Difference between revisions of "Team:HZAU-China/Model"

 
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         <div class="zhengwen">
 
         <div class="zhengwen">
 
             <div id="float01" class="cur">
 
             <div id="float01" class="cur">
                 <div class="h1">Salmonella infection model</div>
+
                 <div class="h1"><i>Salmonella</i> infection model</div>
                 <p>We want to simulate the situation that tumor cells and Salmonella together in a liquid environment.
+
                 <p>We want to simulate the situation that tumor cells and <i>Salmonella</i> together in a liquid
 +
                    environment.
 
                     We used the law of mass action to establish a model for the infection process of <i>Salmonella</i>,
 
                     We used the law of mass action to establish a model for the infection process of <i>Salmonella</i>,
 
                     which
 
                     which
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                 </p>
 
                 </p>
 
                 <p>
 
                 <p>
                     $$S_{almonella} = S_{almonella0} - N_{normal\_cell} - N_{tumor}$$
+
                     $$S_{almonella} = S_{almonella}(t0) - N_{normal\_cell} - N_{tumor}$$
 
                 </p>
 
                 </p>
 
                 <p>
 
                 <p>
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                 <p>
 
                 <p>
 
                     \(N_{normal\_cell}\): The density of normal cells.<br>
 
                     \(N_{normal\_cell}\): The density of normal cells.<br>
                     \(S_{almonella}\): The density of Salmonella in the liquid environment.<br>
+
                     \(S_{almonella}\): The density of <i>Salmonella</i> in the liquid environment.<br>
 
                     \(N_{tumor}\): The density of tumor cells.<br>
 
                     \(N_{tumor}\): The density of tumor cells.<br>
                     \(N_w\): The number of Salmonella in the normal cells.<br>
+
                     \(N_w\): The number of <i>Salmonella</i> in the normal cells.<br>
                     \(N_s\): The number of Salmonella in the tumor cells.<br>
+
                     \(N_s\): The number of <i>Salmonella</i> in the tumor cells.<br>
                     \(A_w\): The affinity between Salmonella and normal cells.<br>
+
                     \(A_w\): The affinity between <i>Salmonella</i> and normal cells.<br>
                     \(A_s\): he affinity between Salmonella and tumor cells.<br>
+
                     \(A_s\): The affinity between <i>Salmonella</i> and tumor cells.<br>
 +
                    \(S_{al\_normal}\): The density of infected normal cells.<br>
 +
                    \(S_{al\_tumor}\): The density of infected tumor cells.<br>
 
                 </p>
 
                 </p>
                 <p>Salmonella begins to replicate two hours after infection<sup>1</sup> .</p>
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                 <p><i>Salmonella</i> begins to replicate two hours after infection<sup>1</sup> .</p>
 
                 <p>
 
                 <p>
                     $$\dfrac {dN_{sal}} {d_{t}} = K_{break} N_{s} + N_{sal} 2^{\dfrac {t} {T}} \ln{2} \dfrac {1} {T} $$
+
                     $$\dfrac {dN_{sal}} {d_{t}}(t) =$$
 
                 </p>
 
                 </p>
 +
                <p>
 +
                        $$\dfrac {dS_{al}} {d_{t}}(t) + S_{al}(t-2) 2^{\dfrac {t-2} {T}} \ln{2}\dfrac {1} {T} $$
 +
            </P>
 +
                <p>
 +
                  \(T\): Cell cycle.<br>
 +
                  </p>
 
                 <div class="h2">Identification of infection time</div>
 
                 <div class="h2">Identification of infection time</div>
                 <p>According to our experimental results, we noticed that Salmonella follows Poisson distribution in
+
                 <p>According to our experimental results, we noticed that <i>Salmonella</i> might follow Poisson
                     normal cells, and an app was designed to judge the distribution of bacteria in the cells. We assume
+
                    distribution in
 +
                     cells, so we use Matlab to judge the distribution of bacteria in the cells. We assume
 
                     that the area less than 1 in the Poisson distribution is a part of cells which are not infected by
 
                     that the area less than 1 in the Poisson distribution is a part of cells which are not infected by
                     Salmonella. According to our experimental results, cells which is infected by only one Salmonella
+
                     <i>Salmonella</i>. According to our experimental results, cells which are infected by only one <i>Salmonella</i>
 
                     can also die of pyroptosis. Based on this feature, we divide cells into uninfected and infected
 
                     can also die of pyroptosis. Based on this feature, we divide cells into uninfected and infected
 
                     cells. When the average number of bacteria in the cell changes, which means that the λ of Possion
 
                     cells. When the average number of bacteria in the cell changes, which means that the λ of Possion
 
                     distribution changes, the ratio of the two kind of cells will change. In summary, when the average
 
                     distribution changes, the ratio of the two kind of cells will change. In summary, when the average
                     number of Salmonella in cells changes, the proportion of dead cells will change.</p>
+
                     number of <i>Salmonella</i> in cells changes, the proportion of dead cells will change.</p>
 
                 <div style="width: 80%; margin: 0px auto">
 
                 <div style="width: 80%; margin: 0px auto">
 
                     <img src="https://static.igem.org/mediawiki/2018/d/df/T--HZAU-China--model1.png" width=100% alt="">
 
                     <img src="https://static.igem.org/mediawiki/2018/d/df/T--HZAU-China--model1.png" width=100% alt="">
 
                 </div>
 
                 </div>
                 <p><b>Figure 1. Poisson distribution and Salmonella infection results.</b> Figure 1a. Based on
+
                 <p><b>Figure 1. Poisson distribution and <i>Salmonella</i> infection results.</b> <b>a.</b> Based on
                     statistics on
+
                     statistics of
                     experimental results, we proved that the Salmonella follows Poisson distribution in normal cells.
+
                     experimental results, we proved that the <i>Salmonella</i> follows Poisson distribution in normal
                     Figure 1b. We assume that the area less than 1 in the Possion distribution is a part of cells which
+
                    cells.
                     are not infected by Salmonella. When the λ of Possion distribution changes, which means the average
+
                     <b>b.</b> We assume that the area less than 1 in the Possion distribution is a part of cells which
                     number of Salmonella in cells changes, the proportion of infected cells changes. Figure 1c. Cells
+
                     are not infected by <i>Salmonella</i>. When the λ of Possion distribution changes, which means the
                     which is infected by only one Salmonella can also die of pyroptosis.</p>
+
                    average
 +
                     number of <i>Salmonella</i> in cells changes, the proportion of infected cells changes. <b>c.</b>
 +
                    Cells
 +
                     which are infected by only one <i>Salmonella</i> can also die of pyroptosis.</p>
 
                 <div class="h2">Infection in tumor cell culture experiments</div>
 
                 <div class="h2">Infection in tumor cell culture experiments</div>
                 <p>We hope that the mathematical model can help the Salmonella infection experiment. In our final
+
                 <p>We hope that the mathematical model can help the <i>Salmonella</i> infection experiment. In our
                     phenotypic experiment, the cell carries the GSDMD gene induced by atc, and we hope that the
+
                    final
                     observed result is that the proportion of atc-induced cell death is more than which is not induced
+
                     phenotypic experiment, the cells carrying the GSDMD gene are induced by ATc, and we hope that the
                     to prove the atc promoter is effective. In this experiment, the error may be big if the proportion
+
                     observed result is that the proportion of ATc-induced cell death is more than that of not induced
                     of cells infected by Salmonella is different. What’s worse is that the experimental results we
+
                     to prove the ATc promoter is effective. In this experiment, the error may be big if the proportion
                    observed may be contrary to the actual situation.</p>
+
                     of cells infected by <i>Salmonella</i> is different.</p>
 
                 <div style="width: 60%; margin: 10px auto">
 
                 <div style="width: 60%; margin: 10px auto">
 
                     <img src="https://static.igem.org/mediawiki/2018/7/73/T--HZAU-China--model2.png" width=100% alt="">
 
                     <img src="https://static.igem.org/mediawiki/2018/7/73/T--HZAU-China--model2.png" width=100% alt="">
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                     <img src="https://static.igem.org/mediawiki/2018/1/10/T--HZAU-China--model3.png" width=100% alt="">
 
                     <img src="https://static.igem.org/mediawiki/2018/1/10/T--HZAU-China--model3.png" width=100% alt="">
 
                 </div>
 
                 </div>
                 <p><b>Figure 2.</b> Results caused by efficiency differences of infection. If the proportions of infections
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                 <p style="width: 100%; text-align: center !important;"><b>Figure 2.</b> Results caused by efficiency
                    are different.</p>
+
                    differences of infection. </p>
                 <p>The experimental results can not prove atc promoter induced pyroptosis. The picture showed that the
+
                <p></p><br>
                    act promoter is induced and caused 90% cells’ death and 70% cells dead because of the promoter
+
                 <p>If the proportions of infection are different, the experimental results may not be able to prove that pyroptosis is induced by atc promoter. Figure 2 showed that the ATc promoter is induced and 90% cells death are caused and 70% cells die because of the promoter disclosure, but the difference of the proportions of infected cells is so big that the experimental results cannot reflect the real situation. We can reduce the difference by letting the infection proportions of the two kinds of cells both close to 100%.</p>
                    disclosure, but the difference of the propotion of infected cells is so big that the experimental
+
                 <p>We solved this problem by predicting the proportion of cells infected with bacteria over time.</p>
                    results are contrary to the truth. However, we can solve this problem by improving the proportions
+
                 <p>Based on these, we designed an App with MatLab (<a href="https://github.com/cccoolll/Pyroptosis.git">https://github.com/cccoolll/Pyroptosis.git</a>)
                    of infected cells as much as possible.</p>
+
                  . In this App, different parameters obtained from measurement experiments can be input to predict the optimal infection time.
                 <p>We solve this problem by predicting the proportion of cells infected with bacteria over time.</p>
+
                     Therefore, this App can provide guidance to the design of our experiments.</p>
                 <p>Based on these, we designed an app with matlab (<a href="">https://github.com/cccoolll/Pyroptosis.git</a>)
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                    . In this
+
                    app, different parameters got from the experiment can be input to predict the experimental results.
+
                     Not only that, the app can povide guidance to our experiments.</p>
+
 
                 <div style="width: 80%; margin: 30px auto">
 
                 <div style="width: 80%; margin: 30px auto">
 
                     <img src="https://static.igem.org/mediawiki/2018/a/a1/T--HZAU-China--model4.png" width=100% alt="">
 
                     <img src="https://static.igem.org/mediawiki/2018/a/a1/T--HZAU-China--model4.png" width=100% alt="">
 
                 </div>
 
                 </div>
                 <p><b>Figure 3.</b> The app we designed. </p>
+
                 <p style="width: 100%; text-align: center !important;"><b>Figure 3.</b> Salmonella infection prediction tool (for tumor cell). </p>              
                <p>1: The change of the number of Salmonella in a single cell of tumor cell and normal cell infection
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                    A1: The predicted <i>Salmonella</i> numbers in tumor cell in the single cell
                    experiments, the red curve is the condition in the cancer cell, and the blue curve is the condition
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                     infection experiment.<br>
                     within the normal cell;<br>
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                     A2: The predicted proportions of infected tumor cells.<br>
                     2: Changes of the proportion of infected cells of tumor cell and normal<br>
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                     A3: The concentration of added <i>Salmonella</i>.<br>
                    cell infection experiments, the blue curve is the condition in the cancer cell, and the red curve
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                     A4: The density of tumor cells;<br>
                    is the condition in the normal cell.<br>
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                     A5: Rate constant of <i>Salmonella</i> infecting tumor cells.<br>
                     3: the concentration of added Salmonella;<br>
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                     A6: The proportion of tumor cells expected to be infected.<br>
                    4: Output value for
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                     A7: The time to reach the wanted proportion of infected cells.<br>                  
                    the result of the optimal infection time.<br>
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                     5:The density of tumor cells;<br>
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                     6: Rate constant of
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                    Salmonella infecting tumor cells.<br>
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                     7: The density of normal cells;<br>
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                     8: Rate constant of Salmonella
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                    infecting normal cells;<br>
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                    9: Changes of the proportion of infected tumor cells of tumor cell
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                    infection experiments.<br> </p>
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                <div class="h2">The parameters Nsal, Tumor and As is obtained from our experiments.</div>
+
                <div class="h3">Guidance for tumor cells infecting experiments</div>
+
 
                 <div style="width: 80%; margin: 30px auto">
 
                 <div style="width: 80%; margin: 30px auto">
                     <img src="https://static.igem.org/mediawiki/2018/a/a1/T--HZAU-China--model5.png" width=100% alt="">
+
                     <img src="https://static.igem.org/mediawiki/2018/3/3a/T--HZAU-China--model4.1.png" width=100% alt="">
 
                 </div>
 
                 </div>
                 <p><b>Figure 4.</b> According to our experimental protocol, the MOI is 100 and the we correspond the MOI to
+
                 <p style="width: 100%; text-align: center !important;"><b>Figure 4.</b> <i>Salmonella</i> infection prediction tool (for mixed culture of tumor cell and normal cell). </p>
                    concentration of cells. The result showed that the infection time is at least 2 hours to eliminate
+
                   
                    unnecessary variables.</p>
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                    B1: The predicted numbers of <i>Salmonella</i> in tumor cell (red) and normal cell (blue) in a single cell infection
                 <div class="h3">Guidance for mixed culture experiments</div>
+
                    experiments.<br>
 +
                    B2: The predicted proportions of infected cells (red for tumor cell and blue for normal cell).<br>
 +
                    B3: The concentration of added <i>Salmonella</i>.<br>
 +
                    B4: The density of tumor cells.<br>
 +
                    B5: Rate constant of <i>Salmonella</i> infecting tumor cells.<br>
 +
                    B6: The density of normal cells.<br>
 +
                    B7: Rate constant of <i>Salmonella</i> infecting normal cells.<br>
 +
                    B8: The predicted optimal infection time.
 +
                    <br> </p><br>
 +
                <p>The parameters Nsal, Tumor and As are measured from our experiments.</p><br>
 +
                <div class="h3">Guidance to tumor cell infection experiments</div>
 +
                <div style="width: 80%; margin: 30px auto">
 +
                    <img src="https://static.igem.org/mediawiki/2018/e/e2/T--HZAU-China--model5.png.png" width=100% alt="">
 +
                </div>
 +
                <p><b>Figure 5 (a part of Figure 3).</b> Guidance for tumor cells infection experiments.<br><br>
 +
                  According to our experiment protocol, the MOI (multiplicity of infection) is 100, corresponding to the concentration of cells. If we want 98% of the tumor cells to be infected, the prediction result show that the infection time should be at least 2 hours to eliminate unnecessary variables.</p>
 +
                 <div class="h3">Guidance to mixed culture experiments</div>
 
                 <div style="width: 60%; margin: 30px auto">
 
                 <div style="width: 60%; margin: 30px auto">
 
                     <img src="https://static.igem.org/mediawiki/2018/b/b7/T--HZAU-China--model6.png" width=100% alt="">
 
                     <img src="https://static.igem.org/mediawiki/2018/b/b7/T--HZAU-China--model6.png" width=100% alt="">
 
                 </div>
 
                 </div>
                 <p><b>Figure 5.</b> In order to reflect the affinity of Salmonella between tumor cells and normal cells, we
+
                 <p><b>Figure 6 (a part of Figure 4).</b> The predicted proportions of infected cells (red for tumor
                     hope that the difference between experimental results of tumor cells and normal cells infecting
+
                    cell and blue for normal cell). <br><br>
                    experiment is obvious. However, the number of bacteria in different cells is difficult to count, we
+
                    In order to reflect the affinity of <i>Salmonella</i> to tumor cells and to normal cells, we
                    can only obvious experimental results by counting the number of infected cells and calculate the
+
                     hope that the difference between experimental results of tumor cells and normal cells is obvious.
                    proportion of infected cells. Therefore, we need to predict the time when the difference of
+
                    However, the numbers of bacteria in different host cells are difficult to count, and we can only count the
                    experimental results are the best.</p>
+
                    number of infected cells and calculate their the proportions. Therefore, we need to
 
+
                    predict the time when the difference is most obvious. Our App just can do this for us. </p>
 
+
  
  
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                 <p>
 
                 <p>
 
                     The Tet repressor protein (tetR) regulates transcription of tetracyclines resistance protein, tetA.
 
                     The Tet repressor protein (tetR) regulates transcription of tetracyclines resistance protein, tetA.
                     The expression of tetA must be strictly regulated since tetA is a membrane-spanning H+-[Tc-Mg]2+
+
                     The expression of tetA must be strictly regulated since tetA is a membrane-spanning H<sup>+-</sup>[Tc-Mg]<sup>2+</sup>
 
                     antiporter which means it can lower the pH environment of cytoplasm. As a result, the natural
 
                     antiporter which means it can lower the pH environment of cytoplasm. As a result, the natural
 
                     circuit of tetracyclines regulation is a negative-feedback circuit<sup>2</sup>. Tc is the inducer,
 
                     circuit of tetracyclines regulation is a negative-feedback circuit<sup>2</sup>. Tc is the inducer,
 
                     which shows
 
                     which shows
 
                     high affinity to tetR protein. The tetR protein binds to tetO sequence on DNA specifically, thus
 
                     high affinity to tetR protein. The tetR protein binds to tetO sequence on DNA specifically, thus
                     inhibits the expression of Tet promoter. When Tc, or other substantial molecule like ATc
+
                     inhibits the expression of Tet promoter. When Tc or other similar molecules like ATc
                     (anhydrotetracycline) diffuse into bacteria, it will bind to tetR protein and unleash the tetR
+
                     (anhydrotetracycline) diffuse into bacteria, they will bind to tetR protein and unleash the tetR
                     protein from DNA, thus release the inhibit and start the expression of Tet promoter.
+
                     protein from DNA, and thus relieve the inhibition and start the expression of Tet promoter.
 
                 </p>
 
                 </p>
 
                 <p> In our project, we choose ATc (anhydrotetracyclines) as the inducer. ATc is less harmful to
 
                 <p> In our project, we choose ATc (anhydrotetracyclines) as the inducer. ATc is less harmful to
 
                     bacteria than Tc and about 100-fold higher affinity to tetR than Tc<sup>2</sup>.
 
                     bacteria than Tc and about 100-fold higher affinity to tetR than Tc<sup>2</sup>.
 
                 </p>
 
                 </p>
                 <p> The ATc model aims to predict and solve two problems: Firstly, how fast dose the circuit react to
+
                 <p> The ATc model aims to predict and solve two problems: first, how fast does the circuit react to
                     ATc; Secondly, how much target gene will express in the bacteria community under a certain
+
                     ATc; second, how many target gene will express in the bacteria community under a certain
 
                     concentration of ATc.
 
                     concentration of ATc.
 
                 </p>
 
                 </p>
 
                 <div class="h2">Hypothesis</div>
 
                 <div class="h2">Hypothesis</div>
 
                 <p>
 
                 <p>
                     There are two tetO sites on the Tet Promoter and both can bind to tetR protein randomly and inhibit
+
                     There are two tetO sites on the Tet promoter and both can bind to tetR protein randomly and inhibit
 
                     the promoter’s expression independently. To make the condition simple, we consider the two tetO
 
                     the promoter’s expression independently. To make the condition simple, we consider the two tetO
                     sites into one as we just want to explain the relationship of the inhibition of the promoter and
+
                     sites into one as we just want to explain the relationship between the promoter inhibition and
                     the tetR protein.
+
                     the tetR protein expression.
 
                 </p>
 
                 </p>
 
                 <p> In our project, the ATc concentration in our incubation environment is uniform, and the diffusion
 
                 <p> In our project, the ATc concentration in our incubation environment is uniform, and the diffusion
                     rate of anhydrotetracycline can be ignored<sup>3</sup> . In spite of this, the degradation rate of
+
                     rate of anhydrotetracycline can be ignored<sup>3</sup>. In spite of this, the degradation rate of
                     ATc under
+
                     ATc under 37℃ must be taken into account as reported<sup>4</sup>.
                    37℃ must be taken into account as reported<sup>4</sup>.
+
 
                 </p>
 
                 </p>
                 <p> Based on these consideration and truth, we give out these hypotheses:
+
                 <p> Based on these facts, we give the following hypotheses:
 
                 </p>
 
                 </p>
                 <p> 1. Regard two tetO operon as one equivalently.<br>
+
                 <p> 1. Regard two tetO operons as one equivalently.<br>
                     2. Ignore the diffusion of ATc<br>
+
                     2. Ignore the diffusion of ATc.<br>
                     3. The reaction time between ATc and tetR, tetR and DNA is much faster than transcription and
+
                     3. The reaction time between ATc and tetR, tetR and DNA is much shorter than transcription and
                     transformation.<br>
+
                     translation.<br>
 
                 </p>
 
                 </p>
 
                 <div class="h2">Description and Equation</div>
 
                 <div class="h2">Description and Equation</div>
Line 892: Line 904:
 
                 <div class="h3">Equations<sup>5</sup>:</div>
 
                 <div class="h3">Equations<sup>5</sup>:</div>
 
                 <p>Based on Hill function, we can determine the amount of activated tetR, tetR<sub>act</sub>:</p>
 
                 <p>Based on Hill function, we can determine the amount of activated tetR, tetR<sub>act</sub>:</p>
                 <p>$$ tetR_{act} + n \times S(t) \rightarrow [tetR \times S(t)_n] $$</p>
+
                 <p>$$ tetR_{act} + n \times S_x(t) \rightarrow [tetR - S_x(t)_n] $$</p>
                 <p>$$ K_X = \dfrac {tetR_{act} \times S^n (t)} {[tetR \times S(t)_n]} $$</p>
+
                 <p>$$ K_X = \dfrac {tetR_{act} \times S_x^n (t)} {[tetR - S_x(t)_n]} $$</p>
                 <p>$$ tetR = tetR_{act} + [tetR \times S(t)_n] $$</p>
+
                 <p>$$ tetR = tetR_{act} + [tetR - S_x(t)_n] $$</p>
                 <p>$$ tetR_{act} = \dfrac {tetR} {1 + \dfrac {S^n (t)} {K_{X}}} $$</p>
+
                 <p>$$ tetR_{act} = \dfrac {tetR} {1 + \dfrac {S_x^n (t)} {K_{X}}} $$</p>
 
                 <p>Based on Hill function, we can determine the amount of activated promoter, with which we can
 
                 <p>Based on Hill function, we can determine the amount of activated promoter, with which we can
 
                     calculate the total transcription speed of all promoters per cell:</p>
 
                     calculate the total transcription speed of all promoters per cell:</p>
                 <p>$$ P_{tet\_act} + n\cdot tetR_{act} \rightarrow [P_{tet} \cdot (tetR_{act})_n] $$</p>
+
                 <p>$$ P_{tet\_act} + n\cdot tetR_{act} \rightarrow [P_{tet} - (tetR_{act})_n] $$</p>
                 <p>$$ P_{tet\_copy} = P_{tet\_act} + [P_{tet} \cdot (tetR_{act})_n] $$</p>
+
                 <p>$$ P_{tet\_copy} = P_{tet\_act} + [P_{tet} - (tetR_{act})_n] $$</p>
                 <p>$$ K_d = \dfrac {P_{tet\_copy} \times tetR^n_{act} } {[P_{tet} \cdot (tetR_{act})_n]} $$</p>
+
                 <p>$$ K_d = \dfrac {P_{tet\_copy} \times tetR^n_{act} } {[P_{tet} - (tetR_{act})_n]} $$</p>
 
                 <p>$$ A_{mRNA} = P_{tet\_act} \times beta $$</p>
 
                 <p>$$ A_{mRNA} = P_{tet\_act} \times beta $$</p>
 
                 <p>$$ A_{mRNA} = \dfrac {P_{tet\_copy} \times beta } { 1 + \dfrac {tetR^n_{act}} {K_{d}}} $$</p>
 
                 <p>$$ A_{mRNA} = \dfrac {P_{tet\_copy} \times beta } { 1 + \dfrac {tetR^n_{act}} {K_{d}}} $$</p>
                 <p>Kinetic equations of transcription and transformation:</p>
+
                 <p>Kinetic equations of transcription and translation:</p>
 
                 <p>$$ \dfrac {dmRNA} {dt} = A_{mRNA} - K_{deg\_mRNA} \times mRNA $$</p>
 
                 <p>$$ \dfrac {dmRNA} {dt} = A_{mRNA} - K_{deg\_mRNA} \times mRNA $$</p>
 
                 <p>$$ \dfrac {dtetR} {dt} = K_{trans\_tetR} \times mRNA - K_{deg\_tetR} \times tetR $$</p>
 
                 <p>$$ \dfrac {dtetR} {dt} = K_{trans\_tetR} \times mRNA - K_{deg\_tetR} \times tetR $$</p>
Line 910: Line 922:
 
                 <p>$$ \dfrac {dS_x(t)} {d_t} = -K_{deg\_ATc} \times S_x(t) $$</p>
 
                 <p>$$ \dfrac {dS_x(t)} {d_t} = -K_{deg\_ATc} \times S_x(t) $$</p>
 
                 <p>$$ \ln(S_x(t)) = \ln(S_x(0)) - K_{deg\_ATc} \times t $$</p>
 
                 <p>$$ \ln(S_x(t)) = \ln(S_x(0)) - K_{deg\_ATc} \times t $$</p>
                 <p>Growth curve of bacteria based on logistics model by P. F. Verhulst:</p>
+
                 <p>Growth curve of bacteria based on logistics model from P. F. Verhulst:</p>
 
                 <p>$$ N(t) = \dfrac {K_{max}} {1 + C \cdot e^{-rt}} $$</p>
 
                 <p>$$ N(t) = \dfrac {K_{max}} {1 + C \cdot e^{-rt}} $$</p>
 
                 <p>Total GSDMD expressed in bacteria community:</p>
 
                 <p>Total GSDMD expressed in bacteria community:</p>
                 <p>$$ GSDMD_{total} = N(t) \cdot GSDMD $$</p>
+
                 <p>$$ GSDMD_{total-amount} = N(t) \cdot GSDMD\cdot{CFU}\cdot{diluted-ratio}\cdot{V_{Bacteria volume}}$$</p>
                 <div class="h3">Illustrations of the symbols in the equations:</div>
+
                 <div class="h3">The symbols in the equations:</div>
                 <p> \(S_x(t)\): concentration of ATc, as a function of time<br>
+
                 <p> \(S_x(t)\): concentration of ATc, as a function of time.<br>
                     \(tetR_{act}\): concentration of activated tetR<br>
+
                     \(tetR_{act}\): concentration of activated tetR.<br>
                     \(tetR \): concentration of total tetR<br>
+
                     \(tetR \): concentration of total tetR.<br>
                     \(GSDMD \): concentration of GSDMD<br>
+
                     \(GSDMD \): concentration of GSDMD.<br>
                     \(A_{mRNA} \): transcription rate constant of the promoter<br>
+
                     \(A_{mRNA} \): transcription rate constant of the promoter.<br>
                     \(P_{tet\_copy} \): plasmid copy number<br>
+
                     \(P_{tet\_copy} \): plasmid copy number.<br>
                     \(K_X \): disassociation rate constant of tetR and ATc<br>
+
                     \(K_X \): disassociation rate constant of tetR and ATc.<br>
                     \(K_d \): disassociation rate constant of tetR and DNA<br>
+
                     \(K_d \): disassociation rate constant of tetR and DNA.<br>
                     \(beta \): original transcription rate constant of the promoter<br>
+
                     \(beta \): original (unrepressed) transcription rate constant of the promoter.<br>
                     \(K_{deg\_mRNA} \): degradation rate constant of mRNA<br>
+
                     \(K_{deg\_mRNA} \): degradation rate constant of mRNA.<br>
                     \(K_{deg\_tetR} \): degradation rate constant of tetR<br>
+
                     \(K_{deg\_tetR} \): degradation rate constant of tetR.<br>
                     \(K_{trans\_tetR} \): translation rate constant of tetR<br>
+
                     \(K_{trans\_tetR} \): translation rate constant of tetR.<br>
                     \(mRNA \): concentration of mRNA<br>
+
                     \(mRNA \): concentration of mRNA.<br>
                     \(K_{deg\_GSDMD} \): degradation rate constant of GSDMD<br>
+
                     \(K_{deg\_GSDMD} \): degradation rate constant of GSDMD.<br>
                     \(K_{trans\_GSDMD} \): transcription rate constant of GSDMD<br>
+
                     \(K_{trans\_GSDMD} \): transcription rate constant of GSDMD.<br>
                     \(K_{deg\_ATc} \): degradation rate constant of ATc<br>
+
                     \(K_{deg\_ATc} \): degradation rate constant of ATc.<br>
                     \(n \): Hill coefficient<br>
+
                     \(n \): Hill coefficient.<br>
                     \(N(t) \): initial OD600 value of the bacteria<br>
+
                     \(N(t) \): initial OD600 value of the bacteria.<br>
                     \(r \): growth rate of the bacteria<br>
+
                     \(r \): growth rate of the bacteria.<br>
                     \(K_{max} \): Maximum OD of the bacteria in cultivation<br>
+
                     \(K_{max} \): maximum OD of the bacteria in cultivation.<br>
 
                 </p>
 
                 </p>
                 <div class="h2">Suggestions to the experiment (Results)</div>
+
                 <div class="h2">Suggestions to our experiments (see <a href="https://2018.igem.org/Team:HZAU-China/Results">Results</a>)</div>
 
                 <p>As is hard to obtain the initial parameters in the equations above on our own without any
 
                 <p>As is hard to obtain the initial parameters in the equations above on our own without any
 
                     experiments, the only way to obtain these parameters is to look up in former research or other
 
                     experiments, the only way to obtain these parameters is to look up in former research or other
 
                     teams work. Fortunately we got a copy of these parameters from team William and Mary iGEM 2016<sup>6</sup>.
 
                     teams work. Fortunately we got a copy of these parameters from team William and Mary iGEM 2016<sup>6</sup>.
 
                     These parameters include \(K_X = 0.36 \), \(K_d = 0.1 \), \(beta = 0.0023 \), \(K_{deg\_mRNA} =
 
                     These parameters include \(K_X = 0.36 \), \(K_d = 0.1 \), \(beta = 0.0023 \), \(K_{deg\_mRNA} =
                     0.009 \), \(K_{deg\_tetR} = 0.631 \), \(K_{trans\_tetR} = 235.5 \) <b>(All unites are combined of
+
                     0.009 \), \(K_{deg\_tetR} = 0.631 \), \(K_{trans\_tetR} = 235.5 \) <b>(All units are combined of
                         nM
+
                         nM and s)</b>. Considering that both <i>Salmonella</i>
                        and s)</b>. Considering
+
                     and <i>E. coli</i> are in <i>Enterobacteriaceae</i>, we assumed that in <i>Salmonella</i> these
                    that both Salmonella
+
                    parameters are the same
                     and E. coli are in Enterobacteriaceae, we assumed that in Salmonella these parameters are the same
+
                     with those in <i>E. coli</i> since we just wanted to figure out a useful instruction to wet lab.</p>
                     with those in E. coli since we just wanted to figure out an abstract instruction to wet lab.</p>
+
                 <p>To gain the parameters in bacterial growth curve, we carried out an experiment to measure the growth
                 <p>To gain the parameters in bacteria growth curve, we carried out an experiment to measure the growth
+
                     of <i>Salmonella</i>. Then we fitted the obtained data into a logistics model. By doing these we
                     of Salmonella. Then we fit the data obtained into the logistics. By doing these we figure out that
+
                    figured out that
 
                     \(r = 60min^{-1} \), \(K_{max} = 0.9997 \) and \(C = 7.2319 \). Results and diagram are shown below
 
                     \(r = 60min^{-1} \), \(K_{max} = 0.9997 \) and \(C = 7.2319 \). Results and diagram are shown below
                     (<b>Figure 5.</b>):</p>
+
                     (<b>Figure 7</b>):</p>
 
                 <div style="width: 80%; margin: 0px auto">
 
                 <div style="width: 80%; margin: 0px auto">
 
                     <img src="https://static.igem.org/mediawiki/2018/2/28/T--HZAU-China--ATC1.png" width=100% alt="">
 
                     <img src="https://static.igem.org/mediawiki/2018/2/28/T--HZAU-China--ATC1.png" width=100% alt="">
 
                 </div>
 
                 </div>
  
                 <p style="width: 100%; text-align: center !important;"><b>Figure 5.</b> Bacteria growth curve</p>
+
                 <p style="width: 100%; text-align: center !important;"><b>Figure 7.</b> Bacterial growth curve.</p><br>
  
                 <p>After complete the works above, we used MATLAB<sup>TM</sup> to solve the equations above and
+
                 <p>After complete the work above, we used MATLAB<sup>TM</sup> to solve the equations above and
                     acquired a series
+
                     acquired a series of diagrams which visually demonstrated the relationships and helped the wet lab
                    of diagrams which visually demonstrated the relationships, which would help the wet lab group get
+
                    group get
                     an abstract view of how ATc influence on the expression of GSDMD. We assumed that \(P_{tet\_copy} =
+
                     an overall view of how ATc influences the expression of GSDMD. We assumed that \(P_{tet\_copy} =
 
                     4
 
                     4
                     \), \(K_{deg\_GSDMD} = 0.8 \), \(K_{trans\_GSDMD} = 200 \), \(K_{deg\_ATc} = 0.0007 \), (All unites
+
                     \), \(K_{deg\_GSDMD} = 0.8 \), \(K_{trans\_GSDMD} = 200 \), \(K_{deg\_ATc} = 0.0007 \) (All unites
 
                     are
 
                     are
                     combined of nM and s) Results are shown below (<b>Figure 6,7,8.</b>):</p>
+
                     combined of nM and s). In this action we didn’t take the growth of bacteria into account. From the diagrams, we can figure out that as ATc added increase, the concentration of GSDMD will also rise. Results are shown below (<b>Figures 8, 9, 10.</b>):</p><br>
 
                 <div style="width: 80%; margin: 0px auto">
 
                 <div style="width: 80%; margin: 0px auto">
 
                     <img src="https://static.igem.org/mediawiki/2018/7/79/T--HZAU-China--ATC2.png" width=100% alt="">
 
                     <img src="https://static.igem.org/mediawiki/2018/7/79/T--HZAU-China--ATC2.png" width=100% alt="">
 
                 </div>
 
                 </div>
                 <p style="width: 100%; text-align: center !important;"><b>Figure 6.</b> Concentration of tetR(nM) - time(s)</p>
+
                 <p style="width: 100%; text-align: center !important;"><b>Figure 8.</b> Concentration of tetR (nM) -
 +
                    time (s).</p><br>
 
                 <div style="width: 80%; margin: 0px auto">
 
                 <div style="width: 80%; margin: 0px auto">
 
                     <img src="https://static.igem.org/mediawiki/2018/9/90/T--HZAU-China--ATC3.png" width=100% alt="">
 
                     <img src="https://static.igem.org/mediawiki/2018/9/90/T--HZAU-China--ATC3.png" width=100% alt="">
 
                 </div>
 
                 </div>
                 <p style="width: 100%; text-align: center !important;">Figure 7. Concentration of GSDMD (nM) - time(s)</p>
+
                 <p style="width: 100%; text-align: center !important;"><b>Figure 9.</b> Concentration of GSDMD (nM) -
 +
                    time (s).</p><br>
 
                 <div style="width: 80%; margin: 0px auto">
 
                 <div style="width: 80%; margin: 0px auto">
 
                     <img src="https://static.igem.org/mediawiki/2018/5/55/T--HZAU-China--ATC4.png" width=100% alt="">
 
                     <img src="https://static.igem.org/mediawiki/2018/5/55/T--HZAU-China--ATC4.png" width=100% alt="">
 
                 </div>
 
                 </div>
  
                 <p style="width: 100%; text-align: center !important;">Figure 8. Max concentration of GSDMD (nM) -
+
                 <p style="width: 100%; text-align: center !important;"><b>Figure 10.</b> Max concentration of GSDMD (nM)
                     time(s)</p>
+
                    -
 +
                     ATc Concentration (nM).</p><br>
  
                 <p>MATLAB<sup>TM</sup>. With this app in hand, you can adjust all the parameters needed in the
+
                 <p> With this software, one can adjust all the parameters needed in the equations above and
                    equations above and
+
                     attain the diagrams which indicates the relations between concentration of GSDMD and time,
                     attain the diagrams which indicates the relation between concentration of GSDMD and time,
+
                     concentration tetR and time and the maximum value of GSDMD and the initial concentration of ATc.
                     concentration tetR and time and the summit value of GSDMD and the initial concentration of ATc. The
+
                    The
                     app will also generate a function describing the relationship between the max concentration of
+
                     program will also generate a function describing the relationship between the maximum concentration
                     GSDMD and the concentration of ATc. With the help of this app, members in wet lab group can
+
                    of
 +
                     GSDMD and the concentration of ATc. With the help of this program, members in wet lab group can
 
                     conveniently decide how much ATc should be added into cultivation environment according to their
 
                     conveniently decide how much ATc should be added into cultivation environment according to their
                     requirements. (Figure 9,10)</p>
+
                     requirements (<b>Figures 11, 12</b>). <b>Be advised that users must multiply the CFU number, bacteria cell volume and diluted ratio to the data obtained from this app to gain a final result.</b></p><br>
 
                 <div style="width: 60%; margin: 30px auto">
 
                 <div style="width: 60%; margin: 30px auto">
 
                     <img src="https://static.igem.org/mediawiki/2018/8/8c/T--HZAU-China--ATC5.png" width=100% alt="">
 
                     <img src="https://static.igem.org/mediawiki/2018/8/8c/T--HZAU-China--ATC5.png" width=100% alt="">
 
                 </div>
 
                 </div>
  
                 <p style="width: 100%; text-align: center !important;">Figure 9. APP Parameters</p>
+
                 <p style="width: 100%; text-align: center !important;"><b>Figure 11.</b> Software parameters.</p><br>
  
 
                 <div style="width: 80%; margin: 30px auto">
 
                 <div style="width: 80%; margin: 30px auto">
Line 998: Line 1,014:
 
                 </div>
 
                 </div>
  
                 <p style="width: 100%; text-align: center !important;">Figure 10. APP Diagrams</p>
+
                 <p style="width: 100%; text-align: center !important;"><b>Figure 12.</b> Software diagrams.</p><br>
  
 
                 <div class="h2">Significance</div>
 
                 <div class="h2">Significance</div>
 
                 <p>The model of ATc induced circuit is very common and well-known to biology researchers. The
 
                 <p>The model of ATc induced circuit is very common and well-known to biology researchers. The
                     common-known significance to this model is that it can demonstrate the relation between
+
                     common-known significance to this model is that it can demonstrate the relationship between
 
                     concentration of target gene and concentration of inducer added, which can instruct the researchers
 
                     concentration of target gene and concentration of inducer added, which can instruct the researchers
                     regulate their circuit precisely. In our project, this model will tell the members in wet lab group
+
                     modulate their circuit precisely. In our project, this model will tell the members in wet lab group
                     that how much GSDMD will be expressed under a certain concentration of ATc in the Salmonella
+
                     that how much GSDMD will be expressed under a certain concentration of ATc in the <i>Salmonella</i>
 
                     community formed in the tumor cell.
 
                     community formed in the tumor cell.
                     Another significance for this model is that, the response time of is very short and the response
+
                     Another significance for this model is that, the response time is very short and the response
 
                     speed is extremely fast. We anticipate that just minutes are needed to induce the fluorescence.
 
                     speed is extremely fast. We anticipate that just minutes are needed to induce the fluorescence.
                     This phenomenon is also verified in the experiment. In less than 10 minutes, fluorescence can be
+
                     This phenomenon is also verified in our experiment. In less than 10 minutes, fluorescence can be
 
                     detected under fluorescence microscope.
 
                     detected under fluorescence microscope.
 
                     Especially, a remarkable significance to our project is that it’s a self-destructive system, which
 
                     Especially, a remarkable significance to our project is that it’s a self-destructive system, which
 
                     means, without any further operation, the process of induction can be self-terminated. As ATc
 
                     means, without any further operation, the process of induction can be self-terminated. As ATc
 
                     degrades, the expression of GSDMD will significantly decreases, thus the process of pyroptosis can
 
                     degrades, the expression of GSDMD will significantly decreases, thus the process of pyroptosis can
                     be inhibited. Based on these considerations, we think that the cytokine storm caused by pyroptosis
+
                     be inhibited. Based on the features, we think that the cytokine storm caused by pyroptosis
 
                     is controllable.
 
                     is controllable.
 
                 </p>
 
                 </p>
                 <div class="h2">The source of the app and scripts used above can be found in this link:</div>
+
                 <div class="h2">The source code of the software and the scripts used above can be found following this link:</div>
                 <p><a href="https://github.com/tom13amy/atc_modelling_software">https://github.com/tom13amy/atc_modelling_software</a></p>  
+
                 <p><a href="https://github.com/tom13amy/atc_modelling_software">https://github.com/tom13amy/atc_modelling_software</a></p>
  
 
             </div>
 
             </div>

Latest revision as of 12:27, 13 November 2018

Salmonella infection model

We want to simulate the situation that tumor cells and Salmonella together in a liquid environment. We used the law of mass action to establish a model for the infection process of Salmonella, which is dimensionless.

$$N_{normal} + S_{almonella} \overset{Aw}{\rightarrow} N_{w} + S_{al\_normal}$$

$$N_{tumor} + S_{almonella} \overset{As}{\rightarrow} N_{s} + S_{al\_tumor}$$

$$S_{almonella} = S_{almonella}(t0) - N_{normal\_cell} - N_{tumor}$$

$$\dfrac {dN_{w}} {d_{t}} = A_{w} S_{almonella} N_{w}$$

$$\dfrac {dN_{s}} {d_{t}} = A_{s} S_{almonella} N_{s}$$

$$\dfrac {dS_{almonella}} {d_{t}} = - \dfrac {dN_{w}} {d_{t}} - \dfrac {dN_{s}} {d_{t}}$$

\(N_{normal\_cell}\): The density of normal cells.
\(S_{almonella}\): The density of Salmonella in the liquid environment.
\(N_{tumor}\): The density of tumor cells.
\(N_w\): The number of Salmonella in the normal cells.
\(N_s\): The number of Salmonella in the tumor cells.
\(A_w\): The affinity between Salmonella and normal cells.
\(A_s\): The affinity between Salmonella and tumor cells.
\(S_{al\_normal}\): The density of infected normal cells.
\(S_{al\_tumor}\): The density of infected tumor cells.

Salmonella begins to replicate two hours after infection1 .

$$\dfrac {dN_{sal}} {d_{t}}(t) =$$

$$\dfrac {dS_{al}} {d_{t}}(t) + S_{al}(t-2) 2^{\dfrac {t-2} {T}} \ln{2}\dfrac {1} {T} $$

\(T\): Cell cycle.

Identification of infection time

According to our experimental results, we noticed that Salmonella might follow Poisson distribution in cells, so we use Matlab to judge the distribution of bacteria in the cells. We assume that the area less than 1 in the Poisson distribution is a part of cells which are not infected by Salmonella. According to our experimental results, cells which are infected by only one Salmonella can also die of pyroptosis. Based on this feature, we divide cells into uninfected and infected cells. When the average number of bacteria in the cell changes, which means that the λ of Possion distribution changes, the ratio of the two kind of cells will change. In summary, when the average number of Salmonella in cells changes, the proportion of dead cells will change.

Figure 1. Poisson distribution and Salmonella infection results. a. Based on statistics of experimental results, we proved that the Salmonella follows Poisson distribution in normal cells. b. We assume that the area less than 1 in the Possion distribution is a part of cells which are not infected by Salmonella. When the λ of Possion distribution changes, which means the average number of Salmonella in cells changes, the proportion of infected cells changes. c. Cells which are infected by only one Salmonella can also die of pyroptosis.

Infection in tumor cell culture experiments

We hope that the mathematical model can help the Salmonella infection experiment. In our final phenotypic experiment, the cells carrying the GSDMD gene are induced by ATc, and we hope that the observed result is that the proportion of ATc-induced cell death is more than that of not induced to prove the ATc promoter is effective. In this experiment, the error may be big if the proportion of cells infected by Salmonella is different.

Figure 2. Results caused by efficiency differences of infection.


If the proportions of infection are different, the experimental results may not be able to prove that pyroptosis is induced by atc promoter. Figure 2 showed that the ATc promoter is induced and 90% cells death are caused and 70% cells die because of the promoter disclosure, but the difference of the proportions of infected cells is so big that the experimental results cannot reflect the real situation. We can reduce the difference by letting the infection proportions of the two kinds of cells both close to 100%.

We solved this problem by predicting the proportion of cells infected with bacteria over time.

Based on these, we designed an App with MatLab (https://github.com/cccoolll/Pyroptosis.git) . In this App, different parameters obtained from measurement experiments can be input to predict the optimal infection time. Therefore, this App can provide guidance to the design of our experiments.

Figure 3. Salmonella infection prediction tool (for tumor cell).

A1: The predicted Salmonella numbers in tumor cell in the single cell infection experiment.
A2: The predicted proportions of infected tumor cells.
A3: The concentration of added Salmonella.
A4: The density of tumor cells;
A5: Rate constant of Salmonella infecting tumor cells.
A6: The proportion of tumor cells expected to be infected.
A7: The time to reach the wanted proportion of infected cells.

Figure 4. Salmonella infection prediction tool (for mixed culture of tumor cell and normal cell).

B1: The predicted numbers of Salmonella in tumor cell (red) and normal cell (blue) in a single cell infection experiments.
B2: The predicted proportions of infected cells (red for tumor cell and blue for normal cell).
B3: The concentration of added Salmonella.
B4: The density of tumor cells.
B5: Rate constant of Salmonella infecting tumor cells.
B6: The density of normal cells.
B7: Rate constant of Salmonella infecting normal cells.
B8: The predicted optimal infection time.


The parameters Nsal, Tumor and As are measured from our experiments.


Guidance to tumor cell infection experiments

Figure 5 (a part of Figure 3). Guidance for tumor cells infection experiments.

According to our experiment protocol, the MOI (multiplicity of infection) is 100, corresponding to the concentration of cells. If we want 98% of the tumor cells to be infected, the prediction result show that the infection time should be at least 2 hours to eliminate unnecessary variables.

Guidance to mixed culture experiments

Figure 6 (a part of Figure 4). The predicted proportions of infected cells (red for tumor cell and blue for normal cell).

In order to reflect the affinity of Salmonella to tumor cells and to normal cells, we hope that the difference between experimental results of tumor cells and normal cells is obvious. However, the numbers of bacteria in different host cells are difficult to count, and we can only count the number of infected cells and calculate their the proportions. Therefore, we need to predict the time when the difference is most obvious. Our App just can do this for us.

Chemical control model
Profile

The Tet repressor protein (tetR) regulates transcription of tetracyclines resistance protein, tetA. The expression of tetA must be strictly regulated since tetA is a membrane-spanning H+-[Tc-Mg]2+ antiporter which means it can lower the pH environment of cytoplasm. As a result, the natural circuit of tetracyclines regulation is a negative-feedback circuit2. Tc is the inducer, which shows high affinity to tetR protein. The tetR protein binds to tetO sequence on DNA specifically, thus inhibits the expression of Tet promoter. When Tc or other similar molecules like ATc (anhydrotetracycline) diffuse into bacteria, they will bind to tetR protein and unleash the tetR protein from DNA, and thus relieve the inhibition and start the expression of Tet promoter.

In our project, we choose ATc (anhydrotetracyclines) as the inducer. ATc is less harmful to bacteria than Tc and about 100-fold higher affinity to tetR than Tc2.

The ATc model aims to predict and solve two problems: first, how fast does the circuit react to ATc; second, how many target gene will express in the bacteria community under a certain concentration of ATc.

Hypothesis

There are two tetO sites on the Tet promoter and both can bind to tetR protein randomly and inhibit the promoter’s expression independently. To make the condition simple, we consider the two tetO sites into one as we just want to explain the relationship between the promoter inhibition and the tetR protein expression.

In our project, the ATc concentration in our incubation environment is uniform, and the diffusion rate of anhydrotetracycline can be ignored3. In spite of this, the degradation rate of ATc under 37℃ must be taken into account as reported4.

Based on these facts, we give the following hypotheses:

1. Regard two tetO operons as one equivalently.
2. Ignore the diffusion of ATc.
3. The reaction time between ATc and tetR, tetR and DNA is much shorter than transcription and translation.

Description and Equation
Reactions implicated:

$$tetR + [tetR - ATc_2] = tetR_{total}$$

$$tetR + 2 \times ATc = [tetR - ATc_2]$$

$$P_{tet} + [tetR_2 - P_{tet}] = [P_{tet}]_{total}$$

$$2 \times tetR + P_{tet} = [tetR_2 - P_{tet}]$$

$$$$

Equations5:

Based on Hill function, we can determine the amount of activated tetR, tetRact:

$$ tetR_{act} + n \times S_x(t) \rightarrow [tetR - S_x(t)_n] $$

$$ K_X = \dfrac {tetR_{act} \times S_x^n (t)} {[tetR - S_x(t)_n]} $$

$$ tetR = tetR_{act} + [tetR - S_x(t)_n] $$

$$ tetR_{act} = \dfrac {tetR} {1 + \dfrac {S_x^n (t)} {K_{X}}} $$

Based on Hill function, we can determine the amount of activated promoter, with which we can calculate the total transcription speed of all promoters per cell:

$$ P_{tet\_act} + n\cdot tetR_{act} \rightarrow [P_{tet} - (tetR_{act})_n] $$

$$ P_{tet\_copy} = P_{tet\_act} + [P_{tet} - (tetR_{act})_n] $$

$$ K_d = \dfrac {P_{tet\_copy} \times tetR^n_{act} } {[P_{tet} - (tetR_{act})_n]} $$

$$ A_{mRNA} = P_{tet\_act} \times beta $$

$$ A_{mRNA} = \dfrac {P_{tet\_copy} \times beta } { 1 + \dfrac {tetR^n_{act}} {K_{d}}} $$

Kinetic equations of transcription and translation:

$$ \dfrac {dmRNA} {dt} = A_{mRNA} - K_{deg\_mRNA} \times mRNA $$

$$ \dfrac {dtetR} {dt} = K_{trans\_tetR} \times mRNA - K_{deg\_tetR} \times tetR $$

$$ \dfrac {dGSDMD} {dt} = K_{trans\_GSDMD} \times - K_{deg\_GSDMD} \times GSDMD $$

Degradation function of ATc by time3:

$$ \dfrac {dS_x(t)} {d_t} = -K_{deg\_ATc} \times S_x(t) $$

$$ \ln(S_x(t)) = \ln(S_x(0)) - K_{deg\_ATc} \times t $$

Growth curve of bacteria based on logistics model from P. F. Verhulst:

$$ N(t) = \dfrac {K_{max}} {1 + C \cdot e^{-rt}} $$

Total GSDMD expressed in bacteria community:

$$ GSDMD_{total-amount} = N(t) \cdot GSDMD\cdot{CFU}\cdot{diluted-ratio}\cdot{V_{Bacteria volume}}$$

The symbols in the equations:

\(S_x(t)\): concentration of ATc, as a function of time.
\(tetR_{act}\): concentration of activated tetR.
\(tetR \): concentration of total tetR.
\(GSDMD \): concentration of GSDMD.
\(A_{mRNA} \): transcription rate constant of the promoter.
\(P_{tet\_copy} \): plasmid copy number.
\(K_X \): disassociation rate constant of tetR and ATc.
\(K_d \): disassociation rate constant of tetR and DNA.
\(beta \): original (unrepressed) transcription rate constant of the promoter.
\(K_{deg\_mRNA} \): degradation rate constant of mRNA.
\(K_{deg\_tetR} \): degradation rate constant of tetR.
\(K_{trans\_tetR} \): translation rate constant of tetR.
\(mRNA \): concentration of mRNA.
\(K_{deg\_GSDMD} \): degradation rate constant of GSDMD.
\(K_{trans\_GSDMD} \): transcription rate constant of GSDMD.
\(K_{deg\_ATc} \): degradation rate constant of ATc.
\(n \): Hill coefficient.
\(N(t) \): initial OD600 value of the bacteria.
\(r \): growth rate of the bacteria.
\(K_{max} \): maximum OD of the bacteria in cultivation.

Suggestions to our experiments (see Results)

As is hard to obtain the initial parameters in the equations above on our own without any experiments, the only way to obtain these parameters is to look up in former research or other teams work. Fortunately we got a copy of these parameters from team William and Mary iGEM 20166. These parameters include \(K_X = 0.36 \), \(K_d = 0.1 \), \(beta = 0.0023 \), \(K_{deg\_mRNA} = 0.009 \), \(K_{deg\_tetR} = 0.631 \), \(K_{trans\_tetR} = 235.5 \) (All units are combined of nM and s). Considering that both Salmonella and E. coli are in Enterobacteriaceae, we assumed that in Salmonella these parameters are the same with those in E. coli since we just wanted to figure out a useful instruction to wet lab.

To gain the parameters in bacterial growth curve, we carried out an experiment to measure the growth of Salmonella. Then we fitted the obtained data into a logistics model. By doing these we figured out that \(r = 60min^{-1} \), \(K_{max} = 0.9997 \) and \(C = 7.2319 \). Results and diagram are shown below (Figure 7):

Figure 7. Bacterial growth curve.


After complete the work above, we used MATLABTM to solve the equations above and acquired a series of diagrams which visually demonstrated the relationships and helped the wet lab group get an overall view of how ATc influences the expression of GSDMD. We assumed that \(P_{tet\_copy} = 4 \), \(K_{deg\_GSDMD} = 0.8 \), \(K_{trans\_GSDMD} = 200 \), \(K_{deg\_ATc} = 0.0007 \) (All unites are combined of nM and s). In this action we didn’t take the growth of bacteria into account. From the diagrams, we can figure out that as ATc added increase, the concentration of GSDMD will also rise. Results are shown below (Figures 8, 9, 10.):


Figure 8. Concentration of tetR (nM) - time (s).


Figure 9. Concentration of GSDMD (nM) - time (s).


Figure 10. Max concentration of GSDMD (nM) - ATc Concentration (nM).


With this software, one can adjust all the parameters needed in the equations above and attain the diagrams which indicates the relations between concentration of GSDMD and time, concentration tetR and time and the maximum value of GSDMD and the initial concentration of ATc. The program will also generate a function describing the relationship between the maximum concentration of GSDMD and the concentration of ATc. With the help of this program, members in wet lab group can conveniently decide how much ATc should be added into cultivation environment according to their requirements (Figures 11, 12). Be advised that users must multiply the CFU number, bacteria cell volume and diluted ratio to the data obtained from this app to gain a final result.


Figure 11. Software parameters.


Figure 12. Software diagrams.


Significance

The model of ATc induced circuit is very common and well-known to biology researchers. The common-known significance to this model is that it can demonstrate the relationship between concentration of target gene and concentration of inducer added, which can instruct the researchers modulate their circuit precisely. In our project, this model will tell the members in wet lab group that how much GSDMD will be expressed under a certain concentration of ATc in the Salmonella community formed in the tumor cell. Another significance for this model is that, the response time is very short and the response speed is extremely fast. We anticipate that just minutes are needed to induce the fluorescence. This phenomenon is also verified in our experiment. In less than 10 minutes, fluorescence can be detected under fluorescence microscope. Especially, a remarkable significance to our project is that it’s a self-destructive system, which means, without any further operation, the process of induction can be self-terminated. As ATc degrades, the expression of GSDMD will significantly decreases, thus the process of pyroptosis can be inhibited. Based on the features, we think that the cytokine storm caused by pyroptosis is controllable.

The source code of the software and the scripts used above can be found following this link:

https://github.com/tom13amy/atc_modelling_software

Reference

1. I. Hautefort, A. Thompson, et al. During infection of epithelial cells Salmonella enterica serovar Typhimurium undergoes a time-dependent transcriptional adaptation that results in simultaneous expression of three type 3 secretion systems. Cellular Microbiology 10(4), 958–984 (2008).

2. Berens, C. & Hillen, W. Gene regulation by tetracyclines: Constraints of resistance regulation in bacteria shape TetR for application in eukaryotes. Eur. J. Biochem. 270, 3109–3121 (2003).

3. Nevozhay, D., Adams, R. M., Murphy, K. F., Josic, K. & Balazsi, G. Negative autoregulation linearizes the dose-response and suppresses the heterogeneity of gene expression. Proc. Natl. Acad. Sci. 106, 5123–5128 (2009).

4. Politi, N. et al. Half-life measurements of chemical inducers for recombinant gene expression. J. Biol. Eng. 8, 1–10 (2014).

5. Alon, U. An Introduction to Systems Biology: Design Principles of Biological Circuits (Chapman & Hall/CRC Mathematical and Computational Biology).Pdf.

6. William and Mary iGEM 2016. A Kinetic Model of Molecular Titration. 1–11 (2016).

Model

Salmonella infection model

Chemical control model

Reference

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