Team:Tsinghua-A/Modeling

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<META NAME="AUTHOR" CONTENT="Pei Xie">
 
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<P ALIGN=LEFT><FONT FACE="Times New Roman, serif"><SPAN LANG="en-US"><FONT COLOR="#548dd4"><FONT FACE="Arial Unicode MS, sans-serif"><FONT SIZE=6 STYLE="font-size: 22pt"><SPAN LANG="en-US"><B>Introduction
 
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to Model&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;</B></SPAN></FONT></FONT></FONT><FONT COLOR="#548dd4">
 
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<IMG SRC="https://static.igem.org/mediawiki/2011/2/2b/000.png" NAME="图形1" ALIGN=BOTTOM WIDTH=23 HEIGHT=23 BORDER=0 ISMAP></FONT><A HREF="https://static.igem.org/mediawiki/2011/9/9a/Modeling_Wiki.pdf" target="_self"><FONT COLOR="#ff950e"><I><U>Download
 
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the full text </U></I></FONT></A></SPAN></FONT>
 
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</P>
 
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<p align=right><IMG SRC="https://static.igem.org/mediawiki/2011/0/08/Thu_matlab.png" NAME="图形1" ALIGN=BOTTOM WIDTH=23 HEIGHT=23 BORDER=0 ISMAP></FONT><A HREF="http://bioinfo.au.tsinghua.edu.cn/member/IGEM/matlab/2011Igem.zip" target="_self"><FONT COLOR="#ff9500"><I><U>Download
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<div style="position:relative;left:30px;display:inline"><A HREF="https://2011.igem.org/Team:Tsinghua-A/Modeling"><img src="https://static.igem.org/mediawiki/2011/b/b7/ThuA_A2.png" width="150px" height="100px"></A></div>
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the source code package (matlab) </U></I></FONT></A></SPAN></FONT>
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</P>
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<div style="position:relative;left:50px;display:inline"><A HREF="https://2011.igem.org/Team:Tsinghua-A/Modeling/P1A"><img src="https://static.igem.org/mediawiki/2011/0/0d/ThuA_B1.png" width="110px" height="100px"></A></div>
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<P ALIGN=LEFT> <FONT FACE="Arial Unicode MS, sans-serif"><SPAN LANG="en-US">&nbsp;&nbsp;In
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<div style="position:relative;left:70px;display:inline"><A HREF="https://2011.igem.org/Team:Tsinghua-A/Modeling/P2A"><img src="https://static.igem.org/mediawiki/2011/b/b6/ThuA_C1.png" width="110px" height="60px"></A></div>
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our project, we designed a quorum-sensing oscillator which consists
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of two types of cells. The expression of the reporter genes (GFP of
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<div style="position:relative;left:90px;display:inline"><A HREF="https://2011.igem.org/Team:Tsinghua-A/Modeling/P3A"><img src="https://static.igem.org/mediawiki/2011/6/6e/ThuA_D1.png" width="110px" height="60px"></A></div>
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one cell type and GFP of another) of the cells of the same type can
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fluctuate synchronously and certain designs were made to adjust the
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<div style="position:relative;left:110px;display:inline"><A HREF="https://2011.igem.org/Team:Tsinghua-A/Modeling/P4"><img src="https://static.igem.org/mediawiki/2011/6/6a/ThuA_E1.png" width="110px" height="120px"></A></div>
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phase and the period of oscillation.</SPAN></FONT></P>
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<P ALIGN=CENTER STYLE="text-indent: 0.74cm"><IMG SRC="https://static.igem.org/mediawiki/2011/0/03/001.png" NAME="图形2" ALIGN=BOTTOM WIDTH=511 HEIGHT=204 BORDER=0></P>
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<div class="slider" style="padding-buttom:20px">
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<P ALIGN=CENTER STYLE="text-indent: 0.74cm"><IMG SRC="https://static.igem.org/mediawiki/2011/b/b0/002.png" NAME="图形3" ALIGN=BOTTOM WIDTH=510 HEIGHT=173 BORDER=0></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 ALIGN=LEFT> <FONT FACE="Arial Unicode MS, sans-serif"><SPAN LANG="en-US">&nbsp;&nbsp;To
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<div class="nav">
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understand the property of our system, we built a mathematical model
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<table id="toc" class="toc">
-
based on ODEs (Ordinary Differential Equations) and DDEs (Delayed
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<tr>
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Differential Equations) to model and characterize this system. The
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<td>
-
simulation results helped us to deepen into further characteristics
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<div id="toctitle">
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of the system.</SPAN></FONT></P>
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<h2>Contents</h2>
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<P ALIGN=LEFT STYLE="text-indent: 0.74cm; border-top: none; border-bottom: 1px solid #000000; border-left: none; border-right: none; padding-top: 0cm; padding-bottom: 0.07cm; padding-left: 0cm; padding-right: 0cm">
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</div>
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<BR><BR>
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<ul>
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</P>
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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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<P ALIGN=LEFT STYLE="border-top: none; border-bottom: 1px solid #000000; border-left: none; border-right: none; padding-top: 0cm; padding-bottom: 0.07cm; padding-left: 0cm; padding-right: 0cm">
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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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<FONT FACE="Times New Roman, serif"><SPAN LANG="en-US"><FONT COLOR="#7030a0"><FONT FACE="Arial Unicode MS, sans-serif"><FONT SIZE=4 STYLE="font-size: 16pt"><SPAN LANG="en-US"><B>Original
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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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Full Model&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;                                                         
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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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      </B></SPAN></FONT></FONT></FONT><A HREF="https://2011.igem.org/Team:Tsinghua-A/Modeling/P1A"><FONT COLOR="#ff950e"><FONT FACE="Times New Roman, serif"><FONT SIZE=3><SPAN LANG="en-US"><I><U><SPAN STYLE="font-weight: normal">Read
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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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more</SPAN></U></I></SPAN></FONT></FONT></FONT></A></SPAN></FONT></P>
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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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<P ALIGN=LEFT><FONT COLOR="#ff0000">  <FONT FACE="Times New Roman, serif"><SPAN LANG="en-US"><FONT FACE="Arial Unicode MS, sans-serif"><SPAN LANG="en-US"><B>&nbsp;&nbsp;Firstly</B></SPAN></FONT></FONT>
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</ul>
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<FONT FACE="Arial Unicode MS, sans-serif"><FONT COLOR="#000000"><SPAN LANG="en-US">we
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</td>
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wanted to describe the system thoroughly without leaving out any
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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>
 +
<h1 id="Introduction">Introduction to Model</h1><hr width="100%" size=2 color=gray>
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<div class="imgbox">
 +
<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.
 +
</p>
 +
 
 +
<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>
 +
<P id="Original Full Model"><h1>Original Full Model</h1></P>
 +
<hr width="100%" size=2 color=gray>
 +
 
 +
<P>Firstly we
 +
described the system thoroughly without leaving out any
seemingly unimportant actions and factors. As a result, the
seemingly unimportant actions and factors. As a result, the
description of the system contains every possible mass actions as
description of the system contains every possible mass actions as
well as some hill kinetics, Henri-Michaelis-Menten kinetics, and the
well as some hill kinetics, Henri-Michaelis-Menten kinetics, and the
parameters were got from literature. The model was represented and
parameters were got from literature. The model was represented and
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simulated in the Matlab toolbox SIMBIOLOGY. We listed all 19 ODEs in
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simulated in the Matlab toolbox SIMBIOLOGY, but too many parameters make
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the attached pdf file, you can see more details there.</SPAN></FONT></SPAN></FONT></P>
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it difficult to do further analyses, So here we only listed all 19 ODEs  
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<P ALIGN=LEFT STYLE="border-top: none; border-bottom: 1px solid #000000; border-left: none; border-right: none; padding-top: 0cm; padding-bottom: 0.07cm; padding-left: 0cm; padding-right: 0cm">
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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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<BR><BR>
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</P>
</P>
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<P ALIGN=LEFT STYLE="border-top: none; border-bottom: 1px solid #000000; border-left: none; border-right: none; padding-top: 0cm; padding-bottom: 0.07cm; padding-left: 0cm; padding-right: 0cm">
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<FONT FACE="Times New Roman, serif"><SPAN LANG="en-US"><FONT COLOR="#7030a0"><FONT FACE="Arial Unicode MS, sans-serif"><FONT SIZE=4 STYLE="font-size: 16pt"><SPAN LANG="en-US"><B>Simplified
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<br>
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DDE Model&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;                                                                                                                       
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<div class="slider">
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</B></SPAN></FONT></FONT></FONT><A HREF="https://2011.igem.org/Team:Tsinghua-A/Modeling/P2A"><FONT COLOR="#ff950e"><FONT FACE="Times New Roman, serif"><FONT SIZE=3><SPAN LANG="en-US"><I><U><SPAN STYLE="font-weight: normal">Read
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<p><A HREF="https://2011.igem.org/Team:Tsinghua-A/Modeling/P2A"><U><I>Read
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more</SPAN></U></I></SPAN></FONT></FONT></FONT></A></SPAN></FONT></P>
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more</I></U></A></P>
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<P ALIGN=LEFT><FONT COLOR="#000000"><FONT FACE="Arial Unicode MS, sans-serif"><SPAN LANG="en-US">&nbsp;&nbsp;The
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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>
original model contains too many factors for analyzing the general
original model contains too many factors for analyzing the general
property of system. To understand the essential characters of the
property of system. To understand the essential characters of the
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oscillator, we <FONT COLOR="#ff0000"><B>simplify the original model</B></FONT><FONT COLOR="#000000">
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oscillator, we simplify the original model according to certain  
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according to certain appropriate assumptions, like Quasi-equilibrium
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appropriate assumptions, like Quasi-equilibrium
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for fast reactions.</SPAN></FONT></P>
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for fast reactions.</P>
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<P ALIGN=LEFT><FONT COLOR="#000000"><FONT FACE="Arial Unicode MS, sans-serif"><SPAN LANG="en-US">&nbsp;&nbsp;After
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<P>After series of derivation based on those assumptions, we came
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series of derivation based on those assumptions (see the attached pdf
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up with the following set of DDEs (Delay Differential Equations)
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file), we came up with the following set of DDEs (Delay Differential
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which contains only 6 equations, see the figure right. And it
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Equations)</SPAN></FONT></P>
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would be much more convenient for us to do some neccessary analyses
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<P ALIGN=CENTER><IMG SRC="https://static.igem.org/mediawiki/2011/b/b0/003.png" NAME="图形4" ALT="说明: C:\Users\Harry\AppData\Roaming\Tencent\Users\806995810\QQ\WinTemp\RichOle\RH%W~L%6JQ)ZX06RH76VYV3.jpg" ALIGN=BOTTOM WIDTH=531 HEIGHT=312 BORDER=0></P>
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and research into the mathematical essence of our model.
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<P ALIGN=LEFT><FONT COLOR="#000000"><FONT FACE="Times New Roman, serif"><SPAN LANG="en-US"><FONT FACE="Arial Unicode MS, sans-serif"><SPAN LANG="en-US">&nbsp;&nbsp;We
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<div class="imgbox2">
 +
<img class="border" style="border-color:#B2B2B2;"src="https://static.igem.org/mediawiki/2011/d/d5/004.png" width = "440px" height="310px"/>
 +
<p class="cite">Figure shows all variables are oscillating</p>
 +
</div>
 +
 
 +
<P>We
coded the system by DDE description in MATLAB and did simulation
coded the system by DDE description in MATLAB and did simulation
analysis accordingly. The result showed that the system could
analysis accordingly. The result showed that the system could
-
oscillate under certain parameters .</SPAN></FONT></SPAN></FONT></P>
+
oscillate under certain parameters.</P>
-
<P ALIGN=CENTER><IMG SRC="https://static.igem.org/mediawiki/2011/d/d5/004.png" NAME="图形5" ALT="说明: D:\b1.bmp" ALIGN=BOTTOM WIDTH=395 HEIGHT=296 BORDER=0></P>
+
<P>To
-
<P ALIGN=LEFT><FONT COLOR="#000000"><FONT FACE="Arial Unicode MS, sans-serif"><SPAN LANG="en-US">&nbsp;&nbsp;To
+
further understand what parameters could make the system oscillate,
further understand what parameters could make the system oscillate,
we did bifurcation analysis on the Hill parameters. What we had to do
we did bifurcation analysis on the Hill parameters. What we had to do
was find the critical points where the system can nearly oscillate
was find the critical points where the system can nearly oscillate
-
but a little disruption may lead to a steady state like that:</SPAN></FONT></P>
+
but a little disruption may lead to a steady state.</P>
-
<P ALIGN=CENTER><IMG SRC="https://static.igem.org/mediawiki/2011/2/20/005.png" NAME="图形6" ALT="说明: C:\Users\Harry\AppData\Roaming\Tencent\Users\806995810\QQ\WinTemp\RichOle\)D6IYP[O_UD]}LB_9)SO(Q5.jpg" ALIGN=BOTTOM WIDTH=557 HEIGHT=260 BORDER=0></P>
+
<P>Depicting
-
<P ALIGN=LEFT><FONT COLOR="#000000"><FONT FACE="Arial Unicode MS, sans-serif"><SPAN LANG="en-US">&nbsp;&nbsp;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&rsquo;s Hiill parameters were located in the
+
oscillate when cellB's Hill parameters were located in the  
-
<FONT COLOR="#00b0f0"><B>area</B></FONT> named <FONT COLOR="#00b0f0">&lsquo;</FONT><FONT COLOR="#00b0f0"><I><B>Bistable</B></I></FONT><FONT COLOR="#00b0f0"><B>&rsquo;</B></FONT>.</SPAN></FONT></P>
+
area named <FONT COLOR="#00b0f0"></FONT><FONT COLOR="#00b0f0"><I><B>Bistable</B></I></FONT><FONT COLOR="#00b0f0"></B></I>.</P>
-
<P ALIGN=CENTER><IMG SRC="https://static.igem.org/mediawiki/2011/1/18/006.png" NAME="图形7" ALT="说明: D:\b2.bmp" ALIGN=BOTTOM WIDTH=438 HEIGHT=329 BORDER=0></P>
+
 
-
<P ALIGN=LEFT><FONT COLOR="#000000"><FONT FACE="Arial Unicode MS, sans-serif"><SPAN LANG="en-US">&nbsp;&nbsp;By
+
<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"/>
 +
<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.</p>
 +
</div>
 +
<div class="imgbox4">
 +
<img class="border" style="border-color:#B2B2B2;"src="https://static.igem.org/mediawiki/2011/1/18/006.png" width = "370px" height="300px"/>
 +
<p class="cite">Our system oscillates when parameters were selected in the area named 'Bistable'</p>
 +
</div>
 +
 
 +
<div class="imgbox5">
 +
<img class="border" style="border-color:#B2B2B2;"src="https://static.igem.org/mediawiki/2011/e/e8/007.png" width = "370px" height="280px"/>
 +
</div>
 +
<br><br>
 +
<div class="temp"><P>By
adjusting certain parameters, we saw that the oscillation&rsquo;s
adjusting certain parameters, we saw that the oscillation&rsquo;s
period and phase could be controlled properly, which is the most
period and phase could be controlled properly, which is the most
Line 197: Line 346:
pBad promoter, in cell type B. After adding araC to our system at
pBad promoter, in cell type B. After adding araC to our system at
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.</SPAN></FONT></P>
+
gradually recover and finally, the phase was changed.</P></div>
-
<P ALIGN=CENTER STYLE="text-indent: 0.74cm"><IMG SRC="https://static.igem.org/mediawiki/2011/e/e8/007.png" NAME="图形8" ALIGN=BOTTOM WIDTH=380 HEIGHT=285 BORDER=0></P>
+
<br><br><br><br><br><br>
-
<P ALIGN=LEFT STYLE="text-indent: 0.74cm; border-top: none; border-bottom: 1px solid #000000; border-left: none; border-right: none; padding-top: 0cm; padding-bottom: 0.07cm; padding-left: 0cm; padding-right: 0cm">
+
 
-
<BR><BR>
+
<div class="slider">
-
</P>
+
<p><A HREF="https://2011.igem.org/Team:Tsinghua-A/Modeling/P3A"><U><I>Read
-
<P ALIGN=LEFT STYLE="border-top: none; border-bottom: 1px solid #000000; border-left: none; border-right: none; padding-top: 0cm; padding-bottom: 0.07cm; padding-left: 0cm; padding-right: 0cm">
+
more</I></U></A></P>
-
<FONT FACE="Times New Roman, serif"><SPAN LANG="en-US"><FONT COLOR="#7030a0"><FONT FACE="Arial Unicode MS, sans-serif"><FONT SIZE=4 STYLE="font-size: 16pt"><SPAN LANG="en-US"><B>Dimensionless
+
</div>
-
model&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;                                                           
+
 
-
</B></SPAN></FONT></FONT></FONT><A HREF="https://2011.igem.org/Team:Tsinghua-A/Modeling/P3A"><FONT COLOR="#ff950e"><FONT FACE="Times New Roman, serif"><FONT SIZE=3><SPAN LANG="en-US"><I><U><SPAN STYLE="font-weight: normal">Read
+
<P id="Dimensionless Model"><h1>Dimensionless Model</h1></P>
-
more</SPAN></U></I></SPAN></FONT></FONT></FONT></A></SPAN></FONT></P>
+
<hr width="100%" size=2 color=gray>
-
<P ALIGN=LEFT>  <FONT FACE="Arial Unicode MS, sans-serif"><SPAN LANG="en-US">&nbsp;&nbsp;In
+
 
 +
<P>In
order to make a further analysis on stability of the system, and
order to make a further analysis on stability of the system, and
sensitivity of parameters, we further simplified the model to make
sensitivity of parameters, we further simplified the model to make
-
<FONT COLOR="#ff0000"><B>them dimensionless</B></FONT>. In addition,
+
them dimensionless. In addition, we tried to introduce feedback to
-
we tried to introduce feedback to our system and made a brief
+
our system and made a brief analysis on different types of  
-
analysis on different types of feedback we introduced.</SPAN></FONT></P>
+
feedback we introduced. Some analyses were similar to the simplified
-
<P ALIGN=LEFT STYLE="border-top: none; border-bottom: 1px solid #000000; border-left: none; border-right: none; padding-top: 0cm; padding-bottom: 0.07cm; padding-left: 0cm; padding-right: 0cm">
+
DDE model, and you can see more details by clicking
-
<BR><BR>
+
<A HREF="https://2011.igem.org/Team:Tsinghua-A/Modeling/P4">read
-
</P>
+
more</A></P>
-
<P ALIGN=LEFT STYLE="border-top: none; border-bottom: 1px solid #000000; border-left: none; border-right: none; padding-top: 0cm; padding-bottom: 0.07cm; padding-left: 0cm; padding-right: 0cm">
+
 
-
<FONT FACE="Times New Roman, serif"><SPAN LANG="en-US"><FONT COLOR="#7030a0"><FONT FACE="Arial Unicode MS, sans-serif"><FONT SIZE=4 STYLE="font-size: 16pt"><SPAN LANG="en-US"><B>Quorum
+
<br>
-
Sensing Effect&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;                                                                                                                 
+
<div class="slider">
-
  </B></SPAN></FONT></FONT></FONT><A HREF="https://2011.igem.org/Team:Tsinghua-A/Modeling/P4"><FONT COLOR="#ff950e"><FONT FACE="Times New Roman, serif"><FONT SIZE=3><SPAN LANG="en-US"><I><U><SPAN STYLE="font-weight: normal">Read
+
<p><A HREF="https://2011.igem.org/Team:Tsinghua-A/Modeling/P4"><U><I>Read
-
more</SPAN></U></I></SPAN></FONT></FONT></FONT></A></SPAN></FONT></P>
+
more</I></U></A></P>
-
<P ALIGN=LEFT>  <FONT FACE="Arial Unicode MS, sans-serif"><SPAN LANG="en-US">&nbsp;&nbsp;What
+
</div>
 +
<P id="Quorum Sensing"><h1>Quorum Sensing Effect</h1></P>
 +
<hr width="100%" size=2 color=gray>
 +
 
 +
<P>What
we have analyzed so far is focused on two-cell oscillation.
we have analyzed so far is focused on two-cell oscillation.
-
<FONT COLOR="#ff0000"><B>Quorum-sensing oscillator</B></FONT> is not
+
Quorum-sensing oscillator is not
simply a matter of expansion in magnitude, but a matter of robustness
simply a matter of expansion in magnitude, but a matter of robustness
in allowing difference of each individual cell. Moreover, we test the
in allowing difference of each individual cell. Moreover, we test the
-
adjustment of phase and period of oscillation in this part.</SPAN></FONT></P>
+
adjustment of phase and period of oscillation in this part.</P>
-
<P ALIGN=LEFT>  <FONT FACE="Arial Unicode MS, sans-serif"><SPAN LANG="en-US">&nbsp;&nbsp;As
+
<P>As
we all know, no two things in this world are exactly the same, so do
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
cells. The major differences between individual cells that we take
-
into consideration include:</SPAN></FONT></P>
+
into consideration include:</P>
-
<P ALIGN=LEFT STYLE="margin-left: 0.64cm"><FONT COLOR="#548dd4">●
+
<P><B>●Each cell's activity of promoter is varied, so each cell has
-
</FONT><FONT FACE="Times New Roman, serif"><SPAN LANG="en-US"><FONT COLOR="#548dd4"><FONT FACE="Arial Unicode MS, sans-serif"><SPAN LANG="en-US"><B>Each
+
different rate to generate AHL.</B></P>
-
cell&rsquo;s activity of promoter is varied, so each cell has
+
<P><B>●The initial amount of AHL may be disproportionally distributed among
-
different rate to generate AHL.</B></SPAN></FONT></FONT></SPAN></FONT></P>
+
cells.</B></P>
-
<P ALIGN=LEFT STYLE="text-indent: 0.64cm"><FONT COLOR="#548dd4">●
+
<P>The rate of generating AHL is closely related to parameter m and n.
-
</FONT><FONT FACE="Times New Roman, serif"><SPAN LANG="en-US"><FONT COLOR="#548dd4"><FONT FACE="Arial Unicode MS, sans-serif"><SPAN LANG="en-US"><B>The
+
-
initial amount of AHL may be disproportionally distributed among
+
-
cells.</B></SPAN></FONT></FONT></SPAN></FONT></P>
+
-
<P ALIGN=LEFT><A NAME="OLE_LINK67"></A><A NAME="OLE_LINK66"></A>  <FONT FACE="Times New Roman, serif"><SPAN LANG="en-US"><FONT FACE="Arial Unicode MS, sans-serif"><SPAN LANG="en-US">&nbsp;&nbsp;The
+
-
rate of generating AHL is closely related to parameter m and n.
+
Therefore, we introduce randomness to both parameters by letting them
Therefore, we introduce randomness to both parameters by letting them
-
obey normal distribution, that is: </SPAN></FONT></SPAN></FONT>
+
obey normal distribution, that is:
</P>
</P>
-
<P ALIGN=CENTER><A NAME="OLE_LINK77"></A><A NAME="OLE_LINK76"></A><FONT FACE="Arial Unicode MS, sans-serif"><SPAN LANG="en-US">m(i)
+
<P ALIGN=CENTER style="text-intent:0em">m(i)= &mu;1+<I>N</I>(0,&sigma;1);</P>
-
= &mu;1+<I>N</I>(0,&sigma;1);</SPAN></FONT></P>
+
<P ALIGN=CENTER style="text-intent:0em">n(i)= &mu;2+<I>N</I>(0,&sigma;2);</P>
-
<P ALIGN=CENTER><FONT FACE="Times New Roman, serif"><SPAN LANG="en-US"><FONT FACE="Arial Unicode MS, sans-serif"><SPAN LANG="en-US">n(i)
+
<P ALIGN=LEFT><A NAME="OLE_LINK63"></A><A NAME="OLE_LINK62"></A>&mu;<SPAN LANG="en-US">1
-
= &mu;2+ </SPAN></FONT><FONT FACE="Arial Unicode MS, sans-serif"><SPAN LANG="en-US"><I>N</I></SPAN></FONT><FONT FACE="Arial Unicode MS, sans-serif"><SPAN LANG="en-US">(0,&sigma;2);</SPAN></FONT>
+
-
</SPAN></FONT>
+
-
</P>
+
-
<P ALIGN=LEFT><A NAME="OLE_LINK63"></A><A NAME="OLE_LINK62"></A> <FONT FACE="Arial Unicode MS, sans-serif"><SPAN LANG="en-US">&nbsp;&nbsp;&mu;<SPAN LANG="en-US">1
+
and &mu;2 are the average ability of generating 30C6HSL and 3012CHSL,
and &mu;2 are the average ability of generating 30C6HSL and 3012CHSL,
and normal distribution-- </SPAN><SPAN LANG="en-US"><I>N</I></SPAN><SPAN LANG="en-US">(0,&sigma;)--describes
and normal distribution-- </SPAN><SPAN LANG="en-US"><I>N</I></SPAN><SPAN LANG="en-US">(0,&sigma;)--describes
Line 256: Line 401:
expanded our equations from 2 cells to a population of cells. Each
expanded our equations from 2 cells to a population of cells. Each
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></FONT></P>
+
environment is proportionally distributed.</SPAN></SPAN></P>
-
<P ALIGN=CENTER><IMG SRC="https://static.igem.org/mediawiki/2011/1/17/008.png" NAME="图形11" ALT="说明: D:\b1.bmp" ALIGN=BOTTOM WIDTH=579 HEIGHT=430 BORDER=0></P>
+
<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 ALIGN=LEFT>  <FONT FACE="Arial Unicode MS, sans-serif"><SPAN LANG="en-US">&nbsp;&nbsp;The
+
<P>The
figures indicate that our system can oscillate synchronically being
figures indicate that our system can oscillate synchronically being
able to tolerate differences at certain range among a population of
able to tolerate differences at certain range among a population of
-
cells.</SPAN></FONT></P>
+
cells.</P>
-
<P ALIGN=LEFT>  <FONT FACE="Arial Unicode MS, sans-serif"><SPAN LANG="en-US">&nbsp;&nbsp;We
+
<P>We
also tested whether the oscillation is dependent on initial
also tested whether the oscillation is dependent on initial
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:</SPAN></FONT></P>
+
letting them follow uniform distribution. That is:</P>
-
<P ALIGN=CENTER><FONT FACE="Arial Unicode MS, sans-serif"><SPAN LANG="en-US">Initial(i)
+
<P ALIGN="center" style="text-intent:0em">Initial(i)= <I>U</I>(0,20);</P>
-
= <I>U</I>(0,20);</SPAN></FONT></P>
+
<P>Based
-
<P ALIGN=LEFT> <FONT FACE="Arial Unicode MS, sans-serif"><SPAN LANG="en-US">&nbsp;&nbsp;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.</SPAN></FONT></P>
+
each cell, we simulated out a figure as follows.</P>
-
<P ALIGN=CENTER STYLE="text-indent: 0.74cm"><IMG SRC="https://static.igem.org/mediawiki/2011/d/d2/009.png" NAME="图形12" ALT="说明: D:\b1.bmp" ALIGN=BOTTOM WIDTH=532 HEIGHT=399 BORDER=0></P>
+
<P ALIGN="center" style="text-intent:0em"><IMG SRC="https://static.igem.org/mediawiki/2011/d/d2/009.png" WIDTH=800 HEIGHT=600 BORDER=0></P>
-
<P ALIGN=LEFT>  <FONT FACE="Arial Unicode MS, sans-serif"><SPAN LANG="en-US">&nbsp;&nbsp;The
+
<P>The
results demonstratively give evidence proving that our system can
results demonstratively give evidence proving that our system can
start to oscillate synchronically given variant initial starting
start to oscillate synchronically given variant initial starting
-
status.</SPAN></FONT></P>
+
status.</P>
-
<P ALIGN=CENTER><A HREF="https://static.igem.org/mediawiki/2011/8/8c/Tsinghua_A_Matlab_Code.zip" target="_self"><FONT COLOR="#ff950e"><U>Click here to download
+
<br>
-
MATLAB code </U></FONT></A></SPAN></FONT>
+
 
-
<P ALIGN=LEFT STYLE="border-top: none; border-bottom: 1px solid #000000; border-left: none; border-right: none; padding-top: 0cm; padding-bottom: 0.07cm; padding-left: 0cm; padding-right: 0cm">
+
<P id="References"><h1>References</h1></P>
-
<BR><BR>
+
<hr width="100%" size=2 color=gray>
-
</P>
+
<P>[1]
-
<P ALIGN=LEFT><FONT FACE="Arial Unicode MS, sans-serif"><SPAN LANG="en-US"><FONT COLOR="#548dd4"><FONT SIZE=6 STYLE="font-size: 22pt"><B>References</B></FONT></FONT></SPAN></FONT></P>
+
-
<P ALIGN=LEFT><FONT FACE="Arial Unicode MS, sans-serif"><SPAN LANG="en-US">[1]
+
Uri Alon, (2007). Network motifs: theory and experimental approaches.
Uri Alon, (2007). Network motifs: theory and experimental approaches.
-
Nature.</SPAN></FONT></P>
+
Nature.</P>
-
<P ALIGN=LEFT><FONT FACE="Arial Unicode MS, sans-serif"><SPAN LANG="en-US">[2]
+
<P>[2]
Chunbo Lou, Xili Liu, Ming Ni, et al. (2010). Synthesizing a novel
Chunbo Lou, Xili Liu, Ming Ni, et al. (2010). Synthesizing a novel
genetic sequential logic circuit: a push-on push-off switch.
genetic sequential logic circuit: a push-on push-off switch.
-
Molecular Systems Biology.</SPAN></FONT></P>
+
Molecular Systems Biology.</P>
-
<P ALIGN=LEFT><FONT FACE="Arial Unicode MS, sans-serif"><SPAN LANG="en-US">[3]
+
<P>[3]
Tal Danino, Octavio Mondragon-Palomino, Lev Tsimring &amp; Jeff Hasty
Tal Danino, Octavio Mondragon-Palomino, Lev Tsimring &amp; Jeff Hasty
-
(2010). A synchronized quorum of genetic clocks. Nature.</SPAN></FONT></P>
+
(2010). A synchronized quorum of genetic clocks. Nature.</P>
-
<P ALIGN=LEFT><FONT FACE="Arial Unicode MS, sans-serif"><SPAN LANG="en-US">[4]
+
<P>[4]
Marcel Tigges, Tatiana T. Marquez-Lago, Jorg Stelling &amp; Martin
Marcel Tigges, Tatiana T. Marquez-Lago, Jorg Stelling &amp; Martin
-
Fussenegger (2009). A tunable synthetic mammalian oscillator. Nature.</SPAN></FONT></P>
+
Fussenegger (2009). A tunable synthetic mammalian oscillator. Nature.</P>
-
<P ALIGN=LEFT><FONT FACE="Arial Unicode MS, sans-serif"><SPAN LANG="en-US">[5]
+
<P>[5]
Sergi Regot, Javio Macia el al. (2010). Distributed biological
Sergi Regot, Javio Macia el al. (2010). Distributed biological
-
computation with multicellular engineered networks. Nature.</SPAN></FONT></P>
+
computation with multicellular engineered networks. Nature.</P>
-
<P ALIGN=LEFT><FONT FACE="Arial Unicode MS, sans-serif"><SPAN LANG="en-US">[6]
+
<P>[6]
-
Martin Fussenegger, (2010). Synchronized bacterial clocks. Nature.</SPAN></FONT></P>
+
Martin Fussenegger, (2010). Synchronized bacterial clocks. Nature.</P>
-
<P ALIGN=LEFT><FONT FACE="Arial Unicode MS, sans-serif"><SPAN LANG="en-US">[7]
+
<P>[7]
Andrew H Babiskin and Christina D Smolke, (2011). A synthetic library
Andrew H Babiskin and Christina D Smolke, (2011). A synthetic library
of RNA control modules for predictable tuning of gene expression in
of RNA control modules for predictable tuning of gene expression in
-
yeast. Molecular Systems Biology.</SPAN></FONT></P>
+
yeast. Molecular Systems Biology.</P>
-
<P ALIGN=LEFT><FONT FACE="Arial Unicode MS, sans-serif"><SPAN LANG="en-US">[8]
+
<P>[8]
Santhosh Palani and Casim A Sarkar, (2011). Synthetic conversion of a
Santhosh Palani and Casim A Sarkar, (2011). Synthetic conversion of a
graded receptor signal into a tunable, reversible switch. Molecular
graded receptor signal into a tunable, reversible switch. Molecular
-
Systems Biology.</SPAN></FONT></P>
+
Systems Biology.</P>
-
<P ALIGN=LEFT><FONT FACE="Times New Roman, serif"><SPAN LANG="en-US"><FONT FACE="Arial Unicode MS, sans-serif"><SPAN LANG="en-US">[9]
+
<P>[9]
Nancy Kopell, (2002). Synchronizing genetic relaxation oscillation by
Nancy Kopell, (2002). Synchronizing genetic relaxation oscillation by
-
intercell signaling. PNS</SPAN></FONT></SPAN></FONT></P>
+
intercell signaling. PNS</P>
<br><br><br><br>
<br><br><br><br>
-
<p align="CENTER"><a href="https://2011.igem.org/Team:Tsinghua-A"><img src="https://static.igem.org/mediawiki/2011/9/92/Killbanner_header.jpg" alt="" width="960"/></p>
+
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Latest revision as of 05:20, 28 October 2011

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Introduction to Model


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


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.

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.

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.

Our system oscillates when parameters were selected in the area named 'Bistable'



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


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