Introduction to the Modelling of the ayeSwitch

This page is an introduction to the different equations and techniques that we used to design and predict the behaviour of the ayeSwitch.

Equations

In this section we describe the process of developing a basic mathematical model for the ayeSwitch based on the promotion and inhibition behaviour necessary for mutual repression. We developed a set of four differential equations, one to model each of the two mRNAs and two proteins that are the active components of our system.

Nullclines and Fixed Points

We used fixed point analysis to predict the equilibrium state(s) of the ayeSwitch system for different parameters. Three or more equilibrium points will give us bistability and the possibility of switching.

Bifurcation

Bifurcation analysis allows us to track the number and relative position of the equilibrium points of our system for different parameters. Parameters below a certain value may only give us one fixed point, but increasing the value of the parameter could give us a bifurcation point and then increasing the value more we can get three fixed points. The further apart the fixed points are from each other, the more difficult it will be to switch between them.

Stochastic Model

There are two ways to model our system - deterministically and stochastically. Both methods have their advantages and disadvantages depending on the system in question. In our system, it was more beneficial to model the system stochastically due to the low numbers of molecules involved.

Stability

Parameter Space Analysis

Determination of the Hill coefficient n₁

Directed Evolution

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