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.

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

In this section we describe how 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. This section describes in detail how bifurcation analysis can help us determine the optimal range for our parameters.

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. This section describes in detail both methods, their advantages and disadvantages and how they are used to model our system.

Stability

Parameter Space Analysis

This section describes in detail how we analysed the parameter space for our system. The results of this will show when bistability is possible and when it is not. Using this information, we can determine the optimal parameter ranges for our system.

Determination of the Hill coefficient n₁

The Hill coefficient relating to the CFP/MS2 stem loop association is assumed to be around 2 due to the number of stem loops present. This section details the method we used to calculate this value more accurately, and what the result means in terms of the parameter space analysis.

Directed Evolution

Here we describe one of the possible ways to improve our switch which we would have been able to attempt given more time. The results show an optimized version of our original system.

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