# Team:Slovenia/ModelingDerivation

### From 2012.igem.org

# Deterministic model derivation - genetic switch

The deterministic model for the mutual repressor switch was derived in the following way.

We assumed the following binding site states were possible:

Active promoter state is a state leading to gene expression.

Construct 1 promoter state transitioning is described by a reaction:

The fractional occupancy of construct 1 promoter, f_{1}, can be expressed as a ratio of active states to all states:

The equations for other constructs take the same form.

Assuming equilibrium of binding and unbinding, it follows that the fractional occupancy for construct 1 promoter is:

To account for non-linearity, an exponent n_{1} was added, and the equation generalized to:

K_{r} is the amount of TAL-A:KRAB it takes for f_{1} to be equal to 50% when n_{1}=k_{1}=1.

Derivation for constructs 2, 3 and 4 was similar except for different transcription factor names.

Fractional occupancies were then used to construct a set of ordinary differential equations representing each protein production. Because each protein can be produced from different constructs, production rates (including leaky rates) were summed together. E.g., because TAL-B:KRAB is produced from both constructs 1 and 3, fractional occupancies f_{1} and f_{3} were used and corresponding terms summed to obtain:

Since construct 5 promoter has no binding sites and is active at all times, fractional occupancy of the promoter is equal to 1.

The positive feedback loop switch model was derived in a similar manner.

**Deterministic model of the mutual repressor switch**

Deterministic model of the positive feedback loop switch

Deterministic model of the positive feedback loop switch