# Team:UCSF/Modeling

### From 2013.igem.org

**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, a Decision Making Circuit (Figure 1). In other words, we want a circuit that exhibits behavior A given a certain input and behavior B when the input is changed. A computer model can help us rapidly prototype by providing an environment where we can test out different promoters and repression strengths in the computer before we spend the time building and testing strains 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. The circuit should express GFP at lower inducer concentrations and RFP at high inducer concentrations, as shown in Figure 1 below.

**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.

**EQUATIONS**

Given these assumptions we have the following equations for the system:

Fluorescent proteins:

d_{x} – rate of degradation of GFP based on the “low” function

d_{y} – rate of degradation of RFP based on the “high” function

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.

p_{x} – conversion factor (mRNA/protein) for x (GFP)

p_{y} – conversion factor (mRNA/protein) for y (RFP)

The mRNA production functions depend on the amount of inducer and the amount of repressor complex as follows:

_{L}) and another for the high sensitivity promoter (f

_{H}) . The other parameters for the production functions are:

B – maximal expression level of promoter

k – Activation coefficient for low/high promoter

k

_{R}– half maximal effective concentration of R:C

n – hill coefficient for induction.

n

_{R}– hill coefficient for repression

I – Inducer concentration

R:C – repressor/dCas9 complex.

_{x}:C and R

_{y}:C). The binding and unbinding of the gRNA and dCas9 complex is represented by the following chemical reactions:

Given these chemical reactions, we can write the following equations for the gRNA/dCas9 Complex:

_{Rxf}represents the forward reaction rate for reaction (5) and K

_{Rxb}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.

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:

Where the amount of dCas9 available in the system is given by:

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:

**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 (Table1).

**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.

**How does the model look with our actual “low” and “high” promoters?**

If the only change in the low and high functions (F

_{H}and F

_{L}) 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.

If we set B

_{L}to 443.7 and B

_{H}to 443.7*1.25, and if we set the half max values to k

_{L}= 11.45 and k

_{H}=17, the promoters have the following profile:

**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.

n_{L} = 2.551

n_{H} = 2.551

n_{L} = 2.551

n_{H} = 1.551

**Based on our model, if certain conditions are met, our synthetic circuit will work well and work as expected.**