/Team:UT-Tokyo/Sudoku_modelingTeam:UT-Tokyo/Sudoku modeling2010-10-27T17:38:06Z<p>Lena: /* Method */</p>
<hr />
<div>__NOTOC__{{UT-Tokyo_CSS3}}{{UT-Tokyo_CSS_p}}{{UT-Tokyo_Head}}<br />
<br />
= '''Sudoku''' =<br />
<br />
[https://2010.igem.org/Team:UT-Tokyo/Sudoku_abstract Introduction] <br />
[https://2010.igem.org/Team:UT-Tokyo/Sudoku_construct System] <br />
Modeling <br />
[https://2010.igem.org/Team:UT-Tokyo/Sudoku_lab_note Lab note] <br />
[https://2010.igem.org/Team:UT-Tokyo/Sudoku_experiments Experiment] <br />
[https://2010.igem.org/Team:UT-Tokyo/Sudoku_result Result]<br />
[https://2010.igem.org/Team:UT-Tokyo/Sudoku_reference Reference]<br />
<br />
== '''4C3leak switch''' ==<br />
<br />
We modeled our 4C3 leak switch as deterministic ODE system.<br />
<br />
<br />
==='''Modelling'''===<br />
<br />
<br />
===='''Assumption'''====<br />
<br />
1.We treated DNA as continuous variabl, since there could be hundreds of plasmid in one E.coli (multicopy plasmid).<br />
There are 32 kinds of variables that stand for DNA concentration corresponding to their different internal states.<br />
<br />
2.We assumed that cre recombinase operate as tetramer, and other recombinase as dimer (but this seems not to be relevant).<br />
<br />
3.We assumed that association and dissociation of DNA recombinase to DNA is sufficiently fast so that equilibrate in the timescale of the whole switch.<br />
<br />
4.We ignored the reverse reaction of DNA recombinase because we used irreversible sequence (lox66, lox71 etc).<br />
<br />
===='''Variable'''====<br />
<br />
Basically, "pn" denotes for protein concentration and "rn" denotes for mRNA concentration<br />
<br />
p1,p2,p3,p4 : concentration of 4 recombinase protein<br />
<br />
r1,r2,r3,r4 : concentration of 4 recombinass mRNA<br />
<br />
u1,u2,u3,u4 : concentration of input mRNA<br />
<br />
pc,rc : concentration of cre recombinase protein and mRNA<br />
<br />
ps,rs : concentration of SP6 polymerase protein and mRNA<br />
<br />
<br />
====''' Parameters '''====<br />
k0 : mRNA translation rate<br />
<br />
k1 : protein degradation rate<br />
<br />
k2 : mRNA translation rate<br />
<br />
k3 : cre protein degradation rate<br />
<br />
l0 : transcription speed per unit concentration of RNA polymearase<br />
<br />
l1 : mRNA degradation rate<br />
<br />
l2 : cre mRNA degradation rate<br />
<br />
K : recombinase binding constant<br />
<br />
v : recombinase reaction rate<br />
<br />
pT7 : concentration o T7 polymerase<br />
<br />
p : terminator leak probability<br />
<br />
<br />
<br />
===='''Equations'''====<br />
<br />
<br />
<br />
==='''Methods'''===<br />
<br />
We used our original python program (4 th order explicit Runge-Kutta algorithm) to solve thess equations.<br />
<br />
<br />
==='''Results'''===<br />
<br />
We input signal 1,2,4 and simulated the time evolution of the system.<br />
<br />
The time course of DNA concentration<br />
<br />
[[Image:Sim_res.JPG]]<br />
<br />
The time course of mRNA concentration. Note that the output of this system is not protein but mRNA.<br />
<br />
[[Image:Rna.JPG]]<br />
<br />
==='''Discussion'''===<br />
<br />
It is confirmed that the system works as expected with a certain combination of parameters.<br />
<br />
In addition, we changed many parameters, especially,l0,v(whose values are unclear) and p(leakiness of terminator) , to see whether the device is able to operate within a large part of parameter space.<br />
<br />
As a result, the circuit can function correctly within the range of 0.001< p < 0.1, 0.001 < v <1.0, 0.01 < l0 < 1.0 , demonstrating its robustness.<br />
This robustness may come from the assumption that DNA recombination is irreversible, which is supported by our design of the system.<br />
<br />
==='''conclusion'''===<br />
<br />
The correct operation of the switch was confirmed. By changing parameter exponentially, the robustness of the system was shown.<br />
<br />
=='''MS2 phage infection'''==<br />
<br />
<br />
'''Modeling'''<br />
<br />
'''Variable'''<br />
<br />
U: uninfected coli<br />
<br />
I: coli which got '1' signal<br />
<br />
Io1: other once infected coli<br />
<br />
I2: coli which got '1''2' signal<br />
<br />
Io2: other second infected coli<br />
<br />
I3: coli which got '1''2''3' signal<br />
<br />
Io3: other third infected coli<br />
<br />
B4: '4' coli which began phage making<br />
<br />
Bo: other coli which began phage making<br />
<br />
P4: free phage of '4'<br />
<br />
Po: other free phage<br />
<br />
<br />
'''Parameters'''<br />
<br />
g: growth rate of coli <br />
<br />
k1: rate of infection<br />
<br />
k2: death rate of uninfected<br />
<br />
k3: lysis rate of infected<br />
<br />
k4: rate of phage producing<br />
<br />
k5: phage death rate<br />
<br />
k6: terminator leak rate<br />
<br />
<br />
'''Equations'''<br />
<br />
We assume that cell lysis caused by MS2 phage is sufficiently rapid than coli death rate.<br />
<br />
'''Method'''<br />
<br />
We used original matlab program.<br />
<br />
We set conditions as following.<br />
-when t = 10(min), signal of number '1' comes.(step input)<br />
-when t = 30(min), signal of number '2' comes.(step input)<br />
-when t = 50(min), signal of number '3' comes.(step input)<br />
<br />
=='''Results'''==<br />
<br />
<br />
=='''Discussion'''==<br />
<br />
<br />
<br />
●conclusion<br />
<br />
=='''Whole system'''==<br />
<br />
[[Image:What_Sudoku_99.png|200px|thumb|What's Sudoku?]]<br />
<br />
<br />
<br />
<br />
<br />
<br />
{{UT-Tokyo_Foot}}</div>Lenahttp://2010.igem.org/Team:UT-Tokyo/Sudoku_modelingTeam:UT-Tokyo/Sudoku modeling2010-10-27T17:37:29Z<p>Lena: /* Modeling */</p>
<hr />
<div>__NOTOC__{{UT-Tokyo_CSS3}}{{UT-Tokyo_CSS_p}}{{UT-Tokyo_Head}}<br />
<br />
= '''Sudoku''' =<br />
<br />
[https://2010.igem.org/Team:UT-Tokyo/Sudoku_abstract Introduction] <br />
[https://2010.igem.org/Team:UT-Tokyo/Sudoku_construct System] <br />
Modeling <br />
[https://2010.igem.org/Team:UT-Tokyo/Sudoku_lab_note Lab note] <br />
[https://2010.igem.org/Team:UT-Tokyo/Sudoku_experiments Experiment] <br />
[https://2010.igem.org/Team:UT-Tokyo/Sudoku_result Result]<br />
[https://2010.igem.org/Team:UT-Tokyo/Sudoku_reference Reference]<br />
<br />
== '''4C3leak switch''' ==<br />
<br />
We modeled our 4C3 leak switch as deterministic ODE system.<br />
<br />
<br />
==='''Modelling'''===<br />
<br />
<br />
===='''Assumption'''====<br />
<br />
1.We treated DNA as continuous variabl, since there could be hundreds of plasmid in one E.coli (multicopy plasmid).<br />
There are 32 kinds of variables that stand for DNA concentration corresponding to their different internal states.<br />
<br />
2.We assumed that cre recombinase operate as tetramer, and other recombinase as dimer (but this seems not to be relevant).<br />
<br />
3.We assumed that association and dissociation of DNA recombinase to DNA is sufficiently fast so that equilibrate in the timescale of the whole switch.<br />
<br />
4.We ignored the reverse reaction of DNA recombinase because we used irreversible sequence (lox66, lox71 etc).<br />
<br />
===='''Variable'''====<br />
<br />
Basically, "pn" denotes for protein concentration and "rn" denotes for mRNA concentration<br />
<br />
p1,p2,p3,p4 : concentration of 4 recombinase protein<br />
<br />
r1,r2,r3,r4 : concentration of 4 recombinass mRNA<br />
<br />
u1,u2,u3,u4 : concentration of input mRNA<br />
<br />
pc,rc : concentration of cre recombinase protein and mRNA<br />
<br />
ps,rs : concentration of SP6 polymerase protein and mRNA<br />
<br />
<br />
====''' Parameters '''====<br />
k0 : mRNA translation rate<br />
<br />
k1 : protein degradation rate<br />
<br />
k2 : mRNA translation rate<br />
<br />
k3 : cre protein degradation rate<br />
<br />
l0 : transcription speed per unit concentration of RNA polymearase<br />
<br />
l1 : mRNA degradation rate<br />
<br />
l2 : cre mRNA degradation rate<br />
<br />
K : recombinase binding constant<br />
<br />
v : recombinase reaction rate<br />
<br />
pT7 : concentration o T7 polymerase<br />
<br />
p : terminator leak probability<br />
<br />
<br />
<br />
===='''Equations'''====<br />
<br />
<br />
<br />
==='''Methods'''===<br />
<br />
We used our original python program (4 th order explicit Runge-Kutta algorithm) to solve thess equations.<br />
<br />
<br />
==='''Results'''===<br />
<br />
We input signal 1,2,4 and simulated the time evolution of the system.<br />
<br />
The time course of DNA concentration<br />
<br />
[[Image:Sim_res.JPG]]<br />
<br />
The time course of mRNA concentration. Note that the output of this system is not protein but mRNA.<br />
<br />
[[Image:Rna.JPG]]<br />
<br />
==='''Discussion'''===<br />
<br />
It is confirmed that the system works as expected with a certain combination of parameters.<br />
<br />
In addition, we changed many parameters, especially,l0,v(whose values are unclear) and p(leakiness of terminator) , to see whether the device is able to operate within a large part of parameter space.<br />
<br />
As a result, the circuit can function correctly within the range of 0.001< p < 0.1, 0.001 < v <1.0, 0.01 < l0 < 1.0 , demonstrating its robustness.<br />
This robustness may come from the assumption that DNA recombination is irreversible, which is supported by our design of the system.<br />
<br />
==='''conclusion'''===<br />
<br />
The correct operation of the switch was confirmed. By changing parameter exponentially, the robustness of the system was shown.<br />
<br />
=='''MS2 phage infection'''==<br />
<br />
<br />
'''Modeling'''<br />
<br />
'''Variable'''<br />
<br />
U: uninfected coli<br />
<br />
I: coli which got '1' signal<br />
<br />
Io1: other once infected coli<br />
<br />
I2: coli which got '1''2' signal<br />
<br />
Io2: other second infected coli<br />
<br />
I3: coli which got '1''2''3' signal<br />
<br />
Io3: other third infected coli<br />
<br />
B4: '4' coli which began phage making<br />
<br />
Bo: other coli which began phage making<br />
<br />
P4: free phage of '4'<br />
<br />
Po: other free phage<br />
<br />
<br />
'''Parameters'''<br />
<br />
g: growth rate of coli <br />
<br />
k1: rate of infection<br />
<br />
k2: death rate of uninfected<br />
<br />
k3: lysis rate of infected<br />
<br />
k4: rate of phage producing<br />
<br />
k5: phage death rate<br />
<br />
k6: terminator leak rate<br />
<br />
<br />
'''Equations'''<br />
<br />
We assume that cell lysis caused by MS2 phage is sufficiently rapid than coli death rate.<br />
<br />
=='''Method'''==<br />
<br />
We used original matlab program.<br />
<br />
We set conditions as following.<br />
-when t = 10(min), signal of number '1' comes.(step input)<br />
-when t = 30(min), signal of number '2' comes.(step input)<br />
-when t = 50(min), signal of number '3' comes.(step input)<br />
<br />
<br />
<br />
=='''Results'''==<br />
<br />
<br />
=='''Discussion'''==<br />
<br />
<br />
<br />
●conclusion<br />
<br />
=='''Whole system'''==<br />
<br />
[[Image:What_Sudoku_99.png|200px|thumb|What's Sudoku?]]<br />
<br />
<br />
<br />
<br />
<br />
<br />
{{UT-Tokyo_Foot}}</div>Lenahttp://2010.igem.org/Team:UT-Tokyo/Sudoku_modelingTeam:UT-Tokyo/Sudoku modeling2010-10-27T17:35:26Z<p>Lena: /* MS2 phage infection */</p>
<hr />
<div>__NOTOC__{{UT-Tokyo_CSS3}}{{UT-Tokyo_CSS_p}}{{UT-Tokyo_Head}}<br />
<br />
= '''Sudoku''' =<br />
<br />
[https://2010.igem.org/Team:UT-Tokyo/Sudoku_abstract Introduction] <br />
[https://2010.igem.org/Team:UT-Tokyo/Sudoku_construct System] <br />
Modeling <br />
[https://2010.igem.org/Team:UT-Tokyo/Sudoku_lab_note Lab note] <br />
[https://2010.igem.org/Team:UT-Tokyo/Sudoku_experiments Experiment] <br />
[https://2010.igem.org/Team:UT-Tokyo/Sudoku_result Result]<br />
[https://2010.igem.org/Team:UT-Tokyo/Sudoku_reference Reference]<br />
<br />
== '''4C3leak switch''' ==<br />
<br />
We modeled our 4C3 leak switch as deterministic ODE system.<br />
<br />
<br />
==='''Modelling'''===<br />
<br />
<br />
===='''Assumption'''====<br />
<br />
1.We treated DNA as continuous variabl, since there could be hundreds of plasmid in one E.coli (multicopy plasmid).<br />
There are 32 kinds of variables that stand for DNA concentration corresponding to their different internal states.<br />
<br />
2.We assumed that cre recombinase operate as tetramer, and other recombinase as dimer (but this seems not to be relevant).<br />
<br />
3.We assumed that association and dissociation of DNA recombinase to DNA is sufficiently fast so that equilibrate in the timescale of the whole switch.<br />
<br />
4.We ignored the reverse reaction of DNA recombinase because we used irreversible sequence (lox66, lox71 etc).<br />
<br />
===='''Variable'''====<br />
<br />
Basically, "pn" denotes for protein concentration and "rn" denotes for mRNA concentration<br />
<br />
p1,p2,p3,p4 : concentration of 4 recombinase protein<br />
<br />
r1,r2,r3,r4 : concentration of 4 recombinass mRNA<br />
<br />
u1,u2,u3,u4 : concentration of input mRNA<br />
<br />
pc,rc : concentration of cre recombinase protein and mRNA<br />
<br />
ps,rs : concentration of SP6 polymerase protein and mRNA<br />
<br />
<br />
====''' Parameters '''====<br />
k0 : mRNA translation rate<br />
<br />
k1 : protein degradation rate<br />
<br />
k2 : mRNA translation rate<br />
<br />
k3 : cre protein degradation rate<br />
<br />
l0 : transcription speed per unit concentration of RNA polymearase<br />
<br />
l1 : mRNA degradation rate<br />
<br />
l2 : cre mRNA degradation rate<br />
<br />
K : recombinase binding constant<br />
<br />
v : recombinase reaction rate<br />
<br />
pT7 : concentration o T7 polymerase<br />
<br />
p : terminator leak probability<br />
<br />
<br />
<br />
===='''Equations'''====<br />
<br />
<br />
<br />
==='''Methods'''===<br />
<br />
We used our original python program (4 th order explicit Runge-Kutta algorithm) to solve thess equations.<br />
<br />
<br />
==='''Results'''===<br />
<br />
We input signal 1,2,4 and simulated the time evolution of the system.<br />
<br />
The time course of DNA concentration<br />
<br />
[[Image:Sim_res.JPG]]<br />
<br />
The time course of mRNA concentration. Note that the output of this system is not protein but mRNA.<br />
<br />
[[Image:Rna.JPG]]<br />
<br />
==='''Discussion'''===<br />
<br />
It is confirmed that the system works as expected with a certain combination of parameters.<br />
<br />
In addition, we changed many parameters, especially,l0,v(whose values are unclear) and p(leakiness of terminator) , to see whether the device is able to operate within a large part of parameter space.<br />
<br />
As a result, the circuit can function correctly within the range of 0.001< p < 0.1, 0.001 < v <1.0, 0.01 < l0 < 1.0 , demonstrating its robustness.<br />
This robustness may come from the assumption that DNA recombination is irreversible, which is supported by our design of the system.<br />
<br />
==='''conclusion'''===<br />
<br />
The correct operation of the switch was confirmed. By changing parameter exponentially, the robustness of the system was shown.<br />
<br />
=='''MS2 phage infection'''==<br />
<br />
=='''Modeling'''==<br />
<br />
'''Variable'''<br />
<br />
U: uninfected coli<br />
<br />
I: coli which got '1' signal<br />
<br />
Io1: other once infected coli<br />
<br />
I2: coli which got '1''2' signal<br />
<br />
Io2: other second infected coli<br />
<br />
I3: coli which got '1''2''3' signal<br />
<br />
Io3: other third infected coli<br />
<br />
B4: '4' coli which began phage making<br />
<br />
Bo: other coli which began phage making<br />
<br />
P4: free phage of '4'<br />
<br />
Po: other free phage<br />
<br />
<br />
'''Parameters'''<br />
<br />
g: growth rate of coli <br />
<br />
k1: rate of infection<br />
<br />
k2: death rate of uninfected<br />
<br />
k3: lysis rate of infected<br />
<br />
k4: rate of phage producing<br />
<br />
k5: phage death rate<br />
<br />
k6: terminator leak rate<br />
<br />
<br />
'''Equations'''<br />
<br />
We assume that cell lysis caused by MS2 phage is sufficiently rapid than coli death rate.<br />
<br />
<br />
=='''Method'''==<br />
<br />
We used original matlab program.<br />
<br />
We set conditions as following.<br />
-when t = 10(min), signal of number '1' comes.(step input)<br />
-when t = 30(min), signal of number '2' comes.(step input)<br />
-when t = 50(min), signal of number '3' comes.(step input)<br />
<br />
<br />
<br />
=='''Results'''==<br />
<br />
<br />
=='''Discussion'''==<br />
<br />
<br />
<br />
●conclusion<br />
<br />
=='''Whole system'''==<br />
<br />
[[Image:What_Sudoku_99.png|200px|thumb|What's Sudoku?]]<br />
<br />
<br />
<br />
<br />
<br />
<br />
{{UT-Tokyo_Foot}}</div>Lenahttp://2010.igem.org/Team:UT-Tokyo/Sudoku_modelingTeam:UT-Tokyo/Sudoku modeling2010-10-27T17:33:02Z<p>Lena: /* MS2 phage infection */</p>
<hr />
<div>__NOTOC__{{UT-Tokyo_CSS3}}{{UT-Tokyo_CSS_p}}{{UT-Tokyo_Head}}<br />
<br />
= '''Sudoku''' =<br />
<br />
[https://2010.igem.org/Team:UT-Tokyo/Sudoku_abstract Introduction] <br />
[https://2010.igem.org/Team:UT-Tokyo/Sudoku_construct System] <br />
Modeling <br />
[https://2010.igem.org/Team:UT-Tokyo/Sudoku_lab_note Lab note] <br />
[https://2010.igem.org/Team:UT-Tokyo/Sudoku_experiments Experiment] <br />
[https://2010.igem.org/Team:UT-Tokyo/Sudoku_result Result]<br />
[https://2010.igem.org/Team:UT-Tokyo/Sudoku_reference Reference]<br />
<br />
== '''4C3leak switch''' ==<br />
<br />
We modeled our 4C3 leak switch as deterministic ODE system.<br />
<br />
<br />
==='''Modelling'''===<br />
<br />
<br />
===='''Assumption'''====<br />
<br />
1.We treated DNA as continuous variabl, since there could be hundreds of plasmid in one E.coli (multicopy plasmid).<br />
There are 32 kinds of variables that stand for DNA concentration corresponding to their different internal states.<br />
<br />
2.We assumed that cre recombinase operate as tetramer, and other recombinase as dimer (but this seems not to be relevant).<br />
<br />
3.We assumed that association and dissociation of DNA recombinase to DNA is sufficiently fast so that equilibrate in the timescale of the whole switch.<br />
<br />
4.We ignored the reverse reaction of DNA recombinase because we used irreversible sequence (lox66, lox71 etc).<br />
<br />
===='''Variable'''====<br />
<br />
Basically, "pn" denotes for protein concentration and "rn" denotes for mRNA concentration<br />
<br />
p1,p2,p3,p4 : concentration of 4 recombinase protein<br />
<br />
r1,r2,r3,r4 : concentration of 4 recombinass mRNA<br />
<br />
u1,u2,u3,u4 : concentration of input mRNA<br />
<br />
pc,rc : concentration of cre recombinase protein and mRNA<br />
<br />
ps,rs : concentration of SP6 polymerase protein and mRNA<br />
<br />
<br />
====''' Parameters '''====<br />
k0 : mRNA translation rate<br />
<br />
k1 : protein degradation rate<br />
<br />
k2 : mRNA translation rate<br />
<br />
k3 : cre protein degradation rate<br />
<br />
l0 : transcription speed per unit concentration of RNA polymearase<br />
<br />
l1 : mRNA degradation rate<br />
<br />
l2 : cre mRNA degradation rate<br />
<br />
K : recombinase binding constant<br />
<br />
v : recombinase reaction rate<br />
<br />
pT7 : concentration o T7 polymerase<br />
<br />
p : terminator leak probability<br />
<br />
<br />
<br />
===='''Equations'''====<br />
<br />
<br />
<br />
==='''Methods'''===<br />
<br />
We used our original python program (4 th order explicit Runge-Kutta algorithm) to solve thess equations.<br />
<br />
<br />
==='''Results'''===<br />
<br />
We input signal 1,2,4 and simulated the time evolution of the system.<br />
<br />
The time course of DNA concentration<br />
<br />
[[Image:Sim_res.JPG]]<br />
<br />
The time course of mRNA concentration. Note that the output of this system is not protein but mRNA.<br />
<br />
[[Image:Rna.JPG]]<br />
<br />
==='''Discussion'''===<br />
<br />
It is confirmed that the system works as expected with a certain combination of parameters.<br />
<br />
In addition, we changed many parameters, especially,l0,v(whose values are unclear) and p(leakiness of terminator) , to see whether the device is able to operate within a large part of parameter space.<br />
<br />
As a result, the circuit can function correctly within the range of 0.001< p < 0.1, 0.001 < v <1.0, 0.01 < l0 < 1.0 , demonstrating its robustness.<br />
This robustness may come from the assumption that DNA recombination is irreversible, which is supported by our design of the system.<br />
<br />
==='''conclusion'''===<br />
<br />
The correct operation of the switch was confirmed. By changing parameter exponentially, the robustness of the system was shown.<br />
<br />
=='''MS2 phage infection'''==<br />
<br />
'''Modeling'''<br />
<br />
Variable<br />
<br />
U: uninfected coli<br />
<br />
I: coli which got '1' signal<br />
<br />
Io1: other once infected coli<br />
<br />
I2: coli which got '1''2' signal<br />
<br />
Io2: other second infected coli<br />
<br />
I3: coli which got '1''2''3' signal<br />
<br />
Io3: other third infected coli<br />
<br />
B4: '4' coli which began phage making<br />
<br />
Bo: other coli which began phage making<br />
<br />
P4: free phage of '4'<br />
<br />
Po: other free phage<br />
<br />
<br />
Parameters<br />
<br />
g: growth rate of coli <br />
k1: rate of infection<br />
k2: death rate of uninfected<br />
k3: lysis rate of infected<br />
k4: rate of phage producing<br />
k5: phage death rate<br />
k6: terminator leak rate<br />
<br />
Equations<br />
<br />
We assume that cell lysis caused by MS2 phage is sufficiently rapid than coli death rate.<br />
<br />
<br />
'''Method'''<br />
<br />
We used original matlab program.<br />
<br />
We set conditions as following.<br />
-when t = 10(min), signal of number '1' comes.(step input)<br />
-when t = 30(min), signal of number '2' comes.(step input)<br />
-when t = 50(min), signal of number '3' comes.(step input)<br />
<br />
<br />
<br />
'''Results'''<br />
<br />
<br />
'''Discussion'''<br />
<br />
<br />
<br />
●conclusion<br />
<br />
=='''Whole system'''==<br />
<br />
[[Image:What_Sudoku_99.png|200px|thumb|What's Sudoku?]]<br />
<br />
<br />
<br />
<br />
<br />
<br />
{{UT-Tokyo_Foot}}</div>Lenahttp://2010.igem.org/Team:UT-Tokyo/Sudoku_modelingTeam:UT-Tokyo/Sudoku modeling2010-10-27T17:31:59Z<p>Lena: /* MS2 phage infection */</p>
<hr />
<div>__NOTOC__{{UT-Tokyo_CSS3}}{{UT-Tokyo_CSS_p}}{{UT-Tokyo_Head}}<br />
<br />
= '''Sudoku''' =<br />
<br />
[https://2010.igem.org/Team:UT-Tokyo/Sudoku_abstract Introduction] <br />
[https://2010.igem.org/Team:UT-Tokyo/Sudoku_construct System] <br />
Modeling <br />
[https://2010.igem.org/Team:UT-Tokyo/Sudoku_lab_note Lab note] <br />
[https://2010.igem.org/Team:UT-Tokyo/Sudoku_experiments Experiment] <br />
[https://2010.igem.org/Team:UT-Tokyo/Sudoku_result Result]<br />
[https://2010.igem.org/Team:UT-Tokyo/Sudoku_reference Reference]<br />
<br />
== '''4C3leak switch''' ==<br />
<br />
We modeled our 4C3 leak switch as deterministic ODE system.<br />
<br />
<br />
==='''Modelling'''===<br />
<br />
<br />
===='''Assumption'''====<br />
<br />
1.We treated DNA as continuous variabl, since there could be hundreds of plasmid in one E.coli (multicopy plasmid).<br />
There are 32 kinds of variables that stand for DNA concentration corresponding to their different internal states.<br />
<br />
2.We assumed that cre recombinase operate as tetramer, and other recombinase as dimer (but this seems not to be relevant).<br />
<br />
3.We assumed that association and dissociation of DNA recombinase to DNA is sufficiently fast so that equilibrate in the timescale of the whole switch.<br />
<br />
4.We ignored the reverse reaction of DNA recombinase because we used irreversible sequence (lox66, lox71 etc).<br />
<br />
===='''Variable'''====<br />
<br />
Basically, "pn" denotes for protein concentration and "rn" denotes for mRNA concentration<br />
<br />
p1,p2,p3,p4 : concentration of 4 recombinase protein<br />
<br />
r1,r2,r3,r4 : concentration of 4 recombinass mRNA<br />
<br />
u1,u2,u3,u4 : concentration of input mRNA<br />
<br />
pc,rc : concentration of cre recombinase protein and mRNA<br />
<br />
ps,rs : concentration of SP6 polymerase protein and mRNA<br />
<br />
<br />
====''' Parameters '''====<br />
k0 : mRNA translation rate<br />
<br />
k1 : protein degradation rate<br />
<br />
k2 : mRNA translation rate<br />
<br />
k3 : cre protein degradation rate<br />
<br />
l0 : transcription speed per unit concentration of RNA polymearase<br />
<br />
l1 : mRNA degradation rate<br />
<br />
l2 : cre mRNA degradation rate<br />
<br />
K : recombinase binding constant<br />
<br />
v : recombinase reaction rate<br />
<br />
pT7 : concentration o T7 polymerase<br />
<br />
p : terminator leak probability<br />
<br />
<br />
<br />
===='''Equations'''====<br />
<br />
<br />
<br />
==='''Methods'''===<br />
<br />
We used our original python program (4 th order explicit Runge-Kutta algorithm) to solve thess equations.<br />
<br />
<br />
==='''Results'''===<br />
<br />
We input signal 1,2,4 and simulated the time evolution of the system.<br />
<br />
The time course of DNA concentration<br />
<br />
[[Image:Sim_res.JPG]]<br />
<br />
The time course of mRNA concentration. Note that the output of this system is not protein but mRNA.<br />
<br />
[[Image:Rna.JPG]]<br />
<br />
==='''Discussion'''===<br />
<br />
It is confirmed that the system works as expected with a certain combination of parameters.<br />
<br />
In addition, we changed many parameters, especially,l0,v(whose values are unclear) and p(leakiness of terminator) , to see whether the device is able to operate within a large part of parameter space.<br />
<br />
As a result, the circuit can function correctly within the range of 0.001< p < 0.1, 0.001 < v <1.0, 0.01 < l0 < 1.0 , demonstrating its robustness.<br />
This robustness may come from the assumption that DNA recombination is irreversible, which is supported by our design of the system.<br />
<br />
==='''conclusion'''===<br />
<br />
The correct operation of the switch was confirmed. By changing parameter exponentially, the robustness of the system was shown.<br />
<br />
=='''MS2 phage infection'''==<br />
<br />
'''Modeling'''<br />
<br />
Variable<br />
U: uninfected coli<br />
I: coli which got '1' signal<br />
Io1: other once infected coli<br />
I2: coli which got '1''2' signal<br />
Io2: other second infected coli<br />
I3: coli which got '1''2''3' signal<br />
Io3: other third infected coli<br />
B4: '4' coli which began phage making<br />
Bo: other coli which began phage making<br />
P4: free phage of '4'<br />
Po: other free phage<br />
<br />
Parameters<br />
g: growth rate of coli <br />
k1: rate of infection<br />
k2: death rate of uninfected<br />
k3: lysis rate of infected<br />
k4: rate of phage producing<br />
k5: phage death rate<br />
k6: terminator leak rate<br />
<br />
Equations<br />
We assume that cell lysis caused by MS2 phage is sufficiently rapid than coli death rate.<br />
<br />
<br />
'''Method'''<br />
We used original matlab program.<br />
<br />
We set conditions as following.<br />
-when t = 10(min), signal of number '1' comes.(step input)<br />
-when t = 30(min), signal of number '2' comes.(step input)<br />
-when t = 50(min), signal of number '3' comes.(step input)<br />
<br />
<br />
<br />
'''Results'''<br />
<br />
<br />
'''Discussion'''<br />
<br />
<br />
<br />
●conclusion<br />
<br />
=='''Whole system'''==<br />
<br />
[[Image:What_Sudoku_99.png|200px|thumb|What's Sudoku?]]<br />
<br />
<br />
<br />
<br />
<br />
<br />
{{UT-Tokyo_Foot}}</div>Lena