# Coiled-coil interaction model

**We designed a two state model that describes the interactions of coiled coils within our inducible system.****The difference of affinities required for a favorable ratio of signal to noise ratio whereas determined using the model.**

Logic operations in biological systems have been tested with several approaches
*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 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 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.

The two systems are connected by the relation between the dissociation constants $Kd_b$ and
$Kd_x$,
\begin{equation}
Kd_x = \frac{Kd_b}{4 * 10^{-3}M}
\end{equation}
This relation (12) 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”)

We plotted 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 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 $Kd_B$) (5.4.2. A). On the other hand, the affinity of the autoinhibitory coil **b** for **A** should be slightly lower than the affinity of **B** ($ Kd_b \gt Kd_B $), but not so low that it would allow too much leakage in the pre-cleavage state (5.4.2. B).

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
*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 (**B**) since it presents few substitutions (*i.e.* Gln and Ser instead of Ala in *b* and *c* positions) which confer a 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).