Team:Tsinghua-A/Modeling/P1A

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<div style="position:relative;left:60px;display:inline"><A HREF="https://2011.igem.org/Team:Tsinghua-A/Modeling"><img src="https://static.igem.org/mediawiki/2011/2/2f/ThuA_A1.png" width="110px" height="105px"></A></div>
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<div style="position:relative;left:80px;display:inline"><A HREF="https://2011.igem.org/Team:Tsinghua-A/Modeling/P1A"><img src="https://static.igem.org/mediawiki/2011/6/63/ThuA_B2.png" width="130px" height="120px"></A></div>
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<div style="position:relative;left:100px;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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<div style="position:relative;left:120px;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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<div style="position:relative;left:140px;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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<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-1"><a href="#Construction"><span class="tocnumber">1</span> <span class="toctext">Construction of ODE equation</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-2"><a href="#Parameters"><span class="tocnumber">2</span> <span class="toctext">Parameters</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-3"><a href="#Results"><span class="tocnumber">3</span> <span class="toctext">Results</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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<h1 id="Construction">Construction of ODE equation</h1><hr width="100%" size=2 color=gray>
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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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<p>At our first step, we wanted to describe 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. We came up a set of ODEs with 19 equations.</p>
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<h1 id="Introduction">Introduction to Model</h1><hr width="100%" size=2 color=gray>
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<p class="cite">Figure 1 designed circuit of cell I</p>
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<p class="cite">Designed gene circuit in cell A and cell B</p>
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<img style="border-color:#B2B2B2;"src="https://static.igem.org/mediawiki/2011/b/b0/002.png" width = "430px" height="150px" />
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<p class="cite">Figure 2 designed circuit of cell II</p>
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<p>Promoter 1 and promoter 2 preceding lasR and luxR genes respectively are constant promoters, which will transcribe and translate into protein PlasR and PluxR. LA1 is the binding association of lasR and 30C12HSL(A2C1) and it can affect the subsequent promoter 2 which can be described by Hill Equation. The same goes to LA2. Gene luxI will be translated into protein PluxI which would generate 30C6HSL(A1C1) through enzymatic reaction. The AHL will diffuse through the membrane to the environment(A1e) and finally enter into Cell 2(A1C2). Protein PtetR which is translated from gene tetR represses promoter 5 which is responsible for transcription of gene lasI. Promoter 6 is constant for translation of protein PlasI. 30C12HSL(A2C2) is generated from Protein PlasI through enzymatic reaction. 30C12HSL in the environment is called A2e which will diffuse to Cell 1. aTc is added manipulatively to change the phase of oscillation by binding the protein PTetR. Therefore, we have these following ODEs:<br><br></p>
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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><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>
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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><A HREF="https://2011.igem.org/Team:Tsinghua-A/Modeling/P2A"><U><I>Read
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<P id="Simplified DDE Model"><h1>Simplified DDE</h1></P>
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<p class="cite">Simplified DDEs</p>
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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><img style="border-color:#B2B2B2;"src="https://static.igem.org/mediawiki/igem.org/9/9a/ThuAModel_1_1.png" width = "707px" height="1115px" /><br></p>
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<p class="cite">Figure shows all variables are oscillating</p>
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<h1 id="Parameters">Parameters</h1><hr width="100%" size=2 color=gray>
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<P>We
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<p>The parameters are inherent factors determining the behaviors, properties of a system. We selected the quantities thoughtfully from previous iGEM teams and some others were found from published papers.</p>
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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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<p align="center"><img style="border-color:#B2B2B2;"src="https://static.igem.org/mediawiki/igem.org/c/c8/ThuAModel_1_2.png" width = "746px" height="1794px" /><br></p>
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<p class="cite">The figure right is the phase trajectory of two signal molecules in environment,
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<p class="cite">Table 1 Parameters of ODEs</p>
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the more it looks like a circle, the more steadily our system will oscillate.</p>
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<h1 id="Results">Results</h1><hr width="100%" size=2 color=gray>
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<p class="cite">Our system oscillates when parameters were selected in the area named 'Bistable'</p>
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<p>We simulated this system by SIMBIOLOGY, a toolbox embedded in MATLAB. However, unaware of the key parameters to which the system is sensitive, we felt difficult to control or adjust properly, and the simulation result of the system came into a damped oscillation. We ascribed the inability of our model to the fact that the precise descriptions contain too many equations and parameters and we felt obliged to establish a simplified model in place of the precise one for simulation and further analysis.</p>
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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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<p><A HREF="https://2011.igem.org/Team:Tsinghua-A/Modeling/P3A"><U><I>Read
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<P id="Dimensionless Model"><h1>Dimensionless Model</h1></P>
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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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<p><A HREF="https://2011.igem.org/Team:Tsinghua-A/Modeling/P4"><U><I>Read
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<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.
+
-
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>
+
-
<P>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:</P>
+
-
<P><B>●Each cell's activity of promoter is varied, so each cell has
+
-
different rate to generate AHL.</B></P>
+
-
<P><B>●The initial amount of AHL may be disproportionally distributed among
+
-
cells.</B></P>
+
-
<P>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:
+
-
</P>
+
-
<P ALIGN=CENTER style="text-intend:0em">m(i)= &mu;1+<I>N</I>(0,&sigma;1);</P>
+
-
<P ALIGN=CENTER style="text-intend:0em">n(i)= &mu;2+<I>N</I>(0,&sigma;2);</P>
+
-
<P ALIGN=LEFT><A NAME="OLE_LINK63"></A><A NAME="OLE_LINK62"></A>&mu;<SPAN LANG="en-US">1
+
-
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
+
-
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.</SPAN></SPAN></P>
+
-
<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>
+
-
<P>The
+
-
figures indicate that our system can oscillate synchronically being
+
-
able to tolerate differences at certain range among a population of
+
-
cells.</P>
+
-
<P>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:</P>
+
-
<P ALIGN="center" style="text-intend:0em">Initial(i)= <I>U</I>(0,20);</P>
+
-
<P>Based
+
-
on this distribution restraining the initial AHL concentration in
+
-
each cell, we simulated out a figure as follows.</P>
+
-
<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>
+
-
<P>The
+
-
results demonstratively give evidence proving that our system can
+
-
start to oscillate synchronically given variant initial starting
+
-
status.</P>
+
-
<br>
+
-
 
+
-
<P id="References"><h1>References</h1></P>
+
-
<hr width="100%" size=2 color=gray>
+
-
<P>[1]
+
-
Uri Alon, (2007). Network motifs: theory and experimental approaches.
+
-
Nature.</P>
+
-
<P>[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.</P>
+
-
<P>[3]
+
-
Tal Danino, Octavio Mondragon-Palomino, Lev Tsimring &amp; Jeff Hasty
+
-
(2010). A synchronized quorum of genetic clocks. Nature.</P>
+
-
<P>[4]
+
-
Marcel Tigges, Tatiana T. Marquez-Lago, Jorg Stelling &amp; Martin
+
-
Fussenegger (2009). A tunable synthetic mammalian oscillator. Nature.</P>
+
-
<P>[5]
+
-
Sergi Regot, Javio Macia el al. (2010). Distributed biological
+
-
computation with multicellular engineered networks. Nature.</P>
+
-
<P>[6]
+
-
Martin Fussenegger, (2010). Synchronized bacterial clocks. Nature.</P>
+
-
<P>[7]
+
-
Andrew H Babiskin and Christina D Smolke, (2011). A synthetic library
+
-
of RNA control modules for predictable tuning of gene expression in
+
-
yeast. Molecular Systems Biology.</P>
+
-
<P>[8]
+
-
Santhosh Palani and Casim A Sarkar, (2011). Synthetic conversion of a
+
-
graded receptor signal into a tunable, reversible switch. Molecular
+
-
Systems Biology.</P>
+
-
<P>[9]
+
-
Nancy Kopell, (2002). Synchronizing genetic relaxation oscillation by
+
-
intercell signaling. PNS</P>
+
-
 
+
-
<br><br><br><br>
+
-
<p style="text-indent:0em" 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"/><a href="https://2011.igem.org"><img src="https://static.igem.org/mediawiki/igem.org/2/29/Killbanner_header2.jpg" alt="" width="960"/></p>
+
 +
<p align="CENTER" style="text-indent:0em"><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"/><a href="https://2011.igem.org"><img src="https://static.igem.org/mediawiki/igem.org/2/29/Killbanner_header2.jpg" alt="" width="960"/></p>
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Contents

Construction of ODE equation

At our first step, we wanted to describe 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. We came up a set of ODEs with 19 equations.

Figure 1 designed circuit of cell I

Figure 2 designed circuit of cell II

Promoter 1 and promoter 2 preceding lasR and luxR genes respectively are constant promoters, which will transcribe and translate into protein PlasR and PluxR. LA1 is the binding association of lasR and 30C12HSL(A2C1) and it can affect the subsequent promoter 2 which can be described by Hill Equation. The same goes to LA2. Gene luxI will be translated into protein PluxI which would generate 30C6HSL(A1C1) through enzymatic reaction. The AHL will diffuse through the membrane to the environment(A1e) and finally enter into Cell 2(A1C2). Protein PtetR which is translated from gene tetR represses promoter 5 which is responsible for transcription of gene lasI. Promoter 6 is constant for translation of protein PlasI. 30C12HSL(A2C2) is generated from Protein PlasI through enzymatic reaction. 30C12HSL in the environment is called A2e which will diffuse to Cell 1. aTc is added manipulatively to change the phase of oscillation by binding the protein PTetR. Therefore, we have these following ODEs:


Parameters

The parameters are inherent factors determining the behaviors, properties of a system. We selected the quantities thoughtfully from previous iGEM teams and some others were found from published papers.


Table 1 Parameters of ODEs




Results

We simulated this system by SIMBIOLOGY, a toolbox embedded in MATLAB. However, unaware of the key parameters to which the system is sensitive, we felt difficult to control or adjust properly, and the simulation result of the system came into a damped oscillation. We ascribed the inability of our model to the fact that the precise descriptions contain too many equations and parameters and we felt obliged to establish a simplified model in place of the precise one for simulation and further analysis.




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