Difference between revisions of "Team:Fudan China/IntModel"

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         <h1>Modeling</h1>
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         <h1>Integrase Model</h1>
 
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             <h2 style="top: 135px;">Parameter</h2>
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             <h2 style="top: -10px;">Overview</h2>
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     <h2 id="origin">The origin model of DNA recombination by integrase</h2>
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     <h2 id="ori">The origin model of DNA recombination by integrase</h2>
 
     <div class="row">
 
     <div class="row">
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         <p>First of all, we use a simple model to describe the main process of the integrase recombination. (figure1.1)
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         <p>First of all, we use a simple model to describe the main process of the integrase recombination. (figure1.1)</p>
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            </p><p>A. Predicting the efficiency of integrase in vivo. It confirmed the feasibility of our circuit when we designed it.
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    </p><p>B. Simulating the diffusion process of the repressor and trying to get a reasonable solution of the differential equations to interpret the leakage of integrases when our wet lab experiment went wrong.
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    </p><p>  All the results is calculated on MATLAB or Mathemetica.
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        </p>
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    </div>
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    <h3>First part [<a href="https://2017.igem.org/Team:Fudan_China/IntModel">Details</a>]</h3>
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    <div class="row">
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        <p>In this part, we set up an origin model of DNA recombination by integrase, the reaction function goes as follows:</p>
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         <div class="figure" style="width:90%;">
 
         <div class="figure" style="width:90%;">
 
             <img src="https://static.igem.org/mediawiki/2017/7/74/T--Fudan_China--Model--integrase.png"/>
 
             <img src="https://static.igem.org/mediawiki/2017/7/74/T--Fudan_China--Model--integrase.png"/>
 
             <b>Figure 1.1</b> | “PB” is the attP and attB sites of DNA, and “LR” is the recombination product, attL and attR sites, which are direction-changed, of the DNA. PBI is the complex of DNA and 4 integrase molecule. LRI1 and LRI2 are two kinds of complex which are conformationally distinct from PBI. The “single arrow” represents the reaction which can reach rapid equilibrium with the equilibrium constant over it. The “double arrow” represents the slow reaction with the reaction rate constant over and below it. “K(bI)” is the equilibrium constant. “k(+r), “k(-r)”, “k(+syn)”, “k(-syn)” is the reaction rate constant. The site direction change occur in the second reaction: “PBI” to “LRI(1)”.
 
             <b>Figure 1.1</b> | “PB” is the attP and attB sites of DNA, and “LR” is the recombination product, attL and attR sites, which are direction-changed, of the DNA. PBI is the complex of DNA and 4 integrase molecule. LRI1 and LRI2 are two kinds of complex which are conformationally distinct from PBI. The “single arrow” represents the reaction which can reach rapid equilibrium with the equilibrium constant over it. The “double arrow” represents the slow reaction with the reaction rate constant over and below it. “K(bI)” is the equilibrium constant. “k(+r), “k(-r)”, “k(+syn)”, “k(-syn)” is the reaction rate constant. The site direction change occur in the second reaction: “PBI” to “LRI(1)”.
 
         </div>
 
         </div>
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         <p>We use “PB” and “LR” to represent the sites of DNA being recombined or not separately and “Int” to represent the integrase. The others represent the intermediates.(figure.1)
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         <p>Based on existing research, we know that the combination and dissociation process between DNA and the integrase are very fast. Therefore, we assume that the first and forth step in our model can reach rapid equilibrium. Both of the reaction share the same equilibrium constant, KbI. [1]
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        </p><p>  Regulating all the reaction process by appropriate mathematical functions, we figure out the efficiency of the integrase recombination in vitro, where there is no substrate production, dilution and degradation.
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    </p><p>   Then, we get the figure1.2 to interpret how the recombination rate changes over time. And as the graph shows, the transformation rate can reach almost 80 percent within 0.5 hour approximately. It confirms the possibility that our circuit based on the serine integrase possess the capacity to make regular response to the clock signal at the hour timescale.
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        </p><p>   Thus, we are able to describe the whole process above with four equations.
 
         </p>
 
         </p>
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        <div class="figure" style="width:80%;">
 +
            <img src="https://static.igem.org/mediawiki/2017/b/b9/T--Fudan_China--Model--integrase_equations.PNG"/>
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        </div>
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        <p>The origin model composed of all the functions above can perfect predict the recombination reaction in vitro[1], where the concentration of integrase is considered to be constant, because the concentration of integrase required for efficient recombination is much higher than the one of the DNA substrate. The simulation results are shown here. (figure1.2)</p>
 
         <div class="figure" style="width:90%;">
 
         <div class="figure" style="width:90%;">
 
             <img src="https://static.igem.org/mediawiki/2017/c/c4/T--Fudan_China--Model--vitro.png"/>
 
             <img src="https://static.igem.org/mediawiki/2017/c/c4/T--Fudan_China--Model--vitro.png"/>
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             <b>Figure 1.2</b> | Figure1.2 It separately describes how the proportion of substrate “PB” and product “LR” in total DNA varies within 10 hours. The red line represents the concentration proportion of the product, “LR”. More calculation details about this graph are attached to the back.
+
             <b>Figure 1.2</b> | It separately describes how the proportion of substrate “PB” and product “LR” in total DNA varies within 10 hours. The red line represents the concentration proportion of the product, “LR”. More calculation details about this graph are attached to the back.
 
         </div>
 
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         <p>Then, we improve the origin model by adding gene expressing and other biochemical reaction to predict the theoretical efficiency of the integrase in vivo.
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         <p>As the graph shows, the transformation rate can reach almost 80 percent in general.</p>
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          </p><p> We are not sure whether the integrase is effective as we expect in the coli, even if it prefers well in vitro, so we try to build a more integrated model. Considering that the factors in the coli are so complicated, we merely select the fairly significant factors as the component of our model, including the gene expression, the change of the promoter’s activity and the dilution and the digestion of the protein.
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    </div>
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     </p><p>Finally, we get the exciting results of the model shown in the figure3. The transformation rate can reach to nearly 100 percent in 5 hours. The mathematical demonstration will explain how the factors in vivo help to shift the equilibrium forward.
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    <h2 id="imp">The improved model with gene expression</h2>
 +
    <div class="row">
 +
        <p>Though the origin model shows great accuracy in vitro, we aim to figure out how the circuits work in vivo experiment with gene expression.
 +
 
 +
        </p><p>  Considering the fact that the combination between the inducer and promoter are much faster than the production of integrase, we suppose that the activity of the promoter we use changes instantaneously.
 +
 
 +
     </p><p>   Furthermore, the concentration of the protein shall decrease as a result of degradation and dilution.
 
         </p>
 
         </p>
 +
        <div class="figure" style="width:30%;">
 +
            <img src="https://static.igem.org/mediawiki/2017/9/9b/T--Fudan_China--Model--gene_expression.PNG"/>
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        </div>
 +
          <p>The maximum promoter activity can be estimated by the following equation</p>
 +
        <div class="figure" style="width:35%;">
 +
            <img src="https://static.igem.org/mediawiki/2017/b/b5/T--Fudan_China--Model--promoter_activity.PNG"/>
 +
        </div>
 +
        <p>For the serine integrase, we suppose only the dilution exists. And the dilution rate can be estimated as follows.</p>
 +
        <div class="figure" style="width:35%;">
 +
            <img src="https://static.igem.org/mediawiki/2017/3/32/T--Fudan_China--Model--dilution_rate.PNG"/>
 +
        </div>
 +
        <p>Last but not least, with the cell division, the new DNA is produced without integrase combination. So the origin model can be adapted to a new form as follows.</p>
 +
        <div class="figure" style="width:50%;">
 +
            <img src="https://static.igem.org/mediawiki/2017/f/ff/T--Fudan_China--Model--improved_integrase_equations_with_gene_expression_and_dilution.PNG"/>
 +
        </div>
 +
        <p>As the graph shows(figure2.1), the transformation rate can reach almost 100 percent in general. The critical difference between the vivo and vitro environment is the replication of DNA which decrease the concentration of <i>LRI<sub>1</sub></i> and <i>LRI<sub>2</sub></i> and increase the one of LR relatively and partially. Such process is a special kind of “transformation” which can skip the slow equilibrium process., ”syn”, to achieve a higher transformation rate in a short time. </p>
 
         <div class="figure" style="width:90%;">
 
         <div class="figure" style="width:90%;">
 
             <img src="https://static.igem.org/mediawiki/2017/e/e3/T--Fudan_China--Model--vivo2.png"/>
 
             <img src="https://static.igem.org/mediawiki/2017/e/e3/T--Fudan_China--Model--vivo2.png"/>
 
             <b>Figure 2.1</b> | The figure separately describes how the proportion of substrate “PB” and product “LR” in total DNA varies within 10 hours in vivo. The red line represents the concentration proportion of the product, “LR”. More calculation details about this graph are attached to the back.
 
             <b>Figure 2.1</b> | The figure separately describes how the proportion of substrate “PB” and product “LR” in total DNA varies within 10 hours in vivo. The red line represents the concentration proportion of the product, “LR”. More calculation details about this graph are attached to the back.
 
         </div>
 
         </div>
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    <h3>Second part [<a href="https://2017.igem.org/Team:Fudan_China/DifModel">Details</a>]</h3>
 
−
    <div class="row">
 
−
        <p>In this part, we use the diffusion model to solve an experimental problem.
 
  
−
        </p><p>  Although the models above can describe the whole process of our system in general, the experimental data still do not perform so well as our expectation because all the promoters are leaked. According to this case, we come up with an assumption that the diffusion of the repressor results in the lower concentration around the promoter than the one in center of the gene site where the repressor protein produced. Therefore, the lower concentration of repressor causes the high probability of separation between the repressor and promoter. That explained why the promoters were leaked seriously.
 
  
−
     </p><p>    In order to describe the change of the concentration with the distance in cell, we build a diffusion model. We find that the concentration of the repressor will decline rapidly due to the diffusion. However, when there is a source in the center, for example the translation of mRNA causes a steady flow of repressor produced, the concentration distribution will get constant in the end. The results are shown in figure3.2.
+
     </div>
  
−
    </p><p>    The average distance (relative) between the promoter and the repressor source is about 0.17 and the corresponding concentration (relative) is 0.6577 which means when the repressor get to the promoter, its concentration decreases more than 33 percent. It gives us an reasonable explanation to the problem why all the promoters are leaked seriously. According to the enlightenment of the model, we take two kinds of measures to ease the problem.
 
−
 
−
    </p><p>    First, we increase the plasmid copy number so as to increase the dense of the point sources and reduce the average distance.
 
−
 
−
    </p><p>    Second, we can use only one kinds of plasmid for both repressor production and the integrase production, so the source of the repressor will get extremely close to the promoter.
 
−
        </p>
 
−
        <div class="figure" style="width:90%;">
 
−
            <img src="https://static.igem.org/mediawiki/2017/e/e7/T--Fudan_China--Model--diffusion_improved.png"/>
 
−
            <b>Figure 3.2</b> | The concentration distribution describes the steady relative concentration of the integrase varies with the relative position to the plasmid which we consider as the point source of repressor production in E.coli. The arrow marks out the average distance between the promoter and the point source as well as the corresponding concentration. We suppose both the average radius of the cell and the concentration of the point source as the unit 1.
 
−
        </div>
 
−
        <p>In general, the model not only helped us to demonstrate and analyze our project, but also gives us a vigorous tool to solve the unexpected problem. You can see more detailed mathematical demonstration is at the back of each part.</p>
 
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    </div>
 
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    <h2 id="para">Parameter</h2>
 
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    <h2 id="sp">Species</h2>
 
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    <div class="figure" style="width:90%;">
 
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        <img src="https://static.igem.org/mediawiki/2017/7/70/T--Fudan_China--Model--species.PNG"/>
 
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Revision as of 06:35, 31 October 2017

The origin model of DNA recombination by integrase

First of all, we use a simple model to describe the main process of the integrase recombination. (figure1.1)

Figure 1.1 | “PB” is the attP and attB sites of DNA, and “LR” is the recombination product, attL and attR sites, which are direction-changed, of the DNA. PBI is the complex of DNA and 4 integrase molecule. LRI1 and LRI2 are two kinds of complex which are conformationally distinct from PBI. The “single arrow” represents the reaction which can reach rapid equilibrium with the equilibrium constant over it. The “double arrow” represents the slow reaction with the reaction rate constant over and below it. “K(bI)” is the equilibrium constant. “k(+r), “k(-r)”, “k(+syn)”, “k(-syn)” is the reaction rate constant. The site direction change occur in the second reaction: “PBI” to “LRI(1)”.

Based on existing research, we know that the combination and dissociation process between DNA and the integrase are very fast. Therefore, we assume that the first and forth step in our model can reach rapid equilibrium. Both of the reaction share the same equilibrium constant, KbI. [1]

Thus, we are able to describe the whole process above with four equations.

The origin model composed of all the functions above can perfect predict the recombination reaction in vitro[1], where the concentration of integrase is considered to be constant, because the concentration of integrase required for efficient recombination is much higher than the one of the DNA substrate. The simulation results are shown here. (figure1.2)

Figure 1.2 | It separately describes how the proportion of substrate “PB” and product “LR” in total DNA varies within 10 hours. The red line represents the concentration proportion of the product, “LR”. More calculation details about this graph are attached to the back.

As the graph shows, the transformation rate can reach almost 80 percent in general.

The improved model with gene expression

Though the origin model shows great accuracy in vitro, we aim to figure out how the circuits work in vivo experiment with gene expression.

Considering the fact that the combination between the inducer and promoter are much faster than the production of integrase, we suppose that the activity of the promoter we use changes instantaneously.

Furthermore, the concentration of the protein shall decrease as a result of degradation and dilution.

The maximum promoter activity can be estimated by the following equation

For the serine integrase, we suppose only the dilution exists. And the dilution rate can be estimated as follows.

Last but not least, with the cell division, the new DNA is produced without integrase combination. So the origin model can be adapted to a new form as follows.

As the graph shows(figure2.1), the transformation rate can reach almost 100 percent in general. The critical difference between the vivo and vitro environment is the replication of DNA which decrease the concentration of LRI1 and LRI2 and increase the one of LR relatively and partially. Such process is a special kind of “transformation” which can skip the slow equilibrium process., ”syn”, to achieve a higher transformation rate in a short time.

Figure 2.1 | The figure separately describes how the proportion of substrate “PB” and product “LR” in total DNA varies within 10 hours in vivo. The red line represents the concentration proportion of the product, “LR”. More calculation details about this graph are attached to the back.