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- | <center><img style="height:40px"; padding:0;" | + | <center><img style="height:60px"; padding:0;" |
| src="https://static.igem.org/mediawiki/2013/8/8d/Conj_EQ1_UCSF.png"> </center></div> | | src="https://static.igem.org/mediawiki/2013/8/8d/Conj_EQ1_UCSF.png"> </center></div> |
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| + | <div id="description" align="justify" style = "width:950px; height:300px"> |
- | <center><img style="height:40px"; padding:0;"
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- | src="https://static.igem.org/mediawiki/2013/e/ee/UCSF_Modeling_Image005.png"> </center></div>
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- | <div id="description" align="justify" style = "width:950px; height:270px"> | + | |
| <font face="calibri" size = "4"> | | <font face="calibri" size = "4"> |
- | In these equations, the amount of fluorescent protein depends on a production term and a degradation term. The parameters for the degradation terms are: <br><p>
| + | where pR is the production term and d(R+T)r is the degradation term. The parameters in this equation are: <br><p> |
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- | d<sub>x</sub> – rate of degradation of GFP based on the “low” function <br>
| + | <br>p – RFP production rate in a single cell <br> |
- | d<sub>y</sub> – rate of degradation of RFP based on the “high” function <br><p> | + | d – RFP degradation rate in a single cell <br><p> |
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- | The production term consists of a function that relates inducer and repressor complex to mRNA production rate and conversion factor that relates mRNA production to protein production rate. <br><p>
| + | <br>And the variables in this equation are: <br><p> |
- | p<sub>x</sub> – conversion factor (mRNA/protein) for x (GFP) <br>
| + | r – concentration of RFP <br> |
- | p<sub>y</sub> – conversion factor (mRNA/protein) for y (RFP) <br>
| + | t – time <br> |
- | The mRNA production functions depend on the amount of inducer and the amount of repressor complex as follows:
| + | R – number of recipient cells <br> |
| + | T – number of transconjugant cells <br> |
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| </div> | | </div> |
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- | <div id="photos">
| + | <div id="description" style = "width:950px; height:45px" margin-top:10px; align="justify"> |
- | <center><img style="height:50px"; padding:0;"
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- | src="https://static.igem.org/mediawiki/2013/4/49/UCSF_Modeling_Image10.png"> </center></div>
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- | <div id="photos">
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- | <center><img style="height:57px"; padding:0;"
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- | src="https://static.igem.org/mediawiki/2013/6/61/UCSF_Modeling_Image011.png"> </center></div>
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- | <div id="description" style = "width:950px; height:45px" align="justify"> | + | |
- | <font face="calibri" size = "4">There is one function for the low sensitivity promoter (f<sub>L</sub>)
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- | and another for the high sensitivity promoter (f<sub>H</sub>) . The other parameters for the production functions are: <br> </div>
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- | <div id="description" style = "width:950px; height:175px;margin-left:40px" align="justify">
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| <font face="calibri" size = "4"> | | <font face="calibri" size = "4"> |
- | A – basal expression level of promoter<br>
| + | <br>In order to simulate the RFP levels, we need to model the population dynamics of our system. Thus, these equations have been to model the three strains present in this system: |
- | B – maximal expression level of promoter <br>
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- | k – Activation coefficient for low/high promoter <br>
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- | k<sub>R</sub> – half maximal effective concentration of R:C <br>
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- | n – hill coefficient for induction. <br>
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- | n<sub>R</sub> – hill coefficient for repression <br>
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- | I – Inducer concentration <br>
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- | R:C – repressor/dCas9 complex. <br><p>
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| </div> | | </div> |
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- | | + | <div id="description" align="justify" style = "width:950px; height:40px"> |
- | <div id="description" style = "width:950px; height:95px" align="justify"> | + | <p><font face="calibri" size = "4">Engineered Donor Cell levels:</font> |
- | <font face="calibri" size = "4"> | + | |
- | The subscripts indicate whether the parameter is a property of the low or high sensitivity promoter (L or H) or related to protein x or y. Now we have to figure out what the equations for the gRNA/Repressor complexes are (R<sub>x</sub>:C and R<sub>y</sub>:C). The binding and unbinding of the gRNA and dCas9 complex is represented by the following chemical reactions:
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| </div> | | </div> |
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| <div id="photos"> | | <div id="photos"> |
- | <center><img style="height:70px; margin-top:10px"; padding:0;" | + | <center><img style="height:60px;margin-top:10px"; padding:0;" |
- | src="https://static.igem.org/mediawiki/2013/d/d5/Binding_unbinding.png"> </center></div> | + | src="https://static.igem.org/mediawiki/2013/8/86/Conj_EQ2_UCSF.png"> </center></div> |
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| <div id="description" align="justify" style = "width:950px; height:40px"> | | <div id="description" align="justify" style = "width:950px; height:40px"> |
- | <p><font face="calibri" size = "4">Given these chemical reactions, we can write the following equations for the gRNA/dCas9 Complex:</font> | + | <p><font face="calibri" size = "4">RFP producing Recipient Cell levels:</font> |
- | </div>
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- | <div id="photos">
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- | <center><img style="height:50px;margin-top:10px"; padding:0;"
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- | src="https://static.igem.org/mediawiki/2013/7/78/UCSF_Modeling_Image019.png"> </center></div>
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- | <div id="photos">
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- | <center><img style="height:50px;margin-top:10px"; padding:0;"
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- | src="https://static.igem.org/mediawiki/2013/4/4e/UCSF_Modeling_Image020.png"> </center></div>
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- | <div id="description" style = "width:950px; height:175px" align="justify">
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- | <font face="calibri" size = "4">Where K<sub>R<sub>x</sub>f</sub> represents the forward reaction rate for reaction (5) and K<sub>R<sub>x</sub>b</sub> represents the reverse reaction rate. The equations for the gRNA/dCas9 complex depend on the rate at which the gRNA complex with dCas9 and also the rate at which the complex breaks apart. As mentioned earlier, we made an assumption that the binding and unbinding of the gRNA/dCas9 complex happens much quicker than the production and degradation of gRNAs and fluorescent proteins. Since the binding and unbinding happen very quickly, we evaluate the equations above at steady state (equal to zero) for the given values of the other parameters. <br><p>
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- | Under that assumption (setting equations (7) and (8) to zero – known as the quasi steady state assumption), we can solve for the complex in terms of the unbound repressor concentrations:
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- | </font> | + | |
| </div> | | </div> |
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| <div id="photos"> | | <div id="photos"> |
| <center><img style="height:60px;margin-top:10px"; padding:0;" | | <center><img style="height:60px;margin-top:10px"; padding:0;" |
- | src="https://static.igem.org/mediawiki/2013/f/f5/UCSF_Modeling_Image023.png"> </center></div> | + | src="https://static.igem.org/mediawiki/2013/2/29/Conj_EQ3_UCSF.png"> </center></div> |
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| <div id="description" align="justify" style = "width:950px; height:40px"> | | <div id="description" align="justify" style = "width:950px; height:40px"> |
- | <p><font face="calibri" size = "4">Where the amount of dCas9 available in the system is given by:</font> | + | <p><font face="calibri" size = "4">Transconjugant Cell levels:</font> |
| </div> | | </div> |
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| <div id="photos"> | | <div id="photos"> |
| <center><img style="height:60px;margin-top:10px"; padding:0;" | | <center><img style="height:60px;margin-top:10px"; padding:0;" |
- | src="https://static.igem.org/mediawiki/2013/2/26/AmtdCas9.png"> </center></div> | + | src="https://static.igem.org/mediawiki/2013/8/8a/Conj_EQ4_UCSF.png"> </center></div> |
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- | <div id="description" style = "width:950px; height:45px" align="justify">
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- | <font face="calibri" size = "4">
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- | The available amount of dCas9 depends on the amount of the two different complexes and also the amount of free dCas9. It has the following parameters:
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- | </font>
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- | </div>
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- | | + | <div id="description" style = "width:950px; height:100px" align="justify"> |
- | <div id="photos"> | + | <font face="calibri" size = "4"><br>The population dynamics for each strain are modeled based on logistic growth and are affected by conjugation. Conjugation, which is the term γDR, negatively affects the recipient cell population while positively contributing to the transconjugant cell population. The engineered donor cell remains unaffected by conjugation. <br><p> |
- | <center><img style="height:50px;margin-top:10px"; padding:0;"
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- | src="https://static.igem.org/mediawiki/2013/9/9f/UCSF_Modeling_image025.png"> </center></div>
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- | <div id="photos">
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- | <center><img style="height:50px;margin-top:10px"; padding:0;"
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- | src="https://static.igem.org/mediawiki/2013/c/c8/UCSF_Modeling_Image026.png"> </center></div>
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- | <div id="description" align="justify" style = "width:950px; height:100px">
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- | <p><font face="calibri" size = "4">The equations for the gRNAs depend on the amount of the gRNAs that is produced, the degradation rate, and also the rate at which the gRNA complexes with dCas9. With the quasi-steady state assumption, the terms for complexing with dCAS9 drop out and the final equations for the gRNAs are similar to equations (1) and (2) for the fluorescent proteins:</font>
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- | </div>
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- | <div id="photos">
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- | <center><img style="height:100px;margin-top:10px"; padding:0;"
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- | src="https://static.igem.org/mediawiki/2013/2/23/GRNAS.png"> </center></div>
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- | <div id="description" align="justify" style = "width:950px; height:90px">
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- | <font face="calibri" size = "4">
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- | <br><b><FONT COLOR="#008000">PARAMETERS</FONT COLOR="#008000"></b> <p> | + | |
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- | <font face="calibri" size = "4">
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- | This model has many parameters, so in order for it to be more useful, we need to reduce the number of parameters that are undetermined. To accomplish this, we gathered some values from literature and also did experiments to find other parameters (Table1).
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| </font> | | </font> |
| </div> | | </div> |
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- | | + | <div id="description" align="justify" style = "width:950px; height:300px"> |
- | <div id="photos"> | + | |
- | <center><img style="height:350px;margin-top:20px"; padding:0;"
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- | src="https://static.igem.org/mediawiki/2013/9/95/UCSF_Modeling_Image088.png"> </center></div>
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- | <div id="description" style = "width:950px; height:55px" align="justify">
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| <font face="calibri" size = "4"> | | <font face="calibri" size = "4"> |
- | <b>How did we fit parameters?</b>
| + | The parameters for these equations are shown below:<br><p> |
- | A few parameters are properties of the promoters (A, B, k, n). By determining the dosage response of a promoter to inducer we are able to fit those parameters using a curve fitting tool to get the following values.
| + | |
- | </font> | + | |
- | </div> | + | |
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- | <div id="photos"> | + | <br>D – number of engineered donor cells <br> |
- | <center><img style="height:130px;margin-top:10px"; padding:0;" | + | R – number of recipient cells <br> |
- | src="https://static.igem.org/mediawiki/2013/d/d7/Known_Parameters.png"> </center></div>
| + | T – number of transconjugant cells <br> |
| + | ϕ – maximum growth rate <br> |
| + | K<sub>max</sub> – carrying capacity <br> |
| + | γ – conjugation rate <br> |
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- | <div id="photos">
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- | <center><img style="height:200px;margin-top:10px"; padding:0;"
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- | src="https://static.igem.org/mediawiki/2013/7/7b/Experimental_Data.png"> </center></div>
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- | <div id="description" style = "width:950px; height:55px" align="justify">
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- | <font face="calibri" size = "4">
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- | <b>How is the hill function affected by the amount of gRNA/dCas9 complex?</b>
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- | The previous plot helped us determine a few parameters, but there are still a few parameters we have no values for from experimental data or from literature. The following plots show how the parameter for amounts of the gRNA/dCas9 complex affects the behavior of the model. </div>
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- | <center><img style="height:330px;margin-top:10px"; padding:0;"
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- | src="https://static.igem.org/mediawiki/2013/6/6f/Hill_Function-1.png"></center></div>
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- | <center><img style="height:330px;margin-top:10px"; padding:0;"
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- | src="https://static.igem.org/mediawiki/2013/9/9d/Hill_Function-2.png"></center></div>
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- | <div id="description" style = "width:950px; height:205px" align="justify">
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- | <font face="calibri" size = "4">
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- | From given values for inducer concentrations and amounts of complex, we can calculate the amount of fluorescent protein that should be present. Our model can help us design an experiment that helps us calculate parameters that are still unknown.
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- | <br><br>
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- | <b>How does the model look with our actual “low” and “high” promoters?</b> <br>
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- | If the only change in the low and high functions (F<sub>H</sub> and F<sub>L</sub>) is the K values (which determine the sensitivity of the promoters), then we don’t get our desired behavior. However, there are other parameters that might give us the desired behavior for the low and high promoters. <br>
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- | If we set B<sub>L</sub> to 443.7 and B<sub>H</sub> to 443.7*1.25, and if we set the half max values to k<sub>L</sub> = 11.45 and k<sub>H</sub>=17, the promoters have the following profile:
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| </div> | | </div> |
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- | <div id="photos">
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- | <center><img style="height:330px;margin-bottom:10px"; padding:0;"
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- | src="https://static.igem.org/mediawiki/2013/3/3d/UCSF_Modeling_Image033.png"></center></div>
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- | <div id="description" style = "width:950px; height:45px" align="justify">
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- | <font face="calibri" size = "4">And they generate the following behavior in the full model:</div>
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- | <div id="photos">
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- | <center><img style="height:330px;margin-top:10px"; padding:0;"
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- | src="https://static.igem.org/mediawiki/2013/d/d6/UCSF_Modeling_Image034.jpg"></center></div>
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- | <div id="description" style = "width:950px; height:35px" align="justify">
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- | <font face="calibri" size = "4">Thus, based on our model, if certain conditions are met, our synthetic circuit will work as expected.
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- | </div>
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- | <div id="description" style = "width:950px; height:105px" align="justify">
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- | <font face="calibri" size = "4">
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- | <br><br>
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- | <b>How does the system change when the hill coefficient is manipulated?</b>
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- | In this first plot, the hill coefficients for both the low and the high function are the same number: 2.551. This number is the one we determined from our experimental data. <br><p>
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- | n<sub>L</sub> = 2.551<br>
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- | n<sub>H</sub> = 2.551<br>
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- | </div>
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- | <div id="photos">
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- | <center><img style="height:330px;margin-top:10px"; padding:0;"
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- | src="https://static.igem.org/mediawiki/2013/0/04/Modeling_SS-1.png"></center></div>
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- | <div id="description" style = "width:950px; height:125px" align="justify">
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- | <font face="calibri" size = "4">
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- | We can see that the switch from GFP to RFP is relatively sharp, and that RFP seems to be expressed in higher concentrations of inducer, while GFP is being expressed in lower concentrations.
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- | In this next plot, the only thing that changed was the hill coefficient for the high function. In the first plot, it was 2.551, in the second plot, it is 1.551. <br><p>
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- | n<sub>L</sub> = 2.551<br>
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- | n<sub>H</sub> = 1.551<br>
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- | </div>
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- | <div id="photos">
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- | <center><img style="height:330px;margin-top:5px"; padding:0;"
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- | src="https://static.igem.org/mediawiki/2013/6/69/Modeling_SSPlot-2.png"></center></div>
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- | <div id="description" style = "width:950px; height:125px" align="justify">
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- | <font face="calibri" size = "4">
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- | It is similar to the first graph in that the switch between GFP and RFP is sharp. However, RFP is being expressed in both low and high concentrations of inducer, while GFP is being expressed in medium concentrations. </div>
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