Team:Tsinghua-A/Modeling
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<li class="toclevel-1 tocsection-1"><a href="#Introduction"><span class="tocnumber">1</span> <span class="toctext">Introduction to Model</span></a></li> | <li class="toclevel-1 tocsection-1"><a href="#Introduction"><span class="tocnumber">1</span> <span class="toctext">Introduction to Model</span></a></li> | ||
- | <li class="toclevel-1 tocsection-2"><a href="#Full Model"><span class="tocnumber">2</span> <span class="toctext">Original Full Model</span></a></li> | + | <li class="toclevel-1 tocsection-2"><a href="#Original Full Model"><span class="tocnumber">2</span> <span class="toctext">Original Full Model</span></a></li> |
- | <li class="toclevel-1 tocsection-3"><a href="#DDE Model"><span class="tocnumber">3</span> <span class="toctext">Simplified DDE Model</span></a></li> | + | <li class="toclevel-1 tocsection-3"><a href="#Simplified DDE Model"><span class="tocnumber">3</span> <span class="toctext">Simplified DDE Model</span></a></li> |
<li class="toclevel-1 tocsection-4"><a href="#Dimensionless Model"><span class="tocnumber">4</span> <span class="toctext">Dimensionless Model</span></a></li> | <li class="toclevel-1 tocsection-4"><a href="#Dimensionless Model"><span class="tocnumber">4</span> <span class="toctext">Dimensionless Model</span></a></li> | ||
<li class="toclevel-1 tocsection-5"><a href="#Quorum Sensing"><span class="tocnumber">5</span> <span class="toctext">Quorum Sensing Effect</span></a></li> | <li class="toclevel-1 tocsection-5"><a href="#Quorum Sensing"><span class="tocnumber">5</span> <span class="toctext">Quorum Sensing Effect</span></a></li> | ||
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- | <p><IMG SRC="https://static.igem.org/mediawiki/2011/2/2b/000.png" NAME="graph1" ALIGN=bottom WIDTH=20 HEIGHT=20 BORDER=0 ISMAP><A HREF="https://static.igem.org/mediawiki/2011/9/9a/Modeling_Wiki.pdf"><U><I>Download the full text </I></U></A><IMG SRC="https://static.igem.org/mediawiki/2011/0/08/Thu_matlab.png" NAME="graph2" ALIGN=BOTTOM WIDTH=20 HEIGHT=20 BORDER=0 ISMAP><A HREF="https://static.igem.org/mediawiki/2011/ | + | <p><IMG SRC="https://static.igem.org/mediawiki/2011/2/2b/000.png" NAME="graph1" ALIGN=bottom WIDTH=20 HEIGHT=20 BORDER=0 ISMAP><A HREF="https://static.igem.org/mediawiki/2011/9/9a/Modeling_Wiki.pdf"><U><I>Download the full text </I></U></A><IMG SRC="https://static.igem.org/mediawiki/2011/0/08/Thu_matlab.png" NAME="graph2" ALIGN=BOTTOM WIDTH=20 HEIGHT=20 BORDER=0 ISMAP><A HREF="https://static.igem.org/mediawiki/2011/c/c6/Thu_A_Matlab_Code.zip"><U><I>Download the source code package(Matlab)</I></U></A></P></div> |
- | <h1 id="Introduction">Introduction to Model</h1><hr width="100%"> | + | <h1 id="Introduction">Introduction to Model</h1><hr width="100%" size=2 color=gray> |
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<img class="border" style="border-color:#B2B2B2;"src="https://static.igem.org/mediawiki/2011/0/03/001.png" width = "430px" height="148px"/> | <img class="border" style="border-color:#B2B2B2;"src="https://static.igem.org/mediawiki/2011/0/03/001.png" width = "430px" height="148px"/> | ||
<img class="border" style="border-color:#B2B2B2;"src="https://static.igem.org/mediawiki/2011/b/b0/002.png" width = "430px" height="148px" /> | <img class="border" style="border-color:#B2B2B2;"src="https://static.igem.org/mediawiki/2011/b/b0/002.png" width = "430px" height="148px" /> | ||
- | <p>Designed gene circuit in cell A and cell B</p> | + | <p class="cite">Designed gene circuit in cell A and cell B</p> |
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<p><A HREF="https://2011.igem.org/Team:Tsinghua-A/Modeling/P1A"><U><I>Read | <p><A HREF="https://2011.igem.org/Team:Tsinghua-A/Modeling/P1A"><U><I>Read | ||
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<P id="Original Full Model"><h1>Original Full Model</h1></P> | <P id="Original Full Model"><h1>Original Full Model</h1></P> | ||
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<P>Firstly we | <P>Firstly we | ||
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<p><A HREF="https://2011.igem.org/Team:Tsinghua-A/Modeling/P2A"><U><I>Read | <p><A HREF="https://2011.igem.org/Team:Tsinghua-A/Modeling/P2A"><U><I>Read | ||
more</I></U></A></P> | more</I></U></A></P> | ||
</div> | </div> | ||
- | <P id="Simplified DDE"><h1>Simplified DDE</h1></P> | + | <P id="Simplified DDE Model"><h1>Simplified DDE</h1></P> |
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<img class="border" style="border-color:#B2B2B2;"src="https://static.igem.org/mediawiki/2011/b/b0/003.png" width = "440px" height="310px"/> | <img class="border" style="border-color:#B2B2B2;"src="https://static.igem.org/mediawiki/2011/b/b0/003.png" width = "440px" height="310px"/> | ||
- | <p | + | <p class="cite">Simplified DDEs</p> |
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<img class="border" style="border-color:#B2B2B2;"src="https://static.igem.org/mediawiki/2011/d/d5/004.png" width = "440px" height="310px"/> | <img class="border" style="border-color:#B2B2B2;"src="https://static.igem.org/mediawiki/2011/d/d5/004.png" width = "440px" height="310px"/> | ||
- | <p | + | <p class="cite">Figure shows all variables are oscillating</p> |
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<P>Depicting | <P>Depicting | ||
all those critical points, as shown in the figure, the system could | all those critical points, as shown in the figure, the system could | ||
- | oscillate when cellB's | + | oscillate when cellB's Hill parameters were located in the |
area named <FONT COLOR="#00b0f0"></FONT><FONT COLOR="#00b0f0"><I><B>Bistable</B></I></FONT><FONT COLOR="#00b0f0"></B></I>.</P> | area named <FONT COLOR="#00b0f0"></FONT><FONT COLOR="#00b0f0"><I><B>Bistable</B></I></FONT><FONT COLOR="#00b0f0"></B></I>.</P> | ||
<div class="imgbox3"> | <div class="imgbox3"> | ||
<img class="border" style="border-color:#B2B2B2;"src="https://static.igem.org/mediawiki/2011/2/20/005.png" width = "540px" height="300px"/> | <img class="border" style="border-color:#B2B2B2;"src="https://static.igem.org/mediawiki/2011/2/20/005.png" width = "540px" height="300px"/> | ||
- | <p | + | <p class="cite">The figure right is the phase trajectory of two signal molecules in environment, |
- | the more it looks like a circle, the more steadily our system will oscillate. | + | the more it looks like a circle, the more steadily our system will oscillate.</p> |
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<img class="border" style="border-color:#B2B2B2;"src="https://static.igem.org/mediawiki/2011/1/18/006.png" width = "370px" height="300px"/> | <img class="border" style="border-color:#B2B2B2;"src="https://static.igem.org/mediawiki/2011/1/18/006.png" width = "370px" height="300px"/> | ||
- | <p | + | <p class="cite">Our system oscillates when parameters were selected in the area named 'Bistable'</p> |
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certain time, the oscillation was interrupted at beginning, but could | certain time, the oscillation was interrupted at beginning, but could | ||
gradually recover and finally, the phase was changed.</P></div> | gradually recover and finally, the phase was changed.</P></div> | ||
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<p><A HREF="https://2011.igem.org/Team:Tsinghua-A/Modeling/P3A"><U><I>Read | <p><A HREF="https://2011.igem.org/Team:Tsinghua-A/Modeling/P3A"><U><I>Read | ||
more</I></U></A></P> | more</I></U></A></P> | ||
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<P id="Dimensionless Model"><h1>Dimensionless Model</h1></P> | <P id="Dimensionless Model"><h1>Dimensionless Model</h1></P> | ||
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<P>In | <P>In | ||
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<P id="Quorum Sensing"><h1>Quorum Sensing Effect</h1></P> | <P id="Quorum Sensing"><h1>Quorum Sensing Effect</h1></P> | ||
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<P>What | <P>What | ||
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obey normal distribution, that is: | obey normal distribution, that is: | ||
</P> | </P> | ||
- | <P ALIGN=CENTER>m(i)= μ1+<I>N</I>(0,σ1);</P> | + | <P ALIGN=CENTER style="text-intent:0em">m(i)= μ1+<I>N</I>(0,σ1);</P> |
- | <P ALIGN=CENTER>n(i)= μ2+<I>N</I>(0,σ2);</P> | + | <P ALIGN=CENTER style="text-intent:0em">n(i)= μ2+<I>N</I>(0,σ2);</P> |
<P ALIGN=LEFT><A NAME="OLE_LINK63"></A><A NAME="OLE_LINK62"></A>μ<SPAN LANG="en-US">1 | <P ALIGN=LEFT><A NAME="OLE_LINK63"></A><A NAME="OLE_LINK62"></A>μ<SPAN LANG="en-US">1 | ||
and μ2 are the average ability of generating 30C6HSL and 3012CHSL, | and μ2 are the average ability of generating 30C6HSL and 3012CHSL, | ||
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cell share a mutual environment in which we assume that AHL in | cell share a mutual environment in which we assume that AHL in | ||
environment is proportionally distributed.</SPAN></SPAN></P> | environment is proportionally distributed.</SPAN></SPAN></P> | ||
- | <P ALIGN= | + | <P ALIGN="center" style="text-intent:0em"><IMG SRC="https://static.igem.org/mediawiki/2011/1/17/008.png" WIDTH=800 HEIGHT=600 BORDER=0></P> |
<P>The | <P>The | ||
figures indicate that our system can oscillate synchronically being | figures indicate that our system can oscillate synchronically being | ||
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distribution of AHL by changing the initial amount drastically by | distribution of AHL by changing the initial amount drastically by | ||
letting them follow uniform distribution. That is:</P> | letting them follow uniform distribution. That is:</P> | ||
- | <P ALIGN= | + | <P ALIGN="center" style="text-intent:0em">Initial(i)= <I>U</I>(0,20);</P> |
<P>Based | <P>Based | ||
on this distribution restraining the initial AHL concentration in | on this distribution restraining the initial AHL concentration in | ||
each cell, we simulated out a figure as follows.</P> | each cell, we simulated out a figure as follows.</P> | ||
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<P>The | <P>The | ||
results demonstratively give evidence proving that our system can | results demonstratively give evidence proving that our system can | ||
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<br> | <br> | ||
- | + | <P id="References"><h1>References</h1></P> | |
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- | <hr width="100%"> | + | |
<P>[1] | <P>[1] | ||
Uri Alon, (2007). Network motifs: theory and experimental approaches. | Uri Alon, (2007). Network motifs: theory and experimental approaches. | ||
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Latest revision as of 05:20, 28 October 2011
Introduction to Model
![](https://static.igem.org/mediawiki/2011/0/03/001.png)
![](https://static.igem.org/mediawiki/2011/b/b0/002.png)
Designed gene circuit in cell A and cell B
In our project, we designed a quorum-sensing oscillator which consists of two types of cells. The expression of the reporter genes (GFP of one cell type and GFP of another) of the cells of the same type can fluctuate synchronously and certain designs were made to adjust the phase and the period of oscillation. To understand the property of our system, we built a mathematical model based on ODEs (Ordinary Differential Equations) and DDEs (Delayed Differential Equations) to model and characterize this system. The simulation results helped us to deepen into further characteristics of the system.
Original Full Model
Firstly we described the system thoroughly without leaving out any seemingly unimportant actions and factors. As a result, the description of the system contains every possible mass actions as well as some hill kinetics, Henri-Michaelis-Menten kinetics, and the parameters were got from literature. The model was represented and simulated in the Matlab toolbox SIMBIOLOGY, but too many parameters make it difficult to do further analyses, So here we only listed all 19 ODEs and a reletive parameter table( see attached pdf file).
Simplified DDE
![](https://static.igem.org/mediawiki/2011/b/b0/003.png)
Simplified DDEs
original model contains too many factors for analyzing the general property of system. To understand the essential characters of the oscillator, we simplify the original model according to certain appropriate assumptions, like Quasi-equilibrium for fast reactions.
After series of derivation based on those assumptions, we came up with the following set of DDEs (Delay Differential Equations) which contains only 6 equations, see the figure right. And it would be much more convenient for us to do some neccessary analyses and research into the mathematical essence of our model.
![](https://static.igem.org/mediawiki/2011/d/d5/004.png)
Figure shows all variables are oscillating
We coded the system by DDE description in MATLAB and did simulation analysis accordingly. The result showed that the system could oscillate under certain parameters.
To further understand what parameters could make the system oscillate, we did bifurcation analysis on the Hill parameters. What we had to do was find the critical points where the system can nearly oscillate but a little disruption may lead to a steady state.
Depicting all those critical points, as shown in the figure, the system could oscillate when cellB's Hill parameters were located in the area named Bistable.
![](https://static.igem.org/mediawiki/2011/2/20/005.png)
The figure right is the phase trajectory of two signal molecules in environment, the more it looks like a circle, the more steadily our system will oscillate.
![](https://static.igem.org/mediawiki/2011/1/18/006.png)
Our system oscillates when parameters were selected in the area named 'Bistable'
![](https://static.igem.org/mediawiki/2011/e/e8/007.png)
By adjusting certain parameters, we saw that the oscillation’s period and phase could be controlled properly, which is the most impressive character of our system. Here we present a figure that the oscillation phase was adjusted by adding araC, which could induce the pBad promoter, in cell type B. After adding araC to our system at certain time, the oscillation was interrupted at beginning, but could gradually recover and finally, the phase was changed.
Dimensionless Model
In order to make a further analysis on stability of the system, and sensitivity of parameters, we further simplified the model to make them dimensionless. In addition, we tried to introduce feedback to our system and made a brief analysis on different types of feedback we introduced. Some analyses were similar to the simplified DDE model, and you can see more details by clicking read more
Quorum Sensing Effect
What we have analyzed so far is focused on two-cell oscillation. Quorum-sensing oscillator is not simply a matter of expansion in magnitude, but a matter of robustness in allowing difference of each individual cell. Moreover, we test the adjustment of phase and period of oscillation in this part.
As we all know, no two things in this world are exactly the same, so do cells. The major differences between individual cells that we take into consideration include:
●Each cell's activity of promoter is varied, so each cell has different rate to generate AHL.
●The initial amount of AHL may be disproportionally distributed among cells.
The rate of generating AHL is closely related to parameter m and n. Therefore, we introduce randomness to both parameters by letting them obey normal distribution, that is:
m(i)= μ1+N(0,σ1);
n(i)= μ2+N(0,σ2);
μ1 and μ2 are the average ability of generating 30C6HSL and 3012CHSL, and normal distribution-- N(0,σ)--describes the fluctuations of AHL generating rate in individual cell. We then expanded our equations from 2 cells to a population of cells. Each cell share a mutual environment in which we assume that AHL in environment is proportionally distributed.
The figures indicate that our system can oscillate synchronically being able to tolerate differences at certain range among a population of cells.
We also tested whether the oscillation is dependent on initial distribution of AHL by changing the initial amount drastically by letting them follow uniform distribution. That is:
Initial(i)= U(0,20);
Based on this distribution restraining the initial AHL concentration in each cell, we simulated out a figure as follows.
The results demonstratively give evidence proving that our system can start to oscillate synchronically given variant initial starting status.
References
[1] Uri Alon, (2007). Network motifs: theory and experimental approaches. Nature.
[2] Chunbo Lou, Xili Liu, Ming Ni, et al. (2010). Synthesizing a novel genetic sequential logic circuit: a push-on push-off switch. Molecular Systems Biology.
[3] Tal Danino, Octavio Mondragon-Palomino, Lev Tsimring & Jeff Hasty (2010). A synchronized quorum of genetic clocks. Nature.
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