Logic operations in biological systems have been tested with several approaches Singh2014 . Our project relies on the reconstitution of split protein promoted by coiled-coil (CC) dimerization. The interaction between CC peptides can be finely tuned Woolfson2005, Gradisar2011, Negron2014 , thereby CCs offers a flexible and versatile platform in terms of designing logic operation in vivo. With the purpose of understanding the relation that underlies the interaction between coiled-coil peptides and therefore using them in logic gates, we designed the following model ( 5.4.1. ). Our system is based on constructs that have been characterized in mammalian cells in the context of logic function design. Two orthogonal CC segment, A and b, fused together in on chain can bind each other and form a stable CC pair. This complex exists in combination with the peptide B, which can also bind the peptide A and has a different affinity from the peptide b. The linker that connects A and b can be cleaved by a generic protease (e.g. TEV), this irreversible reaction shift the equilibrium towards a state in which all of the three peptides are free in solution and therefore compete for binding. In our experiments, a similar system as the generic coils A and B was fused to the split reporter firefly luciferase.

The relationship between the signal before and after cleavage by proteases is represented by the difference [AB] - [AB-b]. In order to understand the optimal combination of dissociation constant required to obtain a good signal we solved two systems of equations set up considering the two state of the reaction scheme, Before cleavage and After cleavage, (1) and (6) respectively, as separate phases of the reaction and additionally, considering cleavage as an irreversible and complete reaction.

Given values for total concentrations and Kd (from 10^-9 to 10^-3 M) the equations, for the reaction constants (2), (3) and (7), (8) and and for mass conservation (4), (5) and (9), (10), (11) were solved for the species at equilibrium.

Before cleavage \begin{equation} \ce{Axb + B <=>[Kd_x] A-b + B <=>[Kd_B] AB-b} \end{equation} \begin{align} Kd_x &= \frac{[A-b]}{[Axb]} \label{1.1-2}\\ Kd_B &= \frac{[A-b] * [B]}{[AB - b]} \\ c_B &= [B] + [AB-b]\\ c_A-b &= [A-b]+[Axb]+[AB-b] \label{2.1-2} \end{align} After cleavage \begin{equation} \ce{Ab + B <=>[Kd_b] A + b + B <=>[Kd_B] AB + b} \end{equation} \begin{align} Kd_b &= \frac{[A] * [b]}{[Ab]} \label{1.3-4}\\ Kd_B &= \frac{[A] * [B]}{[AB]} \\ c_A &= [A]+[AB]+[Ab]\\ c_B &= [B] +[AB]\\ c_b &= [b] + [Ab] \label{2.3-5} \end{align} The two systems are connected by the relation between the dissociation constants $Kd_b$ and $Kd_x$, \begin{equation} Kd_x = Kd_b * 4 * 10^{-3} M^{-1} \end{equation} This relation approximates the higher affinity between the coils A and b when they are covalently linked by a short peptide (as in the system “Before cleavage”) Moran1999, Zhou2004 .

The results have been plotted varying the $_Kd for the interaction of A with both B and b, against the difference [AB] - [AB-b], where [AB] is considered the signal after cleavage and [AB-b] the signal before cleavage (leakage). The system revealed that in order to obtain a high difference between signal and leakage a high affinity of the coil B for the coil A (low $Kd_B$) is required, while on the other hand an excessive destabilization of the autoinhibitory coil b (high $Kd_b$) would prevent the signal to be visible ( 5.4.2. ).

This relationship suggested to try using a different version of the coiled-coils available in the toolset already used by the Slovenian iGEM 2009 team Gradisar2011 .In order to obtain a detectable signal for logic operation in vivo we decided to use an inhibitory coiled-coil, which would be displaced by the second coiled-coil with higher affinity, only once is cleaved off its partner ($ Kd_B \lt Kd_b $). In doing so we selected P3 as B and P3mS as b, these two coiled-coil peptides present only few substitutions and the higher solubility of P3mS (b), which presents Gln and Ser instead of Ala in b and c position of the heptads, would favour the dissociation. We also tried differently destabilized versions of P3 and it turned out that, as in the forehead described model, an excessive destabilization (obtained by substituting a and d positions with Ala) leads to a small difference of the signal before and after cleavage. Using a slightly destabilized coiled-coil (P3mS-2A), which presents only 2 alanines in the second heptad, the signal after cleavage reached its maximum of 16 folds (Logic Figure 10).