Difference between revisions of "Team:USTC/Modeling"

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    <div id="Film-Candidate" class="row">

      <div class="card hoverable">

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            <h4 id="Film-Candidates-overview" class="scrollspy">Overview</h4>

            <p>This part starts with the bacteria movement ability, analyze the interaction between bacteria and the special film, and obtain the requests of the material properties eventually.</p>

            <div class="divider"></div>

            <h4 id="force-of-single-bacterium" class="scrollspy">Force of Single Bacterium</h4>

            <p>Assume the driving force of the movement of single bacteria is \(F_{0}\) in horizontal plain.</p>

            <p>\(F_{0}=f_{1}=kV_{1}\)</p>

            <p>Where rho represents the density of solution, V indicates the extra volume of liquid immersed with bacteria, g means the acceleration of gravity. Because of the same interaction between solution and bacteria, in gravitational field, bacteria share the same k with the horizontal movement constant.</p>

            <p>Assume the motor ability of each bacteria does not change. Then, in accordance with the data in previous research, the speed of bacterial movement((\(V_{1}\))) is approximately ~10\(\mu m\)/s, while the speed of bacterial sedimentation((\(V_{2}\))) ~\(\mu m\)/s.</p>

            <div class="divider"></div>

 

 

            <p>Firstly, let's check the geometric size of our special film I. The film is shown in green edge and the clip that used to fix the film presents in black edge. The film I is a circle with the <strong>radius(r)</strong> of <strong>2cm</strong>. The <strong>thickness(d)</strong> of film is <strong>0.1mm</strong>.</p>

            <p>Assume the <strong>numerical density(σ)</strong> of bacteria is <strong>~\(1000 /mm^{2}\)<strong>, which means a single bacteria occupying the area of approximately <strong>~100 \(\sim 100 \mu m^2\)</strong>.</p>

            <p><strong>\(\Delta P= \frac{\sum F}{S}=\sigma F_{0}\)</strong></p>

            <p>Calculating with these data, we concluded additional pressure <strong>ΔP</strong> is <strong>~0.001Pa</strong>.</p>

            <p>The film is subject to several forces acting, including force f given by bacteria and tensile force F through the film.</p>

            <p>where f is the resultant force of the bacteria, F is the tensile force in the film, h is the deformation distance, r is the radius of the film,Δr is the variation of the radius, which is r, d is the thickness of the film, G is the Young Modulus. Solving these equations, we are able to get Young Modulus, a critical physical parameters to depicting the characteristic of film candidates.</p>

            <p><strong><em>Film II</em></strong></p>

            <p>Similarly, we are able to calculate the characteristic of film.</p>

            <p>Assume that 1% of bacteria are push ahead statistically, then the additional pressure given by bacteria is <strong>\(\Delta P=0.01 \times \sigma F_{0}\)</strong>. After solving these equations above, we ultimately get the Young modulus of material required <strong>G &lt;1GPa</strong> in order to get ~um order deformation.</p>

            <p>There are some common material's Young modulus,</p>

            <table>

              <thead>

                <tr>

                  <th>Material type</th>

                  <th>Young modulus(GPa)</th>

                </tr>

              </thead>

              <tbody>

                <tr>

                  <td>Gray cast iron</td>

                  <td>118~126</td>

                </tr>

                <tr>

                  <td>Carbon steel</td>

                  <td>206</td>

                </tr>

                <tr>

                  <td>Roll copper</td>

                  <td>108</td>

                </tr>

                <tr>

                  <td>Brass</td>

                  <td>89~97</td>

                </tr>

                <tr>

                  <td>Roll aluminium</td>

                  <td>68</td>

                </tr>

                <tr>

                  <td>Roll zinc</td>

                  <td>82</td>

                </tr>

                <tr>

                  <td>Lead</td>

                  <td>16</td>

                </tr>

                <tr>

                  <td>Rubber</td>

                  <td>0.00008</td>

                </tr>

                <tr>

                  <td>Polyamides</td>

                  <td>0.011</td>

                </tr>

                <tr>

                  <td>High pressure polyethylene</td>

                  <td>0.015~0.025</td>

                </tr>

                <tr>

                  <td>Low pressure polyethylene</td>

                  <td>0.49~0.78</td>

                </tr>

                <tr>

                  <td>Polypropylene</td>

                  <td>1.32~1.42</td>

                </tr>

              </tbody>

            </table>

            <p>Therefore, we selected several materials as possible films: low pressure polyethylene, rubbers, and glass as our material of the film.</p>

            <p>Just think about it, if every bacteria pull together, we can detect the deformation even the film is made of steel and even bigger than steel.</p>

          </div>

            <h4 id="bacterial-absorption-simulation" class="scrollspy">Bacterial Absorption Simulation</h4>

            <p>Let's assume variables firstly, </p>

            <p><strong>[C]</strong>: Concentration of bacteria.(/\(m^{3}\))</p>

            <p><strong>S</strong>: Area of film.(\(m^{2}\))</p>

            <p><strong>V</strong>: Average swiming speed of bacteria.(m/s)</p>

            <p><strong>Vz</strong>: Average swiming speed component in the z axis, which is perpendicular to S.(m/s)</p>

            <p><strong>σ</strong>: Density of the adhesive bacteria.(/\(m^{2}\))</p>

            <p><strong>N</strong>: Total number of sticked bacteria.</p>

            <p><strong>m</strong>: Movement percentage of bacteria.(%)</p>

            <p><strong>M</strong>: Exact movement number of bacteria.</p>

            <p><strong>Bacterial Absorption Dynamics Fits Langmuir Equation</strong></p>

            <p>Assuming velocity of bacteria in any direction is the same, let's set it <strong>V</strong>,</p>

            <figure>

              <img src="https://static.igem.org/mediawiki/2015/1/17/008.png">

              </figcaption>

            <p>and irrespective of contacting with each other when swimming, we are able to conclude bacterial movement is free.</p>

            <p>Then we calculate the average velocity in z axis, </p>

            <p>\(V_{z}=\int_{0}^{\frac{\pi }{2}}V\times cos\theta \times \frac{dS}{S}=\int_{0}^{\frac{\pi }{2}}\frac{V}{2}cos\theta sin\theta d\theta =\frac{V}{4}\)</p>

            <p>Consider during the interval <strong>dt</strong>, in area <strong>dS</strong>, there are <strong>dN</strong> bacteria in tiny volume <strong>\(dS \times V_{z}dt\)</strong> hitting the wall whose area is <strong>S</strong>. Schematic image illustrates the process mentioned above,</p>

            <figure>

              <img src="https://static.igem.org/mediawiki/2015/d/d2/009.png">

              </figcaption>

            <p>Consequently we can know that</p>

            <p>\(dN=C\times dS\times V_{z} dt\)</p>

            <p>Because the number of bacteria in solution is much more larger than it on the glass surface whose area is <strong>S</strong>. So the concentration of bacteria (<strong>C</strong>) can be regarded as steady during the measurement period.</p>

            <p>Assuming the number of hitting wall bacteria is stable, the surface can only adhere one layer of bacteria, that is the place which has already adhered bacteria can not stick any more bacteria. This means we could use Langmuir Adsorption Isotherm to solve this problem!</p>

            <p>Let's see how we figure out this issue using Langmuir Equation. Consider in a current area <strong>S</strong>, the density of bacteria on surface is <strong>σ</strong>, and during a interval <strong>dt</strong>, the change of bacteria density is <strong>dσ</strong></p>

            <p>\(Sd\sigma =K_{a}\times (1-\frac{\sigma }{\sigma _{0}})\times dN-K_{d}\times S\sigma dt\)<p>

            <p>Where <strong>Ka</strong> is the adhesive rate of each hit, <strong>Kd</strong> is the drop rate of the adhered bacteria.</p>

            <p>Then, after solving this Ordinary Differentiate Equation, ODE, we got the equation shown below,</p>

            <p>\(\sigma (t)=\frac{K_{a}CV_{z}\sigma _{0}}{KCV_{z}+K_{a}\sigma _{0}}\times (1-e^{-\frac{K_{a}CV_{z}+K_{d}\sigma _{0}}{\sigma _{0}}\times t})\)</p>

            <p>According to this result, the density of bacteria at time t is related to adhesive rate of bacteria <strong>Ka</strong>, concentration of bacteria <strong>c</strong> and velocity of bacteria <strong>Vz</strong>. And we can let <strong>\(\frac{K_{a}CV_{z}}{K_{d}\sigma _{0}+K_{a}CV_{z}}\)</strong> equals an integrative constant, <strong>K</strong>. In addition, we cannot start recording the image data the moment we put the bacteria on the cover glass in reality, so there should be a time delay in our modeling. Thus we got the accurate equation depicting the density of bacteria time t as,</p>

            <p>\(\sigma (t)=K\sigma _{0}\times (1-e^{-\frac{K_{a}CV_{z}}{K\sigma _{0}}\times (t-t_{0})})\)<p>

            <p>In order to fit the data conveniently, we change the equation form into a more general one,</p>

            <p>\(\sigma (t)=ae^{-bt}+c\)<p>

            <p><strong>Bacteria Density Simulation</strong></p>

            <p>With assuming a constant value group:</p>

            <table>

              <thead>

                <tr>

                  <th>Ka</th>

                  <th>Kd</th>

                  <th>C</th>

                  <th>Vz</th>

                  <th>σ0</th>

                </tr>

              </thead>

              <tbody>

                <tr>

                  <td>0.5</td>

                  <td>0.01</td>

                  <td>10^9/m^3</td>

                  <td>5um/s</td>

                  <td>10^10/m^2</td>

                </tr>

              </tbody>

            </table>

            <p>we got the plot expressing the bacteria density variate through time, </p>

              <img src="https://static.igem.org/mediawiki/2015/9/99/20150901024.png">

            <p>This program is used to calculate the amount of bacteria and get the percentage of moving bacteria, which is essential for our further research. More details on our coding using Matlab please refer to <a href="https://github.com/Cintau/2015USTCiGEM/">2015 USTC in Github</a>.</p>

            <p>The principle of programming is told below:</p>

            <p>Using MATLAB® to simulate these data with the function '<strong>f(x)=a exp(-b*x)+c</strong>', here is our analysis,</p>

            <p>Now we know that bacteria can move straight because of its flagellum rotating counterclockwise according to previous research. However, due to the stickiness of PLL, some flagella may stick to film when spinning. Assuming the rate of stick (P) is constant, the movement percentage of bacteria will present in an exponential form changing through time.</p>

            <p>Substituting this function into fitting result, we can get the real starting time of each test. Results are delivered below:</p>

            <p>It is interesting that we could know the "Initiation moment" through our data analysis, which is very cool.</p>

            <div class="divider"></div>

            <h4 id="experiment-guidance" class="scrollspy">Experiment Guidance</h4>

            <p>In Adhesion Assay we finally could know several important things, </p>

              <li><strong>Film-coating time</strong></li>

              <li><strong>Bacteria-film interaction time(Ti)</strong></li>

              <li><strong>Concentration of the bacteria solution,</strong></li>

            <p>All of these can be known through the pre-test results analysis.</p>

            <p><strong><em>Film-coating time</em></strong></p>

            <p>Through the pre-test and data in paper, we use 20ug/ml poly-L-lysine coating the film <strong>over 4 hours or overnight</strong> at the temperature <strong>4℃</strong> eventualy.</p>

            <p><strong><em>Bacteria-film interaction time(Ti)</em></strong></p>

            <p>Because the motility of bacteria will decrease when we do not administrate antibiotics. So we need to balance the total number of bacteria and motility. If we implement the data from "PAO1-PLL-0" assay as sample to analyse the best time of bacteria-film interaction time. We got the best interaction time is the moment when the number of movement bacteria reach the maximum value.</p>

            <p>\(M=S\sigma m\)</p>

            <p>\(M=c(1-e^{-bt})\times m_{0}e^{-kt}\)</p>

            <p>Now we can simulate those data!</p>

            <figure>

              <img src="https://static.igem.org/mediawiki/2015/8/88/20150907052.png">

              </figcaption>

            <p>Thus we highly recommend the <strong>Bacteria-film intraction time(Ti) is approximately 100s after treating with bacteria solution</strong>, which means since you inoculate the bacteria about 100s, you should put the film into your sample solution and determine its antibiotics concentration.</p>

            <p><strong>Concentration of bacteria solution</strong></p>

            <p><strong><em>Observation Moment</em></strong></p>

            <p>If we want to observe the deformation of the film, the bacterial additional pressure on film should reach a stable stage. Assume there are K% of inactivated bacteria move when administrating with antibiotics. Then the movement percentage differential equation will be changed to this:</p>

            <p>\(\frac{dm}{dt}=(1-m)K-bm\)</p>

            <p>After solving this differential equation, we received a <em>m to t</em> function:</p>

            <p>\(m(t)=\frac{K+(m_{0}k-(1-m_{0})K)e^{-(K+k)t}}{K+k}\)</p>

            <p>Let \(m_{0}\)=0.5, and choose different 'K'. We are able to acquire different simulating curve.(According to previous analysis, 'K'=0.0065.)</p>

            <figure>

              <img src="https://static.igem.org/mediawiki/2015/1/1e/20150908055.png">

              </figcaption>

            <p>Very luckily, that's very similar to our experiment data,</p>

            <figure>

              <img src="https://static.igem.org/mediawiki/2015/c/c3/Finalresult.png">

              </figcaption>

            <p>If we want to limit the error probability less than 5%, we need the movement percentage reach over 95% of the maximum or minimun value. According to the simulation result, when 'm' reach 95% of its maximum or minimum value\(\frac{K}{K+k}\), t~100s.</p>

            <p>Consequently, we recommend user to get the results in NDM, <strong>you should wait approximately 100s after putting film into optical path.</strong></p>

          </div>

            <h4 id="pre-experiment" class="scrollspy">Pre-experiment</h4>

             <p>Our experiment is initiated from interference fringes pattern formed naturally. One thing need to know is that <strong>the different clip we implement in our pre-experiment will get different fringes.</strong></p>

            <p><strong>(I)</strong> If we use clip I (the round clip),</p>

            <figure>

              <img src="https://static.igem.org/mediawiki/2015/5/5f/QQ%E6%88%AA%E5%9B%BE20150916223725.png">

            <p> we will get Newton's rings like fringes.</p>

            <p>In the pre-experiment1 (<strong>you can read protocol of our pre-experiment in annex showed below</strong>).</p>

            <p>As a matter of fact, we captured a picture like this,</p>

            <figure>

              <img src="https://static.igem.org/mediawiki/2015/9/98/20150823001.jpg">

              Figure 2: Film I pre-test interference fringes image

              <img src="https://static.igem.org/mediawiki/2015/2/2a/QQ%E6%88%AA%E5%9B%BE20150916223708.png">

              </figcaption>

            <p>In the pre-experiment2 (<strong>you can read protocol of our pre-experiment in annex showed below</strong>).</p>

            <p>As well, we captured a picture like this,</p>

            <figure>

              <img src="https://static.igem.org/mediawiki/2015/3/32/20150917060.png">

              </figcaption>

            <p>That's a typical equal thickness interference fringes image.</p>

            <div class="divider"></div>

            <h4 id="modeling-prinicple" class="scrollspy">Modeling Prinicple</h4>

            <p><strong>(1)</strong> In method 1, we choose clip I.</p>

            <p>Let's concentrate on the deformation of film.</p>

            <p>As the deformation range(h) is much more smaller than the radius(r) of the film (h&lt;&lt;r),</p>

            <figure>

              <img src="https://static.igem.org/mediawiki/2015/c/c0/20150831001.png">

              </figcaption>

            <p>we can consider the light is approximately paraxial spherical.</p>

            <p>Because the area of CCD camera is relatively small, whose size is ~cm x cm, the interference pattern is approximately paraxial spherical as well.</p>

            <p>Ideally, light path sketch should be as follows. (Light shown in <strong>blue</strong> line is the reflected light from <strong>holophote</strong>, light shown in <strong>red</strong> line is the reflected light from <strong>film</strong>).</p>

            <figure>

              <img src="https://static.igem.org/mediawiki/2015/f/f7/20150831002.png">

              </figcaption>

            <p>Where L is the distance form the virtual image to the CCD camera.</p>

            <p>However, in reality, because the virtual image of holophote and the film can not be set strict parallel in actual situation, the light path sketch becomes like below, where light shown in <strong>blue</strong> line is the reflected light from <strong>holophote</strong>, light shown in <strong>red</strong> line is the reflected light from <strong>film</strong>.</p>

            <figure>

              <img src="https://static.igem.org/mediawiki/2015/1/18/02150831003.png">

              </figcaption>

            <p>We could use the method of coordinate transformation to simplify them as follows, where light shown in <strong>blue</strong> line is the reflected light from <strong>holophote</strong>, light shown in <strong>red</strong> line is the reflected light from <strong>film</strong>.</p>

            <figure>

              <img src="https://static.igem.org/mediawiki/2015/a/a2/20150831004.png">

              </figcaption>

            <p>These are the parameters we preset shown in table,</p>

            <table>

              <thead>

                <tr>

                  <th>r</th>

                  <th>h</th>

                  <th>a</th>

                  <th>θ</th>

                </tr>

              </thead>

              <tbody>

                <tr>

                  <td>0.02m</td>

                  <td>5e-6m</td>

                  <td>0.02m</td>

                  <td>5e-4rad</td>

                </tr>

              </tbody>

            </table>

            <p>where <strong>r</strong> is the radius of the film, <strong>h</strong> is the deformation length of the film, <strong>a</strong> is the length of each side of the CCD camera, <strong>θ</strong> is the slip angle between the film and the holophote which we estimate.</p>

            <p>Using these parameters, we simulated the interference fringes image as following,</p>

            <figure>

              <img src="https://static.igem.org/mediawiki/2015/c/cd/0.02-5e-6-5e-4.jpg">

              </figcaption>

            <p>How amazing our model is ! Simulational result just hit the row image we got before!</p>

            <div class="divider"></div>

            <h4 id="fringe-analysis-protocol" class="scrollspy">Fringe Analysis Protocol</h4>

            <p><strong>Methods</strong></p>

            <p>1.Take a series of photos at the same position in a short time.</p>

            <p>2.Superpose these photos to sharp the edge of every object.</p>

            <p>3.Choose two points in multi-image, the point must be on the black fringes.</p>

            <p>4.Scaning these two fringes to find the shortest distance between them.</p>

            <p>5.Calculate the radius and rank of every fringes.</p>

            <p>6.Calculate the deformation of film.</p>

            <p>More details on coding please refer to <a href="https://github.com/Cintau/2015USTCiGEM">Github:2015USTCiGEM</a>.</p>

            <div class="divider"></div>

            <h4 id="annex" class="scrollspy">Annex</h4>

            <p><strong>Pre-experiment Protocol</strong></p>

            <p>Optical path image in pre-experiment is as follows</p>

            <p><img src="https://static.igem.org/mediawiki/2015/d/d9/20150831005.png" alt="Figure 12: Pre-experiment light path"></p>

            <p>Light shown in <strong>red</strong> line is the light from <strong>laser</strong>, light shown in <strong>green</strong> is reflected by <strong>film</strong>, light shown in <strong>purple</strong> is reflected by <strong>holophote</strong>.</p>

            <p>The wave length of our laser is <strong>650nm</strong>.</p>

            <p><strong>The distance between 50% reflector and film is about 10cm.</strong></p>

            <p>The film was covered tightly on a tube's hole, and we give a perturbation on temperature to change the pressure in tube. So the film would deform too, then we could simulate the bacteria force in a physical way.</p>

            <p>In method II:</p>

            <p>Because the interference fringes are similar to the situation that the film does not deform, we can observe the interference with the film directly sticked on the clip.</p>

          </div>

    <div id="Calibration" class="row">

      <div class="card hoverable">

        <div class="col s12 m9">

          <div class="card-content">

            <p>This part wil give the basic method of establish a bio-calibration of the concentration of antibiotics. Using modeling prediction and results developed in previous work, we propose a transform formula between concentration and deformation. We also get the deformation of film from interference fringes by using the interference analyzing program.</p>

 

            <h4 id="basic-hypotheses" class="scrollspy">Basic hypotheses</h4>

            <p><strong>Hypotheses</strong></p>

            <p>Let's assume that <strong>bacteria response on antibiotic  is linear to antibiotic concentration to a supposing n power.</strong> That is \(K=CA^{n}\)</p>

            <p>And let's show the variable lists before we get started:</p>

            <p><strong>[A]</strong>: Concentration of antibiotics.</p>

            <p><strong>m</strong>: Percentage of moving bacteria.(%)</p>

            <p><strong>M</strong>: Exact movement number of bacteria.</p>

            <p><strong>h</strong>: Deformation length of the film.</p>

            <p><strong>m0</strong>:Percentage of moving bacteria at the time we start the test.</p>

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            <h4 id="concentration-motility" class="scrollspy">Concentration→Motility</h4>

            <p>The motility of bacteria can be divided into two parts: original motility and induced motility (by antibiotics). According to our previous work, Adhesion Dynamics, we conclude the relationship between the motility and time. That is</p>

            <p>\(m(t)=\frac{K+(m_{0}k-(1-m_{0})K)\ e^{ -(K+k)t}}{K+k}\)</p>

            <p>More information about the formula please refer to <a href="https://github.com/Cintau/2015USTCiGEM">Github:2015USTCiGEM</a></p>

            <p>Because we assume the effect of antibiotics on motility is n power, we can define \(K=CA^{n}\).</p>

            <p>When the movement percentage reachs a steady state, that means \(m=\frac{K}{K+k}\), we can infer that \(m=\frac{CA^{n}}{CA^{n}+k}\).</p>

            <p>And due to the concentration of bacteria solutions we used are almost the same(OD~0.3), the AA-B-CS will be almost the same as well. That means the total number 'N' and the 'k' will be almost the same, which is very important in our standardized operation.</p>

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            <h4 id="motility-deformation" class="scrollspy">Motility→Deformation</h4>

            <p>According to the analysis in <strong>Film Candidate</strong> we can know the relation between deformation and bacteria's movement. That is</p>

            <p>\(f=MF_{0}\)</p>

            <p>\(f=2F\times \frac{h}{b}\)</p>

            <p>\(\frac{F}{ac}=\frac{\Delta b}{b}G\)</p>

            <p>\(\Delta b=\frac{b}{2}\times (\frac{h}{b})^{2}\)</p>

            <p>So \(f=acG(\frac{h}{b})^{3}\)</p>

            <p>Now we can substitute each equations and predict the relationship between deformation <strong>'h'</strong> and concentration <strong>'A'</strong>:</p>

            <p>\(\frac{CA^{^{n}}}{CA^{n}+k}N=acG(\frac{h}{b})^{3}\)</p>

            <p>\(\frac{acG}{Nb^{3}}(1+\frac{k}{CA^{n}})=\frac{1}{h^{3}}\)</p>

            <p>That means 1/h^3 is linear to 1/A^n.</p>

 

            <p>According to the modeling of fringes analysis we conclude the relationship between deformation <strong>'h'</strong> and the changes of number of fringes on y axis (<strong>\(\Delta N\)</strong>) is: <strong>\(\Delta N=\frac{2h}{\lambda }\)</strong></p>

            <p>\(\frac{acG\lambda ^{3}}{8Nb^{3}}(1+\frac{k}{CA^{n}})=\frac{1}{\Delta N^{3}}\)</p>

            <p>Define \(\frac{acG\lambda ^{3}}{8Nb^{3}}\) as a complex constant \(A_{0}\), \(\frac{ackG\lambda ^{3}}{8NCb^{3}}\) as another complex constant \(B_{0}\), and simplify the formula as below:</p>

            <p>\(A_{0}+\frac{B_{0}}{A^{n}}=\frac{1}{\Delta N^{3}}\)</p>

            <p>So if we measure the two constants <strong>\(A_{0}\)&amp;\(B_{0}\)</strong>, we can build a calibration on concentration detecting. And when we get a solution with unknown concentration, we can test it and find the concentration in calibration!</p>

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            <h4 id="fringes-analysis" class="scrollspy">Fringes analysis</h4>

            <p><strong>Methods</strong></p>

            <p>1.Use matlab recognize the number of fringes in each image.</p>

            <p>2.Collect the output information from program and calculate the change fringe numbers in average.</p>

            <p>3.Find the concentration in calibration.</p>

            <p>More details on coding please refer to <a href="https://github.com/Cintau/2015USTCiGEM">Github:2015USTCiGEM</a></p>

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