Model

The Quorum sensing module is mainly involved in receiving signals from the sender cells. The sender cells secrete some signalling molecules (inducers) which bind to the regulator molecules in the receiver cells, thus activating the transcription of certain genes. The model for this module is presented below.

Chemical Species

• Lux-AHL: 30C6HSL is an acyl homoserine lactone which mainly binds to LuxR.

• LuxR: constitutively expressed regulator protein that can bind Lux-AHL and stimulate transcription of Bxb1.

• RLux: LuxR and Lux-AHL complex which can dimerise.

• DRLux: Dimerised form of RLux.

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

• Las-AHL: 30C12HSL is an acyl homoserine lactone which mainly binds to LasR.

• LasR: constitutively expressed regulator protein that can bind Las-AHL and stimulate transcription of ΦC31.

• RLas: LasR and Las-AHL complex which can dimerise.

• DRLas: Dimerised form of RLas.

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

Modelization of 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 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 to this site.

• SF_{IntegraseName}: flipped DNA binding site. This DNA binding site has been used by a flipping.

The three different states of DBxb1-DNA binding sites.

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.

$$\frac{d[Bxb1]}{dt}=-2 k_{dimBxb1}[Bxb1]^2+ 2 k_{-dimBxb1}[DBxb1]-d_{Bxb1}[Bxb1]$$

$$\frac{d[DBxb1]}{dt}=-k_{DNABxb1}[DBxb1][SI_{Bxb1}]+k_{-DNABxb1}[SA_{Bxb1}]+k_{dimBxb1}[Bxb1]^2-k_{-dimBxb1}[DBxb1a]-d_{DBxb1}[DBxb1]$$

$$\frac{d[SA_{Bxb1}]}{dt}=k_{DNABxb1}[DBxb1][SI_{Bxb1}]-k_{-DNABxb1}[SA_{Bxb1}]$$

Characterization

Data

Assumptions

Parameter fitting

Range of validity of the assumptions

Characterization

Data

Assumptions

Parameter fitting

Range of validity of the assumptions