Difference between revisions of "Team:Slovenia/ModelLogic"

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             }
 
             }
 
         });
 
         });
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</script>
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 +
    </script>
 
     <script type="text/javascript" async
 
     <script type="text/javascript" async
 
             src="//2016.igem.org/common/MathJax-2.5-latest/MathJax.js?config=TeX-AMS-MML_HTMLorMML">
 
             src="//2016.igem.org/common/MathJax-2.5-latest/MathJax.js?config=TeX-AMS-MML_HTMLorMML">
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                 <!-- content goes here -->
 
                 <!-- content goes here -->
 
                 <div class="main ui citing justified container">
 
                 <div class="main ui citing justified container">
−
                     <h3>Introduction</h3>
+
                     <h3><span id="intro" class="section"> &nbsp; </span>Introduction</h3>
 
                     <p>
 
                     <p>
 
                         Engineering and designing biological circuits constitute a central core of synthetic biology. In
 
                         Engineering and designing biological circuits constitute a central core of synthetic biology. In
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                         assumed reactions included in the implementation of described logical functions are represented
 
                         assumed reactions included in the implementation of described logical functions are represented
 
                         in
 
                         in
−
                         <ref>fig:sheme_buffer</ref>, <ref>fig:sheme_nor</ref>, <ref>fig:shemes_imply</ref> and <ref>fig:shemes_nimply</ref>
+
                         <ref>fig:scheme_buffer</ref>
 +
                        ,
 +
                        <ref>fig:scheme_nor</ref>
 +
                        ,
 +
                        <ref>fig:schemes_imply</ref>
 +
                        and
 +
                        <ref>fig:schemes_nimply</ref>
 
                         . All
 
                         . All
 
                         of
 
                         of
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                     </p>
 
                     </p>
  
−
                     <figure>
+
                     <figure data-ref="fig:scheme_buffer">
−
                         <img src="https://static.igem.org/mediawiki/2016/2/20/T--Slovenia--5.5.2.png"
+
                         <img class="ui huge centered image"
−
                            data-ref="fig:sheme_buffer">
+
                            src="https://static.igem.org/mediawiki/2016/2/20/T--Slovenia--5.5.2.png">
 
                         <figcaption>Scheme of the modelled function $f_1$.The output is represented with the emission of
 
                         <figcaption>Scheme of the modelled function $f_1$.The output is represented with the emission of
 
                             light induced
 
                             light induced
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                     </figure>
 
                     </figure>
  
−
                     <figure>
+
                     <figure data-ref="fig:scheme_nor">
−
                         <img src="https://static.igem.org/mediawiki/2016/c/c7/T--Slovenia--5.5.3.png"
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                         <img class="ui huge centered image"
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                            data-ref="fig:sheme_nor">
+
                            src="https://static.igem.org/mediawiki/2016/c/c7/T--Slovenia--5.5.3.png">
 
                         <figcaption>Scheme of the modelled function $f_2$. The output is represented with the emission
 
                         <figcaption>Scheme of the modelled function $f_2$. The output is represented with the emission
 
                             of light induced
 
                             of light induced
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                     </figure>
 
                     </figure>
  
−
                     <figure>
+
                     <figure data-ref="fig:schemes_imply">
−
                         <img src="https://static.igem.org/mediawiki/2016/c/c1/T--Slovenia--5.5.4.png"
+
                         <img class="ui huge centered image"
−
                            data-ref="fig:shemes_imply">
+
                            src="https://static.igem.org/mediawiki/2016/c/c1/T--Slovenia--5.5.4.png">
 
                         <figcaption>Scheme of the modelled function $f_3$. The output is represented with the emission
 
                         <figcaption>Scheme of the modelled function $f_3$. The output is represented with the emission
 
                             of light induced
 
                             of light induced
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                     </figure>
 
                     </figure>
  
−
                     <figure>
+
                     <figure data-ref="fig:schemes_nimply">
−
                         <img src="https://static.igem.org/mediawiki/2016/5/59/T--Slovenia--5.5.5.png"
+
                         <img class="ui huge centered image"
−
                            data-ref="fig:shemes_nimply">
+
                            src="https://static.igem.org/mediawiki/2016/5/59/T--Slovenia--5.5.5.png">
 
                         <figcaption>Scheme of the modelled function $f_4$. The output is represented with the emission
 
                         <figcaption>Scheme of the modelled function $f_4$. The output is represented with the emission
 
                             of light induced
 
                             of light induced
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                     </figure>
 
                     </figure>
  
−
                     <h3>Deterministic modeling</h3>
+
                     <h3><span id="model" class="section"> &nbsp; </span>Deterministic modeling</h3>
 
                     We have established the following ordinary differential equations (ODEs) based model:
 
                     We have established the following ordinary differential equations (ODEs) based model:
−
                     <h3>Projection function $f_1$</h3>
+
                     <h4>Projection function $f_1$</h4>
 
                     \begin{align}
 
                     \begin{align}
 
                     v'(t) =&
 
                     v'(t) =&
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                     \end{align}
 
                     \end{align}
  
−
                     <h3>Logical NOR $f_2$</h3>
+
                     <h4>Logical NOR $f_2$</h4>
 
                     \begin{align}
 
                     \begin{align}
 
                     c'(t) =&
 
                     c'(t) =&
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                     \end{align}
 
                     \end{align}
  
−
                     <h3>Converse implication $f_3$</h3>
+
                     <h4>Converse implication f<sub>3</sub></h4>
 
                     \begin{align}
 
                     \begin{align}
 
                     b'(t) =&
 
                     b'(t) =&
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                     \alpha_2 - \delta_1 * z(t) - \sigma_1 * w(t) * z(t) * l_2(t) + \sigma_2 * p_2(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'(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) +
+
                     k_1'(t) =& -\delta_1 * k_1(t) - \beta_1 * b(t) * k_1(t) + \gamma_2 * k_{12}(t) + \\
−
                    \gamma_2 * k_{12}(t) + \gamma_2 * k_{123}(t) + \beta_2 * k_1b(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_2(t) - \gamma_1 * k_1(t) * k_{23}(t) + \tau * k(t) * p_1(t) +
+
                    & \gamma_1 * k_1(t) * k_{23}(t) + \tau * k(t) * p_1(t) + \tau * k_1k_2(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) +
 
                     k_{12}'(t) =& -\delta_1 * k_{12}(t) - \gamma_2 * k_{12}(t) +
 
                     \gamma_1 * k_1(t) * k_2(t), \\
 
                     \gamma_1 * k_1(t) * k_2(t), \\
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                     \end{align}
 
                     \end{align}
  
−
                     <h3>Mathematical nonimplication $f_4$</h3>
+
                     <h4>Mathematical nonimplication f<sub>4</sub></h4>
 
                     \begin{align}
 
                     \begin{align}
 
                     v'(t) =&
 
                     v'(t) =&
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                     \tau * g_1d(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'(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) -
+
                     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_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), \\
 
                     \gamma_2 * g_1g_2(t) - \gamma_1 * g_1(t) * g_2(t), \\
 
                     g_1d'(t) =&
 
                     g_1d'(t) =&
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                     </table>
 
                     </table>
  
−
                     <h1>Results</h1>
+
                     <h1><span id="results" class="section"> &nbsp; </span>Results</h1>
 
                     <p>We simulated the dynamics of established logic gates with the numerical integration of their
 
                     <p>We simulated the dynamics of established logic gates with the numerical integration of their
 
                         mathematical models
 
                         mathematical models
−
                         described in the previous paragraphs. The results of our simulations are shown in <ref>fig:buffer</ref>, <ref>fig:nor</ref>, <ref>fig:imply</ref> and <ref>fig:nimply</ref>.
+
                         described in the previous paragraphs. The results of our simulations are shown in
 +
                        <ref>fig:buffer</ref>
 +
                        ,
 +
                        <ref>fig:nor</ref>
 +
                        ,
 +
                        <ref>fig:imply</ref>
 +
                        and
 +
                        <ref>fig:nimply</ref>
 +
                        .
 
                         They confirm our
 
                         They confirm our
 
                         assumption that all four types of logic functions offer short delay compared to their
 
                         assumption that all four types of logic functions offer short delay compared to their
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                         seconds compared to
 
                         seconds compared to
 
                         hours
 
                         hours
−
                         that transcription regulation circuits usually require.</p>
+
                         that transcription regulation circuits usually require.
 +
                    </p>
  
−
                     <figure>
+
                     <figure data-ref="fig:buffer">
−
                         <img src="https://static.igem.org/mediawiki/2016/7/7a/T--Slovenia--5.5.6.png" data-ref="fig:buffer">
+
                         <img class="ui huge centered image"
 +
                            src="https://static.igem.org/mediawiki/2016/7/7a/T--Slovenia--5.5.6.png">
 
                         <figcaption><b>$x_1$.</b> The output concentration of the logical function $x_1$ is shown with
 
                         <figcaption><b>$x_1$.</b> The output concentration of the logical function $x_1$ is shown with
 
                             both possible
 
                             both possible
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                     </figure>
 
                     </figure>
  
−
                     <figure>
+
                     <figure data-ref="fig:nor">
−
                         <img src="https://static.igem.org/mediawiki/2016/2/27/T--Slovenia--5.5.7.png" data-ref="fig:nor">
+
                         <img class="ui huge centered image"
 +
                            src="https://static.igem.org/mediawiki/2016/2/27/T--Slovenia--5.5.7.png">
 
                         <figcaption><b>$x_1$ NOR $x_2$.</b> The output concentration of the logical function $x_1$ NOR
 
                         <figcaption><b>$x_1$ NOR $x_2$.</b> The output concentration of the logical function $x_1$ NOR
 
                             $x_2$ is shown
 
                             $x_2$ is shown
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                     </figure>
 
                     </figure>
  
−
                     <figure>
+
                     <figure data-ref="fig:imply">
−
                         <img src="https://static.igem.org/mediawiki/2016/0/06/T--Slovenia--5.5.8.png" data-ref="fig:imply">
+
                         <img class="ui huge centered image"
 +
                            src="https://static.igem.org/mediawiki/2016/0/06/T--Slovenia--5.5.8.png">
 
                         <figcaption><b>$x_2$ imply $x_1$.</b> The output concentration of the logical function $x_2$
 
                         <figcaption><b>$x_2$ imply $x_1$.</b> The output concentration of the logical function $x_2$
 
                             imply $x_1$ is
 
                             imply $x_1$ is
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                     </figure>
 
                     </figure>
  
−
                     <figure>
+
                     <figure data-ref="fig:nimply">
−
                         <img src="https://static.igem.org/mediawiki/2016/1/11/T--Slovenia--5.5.9.png" data-ref="fig:nimply">
+
                         <img class="ui huge centered image"
 +
                            src="https://static.igem.org/mediawiki/2016/1/11/T--Slovenia--5.5.9.png">
 
                         <figcaption><b>$x_1$ nimply $x_2$.</b> The output concentration of the logical function $x_1$
 
                         <figcaption><b>$x_1$ nimply $x_2$.</b> The output concentration of the logical function $x_1$
 
                             nimply $x_2$ is
 
                             nimply $x_2$ is
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                     </p>
 
                     </p>
  
−
                     <figure>
+
                     <figure data-ref="fig:reducedtime">
−
                         <img src="https://static.igem.org/mediawiki/2016/0/08/T--Slovenia--5.5.10.png"
+
                         <img class="ui huge centered image"
−
                            data-ref="fig:reducedtime">
+
                            src="https://static.igem.org/mediawiki/2016/0/08/T--Slovenia--5.5.10.png"
 +
                        >
 
                         <figcaption>Shortened output time due to the addition of degradation tags to the output
 
                         <figcaption>Shortened output time due to the addition of degradation tags to the output
 
                             protein.
 
                             protein.
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                     </figure>
 
                     </figure>
  
−
        <h2 id="ref-title" class="ui centered dividing header">References</h2>
+
                    <h2 id="ref-title" class="ui centered dividing header">References</h2>
−
        <div class="citing" id="references"></div>
+
                    <div class="citing" id="references"></div>
 
                 </div>
 
                 </div>
 
             </div>
 
             </div>

Revision as of 15:44, 17 October 2016

Model Logic

Project Achievements Medal requirements Idea

  Introduction

Engineering and designing biological circuits constitute a central core of synthetic biology. In the context of our iGEM project, one the purpose 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 16 two input binary logic functions based on a protein-protein interaction (coiled coil) and proteolysis system in cells. Designed logic gates based on protein-protein interaction are expected to have a shorter time delay compared to their analogues based on genetic regulatory networks Gaber:2014, Kiani:2014 .

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 not that important.

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 modelling. 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 modelling is omitted. We selected a single function from the other four subgroups, for which a mathematical model was established and analysed. 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.

Scheme of the modelled function $f_1$.The output is represented with the emission of light induced by reconstitution of the split firefly luciferase reporter.
Scheme of the modelled function $f_2$. The output is represented with the emission of light induced by reconstitution of the split firefly luciferase reporter.
Scheme of the modelled function $f_3$. The output is represented with the emission of light induced by reconstitution of the split firefly luciferase reporter.
Scheme of the modelled function $f_4$. The output is represented with the emission of light induced by reconstitution of the split firefly luciferase reporter.

  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 f3

\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 f4

\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$^{-1}$ Mariani:2010, Alon:2006
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) $s^{-1}$ Eden:2011
light inducible split protease dissociation rate $\sigma_2$ Log[2] / (60 * 5.5) s$^{-1}$ Taslimi:2016
light inducible split protease association rate $\sigma_1$ 1 nM$^{-1}$ s$^{-1}$ Alon:2006
protease cleavage rate $\tau$ 1.2 * 10$^-6$ nM$^-1$ s$^{-1}$ Yi:2013
stronger coiled coils association rate $\beta_1$ 3.17 * 10$^{-3}$ nM$^{-1}$ s$^{-1}$ DeCrescenzo:2003
stronger coiled coils dissociation rate $\beta_2$ 2 * 10$^{-4}$ s$^{-1}$ DeCrescenzo:2003
weaker coiled coils association rate $\gamma_1$ 7.3 * 10$^{-6}$ nM$^{-1}$ s$^{-1}$ DeCrescenzo:2003
weaker coiled coils dissociation rate $\gamma_2$ 1.67 * 10$^{-1}$ s$^{-1}$ DeCrescenzo:2003
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 short 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.

$x_1$. The output concentration of the logical function $x_1$ is shown with both possible inputs in the following order 0, 1.
$x_1$ NOR $x_2$. The output concentration of the logical function $x_1$ NOR $x_2$ is shown with all four possible inputs in the following order (0,0), (0,1), (1,0), (1,1).
$x_2$ imply $x_1$. The output concentration of the logical function $x_2$ imply $x_1$ is shown with all four possible inputs in the following order (0,0), (0,1), (1,0), (1,1).
$x_1$ nimply $x_2$. The output concentration of the logical function $x_1$ nimply $x_2$ is shown with all four possible inputs in the following order (0,0), (0,1), (1,0), (1,1).

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.

Shortened output time due to the addition of degradation tags to the output protein.

References