## Model

In our design, integrases compute the output of the logic gates. Integrases allow to flip one fragment of DNA. The model we developed is described here.

### Chemical Species

Name
| Description |
---|---|

Bxb1
| Serine integrase that can fold into two conformations - Bxb1a and Bxb1b. We chose to use a common connotation for both conformations - Bxb1. |

ΦC31
| Serine integrase that can fold into two conformations - ΦC31a and ΦC31b. We chose to use a common connotation for both conformations - ΦC31. |

DBxb1
| Dimerized form of Bxb1. We chose to use a common connotation for both homodimers, DBxb1a and DBxb1b. |

DΦC31
| Dimerized form of ΦC31. We chose to use a common connotation for both homodimers, DΦC31a and DΦC31b. |

### Modeling DNA-binding sites

Each dimer of integrases can specifically bind to a DNA binding site. As the flipping is irreversible, these DNA binding sites can be in three possible states:

- SI
_{IntegraseName}: inactive DNA binding site. No dimer is bound to this site, which has never been flipped.

- SA
_{IntegraseName}: active DNA binding site. A dimer is bound to this site.

- SF
_{IntegraseName}: flipped DNA binding site. This site has been irreversibly flipped.

### Reactions

- For Bxb1

$$ \begin{align} Bxb1 + Bxb1 &\leftrightarrow DBxb1 \\ DBxb1 + SI_{Bxb1} & \leftrightarrow SA_{Bxb1}\\ Bxb1 &\rightarrow \\ DBxb1 &\rightarrow \end{align}$$

- For ΦC31

\begin{align} \phi C31 + \phi C31 &\leftrightarrow D\phi C 31 \\ D\phi C 31 + SI_{\phi C31} & \leftrightarrow SA_{\phi C31}\\ \phi C31 &\rightarrow \\ D\phi C31 &\rightarrow \end{align}

### Differential Equations

Applying mass action kinetic laws, we obtain the following set of differential equations for Bxb1.

$$\frac{d[Bxb1]}{dt}=-2 k_{DBxb1}[Bxb1]^2+ 2 k_{-DBxb1}[DBxb1]-d_{Bxb1}[Bxb1]$$

$$\frac{d[DBxb1]}{dt}=-k_{SABxb1}[DBxb1][SI_{Bxb1}]+k_{-SABxb1}[SA_{Bxb1}]+k_{DBxb1}[Bxb1]^2-k_{-DBxb1}[DBxb1]-d_{DBxb1}[DBxb1]$$

$$\frac{d[SA_{Bxb1}]}{dt}=k_{SABxb1}[DBxb1][SI_{Bxb1}]-k_{-SABxb1}[SA_{Bxb1}]$$

Replacing every occurence of Bxb1 by ΦC31 gives the set of differential equations for ΦC31.