Model

After binding to DNA, integrases can flip the fragment and thus compute the output of the XOR logic gate.

XOR Logic Gate

We consider a binary exclusive or (XOR) logic gate, with two inputs and one output.

Figure 1 Truth table of the XOR logic gate.

Biological Principles

The fragment integrases can flip is a terminator. Thus, the terminator can either be on or off.

• T_on: terminator is on, transcription is blocked.

• T_off: terminator is off, transcription is active. It corresponds to one flipping of the terminator.

In our design, we are interested in a double flipping. That is to say that two pairs of binding sites surrounds the fragment to be flipped. One pair of binding sites can be bound by DBxb1 and the other one by ΦC31.

The terminator can be flipped once if either DBxb1 or ΦC31 is present. The state T_off can be reached via two possible transitions. We further decompose it into two different states: T_offBxb1(flipping due to presence of Bxb1) and T_offΦC31(flipping due to presence of ΦC31).

Figure 2 Decomposition of the on output into two terminator states.

Flipping by integrases is irreversible. The initial state, in which the terminator is on, is different from the state after two switches. From this last state, no further evolution of the system is possible. Therefore, we decompose the T_on into two different states: T_on,i(initial state of the system, no flipping) and T_on,f(final state of the system after two flips).

Figure 3 Decomposition of the off output into two terminator states.

XOR biologic gate

Figure 4 Truth table of the XOR biologic gate: Summary of the model, coupled with the biological explanation.

Other Chemical Species

Reactions

The following reactions are valid for the strain producing LasAHL as output (It corresponds to the blue cells here).

To have the equivalent for the strain producing LuxAHL as output, it suffices to replace every occurence of LasI by LuxI.

Transfer Function

Determistic Differential Equations

We modeled the XOR gate module deterministically. Thus, we considered a continuous approximation of variables, like the number of integrases-DNA binding sites or the number of terminator. We wrote them as concentration. We do not take the species LasI and mRNA_LasI into account because they are only produced by the XOR biologic gate and GFP is another product of this biologic gate that can be read out. $$\begin{align*} \frac{d[T_{on,i}]}{dt}&= -k_{ToffBxb1}[T_{on,i}][SA_{Bxb1}]^2 -k_{Toff\phi C31} [T_{on,i}][SA_{\phi C31}]^2 \\ \frac{d[T_{offBxb1}]}{dt}&=k_{ToffBxb1} [T_{on,i}][SA_{Bxb1}]^2 - k_{-ToffBxb1} [T_{offBxb1}] [SA_{\phi C31}]^2\\ \frac{d[T_{off\phi C31}]}{dt}&=k_{Toff\phi C31} [T_{on,i}][SA_{\phi C31}]^2 - k_{-Toff\phi C31} [T_{offBxb1}] [SA_{Bxb1}]^2\\ \frac{d[T_{on,f}]}{dt}&= k_{-ToffBxb1} [T_{offBxb1}] [SA_{\phi C31}]^2 + k_{-Toff\phi C31} [T_{offBxb1}] [SA_{Bxb1}]^2\\ \frac{d[mRNA_{GFP}]}{dt}&= k_{mRNA_{GFP}} (\frac{[T_{offBxb1}]}{\theta + [T_{offBxb1}]} + \frac{[T_{off\phi C31}]}{\theta + [T_{off\phi C31}]}) - d_{mRNA_{GFP}}[mRNA_{GFP}]\\ \frac{d[GFP]}{dt}&= k_{GFP} [mRNA_{GFP}] - d_{GFP}[GFP]\\ \end{align*}$$

As there is a strong codon bias in both integrases' seuquences ^[14], the flipping parameters were assumed such that they would not be a rate limiting step. (see the parameter pages)

Deterministic simulation

At steady state, the transfer function obtained by simulation is shown on the next figure. It is remarkable to see that even a little amount of one integrase is sufficient to switch the XOR gate on.

Figure 5 The behaviour of XOR module as a function of activated Bxb1 sites (SABxb1) and ΦC31 sites (SAΦC31). The XOR behaviour is continuous since we modelled it deterministically.