Difference between revisions of "Team:Slovenia/ModelLogic"

(Undo revision 357935 by Lidijam (talk))
(Undo revision 357872 by Lidijam (talk))
Line 27: Line 27:
 
             }
 
             }
 
         })
 
         })
 
 
 
 
 
 
      
 
      
 
     </script>
 
     </script>
Line 91: Line 86:
 
                         </div>
 
                         </div>
 
                     </div>
 
                     </div>
                     <div class="ui segment"><span id="intro" class="section colorize"> &nbsp; </span>
+
                     <div>
                        <p>
+
                        <div class="ui segment"><span id="intro" class="section colorize"> &nbsp; </span>
                            Engineering and designing biological circuits constitute a central core of synthetic
+
                            <p>
                            biology. In
+
                                Engineering and designing biological circuits constitute a central core of synthetic
                            the context of our
+
                                biology. In
                            iGEM
+
                                the context of our
                            project, one of the challenges was to create, tune and regulate novel pathways in living
+
                                iGEM
                            cells
+
                                project, one of the challenges was to create, tune and regulate novel pathways in living
                            using a
+
                                cells
                            fast-relay system.
+
                                using a
                            The
+
                                fast-relay system.
                            <a href="https://2016.igem.org/Team:Slovenia/Protease_signaling/Orthogonality">toolset of
+
                                The
                                orthogonal proteases</a>
+
                                <a href="https://2016.igem.org/Team:Slovenia/Protease_signaling/Orthogonality">toolset of
                            that we developed worked as input for <a
+
                                    orthogonal proteases</a>
                                href="https://2016.igem.org/Team:Slovenia/Protease_signaling/Logic">logic
+
                                that we developed worked as input for <a
                            function in mammalian cells</a>. Therefore, here we propose schemes for implementation
+
                                    href="https://2016.igem.org/Team:Slovenia/Protease_signaling/Logic">logic
                            of
+
                                function in mammalian cells</a>. Therefore, here we propose schemes for implementation
                            all 16
+
                                of
                            two input
+
                                all 16
                            binary logic functions based on a protein-protein interaction (coiled coil) and
+
                                two input
                            proteolysis
+
                                binary logic functions based on a protein-protein interaction (coiled coil) and
                            system in cells. Designed logic gates based on
+
                                proteolysis
                            protein-protein interaction are
+
                                system in cells. Designed logic gates based on
                            expected to have a shorter time delay compared to their analogues based on transcription
+
                                protein-protein interaction are
                            activation
+
                                expected to have a shorter time delay compared to their analogues based on transcription
                            <x-ref>Gaber:2014, Kiani:2014</x-ref>
+
                                activation
                            .
+
                                <x-ref>Gaber:2014, Kiani:2014</x-ref>
                        </p>
+
                                .
 +
                            </p>
  
                        <p>The main post-translational modification on which signaling and information processing
+
                            <p>The main post-translational modification on which signaling and information processing
                            systems
+
                                systems
                            are based is
+
                                are based is
                            protein
+
                                protein
                            phosphorylation, which enables reversibility and fast response. Proteolysis is on the
+
                                phosphorylation, which enables reversibility and fast response. Proteolysis is on the
                            other
+
                                other
                            hand
+
                                hand
                            irreversible,
+
                                irreversible,
                            which
+
                                which
                            imposes some limitations with respect to phosphorylation. However for many applications
+
                                imposes some limitations with respect to phosphorylation. However for many applications
                            fast
+
                                fast
                            activation is most
+
                                activation is most
                            important, while the time to reset the system in the resting state is of secondary
+
                                important, while the time to reset the system in the resting state is of secondary
                            importance.</p>
+
                                importance.</p>
  
                        <p>Our protein-based system is designed in such a way that it works through coiled coil
+
                            <p>Our protein-based system is designed in such a way that it works through coiled coil
                            interactions, where each
+
                                interactions, where each
                            coiled
+
                                coiled
                            coil in the system is either free or bound to its partner depending on the proteolytic
+
                                coil in the system is either free or bound to its partner depending on the proteolytic
                            activity.
+
                                activity.
                            Furthermore,
+
                                Furthermore,
                            the
+
                                the
                            signal
+
                                signal
                            output is represented by reconstitution of a split protein (<i>i.e.</i> luciferase or
+
                                output is represented by reconstitution of a split protein (<i>i.e.</i> luciferase or
                            protease),
+
                                protease),
                            which
+
                                which
                            is fused
+
                                is fused
                            separately
+
                                separately
                            to
+
                                to
                            different coiled coil segments. To prove the feasibility of this design, we simulated
+
                                different coiled coil segments. To prove the feasibility of this design, we simulated
                            the
+
                                the
                            system's behavior
+
                                system's behavior
                            using
+
                                using
                            deterministic modeling. The simulations were run in Wolfram Mathematica, using
+
                                deterministic modeling. The simulations were run in Wolfram Mathematica, using
                            xCellerator's
+
                                xCellerator's
                            xlr8r
+
                                xlr8r
                            libraries.</p>
+
                                libraries.</p>
  
                        <p>The designed binary logic gates can be divided into 5 subgroups, based on the position of
+
                            <p>The designed binary logic gates can be divided into 5 subgroups, based on the position of
                            the
+
                                the
                            protease cleavage
+
                                protease cleavage
                            sites:</p>
+
                                sites:</p>
                        <ul>
+
                            <ul>
                            <li>a) cleavage site between coiled-coils: conjunction, disjunction and both projection
+
                                <li>a) cleavage site between coiled-coils: conjunction, disjunction and both projection
                                functions;
+
                                    functions;
                            </li>
+
                                </li>
                            <li>b) cleavage site between the coiled-coil and split protease: logical NAND, logical
+
                                <li>b) cleavage site between the coiled-coil and split protease: logical NAND, logical
                                NOR
+
                                    NOR
 +
                                    and
 +
                                    both
 +
                                    negations;
 +
                                </li>
 +
                                <li>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;
 +
                                </li>
 +
                                <li>
 +
                                    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;
 +
                                </li>
 +
                                <li>e) no cleavage sites: tautology and contradiction.</li>
 +
                            </ul>
 +
 
 +
                            <p>
 +
                                For applications that require fast response (<i>e.g.</i> protein secretion), which are
 +
                                the purpose
 +
                                of
 +
                                our attempt, only
 +
                                falsity
 +
                                preserving gates are appropriate, as biological systems usually require fast activation
 
                                 and
 
                                 and
                                 both
+
                                 not
                                 negations;
+
                                fast
                             </li>
+
                                deactivation.
                             <li>c) cleavage sites between coiled-coils as well as between the coiled-coil and split
+
                                The
                                 protease
+
                                following functions correspond to the desired condition: both projection functions,
                                 in
+
                                 conjunction,
                                 the same construct: material implication and converse implication;
+
                                disjunction,
                             </li>
+
                                exclusive
                             <li>
+
                                disjunction, material nonimplication, converse nonimplication and true.
                                 d) cleavage sites between coiled-coils as well as between the coiled-coil and split
+
                             </p>
                                 protease
+
                             <p>
 +
                                Since the dynamics of both functions in subgroup e) is trivial, <i>i.e.</i> 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).
 +
                             </p>
 +
                             <p>
 +
                                 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
 
                                 in
                                 different constructs: exclusive disjunction, logical biconditional, material
+
                                 <ref>fig:scheme_buffer</ref>
                                 nonimplication
+
                                ,
 +
                                <ref>fig:scheme_nor</ref>
 +
                                ,
 +
                                 <ref>fig:schemes_imply</ref>
 
                                 and
 
                                 and
                                 converse
+
                                 <ref>fig:schemes_nimply</ref>
                                 nonimplication;
+
                                 . All
                             </li>
+
                                of
                             <li>e) no cleavage sites: tautology and contradiction.</li>
+
                                them ignore the leakage due to the binding of the coiled-coils before cleavage, which
                        </ul>
+
                                could
 +
                                be
 +
                                solved by setting
 +
                                the
 +
                                building elements with appropriate parameters as demonstrated in the experimental
 +
                                section on
 +
                                the
 +
                                <a
 +
                                        href="https://2016.igem.org/Team:Slovenia/Protease_signaling/Logic">CC-based
 +
                                    logic
 +
                                    design. </a>
 +
                             </p>
 +
                             <div style="float:left; width:100%">
 +
                                <figure data-ref="fig:scheme_buffer">
 +
                                    <img
 +
                                            src="https://static.igem.org/mediawiki/2016/2/20/T--Slovenia--5.5.2.png">
 +
                                    <figcaption><b>Scheme of the modeled function $f_1$.</b>
 +
                                        The output is represented with the
 +
                                        emission of
 +
                                        light induced
 +
                                        by
 +
                                        reconstitution of the split firefly luciferase reporter.
  
                        <p>
+
                                    </figcaption>
                            For applications that require fast response (<i>e.g.</i> protein secretion), which are
+
                                </figure>
                            the purpose
+
                             </div>
                            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.
+
                        </p>
+
                        <p>
+
                             Since the dynamics of both functions in subgroup e) is trivial, <i>i.e.</i> 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).
+
                        </p>
+
                        <p>
+
                            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
+
                            <ref>fig:scheme_buffer</ref>
+
                            ,
+
                            <ref>fig:scheme_nor</ref>
+
                            ,
+
                            <ref>fig:schemes_imply</ref>
+
                            and
+
                            <ref>fig:schemes_nimply</ref>
+
                            . 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
+
                            <a
+
                                    href="https://2016.igem.org/Team:Slovenia/Protease_signaling/Logic">CC-based
+
                                logic
+
                                design. </a>
+
                        </p>
+
                        <div style="float:left; width:100%">
+
                            <figure data-ref="fig:scheme_buffer">
+
                                <img
+
                                        src="https://static.igem.org/mediawiki/2016/2/20/T--Slovenia--5.5.2.png">
+
                                <figcaption><b>Scheme of the modeled function $f_1$.</b>
+
                                    The output is represented with the
+
                                    emission of
+
                                    light induced
+
                                    by
+
                                    reconstitution of the split firefly luciferase reporter.
+
  
                                 </figcaption>
+
                            <div style="float:left; width:100%">
                            </figure>
+
                                 <figure data-ref="fig:scheme_nor">
                        </div>
+
                                    <img
 +
                                            src="https://static.igem.org/mediawiki/2016/c/c7/T--Slovenia--5.5.3.png">
 +
                                    <figcaption><b>Scheme of the modeled function $f_2$.</b>
 +
                                        The output is represented with the
 +
                                        emission
 +
                                        of light induced
 +
                                        by
 +
                                        reconstitution of the split firefly luciferase reporter.
 +
                                    </figcaption>
 +
                                </figure>
 +
                            </div>
  
                        <div style="float:left; width:100%">
+
                            <div style="float:left; width:100%">
                            <figure data-ref="fig:scheme_nor">
+
                                <figure data-ref="fig:schemes_imply">
                                <img
+
                                    <img
                                        src="https://static.igem.org/mediawiki/2016/c/c7/T--Slovenia--5.5.3.png">
+
                                            src="https://static.igem.org/mediawiki/2016/c/c1/T--Slovenia--5.5.4.png">
                                <figcaption><b>Scheme of the modeled function $f_2$.</b>
+
                                    <figcaption><b>Scheme of the modeled function $f_3$.</b>
                                    The output is represented with the
+
                                        The output is represented with the
                                    emission
+
                                        emission
                                    of light induced
+
                                        of light induced
                                    by
+
                                        by
                                    reconstitution of the split firefly luciferase reporter.
+
                                        reconstitution of the split firefly luciferase reporter.
                                </figcaption>
+
                                    </figcaption>
                            </figure>
+
                                </figure>
                        </div>
+
                            </div>
 
+
                            <div style="float:left; width:100%">
                        <div style="float:left; width:100%">
+
                                <figure data-ref="fig:schemes_nimply">
                            <figure data-ref="fig:schemes_imply">
+
                                    <img
                                <img
+
                                            src="https://static.igem.org/mediawiki/2016/5/59/T--Slovenia--5.5.5.png">
                                        src="https://static.igem.org/mediawiki/2016/c/c1/T--Slovenia--5.5.4.png">
+
                                    <figcaption><b>Scheme of the modeled function $f_4$.</b>
                                <figcaption><b>Scheme of the modeled function $f_3$.</b>
+
                                        The output is represented with the
                                    The output is represented with the
+
                                        emission
                                    emission
+
                                        of light induced
                                    of light induced
+
                                        by
                                    by
+
                                        reconstitution of the split firefly luciferase reporter.
                                    reconstitution of the split firefly luciferase reporter.
+
                                    </figcaption>
                                </figcaption>
+
                                </figure>
                            </figure>
+
                            </div>
                        </div>
+
                            <p style="clear:both"></p>
                        <div style="float:left; width:100%">
+
                            <figure data-ref="fig:schemes_nimply">
+
                                <img
+
                                        src="https://static.igem.org/mediawiki/2016/5/59/T--Slovenia--5.5.5.png">
+
                                <figcaption><b>Scheme of the modeled function $f_4$.</b>
+
                                    The output is represented with the
+
                                    emission
+
                                    of light induced
+
                                    by
+
                                    reconstitution of the split firefly luciferase reporter.
+
                                </figcaption>
+
                            </figure>
+
 
                         </div>
 
                         </div>
                        <p style="clear:both"></p>
 
 
                     </div>
 
                     </div>
 
                     <div>
 
                     <div>
Line 684: Line 681:
 
         </div>
 
         </div>
 
     </div>
 
     </div>
    <div>
+
</div>
        <a href="//igem.org/Main_Page">
+
<div>
            <img border="0" alt="iGEM" src="//2016.igem.org/wiki/images/8/84/T--Slovenia--logo_250x250.png" width="5%"
+
    <a href="//igem.org/Main_Page">
                style="position: fixed; bottom:0; right:1%;">
+
        <img border="0" alt="iGEM" src="//2016.igem.org/wiki/images/8/84/T--Slovenia--logo_250x250.png" width="5%"
        </a>
+
            style="position: fixed; bottom:0; right:1%;">
    </div>
+
    </a>
 +
</div>
 
</body>
 
</body>
 
</html>
 
</html>

Revision as of 00:29, 19 October 2016

Model Logic

  Modeling logic gates

  • Design of several logic gates were modeled.
  • Fast response of the proposed design was proved via simulation.

 

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 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 transcription activation 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 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.

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

\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$^{-1}$ 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 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.

$x_1$. 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