Team:UCSF/Modeling2

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  <p><font face="calibri" size = "4">Where the  amount of dCas9 available in the system is given by:</font>
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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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The parameters for these equations are shown below:<br><p>
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<br>D – number of engineered donor cells <br>
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R – number of recipient cells <br>
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T – number of transconjugant cells <br>
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ϕ – maximum growth rate <br>
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K<sub>max</sub> – carrying capacity <br>
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γ – conjugation rate <br>
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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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<br><b><FONT COLOR="#008000">PARAMETERS</FONT COLOR="#008000"></b> <p> 
 
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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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<b>How did we fit parameters?</b>
 
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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.
 
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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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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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<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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<font face="calibri" size = "4">And they generate the following behavior in the full model:</div>
 
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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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<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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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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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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Revision as of 02:07, 29 October 2013

Modeling: Conjugation

The primary goal of the modeling portion for the CRISPRi conjugation project is to create a model that will help us identify the behavior of our transferable targeting system and identify critical parameters, given our assumptions. We used this model to check whether our CRISPRi conjugation system would have the desired behavior under biologically relevant parameters.

This system is designed to be transferred from cell to cell, targeting a specific gene of interest within the community. We model the activity of three strains of E. coli, an engineered donor strain containing a conjugative plasmid with genes coding for dCas9 and a gRNA, a recipient strain containing the RFP gene (target gene) in its genome, and upon successful conjugation, a transconjugate strain that no longer expresses RFP as well as the community's total RFP concentration.

Our objective is to simulate this community over time to see if our engineered system causes RFP levels to decrease.


ASSUMPTIONS: While creating the model for our system, we made four assumptions in order to simplify some of the aspects of the model. Many similar assumptions have been made in the literature.

1) RFP, dCas9, and gRNA are produced and degraded at a constant rate;
2) conjugation rate is linearly dependent on the concentration of donor and recipient cells (mass action);
3) CRISPR expression efficiency in a cell is 100%;
4) The growth rates of the three strains are equal.

EQUATIONS

Given these assumptions we have the following equations for the system:
Change in RFP levels over time:

where pR is the production term and d(R+T)r is the degradation term. The parameters in this equation are:


p – RFP production rate in a single cell
d – RFP degradation rate in a single cell


And the variables in this equation are:

r – concentration of RFP
t – time
R – number of recipient cells
T – number of transconjugant cells

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:

Engineered Donor Cell levels:

RFP producing Recipient Cell levels:

Transconjugant Cell levels:


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.

The parameters for these equations are shown below:


D – number of engineered donor cells
R – number of recipient cells
T – number of transconjugant cells
ϕ – maximum growth rate
Kmax – carrying capacity
γ – conjugation rate

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