Team:UCSF/Modeling2

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If we can get this result from our model, then it would help us figure out how to change our parameters in order to generate the desired behavior.
 
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The first step in modeling our system is to come up with a way to represent our synthetic circuit mathematically. It’s essentially the same diagram as the one shown on the synthetic circuit page, and we represent each component in the system with a different variable.
 
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<br><b><FONT COLOR="#008000">ASSUMPTIONS: </FONT COLOR="#008000"></b>While creating the model for our system, we made five assumptions in order to simplify some of the aspects of the model:
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<br><b><FONT COLOR="#008000">ASSUMPTIONS: </FONT COLOR="#008000"></b>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.
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<FONT COLOR="#008000">1)</font> protein degradation is linear; <br>
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<br><FONT COLOR="#008000">1)</font> RFP, dCas9, and gRNA are produced and degraded at a constant rate; <br>
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<FONT COLOR="#008000">2) </font> protein production is based on a hill function and also depends on inducer concentration; <br>
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<FONT COLOR="#008000">2) </font> conjugation rate is linearly dependent on the concentration of donor and recipient cells (mass action); <br>
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<FONT COLOR="#008000">3)</font> repression is governed by a hill function and depends on the concentration of dCas9 and gRNA complex; <br>
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<FONT COLOR="#008000">3)</font> CRISPR expression efficiency in a cell is 100%; <br>
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<FONT COLOR="#008000">4)</font> that the binding and unbinding of dCas9 and gRNA complex happens much faster than the production/degradation of gRNA and fluorescent proteins (the complex is at <a href="http://en.wikipedia.org/wiki/Steady_State_theory#Quasi-steady_state" target="_blank">Quasi Steady State</a><span>). <br>
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<FONT COLOR="#008000">4)</font> The growth rates of the three strains are equal. <br>
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<FONT COLOR="#008000">5)</font> everything diffuses quickly throughout the cell so that our differential equations depends on the concentration at any given time.
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Given these assumptions we have the following equations for the system: <br>
Given these assumptions we have the following equations for the system: <br>
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Fluorescent proteins: <br></div>
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Change in RFP levels over time:<br></div>
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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>
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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>
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<br>p – RFP  production rate in a single cell <br>
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d<sub>y</sub> – rate of degradation of RFP based on the “high” function <br><p>
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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>
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<br>And the variables in this equation are: <br><p>
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p<sub>x</sub> – conversion factor (mRNA/protein) for x (GFP) <br>
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r – concentration of RFP <br>
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p<sub>y</sub> – conversion factor (mRNA/protein) for y (RFP) <br>
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t – time <br>
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The mRNA production functions depend on the amount of inducer and the amount of repressor complex as follows:
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R – number of recipient cells <br>
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T – number of transconjugant cells <br>
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<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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A – basal expression level of promoter<br>
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<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:
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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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  <p><font face="calibri" size = "4">Engineered Donor Cell levels:</font>
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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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   <p><font face="calibri" size = "4">Given these chemical reactions, we can write the following equations for the gRNA/dCas9 Complex:</font>
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   <p><font face="calibri" size = "4">RFP producing Recipient Cell levels:</font>
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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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   <p><font face="calibri" size = "4">Where the  amount of dCas9 available in the system is given by:</font>
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   <p><font face="calibri" size = "4">Transconjugant Cell levels:</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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<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>
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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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The parameters for these equations are shown below:<br><p>
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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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<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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<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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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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Latest revision as of 02:11, 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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