Team:Slovenia/ModelLogic
Modeling logic gates
Engineering and designing biological circuits constitute a central core of synthetic
biology. In
the context of our
iGEM
project, one of the challenges was to create, tune and regulate novel pathways in living
cells
using a
fast-relay system.
The
toolset of
orthogonal proteases
that we developed worked as input for logic
function in mammalian cells. Therefore, here we propose schemes for implementation
of
all 14 non-trivial
two input
binary logic functions based on a protein-protein interaction (coiled coil) and
proteolysis
system in cells (fig:logicfunctions). Designed logic gates based on
protein-protein interaction are
expected to have a shorter time delay compared to their analogues based on transcription
activation
Implementation of all 14 two input non-trivial logic operations
based on the proteolysis and coiled coil displacement. Note that in addition to the previously publishes coiled coil-based protease
sensor
The main post-translational modification on which signaling and information processing systems are based is protein phosphorylation, which enables reversibility and fast response. Proteolysis is on the other hand irreversible, which imposes some limitations with respect to phosphorylation. However for many applications fast activation is most important, while the time to reset the system in the resting state is of secondary importance.
Our protein-based system is designed in such a way that it works through coiled coil interactions, where each coiled coil in the system is either free or bound to its partner depending on the proteolytic activity. Furthermore, the signal output is represented by reconstitution of a split protein (i.e. luciferase or protease), which is fused separately to different coiled coil segments. To prove the feasibility of this design, we simulated the system's behavior using deterministic modeling. The simulations were run in Wolfram Mathematica, using xCellerator's xlr8r libraries.
The designed binary logic gates can be divided into 5 subgroups, based on the position of the protease cleavage sites:
- a) cleavage site between coiled coils: conjunction, disjunction and both projection functions;
- b) cleavage site between the coiled coil and split protease: logical NAND, logical NOR and both negations;
- c) cleavage sites between coiled coils as well as between the coiled coil and split protease in the same construct: material implication and converse implication;
- d) cleavage sites between coiled coils as well as between the coiled coil and split protease in different constructs: exclusive disjunction, logical biconditional, material nonimplication and converse nonimplication;
- e) no cleavage sites: tautology and contradiction.
For applications that require fast response (e.g. protein secretion), which are the purpose of our attempt, only falsity preserving gates are appropriate, as biological systems usually require fast activation and not fast deactivation. The following functions correspond to the desired condition: both projection functions, conjunction, disjunction, exclusive disjunction, material nonimplication, converse nonimplication and true.
Since the dynamics of both functions in subgroup e) is trivial, i.e. output is a constant, their modeling is omitted. We selected a single function from the other four subgroups, for which a mathematical model was established and analyzed. We selected the following functions $f_1(x_1, x_2) = x_1$ from subgroup a), $f_2(x_1, x_2) = \neg(x_1 \vee x_2)$ from b), $f_3(x_1, x_2) = x_2 \Rightarrow x_1$ from c) and $f_4(x_1, x_2) = \neg(x_1 \Rightarrow x_2)$ from d).
Inducible proteases were assumed as the two input variables for each function. The logical values true and false were in all the cases presented with high and low amounts of output proteins or input proteases, respectively. Where the output signal is presented with several different proteins, the sum of their concentrations was observed. The schemes of the assumed reactions included in the implementation of described logical functions are represented in fig:scheme_buffer , fig:scheme_nor , fig:schemes_imply and fig:schemes_nimply . All of them ignore the leakage due to the binding of the coiled coils before cleavage, which could be solved by setting the building elements with appropriate parameters as demonstrated in the experimental section on the CC-based logic design.
Deterministic modeling
We have established the following ordinary differential equations (ODEs) based model:Projection function $f_1$
\begin{align} v'(t) =& \alpha_2 - \delta_1 * v(t) - \sigma_1 * v(t) * u(t) * l(t) + \sigma_2 * p_1(t), \\ u'(t) =& \alpha_2 - \delta_1 * u(t) - \sigma_1 * v(t) * u(t) * l(t) + \sigma_2 * p_1'(t), \\ g'(t) =& \alpha_1- \delta_1 * g(t) - \tau * g(t) * p_1(t), \\ g_1'(t) =& -\delta_1 * g_1(t) + \gamma_2 * g_1g_2(t) + \beta_2 * g_1i(t) - \gamma_1 * g_1(t) * g_2(t) - \beta_1 * g_1(t) * i(t), \\ g_1g_2'(t) =& -\gamma_2 * g_1g_2(t) + \gamma_1 * g_1(t) * g_2(t) + \tau * g(t) * p_1(t), \\ g_1i'(t) =& -\delta_1 * g_1i(t) - \beta_2 * g_1i(t) + \beta_1 * g_1(t) * i(t), \\ g_2'(t) =& \gamma_2 * g_1g_2(t) - \delta_1 * g_2(t) - \gamma_1 * g_1(t) * g_2(t), \\ i'(t) =& \alpha_1+ \beta_2 * g_1i(t) - \delta_1 * i(t) - \beta_1 * g_1(t) * i(t),\\ p_1'(t) =& \sigma_1 * v(t) * u(t) * l(t) - \sigma_2 * p_1(t) \end{align}Logical NOR $f_2$
\begin{align} c'(t) =& \alpha_1- \delta_1 * c(t) + \beta_2 * cd(t) - \beta_1 * c(t) * d(t) - \tau * c(t) * p_1(t), \\ c_1'(t) =& -\delta_1 * c_1(t) + \tau * c(t) * p_1(t) + \tau * cd(t) * p_1(t), \\ c_2'(t) =& -\delta_1 * c_2(t) + \tau * c(t) * p_1(t), \\ c_2d'(t) =& \tau * cd(t) * p_1(t), \\ cd'(t) =& -\delta_1 * cd(t) - \beta_2 * cd(t) + \beta_1 * c(t) * d(t) - \tau * cd(t) * p_1(t) - \tau * cd(t) * p_2(t), \\ cd_2'(t) =& \tau * cd(t) * p_2(t), \\ v'(t) =& \alpha_2 - \delta_1 * v(t) - \sigma_1 * v(t) * u(t) * l_1(t) + \sigma_2 * p_1(t), \\ w'(t) =& \alpha_2 - \delta_1 * w(t) - \sigma_1 * w(t) * z(t) * l_2(t)+ \sigma_2 * p_2(t), \\ u'(t) =& \alpha_2 - \delta_1 * u(t) - \sigma_1 * v(t) * u(t) * l_1(t) + \sigma_2 * p_1(t), \\ z'(t) =& \alpha_2 - \delta_1 * z(t) - \sigma_1 * w(t) * z(t) * l_2(t) + \sigma_2 * p_2(t), \\ d'(t) =& \alpha_1+ \beta_2 * cd(t) - \delta_1 * d(t) - \beta_1 * c(t) * d(t) - \tau * d(t) * p_2(t), \\ d_1'(t) =& -\delta_1 * d_1(t) + \tau * cd(t) * p_2(t) + \tau * d(t) * p_2(t), \\ d_2'(t) =& -\delta_1 * d_2(t) + \tau * d(t) * p_2(t), \\ p_1'(t) =& \sigma_1 * v(t) * u(t) * l_1(t) - \sigma_2 * p_1(t), \\ p_2'(t) =& \sigma_1 * w(t) * z(t) * l_2(t) - \sigma_2 * p_2(t) \end{align}Converse implication $f_3$
\begin{align} b'(t) =& \alpha_1- \delta_1 * b(t) - \beta_1 * b(t) * k_1(t) + \beta_2 * k_1b(t), \\ v'(t) =& \alpha_2 - \delta_1 * v(t) - \sigma_1 * v(t) * u(t) * l_1(t) + \sigma_2 * p_1(t), \\ w'(t) =& \alpha_2 - \delta_1 * w(t) - \sigma_1 * w(t) * z(t) * l_2(t) + \sigma_2 * p_2(t), \\ u'(t) =& \alpha_2 - \delta_1 * u(t) - \sigma_1 * v(t) * u(t) * l_1(t) + \sigma_2 * p_1(t), \\ z'(t) =& \alpha_2 - \delta_1 * z(t) - \sigma_1 * w(t) * z(t) * l_2(t) + \sigma_2 * p_2(t), \\ k'(t) =& \alpha_1- \delta_1 * k(t) - \tau * k(t) * p_1(t) - \tau * k(t) * p_2(t), \\ k_1'(t) =& -\delta_1 * k_1(t) - \beta_1 * b(t) * k_1(t) + \gamma_2 * k_{12}(t) + \\ & \gamma_2 * k_{123}(t) + \beta_2 * k_1b(t) - \gamma_1 * k_1(t) * k_2(t) - \gamma_1 * k_1(t) * k_{23}(t) \\ &+ \tau * k(t) * p_1(t) + \tau * k_1k_2(t) * p_1(t), \\ k_{12}'(t) =& -\delta_1 * k_{12}(t) - \gamma_2 * k_{12}(t) + \gamma_1 * k_1(t) * k_2(t), \\ k_{123}'(t) =& -\gamma_2 * k_{123}(t) + \gamma_1 * k_1(t) * k_{23}(t), \\ k_1b'(t) =& \beta_1 * b(t) * k_1(t) - \delta_1 * k_1b(t) - \beta_2 * k_1b(t), \\ k_1k_2'(t) =& -\tau * k_1k_2(t) * p_1(t) + \tau * k(t) * p_2(t), \\ k_2'(t) =& \gamma_2 * k_{12}(t) - \delta_1 * k_2(t) - \gamma_1 * k_1(t) * k_2(t) + \tau * k_1k_2(t) * p_1(t) + \tau * k_{23}(t) * p_2(t), \\ k_{23}'(t) =& \gamma_2 * k_{123}(t) - \delta_1 * k_{23}(t) - \gamma_1 * k_1(t) * k_{23}(t) + \tau * k(t) * p_1(t) - \tau * k_{23}(t) * p_2(t), \\ k_3'(t) =& -\delta_1 * k_3(t) + \tau * k(t) * p_2(t) + \tau * k_{23}(t) * p_2(t), \\ p_1'(t) =& \sigma_1 * v(t) * u(t) * l_1(t) - \sigma_2 * p_1(t), \\ p_2'(t) =& \sigma_1 * w(t) * z(t) * l_2(t) - \sigma_2 * p_2(t) \end{align}Mathematical nonimplication $f_4$
\begin{align} v'(t) =& \alpha_2 - \delta_1 * v(t) - \sigma_1 * v(t) * u(t) * l_1(t) + \sigma_2 * p_1(t), \\ w'(t) =& \alpha_2 - \delta_1 * w(t) - \sigma_1 * w(t) * z(t) * l_2(t) + \sigma_2 * p_2(t), \\ u'(t) =& \alpha_2 - \delta_1 * u(t) - \sigma_1 * v(t) * u(t) * l_1(t) + \sigma_2 * p_1(t), \\ z'(t) =& \alpha_2 - \delta_1 * z(t) - \sigma_1 * w(t) * z(t) * l_2(t) + \sigma_2 * p_2(t), \\ d'(t) =& \alpha_1- \delta_1 * d(t) - \beta_1 * d(t) * g_1(t) + \beta_2 * g_1d(t) - \tau * d(t) * p_2(t), \\ d_1'(t) =& -\delta_1 * d_1(t) - \gamma_1 * d_1(t) * g_1(t) + \gamma_2 * g_1d_1(t) + \tau * d(t) * p_2(t), \\ d_2'(t) =& -\delta_1 * d_2(t) + \tau * d(t) * p_2(t) + \tau * g_1d(t) * p_2(t), \\ g'(t) =& \alpha_1- \delta_1 * g(t) - \tau * g(t) * p_1(t), \\ g_1'(t) =& -\delta_1 * g_1(t) - \beta_1 * d(t) * g_1(t) - \gamma_1 * d_1(t) * g_1(t) \\ & + \beta_2 * g_1d(t) + \gamma_2 * g_1d_1(t) + \gamma_2 * g_1g_2(t) - \gamma_1 * g_1(t) * g_2(t), \\ g_1d'(t) =& \beta_1 * d(t) * g_1(t) - \delta_1 * g_1d(t) - \beta_2 * g_1d(t) - \tau * g_1d(t) * p_2(t), \\ g_1d_1'(t) =& \gamma_1 * d_1(t) * g_1(t) - \gamma_2 * g_1d_1(t) + \tau * g_1d(t) * p_2(t), \\ g_1g_2'(t) =& -\gamma_2 * g_1g_2(t) + \gamma_1 * g_1(t) * g_2(t) + \tau * g(t) * p_1(t), \\ g_2'(t) =& \gamma_2 * g_1g_2(t) - \delta_1 * g_2(t) - \gamma_1 * g_1(t) * g_2(t), \\ p_1'(t) =& \sigma_1 * v(t) * u(t) * l_1(t) - \sigma_2 * p_1(t), \\ p_2'(t) =& \sigma_1 * w(t) * z(t) * l_2(t) - \sigma_2 * p_2(t) \end{align}The function of light presence, denoted with $l(t)$, $l_1(t)$ or $l_2(t)$, is a piecewise function which equals 1 if the light is present and 0 otherwise. Functions $p_1$, $p_2$, $g$, $g_1$, $g_1d$, $g_1d_1$, $g_1g_2$, $g_1i$, $g_2$, $c$, $c_1$, $c_2$, $c_2d$, $cd$, $cd_2$, $w$, $z$, $d$, $d_1$, $d_2$, $k$, $k_1$, $k_{12}$, $k_{123}$, $k_1b$, $k_1k_2$, $k_2$, $k_{23}$, $k_3$, $i$, $b$, $k$, $v$, $u$, $w$, $z$ present concentrations of the equally labelled proteins. The constants used for the model are described in tab:refs .
| Description | Name | Rate | Reference |
|---|---|---|---|
| protein production rate | $\alpha$ | 3.5 * 20$^{-2}$ nMs$^{-1}$ |
|
| light inducible split protease production rate | $\alpha_2$ | 7 * 10$^{-1}$ nMs$^{-1}$ | protein:protease DNA ratio is 1:20 |
| protein degradation rate | $\delta_1$ | Log[2] / (3600 * 9) $s^{-1}$ |
|
| light inducible split protease dissociation rate | $\sigma_2$ | Log[2] / (60 * 5.5) s$^{-1}$ |
|
| light inducible split protease association rate | $\sigma_1$ | 1 nM$^{-1}$ s$^{-1}$ |
|
| protease cleavage rate | $\tau$ | 1.2 * 10$^-6$ nM$^-1$ s$^{-1}$ |
|
| stronger coiled coils association rate | $\beta_1$ | 3.17 * 10$^{-3}$ nM$^{-1}$ s$^{-1}$ |
|
| stronger coiled coils dissociation rate | $\beta_2$ | 2 * 10$^{-4}$ s$^{-1}$ |
|
| weaker coiled coils association rate | $\gamma_1$ | 7.3 * 10$^{-6}$ nM$^{-1}$ s$^{-1}$ |
|
| weaker coiled coils dissociation rate | $\gamma_2$ | 1.67 * 10$^{-1}$ s$^{-1}$ |
|
| time of light exposure | / | 60 s | estimated from experimental results |
Results
We simulated the dynamics of established logic gates with the numerical integration of their mathematical models described in the previous paragraphs. The results of our simulations are shown in fig:buffer , fig:nor , fig:imply and fig:nimply . They confirm our assumption that all four types of logic functions offer shorter delay compared to their equivalents based on genetic regulatory networks. The rise and fall times of our gates are simulated to be at around 70 seconds compared to hours that transcription regulation circuits usually require.
Our system also allows us to shorten the lifetime of the output signal without significantly reducing its concentrations by adding degradation tags to the output protein. The high output times achieved can even be similar to the input light induction time of 1 minute. These two characteristics can importantly influence several sequential induction of logic gates and the further development of several layered logic circuits.
References
