(61 intermediate revisions by 6 users not shown) | |||
Line 21: | Line 21: | ||
inlineMath: [['$', '$'], ['\\(', '\\)']], | inlineMath: [['$', '$'], ['\\(', '\\)']], | ||
processEscapes: true | processEscapes: true | ||
+ | }, | ||
+ | TeX: { | ||
+ | extensions: ["mhchem.js"], | ||
+ | equationNumbers: {autoNumber: "all" } | ||
} | } | ||
}) | }) | ||
Line 38: | Line 42: | ||
</a> | </a> | ||
<div class="ui vertical sticky text menu"> | <div class="ui vertical sticky text menu"> | ||
+ | <a class="item" href="//2016.igem.org/Team:Slovenia/Software"> | ||
+ | <i class="chevron circle left icon"></i> | ||
+ | <b>CaPTURE</b> | ||
+ | </a> | ||
+ | <a class="item" href="" style="color:#DB2828;"> | ||
+ | <i class="selected radio icon"></i> | ||
+ | <b>Modeling logic gates</b></a> | ||
+ | <a class="item" href="#achieve" style="margin-left: 10%"> | ||
+ | <i class="selected radio icon"></i> | ||
+ | <b>Achievements</b> | ||
+ | </a> | ||
<a class="item" href="#intro" style="margin-left: 10%"> | <a class="item" href="#intro" style="margin-left: 10%"> | ||
<i class="selected radio icon"></i> | <i class="selected radio icon"></i> | ||
− | <b> | + | <b>Introduction</b> |
</a> | </a> | ||
<a class="item" href="#model" style="margin-left: 10%"> | <a class="item" href="#model" style="margin-left: 10%"> | ||
<i class="selected radio icon"></i> | <i class="selected radio icon"></i> | ||
− | <b>Deterministic | + | <b>Deterministic modeling</b> |
</a> | </a> | ||
<a class="item" href="#results" style="margin-left: 10%"> | <a class="item" href="#results" style="margin-left: 10%"> | ||
Line 50: | Line 65: | ||
<b>Results</b> | <b>Results</b> | ||
</a> | </a> | ||
− | <a class="item" href=" | + | <a class="item" href="//2016.igem.org/Team:Slovenia/CoiledCoilInteraction"> |
− | + | ||
− | + | ||
− | + | ||
− | + | ||
<i class="chevron circle right icon"></i> | <i class="chevron circle right icon"></i> | ||
<b>Coiled Coil interaction model</b> | <b>Coiled Coil interaction model</b> | ||
Line 64: | Line 75: | ||
<!-- content goes here --> | <!-- content goes here --> | ||
<div class="main ui citing justified container"> | <div class="main ui citing justified container"> | ||
− | <h1><span id=" | + | <div> |
− | + | <h1><span id="achieve" class="section colorize"> </span>Modeling logic gates</h1> | |
− | + | <div class="ui segment" style="background-color: #ebc7c7; "> | |
− | Engineering and designing biological circuits constitute a central core of synthetic | + | <p><b> |
− | + | <ul> | |
− | + | <li> Fourteen coiled-coil-based logic operations were designed and modeled in order to function <i>in vivo</i>. | |
− | + | <li> Fast response was obtained upon reconstitution of light inducible proteases used as input. | |
− | + | </ul> | |
− | + | </b></p> | |
− | + | </div> | |
− | + | </div> | |
− | + | <div class="ui segment"> | |
− | + | <h4><span id="intro" class="section colorize"> </span></h4> | |
− | + | <p> | |
− | + | 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 | |
− | + | <a href="https://2016.igem.org/Team:Slovenia/Protease_signaling/Orthogonality">toolset of | |
− | + | orthogonal proteases</a> | |
− | . | + | that we developed worked as input for <a |
− | + | href="https://2016.igem.org/Team:Slovenia/Protease_signaling/Logic">logic | |
+ | function in mammalian cells</a>. 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 (<ref>fig:logicfunctions</ref>). Designed logic gates based on | ||
+ | protein-protein interaction are | ||
+ | expected to have a shorter time delay compared to their analogues based on transcription | ||
+ | activation | ||
+ | <x-ref>Gaber:2014, Kiani:2014</x-ref> | ||
+ | . | ||
+ | </p> | ||
+ | <div style="float:left; width:100%"> | ||
+ | <figure data-ref="fig:logicfunctions"> | ||
+ | <img | ||
+ | src="https://static.igem.org/mediawiki/2016/d/d2/T--Slovenia--logic-functions.png"> | ||
+ | <figcaption><b>Scheme of all non-trivial two input logic functions.</b> | ||
+ | <p style="text-align:justify">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 <x-ref>Shekhawat2009</x-ref> we introduce an additional site for the proteolytic cleavage which enables implementation of | ||
+ | all logic functions in a single layer.</p> | ||
+ | </figcaption> | ||
+ | </figure> | ||
+ | </div> | ||
+ | |||
+ | <p>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.</p> | ||
− | + | <p>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>i.e.</i> 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.</p> | ||
− | + | <p>The designed binary logic gates can be divided into 5 subgroups, based on the position of | |
− | + | the | |
− | + | protease cleavage | |
− | + | sites:</p> | |
− | + | <ul> | |
− | + | <li>a) cleavage site between coiled coils: conjunction, disjunction and both projection | |
− | + | functions; | |
− | + | </li> | |
− | + | <li>b) cleavage site between the coiled coil and split protease: logical NAND, logical | |
− | + | 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 |
− | + | 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 | in | ||
− | + | <ref>fig:scheme_buffer</ref> | |
− | + | , | |
− | + | <ref>fig:scheme_nor</ref> | |
− | + | , | |
− | + | <ref>fig:schemes_imply</ref> | |
− | + | ||
− | + | ||
− | + | ||
and | 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> | |
− | + | </figure> | |
− | + | </div> | |
− | + | ||
− | + | ||
− | + | ||
− | + | ||
− | + | ||
− | + | ||
− | + | ||
− | + | ||
− | + | ||
− | + | ||
− | + | ||
− | + | ||
− | + | ||
− | + | ||
− | + | ||
− | + | ||
− | + | ||
− | + | ||
− | + | ||
− | + | ||
− | + | ||
− | + | ||
− | + | ||
− | + | ||
− | + | ||
− | + | ||
− | + | ||
− | + | ||
− | + | ||
− | + | ||
− | + | ||
− | + | ||
− | + | ||
− | + | ||
− | + | ||
− | + | ||
− | + | ||
− | + | ||
− | + | ||
− | + | ||
− | + | ||
− | + | ||
− | + | ||
− | + | ||
− | + | ||
− | + | ||
− | + | ||
− | + | ||
− | + | ||
− | + | ||
− | + | ||
− | + | ||
− | + | ||
− | + | ||
− | + | ||
− | + | ||
− | + | ||
− | + | ||
− | + | ||
− | + | ||
− | + | ||
− | + | ||
− | + | ||
− | + | <div style="float:left; width:100%"> | |
− | + | <figure data-ref="fig:scheme_nor"> | |
− | + | <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%"> | |
− | + | <figure data-ref="fig:schemes_imply"> | |
− | + | <img | |
− | + | src="https://static.igem.org/mediawiki/2016/c/c1/T--Slovenia--5.5.4.png"> | |
− | + | <figcaption><b>Scheme of the modeled function $f_3$.</b> | |
− | + | The output is represented with the | |
− | + | emission | |
− | + | of light induced | |
− | + | by | |
− | </p></ | + | reconstitution of the split firefly luciferase reporter. |
− | </ | + | </figcaption> |
+ | </figure> | ||
+ | </div> | ||
+ | <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> | ||
+ | <p style="clear:both"></p> | ||
+ | </div> | ||
− | + | <div class="ui segment"> | |
− | + | <h3><span id="model" class="section colorize"> </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: | ||
<h4>Projection function $f_1$</h4> | <h4>Projection function $f_1$</h4> | ||
Line 395: | Line 454: | ||
<p> | <p> | ||
− | The function of light presence, denoted with $l(t)$, $l_1(t)$ or $l_2(t)$, is a piecewise | + | 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 $p_1$, $p_2$, | function which equals 1 if the light is present and 0 otherwise. Functions $p_1$, $p_2$, | ||
$g$, | $g$, | ||
Line 510: | Line 570: | ||
</table> | </table> | ||
</div> | </div> | ||
− | <h1><span id="results" class="section"> </span>Results</h1> | + | <div> |
− | + | <h1><span id="results" class="section colorize"> </span>Results</h1> | |
− | + | <div class="ui segment"> | |
− | + | <p>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 | |
− | + | <ref>fig:buffer</ref> | |
− | + | , | |
− | + | <ref>fig:nor</ref> | |
− | + | , | |
− | + | <ref>fig:imply</ref> | |
− | + | and | |
− | + | <ref>fig:nimply</ref> | |
− | + | . | |
− | + | 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. | ||
+ | </p> | ||
− | + | <div style="float:left; width:100%"> | |
− | + | <figure 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 | |
− | + | both possible | |
− | + | inputs | |
− | + | in the | |
− | + | following order 0, 1. | |
+ | </figcaption> | ||
+ | </figure> | ||
+ | </div> | ||
− | + | <div style="float:left; width:100%"> | |
− | + | <figure data-ref="fig:nor"> | |
− | + | <img | |
− | + | 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 | |
− | + | $x_2$ is shown | |
− | + | with | |
− | + | all | |
− | + | four | |
− | + | possible inputs in the following order (0,0), (0,1), (1,0), (1,1). | |
− | + | </figcaption> | |
+ | </figure> | ||
+ | </div> | ||
− | + | <div style="float:left; width:100%"> | |
− | + | <figure data-ref="fig:imply"> | |
− | + | <img | |
− | + | 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 $x_2$ | |
− | + | imply $x_1$ is | |
− | + | shown | |
− | + | with all | |
− | + | four | |
− | + | possible inputs in the following order (0,0), (0,1), (1,0), (1,1). | |
− | + | </figcaption> | |
+ | </figure> | ||
+ | </div> | ||
− | + | <div style="float:left; width:100%"> | |
− | + | <figure data-ref="fig:nimply"> | |
− | + | <img | |
− | + | 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 | |
− | + | $x_1$ | |
− | + | nimply $x_2$ is | |
− | + | shown | |
− | + | with | |
− | + | all four possible inputs in the following order (0,0), (0,1), (1,0), (1,1). | |
− | + | </figcaption> | |
+ | </figure> | ||
+ | </div> | ||
− | + | <p> | |
− | + | 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. | |
− | + | </p> | |
− | + | <div style="width:60%"> | |
− | + | <figure data-ref="fig:reducedtime"> | |
− | + | <img 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 | |
− | + | protein. | |
− | + | </figcaption> | |
+ | </figure> | ||
+ | </div> | ||
+ | </div> | ||
</div> | </div> | ||
− | + | <h3 class="ui left dividing header"><span id="ref-title" class="section colorize"> </span>References | |
− | + | </h3> | |
− | + | <div class="ui segment citing" id="references"></div> | |
</div> | </div> | ||
</div> | </div> | ||
</div> | </div> | ||
</div> | </div> | ||
− | </div> | + | </div> |
+ | <div> | ||
+ | <a href="//igem.org/Main_Page"> | ||
+ | <img border="0" alt="iGEM" src="//2016.igem.org/wiki/images/8/84/T--Slovenia--logo_250x250.png" width="5%" | ||
+ | style="position: fixed; bottom:0; right:1%;"> | ||
+ | </a> | ||
+ | </div> | ||
</body> | </body> | ||
</html> | </html> |
Latest revision as of 17:49, 19 October 2016
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
![](https://static.igem.org/mediawiki/2016/d/d2/T--Slovenia--logic-functions.png)
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.
![](https://static.igem.org/mediawiki/2016/2/20/T--Slovenia--5.5.2.png)
![](https://static.igem.org/mediawiki/2016/c/c7/T--Slovenia--5.5.3.png)
![](https://static.igem.org/mediawiki/2016/c/c1/T--Slovenia--5.5.4.png)
![](https://static.igem.org/mediawiki/2016/5/59/T--Slovenia--5.5.5.png)
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.
![](https://static.igem.org/mediawiki/2016/7/7a/T--Slovenia--5.5.6.png)
![](https://static.igem.org/mediawiki/2016/2/27/T--Slovenia--5.5.7.png)
![](https://static.igem.org/mediawiki/2016/0/06/T--Slovenia--5.5.8.png)
![](https://static.igem.org/mediawiki/2016/1/11/T--Slovenia--5.5.9.png)
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.
![](https://static.igem.org/mediawiki/2016/0/08/T--Slovenia--5.5.10.png)