Model

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

Chemical Species

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 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}

Parameters

Remplacing every occurence of Bxb1 by ΦC31 gives the set of parameters for ΦC31. The same status can be applied to those parameters.

Even if degradation rates were not determined specifically for the serine integrases and their dimerized form, degradation rates of proteins in E. coli are available. To characterize integrases behavior, we focus on finding the parameters for dimerization and DNA-binding.

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.

Characterization: K_SABxb1

Data

The parameter fitting is based on [[File:ETH Zurich Bonnet Amplifying Genetic Logic Gates 2013.pdf Bonnet's paper]. Their experimental setup is different from ours. However, they experimentally retrieve a transfer function between aTc and Bxb1 switching rate. Here is the figure of interest in our case.

Transfer function from aTc to Bxb1. The experimental data corresponds to the points. They fitted this data with their own model. Supplementary figure S4 of Bonnet's paper Amplifying Genetic Logic Gates

To use their experimental data, we have to model their induction mechanism using aTc. We use a simplify version of it proposed by the 2013 iGEM team UCSF. They modeled the induction with aTc as a leaky Hill function. In the following table are the set of parameters introduced to model induction.

We obtain the following set of differential equations:

$$\begin{align*} \frac{d[Bxb1]}{dt} &= a_{}*(A_L + B_L * \frac{[aTc]^n}{[aTc]^n+K_L^n})- 2 k_{DBxb1}*[Bxb1]^2 + 2k_{-DBxb1}*[DBxb1] - d_{Bxb1}*[Bxb1] \\ \frac{d[DBxb1]}{dt} &= k_{DBxb1}*[Bxb1]^2 - k_{-DBxb1}*[DBxb1] - k_{SABxb1}*[DBxb1]*[SI_{Bxb1}] + k_{-SABxb1}*[SA_{Bxb1}] - d_{DBxb1}*[DBxb1] \\ \frac{d[SA_{Bxb1}]}{dt} &= k_{SABxb1}*[DBxb1]*[SI_{Bxb1}] - k_{-SABxb1}*[SA_{Bxb1}] \end{align*}$$

Assumptions

Parameter fitting

Range of validity of the assumptions

Characterization

Data

Assumptions

Parameter fitting

Range of validity of the assumptions