# Team:Slovenia/ModelingDerivation

### From 2012.igem.org

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<p>Construct 1 promoter state transitioning is described by a reaction:<img src="https://static.igem.org/mediawiki/2012/f/ff/Svn12_derivation1_eq1.png"/> | <p>Construct 1 promoter state transitioning is described by a reaction:<img src="https://static.igem.org/mediawiki/2012/f/ff/Svn12_derivation1_eq1.png"/> | ||

</p> | </p> | ||

- | + | <!-- | |

<p>Assuming equilibrium of binding and unbinding, we can write: | <p>Assuming equilibrium of binding and unbinding, we can write: | ||

<img src="https://static.igem.org/mediawiki/2012/f/f7/Svn12_derivation1_eq2.png"/> | <img src="https://static.igem.org/mediawiki/2012/f/f7/Svn12_derivation1_eq2.png"/> | ||

- | </p> | + | </p>--> |

<p> | <p> | ||

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<p> | <p> | ||

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

<img src="https://static.igem.org/mediawiki/2012/b/ba/Svn12_derivation1_eq4.png" /> | <img src="https://static.igem.org/mediawiki/2012/b/ba/Svn12_derivation1_eq4.png" /> | ||

</p> | </p> | ||

- | <p>To account for non-linearity, an exponent | + | <p>To account for non-linearity, an exponent n<sub>1</sub> was added, and the equation generalized to: |

<img src="https://static.igem.org/mediawiki/2012/a/a9/Svn12_derivation1_eq5.png" /> | <img src="https://static.igem.org/mediawiki/2012/a/a9/Svn12_derivation1_eq5.png" /> | ||

</p> | </p> | ||

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<p> | <p> | ||

- | K<sub>r</sub> is the amount of TAL-A:KRAB it takes for f<sub>1</sub> to be equal to 50%. | + | K<sub>r</sub> is the amount of TAL-A:KRAB it takes for f<sub>1</sub> to be equal to 50% when n<sub>1</sub>=k<sub>1</sub>=1. |

</p> | </p> | ||

## Latest revision as of 18:02, 3 June 2013

# 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