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      <h1></br></br>E-oscillators</h1></br>
      <p>Two genetic oscillators, dual-feedback oscillator and quorum-sensing oscillator, are built in our wetlab, so we first analyze the important biological processes and construct ODE sets to simulate these two oscillators separately. However, due to the time interval between the expression of regulating protein and their binding to promoters, we improve our models more precise by DDEs. Once the better genetic oscillator is selected through experiments, we will adopt the corresponding one as the virtual part.</p></br>
      <h3>Quorum-sensing oscillator</h3>
</br>
<img src="https://static.igem.org/mediawiki/2015/b/bf/Team_HZAU-China_Fig_W1.png" width="335px" height="200px">
</br>
      <p><strong>ODEs</strong></p>
      <p>As for quorum-sensing oscillator, luxI proteins could generate AHL from some substrates consisting of acyl-ACPs and Sadenosylmethionine (SAM). We assume that the amount of substrates is sufficient.</p>

<img src="https://static.igem.org/mediawiki/2015/0/08/Hzau_image002.png">

      <p>AHL(Acyl homoserine lactones) is a kind of auto-inducer, which could combine with luxR protein and activate the promoter. In addition, the lactonic ring of the AHL will be hydrolyzed once in presence of AiiA protein.</p>

<img src="https://static.igem.org/mediawiki/2015/a/af/Hzau_image003.png">
<img src="https://static.igem.org/mediawiki/2015/9/9c/Hzau_image004.png" width="200px" height="80px">

      <p>For transcription activation, we use Hill function to describe the rate of production.<span><img src="https://static.igem.org/mediawiki/2015/d/db/Hzau_image005.png"></span>is the maximal transcription rate. When the promoter is bound by the transcription factor, gene will be transcribed at the maximal transcription rate. So the Hill function is</p>
<img src="https://static.igem.org/mediawiki/2015/1/18/Hzau_image006.png">
      <p>In addition, many genes have a non-zero minimal expression level, namely basal expression level. It can be described by adding a term <span><img src="https://static.igem.org/mediawiki/2015/9/9b/Hzau_image007.png"></span>.</p>
      <p>The translation and degradation processes are governed by the following set of reactions:</p>
<img src="https://static.igem.org/mediawiki/2015/d/dc/Hzau_image008.png">     
      <p>Y refers to luxI and AiiA.<span><img src="https://static.igem.org/mediawiki/2015/c/c3/Hzau_image009.png"></span>is the translation rate. The degradation of mRNA is a much swifter process compared to protein, so we use parameter<span><img src="https://static.igem.org/mediawiki/2015/a/a4/Hzau_image010.png"></span>representing the degradation rate. LuxI and AiiA protein added with LVA-tag can be degraded by the same enzymatic reaction, so they can be modeled to follow Michaelis-Menten enzyme kinetics. Different values of<span><img src="https://static.igem.org/mediawiki/2015/f/ff/Hzau_image011.png"></span>and<span><img src="https://static.igem.org/mediawiki/2015/4/43/Hzau_image012.png"></span>represent different preferential binding dynamics of LuxI and AiiA to degrading enzyme.</p> 
    <img src="https://static.igem.org/mediawiki/2015/c/c0/Hzau_image013.png">
      <p>According to the above processes，we can describe the genetic circuit by the following equations:</p>
      <img src="https://static.igem.org/mediawiki/2015/9/98/Hzau_image014.png" style="padding-left:20%;">
      <p><span><img src="https://static.igem.org/mediawiki/2015/8/85/Hzau_image015.png"></span>refers to mRNA while<span><img src="https://static.igem.org/mediawiki/2015/0/05/Hzau_image016.png"></span>is protein. The internal and external AHL are described by AHL and Ae. LuxR is constitutively produced at a constant level and the protein is degraded at at a proportion<span><img src="https://static.igem.org/mediawiki/2015/1/1e/Hzau_image017.png"></span>, since it is not tagged for fast degradation.</p>

      </br></br>
      <p><strong>DDEs</strong></p>
      <p>Due to the time interval between expression of regulating protein and transcriptional activation by the luxR and AHL complex which depends on the past concentration of internal AHL, we decide to use DDEs model. Transcription, translation, and maturation rate of proteins are combined into a single time-delay parameter<span><img src="https://static.igem.org/mediawiki/2015/5/5f/Hzau_image018.png"></span>. We modeled based on the paper “Rapid and tunable post-translational coupling of genetic circuits”.</p>
      <img src="https://static.igem.org/mediawiki/2015/d/dc/Hzau_image019.png"/>
      <p>The simulation is carried out using custom written software in MATLAB. We get the simulation result of oscillator, as shown in Figure 1.</p>
      <img src="https://static.igem.org/mediawiki/2015/a/ab/Hzau_image020.png.png" width="700px" height="400px">
      <p class="zhushi">Figure 1. Simulation result of quorum-sensing oscillator.</p>

    <h3>Dual-feedback oscillator</h3>
<img src="https://static.igem.org/mediawiki/2015/e/ea/Hzau_image021.png">
      <p>ODEs</p>
      <p>In the genetic circuit of Dual-feedback oscillator, the three promoters are the same hybrid promoter composed of an activation site and a repression site. They can be activated by the araC protein in the presence of arabinose and repressed by the LacI protein in the absence of IPTG. The mechanism of dual-feedback makes oscillatory behaviour stable and adjustable.</p>
     <p>There are four states of promoter: original promoter without inducer, bound with araC dimers (<span><img src="https://static.igem.org/mediawiki/2015/7/71/Hzau_image022.png"></span>), bound with lacI tetramers (<span><img src="https://static.igem.org/mediawiki/2015/2/2d/Hzau_image023.png"></span>), bound with araC and lacI tetramers together.<span><img src="https://static.igem.org/mediawiki/2015/9/90/Hzau_image024.png"></span>stands for the proportion of the four states of promoters.</p>
<img src="https://static.igem.org/mediawiki/2015/6/62/Hzau_image025.png">
<p>when reached the quasi steady state, we can get</p>
<img src="https://static.igem.org/mediawiki/2015/e/e7/Hzau_image026.png">
<p>Because of<span><img src="https://static.igem.org/mediawiki/2015/e/e9/Hzau_image027.png"></span>, we get</p>
<img src="https://static.igem.org/mediawiki/2015/0/00/Hzau_image028.png">
<p>We assume that araC dimers and lacI tetramers combine with promoter separately without interference. So it means that<span><img src="https://static.igem.org/mediawiki/2015/d/d5/Hzau_image029.png"></span></p>
<img src="https://static.igem.org/mediawiki/2015/0/00/Hzau_image030.png">
<p>AraC dimers and lacI tetramers are generated through their monomeric versions.</p>
<p class="zhushi"><span><img src="https://static.igem.org/mediawiki/2015/4/4c/Hzau_image031.png"><img src="https://static.igem.org/mediawiki/2015/2/26/Hzau_image032.png"><img src="https://static.igem.org/mediawiki/2015/4/4a/Hzau_image033.png"></span></p>
<p>When reached the quasi steady state, we can get</p>
<p class="zhushi"><span><img src="https://static.igem.org/mediawiki/2015/4/44/Hzau_image034.png"><img src="https://static.igem.org/mediawiki/2015/f/f6/Hzau_image035.png"></span></p>
<p>We define that<span><img src="https://static.igem.org/mediawiki/2015/1/13/Hzau_image036.png"></span>，<span><img src="https://static.igem.org/mediawiki/2015/3/39/Hzau_image037.png"></span>，and plug them into D~D3, then we have</p>
<img src="https://static.igem.org/mediawiki/2015/5/5e/Hzau_image038.png">
<p>In addition, from the consequence of Jesse Stricker(2008), we know that the binding rates of the activator dimer and repressor tetramer to the operator sites are dependent upon the concentrations of arabinose and IPTG.</p>
<img src="https://static.igem.org/mediawiki/2015/f/f5/Hzau_image039.png">
<p><span><img src="https://static.igem.org/mediawiki/2015/7/7e/Hzau_image040.png"></span>are the maximum and minimum affinities of AraC dimers to the binding sites, [ara] is the concentration of arabinose (in % w/v), [IPTG] is the IPTG concentration (in mM),<span><img src="https://static.igem.org/mediawiki/2015/4/4e/Hzau_image041.png"></span>,<span><img src="https://static.igem.org/mediawiki/2015/5/57/Hzau_image042.png"></span>,<span><img src="https://static.igem.org/mediawiki/2015/e/ed/Hzau_image043.png"></span>,<span><img src="https://static.igem.org/mediawiki/2015/0/05/Hzau_image044.png"></span>.</p>
<img src="https://static.igem.org/mediawiki/2015/d/d7/Hzau_image045.png">
<p><span><img src="https://static.igem.org/mediawiki/2015/0/0e/Hzau_image046.png"></span>are the maximum and minimum affinities of LacI tetramers to the binding site,<span><img src="https://static.igem.org/mediawiki/2015/c/c4/Hzau_image047.png"></span>,<span><img src="https://static.igem.org/mediawiki/2015/c/c8/Hzau_image048.png"></span>.</p>
<p>We put the relationships between<span><img src="https://static.igem.org/mediawiki/2015/1/1a/Hzau_image_j2.png
"></span>and<span><img src="https://static.igem.org/mediawiki/2015/3/32/Hzau_image_j1.png"></span>into the above equations.</p>
<p>The transcription, translation and degradation processes are listed by the following reactions:</p>
<img src="https://static.igem.org/mediawiki/2015/f/fc/Hzau_image_j3.png">
<p>Based on the above analysis,We could construct a set of ODEs to describe the dual-feedback oscillator:</p>
<img src="https://static.igem.org/mediawiki/2015/c/c6/Hzau_image049.png">
<p><span><img src="https://static.igem.org/mediawiki/2015/6/65/Hzau_image050.png"></span>stand for the number of plasmid copies. The degradation rate of protein<span><img src="https://static.igem.org/mediawiki/2015/4/48/Hzau_image051.png"></span>,<span><img src="https://static.igem.org/mediawiki/2015/b/b1/Hzau_image052.png"></span>，<span><img src="https://static.igem.org/mediawiki/2015/4/47/Hzau_image053.png"></span>,<em>X</em> is the total number of tags in the system.</p>
</br></br>

<p><strong>DDEs</strong></p>
<p>In consideration of the delay of transcription, translation and maturation, DDEs may be more capable. Based on the project of HUST-China 2013, we simulate the genetic circuit by DDE sets</p>
<img src="https://static.igem.org/mediawiki/2015/c/cc/Hzau_image054.png">
<p>Due to the delay, we denote<span><img src="https://static.igem.org/mediawiki/2015/8/8a/Hzau_image055.png"></span>as the time interval parameters of araC dimers and lacI tetramers. And attaching delay to three parameters:</p>
<img src="https://static.igem.org/mediawiki/2015/2/2a/Hzau_image056.png">
<p>Then we code in MATLAB, and the simulation result is shown in Figure 2.</p>
<img src="https://static.igem.org/mediawiki/2015/0/04/Hzau_imagela.png" width="800px" height="400px;">
<p class="zhushi">Figure 2. Simulation result of dual-feedback oscillator.</p>

</br></br>
<p><strong>Parameters</strong></p>
<p>Parameters play an important role in simulating, which would determine the behaviors of the system. In our modeling part, parameters in quorum-sensing e-oscillator are from <a href="https://2011.igem.org/Team:Tsinghua-A/Modeling/P1A">parameter part of Original Full Model by Tsinghua-A iGEM 2011</a> and the paper “Rapid and tunable post-translational coupling of genetic circuits” by Arthur Prindle et al. And parameters used in dual-feedback e-oscillator are based on the paper “A fast, robust and tunable synthetic gene oscillator” and <a href="https://2013.igem.org/Team:HUST-China/Modelling/DDE_Model">parameter table in HUST-China iGEM 2013</a>. Although these previous work give us some experience, we still adjust some parameters more reasonable in terms of our system and some experimental tests.</p>
</br></br>
<p><strong>Reference</strong></p>
<p>1.Arthur P, Jangir S, Howard L, et al. Rapid and tunable post-translational coupling of genetic circuits.[J]. Nature, 2014, 508(7496):387-391.</p>

<p>2.Stricker J, Cookson S, Bennett M R, et al. A fast, robust and tunable synthetic gene oscillator[J]. Nature, 2008, 456(7221):516-519.</p>

<p>3.O’Brien E L, Itallie E V, Bennett M R. Modeling synthetic gene oscillators[J]. Mathematical Biosciences, 2012, 236(1):1–15.</p>

<p>4.Tabor J J, Salis H M, Simpson Z B, et al. A synthetic genetic edge detection program.[J]. Cell, 2009, 137(7):1272-1281.</p>

<p>5.Hubler A, Gintautas V. Experimental evidence for mixed reality states[J]. Complexity, 2011, 13(6):7–10.</p>

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