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<P><A HREF="https://2011.igem.org/Team:Tsinghua-A/Modeling"><FONT FACE="Arial, sans-serif"><SPAN LANG="en-US"><FONT SIZE=5 STYLE="font-size: 20pt">Modeling</A>::</FONT><FONT COLOR="#0099ff"><FONT SIZE=5 STYLE="font-size: 20pt">Dimensionless Model</FONT></FONT></SPAN></FONT></P>
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<h2>Contents</h2>
 
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<ul>
 
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<li class="toclevel-1 tocsection-1"><a href="#Dimensionless process"><span class="tocnumber">1</span> <span class="toctext">Dimensionless process</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="#Results"><span class="tocnumber">3</span> <span class="toctext">Results</span></a>
 
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<ul>
 
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<li class="toclevel-2 tocsection-1"><a href="#Sensitivity analysis"><span class="tocnumber">3.1</span> <span class="toctext">Sensitivity analysis</span></a></li>
 
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<li class="toclevel-2 tocsection-2"><a href="#Stability analysis"><span class="tocnumber">3.2</span> <span class="toctext">Stability analysis</span></a></li>
 
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<li class="toclevel-2 tocsection-3"><a href="#Feedback analysis"><span class="tocnumber">3.3</span> <span class="toctext">Feedback analysis</span></a></li>
 
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<br>
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<h1 id="Dimensionless process">Dimensionless process</h1><hr width="100%" size=2 color=gray>
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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>In order to make a further analysis on stability of the system, sensitivity of parameters,
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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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feedback factors-we manipulate all the arguments and parameters to make them dimensionless.
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<P><B>●Each cell's activity of promoter is varied, so each cell has
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Analysis of this part is crucial since parameters in vivo experiment may be different and
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different rate to generate AHL.</B></P>
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even at odds with modeling ones but a proper dimensionless can reveal the mathematical essence
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<P><B>●The initial amount of AHL may be disproportionally distributed among
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of our model.</p>
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cells.</B></P>
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<p> Considering the Hill equation in the simplification DDEs, A1<sub>c1</sub> and K<sub>M1</sub>/ρ<sub>1</sub>  should be the same order of magnitude, thus K<sub>M1</sub>/ρ<sub>1</sub> is a well measurement of quantities of A1<sub>c1</sub>. We have:</p>
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<P>The rate of generating AHL is closely related to parameter m and n.
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<p align="LEFT" style="text-indent:1.2em"><img src="https://static.igem.org/mediawiki/2011/b/be/Part3-1.png" width="705px" height="1258px"></p>
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Therefore, we introduce randomness to both parameters by letting them
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<h1 id="Parameters">Parameters</h1><hr width="100%" size=2 color=gray>
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obey normal distribution, that is:
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<p>Parameters in equations are listed below.</p>
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</P>
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<p align="CENTER" style="text-indent:0em"><img src="https://static.igem.org/mediawiki/2011/2/22/Part3-2.png" width="753px" height="355px"></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-indent:0em" class="cite">Table 4 Parameters in Dimensionless Model</p>
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<P ALIGN=CENTER>n(i)= &mu;2+<I>N</I>(0,&sigma;2);</P>
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<h1 id="Results">Results</h1><hr width="100%" size=2 color=gray>
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<P>&mu;1 and &mu;2 are the average ability of generating 30C6HSL and 3012CHSL,
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<h2 id="Sensitivity analysis">Sensitivity analysis</h2>
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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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<p>In order to find out the key parameters which will affect stability of the system at most, we need to make a sensitivity analysis on each. At first, we did brief and instinctive analyses on each parameter as follows. τ<sub>1</sub><sup>*</sup> and τ<sub>2</sub><sup>*</sup> represent time delay in cell 1 and cell 2 respectively, which have been discussed in part 2, have little influence on stability of system but intend to affect the oscillation period merely. Parameters a and b refer to feedback factors indirectly, which have not been discussed before, we will see how a and b affect our system later. We have clarified that parameter u is equivalent to μ/γ, thus, u is directly decided by the dilution rate of signal molecules 3OC12HSL and 3OC6HSL in environment and will inevitably influence stability of oscillation. As for m and n, they are inseparably connected to the Hill parameters whose sensitivity have been analyzed in part 2, so we can deduce that m and n are both sensitive parameters to our system.</p>
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<p>Here we mainly did sensitivity analyses on parameters m, n and u. Parameters were set fundamentally as Table 4 shows.</p>
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<p align="center" style="text-indent:0em"><img src="https://static.igem.org/mediawiki/igem.org/0/00/Part4-1.png" width = "900px" height="675px"></p>
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<p>Simulation results reveal that the system can oscillate stably only when u<5.3(fixed the other two sensitive parameters), 11.9<m<71.3 and n>34.4. In other words, to ensure the stability of oscillation, the dilution rate cannot be too high, while the promoter 2 and 4 which affect m and n should be chosen appropriately.</p>
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<p align="center" style="text-indent:0em"><b>Figure 16 100 Cells Varied in parameter m and n</b></p>
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<p align="CENTER" style="text-indent:0em"><img src="https://static.igem.org/mediawiki/igem.org/5/59/Part3-3.png" width="561px" height="421px"></p>
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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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<p align="center" class="cite">Figure 11 Sensitivity analyses results (On u)</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-indent:0em"><img src="https://static.igem.org/mediawiki/igem.org/5/5c/Part3-4.png" width="561px" height="421px"></p>
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<P ALIGN=CENTER>Initial(i)= <I>U</I>(0,20);</P>
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<p align="center" class="cite">Figure 12 Sensitivity analyses results (On m)</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 align="CENTER" style="text-indent:0em"><img src="https://static.igem.org/mediawiki/igem.org/6/60/Part3-5.png" width="561px" height="421px"></p>
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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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<p align="center" class="cite">Figure 13 Sensitivity analyses results (On n)</p>
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<p align="center" style="text-indent:0em"><img src="https://static.igem.org/mediawiki/igem.org/2/2d/Part4-2.png" width = "900px" height="675px"></p>
 +
<p align="center" style="text-indent:0em"><b>Figure 17 100 Cells Varied in initial AHL concentration</b></p>
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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>
 +
<br><br>
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<h2 id="Stability analysis">Stability analysis</h2>
 
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<p>Although we have done sensitivity analyses on some predominant parameters and acquired fabulous results, these analyses were all based on unary composites, holding only a single subject. We are not content with only doing sensitivity analyses, which merely care about single-in-single-out outcomes but not considering binary relation in systematic concept. So we made a bifurcation analysis on binary parameter (u,m) adopting the same method as what we have done in part 2.</p>
 
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<p align="CENTER" style="text-indent:0em"><img src="https://static.igem.org/mediawiki/igem.org/6/69/Part3-6.png" width="705px" height="101px"></p>
 
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<p align="center" class="cite">Table 3 critical points (u,m) for oscillation</p>
 
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<p>  Depicting those points into an axis, we got the bifurcation line, which indicates the parameters’ value range when our system can oscillate stably is in the area marked by ‘bistable’ as follows.</p>
 
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<p align="CENTER" style="text-indent:0em"><img src="https://static.igem.org/mediawiki/igem.org/7/73/Part3-7.png" width="900px" height="675px"></p>
 
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<p align="center" class="cite">Figure 14 Bifurcation analysis on (u,m)</p>
 
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<h2 id="Feedback analysis">Feedback analysis</h2>
 
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<p>By changing parameters a and b, which is equivalent to varying types of feedback introduced, we got simulation results as follows (In order to manifest more clearly, the parameter v was set larger, thus each cell’s feedback effect would put greater influence on the whole system).</p>
 
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<p align="CENTER" style="text-indent:0em"><img src="https://static.igem.org/mediawiki/igem.org/d/d4/Part3-8.png.jpg" width="619px" height="606px"></p>
 
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<p align="center" class="cite">Figure 15 System with feedback</p>
 
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<p>  The analysis shows that only with a negative feedback mechanism could the overall system be working as an oscillator. When a=b=0, the system contains no artificial negative feedback, but there may be some inherent negative feedback within the system.</p>
 
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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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