Team:UCSF/Modeling

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Modeling: Decision Making Circuit

The primary goal of the modeling portion for the synthetic circuit project is to create a model that will help us figure out the right parameters, given our assumptions, which will generate the desired result. The model can help us test out different promoters and repression strengths in the computer without wasting time trying to do all of that in the lab. The circuit is designed to produce different outputs according to different levels of inducer by utilizing the CRISPRi system. In lower concentrations of inducer, the guide RNA (gRNA) will be made to repress RFP. In higher concentrations of inducer, another gRNA will be made to repress GFP. Since our circuit should express GFP at lower inducer concentrations and RFP at high inducer concentrations, we should expect the graph to look something like the one below:


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.
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, but we added letters to represent each variable for our model. (R:C – gRNA/dCas9 complex; R – gRNA; C – dCas9; L – low inducible promoter; H – high inducible promoter)


ASSUMPTIONS: While creating the model for our system, we made five assumptions in order to simplify some of the aspects of the model:
1) protein degradation is linear;
2) protein production is based on a hill function and also depends on inducer concentration;
3) repression is governed by a hill function and depends on the concentration of dCas9 and gRNA complex;
4) 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 Quasi Steady State).
5) everything diffuses quickly throughout the cell so that our differential equations depends on the concentration at any given time.

VARIABLES:


EQUATIONS:

For fluorescent proteins

These equations show that the amount of fluorescent proteins depends on the production of, as well as the degradation of, the proteins.

Protein Production Equations Depend on Inducer & Repressor Complex:

These next two equations describe the binding and unbinding of the gRNA and dCas9 complex:

Repressor (gRNA) & Repressor/dCas9 Complex:

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, the equations above will be at steady state for the given values of the other parameters.

gRNAs:

Amount of dCas9 Available in the System:

The equations for the gRNA/dCas9 complex depends on the rate at which the gRNA complex with dCas9 and also the rate at which the complex breaks apart.
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.
The available amount of dCas9 depends on the amount of the two different complexes and also the amount of free dCas9.
PARAMETERS:

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.
How did we fit parameters? 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.
How is the hill function affected by the amount of gRNA/dCas9 complex? 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.
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.

How does the system change when the hill coefficient is manipulated? 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.

nL = 2.551
nH = 2.551

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. 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.

nL = 2.551
nH = 1.551

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.

How does the model look with our actual “low” and “high” promoters?
If the only change in the low and high functions (FH and FL) 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.

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