# Repressilator

A repressilator is a feedback loop in which each gene suppresses the next one. The following diagram shows an implementation of a repressilator.

Questions about the repressilator

1. When does the repressilator oscillate (or not)?
2. What are the design requirements?

## Model

The reactions are the following:

We will simplify these equations by writing them in terms of dimensionless variables. So instead of having the derivatives in terms of $d/dt$ let’s use $d/d(\mu t)$.

After some algebra we get the following dimensionless rate equations

We can make these equations a bit cleaner my making the following definitions

So the reduced system is…

Assume that the mRNA dynamics are faster than the protein dynamics.

Let p, q, and r represent the rescaled protein concentrations.

Like in previous lectures, we will now look for fixed points in the reduced system.

By symmetry all the fixed points are equal to one another.

Assume that $\alpha_0 \to 0$.

### Is the fixed point stable or unstable?

First, assume a small deviation from s in order to linearize the system.

Let’s find the eigenvalues of the matrix J. Since J is a 3 x 3 matrix it will have three eigenvalues.

The eignevalues must have the form described below, where $a$, $b$, and $c$ are real numbers.

#### If $c \neq 0$

Two of the eignevalues are complex.

The unstable case can exhibit several different behaviors.

• Growing oscillations
• Limit cycle
• Chaos

In the case of the repressilator we get a limit cycle.

Let’s find the eigenvalues for this case.

We can write the factor of negative one in terms of a complex exponential.

#### For the case of oscillations

Therefore $n>2$ is required, i.e. we require cooperative repression. Recall that the expression for $\alpha$ is

So the above constraint leads to the following inequalities which will constrain the possible values of $\alpha$.

This means that repression must be sufficiently strong in order for oscillations to occur.

So the hill function must be sufficiently steep and tall in order to get oscillations.

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Written on April 5, 2015