Team:iHKU/modeling

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                         <td width="80%" align="left"><h1>Modeling</h1>
                         <td width="80%" align="left"><h1>Modeling</h1>
                           <h2><a name="top" id="top"></a>CONTENTS:</h2>
                           <h2><a name="top" id="top"></a>CONTENTS:</h2>
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                           <ul class="style24">
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                           <ul class="style24 ">
                             <li><a href="#1">Cell Movement in Microscopic and Macroscopic Aspects</a></li>
                             <li><a href="#1">Cell Movement in Microscopic and Macroscopic Aspects</a></li>
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                             <li><a href="#2">Front propagation for Cell Growth </a></li>
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                             <li><a href="#2">Front propagation for Cell Growth</a></li>
                             <li><a href="#3">Density Dependent Motility</a></li>
                             <li><a href="#3">Density Dependent Motility</a></li>
                             <li><a href="#4">Full Model of Density  Dependent Motility</a></li>
                             <li><a href="#4">Full Model of Density  Dependent Motility</a></li>
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                           </ul>
                           </ul>
                           <p>&nbsp;</p>
                           <p>&nbsp;</p>
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                           <p align="center" class="style26"><img src="/wiki/images/7/76/Modelling_pic4.JPG" width="180" height="52" /> , </p>
                           <p align="center" class="style26"><img src="/wiki/images/7/76/Modelling_pic4.JPG" width="180" height="52" /> , </p>
                           <p class="style26"><em>v</em>=<em> l</em>/<em>τ</em> was the average speed of the cell movement, and <em>f</em>=1/<em>τ </em>was the tumbling  frequency of <em>E.coli</em>. </p>
                           <p class="style26"><em>v</em>=<em> l</em>/<em>τ</em> was the average speed of the cell movement, and <em>f</em>=1/<em>τ </em>was the tumbling  frequency of <em>E.coli</em>. </p>
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                           <p class="style26">In two dimensions, the square of the distance  from the origin to the point (x, y) was <em>r</em>2=<em>x</em>2+<em>y</em>2; therefore</p>
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                           <p class="style26">In two dimensions, the square of the distance  from the origin to the point (x, y) was <span class="STYLE14">r<sup>2</sup>=x<sup>2</sup>+y<sup>2</sup></span>; therefore</p>
                           <p align="center" class="style26"><img src="/wiki/images/2/2e/Modelling_pic5.JPG" width="108" height="45" /></p>
                           <p align="center" class="style26"><img src="/wiki/images/2/2e/Modelling_pic5.JPG" width="108" height="45" /></p>
                           <p class="style26">For a macroscopic view, <em>D</em> was defined as the diffusion coefficient. For a simple diffusion process, it is easy to write an diffusion equation to describe the density <em>ρ</em> distribution.</p>                           
                           <p class="style26">For a macroscopic view, <em>D</em> was defined as the diffusion coefficient. For a simple diffusion process, it is easy to write an diffusion equation to describe the density <em>ρ</em> distribution.</p>                           
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                           <img src="/wiki/images/d/d9/ModelEq14.gif" width="121" height="61">...................................................................................................(1.4)
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                           <img src="/wiki/images/d/d9/ModelEq14.gif" width="100" height="50">...................................................................................................(1.4)
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                           <p class="style26">Normally, as the  swimming speed of <em>E.coli</em> is about 20<em>um</em>/<em>s</em> and the tumbling frequency is about 1 <em>Hz</em>,  the diffusion coefficient is about 200<em>um</em>2/<em>s </em>[<a href="#ref">3</a>].</p>
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                           <p class="style26">Normally, as the  swimming speed of <em>E.coli</em> is about 20<em>um</em>/<em>s</em> and the tumbling frequency is about 1 <em>Hz</em>,  the diffusion coefficient is about 200<em>um</em><sup>2</sup>/<em>s </em>[<a href="#ref">3</a>].</p>
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                           <p>&nbsp;</p>
                           <p>&nbsp;</p>
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                           <h3><span class="style7"><strong><a name="2" id="2"></a>Front propagation for cell growth</strong></span> <br />
                           <h3><span class="style7"><strong><a name="2" id="2"></a>Front propagation for cell growth</strong></span> <br />
                           </h3>
                           </h3>
                           <p class="style26">If we only considered the wild type E.coli that has the diffusion effect and  growth effect, it came up with the model Fisher-Kolmogorov  equation which was firstly developed by R. A. Fisher and A. N. Kolmogorov [<a href="#ref">4</a>]. </p>
                           <p class="style26">If we only considered the wild type E.coli that has the diffusion effect and  growth effect, it came up with the model Fisher-Kolmogorov  equation which was firstly developed by R. A. Fisher and A. N. Kolmogorov [<a href="#ref">4</a>]. </p>
                           <div align="center" class="style26"><img src="/wiki/images/8/8d/Modelling_pic6.JPG" width="387" height="41" /> (2.1)                        </div>
                           <div align="center" class="style26"><img src="/wiki/images/8/8d/Modelling_pic6.JPG" width="387" height="41" /> (2.1)                        </div>
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                           <p class="style26">where <em>ρ </em>was the cell density, <em>D</em> was the diffusion coefficient, <em>γ</em>0 was growth rate, <em>ρ</em>s was saturation density. </p>                           
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                           <p class="style26">where <em>ρ </em>was the cell density, <em>D</em> was the diffusion coefficient, <em>γ</em><sub>0</sub>was growth rate, <em>ρ</em><sub>s </sub>was saturation density. </p>                           
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                           <p class="style26">Here, due to the symmetry, we investigated  equation (2.1) in case of cylindrical domains with u depending only on radius.  Numerical study with an initially condition of  Gaussian function                                                                      as Fig.2 can  give a solution shown in Fig.2, while analytical solution can refer to  reference [<a href="#ref">5</a>]. Based on the solution, it was found that the speed of the front  propagation of the pattern was proportional to the square root of the product of  diffusion coefficient    <em>D</em> and growth rate <em>γ</em>. As this migration speeds of the cell pattern at  different <em>cheZ</em> expression levels can  be obtained from experiments, and the growth rate was also measured, the  diffusion coefficient      <em>D</em> at different <em>cheZ</em> expression levels were also known  then. And the cell density can be transformed to the brightness that we observed  in the experiments <a href="/wiki/Team:iHKU/result">(see the brightness model part)</a>, the quantity <em>ρ</em>    can be compared with the experiments. Therefore, each  quantity in equation (4) can be compared with the experiments. </p>
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                           <p class="style26">Here, due to the symmetry, we investigated  equation (2.1) in case of cylindrical domains with u depending only on radius.  Numerical study with an initially condition of  Gaussian function                                                                      as Fig.2 can  give a solution shown in Fig.2, while analytical solution can refer to  reference [<a href="#ref">5</a>]. Based on the solution, it was found that the speed of the front  propagation of the pattern was proportional to the square root of the product of  diffusion coefficient    <em>D</em> and growth rate <em>γ</em>. As this migration speeds of the cell pattern at  different <em>cheZ</em> expression levels can  be obtained from experiments, and the growth rate was also measured, the  diffusion coefficient      <em>D</em> at different <em>cheZ</em> expression levels were also known  then. And the cell density can be transformed to the brightness that we observed  in the experiments <a href="https://2008.igem.org/Team:iHKU/protocol#p4">(see the brightness model part)</a>, the quantity <em>ρ</em>    can be compared with the experiments. Therefore, each  quantity in equation (4) can be compared with the experiments. </p>
                           <p class="style26"> If we  looked at the wild type <em>E.coli </em>which  only had random walk and growth, the equation (2.1) was successful to describe its behavior. As shown in Fig.2, a droplet of wild type <em>E.coli</em> in the center of the plate will  from a round expanding pattern. And most regions in this pattern except for  some near in boundary seems to be uniform, which was the same as the  result of the model (Fig.2).</p>
                           <p class="style26"> If we  looked at the wild type <em>E.coli </em>which  only had random walk and growth, the equation (2.1) was successful to describe its behavior. As shown in Fig.2, a droplet of wild type <em>E.coli</em> in the center of the plate will  from a round expanding pattern. And most regions in this pattern except for  some near in boundary seems to be uniform, which was the same as the  result of the model (Fig.2).</p>
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                          </table>  <div align="center"><span class="style27">Fig. 2</span> The movie of the wild type pattern obtained by model in 2D(left) and 3D(right)</div>
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                          <p>&nbsp;</p><p align="center"><a href="https://2008.igem.org/Team:iHKU/result#mgvideo"><img src="https://static.igem.org/mediawiki/2008/9/95/Resultnew2.png" width="400" height="400" border="0" /></a></p>
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                          <div align="center"><a href="https://2008.igem.org/Team:iHKU/result#mgvideo" class="STYLE31">(Click here to see the movie obtained by experiments
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                          <p align="right"><a href="#top">[Back to Top]</a></p>
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                          <p>&nbsp;</p>
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                          <p class="style7">&nbsp;</p><p class="style7">&nbsp;</p><p class="style7">&nbsp;</p><p class="style7">&nbsp;</p>
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                          <h3 class="style3"><strong><a name="3" id="3"></a><span class="style7">Density Dependent Motility</span></strong></h3>
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                          <p align="center"><img src="/wiki/images/0/0a/Modelling_pic7.JPG" alt="" width="265" height="114" /></p>
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                          <div align="center"><span class="style27">Fig. 3</span> Designed genetic circuit</div>
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                          <p class="style26">In experiments, we designed a circuit  that the cell motility was repressed by cell density                                                                      <em>ρ</em>. When the cell density    <em>ρ</em> was high, the  diffusion coefficient    <em>D</em> became small.  Therefore, the fisher’s equation as equation (2.1) was not valid  in this  case any more. </p>
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                          <p class="style26">In  order to be simple, we firstly considered  the one dimension problem again. We assumed the  cell density at point <em>x</em> was    <em>ρ</em>(x) at time <em>t</em>. In a very short time <em>τ</em>, there would be two groups of<em> cell </em>at <em>x</em> moving into its nearby points <em>x</em>-<em>δ</em>, <em>x</em>+<em>δ,</em> due to the random walk. And the amount of cell in each group  were proportional to the product of <em>D</em>(<em>ρ</em>(<em>x</em>)) and <em>ρ</em>(<em>x</em>). Therefore,</p>
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                          <img src="/wiki/images/9/91/ModelEq8.gif" width="500" height="97">(3.1)                         
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                          <p>In the limit <em>τ</em>--&gt;0 and<em> δ</em>--&gt;0, we obtain</p>
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                          <img src="/wiki/images/d/df/ModelEq9.gif" width="500" height="69">(3.2)
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<p align="center" class="style26">&nbsp;</p>
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<p class="style26">And the function<em>D<sub>ρ</sub></em>(<em>ρ</em>)  was a decrease  function. For a simplest case, we considered a Heaviside Function(Fig.4). There was a threshold of cell density above which the cell can not swim forward but always tumble in the same place. Considering five points with the different cell denisty, among which there were group cell swimming to the nearby sites, we assumed that only the cell density in the middle of which was larger than the threshold. Then the cell in the middle would not go out of it. At the same time, the cell nearby would incessantly come into this point. As a result, the cell density here would increase, while the cell density nearby would continually decrease until zero.</p>                           
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<p align="center" class="style26">&nbsp;</p>
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                            <p class="style26">The numerical simulations gave us the results  shown in Fig.5. The cell density showed a periodical-narrow-peak structure.  These peaks were what we wanted, as they produced some regions of low cell density,  though the whole pattern was not quite similar with that of experiments. And the exact pattern would come out when we took account of the other parts in the whole genetic circuit(See <a href="#4">Full Model</a>).</p>
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                            <p><img src="/wiki/images/d/dc/Modelnew2.png" width="561" height="272" /></p>
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                            <div align="center" class="style27">                              Fig. 5 Periodical-narrow-peak pattern in 2D(left); the cell density distribution along the radius(right) </div>
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                          <p align="right"><a href="#top">[Back to Top]</a></p>
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                            <p>&nbsp;</p><p class="style7">&nbsp;</p>
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                            <p class="style7">&nbsp;</p><p class="style7">&nbsp;</p><p class="style7">&nbsp;</p>
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                            <h3><a name="4" id="4"></a><span class="style7">Full  Model of Density Dependent Motility</span></h3>
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                            <div align="center">
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                              <p><img src="/wiki/images/0/01/Modelling_pic10.png" width="386" height="242" /></p>
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                              <span class="style27">Fig. 6</span> The entire designed genetic circuit </div>
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                            <p>Actually, in our genetic circuit, we transformed a  plasmid which can secrete AHL to environment (Fig.6). When the AHL density <em>h</em> of  environment was high, the AHL came into the cell. Then AHL combining with <em>LuxR</em> repressed the expression level of <em>cheZ</em> which controled the motility of <em>E.coli.</em> So it was necessary to take  account of the AHL effect. </p>
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                            <p>First,  the AHL was synthesized by the <em>E.coli</em> cell at the rate <em>λ</em>. And the  degradation rate<em> β</em> of AHL whose half life is normally about 15 to 30 minutes. Considering the diffusion of AHL, we obtained </p>                         
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                            <img src="/wiki/images/e/ee/HkumodelEq5.gif" width="500" height="69">(4.1)
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                            <p>where <em>D<sub>h</sub></em> is the diffusion coefficient of AHL, which is about 0.001mm<sup>2</sup>/min[6].<br>
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                          And the cell diffusion coefficient of <em>E.coli</em> is determined by the density of AHL. Therefore                            </p>
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                            <img src="/wiki/images/6/60/ModelEq6.gif" width="500" height="69">(4.2)
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                            <p>where<em>D<sub>ρ</sub></em>(<em>h</em>) is a decreasing function of <em>h</em>.</p>
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                            <p>Furthermore, the nutrient consumption influences  the growth rate of <em>E.coli</em>. Therefore</p>
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                            <img src="/wiki/images/0/00/HkumodelEq7.gif" width="500" height="69"> (4.3)
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                          <p>where n is the
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                          nutrient concentration.  </p> <div align="left">
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<p><img src="https://static.igem.org/mediawiki/igem.org/1/1f/Fig7.gif" width="164" height="140" /></p>
 +
</a>Fig. 7 The growth rate relates to nutrient concentration</strong></div>
 +
                              </div>
 +
                                </blockquote>
 +
                          </blockquote>
 +
                            <p>For the nutrient, we made an assumption that  the amount of nutrient consumed were proportional to the amount of cell increasing.  Taking account of the nutrient diffusion, we had</p>                           
 +
                            <div align="left"><img src="/wiki/images/5/58/Modelling_pic12.JPG" width="500" height="58" /> (4.4)                          </div>
 +
                            <span class="style26">where <em>k</em> was the ratio that nutrient converts to  cell mass, <em>D</em><sub>n</sub> is nutrient diffusion coefficient which is about that of small molecule[<a href="*ref">7</a>] .                            </span>
 +
                            <p>We used several possible forms of function<em>D<sub>ρ</sub></em>(<em>h</em>)(see <a href="#Drho">Results Section2</a>). The  results showed that if the cell diffusion coefficient decreases fast near the AHL  density threshold, the pattern came out as a multiple-ring one(<a href="#Drho">Results Section2 (a),(e)</a>). On the contrary, there was only one-ring pattern. </p>
 +
                            <p>Therefore, in order to to make this curve<em>D<sub>ρ</sub></em>(<em>h</em>) decreasing sharply near the threshold, it followed a prediction that we can get a multiple-ring pattern by making an auto-activate genetic  circuit which the combination of LuxR and AHL can activate the expression level  of itself</span>. </p>
 +
                            <div align="center">
 +
                              <p><img src="/wiki/images/f/f4/Modelling_pic13.png" width="329" height="221" />                            </p>
 +
                              <span class="style27">Fig.8</span> The designed genetic circuit for predicted pattern</div>
 +
                            <p>Furthermore, we did some  two-spot patterns which initially have two points of <em>E.coli</em> cell, so as to compare with the experiments culture (<a href="#result3">Results Section3</a>). </p>
 +
                          <p align="right"><a href="#top">[Back to Top]</a></p>
 +
                            <p>&nbsp;</p><p class="style7">&nbsp;</p>
 +
                            <p class="style7">&nbsp;</p><p class="style7">&nbsp;</p><p class="style7">&nbsp;</p><p class="style7">&nbsp;</p>
 +
                            <h3 align="left"><strong><a name="5" id="5"></a><span class="style7">Modeling Result</span></strong><span class="style7">s</span></h3>
 +
                            <p> In the simulation, we tried different values of undetermined parameters which were possible to change in experiments, to see how they influenced the pattern. Generally, we can vary the  parameters of cell growth and cell motility which, in experiments, are easy to  change. So we list the  results of varying several parameters in the cell growth part and cell motility  part. </p>
 +
                            <p><span class="style7"><a href="#resg">Cell Growth</a></span></span></p>
 +
                            <ul>
 +
                              <li><span class="STYLE23"><a href="#res1">Maximum growth rate <em>γ</em><sub>0</sub></a></span></li>
 +
                              <li class="STYLE23"><a href="#res2"> Initial nutrient concentration <strong>n</strong></a><span class="STYLE29"><sub>i</sub></span></li>
 +
                              <li class="STYLE23"><em class="style26"><a href="#res3">κ</a></em></li>
 +
                              <li><span class="STYLE23"><a href="#res4">k</a></span></li>
 +
                          </ul>
 +
                            <p><span class="STYLE23"><span class="style7"><a href="#Drho">Cell movement</a></span></span></span></p>
 +
                            <p><span class="style7 style7"><a href="#result3" class="STYLE23">Multiple initial Spots</a></span></p>
 +
                          <p><a name="resg"></a><span class="style7">1. Cell Growth</span></p>
 +
                            <p>In the cell growth term, we  considered the growth rate was related to nutrient concentration. When the  nutrient was consumed by the cell, the grow rate of the cell would decrease  (<a href="#F7">Fig. 7</a>). There was a maximum growth rate <em>γ</em><sub>0</sub>,  when the nutrient was rich. The parameter <em>κ </em>described the sharp of this curve(<a href="#F7">Fig. 7</a>). The larger <em>κ</em> gave us more smooth increasing curve. And we made an assumption  that one unit of nutrient would change to cell number with a ratio k. Also in  the initial condition, the nutrient concentration <span class="STYLE23">n</span><sub>i</sub> can influence  the patterns. Therefore, the effects of all the four parameters were studied in  our model.</p>
 +
                            <p align="center"><a name="res1"></a><img src="/wiki/images/b/b7/Modelnew3_v2.PNG" width="541" height="441" /></p>
 +
                            <p>Above reuslts indicated that, the  faster cell growth (larger <em>γ<span class="STYLE13"><sub>0</sub></span> </em>and  smaller doubling time) formed a smaller and a little more unclear ring pattern.  And the inner ring diameter would become a litter more large when the growth  rate increased. </p>
 +
                            <div align="center"><a name="res2"></a><img src="/wiki/images/6/6d/Modelnew4.png" width="535" height="441" /></span> </div>
 +
                            <p>From  above figures, it was found that the initial nutrient concentration ni had  little influence to the pattern, especially when it was large. The only effect  we observed was that smaller initial nutrient concentration ni  made the ring slightly wider and more clear.</p>
 +
                            <div align="center"><a name="res3"></a><img src="/wiki/images/6/6c/Modelnew5.png" width="543" height="444" />                            </div>
 +
                            <p>Here,  the value of <em>κ </em>was related to the  initial nutrient concentration n<sub>i</sub> . When initial nutrient  concentration n<sub>i</sub> was fixed, a smaller <em>κ </em>parameter gave us the similar results with that of faster cell  growth. </p>
 +
                            <div align="center"><a name="res4"></a><img src="/wiki/images/5/51/Modelnew6.png" width="545" height="447" /></div>
 +
                            <p>When  we increased the value of <em>k</em>, the  patterns did not change a lot, but only became a little more clearly.</p>
 +
                            <p>In conclusion, the simulation  results showed that the parameters of maximum growth rate <em>γ<span class="STYLE13">o</span></em>and <em>κ</em> had a more important role on the ring-like pattern. And the other two parameters <em>k </em>and n<sub>i</sub>  did not influence the  pattern much. </p>
 +
                            <p>&nbsp;</p>
 +
                            <p class="style7 style7 style7">2. Cell Motility <em>D<sub>ρ</sub></em>(<em>h</em>)</p>
 +
                            <p><a name="Drho" id="Drho"></a><span class="STYLE23">Forms of <em>D<sub>ρ</sub></em>(<em>h</em>)</span></p>
 +
                            <p>As <em>D<sub>ρ</sub></em>(<em>h</em>) was a decreasing function of <em>h</em>, the possible formed of it can be</p>
 +
<table width="200" border="1">
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                              <tr>
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                                <td height="209"><p><a name="a"></a>A</p>
 +
                                  <p><img src="/wiki/images/e/e4/Modelnew7.JPG" alt="" width="145" height="205" /></p></td>
 +
                                <td width="200"><p>&nbsp;
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                              <tr>
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                                <td height="209"><p><a name="b"></a>B</p>
 +
                                <p><img src="/wiki/images/6/67/Modelnew8.JPG" alt="" width="142" height="192" /></p></td>
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                                <td height="209"><p><a name="c"></a>C</p>
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                                <p><img src="/wiki/images/3/31/Modelnew9.JPG" alt="" width="142" height="192" /></p></td>
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                                <td height="209"><p><a name="d"></a>D</p>
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                                <td height="209"><p><a name="e"></a>E</p>
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                            <p>The  above movies showed that if the cell diffusion coefficient decreases fast near the AHL  density threshold, the pattern came out as a multiple-ring one(<a href="https://2008.igem.org/Team:iHKU/modeling/movie#a">Results Section2 (a),(e)</a>). On the contrary, there was only one-ring pattern<a href="https://2008.igem.org/Team:iHKU/modeling/movie#b">(Results Section2 (b),(c),(d))</a>. </span>                            </p>
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                          <p>&nbsp;</p>
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                            <p class="style7"><a name="result3" id="result3"></a>3. Mulitple initial spots</p>
 +
                            <p>With the initial conditions of two or more spots, we obtained the below funny patterns which are amazing similar with that of experiments<a href="/Team:iHKU/result#fun">(results)</a>.</p>
 +
                          <p align="center"><img src="/wiki/images/8/85/Modelnew12.png" width="384" height="508" /></p>
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                            <p class="STYLE28"><a name="ref" id="ref"></a>Reference:</p>
 +
                        <ul>
 +
                              <li class="style25">[1] P.K.Pathria, <em>Statistical  Mechanics</em> (Pergamon Press, Headington Hill Hall, Oxford; 1972)</li>
 +
                              <li class="style25">[2] Howard C.Berg, <em>Random  Walks in Biology</em> (Princeton University Press, Princeton, Hew Jersey; 1993)</li>
 +
                              <li class="style25">[3] Nikhil Mittal, et al, Proc Natl Acad Sci <strong>100</strong>, 13259 (2003)</li>
 +
                              <li class="style25">[4] R. A. Fisher, Ann. Eugenics <strong>7</strong>, 353 (1937)</li>
 +
                              <li class="style25">[5] A.H.Bokhari, et al., Nonlinear Analysis (In Press)</li>
 +
                              <li class="style25">[6]Basu S, et al., Nature <strong>434</strong>, 1130(2005)</li>
 +
                              <li class="style25">[7]J.W. Costerton, Naomi Balaban,<em> Control of Biofilm Infections by Signal Manipulation</em> (Springer,2008)</li>
 +
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Latest revision as of 11:52, 30 October 2008

 

Modeling

CONTENTS:

 

 

 

 

 

 

Cell Movement in Microscopic and Macroscopic Aspect

Basically, we considered the movement of a cell as a process of random walk, since an individual E.coli cell was always in the states between moving and tumbling (Fig.1).Similar to Brownian particles, the random walk of E.coli followed the Einstein-Smoluchowski theory [1].

Fig . 1 Genetic circuit related to cell movementin(left) and cell random walk(right)

Firstly, for the one dimension of random walk, let x (t) denote the position of an E.coli cell at time t, given that its position coincided with the point x=0 at time t=0. And we assumed an E.coli cell moves an average distance l—in either the positive or negative direction of the x-axis—each step (during a time (τ) between two tumbling states). The probability that the cell was found at the point x at time t was now equal to the probability that, in a series of n (=t/τ) successive moving steps, the cell made m more steps in the positive direction of the axis than in the negative direction. The desired probability was given by the binomial expression [1, 2]

(1.1)

To simplify equation (1.1), for m<<n, we obtain

(1.2)

Taking x to be a continuous variable, we can obtain

(1.3)

where

,

v= l/τ was the average speed of the cell movement, and f=1/τ was the tumbling frequency of E.coli.

In two dimensions, the square of the distance from the origin to the point (x, y) was r2=x2+y2; therefore

For a macroscopic view, D was defined as the diffusion coefficient. For a simple diffusion process, it is easy to write an diffusion equation to describe the density ρ distribution.

...................................................................................................(1.4)

Normally, as the swimming speed of E.coli is about 20um/s and the tumbling frequency is about 1 Hz, the diffusion coefficient is about 200um2/s [3].

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Front propagation for cell growth

If we only considered the wild type E.coli that has the diffusion effect and growth effect, it came up with the model Fisher-Kolmogorov equation which was firstly developed by R. A. Fisher and A. N. Kolmogorov [4].

(2.1)

where ρ was the cell density, D was the diffusion coefficient, γ0was growth rate, ρs was saturation density.

Here, due to the symmetry, we investigated equation (2.1) in case of cylindrical domains with u depending only on radius. Numerical study with an initially condition of Gaussian function as Fig.2 can give a solution shown in Fig.2, while analytical solution can refer to reference [5]. Based on the solution, it was found that the speed of the front propagation of the pattern was proportional to the square root of the product of diffusion coefficient D and growth rate γ. As this migration speeds of the cell pattern at different cheZ expression levels can be obtained from experiments, and the growth rate was also measured, the diffusion coefficient D at different cheZ expression levels were also known then. And the cell density can be transformed to the brightness that we observed in the experiments (see the brightness model part), the quantity ρ can be compared with the experiments. Therefore, each quantity in equation (4) can be compared with the experiments.

If we looked at the wild type E.coli which only had random walk and growth, the equation (2.1) was successful to describe its behavior. As shown in Fig.2, a droplet of wild type E.coli in the center of the plate will from a round expanding pattern. And most regions in this pattern except for some near in boundary seems to be uniform, which was the same as the result of the model (Fig.2).

Fig. 2 The movie of the wild type pattern obtained by model in 2D(left) and 3D(right)

 

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Density Dependent Motility

Fig. 3 Designed genetic circuit

In experiments, we designed a circuit that the cell motility was repressed by cell density ρ. When the cell density ρ was high, the diffusion coefficient D became small. Therefore, the fisher’s equation as equation (2.1) was not valid in this case any more.

In order to be simple, we firstly considered the one dimension problem again. We assumed the cell density at point x was ρ(x) at time t. In a very short time τ, there would be two groups of cell at x moving into its nearby points x-δ, x+δ, due to the random walk. And the amount of cell in each group were proportional to the product of D(ρ(x)) and ρ(x). Therefore,

(3.1)

In the limit τ-->0 and δ-->0, we obtain

(3.2)

 

And the functionDρ(ρ) was a decrease function. For a simplest case, we considered a Heaviside Function(Fig.4). There was a threshold of cell density above which the cell can not swim forward but always tumble in the same place. Considering five points with the different cell denisty, among which there were group cell swimming to the nearby sites, we assumed that only the cell density in the middle of which was larger than the threshold. Then the cell in the middle would not go out of it. At the same time, the cell nearby would incessantly come into this point. As a result, the cell density here would increase, while the cell density nearby would continually decrease until zero.

 

The numerical simulations gave us the results shown in Fig.5. The cell density showed a periodical-narrow-peak structure. These peaks were what we wanted, as they produced some regions of low cell density, though the whole pattern was not quite similar with that of experiments. And the exact pattern would come out when we took account of the other parts in the whole genetic circuit(See Full Model).

Fig. 5 Periodical-narrow-peak pattern in 2D(left); the cell density distribution along the radius(right)

[Back to Top]

 

 

 

 

 

Full Model of Density Dependent Motility

Fig. 6 The entire designed genetic circuit

Actually, in our genetic circuit, we transformed a plasmid which can secrete AHL to environment (Fig.6). When the AHL density h of environment was high, the AHL came into the cell. Then AHL combining with LuxR repressed the expression level of cheZ which controled the motility of E.coli. So it was necessary to take account of the AHL effect.

First, the AHL was synthesized by the E.coli cell at the rate λ. And the degradation rate β of AHL whose half life is normally about 15 to 30 minutes. Considering the diffusion of AHL, we obtained

(4.1)

where Dh is the diffusion coefficient of AHL, which is about 0.001mm2/min[6].
And the cell diffusion coefficient of E.coli is determined by the density of AHL. Therefore

(4.2)

whereDρ(h) is a decreasing function of h.

Furthermore, the nutrient consumption influences the growth rate of E.coli. Therefore

(4.3)

where n is the nutrient concentration.

Fig. 7 The growth rate relates to nutrient concentration

For the nutrient, we made an assumption that the amount of nutrient consumed were proportional to the amount of cell increasing. Taking account of the nutrient diffusion, we had

(4.4)
where k was the ratio that nutrient converts to cell mass, Dn is nutrient diffusion coefficient which is about that of small molecule[7] .

We used several possible forms of functionDρ(h)(see Results Section2). The results showed that if the cell diffusion coefficient decreases fast near the AHL density threshold, the pattern came out as a multiple-ring one(Results Section2 (a),(e)). On the contrary, there was only one-ring pattern.

Therefore, in order to to make this curveDρ(h) decreasing sharply near the threshold, it followed a prediction that we can get a multiple-ring pattern by making an auto-activate genetic circuit which the combination of LuxR and AHL can activate the expression level of itself.

Fig.8 The designed genetic circuit for predicted pattern

Furthermore, we did some two-spot patterns which initially have two points of E.coli cell, so as to compare with the experiments culture (Results Section3).

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Modeling Results

In the simulation, we tried different values of undetermined parameters which were possible to change in experiments, to see how they influenced the pattern. Generally, we can vary the parameters of cell growth and cell motility which, in experiments, are easy to change. So we list the results of varying several parameters in the cell growth part and cell motility part.

Cell Growth

Cell movement

Multiple initial Spots

1. Cell Growth

In the cell growth term, we considered the growth rate was related to nutrient concentration. When the nutrient was consumed by the cell, the grow rate of the cell would decrease (Fig. 7). There was a maximum growth rate γ0, when the nutrient was rich. The parameter κ described the sharp of this curve(Fig. 7). The larger κ gave us more smooth increasing curve. And we made an assumption that one unit of nutrient would change to cell number with a ratio k. Also in the initial condition, the nutrient concentration ni can influence the patterns. Therefore, the effects of all the four parameters were studied in our model.

Above reuslts indicated that, the faster cell growth (larger γ0 and smaller doubling time) formed a smaller and a little more unclear ring pattern. And the inner ring diameter would become a litter more large when the growth rate increased.

From above figures, it was found that the initial nutrient concentration ni had little influence to the pattern, especially when it was large. The only effect we observed was that smaller initial nutrient concentration ni made the ring slightly wider and more clear.

Here, the value of κ was related to the initial nutrient concentration ni . When initial nutrient concentration ni was fixed, a smaller κ parameter gave us the similar results with that of faster cell growth.

When we increased the value of k, the patterns did not change a lot, but only became a little more clearly.

In conclusion, the simulation results showed that the parameters of maximum growth rate γoand κ had a more important role on the ring-like pattern. And the other two parameters k and ni did not influence the pattern much.

 

2. Cell Motility Dρ(h)

Forms of Dρ(h)

As Dρ(h) was a decreasing function of h, the possible formed of it can be

A

 

[Back to Top]

B

[Back to Top]

C

[Back to Top]

D

[Back to Top]

E

[Back to Top]

The above movies showed that if the cell diffusion coefficient decreases fast near the AHL density threshold, the pattern came out as a multiple-ring one(Results Section2 (a),(e)). On the contrary, there was only one-ring pattern(Results Section2 (b),(c),(d)).

 

3. Mulitple initial spots

With the initial conditions of two or more spots, we obtained the below funny patterns which are amazing similar with that of experiments(results).

Reference:

  • [1] P.K.Pathria, Statistical Mechanics (Pergamon Press, Headington Hill Hall, Oxford; 1972)
  • [2] Howard C.Berg, Random Walks in Biology (Princeton University Press, Princeton, Hew Jersey; 1993)
  • [3] Nikhil Mittal, et al, Proc Natl Acad Sci 100, 13259 (2003)
  • [4] R. A. Fisher, Ann. Eugenics 7, 353 (1937)
  • [5] A.H.Bokhari, et al., Nonlinear Analysis (In Press)
  • [6]Basu S, et al., Nature 434, 1130(2005)
  • [7]J.W. Costerton, Naomi Balaban, Control of Biofilm Infections by Signal Manipulation (Springer,2008)

[Back to Top]