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 segments, A and b, fused together in one chain can bind each other and form a stable CC pair. This complex exists in equilibrium 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. TEVp). This irreversible reaction shifts the equilibrium towards a state in which all 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 that describe the two separate states of the system, Before cleavage (eq. 1) and After cleavage (eq. 6). The two states are modeled as separate equilibria, with proteolytic cleavage considered 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 .

We plotted the the difference [AB] - [AB-b], where [AB] is considered the signal after cleavage and [AB-b] the signal before cleavage (leakage), against different combinations of Kd for the interaction of A with both B and b ($Kd_B$ and $Kd_b$). Our calculations (Figure 2) show that in order to obtain a large difference between signal and leakage the affinity of coil B for coil A needs to be strong (low $Kd_B$). On the other hand, the affinity of the autoinhibitory coil b should be slightly lower, ($Kdb$ \gt $Kd_B$), but not so low that it would allow too much leakage in the pre-cleavage state (5.4.2., right panel).

Based on these results, we decided to use as B one of the peptides from the previously characterized coiled coil toolset used by the Slovenian iGEM 2009 team Gradisar2011, P3. 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 P3mS as b, this coiled-coil peptide binds AP4 (A) with lower affinity than P3 since it presents few substitutions (i.e. Gln and Ser instead of Ala in b and c positions) which confer an higher solubility than P3 (b). We also tried differently destabilized versions of P3mS and it turned out that, as in the model described above, 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).