Team:Tsinghua-A/Modeling/P1A

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<li><a href="https://2011.igem.org/Team:Tsinghua-A/Team">Team</a></li>
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<P><FONT COLOR="#000000"><FONT FACE="Times New Roman, serif"><SPAN LANG="en-US"><FONT FACE="Arial, sans-serif"><FONT SIZE=5 STYLE="font-size: 20pt"><SPAN LANG="en-US"><A HREF="https://2011.igem.org/Team:Tsinghua-A/Modeling">Modeling</A></SPAN></FONT></FONT></FONT><FONT FACE="Arial, sans-serif"><FONT SIZE=5 STYLE="font-size: 20pt"><SPAN LANG="en-US">::</SPAN></FONT></FONT><FONT COLOR="#666600"><FONT FACE="Arial, sans-serif"><FONT SIZE=5 STYLE="font-size: 20pt"><SPAN LANG="en-US">Quorum-sensing Effect</SPAN></FONT></FONT></FONT></SPAN></FONT></P><br>
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<div class="slider">
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<p>What we have done insofar 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 amplitude of oscillation in this part.</p>
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<p><IMG SRC="https://static.igem.org/mediawiki/2011/1/13/ModelBack.jpg" ALIGN=bottom WIDTH=750 HEIGHT=175 BORDER=0 ISMAP></div>
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<p>As we all know, no two things in this world are the same, so do cells. The major difference of individual cell that we take into considerations is twofold:</p>
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<div class="nav">
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<table id="toc" class="toc">
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<tr>
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<td>
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<div id="toctitle">
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<h2>Contents</h2>
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</div>
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<ul>
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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>
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<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>
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<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>
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<li class="toclevel-1 tocsection-4"><a href="#Dimensionless Model"><span class="tocnumber">4</span> <span class="toctext">Dimensionless Model</span></a></li>
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<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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<li class="toclevel-1 tocsection-6"><a href="#References"><span class="tocnumber">6</span> <span class="toctext">References</span></a></li>
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</ul>
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</td>
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</tr>
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</table>
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</div>
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<br>
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<div class="slider">
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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/c/c6/Thu_A_Matlab_Code.zip"><U><I>Download the source code package(Matlab)</I></U></A></P></div>
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<h1 id="Introduction">Introduction to Model</h1><hr width="100%" size=2 color=gray>
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<div class="imgbox">
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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"/>
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<img class="border" style="border-color:#B2B2B2;"src="https://static.igem.org/mediawiki/2011/b/b0/002.png" width = "430px" height="148px" />
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<p class="cite">Designed gene circuit in cell A and cell B</p>
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</div>
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<p>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.
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</p>
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<br><br>
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<div class="slider">
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<p><A HREF="https://2011.igem.org/Team:Tsinghua-A/Modeling/P1A"><U><I>Read
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more</I></U></A></P>
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</div>
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<P id="Original Full Model"><h1>Original Full Model</h1></P>
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<hr width="100%" size=2 color=gray>
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<P>Firstly we
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described the system thoroughly without leaving out any
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seemingly unimportant actions and factors. As a result, the
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description of the system contains every possible mass actions as
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well as some hill kinetics, Henri-Michaelis-Menten kinetics, and the
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parameters were got from literature. The model was represented and
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simulated in the Matlab toolbox SIMBIOLOGY, but too many parameters make
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it difficult to do further analyses, So here we only listed all 19 ODEs
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and a reletive parameter table( see <A HREF="https://static.igem.org/mediawiki/2011/9/9a/Modeling_Wiki.pdf">attached pdf file</A>).
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</P>
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<br>
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<div class="slider">
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<p><A HREF="https://2011.igem.org/Team:Tsinghua-A/Modeling/P2A"><U><I>Read
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more</I></U></A></P>
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</div>
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<P id="Simplified DDE Model"><h1>Simplified DDE</h1></P>
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<hr width="100%" size=2 color=gray>
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<div class="imgbox">
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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"/>
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<p class="cite">Simplified DDEs</p>
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</div>
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<P>
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original model contains too many factors for analyzing the general
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property of system. To understand the essential characters of the
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oscillator, we simplify the original model according to certain
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appropriate assumptions, like Quasi-equilibrium
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for fast reactions.</P>
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<P>After series of derivation based on those assumptions, we came
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up with the following set of DDEs (Delay Differential Equations)
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which contains only 6 equations, see the figure right. And it
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would be much more convenient for us to do some neccessary analyses
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and research into the mathematical essence of our model.
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</P>
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<div class="imgbox2">
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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"/>
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<p class="cite">Figure shows all variables are oscillating</p>
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</div>
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<P>We
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coded the system by DDE description in MATLAB and did simulation
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analysis accordingly. The result showed that the system could
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oscillate under certain parameters.</P>
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<P>To
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further understand what parameters could make the system oscillate,
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we did bifurcation analysis on the Hill parameters. What we had to do
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was find the critical points where the system can nearly oscillate
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but a little disruption may lead to a steady state.</P>
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<P>Depicting
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all those critical points, as shown in the figure, the system could
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oscillate when cellB's Hill parameters were located in the
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area named <FONT COLOR="#00b0f0"></FONT><FONT COLOR="#00b0f0"><I><B>Bistable</B></I></FONT><FONT COLOR="#00b0f0"></B></I>.</P>
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<div class="imgbox3">
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<img class="border" style="border-color:#B2B2B2;"src="https://static.igem.org/mediawiki/2011/2/20/005.png" width = "540px" height="300px"/>
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<p class="cite">The figure right is the phase trajectory of two signal molecules in environment,
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the more it looks like a circle, the more steadily our system will oscillate.</p>
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</div>
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<div class="imgbox4">
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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"/>
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<p class="cite">Our system oscillates when parameters were selected in the area named 'Bistable'</p>
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</div>
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<div class="imgbox5">
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<img class="border" style="border-color:#B2B2B2;"src="https://static.igem.org/mediawiki/2011/e/e8/007.png" width = "370px" height="280px"/>
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</div>
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<br><br>
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<div class="temp"><P>By
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adjusting certain parameters, we saw that the oscillation&rsquo;s
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period and phase could be controlled properly, which is the most
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impressive character of our system. Here we present a figure that the
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oscillation phase was adjusted by adding araC, which could induce the
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pBad promoter, in cell type B. After adding araC to our system at
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certain time, the oscillation was interrupted at beginning, but could
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gradually recover and finally, the phase was changed.</P></div>
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<br><br><br><br><br><br>
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<div class="slider">
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<p><A HREF="https://2011.igem.org/Team:Tsinghua-A/Modeling/P3A"><U><I>Read
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more</I></U></A></P>
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</div>
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<P id="Dimensionless Model"><h1>Dimensionless Model</h1></P>
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<hr width="100%" size=2 color=gray>
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<P>In
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order to make a further analysis on stability of the system, and
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sensitivity of parameters, we further simplified the model to make
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them dimensionless. In addition, we tried to introduce feedback to
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our system and made a brief analysis on different types of
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feedback we introduced. Some analyses were similar to the simplified
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DDE model, and you can see more details by clicking
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<A HREF="https://2011.igem.org/Team:Tsinghua-A/Modeling/P4">read
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more</A></P>
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<br>
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<div class="slider">
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<p><A HREF="https://2011.igem.org/Team:Tsinghua-A/Modeling/P4"><U><I>Read
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more</I></U></A></P>
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</div>
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<P id="Quorum Sensing"><h1>Quorum Sensing Effect</h1></P>
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<hr width="100%" size=2 color=gray>
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<P>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.</P>
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<P>As
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we all know, no two things in this world are exactly the same, so do
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cells. The major differences between individual cells that we take
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into consideration include:</P>
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<P><B>●Each cell's activity of promoter is varied, so each cell has
<P><B>●Each cell's activity of promoter is varied, so each cell has
different rate to generate AHL.</B></P>
different rate to generate AHL.</B></P>
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obey normal distribution, that is:
obey normal distribution, that is:
</P>
</P>
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<P ALIGN=CENTER style="text-intend:0em">m(i)= &mu;1+<I>N</I>(0,&sigma;1);</P>
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<P ALIGN=CENTER>m(i)= &mu;1+<I>N</I>(0,&sigma;1);</P>
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<P ALIGN=CENTER style="text-intend:0em">n(i)= &mu;2+<I>N</I>(0,&sigma;2);</P>
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<P ALIGN=CENTER>n(i)= &mu;2+<I>N</I>(0,&sigma;2);</P>
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<P ALIGN=LEFT><A NAME="OLE_LINK63"></A><A NAME="OLE_LINK62"></A>&mu;<SPAN LANG="en-US">1
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<P>&mu;1 and &mu;2 are the average ability of generating 30C6HSL and 3012CHSL,
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and &mu;2 are the average ability of generating 30C6HSL and 3012CHSL,
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and normal distribution--(0,&sigma;)--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.</P>
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and normal distribution-- </SPAN><SPAN LANG="en-US"><I>N</I></SPAN><SPAN LANG="en-US">(0,&sigma;)--describes
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the fluctuations of AHL generating rate in individual cell. We then
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<p align="center" style="text-intend:0em"><img src="https://static.igem.org/mediawiki/igem.org/0/00/Part4-1.png" width = "900px" height="675px"></p>
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expanded our equations from 2 cells to a population of cells. Each
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<p><b>Figure 16 100 Cells Varied in parameter m and n</b></p>
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cell share a mutual environment in which we assume that AHL in
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<p>The figures indicate that our system can oscillate synchronically being able to tolerate differences among a population of cells. Furthermore, the figures prove that different ability of generating AHLs of cells have nothing to do with the period and phase of the oscillation. We can also see that the oscillation amplitude of each cell is to a greater extent varied when the Variance of interruption is enlarged.</p>
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environment is proportionally distributed.</SPAN></SPAN></P>
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<p>Moreover, we test whether the oscillation is dependent on initial distribution of AHL by changing the initial amount drastically by letting them follow uniform distribution. That is:</p>
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<P ALIGN="center" style="text-intend:0em"><IMG SRC="https://static.igem.org/mediawiki/2011/1/17/008.png" WIDTH=800 HEIGHT=600 BORDER=0></P>
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<P ALIGN=CENTER>Initial(i)= <I>U</I>(0,20);</P>
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<p ALIGN="center" style="text-intend:0em"><B>Figures were simulated out<br>by given different variance of noise</B></p>
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<p>The results would give evidence to prove that our system can start to oscillate synchronically given variant initial starting numbers.</p>
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<P>The
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<p>Based on this distribution restraining the initial AHL concentration in each cell, we simulated out a figure as follows.</p>
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figures indicate that our system can oscillate synchronically being
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able to tolerate differences at certain range among a population of
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<p align="center" style="text-intend:0em"><img src="https://static.igem.org/mediawiki/igem.org/2/2d/Part4-2.png" width = "900px" height="675px"></p>
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cells.</P>
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<p><b>Figure 17 100 Cells Varied in initial AHL concentration</b></p>
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<P>We
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<p>The results demonstratively give evidence proving that our system can start to oscillate synchronically given variant initial starting numbers.</p>
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also tested whether the oscillation is dependent on initial
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<br><br>
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distribution of AHL by changing the initial amount drastically by
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letting them follow uniform distribution. That is:</P>
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<P ALIGN="center" style="text-intend:0em">Initial(i)= <I>U</I>(0,20);</P>
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<P>Based
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on this distribution restraining the initial AHL concentration in
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each cell, we simulated out a figure as follows.</P>
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<P ALIGN="center" style="text-intend:0em"><IMG SRC="https://static.igem.org/mediawiki/2011/d/d2/009.png" WIDTH=800 HEIGHT=600 BORDER=0></P>
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<p ALIGN="center" style="text-intend:0em"><B>Figures were simulated out<br>under varied initual AHL concentration</B></p>
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<P>The
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results demonstratively give evidence proving that our system can
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start to oscillate synchronically given variant initial starting
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status.</P>
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<br>
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<P id="References"><h1>References</h1></P>
 
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<hr width="100%" size=2 color=gray>
 
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<P>[1]
 
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Uri Alon, (2007). Network motifs: theory and experimental approaches.
 
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Nature.</P>
 
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<P>[2]
 
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Chunbo Lou, Xili Liu, Ming Ni, et al. (2010). Synthesizing a novel
 
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Modeling::Quorum-sensing Effect


What we have done insofar 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 amplitude of oscillation in this part.

As we all know, no two things in this world are the same, so do cells. The major difference of individual cell that we take into considerations is twofold:

●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--(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.

Figure 16 100 Cells Varied in parameter m and n

The figures indicate that our system can oscillate synchronically being able to tolerate differences among a population of cells. Furthermore, the figures prove that different ability of generating AHLs of cells have nothing to do with the period and phase of the oscillation. We can also see that the oscillation amplitude of each cell is to a greater extent varied when the Variance of interruption is enlarged.

Moreover, we test 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);

The results would give evidence to prove that our system can start to oscillate synchronically given variant initial starting numbers.

Based on this distribution restraining the initial AHL concentration in each cell, we simulated out a figure as follows.

Figure 17 100 Cells Varied in initial AHL concentration

The results demonstratively give evidence proving that our system can start to oscillate synchronically given variant initial starting numbers.



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