Model

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

XOR Logic Gate

The truth table of an XOR gate in binary

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 caused by two reasons. 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).
Insert Image of two 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).
Insert Image of two states

XOR biologic gate

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 remplace every occurence of LasI by LuxI. $$\begin{align*} SA_{Bxb1}+SA_{Bxb1}+T_{on,i}& \rightarrow T_{offBxb1}+ SF_{Bxb1}+SF_{Bxb1}\\ T_{offBxb1} &\rightarrow T_{offBxb1} + mRNA_{GFP} + mRNA_{LasI} \\ SA_{\phi C31}+SA_{\phi C31}+T_{on,i}& \rightarrow T_{off\phi C31}+SF_{\phi C31}+SF_{\phi C31}\\ T_{off\phi C31} &\rightarrow T_{off\phi C31} + mRNA_{GFP} + mRNA_{LasI} \\ SA_{Bxb1}+SA_{Bxb1}+T_{off\phi C31}& \rightarrow T_{on,f}+SF_{Bxb1}+SF_{Bxb1}\\ SA_{\phi C31}+SA_{\phi C31}+T_{offBxb1}& \rightarrow T_{on,f}+SF_{\phi C31}+SF_{\phi C31}\\ mRNA_{GFP} &\rightarrow GFP\\ mRNA_{LasI} &\rightarrow LasI\\ mRNA_{GFP} &\rightarrow \emptyset\\ mRNA_{LasI}&\rightarrow \emptyset\\ GFP &\rightarrow \emptyset\\ LasI&\rightarrow \emptyset \end{align*}$$